Arts, Faculty of
Vancouver School of Economics
DSpace
UBCV
Guzman, Federico Ezequiel
Doctor of Philosophy - PhD
University of British Columbia
Chapter 2 studies the interaction between a discretionary central bank and a fiscal authority. The analysis focuses on repeated liquidity trap episodes requiring the central bank to rely on forward guidance. Confirming earlier literature, I show that forward guidance policies can be made credible using reputation built during repeated liquidity traps. The key contribution of this chapter is to investigate how the presence of fiscal stabilization policy affects the credibility of the central bank. I show that an increase in the effectiveness of fiscal stabilization policy reduces the range of credible forward guidance announcements that the central bank can implement. Finally, I show that forward guidance can crowd out fiscal effort and result in a loose monetary-tight fiscal policy mix during recessions. Chapter 3 investigates the role of debt maturity within a model of self-fulfilling debt crises. Using one-period bonds, we characterize an interval of debt levels where creditors' panic can force a government to default, and numerically show that the government optimally lowers debt to reduce its vulnerability to this type of crises. After switching to a model with long-term debt repayable with an infinite stream of coupons, we show that the bounds of the interval where crises are self-fulfilling shift upward with higher debt maturity. Finally, we numerically show that the government decreases debt levels faster with long-term debt, therefore rising the economy's welfare compared to the short-term debt case. Fiscal devaluations are policies that can serve as stabilization tools when countries cannot implement nominal devaluations – e.g., within a monetary union. While it has been argued that fiscal policy lags can affect the impact of fiscal devaluations, chapter 4 specifically explores the incidence of implementation lags in fiscal policy in the Euro Area. My findings suggest that fiscal policy in the Eurozone is subject to considerable lags during recessions and expansions. Analysis carried out for Spain shows similar patterns. When fiscal policy cannot adjust at the business cycle frequency, I use a two-period model of tradable and nontradable goods with nominal rigidities and show how lags reduce welfare attained via fiscal devaluations.
Essays on Fiscal and Monetary PolicyDuring Economic CrisesbyFederico Ezequiel GuzmanB.A., Universidad Nacional de Tucuman, 2012M.A., Universidad Torcuato Di Tella, 2016A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Economics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2023© Federico Ezequiel Guzman, 2023The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the dissertation entitled:Essays on Fiscal and Monetary Policy During Economic Crisessubmitted by Federico Ezequiel Guzmanin partial fulfilment of the requirements for the degree of Doctor of Philosophy in Economics.Examining Committee:Michael B. Devereux, Professor, Economics, UBCCo-SupervisorMichal Szkup, Assistant Professor, Economics, UBCCo-SupervisorViktoriya Hnatkovska, Associate Professor, Economics, UBCSupervisory Committee MemberGiovanni Gallipoli, Professor, Economics, UBCUniversity ExaminerAli Lazrak, Associate Professor, Sauder School of Business, UBCUniversity ExaminerLucas Herrenbrueck, Associate Professor, Economics, Simon Fraser UniversityExternal ExamineriiAbstractChapter 2 studies the interaction between a discretionary central bank and a fiscal authority. Theanalysis focuses on repeated liquidity trap episodes requiring the central bank to rely on forwardguidance. Confirming earlier literature, I show that forward guidance policies can be made credibleusing reputation built during repeated liquidity traps. The key contribution of this chapter isto investigate how the presence of fiscal stabilization policy affects the credibility of the centralbank. I show that an increase in the effectiveness of fiscal stabilization policy reduces the rangeof credible forward guidance announcements that the central bank can implement. Finally, I showthat forward guidance can crowd out fiscal effort and result in a loose monetary-tight fiscal policymix during recessions.Chapter 3 investigates the role of debt maturity within a model of self-fulfilling debt crises.Using one-period bonds, we characterize an interval of debt levels where creditors’ panic can forcea government to default, and numerically show that the government optimally lowers debt to reduceits vulnerability to this type of crises. After switching to a model with long-term debt repayablewith an infinite stream of coupons, we show that the bounds of the interval where crises are self-fulfilling shift upward with higher debt maturity. Finally, we numerically show that the governmentdecreases debt levels faster with long-term debt, therefore rising the economy’s welfare comparedto the short-term debt case.Fiscal devaluations are policies that can serve as stabilization tools when countries cannotimplement nominal devaluations – e.g., within a monetary union. While it has been argued thatfiscal policy lags can affect the impact of fiscal devaluations, chapter 4 specifically explores theincidence of implementation lags in fiscal policy in the Euro Area. My findings suggest that fiscalpolicy in the Eurozone is subject to considerable lags during recessions and expansions. Analysiscarried out for Spain shows similar patterns. When fiscal policy cannot adjust at the business cyclefrequency, I use a two-period model of tradable and nontradable goods with nominal rigidities andshow how lags reduce welfare attained via fiscal devaluations.iiiLay SummaryThis dissertation consists of three core chapters analyzing economic interactions that affect mone-tary and fiscal policy. The first chapter shows how fiscal policy affects forward guidance credibilityduring recessions that feature low nominal interest rates. While a central bank’s reputation canmake forward guidance credible, I show that if fiscal policy mitigates a recession, it can limit theability of the central bank to keep its reputation and, therefore, conduct credible forward guidanceduring a recession. The second chapter studies the effect of lengthening the maturity of debt whencrises are triggered by creditors’ beliefs. We show that lengthening the maturity of debt can de-crease the risk of such crises and also discuss its implications for optimal debt policy. The thirdchapter analyses the presence of fiscal policy lags in Eurozone countries and argues that delays infiscal policy can affect the effectiveness of fiscal devaluations.ivPrefaceChapters 2 and 4 are pieces of original, unpublished and independent work. Chapter 3 is an ongoingjoint work with Professor Michal Szkup (University of British Columbia). I have been involved inall the stages of the research project in chapter 3.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Forward Guidance Credibility and Fiscal Policy . . . . . . . . . . . . . . . . . . . 142.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Private sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Monetary and fiscal authorities . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.3 Exogenous shock and competitive equilibrium . . . . . . . . . . . . . . . . . 222.2.4 Discussion of assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Scenario with no forward guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.1 Equilibrium definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.2 Equilibrium characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.3 Numerical exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.4 Forward guidance and fiscal policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4.1 The policy game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.4.2 Sustainable equilibrium characterization . . . . . . . . . . . . . . . . . . . . 452.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.6 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.7 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64vi3 Maturity Structure of Debt and Self-fulfilling Debt Crises . . . . . . . . . . . . . 693.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.2 Baseline model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.2.2 An equilibrium with short-term debt . . . . . . . . . . . . . . . . . . . . . . 723.3 The Crisis region under short-term debt . . . . . . . . . . . . . . . . . . . . . . . . . 733.3.1 Debt regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.3.2 The optimal price q under short-term debt . . . . . . . . . . . . . . . . . . . 743.3.3 Determining the region boundaries . . . . . . . . . . . . . . . . . . . . . . . 743.3.4 Characterization of the lower bound of the Crisis region . . . . . . . . . . . . 753.3.5 Characterization of the upper bound B¯ of the Crisis region . . . . . . . . . . 753.3.6 Crisis equilibria under short-term debt . . . . . . . . . . . . . . . . . . . . . 783.4 A model with long-term debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.4.1 Law of motion of long-term debt and new budget constraint . . . . . . . . . 823.4.2 Government value function and new price of debt . . . . . . . . . . . . . . . 833.5 The Crisis region under long-term debt . . . . . . . . . . . . . . . . . . . . . . . . . 843.5.1 Optimal price under long-term debt and determination of the region bounds 843.5.2 Zero- and positive-probability crisis equilibrium . . . . . . . . . . . . . . . . 873.6 Welfare analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.8 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.9 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934 Fiscal Devaluations and Lags in Fiscal Policy . . . . . . . . . . . . . . . . . . . . . 984.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.2 Data and empirical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.2.2 Empirical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.2.3 Discussion of empirical evidence . . . . . . . . . . . . . . . . . . . . . . . . . 1114.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.3.1 Structure of the economy and equilibrium . . . . . . . . . . . . . . . . . . . 1144.3.2 Non-stochastic steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.3.3 A negative shock to the economy . . . . . . . . . . . . . . . . . . . . . . . . 1194.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.5 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.6 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143viiBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146AppendicesA Appendix to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153A.1 Scenario without forward guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153A.2 Scenario with forward guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168B Appendix to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177B.1 Short-term debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177B.1.1 Determination of the equilibrium price . . . . . . . . . . . . . . . . . . . . . 177B.1.2 Determination of the lower bound of the Crisis region . . . . . . . . . . . . . 177B.1.3 Determination of the upper bound of the Crisis region . . . . . . . . . . . . 178B.2 Long-term debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189B.2.1 Determination of the equilibrium price . . . . . . . . . . . . . . . . . . . . . 189B.2.2 Value of repayment and determination of the lower bound of the Crisis regionunder long-term debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191B.2.3 Determination of the upper bound of the Crisis region . . . . . . . . . . . . 193B.2.4 Welfare analysis results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197viiiList of Tables2.1 Parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.2 Welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.1 Welfare under short-term and long-term debt . . . . . . . . . . . . . . . . . . . . . . 924.1 Summary statistics: Total tax changes (Euro Area – Period 1999-2018) . . . . . . . 1304.2 Summary statistics: Lags (Euro Area – Period 1999-2018) . . . . . . . . . . . . . . . 1304.3 Summary statistics: Restricted lags (Euro Area – Period 1999-2018) . . . . . . . . . 1314.4 Summary statistics: Lags, Business cycles (Euro Area – Period 1999-2018) . . . . . . 1324.5 Summary statistics: Restricted Lags (Spain – Period 1999-2018) . . . . . . . . . . . 1324.6 Summary statistics: Lags, Business cycles (Spain – Period 1999-2018) . . . . . . . . 1334.7 Summary statistics: Tax measures (Spain – Period 1999-2018) . . . . . . . . . . . . 1334.8 Summary statistics: Direction of tax changes (Spain – Period 1999-2018) . . . . . . . 1344.9 Numerical exercise: Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.10 Numerical exercise: Equilibrium outcomes . . . . . . . . . . . . . . . . . . . . . . . . 1354.11 Numerical exercise: Welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135ixList of Figures2.1 Example: States of the world . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.2 Sequence of events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.3 Simulated time series without forward guidance . . . . . . . . . . . . . . . . . . . . . 652.4 Present discounted value of central bank losses . . . . . . . . . . . . . . . . . . . . . 652.5 The credibility region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.6 The credibility region: Comparative statics . . . . . . . . . . . . . . . . . . . . . . . 662.7 Simulated time series with forward guidance . . . . . . . . . . . . . . . . . . . . . . . 672.8 Equilibrium outcomes and FG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.9 Welfare for different levels of fiscal policy effectiveness. . . . . . . . . . . . . . . . . . 683.1 Bond yields and outright monetary transactions . . . . . . . . . . . . . . . . . . . . . 933.2 A zero-probability crisis equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.3 A positive-probability crisis equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 943.4 Optimal short-term debt choices in the crisis region (different initial debt levels) . . 953.5 Maturing debt accumulated over periods. . . . . . . . . . . . . . . . . . . . . . . . . 953.6 A positive-probability crisis equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 963.7 Optimal long-term debt choices in the crisis region (different initial debt levels) . . . 963.8 Comparison between short-term and long-term debt . . . . . . . . . . . . . . . . . . 973.9 Value functions for short-term and long-term debt . . . . . . . . . . . . . . . . . . . 974.1 Distribution of tax changes (Euro Area countries – Period 1999-2018) . . . . . . . . 1364.2 Distribution of tax changes (Non Euro Area countries – Period 1999-2018) . . . . . . 1364.3 Distribution of lags (Subsample – 1999-2018) . . . . . . . . . . . . . . . . . . . . . . 1374.4 Distribution of lags (Euro Area – Period 1999-2018) . . . . . . . . . . . . . . . . . . 1374.5 Distribution of lags: Business cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1384.6 Distribution of tax changes (Spain – Period 1999-2018) . . . . . . . . . . . . . . . . . 1394.7 Direction of tax changes (Spain – Period 1999-2018) . . . . . . . . . . . . . . . . . . 1404.8 Distribution of lags: Business cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.9 Tax measures (Spain – Period 1999-2018) . . . . . . . . . . . . . . . . . . . . . . . . 142B.1 Value function and envelope curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199B.2 Debt regions and maximum V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200B.3 Envelope functions for short- and long-term debt for a lower probability of crises . . 201xB.4 Envelope functions for short- and long-term debt for a higher probability of crises . . 201xiAcknowledgementsI would like to thank my supervisory committee members, Michael B. Devereux, Michal Szkupand Viktoria Hnatkovska for their encouragement, support, and expert guidance throughout mytime at the University of British Columbia (UBC). I would also like to thank Amartya Lahiri,Giovanni Gallipoli, Gorkem Bostanci, Henry Siu, Jesse Perla and all the participants at the macro-international lunch talks at the Vancouver School of Economics (VSE) for their feedback andinsightful comments.I am grateful to have had the opportunity to share my PhD journey with some wonderful col-leagues at VSE who contributed to making this journey truly unforgettable. I would like to speciallythank Anand Chopra, Arkadev Ghosh, Leo Ma, Ronit Mukherjee, Clemens Possnig, Catherine vander List, Nadhanael Gemaliel Vimala and Dongxiao Zhang.I would also like to express my gratitude to all those professors that guided me and mentoredme prior to my years at UBC. Specially, I would like to thank David Kohn and Juan Mario Jorratfor their guidance in different moments of my academic life. Special thanks are to my friends frommy hometown and from many other places in the world who have always been there for me.I would like to thank my parents. They always believed in me, encouraged me to follow mypassion and supported me in all my decisions; I would not be here today if it weren’t for them.Finally, I would like to thank my wife, Emi, for her love, patience and companionship, every dayand every night of this journey. Her unconditional support made this project possible, and she hasalways been there to bring a smile to my life.Thank you to our families too, who live far away but their support can always be felt.xiiChapter 1IntroductionThis thesis consists of three chapters in macroeconomics, and its central theme is the analysis ofmonetary and fiscal policies in constrained environments.In the last two decades, macroeconomic stabilization policies in many countries experienceddifferent types of constraints. One of these constraints relates to nominal interest rates. Duringthe 2008 Great Financial Crisis, central banks in many advanced economies could not lower furthertheir policy rates to increase consumption – scenario known as a liquidity trap. Another type ofconstraint can operate on the financing of fiscal policy. During the 2012 European debt crisis,this type of constraint was observed when some European countries started to experience financialconstraints due to the sudden rise in their bond yields. While this event was seemingly disconnectedfrom changes in economic fundamentals, it was suggested that this episode could be linked to defaultfears from creditors – situation known as belief driven (or self-fulfilling) debt crisis. Finally, a thirdtype of constraint affecting macro policymaking is the lack of access to monetary policy in countriesbelonging to currency unions. An example of this are Euro Area countries that have limited access tomonetary stabilization policy as they cannot use their own currencies to mitigate recessions. In eachof these three contexts where traditional fiscal or monetary stabilization policies face constraints,the economic literature has proposed different solutions. In the context of a liquidity trap, it hasbeen stressed the use of unconventional monetary policies such as forward guidance to overcomeoutput collapse (Krugman et al., 1998; Eggertsson and Woodford, 2003). In the context of a belief-driven crisis, early literature suggested that countries reduce the financial impact of creditors’default fears by lowering the countries’ total outstanding debt (Cole and Kehoe, 2000). Finally, inthe context of a currency union, fiscal devaluations were proposed as an alternative policy solutionfor the members of the currency union (Schmitt-Grohe´ and Uribe, 2012; Farhi et al., 2014).In this thesis, I analyze relevant economic interactions pertaining to the use of each of thesesolutions. The first core chapter (chapter 2) studies the interaction between forward guidance(in particular, its credibility) and fiscal policy during liquidity traps. The second core chapter(chapter 3) explores the interaction between the policy of lowering the total outstanding debt andthe maturity of this debt during belief driven crises. Finally, the third core chapter (chapter 4)studies the interaction between fiscal devaluations and fiscal implementation lags in the context ofa currency union.The first chapter (chapter 2) investigates how fiscal and monetary policy strategically interactduring recessions. Since the 2008 Great Financial Crisis, central banks in many advanced economieswitnessed close-to-zero nominal interest rates. It has been acknowledged the potential problems1associated to such scenarios: Under low nominal interest rates, central banks have a limited abilityto use conventional monetary policy as a stabilization tool – e.g., see Krugman et al. (1998). Inview of that scenario, central banks implemented alternative policies. One of those policies involvedcentral bank promises stating that the monetary authority would implement actions in the future toimprove the current economic conditions. These promises were called forward guidance, and earlyliterature has shown that forward guidance operates by affecting private sector expectations andcan be an effective policy to mitigate recessions – see, for example, Eggertsson and Woodford (2003)and Jung et al. (2005). In practice, forward guidance turned into a popular policy and becamepart of the toolkit of several central banks – e.g., the US Federal Reserve, the Bank of England,the European Central Bank, to name a few. But just like for monetary policy, the 2008 GreatFinancial Crisis also provided evidence of fiscal responses to the crisis too, with fiscal authoritiesimplementing large fiscal stimulus packages to fight the recession. Given that both policies are usedto mitigate output collapse and, in particular, forward guidance needs credibility to be an effectivepolicy, it is important to understand how these two policies relate and whether they complementeach other. Therefore, in chapter 2, I ask whether fiscal policy can have an effect on the ability ofa central bank to implement forward guidance by affecting forward guidance’s credibility.In this chapter I answer to this question from a theoretical perspective. To that end, I followNakata (2018) who presents a policy game between a central bank and a private sector to analyzeforward guidance announcements, and that utilizes the widely used framework featuring liquiditytraps from Eggertsson and Woodford (2003). The private sector responds to monetary policy, andthe central bank cares about output distortions and price stability. In addition, the central bankhas reputational concerns (via trigger strategies), and it is credible if the implemented forwardguidance promises have been consistent with the expectations of the private sector. Using thismodel as a starting point, the novelty in this chapter is the incorporation of a fiscal agency (atreasury) that uses fiscal policy to mitigate a recession too. Also, in my model, the private sectornow responds to fiscal policy, where a fiscal policy parameter captures the effectiveness of thispolicy. This departure from the standard model will help capture and analyze the impact of fiscalpolicy on forward guidance credibility and, as a result of this modelling approach, in this chapterwe can arrive to two key findings that I explain next.The first finding is that active fiscal policy can limit the credibility of forward guidance policies.The mechanism is the following. Forward guidance improves recessionary outcomes by promisinglow rates when the economy is away from a recession. Away from such recession, forward guidanceis credible as long as a central bank keeps its reputation of implementing that promise. In modelswithout a treasury, the only requirement from the central bank to promise low nominal interestrates is that future recessions repeat – otherwise, if recessions did not repeat, the central bankwould renege on its promises. But when the economy features fiscal policy, I show that a treasurythat mitigates recessions can affect the credibility of some of these forward guidance promises ofthe central bank. This occurs because fiscal policy may fight a recession that the reputationalmechanism of the central bank ‘uses’ to make forward guidance credible. As a result, fiscal policy2can limit the credibility of forward guidance based on reputation.Second, in this chapter I conduct a policy experiment that captures that misaligned policies canarise during recessions. In particular, the model shows that forward guidance based on reputationcan crowd out a portion of fiscal effort even in the event of a crisis. The mechanism behindthis result hinges on the fact that forward guidance operates by affecting future variables whilethe economy is in a recession. Provided that forward guidance is credible, this policy triggers apositive impact in today’s variables given that the private sector is forward looking. But in oursetup with two government agencies, both policies incorporate the forward-looking behavior of theprivate sector, as well as the other agency’s actions. Since fiscal policy therefore observes futurevariables (which are affected by credible forward guidance), the treasury may respond today duringa recession with less fiscal effort relative to the case where forward guidance is absent. As a result,seemingly misaligned fiscal and monetary policies, and in particular the interesting case of ‘loosemonetary-tight fiscal’ policy mixes may arise even during a recession.This work expands an existing theoretical framework by incorporating fiscal stabilization policiesto re-examine the credibility issues inherent to forward guidance. Specifically, it expands theliterature that analyzes reputation-based forward guidance by adding a fiscal dimension to it. Asa result, the importance of the research conducted in this chapter is twofold. First, it studiesthe interaction between monetary and fiscal policy, discussion that has had a large tradition inthe economics literature – see, for example, Sargent et al. (1981), Leeper (1991). In this context,this chapter shows another way in which one type of unconventional monetary policy (in thiscase, forward guidance) can be limited by fiscal activism. Second, this chapter provides insightsinto policy discussions where fiscal policy has appeared to be uncoordinated with monetary policyduring recessions – e.g., Greenwood et al. (2014).This chapter contributes to the growing body of work that has analyzed alternative stabilizationpolicies when the nominal interest rate hits the zero lower bound (bound often referred to as theZLB). On the one hand, a branch within that literature has studied whether forward guidance canmitigate crises in economies subject to liquidity traps.1 On the other hand, a large set of papershas analyzed fiscal policy as a stabilization tool when the monetary authority is assumed to onlyhave access to its standard policy instrument – but assumed to be constrained by the ZLB. As itwill become clear shortly, this chapter seeks to build a linkage between these two strands of theliterature.The first branch related to forward guidance has highlighted that promises of monetary ex-pansions can help increase consumption and fight deflation when the public rationally expects thecentral bank to deliver loose monetary policies in the future. Stemming from Krugman et al. (1998)seminal paper, this theoretical result has been studied in full fledged dynamic stochastic generalequilibrium settings where the economy starts in a liquidity trap and it can escape from it withsome probability. Eggertsson and Woodford (2003), in particular, has become one of the workhorse1It is important to mention that there are also other monetary policy instruments used by central banks tocircumvent the restrictions imposed by the ZLB – e.g., quantitative easing and asset purchases. See Woodford (2012)for an account of these alternative policies.3models in this literature characterizing optimal monetary policies in discrete time, and it is also thesetup I use in this chapter.2 Forward guidance involves making promises of future paths of nominalinterest rates, and in this context Eggertsson and Woodford (2003) show that when the centralbank has no credibility issues (i.e., it can credibly commit to any policy it announces, the Ram-sey policy), monetary policy promises effectively affect macro variables by shaping private sectorexpectations. Since then, several applications with similar setups to that in Eggertsson and Wood-ford (2003) have emerged analyzing perfect foresight (Jung et al., 2005), recurrent liquidity trapsepisodes (Adam and Billi, 2006; Nakov, 2008), continuous time (Werning, 2012), to name a few.But the commitment assumption has previously been known to be critical by the literature (seeKydland and Prescott, 1977, for a seminal exposition on time-consistency, and Clarida et al., 1999,for an early application to monetary policy).3 A main lesson is that the commitment assumption istime-inconsistent since past promises do not survive revisions when the future becomes the present.A corollary of this, in the context of models with liquidity traps, is that lack of commitment (alsoknown as central bank discretion) can render forward guidance ineffective when the central bankcannot convince the public that it will implement its promised monetary policies. To circumventthe lack of credibility that is intrinsic to discretionary policymakers, the framework introducedby Chari and Kehoe (1990) has been a popular way in the macroeconomics literature to conceivecredible forward guidance policies, with credibility based on reputation and where such policiescan potentially lead to outcomes better than those under discretion. Featuring an environmentwith repeated interactions between a non-committed policymaker and infinitely many competitiveprivate agents, policies are deemed credible as long as the policymaker prefers them, rather thandeviating but being punished with a reversion to the discretionary outcome. Chari and Kehoe’sreputational equilibrium concept (called sustainable equilibrium, or simply SE) was early imple-mented by Kurozumi (2008) to analyze under what circumstances the Ramsey monetary policycan be made credible in general setups without the ZLB.4 This analysis was later extended toeconomies where the ZLB is present (Nakata, 2018) to analyze whether forward guidance followingRamsey paths can be made credible. Recent contributions continued investigating the credibilityof reputation-based forward guidance analyzing how the length of the punishment periods affectscredibility of reputation-based forward guidance (Nakata and Sunakawa, 2019), and what the fullset of sustainable forward guidance equilibria is when the punishment is endogenously determined(Dong and Young, 2019). This body of work has modelled how forward guidance can becomecredible and emerge in equilibrium in discretionary setups, and also analyzed the sensitivity offorward guidance to key parameters of the economy. But although liquidity trap episodes have2Jung et al. (2005) do this in a deterministic setup. For other studies using this modelling approach, see forexample Nakov (2008), Eggertsson (2006), and Christiano et al. (2011).3Even absent the ZLB constraint, Clarida et al. (1999) showed how central bank’s inability to commit and,therefore, affect private sector expectations matters when monetary policy cannot smooth out the impact of a shock.This assumption also highlights the long discussed time-inconsistency problem in the choice of monetary policydocumented by Kydland and Prescott (1977).4Chari et al. (1998) is an early use of the sustainable equilibrium apparatus showing how the Ramsey outcomecan be achieved when the economy faces expectation traps. See also Ireland (1997) and Chang (1998).4been accompanied by expansionary fiscal policies in the 2008 world recession, this track of theliterature typically abstracts from modelling fiscal policy, thus leaving unexplored any potentiallinkages between fiscal policy and reputation-based forward guidance.5 Such linkage is relevant tounderstand whether fiscal policy can erode the credibility of forward guidance, and this is the steptaken in chapter 2.A second branch of the literature analyzing effective macro policies during liquidity traps hasfocused on fiscal policy as a stabilization tool. A popular modelling approach within this branch isthe so-called Ricardian setup. Using this setup it has been shown that fiscal policy is effective at theZLB by analyzing the multipliers of distortionary and lump-sum taxation, as well as governmentspending (see, for example, Christiano et al., 2011; Eggertsson, 2011b; Woodford, 2011). Butone departure from this approach has been embodied in the growing literature investigating therole of debt-financed fiscal policies from a non-Ricardian perspective. This approach has beenlargely motivated by the observed developments in fiscal policy practice and the change in thefiscal position of countries during the Great Financial Crisis. In particular, the large fiscal stimuluspackages backed with debt and observed during the 2008 crisis triggered investigations about theeffect of debt financing in non-Ricardian models featuring liquidity traps.6 Such studies revisitedthe effect of debt-financed spending on fiscal multipliers (Devereux, 2010), how optimal governmentspending responds to different tax instruments (Nakata, 2017), and the impact of a risk premiumon government debt (Smets and Trabandt, 2018), to name a few. In this respect, the presenceof debt in non-Ricardian environments contributes to new margins of gains coming from fiscalpolicy, either because of the wealth effect that debt creates when households are finitely lived(Devereux, 2010; Smets and Trabandt, 2018), or because of the discrepancy between initial andsteady-state debt levels that provide incentives for fiscal policy to introduce gains at the ZLB(Nakata, 2017). This strand of the literature can show that active fiscal policy is likely effective tomitigate recessions during liquidity traps, but a simplifying assumption in this class of models isto usually summarize monetary policy with a fixed rule (e.g., a Taylor rule), or with a committedcentral bank (implementing the Ramsey policy).The explorations carried out by both strands of the literature yield several lessons regarding theresponse of economic outcomes to stabilization policies. But it is important to stress that duringthe 2008 financial crisis, neither central banks nor treasuries acted in isolation. On the contrary, aninterplay between fiscal and monetary policy was observed where both agencies used their policies tofight the recession. In this respect, and to the best of my knowledge, this chapter provides the firstanalysis of whether and how forward guidance credibility can be affected by the existence of anotherbig player in the economy - here, the treasury.7 Therefore, this chapter attempts to contribute to5Recent studies also analyze credible forward guidance outcomes that are sub-optimal relative to Ramsey butbetter than those under discretion (Walsh, 2018), and the optimal duration of forward guidance (Bilbiie, 2019).6For a textbook exposition with a survey of fiscal policies, see Garın et al. (2018). For the management of debtduring the 2008 financial crisis, see Greenwood et al. (2014).7In this context, a recent study linking fiscal and monetary policy responses during liquidity traps belongs toWoodford and Xie (2022), which also considers a non-Ricardian environment and it discusses the stabilization powerof forward guidance and countercyclical fiscal transfers. However, the focus in this chapter is different in that I study5the literature on credible forward guidance by linking this strand of the macro literature with thestrand focused on the conduct of fiscal stabilization policies during liquidity traps, in order toexplore whether forward guidance credibility can be affected by fiscal action.Finally, this chapter also relates to the large body of work discussing the institutional arrange-ment between a central bank and a treasury. This literature can be traced back to the earlyanalyses of Sargent et al. (1981) on fiscal- and monetary-dominance, and the active- and passive-policy description from Leeper (1991). In this context, Dixit and Lambertini (2003) and Adamand Billi (2008) are examples that complement this literature as they have further investigatedthese interactions considering strategic monetary and fiscal policymakers. A particularly impor-tant consideration in these studies involves the assumption regarding the timing protocol betweenpolicymakers (e.g., a simultaneous versus a leader-follower sequence of moves) and the real outcomesachievable in each case. While there is little consensus about the proper sequence of moves, stud-ies usually assume a specific timing-protocol to derive fiscal and monetary policy lessons, such assimultaneous games to analyze the limitations of monetary policy (Davig and Gu¨rkaynak, 2015) orfiscal multipliers (Eggertsson, 2011a); central bank leadership to study welfare gains from monetarycommitment when a central bank moves before a discretionary treasury (Gnocchi and Lambertini,2016); the design of monetary rules comprehensive of treasury deviations (Camous and Matveev,2022), among others. From this perspective, this chapter relates to this literature by presenting theinstitutional framework where a central bank leads a treasury, but for a specific pair of policies: Acentral bank’s reputation-based forward guidance and a treasury’s fiscal policy with wealth effectsarising from debt-financing.8 Moreover, the interaction presented here allows us to draw lessonsregarding the relationship (i.e., substitutability or complementarity) between these two specificpolicies.So far, the first chapter focused on a monetary policy that emerged in response to liquidity trapconstraints, and its interaction with fiscal policy. In the second and third core chapters (chapters 3and 4) we focus our attention to fiscal policy. Specifically, chapter 3 focuses on how debt maturityaffects the link between the optimal choice of debt and belief-driven debt crises. Finally, chapter 4analyzes how fiscal policy lags can affect the use of fiscal devaluations in countries that belong toa currency union and, therefore, lack access to monetary policy.In the second core chapter (chapter 3) we explore how the maturity of debt mitigates defaultrisk of sovereign bonds when economies are subject to self-fulfilling debt crises. During the 2012European debt crisis, some countries experienced a sudden rise in their bond yields which seemed tobe disconnected from changes in these countries’ economic fundamentals. During this scenario, theEuropean Central Bank announced a liquidity program called Outright Monetary Transactions,which saw a subsequent sharp decline in bond yields despite no money from that program wasspent.One candidate interpretation to the previous episode suggested the possibility of a debt crisishow fiscal policy can affect the credibility of forward guidance.8This choice has been performed for practical purposes, since it will simplify setting up the equilibrium conceptin this chapter.6triggered by investors’ sentiments, also known as self-fulfilling debt crises. The self-fulfilling crisesliterature proposes that defaults can be triggered when creditors suddenly perceive that a country’sdebt is high. When governments have access to one-period bonds, Cole and Kehoe (2000) provideda theoretical framework that identified a region of debt levels subject to these belief-driven crises(a crisis zone), while at the same time it characterized an optimal debt policy that prescribed togradually escape that region. In reality, however, governments have access to debt instruments oflonger maturity too. These instruments may give rise to different crises regions, thus prescribingdifferent debt policies relative to the short-term debt case. In chapter 3, we therefore ask how theuse of long term debt changes the occurrence of self-fulfilling debt crises.In chapter 3 we use the aforementioned self-fulfilling debt crises framework and depart from thisenvironment by implementing a definition of long-term debt that follows Hatchondo and Martinez(2009). Cole and Kehoe (2000) discussed the impact of lengthening the maturity of debt on the crisisregion and arrived to similar results. However, the definition of long-term debt from Hatchondoand Martinez (2009) has become widely used in more recent literature on sovereign debt - see, forexample, Arellano and Ramanarayanan (2012), Chatterjee and Eyigungor (2012), and Aguiar et al.(2022). In particular, their definition describes long-term debt as consols with coupon paymentsthat decay at a geometrical rate, and it has several appealing features. First, it is a tractablepresentation of long-term bonds that largely reduces the number of state variables in the model.Also, it provides a unique parameter (the geometrical rate of declining coupon payments) thatsummarizes the bond duration. Finally, it provides a bond specification that has been argued tomatch empirical patterns — e.g., see the discussion in Hatchondo and Martinez (2009), and alsoArellano and Ramanarayanan (2012).In the standard model, self-fulfilling debt crises arise when lenders suddenly expect default tooccur for some levels of total maturing debt. Since such lenders’ expectations cause a dry-up ofliquidity in the bond market, the government is effectively unable to roll over its maturing debt.As a result, the government is eventually forced to default — therefore justifying lenders’ initialpessimistic expectations. In this chapter, we adjust the standard setup to incorporate long-termbonds, analytically restate propositions and perform model simulations. Finally we draw threeimportant lessons regarding the behavior of the new crisis zone and the optimal management ofdebt in the context of both belief-driven crises and long-term debt a` la Hatchondo and Martinez(2009). We find that debt levels that were subject to self-fulfilling debt crises under short-termdebt can now become immune to these crises when the government issues consols. The intuitionis that with this long-term bonds there is a lower quantity of debt due per period relative to totaldebt. Hence, since creditors also anticipate that there is less debt the government has to repay inevery period, only higher total debt levels can be affected by self-fulfilling debt crises. Because ofthis result, we can also show that it takes a shorter period of time for a government to escape thepossibility of pessimistic expectations about debt repayment. Finally, we find that the introductionof long-term debt can be welfare enhancing given that levels of total outstanding debt that wereonce affected by belief driven crises are no longer constrained by it. Hence, a government can7finance levels of spending without this coordination failure for newer debt levels, and thereforeguarantee a higher consumption pattern.Chapter 3 relates to three key strands of the literature. First, it contributes to the literature onself-fulfilling debt crises. Cole and Kehoe (1996, 2000) explore governments’ incentives to repudiatedebt when default choices take place after borrowing. They characterize an interval of debt levels(the Crisis region) where creditor’s panic can force the government to default, even with unchangedeconomic fundamentals. The authors analytically show that the government can optimally reducedebt levels to escape that region in order to avoid this type of crises. This framework with coordi-nation failures has been analyzed by several other works in the literature to explore policy optionsavailable to governments in order to prevent defaults. For example, Conesa and Kehoe (2017)extend Cole and Kehoe’s environment allowing a government to wait for an economic recoveryto occur while keeping large debt levels in the Crisis region. Roch and Uhlig (2018) analyze theimpact of a bailout agency in the probabilities of sovereign default. More recently, Szkup (2022)uses global games to explore how austerity and fiscal stimulus affect the probability of self-fulfillingdebt crises. Also, Aguiar et al. (2022) quantitatively study how governments can prevent default byauctioning bonds at low prices. In line with these models, this chapter contributes to the literatureof self-fulfilling debt crises by analyzing the impact that the management of debt maturity usingbonds a` la Hatchondo and Martinez (2009) can have on welfare.This chapter also contributes to the extensive literature on endogenous default started by Eatonand Gersovitz (1981). For example, Aguiar and Gopinath (2006) and Arellano (2008) explorethe link between an economy’s macroeconomic variables and the probability of sovereign default.Hatchondo and Martinez (2009) and Arellano and Ramanarayanan (2012) introduce different debtmaturities to interact them with a government’s default choice. In these papers, crises happen as aresult of poor economic fundamentals. On the contrary, in this chapter we adopt the self-fulfillingdebt crises approach from Cole and Kehoe (2000) to analyze the role of debt maturity.Finally, this chapter also relates to the sovereign default literature that explores different spec-ifications of debt maturity. Niepelt (2014) uses long-term debt with a bond that matures in twoperiods. In contrast, Hatchondo and Martinez (2009) model long-term debt with a bond that payscoupons that decay at a fixed rate over time. This bond specification renders a tractable state-space, and is the definition of debt that we will use in our chapter. Other works also implemented asimilar bond structure. Arellano and Ramanarayanan (2012) combine the bond specification fromHatchondo and Martinez (2009) with a one-period bond. Chatterjee and Eyigungor (2012) definea bond that either matures or pays a coupon with some fixed probability in each period. In morerecent works, Aguiar et al. (2022) and Roch and Uhlig (2018) also feature long-term bonds withinfinite coupon payments.The last chapter (chapter 4) studies the interaction between fiscal policy and fiscal policyimplementation lags. During the 2012 European Debt Crisis, countries belonging to the EuroArea could not use monetary policy to introduce nominal exchange rate devaluations. In thiscontext, a growing literature started to analyze whether the real outcomes of a nominal devaluation8could be attained with fiscal reforms within fixed exchange rate regimes – policies so-called fiscaldevaluations. For example, Schmitt-Grohe´ and Uribe (2012) analytically showed how a governmentcan use fiscal policy to stabilize a currency-peg economy. In a more recent work, Farhi et al. (2014)presented a theoretical equivalence characterizing fiscal devaluations with combinations of taxesand subsidies that replicate the outcomes of nominal devaluations. Fiscal devaluations, therefore,emerged as theoretical candidate policies for countries that belong to currency unions. But, at thesame time, a body of work has also documented the existence of lags in fiscal policy – e.g., see Yang(2005) and Mertens and Ravn (2012). If, as in nominal devaluations, the ability to adjust at thebusiness cycle frequency matters for fiscal devaluations, a natural question that arises is whether(and to what extent) implementation lags can compromise fiscal devaluation’s effectiveness.In this chapter, I first use the Tax Policy Reform Database – see Amaglobeli et al. (2018).The Tax Policy Reform Database (TPRD) is a dataset with information about tax policy changescategorized across several dimensions for 23 advanced economies. In this chapter, I use 1999-2018data to focus on the 9 countries from the dataset that belong to the Euro Area. Two of the keycharacteristics collected in the TPRD are the announcement and implementation dates of tax policychanges, whose difference reports delays in fiscal policy measured in days. Using this dataset, Iperform an analysis similar to that in Mertens and Ravn (2012) and Amaglobeli et al. (2018) whereI present empirical evidence that captures the presence of fiscal policy lags with a focus on the EuroArea as a whole. The choice of Euro Area countries is motivated by the fact that their monetarypolicy is set by the European Central Bank, but they have autonomy on their fiscal policy. Thus,I specifically focus on the amount of time it takes for their fiscal policy to be implemented in orderto consider fiscal devaluations a stabilization tool.The chapter is organized in two parts. In the first part I conduct an empirical analysis onapproximately 500 fiscal policy measures implemented during the 1999-2018 period for the subset ofcountries reported in the TPRD. The key findings are as follows. First, across the full sample of nineEuro Area countries, the implementation of different tax instruments had an average and mediandelay of approximately four and three months, respectively. The existence of these lags might beexplained by several reasons: For example, the parliamentary discussions that take place before afiscal policy is implemented, the existence of tax changes with deferred implementation dates, toname a few. Second, following the literature, I use a 90-day cutoff metric to categorize tax changesas anticipated or unanticipated (e.g., see Mertens and Ravn, 2012), and I find that more than halfof the tax changes implemented in the Euro Area can be catalogued as anticipated fiscal policies.This means that a large share of fiscal policy measures can potentially be anticipated by the privatesector, which opposes to the potentially faster speed of adjustment of monetary policy. Finally, Ido a business cycle analysis to assess whether lags in fiscal policy respond differently to recessionsor expansions – see a similar analysis in Amaglobeli et al. (2018). While lags in the Euro Areaare present in every economic phase, I document shorter averages and medians in two out of threerecessionary episodes relative to their following economic expansions. The fact that one recessionfeatures longer lags than the subsequent recovery suggests that, a priori, it cannot be expected9that fiscal policy necessarily responds faster during recessions. Overall, these considerations arerelevant for macro policymaking as they point toward the existence of considerable implementationlags that are present across different dimensions of fiscal policy.In addition, in this chapter I also perform a similar empirical analysis focusing on Spain. Thisis motivated by the numerical exercise from Farhi et al. (2014), who calibrate their model to theSpanish economy to quantify the welfare impact of fiscal devaluations. In this context, I use theTPRD to empirically analyze some characteristics of fiscal policy in Spain, and one of the takeawaysis that tax instruments took approximately five months on average to be implemented during the1999-2018 period. Also, I find evidence of different lags conditional on the type of tax measure. Forexample, I document larger averages and medians for corporate income taxes (roughly more than3 months) and for personal income taxes (approximately 6 months), relative to the lags of VATand social contributions. This evidence suggests that there can be asymmetric delays dependingon the types of tax measures in place. Overall, these findings highlight that fiscal policy lags canmanifest (and also vary) along different dimensions.The previous section suggests that different fiscal policy aspects might need to be consideredfor a comprehensive assessment of the welfare implications of lags when a country conducts fiscaldevaluations. In particular, the existence of delays can be important for the effectiveness of fiscaldevaluations when economies feature nominal rigidities. To rationalize this, in the second part ofthis chapter I analytically illustrate the impact of fiscal policy lags on tax reforms using a modelbased on Schmitt-Grohe´ and Uribe (2012). Their two-period exposition is rich enough to capturethe relationship between fiscal and nominal devaluations presented in Farhi et al. (2014), butwithout needing computational methods to obtain the equilibrium outcomes. The model consistsin a small open economy with households, exogenous endowment of tradable goods, firms thatproduce nontradable goods, and a government, and this economy also features downward nominalwage rigidity.In the first period, the economy observes a negative shock to the world nominal interest rateand a decline in the consumption of tradable goods. In response to that shock (and since wages aredownward rigid) monetary policy can introduce a devaluation and preserve pre-shock consumptionlevels of nontradables. With a similar logic, fiscal policy can implement a wage subsidy thatalleviates the real wage cost faced by firms. However, my setup departs from the standard model inthat I relax the assumption that fiscal stabilization policy responds at the business cycle frequency.Formally (and unlike monetary policy which carries a nominal devaluation in the first period), fiscalpolicy can only be implemented after the shock subsided in the second period. Despite its simplicityand that there is no role for adjustments via a rational expectations channel, the model illustratesthe relevance of the timing of fiscal policy in countering a recession. In particular, the model showsthat if fiscal policy only becomes operative in the second period, the ability of a fiscal devaluationto improve real allocations declines. Furthermore, this finding can be roughly understood as anextreme case of Farhi et al. (2014) where, if fiscal policy takes too long to be implemented, thenwelfare improvement from a fiscal devaluation vanishes.10The contribution of this chapter is twofold. First, the empirical findings suggest that a con-siderable share of fiscal policies in the Euro Area as a whole, as well as in Spain in particular,exhibit implementation delays across several dimensions. As a result of this empirical evidence,implementation lags may limit the extent to which Euro Area countries can expect fiscal devalu-ations to have the same effects of a policy that is implemented at the business cycle frequency.9Second, I analytically illustrate the impacts of delays on fiscal policy effectiveness and the welfareimplications of lags in a theoretical model. In this context, and while the model is a simple exampleof how welfare gains can vanish under the presence of lags, it still highlights that understandingfiscal policy lags can be relevant to assess the potency of fiscal policies during recessions.Chapter 4 relates to three strands of the macroeconomics literature. First, it relates to the largebody of work that has investigated fiscal and monetary responses to economic fluctuations – e.g.,see textbook treatments and the literature surveyed in Woodford (2003), Gal´ı (2008), and Walsh(2010), and Krugman and Obstfeld (2009) (for the open economy case). Within this body of work,this chapter contributes to the literature on fiscal and nominal devaluations when countries havecurrency pegs within currency unions. When a country cannot nominally devalue the exchange rateusing monetary policy, Schmitt-Grohe´ and Uribe (2012) present how fiscal policies can still mitigatenegative shocks in a tradable-nontradable model environment with nominal rigidities and where acountry follows a currency peg.10 A more recent contribution in this field has been Farhi et al. (2014)(henceforth FGI) who analytically characterized combinations of fiscal instruments that replicatethe real effects of nominal devaluations. Further theoretical analyses studied fiscal devaluations asan alternative to nominal devaluations. For example, Engler et al. (2017) present a two-countrymodel that captures how well a fiscal devaluation performs in a country that seeks to improve itscompetitiveness, and find some positive real effects in the economy’s variables. Similarly, Engleret al. (2018) present a two-country monetary union when there is a third non-member country, andfind fiscal devaluations gains for the less competitive country in the currency union. While someof the previous works also recognize that delays can affect fiscal policy, this chapter explores howFGI’s equivalence result between fiscal and nominal devaluations (and the associated welfare gain)responds to fiscal policy lags.11 Specifically, this chapter contributes to the progresses made in thisliterature by performing a simple exercise that shows how welfare may respond in the extreme casewhere fiscal policy does not adjust at the business cycle frequency.Second, this chapter relates to the literature analyzing the characteristics of tax changes indifferent countries and across time. In this context, discussions about fiscal policy lags have had along tradition in the literature – see for example early works from Friedman (1948) and Friedman9Amaglobeli et al. (2018) discuss empirical evidence and investigate characteristics of fiscal reforms for all theavailable countries in the dataset. The analysis in this chapter, instead, uses the information available in the TPRDto focus on the fiscal policy characteristics of the subset of countries belonging to a currency union.10Here I focus on Schmitt-Grohe´ and Uribe (2012) since in future sections I will present a two-period model followingtheir exposition. However, an infinite-horizon version of the model can be found in Schmitt-Grohe´ and Uribe (2016),where the authors also provide empirical evidence on downward wage rigidity in emerging countries.11For example, Schmitt-Grohe´ and Uribe (2012) discuss that fiscal policy may not be able to adjust at the businesscycle frequency due to delays in their implementation. In this vein, FGI provide a numerical example of an anticipatedfiscal devaluation where there are non-negligible (yet decreasing) welfare gains in the presence of lags.11(1968). Romer and Romer (2009) provide a narrative measurement that collects tax change episodesin the US for the 1945-2007 period. Their methodology has been used by more recent papersdocumenting tax policy changes in different countries. Cloyne (2013) constructs a dataset on taxreforms for the UK. Leigh et al. (2011) also build a dataset on fiscal measures for 17 OECDcountries, which has been further extended and revised by Alesina et al. (2017). These datasetscollect different fiscal aspects: For example, they document exact variations in tax bases and rates,or ad-hoc classifications of tax changes. In this context, Amaglobeli et al. (2018) is a novel databasethat collects information on announcement and implementation dates of tax policy changes for 23advanced and emerging market economies and which has been recently made available. Specifically,the TPRD accounts for announcement and implementation dates that serve for the calculation oflags in tax changes. I add to the existing literature by doing an intensive use of the TPRD toexplore alternative ways to intersect the different dimensions included in the database and discussthe existence of lags in the Euro Area and in Spain. This chapter therefore complements thefindings in Amaglobeli et al. (2018) and documents evidence that fiscal policy in the Euro Areaas a whole persistently exhibits lags in the application of fiscal policy measures. Also, it discussesstylized facts showing recessions that feature different lag lengths. A similar analysis is conductedfor Spain, and it includes the observed delays conditional on the type of tax measure. Overall, theseinsights offer an additional rationale for revisiting Farhi et al. (2014) under different calibrationswhich I plan to explore in future work to understand how welfare and macro variables respond todifferent characteristics of fiscal policy lags.12Finally, this chapter relates to the literature on the macroeconomic implications of tax policy,with a particular interest in anticipated and unanticipated tax changes. If agents are forward-looking, the anticipation of tax changes is relevant as it may affect economic outcomes before theactual implementation of the tax takes place. From an empirical perspective, Mertens and Ravn(2012) (also related to Romer and Romer, 2009) classify anticipated and unanticipated tax changesin the US using an ad-hoc threshold of 90 days – beyond that threshold, tax changes are consideredto be anticipated. They find that a fiscal stimulus to the economy gives a large stimulus to keymacro variables (e.g., output, investment, consumption) when tax changes are unanticipated, butthey also document anticipation effects during the preimplementation period that affects variablesdifferently when tax reforms are foreseen. In this chapter, I use this literature’s 90-day metricto identify anticipated tax changes in the Euro Area, and provide a summary of key statistics ofthis data evidencing the incidence of lags in fiscal policy for this subset of countries. Specifically,a considerable share of tax changes in the Euro Area also falls under the category of anticipatedtax policy changes – and, interestingly, this occurs both during recessions and expansions. From amodelling perspective, Yang (2005) analyzes the performance of a real business cycle model with12For alternative uses of the TPRD, see Amaglobeli et al. (2022) (for analysis on the relationship between taxchanges and revenue responses), Bellon (2021) (for a study of the VAT pass-through to consumer prices consideringmarket structure), and David and Sever (2022) (for analysis on the cyclical behavior of tax reforms with respect toelections). Also, see Niemann and Pichler (2020) for another use of the database to identify fiscal policy reforms inthe context of roll-over crises.12different assumptions on agents’ foresight, and shows that the estimation of total tax effects witha one-quarter foresight in agents already affects the estimation of fiscal policy relative to the no-foresight scenario. Mertens and Ravn (2011) present a dynamic stochastic general equilibriummodel and show the model’s accuracy in accounting for the response of macro variables in theUS economy when there are anticipation effects associated to fiscal policy. The model in thischapter relates to this literature as it illustrates how fiscal policy lags can have large impacts on theeffectiveness of fiscal devaluations in the context of nominal wage rigidities when a country belongsto a currency union.13Chapter 2Forward Guidance Credibility andFiscal Policy2.1 IntroductionFor almost two decades now, liquidity trap episodes in OECD and Euro Area economies haveconstrained the ability of central banks to counter recessions with their conventional instruments.In view of such monetary policy constraint, and to mitigate the collapse in output, governmentsresorted to fiscal stabilization policies backed with debt. In addition, central banks also resorted tonew monetary policy tools in order to regain their ability to stimulate the economy. The interactionbetween these new monetary policies (also known as ‘unconventional’) and fiscal policy is the subjectof this study.This chapter focuses on one of the most popular unconventional monetary policies, known asforward guidance, and its interaction with fiscal stabilization policies. Forward guidance (FG here-after) involves disclosing intended future paths for nominal interest rates. The primary objectiveof such policy is to influence the expectations of the public about the future path of the economy,in turn influencing their current actions. But like any other promise, a critical aspect of FG is thatit needs to be credible, i.e., the private sector needs to believe that the central bank will followthrough with its promises. To overcome this limitation and explore the effectiveness of FG, earlyanalyses studied central banks that can commit to any announcement (often called a committedcentral bank). Under commitment, the central bank promises policies that the private sector alwaysexpects, and credibility concerns are eliminated.13 Without commitment, the central bank acts un-der discretion, and credibility becomes important. This idea is closely related to time-inconsistencyof optimal policies due to Kydland and Prescott (1977)’s seminal work.Works that followed showed that, when the central bank is discretionary and liquidity trapsoccur repeatedly, reputation can help restore forward guidance credibility and, therefore, its effec-tiveness.14 Away from liquidity traps, a central bank with reputational concerns has incentives toalign its actions to what the private sector expects (i.e., the implementation of FG promises) ifsuch actions improve liquidity trap outcomes relative to the outcomes attained without FG. These13In early analyses, Krugman et al. (1998) and Eggertsson and Woodford (2003) initially showed that promises offuture expansionary monetary policies can improve contemporaneous consumption and inflation when a central bankcan commit to such policies.14See Nakata (2018), Dong and Young (2019), Nakata and Sunakawa (2019) for examples of ways to restorecredibility when the central bank acts under discretion.14latter outcomes without FG take place if the central bank ever deviates, in which case the privatesector punishes the central bank by never believing the monetary authority’s promises again. Thisresult, however, is obtained in the environment where the central bank is the sole active policyactor in the economy. However, as previous recessions provide evidence of joint fiscal and monetarypolicy responses to crises, understanding the effects that the presence of an active fiscal authoritymay have on the credibility of central bank’s forward guidance becomes important. To see whyfiscal policy matters for reputation-based FG, it is straightforward to observe that, while fiscalstabilization policies can counter recessions, reputation-based FG relies on recessions to restore FGcredibility. The existing literature, however, has largely been silent on this question – a gap thatthis chapter attempts to fill in.This chapter studies how uncoordinated fiscal and monetary authorities interact during repeatedliquidity traps. Specifically, I analyze whether and how one type of fiscal policy (a debt-financedfiscal expansion) can impact the credibility of reputation-based forward guidance. In addition, Iexplore whether this type of fiscal stabilization policy and forward guidance exhibit complementar-ity or substitutability during liquidity traps. To answer these questions, I modify the frameworkpresented in Nakata (2018), who models a discretionary central bank with reputational concernsthat repeatedly interacts with a competitive private sector, and then studies whether the commit-ment monetary policy can be sustained under trigger strategies. In this chapter, I extend Nakata’sanalysis in two ways. First, I incorporate fiscal stabilization policy. In particular, after the centralbank chooses its policy, but before the private sector responds to the central bank by forming fu-ture expectations, I introduce a fiscal agency whose response takes into account the central bank’spolicy that is already in place. This modification allows us to analyze how a fiscal agency can affectthe ability of a discretionary central bank to fight recessions using FG.15 Second, Nakata’s studyfocuses on the sustainability of the optimal monetary policy. Following recent papers that look intopolicies that are sub-optimal relative to the commitment policy (but easier to communicate, as inWalsh, 2018), I focus on all the FG policies that the central bank can sustain.16The model economy consists of three types of agents: (i) households, who are non-Ricardiandue to finite planning horizon as in Devereux (2010),17 which generates wealth effects from debt-financed fiscal policy; (ii) a discretionary central bank, who minimizes a loss function that penalizesvariations of the output gap and the inflation rate of this economy; and (iii) a fiscal authoritythat cares about the business cycle and tax collection costs. The central bank uses the nominalinterest rate to stabilize the economy, and this policy rate is constrained by the zero lower bound(ZLB). The treasury, instead, stimulates aggregate demand via the wealth effect that private agents15Dixit and Lambertini (2003) is an often-cited paper analyzing fiscal and monetary interactions through the lens ofa game. Other examples include Adam and Billi (2008), Eggertsson (2011a), Davig and Gu¨rkaynak (2015), Gnocchiand Lambertini (2016), Camous and Matveev (2022).16Walsh (2018) and Bilbiie (2019) discuss aspects of the communication of FG to the private sector.17Literature analyzing the impact of fiscal stimuli packages financed with debt (like those present in past recessions)has provided microfoundations of this wealth effect. Other examples of these non-Ricardian economies when monetarypolicy is constrained by the zero-lower-bound (ZLB) include Smets and Trabandt (2018) and Woodford and Xie (2022).For an analysis without the ZLB, see Rigon and Zanetti (2018).15experience when fiscal expansions are financed with debt. Finally, the private sector structurefollows Eggertsson and Woodford (2003), where liquidity trap states occur due to negative shocksto the natural interest rate (explained by a sudden rise in households’ desire to save), and wherethe economy remains in a liquidity trap with positive probability in every period until it leaves itforever. To build reputation, I also include a state (which I call the ‘economic recovery’ state) wherethe economy transitorily exits the ZLB and, after which, the economy may return to the liquiditytrap state with a fixed probability (otherwise, the economy returns to Eggertsson and Woodford’sabsorbing state without liquidity traps). This modification gives incentives for the central bank tocare about its reputation and build credible FG using a trigger-strategy equilibrium in the spiritof Nakata (2018).The chapter consists of two parts. In the first part I characterize an equilibrium (which I callthe ‘No-FG equilibrium’) where the central bank can only use the nominal interest rate and ithas no access to forward guidance during the liquidity trap state. As a result, when the centralbank is constrained by the ZLB, the mitigation of the economic downturn depends solely uponfiscal stabilization policy. Indeed, I numerically show that fiscal policy can improve economicoutcomes on its own, both during the liquidity trap and the recovery states. Since in this No-FGequilibrium the central bank cannot mitigate the recession with promises that affect private sectorexpectations, this equilibrium describes the case where the central bank lacks access to credibility.More importantly, when we explore trigger-strategies in the second part of the chapter, the No-FGequilibrium will be the equilibrium towards which the economy reverts upon central bank deviationsfrom its FG policy.In the second part of the chapter, I construct a reputation-based sustainable equilibrium withFG following Chari and Kehoe (1990).18 Specifically, the central bank, the treasury, and the privatesector will repeatedly interact, but now their past and current choices (i.e., the history of actions)will be part of the equilibrium characterization. Using trigger-strategies, the central bank caresabout its reputation of implementing promised policies, and if the central bank’s actions do notconform with its past promises (and, therefore, with private sector expectations), the economy fallsinto the No-FG equilibrium forever.19I first show that, during recessions, forward guidance announcements can be made credible ifthe central bank keeps its reputation of implementing these announcements. The standard oper-ating mechanism from the literature that makes FG credible needs the private sector to rationallyanticipate that the central bank indeed prefers to implement the FG promise. Such mechanisminvolves the central bank balancing its losses from FG versus its optimal discretionary policy foreach pair of states of the economy, respectively – the liquidity trap state and the recovery state.Intuitively, on the one hand FG creates large central bank losses in the recovery state (since the FGrate triggers larger inflation and output gap variations, relative to the higher discretionary rate)but also FG compensates these losses during a liquidity trap (since there is an aggregate demand18See, for example, Nakata (2018); Dong and Young (2019), and Nakata and Sunakawa (2019).19Nakata and Sunakawa (2019) considers finite-time punishments. I do not study this scenario here.16stimulus that FG induces via private sector expectations that counters deflation and the recessionat the ZLB). On the other hand, the discretionary policy involves low central bank losses duringthe recovery state (since the discretionary policy is not constrained by the ZLB and it is optimal)but larger losses during a liquidity trap (i.e., when the discretionary policy is constrained by theZLB and the central bank cannot affect private sector expectations with FG). If the FG lossesare smaller relative to the optimal discretionary policy losses, FG will be credible. But in thisstandard mechanism, fiscal policy is absent. In this chapter, instead, I suggest that active fiscalpolicy can be relevant to this mechanism (and, therefore, to the characterization of credible FG)since fiscal policy can improve outcomes in every state of the economy (as argued in the first partof the chapter). In particular, if fiscal policy can mitigate the recession during the liquidity trapstate, fiscal policy can prevent the implementation of FG as it alters the costs the central bankbalances when it tries to make FG credible. In this chapter, I use this intuition to argue that wecan still quantitatively characterize credible FG announcements even under the presence of fiscalstabilization policy – as long as fiscal policy does not fully counter the recession.Next, I characterize a credibility region collecting which FG policies remain credible whenthe effectiveness of fiscal policy against recessions varies. The effectiveness is measured by themagnitude of the wealth effect that households enjoy when fiscal policy is debt-financed. Whenfiscal stabilization policy becomes more effective, I find that (i) it reduces the range of credibleforward guidance announcements available for the central bank; and (ii) it only leaves the centralbank with access to more conservative FG announcements. This can be explained by the policiesat the extremes of the credibility region, namely, the least and the most expansionary FG policies.The least expansionary FG policy reflects the discretionary monetary policy followed in the No-FG equilibrium. There, the central bank is (trivially) promising the policy rate that generates thediscretionary outcomes. The most expansionary FG policy, instead, reflects the policy that exhauststhe credibility gains. When fiscal policy becomes more effective against recessions, it reduces theoutput gap and it lowers inflation in the No-FG equilibrium. But when the central bank can usereputation, this improvement in outcomes from the No-FG equilibrium implies a milder punishmentthat the central bank faces if it ever deviates from the FG policy. This contributes to the overallerosion of the credibility of FG (i.e., the reduction of the credibility region), and can be capturedby two effects. The first effect is that a more effective fiscal policy makes recessions milder during aliquidity trap state. As a result, it is more difficult for the central bank to sustain large FG policiesusing reputation and, therefore, the most expansionary FG policies available to the central bankdrop out from the credibility region – i.e., the most expansionary policy rate from the credibilityregion rises. The second effect of more effective fiscal policy is that it also mitigates the economy’sbad outcomes during the recovery state. This results in a rise in the least expansionary policy ratefrom the credibility region. This is because in the recovery state, more effective fiscal policy givesthe central bank more leeway to set nominal interest rates closer to the natural interest rate of theeconomy. These two effects together explain that the credible FG levels available to the centralbank become more conservative, as fiscal policy increases both extremes of the credibility region.17Finally, since the rising of the extremes of the credibility region gradually approaches them to thenatural interest rate of the economy (which is positive in the recovery state, and which the centralbank tries to fully absorb) the range of FG policies available to the central bank must shrink.Finally, I conduct a policy experiment to compare the equilibrium outcomes that result when wetransition from the No-FG equilibrium to one where a central bank has access to forward guidance.I find that the central bank’s ability to conduct forward guidance can induce the treasury to conductless expansionary fiscal policies compared to the case without forward guidance. In particular, thesalient aspect of this result is that a loose monetary-tight fiscal policy mix can emerge in equilibriumduring recessions. This result is of relevance as portrayed by some macro developments frompast recessions. On 2013, the US Federal Open Market Committee repeatedly issued statementsmentioning tighter fiscal policy, and signalling monetary action in the form of low rates for aconsiderable period of time.20,21 The model in this chapter formally captures this narrative evidencethrough the expectations mechanism present in the policy responses from the central bank and thetreasury. Intuitively, since forward guidance involves promises of future outcomes, then duringa liquidity trap credible FG may crowd out fiscal effort (relative to the case with no FG) if thebest response function of fiscal policy depends on those future variables that FG affects, too. Thisresult can highlight the importance that portraying fiscal and monetary interactions from a game-theoretic approach has, as it provides insights of cases where substitutability between fiscal policyand FG can emerge even during recessions.The remainder of the chapter is organized as follows. Section 2.2 describes the model setup,and Section 2.3 outlines the equilibrium when the central bank has no access to forward guidancepolicies. Section 2.4 then presents the equilibrium when the central bank can conduct forwardguidance and introduces the key results. Section 2.5 concludes.2.2 The ModelIn this section I present an economy consisting of a Private Sector, a central bank and a treasury.The description of the Private Sector uses log-linear approximations of the equilibrium conditionsof the standard New Keynesian model,22 augmented to capture the impact of debt-financed fiscalpolicy. The structure of the Private Sector closely resembles those of Kirsanova et al. (2005),Devereux (2010) and Rigon and Zanetti (2018), which incorporate debt in the dynamic IS equation.The description of the central bank and the treasury uses a linear-quadratic setup stemming fromthe standard models of Barro (1979), Barro and Gordon (1983) and Rogoff (1985), which are stillwidely used in the macro literature to describe the preferences of monetary and fiscal authorities(see, for example, Clarida et al., 1999; Dixit and Lambertini, 2003; Jung et al., 2005; Eggertsson,20For more details about this episode, see Davig and Gu¨rkaynak (2015).21For example, see https://www.federalreserve.gov/newsevents/pressreleases/monetary20130619a.htm(accessed on November 6, 2022.)22See, for example, Eggertsson and Woodford (2003). For a textbook presentation of the equilibrium conditions,the reader is referred to Woodford (2003) and Gal´ı (2008).182011a; Davig and Gu¨rkaynak, 2015; Camous and Matveev, 2019). There is an explicit zero-lower-bound (ZLB) constraint on nominal interest rates (the monetary policy instrument), and liquiditytraps are described in the tradition of the widely used approach of Eggertsson and Woodford (2003),modified to incorporate repeated liquidity traps episodes that allow building reputation. In whatfollows, I present the building blocks of the model, and then turn to a detailed discussion of theassumptions before presenting the equilibrium outcomes.2.2.1 Private sectorThe environment summarizes a Private Sector (thereafter PS) with two equilibrium conditions: ANew Keynesian Phillips Curve (NKPC) and a Dynamic Investment-Savings equation (DIS). Forevery period t ≥ 0, the NKPC and DIS arepit = κyt + βEtpit+1 (2.1)andyt = Etyt+1 − (it − Etpit+1 − rt) + γbt (2.2)Operator Et denotes rational expectations of the Private Sector over variables in t+1 given informa-tion from period t, parameters κ, γ and β are positive constants, and variable rt is the exogenouslygiven natural interest rate, or the shock for short – described later in this section. Variable pit isthe inflation rate, namely, the log-deviation of inflation from its steady state. Likewise, variable ytrepresents the output gap capturing the log deviation of output from its steady state level. Finally,variables it and bt capture the impact of monetary and fiscal policy, respectively: Variable it is thecentral bank’s monetary policy instrument representing the nominal interest rate, and variable btrepresents real debt-to-GDP ratio fluctuations around an initial debt-to-GDP ratio – set at 0 forsimplicity; see Devereux (2010) for a similar treatment. Equations (2.1) and (2.2) characterize theoptimal behavior of firms and households. As it is common in the literature (see, for instance, thetextbook treatment in Woodford, 2003, pp. 405, 406), I assume that the log-linear approximationsof these equilibrium conditions have been performed around a zero steady state level of inflation.To facilitate the exposition, I further assume that there is no discrepancy between steady stateoutput and the efficient level of output, and I shall state here (and argue later) that the steadystate level of output gap is zero. Finally, it is worth to point out that the nominal interest rate isnot defined in deviation terms, so that the zero lower bound constraint (to be displayed later) onlyrequires it to be nonnegative.Equation (2.1) shows that current period inflation is increasing in the output gap in proportionκ (the slope of the Phillips Curve), as well as in next period’s expectations over inflation (discountedby factor β ∈ (0, 1)) due to forward looking firms facing price rigidities. Equation (2.2) expresses thelog-linear version of the Euler equation relating households’ optimal present and future consumptionchoices as a function of expected inflation, the exogenous shock, and fiscal and monetary policies.Since market clearing holds, I adopt the convention of interpreting the DIS curve in output gap19terms instead of consumption. Eq. (2.2) then shows that the nominal interest rate, it, and debtfinanced fiscal policy, bt, can stabilize the output gap by absorbing (i) the shock to the naturalinterest rate (rt), and (ii) changes in inflation and output gap expectations (Etpit+1 and Etyt+1).Variable bt captures the fluctuation in aggregate real bonds held by households, which correspondto treasury’s debt from period t (to be repaid in t + 1), and issued to finance time-t fiscal policy(described below). For this reason, in what follows I will also refer to variable bt as debt-financedfiscal policy – the rationale for bt in the DIS is relegated to Section 2.2.4.2.2.2 Monetary and fiscal authoritiesThe economy features two government agencies: a monetary authority and a fiscal authority. Alongthese lines, I will also refer to these agencies as the central bank and the treasury, or simply usinglabels CB (for central bank) and Tr (for treasury), respectively.The central bank is concerned with rates of inflation and output gaps. Hence, at period t, thecentral bank preferences areLCBt ≡ Et∞∑k=0βk(lCBt+k (yt+k, pit+k))(2.3)where lCBt+k denotes the central bank’s per-period loss function,lCBt+k ≡ −12(αCBy y2t+k + αCBpi pi2t+k)(2.4)and LCBt is the expected present discounted value of the central bank’s losses, where parameterβ ∈ (0, 1) is the central bank’s discount factor, and expectations are conditional on the informationavailable in period t. I further assume that the expectations operator and the discount factor ofthe central bank coincide with those of the Private Sector – below I shall assume the same for thetreasury. It is important to point out that one implication of Eq. (2.4) is that the central bank’starget level of inflation coincides with the economy’s steady state level of inflation. Furthermore,the central bank has a zero targeted level of the output gap, which will match with the steady statelevel of the output gap. This also rules out surprise inflation – see Jung et al., 2005 for a detaileddiscussion. Finally, positive parameters αCBpi and αCBy describe a central bank that prefers pricestability and dislikes output gap distortions, respectively. In line with other works in the literature(e.g., Adam and Billi, 2008), in this setup I assume that the central bank cares more about inflationstabilization than output gap stabilization, namely, αCBpi > αCBy .As stated above, the central bank uses the nominal interest rate, it, to absorb shocks to thenatural interest rate, and it features a zero-lower bound constraint,it ≥ 0,∀t (2.5)In addition to the central bank, there is a treasury that controls fiscal instruments that can mitigate20recessions. The fiscal authority has access to debt, bt, as defined before, and it performs lump-sumtransfers to (it collects lump-sum taxes from) currently alive households. Specifically, variables `tand τt denote the real lump-sum-transfers-to-GDP and real tax-collections-to-GDP ratios relativeto initial ratio levels, respectively – both initial ratio levels set at zero for simplicity. The economythen features a treasury with expected present discounted value of its per-period losses at time tgiven byLTrt ≡ Et∞∑k=0βk(lTrt+k (yt+k, τt+k))(2.6)with per-period losses lTrt+k defined bylTrt+k ≡ −12(αTry y2t+k + αTrτ τ2t+k)(2.7)with positive parameters αTry and αTrτ , and where the treasury’s discount factor β and expectationsEt coincide with those of the central bank and the Private Sector. Eq. (2.7) indicates that thetreasury wants to minimize output gap distortions (term αTry y2t+k) and tax collection costs (termαTrτ τ2t+k). The treasury’s intertemporal budget constraint at time t is`t + (1 + r) bt−1 = bt + τt (2.8)This specification represents a linear approximation of a standard treasury budget constraint in thecontext of a New Keynesian model. The right-hand-side variables represent the sources of incomefor the treasury: The treasury issues bt (time-t) units of one-period real bond holdings that pay(1 + r) per unit at time t + 1, with r the steady state level of the natural interest rate,23 and itcollects lump-sum taxes (or taxes, for short), τt . Conversely, the left-hand-side variables in Eq.(2.8) represent treasury expenditures: Payments of outstanding debt from period t− 1 (carried upto period t and yielding interest (1 + r)), and lump-sum transfers to the private sector, `t. Notethat I abstract from seigniorage revenue transfers from the central bank to the treasury. This isbecause I want to keep track of the simplest exposition of the model where forward guidance onlyimpacts the economy’s variables via PS expectations. For a brief discussion about the absence ofseigniorage revenues in this type of models, see Eggertsson (2006).The treasury also faces two additional constraints on bonds, lump-sum transfers and taxes.There is a finitely large (exogenous) limit to the total debt the treasury can borrow to financeexpansionary fiscal policy given by0 ≤ bt ≤ b¯ <∞,∀t (2.9)23Debt from past periods yields the steady state value of rt, namely r, when the linear approximation of theTreasury’s budget constraint is performed around an initial steady state with a debt–to-GDP ratio equal to zero –for example, see Devereux (2010).21and there is also a non-negativity constraint on lump-sum transfers and lump sum taxes,`t ≥ 0,∀t (2.10)τt ≥ 0, ∀t (2.11)These restrictions ensure that the treasury always transfers (or collects) positive or zero amountsto the (from the) private sector.2.2.3 Exogenous shock and competitive equilibriumConsider a state of the world, st, governed by a Markov chain. States take values from the setS = {Z,R, S}. State Z is the zero lower bound state, or ZLB, and is alternatively called a LiquidityTrap state or a crisis state. States R and S are the Economic Recovery state (or just Recoverystate) and the Steady-State, respectively – refer to section 2.2.4 for an explanation of this three-state structure. Associated to each state of the world there is a value that the natural interestrate, rt, can take. Specifically, rt maps states of the world into two values: A negative naturalinterest rate, r < 0, or the positive steady-state natural interest rate, r > 0. When the state stis R or S, then rt becomes rt (R) = rt (S) = r. For simplicity I shall denote rt (R) ≡ rR (= r)and rt (S) ≡ rS (= r). This notation will be helpful to avoid ambiguities and identify the state ofthe variables in the optimization problems of the central bank and the treasury. However, I willsimplify notation {rR, rS} with r when the state is no longer relevant. In addition, in this modelthe desire to save from agents can suddenly increase, which is captured by the exogenous shock tothe natural interest rate. Formally, when state st = Z, then rt (Z) ≡ rZ = r < 0. The rationalebehind this shock to the natural interest rate could be seen as a negative demand shock – which isa common structure in the literature; see, for example, Eggertsson (2006, 2011b). As I shall arguein the solution to the central bank problem in state Z, I shall further assume that r is negativeenough to make the ZLB constraint on nominal interest rates bind in equilibrium.The transition between states of the economy is the following. If the economy is in state Z, itmoves to the Recovery state R with probability 1. If the economy is in state R, it transitions tostate Z with positive probability p ∈ (0, 1) (i.e., p = Pr (st+1 = Z| st = R)), and it arrives state Swith complementary probability, 1− p (i.e., 1− p = Pr (st+1 = S| st = R)). Finally, if the economyis in state S, it remains in that state with probability 1.Figure 2.1 shows all possible histories starting when the state is Z at time 0, and until period5 for the given transition probabilities. The upper-most branch shows the case where the economyreverts to the (absorbing) steady state at time t = 2 with probability 1 − p, right after the firstliquidity trap episode. With complementary probability p, the economy falls into a liquidity trapagain at t = 2, period after which the economy may return to the absorbing steady state in periodt = 4 (or the ZLB episode may take place again, and this cycle repeats).Having described the states of the economy, I revisit the treasury’s constraints to make the22following simplifying assumptions. When the economy is in a liquidity trap (state Z), we assumethat the treasury can only use lump-sum transfers (instead of lump-sum tax rebates) to mitigaterecessions. The impact of this policy will be reflected in the DIS equation, where lump-sum transfersfinanced with debt have a positive impact measured by γbt. On the contrary, when the economyrecovers, or when it reaches its steady state (states R,S), the treasury retires debt by collectingtaxes, and leaves the central bank regain control of stabilizing the economy by adjusting its nominalinterest rate. Therefore, we add the following equations,bt × 1{r 6=r} = 0, `t × 1{r 6=r} = 0, τt × 1{r=r} = 0,∀t (2.12)The role of the indicator functions will be to identify the fiscal instruments that the treasuryis allowed to use in each state. For instance, the indicator variable prevents `t and τt from beingsimultaneously positive in Z states. Also, indicator functions will prevent debt from being positivein the recovery state or the steady state – a rationale for these assumptions is provided in section2.2.4.We are now ready to define a competitive equilibrium for the model economy. The followingequilibrium definition is standard (see, for example, Eggertsson, 2006, 2011b and Nakata, 2018),and it presents the conditions that the private sector must satisfy in equilibrium, for any state andgiven any fiscal and monetary policy choices.Definition (Private sector competitive equilibrium - PSCE). Given a stochastic process for thenatural interest rate, {rt}∞t=0, an initial debt level b0 and monetary and fiscal policies {it}∞t=0 and{τt, bt, `t}∞t=0, a private sector competitive equilibrium (PSCE) is a set of inflation levels {pit}∞t=0and output gaps {yt}∞t=0 that satisfy the NKPC (Eq. (2.1)), the DIS (Eq. (2.2)), the zero-lowerbound constraint (2.5), together with the treasury’s budget constraint (Eq. 2.8) plus the additionalrestrictions and assumptions on fiscal variables (Eqs. (2.9), (2.10) and (2.12)).2.2.4 Discussion of assumptionsOur goal is to summarize key aspects of fiscal and monetary policy interactions observed duringliquidity traps; specifically, to show how fiscal policy matters for the credibility of forward guidance.To that end, the model presented in this chapter needs to allow equilibrium characterizations offorward guidance based on reputation while describing an economy that faces a liquidity trap. Idiscuss below key simplifying assumptions that will let us keep the model tractable and obtainsolutions in closed form.Wealth effect of debt. Parameter γ is the coefficient on government debt in the DIS curve.Therefore, term γbt shows how a treasury that implements expansionary fiscal policies financed withreal units of debt affects the output gap, yt. Term γbt is characteristic of non-Ricardian setupsthat capture a wealth effect from expansionary fiscal policies, and the rationale behind it can beexplained by the assumption that households face a survival probability in each period. Specifically,those that die after receiving fiscal aid financed with debt seize a wealth effect of magnitude γbt23– as they will not pay back the fiscal aid they received.24 Term γbt matters to our analysis as itportrays the impact of deficit-financed fiscal policies. In the context of this chapter, a positive γwill be referred to as a case where fiscal policy is ‘active’, and γ will be analytically relevant as itallows us to control the channel via which fiscal policy affects monetary policy. (In the limitingcase where γ is zero, the economy reverts to standard New Keynesian textbook representations;see, for instance, Gal´ı, 2008).Preferences of the fiscal authority. Similar treasury’s preferences portrayed in Eq. (2.7)have been adopted in other studies (see, for instance, Dixit and Lambertini, 2003; and Adam andBilli, 2008), where some form of social loss function from the Private Sector is usually assigned tothe fiscal authority. Before proceeding, a few further comments on Eq. (2.7) are in order. First,and following the standard specification of monetary policy in charge of an independent centralbank (see, for example, the large literature after Rogoff, 1985), the fiscal authority does not exhibitconcerns over price stability. Second, the treasury faces tax collection costs as measured by thesecond term in Eq. (2.7).25 However, note also that this term indicates that the treasury facesasymmetric costs to τt relative to `t: Specifically, the fiscal authority enjoys zero costs from givingout money via lump-sum transfers, but it faces a cost from collecting money back from householdsin the form of taxes. The rationale of this term is to capture the differential impact that giving outmoney relative to collecting money can have within the population of currently alive cohorts.Central bank independence. This chapter presents a central bank independent from thetreasury. This assumption about separate government agencies has had a large tradition in theliterature (for example, Rogoff, 1985; and for recent contributions see Dixit and Lambertini, 2003;Adam and Billi, 2008; Basso, 2009; Gnocchi and Lambertini, 2016; and Camous and Matveev,2022). This assumption can be easily mapped into the institutional arrangements observed inmany OECD and Euro Area economies from the early 2000’s onward, where monetary policyis delegated to an independent central bank, and which therefore approximates the institutionalarrangement that was in place when liquidity traps started to be observed. In the context of thismodel, the formal version of the independent central bank assumption adopted here closely followsEggertsson (2011a), where the central bank (1) has a loss function different from the treasury losses;and (2) chooses monetary policy independently from the treasury’s budget constraint and any otherrestrictions on fiscal variables.Exogenous states. The Markov process is based on the popular simple stochastic modeldescribed in Eggertsson and Woodford (2003), where the shock to rt follows a Poisson processand that it vanishes at a stochastic time where the economy reverts to an absorbing zero inflationsteady state. However, in this chapter I depart from their exposition in two dimensions. First, Iconsider one-period liquidity traps. This allows me to have debt that does not build up indefinitely24See, for example, Kirsanova et al. (2005) for another application. For an analysis on how this wealth effect ofdebt-financed fiscal policy can formally emerge in the context of a New Keynesian model, see for example Devereux(2010), Smets and Trabandt (2018), and Rigon and Zanetti (2018), who present a microfounded model under theviolation of the representative agent assumption.25See, for example, Barro (1979). A related micro-founded version can be found in Eggertsson (2011a). Also, seea model with a similar preference specification presented in Davig and Gu¨rkaynak (2015).24– thus ruling out the possibility of exploding debt levels while, at the same time, simplifying themodel exposition. Second, I implement an intermediate state following the liquidity trap wherethe shock vanishes, but that it is transitory – unlike the absorbing deterministic steady state inEggertsson and Woodford (2003). This transitory intermediate state allows me to have a stochasticmodel in an identical fashion to Nakov (2008) and Adam and Billi (2008) (where ZLB episodes arerecurrent), but it is still simple enough to allow us build a reputational equilibrium as in Nakata(2018) and Walsh (2018) while keeping the ability to compute results analytically.26 A sequence ofZ and R states, however, may not be sufficient to characterize an equilibrium with liquidity traps.As Nakata (2018) points out, we may reach some types of equilibria where the ZLB constraintbinds in every state when the frequency of the negative shock is too high. The intuition behind thisunwanted outcome is straightforward. If after a state without a liquidity trap it follows a liquiditytrap state, the prospect of deflation in the latter state triggers deflation in the former state (due tothe forward-looking NKPC). But deflation in that no-liquidity trap state calls for lower rates and,in some cases, if deflation in the future is very likely (p is high in the context of our model), then theZLB might be binding even away from the liquidity trap. To circumvent this issue, some optionsare available, and each carry their own costs. A first option is to make the non-liquidity-trap statepersistent, namely, to make state R display a stochastic duration. But this comes at the cost of anadditional parameter that needs to be monitored to assure the model does not exhibit liquidity trapswhen there is no shock – i.e., liquidity traps generated when deflation feeds itself via expectationsand absent a shock, as we argued before. Another option to avoid that the ZLB binds in a non-liquidity-trap state is to keep the duration of state R fixed and append a reversion to a steady statewithout inflation. In this way, Private Sector’s expectations are anchored to a state that disciplinesdeflation when the economy is away from liquidity traps to be zero (i.e., nonnegative). Althoughthis trick adds an additional state, it largely simplifies the model in several dimensions. First, itallows us to think of the economy in the similar Poisson process from Eggertsson and Woodford(2003), where each pair of consecutive Z and R states are our repeated tuples. As such, it displays avery simple model, but with a small departure from Eggertsson and Woodford (2003) that ensuresthe existence of a reputational equilibrium. Second, it focuses our analysis to one-period FG, anddisregards considerations that are not part of this chapter like the duration of FG. And third, itkeeps the model tractable as it summarizes in parameter p all the relevant information that has tobe monitored to confirm that our equilibrium features a liquidity trap only in state Z.2.3 Scenario with no forward guidanceThe objective of this section is to build an equilibrium when the central bank only solves its prob-lem under discretion and it has no access to forward guidance. I will interchangeably refer tothis problem as the discretionary problem, or the problem without forward guidance. The use of26As it will be seen later, the shock is necessary because if, on the contrary, the state without the shock is absorbing,then reputation cannot be sustained – although the liquidity trap is of stochastic duration, reputation is ruled out ifthe ZLB is a one-time event occurring that never repeats when it is over.25this equilibrium is twofold: First, it will allows us build the reputational equilibrium from Section2.4; and second, it will serve as a benchmark for comparison when we analyze the effects of credi-ble forward guidance on the economy’s outcomes relative to the no-forward-guidance equilibrium.Hence, I first describe the policy problems given the timing of events within each period, and thenI introduce the equilibrium definition and characterize optimal monetary and fiscal policies.2.3.1 Equilibrium definitionBefore presenting the equilibrium outcomes of this economy, I need to establish the within-periodtiming protocol. In the literature that characterizes sustainable policies with a discretionary pol-icymaker (Kurozumi, 2008, Walsh, 2018, Nakata and Sunakawa, 2019), the implemented timingstructure usually assumes a policymaker moving first and internalizing PS moves as in Chari andKehoe (1990). In this model, however, there are two policymakers and, consequently, the timingprotocol admits different arrangements where monetary-fiscal interaction is described with a si-multaneous or a leader-follower setup. With the final goal of characterizing how the credibility offorward guidance responds to fiscal policy parameters, the focus of this chapter is, primarily, on FGas an effective stabilization tool for the monetary authority. Driven by practical purposes, I shallassume a leading monetary authority and a follower fiscal authority. The leader-follower struc-ture simplifies the computation of deviations during the proof of forward guidance as a sustainableequilibrium.27In the presence of a monetary leadership structure, the following events take place in everyperiod t ≥ 0. At the beginning of the period, the shock is realized. After the shock, the Centralbank, the treasury and the Private Sector make their decisions in the following order. First, theCentral bank makes its decision after observing the state and sets the nominal interest rate. Then,having observed both the shock and the monetary policy in place, the treasury follows and choosesfiscal policy. Finally, with these policies in place (together with the observed shock) the privatesector satisfies the DIS and the NKPC by determining output and inflation while it forms rationalexpectations. The description of the events is summarized in Figure 2.2.After the shock to the natural interest rate is realized, rt, the central bank and the treasurychoose their policies. In our setup neither the central bank, nor the treasury have access to acommitment technology, and it is standard that we look for a Markov-perfect equilibrium – see, forinstance, Nakata (2018). Therefore, I solve the problem with backward induction, starting fromthe sequential problem of the treasury, and then turn to the central bank’s sequential problem inperiod t.First, having observed the shock and the choices of the central bank and the treasury, the privatesector forms expectations of future output gap and inflation in the last stage of periods t ≥ 0, andchooses the remaining variables of the economy (pit, yt) according to the optimality conditions (the27See Dixit and Lambertini (2003) and Adam and Billi (2008) for a detailed description of the timing of eventsin fiscal-monetary policy setups. Recent applications (absent the ZLB) also feature a leading central bank (Gnocchiand Lambertini, 2016; Camous and Matveev, 2022)), and cases of two strategic policymakers (Basso, 2009).26Dynamic IS relation (Eq. (2.2)), and the New-Keynesian Phillips curve (Eq. (2.1)).After observing the shock and the central bank’s monetary policy instrument, the treasury’sproblem in period t ismax{yt,τt,bt,`t}∞t−12Et{ ∞∑k=0βk[αTry y2t+k + αTrτ τ2t+k]}(2.13)subject toyt =Etyt+1 − (it − Epit+1 − rt) + γbt,∀tbt =`t + (1 + r) bt−1 − τt, ∀t`t ≥0, τt ≥ 0, 0 ≤ bt ≤ b¯ <∞,∀tfor an initial bt−1, shock rt, taking future paths {yt+k, pit+k, it+k, τt+k, bt+k, `t+k} (with k ≥ 1) andcurrent central bank’s choice it as given, and expressions (2.12).In this case, problem (2.13) shows how the treasury determines optimal fiscal policy, where itslosses are functions of output gap (with path determined by the DIS), and tax distortions (subjectto the treasury’s budget constraint and additional restrictions on the fiscal policy instruments).The problem depends on the other agency’s choices (the central bank’s nominal interest rate), andsince the treasury has no access to a commitment technology it takes as given the future paths ofthe variables of the economy. Finally, it is important to point out that the debt limit will neverbind in the equilibrium characterization.After the period-t shock, the central bank chooses its nominal interest rate, internalizing thetreasury’s choice of fiscal instruments. Therefore, its problem in every period t becomesmax{yt,pit,it}∞t−12Et{ ∞∑k=0βk[αCBy y2t+k + αCBpi pi2t+k]}(2.14)subject topit =κyt + βEtpit+1,∀tyt =Etyt+1 − (it − Epit+1 − rt) + γbt,∀tit ≥0, ∀tand for an initial bt−1, shock rt, taking future paths {yt+k, pit+k, it+k, τt+k, bt+k, `t+k} (with k ≥ 1)as given, and while it also internalizes the treasury’s choices, {τt, bt, `t} that are solution to problem(2.13) and expressions (2.12).Basically, problem (2.14) displays a discretionary central bank that minimizes its losses bychoosing a nominal interest rate, it, subject to the zero bound. The central bank losses are func-tions of inflation and output gap, which also respond to the paths described by the NKPC and theDIS that belong to the private sector – and which are constraints in the CB’s problem. Moreover,problem (2.14) also depends on the treasury’s fiscal policy choices (which the central bank inter-27nalizes as they are made after the central bank chooses monetary policy). Finally, the central banktakes as given the future paths of the variables of the economy as it has no access to a commitmenttechnology that can affect them. With these results, we can now proceed to portray our definitionof equilibrium with a central bank with no forward guidance (No-FG) – which, for simplicity, I willinterchangeably call the discretionary central bank equilibrium.Definition (Equilibrium with no forward guidance (No-FG)). For every period t, an equilibriumwithout forward guidance consists of a set policy functions, {it, τt, `t, bt}∞t=0, and private sectorallocations, {yt, pit}∞t=0, such that:1. The central bank solves its optimization problem ((2.14) choosing a monetary policy {it}∞t=0,2. The treasury solves its optimization problem (2.13) choosing fiscal policy {τt, `t, bt}∞t=0, and3. Sequences {yt, pit}∞t=0 constitute a PSCE.2.3.2 Equilibrium characterizationIn this section I present the solutions to the problems outlined in Section 2.3.2. The solutionmethod builds on a standard guess-and-verify mechanism (see similar applications in Christianoet al., 2011), where the equilibrium variables of the economy are supposed to have a value associatedto each of the three states. Specifically, I shall argue that every variable xt in this economy adoptsthree different values, {xZ,xR, xS}. To pin down these values, I cast every optimization problem interms of each state, and linking the resulting optimality conditions of each problem yields a systemof equations whose solution verifies that the initial guess is correct.Throughout the equilibrium characterization, I assume that the first period is t = 0, and theinitial state of the world is s0 = Z. After this period, the shock vanishes, and the recovery statefollows with probability 1. If the ZLB episode does not occur again after the recovery state, theeconomy returns to the absorbing steady state (state S) with probability 1−p. With complementaryprobability, p, the ‘ZLB-Recovery’ tuple repeats for any pair of periods t = {k, k + 1}, with k even– Appendix A.1 describes details of the calculations in this section.Fiscal and monetary policies in each stateTreasury – State R. I first characterize the treasury’s solution. In every state, the treasury solvesa one-period minimization problem choosing fiscal instruments and the amount of debt to repay inthe next period, taking as given the central bank’s monetary policy. Throughout my exposition, Iassume that the treasury repays its debt one period after debt issuance.In the Recovery times state the treasury is assumed to: (i) Set zero lump-sum transfers, and(ii) retire past debt with taxation so as to carry no debt to the next period. Therefore, duringRecovery times, τt, is not controlled by the treasury (taxes only absorb past debt); i.e., bt and `tare fixed at 0. Specifically, in state R, the treasury’s problem (2.13) can be compactly re-expressedas28max{bR,`R,τR}−12(αTry (yR)2 + αTrτ (τR)2)subject toyR =yeR − (iR − pieR − rR) + γbRbR = (1 + r) bZ − τR + `R`R =0, bR = 00 ≤bR ≤ b¯, `R ≥ 0, τR ≥ 0where {iR, bZ , yeR, pieR, rR, rS} are given. Plugging the assumptions imposed on `R and bR, fiscalpolicy is determined bybR = 0 (2.15)`R = 0 (2.16)andτR = (1 + r) bZ (2.17)For positive levels of debt during the liquidity trap (i.e., bZ > 0), the last expression shows thatthe budget constraint instructs the treasury to use taxes during Recovery times to retire outstandingdebt plus interest. Combining this with the fact that yt is determined by the optimality conditionsof the private sector (DIS and NKPC) together with expectations of future variables (yet and piet ),and the central bank’s choice of optimal monetary policy (it), the minimization problem for thetreasury in state R is trivial. Thus, at the recovery state, the treasury’s losses becomelTrR ≡ −12(αTry (yeR − iR + pieR + rR)2 + αTrτ ((1 + rR) bZ)2)Central bank – State R. In the Recovery state, the problem for the central bank ismax{yR,piR,iR}−12(αCBy (yR)2 + αCBpi (piR)2)(2.18)subject topiR =κyR + βpieRyR =yeR − (iR − pieR − rR) + γbRiR ≥0and also subject to the treasury’s choice of debt that the central bank internalizes, given by Eq.(2.15). In addition, the central bank takes as given private sector expectations (which I denote ra-29tional expectations with pieζ ≡ Et {pit+1| st = ζ} and yet ≡ Et {yt+1| st = ζ}, where Et {xt+1| st = ζ}is the expectation of variable xt+1.). Calling λ3R the multiplier of the inequality constraint (the zerolower-bound on iR), F.O.C.s yield the well-known ‘output-gap targeting’ rule,yR = −αCBpi καCBypiR − 1αCByλ3R (2.19)with the complementary slackness condition λ3RiR = 0 – the solution is relegated to the Appendixsection A.1. Recalling that the central bank prefers both low inflation and output gap, Eq. (2.19)indicates that the central bank prevents large losses by stabilizing any changes in inflation withopposite changes in the output gap. Abstracting momentarily from the role of multiplier λ3R, Eq.(2.19) tells that the larger the importance of output gap relative to the importance of price stability(i.e., an increasing αCBy relative to αCBpi ), then the central bank prefers a lower output gap distortionyR relative to inflation piR. The nature of this response comes from the preference structure of themonetary authority: The quadratic nature of the central bank’s preferences exponentially penalizesthe distortions in each variable in response to the shock. It is in the interest of the central bank,then, to exploit the given economy’s environment (the NKPC) to transfer a portion of the increase inthe output gap to current inflation, so as to balance the dispersion by allocating it in two variables,pit and yt, rather than in the output gap alone. This will eventually decrease the magnitude of thedispersion of both variables and, hence, in central bank’s losses. Moreover, the central bank willbe able to further absorb that shock with a variable whose value is not relevant for the monetaryauthority, the nominal interest rate, as long as it does not hit the zero floor: When that is the case,the equivalence in Eq. (2.19) breaks down, as we shall see in the analysis of the economy in Z.The previous relation between output gap and inflation can be maintained with strict equalityas long as the ZLB constraint is not binding (λ3R = 0), which is the case when the economy is atthe Recovery state. Away from a liquidity trap, the optimality condition becomesyR = −αCBpi καCBypiR (2.20)The non-binding ZLB constraint means that at the prevailing rR the central bank can exactlyoffset changes in inflation with changes in output gap. The central bank achieves so by controllingthe nominal interest rate (yet to be determined but denoted iR now), which absorbs the currentrR to keep central bank losses low as we referred above. Specifically, using the NKPC inflation andoutput gap levels becomepiR =αCByαCBy + αCBpi κ2βPSpieR, yR = −αCBpi καCBy + αCBpi κ2βPSpieR (2.21)Plugging the optimality condition (2.21) in the DIS constraint yields the nominal interest rate30in the Recovery state implied by the targeting rule,iR = γ=0︷︸︸︷bR︸ ︷︷ ︸Fiscal policy+ yeR +(αCBy + αCBpi κ2 + αCBpi κβαCBy + αCBpi κ2)pieR︸ ︷︷ ︸Expectations effect+ rR︸︷︷︸Shock, iR ≥ 0 (2.22)The previous expression holds for given pieR, yeR, and bR (variable bR denotes the treasury’schoice of debt in state R to be paid in the next period), and where the latter inequality indicatesthat this nominal interest rate is not constrained by the ZLB. Finally, and in order to proceed withinterpretations below, observe that Eq. (2.22) explicitly shows bR (although bR is zero in state R).The central bank’s choice given by Eq. (2.22) shows three channels affecting the central bank’schoice of interest rates, iR. The first channel shows that that a higher natural interest rate risesnominal interest rates. For instance, a shock that drives up PS’s willingness expand consump-tion increases aggregate demand, and will therefore put upward pressure on prices. To preserveprice stability, the central bank rises nominal interest rates and, thus, it absorbs that response inaggregate demand.Second, fiscal policy can also affect positively the optimal choice of the nominal interest rate.The intuition is that if the treasury conducts an expansionary fiscal policy, then debt-financedlump-sum transfers will benefit living cohorts via a wealth effect captured with term γbR, whichincreases consumption – and, therefore, yR.28 But, at the same, if that level of output gap wereto exceed the level the central bank prefers, then the central bank rises iR to moderate the outputgap distortion.Finally, the third channel shows that the forward looking private sector’s expectations overoutput gap and inflation can pose inflationary pressures today. When prices are expected to risein the future, then the PS seizes a relatively larger purchasing power today that triggers currentconsumption today. But a larger consumption today translates into higher prices, and thereforethe central bank rises the nominal interest rate.Treasury – State Z. At the ZLB state, the treasury takes as given the central bank choicesand private sector expectations. In this state, however, the treasury finances lump-sum transferswith debt issuance instead of taxation – i.e., τZ = 0. For any k ≥ 1, call T = 2k the random dateat which the economy reverts back to the absorbing state S. Then, for all even periods t with0 ≤ t < T , I derive optimal fiscal policy for the liquidity trap state implementing the assumptionsfrom states Z and R, and adjusting the constraints accordingly in order to cast the treasury’sproblem in shorter form as a function of debt bZ . Therefore, the treasury’s problem (2.13) can becompactly re-expressed asmax{bZ ,`Z ,τZ}−12αTry (yeZ − iZ + pieZ + rZ + γbZ)2−βTr12(αTry (yeR − iR + pieR + rR)2 + αTrτ ((1 + rR) bZ)2)(2.23)28I refer to output gap and consumption interchangeably since market clearing holds in the DIS and NKPC.31subject to0 ≤bZ ≤ b¯lTrt+1 ≡lTrt+1 (yt+1, τt+1) ≡ lTrt+1(it+1, bt, bt+1, `t+1, yet+1, piet+1, rt+1)and given {rZ , iZ , yeZ , pieZ} (for the first term of the objective) and {iR, bR, `R, τR, yeR, pieR, rR} (forthe second term of the objective). It is worth pointing out that the choice of bZ could only affectperiods t and t + 1, and that (i) debt is assumed to be honoured in every period, but also that(ii) the treasury is discretionary (additional terms in the objective function belonging to periodst+ 2, t+ 3, ... are omitted).F.O.C. w.r.t. bZ yields the treasury’s optimal choice of debt at the ZLB state,bZ = Φ (iZ︸︷︷︸−Monet. pol.PS’s Expect.︷ ︸︸ ︷(yeZ + pieZ)− rZ)︸︷︷︸Shock(2.24)with Φ ≡ αTry γαTry γ2+βTrαTrτ (1+rR)2 , and where taxes are set at zero by assumption,τZ = 0 (2.25)Finally, from the treasury’s budget constraint we can recover the lump sum transfers, whichhave functional form exactly equal to bZ ,`Z = Φ (iZ − (yeZ + pieZ)− rZ) (2.26)The optimality condition (2.24) for the treasury depends on the interaction of three differentterms.Abstracting from the zero floor on nominal interest rates, note first that the interpretation of thelast negative term is straightforward: The more the PS wants to save today, the more expansionaryfiscal policies will be.Second, the positive sign on iZ (although iZ will be zero in equilibrium) implies that a moreexpansionary monetary policy during a recession (a lower iZ) triggers a less expansionary fiscalpolicy from the treasury. Intuitively, since a monetary expansion increases both inflation andoutput gap, the treasury needs to exert less fiscal effort after a shock to mitigate output drops thataffect its preferences. The monetary response, however, will have a shock portrayed by the ZLB.Finally, observe that PS’ expectations affect negatively debt choices. Since DIS and NKPC areforward looking, when the private sector expects inflation and/or output gap to be increasing inthe next period, then output today (which is the variable that matters for the treasury in its losses)deviates from its target in the same direction. In response to this, expression (2.24) instructs thetreasury to counter expectations of inflation and output growth with contractionary fiscal policy32in order to close the output gap today – and mitigate losses.Further interpretations can be drawn from analyzing the parameters involved in the coefficientof the treasury’s optimal choice of debt. Abstracting from PS’s expectations and CB’s policy choice,without loss of generality we can first multiply both sides of Eq. (2.24) by γ to obtainγbZ = γΦ (iZ − yeZ − pieZ − rZ)whereγΦ ≡ αTry γ2αTry γ2 + βTrαTrτ (1 + rR)2Since the LHS is the same term that enters in the state-Z Dynamic IS, the above expressionallows us to motivate an interpretation in terms of output gap – note that γbZ measures the impactof fiscal policy translated in terms of yZ . For any negative rZ , the numerator of the above expressionshows that the response of the LHS, γbZ , (measured in yZ terms as per the DIS) is increasing bothon (i) the treasury’s weight on output, αTry , and (ii) the magnitude of the wealth effect generatedby debt financed fiscal policy, γ2 – every unit of additional debt bZ maps into yZ in increments ofγ, and it is adjusted to the power of 2 since preferences are quadratic. However, the expansionaryimpact of fiscal policy generate a cost. In particular, this cost is portrayed by the denominator,where term αTry γ2 +βTrαTrτ (1 + rR)2 disciplines the magnitude of bZ since fiscal policy will have tobe repaid in Recovery times. Specifically, the second additive term in the denominator highlightsthat tomorrow’s (discounted) debt yields a quadratic cost (1 + rR)2, which will introduce furtherdistortions in the treasury’s loses tomorrow. Hence, debt-financed fiscal policy is less desirablethe more averse to tax collection costs the treasury is: If αTry is extremely high, then negative rZyields close to zero response of debt financed fiscal policy (and, therefore, the treasury does notmitigate a recession). On the contrary, if the treasury cares little about tax collection costs in theeconomy (αTrτ tends to 0), then the above expression simply becomes γbZ = iZ − yeZ − pieZ − rZand the treasury absorbs a larger proportion of the shock. In summary, Eq. (2.24) indicates thatthe choice of debt in one period inevitably affects relevant treasury’s outcomes in the next period,and because of that the treasury absorbs the shock by balancing the amount of distortion that itgenerates between these two periods.Central bank – State Z. When there is a large negative shock to the natural interest rate,the problem for the central bank in state Z is similar to problem (2.18). The differences, however,are that (i) now we are looking at the case where the ZLB constraint binds; λ3Z > 0, and (ii) thetreasury’s choice of debt is different from zero, with expression given by the optimality condition;(2.24).By the complementary slackness condition, a positive Lagrange multiplier implies thatiZ = 0 (2.27)33during the liquidity trap state, and the F.O.C. becomes(1− γΦ) (αCBy yZ + αCBpi κpiZ) < 0 (2.28)Eq. (2.28) states that the targeting rule of the Central bank fails to both absorb and balancethe fluctuations in inflation and the output gap when a shock to the natural interest rate drives thenominal interest rate to the ZLB. Linking the ZLB constraint in the DIS and the NKPC rendersthe associated inflation and output gap values in state Z,piZ = ((1− γΦ)κ) yeZ +((1− γΦ)κ+ βPS)pieZ + κ (1− γΦ) rZyZ = (1− γΦ) yeZ + (1− γΦ)pieZ + (1− γΦ) rZThe expressions for piZ and yZ described above are in stark contrast with our previous resultsfor piR and yR in (2.21): While those were only a function of PS expectations, the above expressionsfor piZ and yZ indicate that inflation and the output gap in state Z are also affected by the negativediscount factor shock, rZ , as well as parameters from fiscal policy embodied in Φ.Finally, and in order for the ZLB to bind in equilibrium, the shock needs to satisfy the followingtechnical condition (which is assumed to hold from now on).Condition (C1). The nominal interest rate in state Z, iZ , is zero ifrZ < −yeZ −(1 +αCBpi κβPSαCBy + αCBpi κ211− γΦ)pieZ . (2.29)Condition C1 characterizes which values of rZ make the ZLB bind. It is worth to point outthat moving all terms to the left-hand-side of the inequality yields the exact same expression of theoptimal nominal interest rate choice as in Eq. (2.22), with the only difference that it is evaluatedat the Z state instead of R,yeZ +(αCBy + αCBpi κ2 + αCBpi κβαCBy + αCBpi κ2)pieZ + rZ + γ=bZ︷ ︸︸ ︷Φ (−yeZ − pieZ − rZ) ≡ i˜Z (2.30)Expression (2.30) represents an implicit optimal nominal interest rate. It is implicit as it defineswhat the CB’s nominal interest rate choice would be in state Z, absent the ZLB constraint. Inequilibrium, however, i˜Z will not be allowed to happen: For given yeZ and pieZ , the shock, rZ , willbe negative enough to satisfy expression (2.29) which, in turn, means that iZ = 0 prevails. Tocomplete the requirement prescribed in Eq. (2.29), we need to calculate the equilibrium outcomesye, and pieZ – in my numerical exercise I verify that this condition holds in equilibrium.Treasury – State S. In state S the minimization problem of the treasury is trivial. Inparticular, during state S the treasury still pays past debt with taxation, and carries no debt tothe next period. Therefore, at any period t with state S that follows after an S- or R-state, fiscalpolicy is given bybS = 0 (2.31)34`S = 0 (2.32)τS = 0 (2.33)Central bank – State S. When the economy is at state S, it reaches an absorbing state(and there is no further uncertainty onward). Hence, at state S, the solution to the central bank’sgeneral problem (2.14) is similar to the solution in state R; i.e.,piS =αCByαCBy + αCBpi κ2βPSpieS , yS = −αCBpi καCBy + αCBpi κ2βPSpieS (2.34)andiS = yeS +(αCBy + αCBpi κ2 + αCBpi κβαCBy + αCBpi κ2)pieS + rS , iS ≥ 0 (2.35)Private sector – States {Z,R, S}. In each state, the private sector forms rational expectationsthat close the determination of the economy’s variables (pit, yt). Specifically, given the Markovstructure for the states of the world, when the PS makes its decisions in state R it forms rationalexpectations,yeR = pyZ + (1− p) yS and pieR = ppiZ + (1− p)piS (2.36)Eqs. (2.36) use that the economy transitions from state R to Z with probability p. (Afteranalyzing state S, each equality will later feature the fact that the value of the output gap andinflation rate in state S are zero.) Likewise, in states R and S the rational expectations of theprivate sector are such thatyeZ = yR and pieZ = piR (2.37)andyeS = yS and pieS = piS (2.38)Equilibrium without forward guidanceWe are now ready to characterize the equilibrium of the economy when the central bank is dis-cretionary and the treasury chooses debt-financed fiscal policy during liquidity traps to mitigaterecessions and deflations. Using a backward induction logic, I start characterizing the economy’sequilibrium at the steady state, and then implement this result into the equilibrium variables fromstates R and Z. In what follows, recall that I am attempting a solution to a discrete stochasticproblem using the linearized version of the equations of the economy (see, for example, Eggertsson,2011b). I use stars to denote equilibrium outcomes without forward guidance.Consider T a random period of time where the shock to the natural interest rate returns to itssteady state, rt = rS . Since at any period t ≥ T there is no uncertainty and the problem becomes35deterministic, the economy achieves the steady state equilibrium. In line with Eggertsson (2011a),our assumptions on fiscal and monetary policies imply that fiscal and monetary instruments areperfectly correlated with the shock. Hence, in every t ≥ T fiscal policy is set at (τt, `t, bt) = (0, 0, 0).It is worth pointing out that setups with Taylor rules augmented with a ZLB on nominal interestrates can lead to the existence of multiple equilibria – see, for example, Benhabib et al. (2001).These setups depict polar cases characterized by the zero-inflation and zero-output gap, and by aself-fulfilling deflationary equilibrium as the other one, namely, a perpetual liquidity trap. Moreover,this result is also present in a discretionary-central-bank setup like ours – See Appendix A.1. Tofocus our analysis of fiscal and monetary interactions only during liquidity traps, I assume that thecentral bank has the ability to guarantee the zero-inflation steady state – for a similar assumption,see Werning (2012); for a selection mechanism to rule out the self-fulfilling deflationary equilibrium,see Jung et al. (2005). Therefore, when the economy reaches the absorbing steady state, the naturalinterest rate rt = rS > 0, and the system of equations formed with the monetary and fiscal optimalpolicies in the S state (resp. Eqs. (2.35), (2.31), (2.33) and (2.32)), together with the economy’svariables yt and pit determined in S by the private sector’s optimality conditions DIS and NKPCand expectations (Eq. (2.38)) yields a steady state characterized by equilibrium outcomespi∗S = 0, y∗S = 0, i∗S = rS , τ∗S = 0, `∗S = 0, b∗S = 0, λ1,∗S = 0, λ2,∗S = 0, λ3,∗S = 0 (2.39)(where the PS rationally expects (piet , yet ) = (piS , yS) = (0, 0)) – see Appendix section A.1. It isworth pointing out that the solution highlights that the central bank achieves zero inflation andzero output gap by fully absorbing the shocks to the economy with the nominal interest rate.Finally, given our model assumptions together with Condition (2.29), the system of equationsformed with the steady state outcomes (2.39) and the monetary and fiscal optimal policies in theZLB state and in the Recovery state (resp. Eqs. (2.27), (2.24), (2.25), (2.26) and Eqs. (2.22),(2.15), (2.17) (2.16)), together with the economy’s variables yt and pit determined by the privatesector’s optimality conditions DIS and NKPC in the ZLB state and in the Recovery state, andprivate sector expectations in the ZLB state and in the Recovery state (resp. Eq. (2.37) andEq. (2.36)) render the equilibrium outcomes of the model economy when the central bank cannotimplement credible forward guidance.Proposition 2.3.1. The solution to the system formed with the equations characterizing the econ-omy when the central bank cannot implement credible forward guidance during a liquidity trap isdefined by the following equilibrium outcomes: In states Z, R, and S,pi∗Z = θpiZrZ , y∗Z = θyZrZ , i∗Z = 0, b∗Z = θbZrZ , `∗Z = θ`ZrZ , τ∗Z = 0,pi∗R = θpiRrZ , y∗R = θyRrZ , i∗R = θiRrZ + rR, b∗R = 0, `∗R = 0, τ∗R = θτRrZ ,pi∗S = 0, y∗S = 0, i∗S = rS , b∗S = 0, `∗S = 0, τ∗S = 0,with state-Z coefficients θpiZ ≡[αCBy + αCBpi κ2]κυ/φ, θyZ ≡ [αCBy (1−(βPS)2p) + αCBpi κ2]υ/φ, θbZ ≡36−αTry γ(αCBy (1−p(βPS)2)+κ2αCBpi )/φ, and θ`Z = θbZ , and state-R coefficients θpiR ≡[αCBy]κβPSpυ/φ,θyR ≡ −[αCBpi κ2]βPSpυ/φ, θiR ≡ (αCBy ((1−(βPS)2p) + κ) + αCBpi κ2(1 + κ+ βPS))pυ/φ, andθτR ≡(−αCBy((1− p (βPS)2) γ2αTry + (1− pβPS (βPS + κ)) υ)− αCBpi κ2 (γ2αTry + υ)) (1 + rR) γαTryφ(αTry γ2 + υ)where υ ≡ αTrτ βTr(1 + rR)2 and φ ≡ αCBy ((1 − p(βPS)2)γ2αTry + (1 − pβPS(βPS + κ))υ) +αCBpi κ2(γ2αTry + (1 + βPSp)υ).Proof. See proof in Appendix A.1.The previous result characterizes the equilibrium outcomes of the economy in closed form. TheProposition also proves our initial guess from our solution method that there exist solutions forthe values of the variables at different states. The following remarks from Proposition 2.3.1 arein order. First, observe that the numerator of variables yR and piR depend on the probability ofreversion to the ZLB, p, while variables yZ and piZ do not. In particular, observe that yR and piRdrops to zero as this probability approaches zero from above. This result is compatible with theliterature (e.g., see Eggertsson (2011b)), and it implies that, when p → 0, then the economy atstate R replicates the steady state outcomes. Intuitively, this means that households and firms inthe Private Sector expect the shock to vanish, which makes them expect the central bank to bealways able to absorb shocks with the nominal interest rate – and, therefore, zero inflation andoutput gap are realized.Second, observe also that, apart from their difference in p, the numerators of inflation andoutput gap, piR and yR, have: (1) A common factor in the numerator, αTrτ βTr (1 + rR)2; and (2) adiscrepancy captured by the parameters in square brackets. These two factors reveal that inflationand output gap outcomes at each period t feature a combination of a ‘fiscal policy content’ (termαTrτ βTr (1 + rR)2) and a ‘monetary policy content’ (the non-common factor in square brackets),where the latter differs with the state. When monetary policy is away from the ZLB, the equilibriumnominal interest rate, i∗R, is chosen by the central bank to absorb the current natural interest rateshock. This way, the central bank can optimally balance the contemporary feedback channel thatexists between inflation and output gap at the current period (given by the slope, κ, of the NKPC),where the only feedback channel between inflation and output gap that the CB cannot affect is theexpectations channel (due to the discretionary nature of the monetary authority). On the contrary,at the ZLB, the equilibrium nominal interest rate, i∗Z , falls short of absorbing the exogenous shock,rZ , and therefore the monetary policy content of inflation and output gap is a function of bothinflation and output gap parameters simultaneously. This is because the CB exhausts the nominalinterest rate when it reaches the zero floor in its attempt to prevent the contemporary feedbackchannel from the Phillips curve: The nominal interest rate becomes zero, and the unabsorbedportion of the shock to rt cascades back to y∗Z , and therefore the CB cannot even prevent thecontemporaneous feedback existing between yt and pit via the NKPC that adds up an additional37distortion to its losses – and which also adds up to the already-existing distortion in inflation andoutput gap due to the expectations channel.Finally, observe that the wealth effect from debt-financed fiscal policy that enters in the DIS,γ, directly affects the choice of debt from the treasury, b∗Z , as well as the equilibrium outcomes forinflation and the output gap. In particular, observe that, if γ = 0, then b∗Z = 0. The intuition is that,if there is no wealth effect to the PS coming from fiscal policy, then it is not in the treasury’s interestto worsen its losses (via taxes) by issuing lump-sum transfers backed with debt during a liquiditytrap. This conclusion is compatible with Ricardian economies, and it would be equivalent to havingan economy where only the central bank can stabilize macro variables. But, by the same token, asγ approaches zero, output gap and inflation also stop being sensitive to fiscal policy. Specifically,as γ → 0, then φ → [αCBy (1− pβPS (βPS + κ))+ αCBpi κ2 (1 + βPSp)] {αTrτ βTr (1 + rR)2}, andthe factor in braces cancels out with the same factor present in each numerator of the equilibriumoutcomes – the fiscal policy content of (y∗Z , pi∗Z , y∗R, pi∗R). As a result of this, variables in Proposition2.3.1 are independent of any fiscal parameter, and our equilibrium characterization matches withthat in the literature where a discretionary central bank faces deflation both at the ZLB (Eggertssonand Woodford (2003)), and away from the ZLB (Nakov (2008); Nakata (2018); Walsh (2018)).When γ > 0, instead, the impact of fiscal policy will be relevant to the economy’s variables.The next section introduces a graphical analysis of these results that will help identify the responseof the equilibrium outcomes to a large negative shock when the central bank cannot implementcredible forward guidance.2.3.3 Numerical exerciseAssuming that the central bank can only use the nominal interest rate and has no access to forwardguidance, I now proceed to show paths for the equilibrium outcomes in periods 0 and 1 – whichrepresent the equilibrium outcomes for any pair of consecutive periods with states (Z,R). I plotfigures associated to a numerical example with quarterly parameters set to standard values inthe literature – it is worth pointing out that the following parameterization does not represent acalibration as some of the economy’s parameters are freely chosen. See table 2.1 for more details.Weights αCBpi and αTrτ are set to 1.5 and 0.5 relative to an output gap weight normalized at 1.This choice aims to capture a conservative monetary authority (i.e., a central bank relatively moreconcerned with price stability), and a fiscal authority less concerned about tax distortions relativeto output gap distortions. I set the discount factor β to 0.99, which is equivalent to a steady statenatural interest rate level (annualized) of approximately 4 percent. In order to generate a recessionand deflation at the ZLB and along the lines of Christiano et al. (2011), I assume a shock thatincreases households’ discount factor and, therefore, lowers the natural interest rate, r, to −0.02(annualized). The shock lasts for one period, and after that period the natural interest rate revertsback to r ≡ − lnβ in states {R,S}. In the model I also set the coefficient of government debt inthe dynamic IS curve, γ, at values 0 (inactive fiscal policy) and 0.5 (active fiscal policy case),29 and29The parameterization of γ was chosen for convenience in order to magnify the responses. The effects are qualita-38κ to approximately 0.033. Finally, in state R, the ZLB occurs again with probability p set at 0.2.Starting from the zero inflation and zero output gap steady state, Figure 2.3 shows the equilib-rium outcomes for a sudden negative shock to the natural interest rate at time 0, which reverts backto r in period 1 (Recovery state). As portrayed in period 0 from Panel C and Panel D, the negativedemand shock triggers an immediate expansionary response from fiscal and monetary authorities.Note, however, that Condition (2.29) holds, and so the negative demand shock drives the economyto a liquidity trap. Therefore, in period 0 we observe that the Central bank sets zero nominalinterest rates (Panel C), and the treasury finds it optimal to implement debt-financed lump-sumtransfers to mitigate the fall in output (Panels D and F). Note, however, that the counterpartof expansionary fiscal policy is a future rise in taxes (Panel E), and when the treasury suffers apositive cost αTrτ from distortionary fiscal policy, the next-period rise in taxes triggers losses tothe treasury. This means that, in equilibrium, the treasury does not counter the entire decline inoutput with expansionary fiscal policy, and this translates into deflation and recession at the timeof the shock (Panels A and B).The simulations in Figure 2.3 show that, under inactive fiscal policy (γ = 0), the drop ininflation and output gap are worse relative to their active fiscal policy counterpart (γ > 0). Instate Z, and after a negative shock to rt, the blue solid and the green dashed lines in Figure 2.3show that the existence of the ZLB constraint contributes to deflation during a liquidity trap.30In fact, the deflationary outcome at the ZLB is explained by the optimal CB response: When anegative shock to the natural interest rate forces the central bank’s corner solution (i.e., i∗Z = 0),variable pi∗Z inevitably absorbs part of the shock – and deflation emerges in equilibrium. This resultaligns with the literature that observes deflation not as the result of a monetary policy mistake, butrather the result of a central bank that optimizes subject to credibility constraints (see Eggertsson,2006). Analytically, result i∗Z = 0 is because our parameterization satisfies condition C1 (Expression(2.29)): The shock to the natural interest rate is negative enough to leave the monetary authoritywithout ammunition to prevent the entire shock, and therefore the portion of the shock that isnot absorbed creates recession and deflation.31 Eventually, there is a fraction of this portion ofthe shock that the treasury would like to mitigate when its policy is effective – as we shall seewhen γ > 0. Note also that in state R deflation is observed again, and it is linked to the centralbank’s inability to commit to a certain monetary policy – also labelled by Nakov (2008) as the‘deflationary bias in expectations’. Specifically, the central bank cannot commit to inflate in otherperiods, and this happens due to the forward-looking behavior of the PS that anticipates that theZLB (and, therefore, deflation) may repeat in the future. As with some probability there mightbe a shock tomorrow, this shock may trigger a larger desire to save in that period. But since thetively the same when carried out using parameterizations from the literature (e.g., 0.011 from Devereux (2010); and0.01 from Kirsanova et al. (2005)).30This phenomenom so-called deflation bias of optimal monetary policy has been formalized in the literature (seeEggertsson (2006)), and it has been shown to also be present both during and after liquidity traps (see, for instance,Nakov (2008); Nakata and Schmidt (2019); Walsh (2018)).31This result aligns with the literature in that deflation is not the result of a monetary policy mistake, but ratherthe result of a CB optimizing subject to constraints – see Eggertsson (2006).39shock is large, the PS knows that the economy will be again in a liquidity trap with its associateddeflation. This further deters the PS from spending even when the economy is away from the ZLB,which triggers deflation even at the Recovery state.The difference between the green dashed and the blue solid lines is captured by parameter γ.When γ > 0, the blue solid lines show the impact of fiscal policy in the economy. The first takeawayis that expansionary fiscal policy introduces changes in the outcomes of both states. Specifically,in state Z, fiscal policy mitigates a portion of the deflation in state Z (blue solid line above greendashed line; i.e., less negative inflation), and it closes output gap distortions (blue solid line abovegreen dashed line in Z, meaning less recession). From the treasury’s perspective, however, the fiscalauthority will not counter the shock in full extent as it has to balance its preference for outputdrops with its dislike for tax distortions. This is because the treasury bears the total cost (in termsof tax distortions) of any expansionary fiscal policy implemented. Finally, in state R, inflationoutcomes improve again (blue solid line above green dashed line; i.e., less negative inflation), aswell as output gap (blue solid line below green dashed line in R, meaning a smaller distortionrelative to the targeted 0 output gap target). However, we observe that this circumstance emergeswith a simultaneous rise in the nominal interest rate, compared to the case of no fiscal policy, γ = 0.The intuition is that in Z states active fiscal policy mitigates the recession and deflation, which alsoimprove outcomes in R states via expectations. But as fiscal policy introduces gains in terms ofinflation and output gap in state R, it eventually releases the pressure on the nominal interest rateinstrument. Eventually, this release of pressure frees the central bank from using i∗R to operate inview of the possibility of deflation in state Z, and it thus allows the monetary authority to absorba larger portion of the state-R natural interest rate to prevent that a larger portion of rR cascadesback to y∗R – and, eventually, to pi∗R.2.4 Forward guidance and fiscal policyIn the previous section I characterized an economy with a central bank that can only use the nominalinterest rate as the monetary policy instrument during a recession. In this section I analyze the samesetup but when a central bank has the ability to conduct credible forward guidance. In particular,I study whether the central bank can implement credible forward guidance in the presence of fiscalpolicy, and how forward guidance interacts with fiscal policy compared to the equilibrium from theprevious section.In this chapter I define forward guidance as a nominal interest rate that the central bankpromises in states where there is a liquidity trap to implement during Recovery times. Specifically,in the equilibrium I seek to build, I want to characterize credible nominal interest rates for Recoverytimes that are lower relative to the nominal interest rate without access to forward guidance. Bycredible nominal interest rates I mean rates that the central bank announces and that the PrivateSector rationally expects to be implemented after a liquidity trap ends. Since the central bankacts under discretion, forward guidance will be made credible using an equilibrium concept that40exploits the repeated structure of the fiscal-monetary interaction called a sustainable equilibrium.Firstly introduced by Chari and Kehoe (1990), this concept has been widely used in the literaturethat studies equilibria when the central bank cannot access commitment technologies but it can usereputation to implement monetary policy. This is why the equilibrium is interchangeably referredto as reputational equilibrium. In a nutshell, now past actions matter, and the objective is to allowthe central bank to conduct forward guidance based on a trigger strategy that involves revertingto the no forward guidance scenario from the previous section (i.e., where the central bank lackscredibility) if the central bank ever deviates. In this model I assume the PS punishes the centralbank forever in case it deviates. Hence, if the central bank makes an announcement but it thendeviates to the discretionary solution (i.e., it chooses the policy prescribed by the solution to itsloss minimization problem instead of implementing the forward guidance level), then the centralbank loses credibility forever – and the PS no longer believes the central bank, meaning that theeconomy reverts to the solution without forward guidance from Proposition 2.3.1 onward.This section is organized as follows. First, I introduce relevant concepts to characterize reputa-tional equilibria. This part will be similar to other works in the literature (e.g., Kurozumi (2008)).Specifically, it will follow very closely the exposition from Nakata (2018), but it differs from thiswork in that I analyze the impact of fiscal parameters on the credibility of monetary policy. Sec-ond, I will numerically explore whether an equilibrium with credible forward guidance exists whenthere is fiscal policy mitigating a recession. Since the answer will be that there exists such credibleforward guidance equilibrium, I shall proceed to analyze how sensitive the credibility of forwardguidance is to fiscal policy parameters. Finally, I will analyze whether there are any relevant fiscaland monetary policy interactions that describe the type of response (namely, substitutability orcomplementarity) of a central bank and a treasury to recurring liquidity traps events.2.4.1 The policy gameIn what follows, I borrow the sustainable equilibrium design portrayed in Kurozumi (2008) andNakata (2018), and follow their exposition very closely to characterize sustainable equilibria. It isworth to point out, however, that in this chapter I depart from the setup in Nakata (2018) (whomodel the repeated interaction between a central bank and infinitely many small private agentsthat are strategically negligible) in that I append a treasury to this description – see Basso (2009)for a similar case with two big policymakers.32Actions and histories. I shall denote with at =(aCBt , aTRt , aPSt)the vector of actions takenby the central bank, the treasury and the private sector, respectively, where aCBt is defined byaCBt ≡ it ∈ R≥0, aTrt is equivalent to aTrt ≡ (bt, τt, `t) ∈ R3≥0, and aPSt denotes aPSt ≡ (yt, pit) ∈ R2.33Since now competitive equilibrium allocations will depend on these fiscal and monetary policyactions as well as on the exogenous state variable rt (linking states of the world {Z,R, S} to values32For the sake of exposition, I also borrow the notation used in Kurozumi (2008).33Private sector actions will be skipped from histories since the PS is strategically negligible – for a further discus-sion, see Chari and Kehoe (1990).41r < 0 and r > 0), I also need to define histories that keep track of past actions and rt. Forevery t ≥ 0 I formally define a history of the game, ht, as ht =(ht−1, aCBt−1, aTrt−1, rt)– where Iassume h−1 = ∅, aCB−1 = ∅, and aTr−1 = b−1 = 0, and h0 = r0. Given the recursive structure ofht, it can be shown that ht =((aCB−1 , aCB0 , ..., aCBt−1),(aTr−1, aTr0 , ..., aTrt−1), (r0, ..., rt)); i.e., historiesare a function of (i) past monetary and fiscal actions, and (ii) past and present exogenous states.Finally, due to the timing of decisions, we define a current history for k = {CB, Tr, PS} as thevector formed with histories and contemporary events observed by k, and up to the time k makesa decision. Specifically, I shall denote a current history with hkt , where hCBt ≡ ht, hTrt ≡(ht, aCBt),and hPSt ≡(ht, aCBt , aTrt).34Strategies, continuation strategies and future histories. Be{σCB, σTr, σPS}the strate-gies for the central bank, the treasury and the private sector. A strategy σk represents a sequenceof functions mapping histories for k = {CB} (or mapping histories and contemporary actionsfor k = {Tr, PS}) into time-t actions. Formally, for k = {CB, Tr, PS}, a strategy for every tis defined by σk ={σkt (·)}∞t=0with element σkt (·) of the sequence of functions described as fol-lows. First, given the CB’s current history, hCBt ≡ ht, the Central bank uses σCB to (i) setthe current nominal interest rate; σCBt(hCBt)= aCBt ≡ it, and (ii) set future nominal interestrates for every future history;{σCBt′(hCBt)}t′>t – future histories are defined later. Second, giventhe treasury’s current history, hTrt ≡(ht, aCBt), the treasury uses σTr to (i) set the current fis-cal policy; σTrt(hTrt)= aTrt = (bt, τt, `t), and (ii) set future fiscal policy for every future history;{σTrt′(hTrt)}t′>t . Third, given the PS’s current history, hPSt ≡(ht, aCBt , aTrt), the PS uses σPS to(i) set the current inflation and output gap; σPSt(hPSt)=(σPS,pit(hPSt), σPS,yt(hPSt))= aPSt =(pit, yt), and (ii) set future inflation and output gap for every future history;{σPSt′(hPSt)}t′>t.In summary, the element σkt (·) of sequences σk is defined as σkt(hkt)= akt . In addition, givenstrategies{σCB, σTr, σPS}, we call σkt the continuation strategy from a current history hkt fork = {CB, Tr, PS}. Formally, from any current history hkt for k = {CB, Tr, PS}, these are de-fined as{σCBt , σTrt , σPSt}={{σCBt′(hCBt)}t′≥t ,{σTrt′(hTrt)}t′≥t ,{σPSt′(hPSt)}t′≥t}for every fu-ture history. Continuation strategies determine current and future monetary policy, fiscal policyand output gaps and inflation rates for every future history. Finally, I define future histories ashistories generated by{σCB, σTr}. Formally, ht+1 =(ht, σCBt(hCBt), σTrt(hTrt), rt+1), for everyt ≥ 0.The previous notation makes explicit fiscal, monetary and private sector actions as functionsof past (or past and current) actions and past and current exogenous states. Having definedthese objects, I now present the central bank, treasury and private sector problems using thisnotation, and then introduce the sustainable equilibrium concept. In particular, I will now showhow{σCB, σTr, σPS}are determined. First, I shall start with σPS for given{σCB, σTr}. To dothis, I focus on the determination of the continuation strategy σPSt standing at any current historyhPSt for every future history. Rewriting the continuation strategy σPSt as σPSt(hPSt) ∪ σPSt+1, then34It is important to stress that the Central Bank, the Treasury and the Private Sector observe different historiesrelative to each other; note that Dong and Young (2019)call this an ‘expanded history’. See also Gnocchi andLambertini (2016) for a similar exposition.42given any current history hPSt , σPSt must be such that the current strategy at t, σPSt(hPSt) ≡{σPS ,pit(hPSt), σPS,yt(hPSt)}, satisfies the optimality conditions depicted by the NKPC and theDIS (Eqs. (2.1) and (2.2)),σPS ,pit(hPSt)= κσPS ,yt(hPSt)+ βEt[σPS ,pit+1(hPSt+1)](2.40)σPS,yt(hPSt)= Et[σPS ,yt+1(hPSt+1)]− (it − Et [σPS ,pit+1 (hPSt+1)]− rt)+ γbt (2.41)for all future histories ht+1 induced by{σCB, σTr}, and the continuation strategy σPSt+1 ≡{σPS,pis(hPSs), σPS,ys(hPSs)}s≥t+1satisfiesσPS ,pis(hPSs)= κσPS ,ys(hPSs)+ βEs[σPS ,pis+1(hPSt+1)](2.42)σPS,ys(hPSs)= Es[σPS ,ys+1(hPSs+1)]− (σCBs (hCBs )− Es [σPS ,pis+1 (hPSs+1)]− rs)+ γσTrs (hTrs ) (2.43)for all future histories hs induced by{σCB, σTr}with s ≥ t+ 1.For the treasury, standing at any t, the fiscal policy continuation strategy σTrt ={σTrs(hTrs)}∞s=tsolves the following problem: Given a current history hTrt and strategies{σPS , σCB}, then thetreasury doesmax{σ˜Tr,τk ,σ˜Tr,bk}∞t{σ˜Tr,`k}∞t−12Et{ ∞∑k=tβk−t[αTry[σPS ,yk(hk,(σCBk(hCBk), σ˜Trk(hTrk)))]2+ αTrτ[σ˜Tr,τk(hTrk)]2]}(2.44)subject to the dynamic IS curve,σPS ,yk(hk,(σCBk(hCBk), σ˜Trk(hTrk)))=Ek[σPS ,yk+1(hk+1,(σCBk+1(hCBk+1), σ˜Trk+1(hTrk+1)))]−(σCBk(hCBk)− Ek [σPS ,pik+1 (hk+1, (σCBk+1 (hCBk+1) , σ˜Trk+1 (hTrk+1)))]−rk) + γσ˜Tr,bk(hTrk)the treasury’s budget constraintσ˜Tr,bk(hTrk)= σ˜Tr,`k(hTrk)+ (1 + r) σ˜Tr,bk−1(hTrk−1)− σ˜Tr,τk (hTrk )and constraints0 < σ˜Tr,bk(hTrk) ≤ b¯ <∞, σ˜Tr,`k (hTrk ) ≥ 0, σ˜Tr,τk (hTrk ) ≥ 0σ˜Tr,`k(hTrk)1{r 6=r} = 0, σ˜Tr,bk(hTrk)1{r 6=r} = 0, σ˜Tr,τk(hTrk)1{r=r} = 0for all future histories hk induced by{σCB, σTr}with k > t.43Finally for the central bank, standing at any t, the monetary policy continuation strategyσCBt ={σCBs(hCBs)}∞s=tsolves the following problem: Given a current history hCBt and strategies{σPS , σTr}, then the CB doesmax{σ˜CB,ik }∞k=t− 12Et{ ∞∑k=tβk−t[αCBy[σPS ,yk(hk, σ˜CBk(hCBs), σTrk(hTrs))]2+ (2.45)αCBpi[σPS ,pik(hk, σ˜CBk(hCBs), σTrk(hTrs))]2]}subject toσPS ,pik(hk,(σ˜CBk(hCBk), σTrk(hTrk)))=κσPS ,yk(hk,(σ˜CBk(hCBk), σTrk(hTrk)))+βEk[σPS ,pik+1(hk+1,(σ˜CBk+1(hCBk+1), σTrk+1(hTrk+1)))]σPS ,yk(hk,(σ˜CBk(hCBk), σTrk(hTrk)))=Ek[σPS ,yk+1(hk+1,(σ˜CBk+1(hCBk+1), σTrk+1(hTrk+1)))](σ˜CBk(hCBk)− Ek [σPS ,pik+1 (hk+1, (σ˜CBk+1 (hCBk+1) , σTrk+1 (hTrk+1)))]−rk) + γσTr,bk(hTrk)for all future histories hk induced by{σCB, σTr}with k > t.It is important to note that the solutions to problems (2.44) and (2.45) are sequences of functions(i.e., strategies) for the central bank and the treasury with: (1) Arguments depending on histories,and (2) solutions involving elements dated at future events. As a result, the solutions depend onthe history restrictions observed by the optimizing policymakers. For example, we may consider thepolar cases of histories based only on current occurrences of the shock (Markov), or histories withsome finite memory. Also, the solutions to these problems will vary depending on the policymakers’ability to affect either future expectations for every future period (i.e., for every future element ofthe sequences forming the policymakers’ strategies), or for a limited number of periods.Having described the strategies for each player and structure of histories, I now define a sus-tainable equilibrium for the model economy.Definition. A sustainable equilibrium (SE) of the model (denoting sequences and their corre-sponding elements with super-index S) is a triple{σCB,S , σTr,S , σPS,S}such that for every historyht the following holds:SE1. given{σCB,S , σTr,S}, the continuation strategy of the private sector, σPS,St , satisfies rationalexpectations and the NKPC and the DIS (Eqs. (2.40)-(2.43)) for every current history hPSt ,SE2. given{σPS,S , σTr,S}, the continuation strategy of the central bank, σCB,St , solves the centralbank’s problem (2.45) for every current history hCBt , andSE3. given{σPS,S , σCB,S}, the continuation strategy of the treasury, σTr,St , solves the treasury’sproblem (2.44) for every current history hTrt .44Intuitively, the definition states that, for any history ht, then eachcontinuation strategy fork = {CB, Tr, PS} associated to k’s own strategy prescribes a best response to the strategies fromthe other actors in the economy. The definition is in the spirit of Chari and Kehoe (1990), and itadapts Kurozumi (2008) to depict sustainable equilibria in the presence of another policymaker –see, for example, Basso (2009). For early applications of sustainable equilibria applied to crediblemonetary policy, see for example Chari et al. (1998); for more recent examples, see Nakata (2018)and Walsh (2018).Finally, and before turning to the characterization of sustainable equilibrium from the nextsection, it will be useful to introduce the following object. Standing at any t, and given a currenthistory ht and strategies{σPS , σTr}, callV CB,σ˜CBt ≡ −12Et{ ∞∑k=tβk−t[αCBy[σPS ,yk(hk,(σ˜CBk (hk) , σTrk (hk)))]2+ (2.46)αCBpi[σPS ,pik(hk,(σ˜CBk (hk) , σTrk (hk)))]2]}whereσPS ,pik(hk,(σ˜CBk(hCBk), σTrk(hTrk)))=κσPS ,yk(hk,(σ˜CBk(hCBk), σTrk(hTrk)))+βEk[σPS ,pik+1(hk+1,(σ˜CBk+1(hCBk+1), σTrk+1(hTrk+1)))]the value of a monetary policy continuation strategy σ˜CBt ={σ˜CB,is (hs)}∞s=tfor the central bank,for any future history hk at periods k > t that is induced by {{σCBs (hs)}∞s=t,{σTrs (hs)}∞s=t}.2.4.2 Sustainable equilibrium characterizationI now proceed to characterize credible FG in the model economy. To that end, I introduce the nextProposition.Proposition 2.4.1. The No-FG equilibrium is the worst sustainable equilibrium.Proof. See proof in Appendix A.2The above proposition is part of a three-step approach commonly followed in the literature ofsustainable equilibria – see, for instance, Chari and Kehoe (1990), Kurozumi (2008), Basso (2009),and Nakata (2018). Therefore, first, Proposition 2.4.1 argues that the discretionary central bankequilibrium from the previous section is also a sustainable equilibrium. Moreover, Proposition 2.4.1argues that it is the worst SE. Next, and as part of the second step, I proceed to define a triggerstrategy that uses such discretionary equilibrium as the scenario towards which the economy revertsupon central bank deviation from its promised FG level. In the last step, this strategy will finallybe used to characterize credible FG policies.45Credible forward guidance will impact differently on the economy’s outcomes, and will be imple-mentable as long as they are credible. To characterize equilibrium outcomes associated to credibleforward guidance, I use a trigger strategy equilibrium along the lines of Nakata (2018). Specifically,I describe a trigger strategy first, and then present a proposition that allows me to characterize theequilibrium outcomes associated with this trigger strategy. For simplicity, I use the name revert-to-discretion strategy for this trigger strategy, similar to that in Nakata (2018), and denote it with{σCB,fg, σTr,fg, σPS,fg}. Sequence{σCB,fg, σTr,fg, σPS,fg}is described as follows: The centralbank chooses(CB1) σCB,fgt(hCB0)= ifg0(hCB0)for every initial state in S(CB2) σCB,fgt(hCBt)=ifgt(hCBt), if ik = ifgk(hCBk)for every k ≤ t− 1i∗t(hCBt), otherwiseThe revert-to-discretion strategy also instructs the treasury to perform the following actions:(Tr1) σTr,fgt(hTrt)=(bfgt(hTrt), τ fgt(hTrt), `fgt(hTrt)), if ik = ifgk(hTrk)for every k ≤ t− 1(b∗t(hTrt), τ∗t(hTrt), `∗t(hTrt)), otherwiseFinally, the Private Sector strategy is(PS1) σPS,fgt(hPSt)=(yfgt(hPSt), pifgt(hPSt)), if ik = ifgk (hk) for every k ≤ t(y∗t(hPSt), pi∗t(hPSt)), otherwiseIn the No-FG equilibrium, the central bank solved its sequential problem in each state: Inparticular, the central bank chose zero interest rates in the liquidity trap state Z, and it was free tochoose nominal interest rates in state R. In that equilibrium, however, the central bank was unableto affect Private Sector expectations. The trigger strategy of this section now instructs the centralbank to give up its ability to implement the discretionary solution in state R, while at the sametime it gains the ability to make promises about future paths of nominal interest rates that thePrivate Sector can rationally expect will be followed by the monetary authority. For the PrivateSector, its trust on the discretionary CB’s promises lasts as long as the CB keeps its reputation ofimplementing its present and past promises, and it stops trusting the CB otherwise. In particular,when the latter happens, the Private Sector instantaneously expects the discretionary outcome tohold in the future, and this scenario follows in the model economy thereafter.Equipped with the previous definitions and objects, the next Proposition provides conditionsthat characterize economy’s outcomes associated to the sustainable equilibria that use trigger strat-egy outlined above.Proposition 2.4.2. Be an arbitrary set of fiscal and monetary policies, allocations and inflationrates,(iSt , bSt , τSt , `St , ySt , piSt). This set is the outcome of an SE if and only if for t ≥ 0,1.{ySt , piSt}is a PSCE,462.{bSt , τSt , `St}solves the optimization problem of the treasury, and3.{iSt , bSt , τSt , `St , ySt , piSt}satisfies the following sustainability constraint (SC)V CB,St(iSt , bSt , τSt , `St , ySt , piSt) ≥ V CB,∗t (i∗t , b∗t , τ∗t , `∗t , y∗t , pi∗t ) (2.47)where V CB,St represents the time-t present discounted value (PDV) of the central bank’s lossesunder the sustainable equilibrium (i.e., under credible FG), and V CB,∗t represents the time-t PDV of the central bank losses under the discretionary equilibrium (i.e., in the No-FGequilibrium).Proof. See proof in Appendix A.2The first and second requirements call the equilibrium to be a competitive outcome, and tobe consistent with the treasury’s optimization problem. The last requirement from Proposition2.4.2 prescribes that the central bank will follow an arbitrary path for nominal interest rates aslong as that path allows the central bank to attain payoffs that are greater (or at least equalto) the payoffs under discretion. This requirement is particularized by the satisfaction of theSustainability Constraint (SC for simplicity) given in expression (2.47). Next, I focus on buildingthe SC constraint.It is important to note first that the only relevant Sustainability Constraint (Eq. 2.47) neededto characterize sustainable equilibria in this model is the SC at state R. The reason for this is that,in periods where the state is Z, the central bank has no other choice than to set iZ = 0. Specifically,the shock parameterization will always be large enough to make the nominal interest rate in stateZ (absent any ZLB constraint) to become negative. But the CB cannot choose negative nominalinterest rates because the ZLB binds. Thus, in period Z, we are assured that the discretionaryCB will have no incentives to deviate. Furthermore, if the CB cannot deviate (and absent anydeviation in the past), then the private sector can only rationally expect the CB to preserve itspromise. Hence, SC only matters for periods where the state is R (and, in that case, the CB canindeed deviate or not).To build the Sustainability Constraint (Eq. 2.47), observe first that in the discretionary equilib-rium the strategy profile{σCB, σPS , σTr}is given by{σCB,∗, σPS,∗, σTr,∗}, with actions prescribedby Proposition (2.3.1). Now, given these actions, the per-period losses for the central bank underdiscretion are time invariant in each state of the world. In other words, we can present a recursiveversion of the discounted value of the losses, V CB,∗t , standing at any period t and for each stateof the economy. Since we are interested in evaluating the optimal policies that can be sustainedduring state R (which is the state where the central bank can deviate since the ZLB constraint isno longer binding) we can write the value of the losses for the discretionary central bank at R as47follows (the step-by-step procedure is detailed in the Appendix),V CB,∗R ≡ lCBR (y∗R (i∗R) , pi∗R (i∗R)) + βplCBZ (y∗Z (i∗R) , pi∗Z (i∗R)) + βlCBR (y∗R (i∗R) , pi∗R (i∗R))1− β2p (2.48)Expression (2.48) particularizes the right-hand-side of the SC. Furthermore, since alternativeequilibrium outcomes that we attempt to sustain will depend on forward guidance levels (i.e., onnominal interest rates at state R, iSR), then expression (2.48) has also been written as an explicitfunction of the discretionary nominal interest rate at R, i∗R.Just like we did for the value of the losses for the CB at R under discretion, we build the samevalue for any arbitrary forward guidance level, ifgR – and note that I use superscript fg instead ofthe generic S from the Proposition because we are focusing on our revert-to-discretion strategy.Formally,V CB,fgR ≡ lCBR (yR(ifgR ), piR(ifgR )) + βplCBZ (yZ(ifgR ), piZ(ifgR )) + βlCBR (yR(ifgR ), piR(ifgR ))1− β2p (2.49)A forward guidance policy the central bank implements in the Recovery state constitutes asustainable equilibrium if and only if it satisfies the SC defined asV CB,fgR︷ ︸︸ ︷lCBR(yR(ifgR ), piR(ifgR ))+ βplCBZ(yZ(ifgR ), piZ(ifgR ))+ βlCBR(yR(ifgR ), piR(ifgR ))1− β2p ≥ (2.50)lCBR (yR (i∗R) , piR (i∗R)) + βplCBZ (yZ (i∗R) , piZ (i∗R)) + βlCBR (yR (i∗R) , piR (i∗R))1− β2p︸ ︷︷ ︸V CB,∗RThe previous result shows that the Central bank can credibly announce (and maintain) somelevel of nominal interest rate, ifgR , to hold during the Recovery state as long as the the present valueof such policy exceeds the gains from (i) deviating to the nominal interest rate level that solves thesequential problem for the central bank in the Recovery state, i∗R, and (ii) foregoing the ability toconduct forward guidance in the future. V CB,∗ represents the payoff the central bank obtains whenit forgoes its ability to make promises and it instead optimizes in every period. V CB,fg representsthe payoff the central bank obtains when it keeps its ability to control PS expectations and makecredible promises (but giving up its discretionary optimal choice).To gain insight, we can rearrange Eq. (2.50) to obtain,βp[lCBZ(yZ(ifgR ), piZ(ifgR ))− lCBZ (yZ (i∗R) , piZ (i∗R))]≥ lCBR (yR (i∗R) , piR (i∗R))−lCBR(yR(ifgR ), piR(ifgR ))(2.51)48Recall first that the Sustainability Constraint involves the evaluation of gains standing at timeR. The right-hand side of Expression (2.51) then shows the gains the central bank can seize fromdeviating when the state is R. Specifically, the terms in the RHS indicate the discretionary outcomelosses in stateR (as captured by lCBR (yR (i∗R) , piR (i∗R))) relative to the forward guidance policy (termlCBR(yR(ifgR ), piR(ifgR ))). A positive RHS means that the central bank can seize instantaneous gainsfrom reneging its promise of low nominal interest rates. But, at the same time, the left-hand sideshows the (discounted by β) gains the central bank may seize (with probability p) if in state Z thePS believes in the central bank’s promises of expansionary monetary policies to be implemented instate R – i.e., FG. Note that these gains that the central bank can attain in Z do not come fromthe current nominal interest rate (it is already at zero), and instead they only arise if the CB canaffect PS’ expectations. A positive LHS means that, when the Central bank honors its promises inR, then its gains strictly manifest in state Z since the drop in inflation and output gap is mitigatedonly by the expectation of expansionary monetary policy from the next state, R. In summary, theweak inequality in Expression (2.51) then highlights that FG is sustainable when the future gainsFG yields in state Z is at least as good as the instantaneous gains from deviating from FG.A final observation that arises from the sustainability constraint is the constraint may hold withstrict inequality under some parameterizations. In other words, when the sustainability constraintholds with strict inequality, it characterizes a non-empty range of sustainable forward guidancepolicies that the central bank can conduct and that are self-rewarding (i.e., deliver outcomes atleast as good as the discretionary equilibrium), and enforceable (namely, outcomes that can besustained under the threat of a perpetual punishment coming from the private sector expectations).This opens the possibility for the existence of multiple equilibria, which together with the relationbetween credibility of forward guidance and fiscal policy will be the subjects of the next analysis.2.4.3 ResultsIn this section I first ask whether we can characterize credible forward guidance in the presence offiscal policy. Provided that we can, then we would like to analyze what the impact of the changein fiscal parameters is on the existing levels of forward guidance. Finally, I analyze the welfareimpact of forward guidance, and I investigate whether there are relevant policy trade-offs (e.g.,complementarity or substitutability between fiscal and monetary policy) that we can identify whena central bank uses reputation-based forward guidance and it faces a fiscal policy that seeks tomitigate a liquidity trap.Sustainable levels of forward guidanceParameter γ connects fiscal policy and monetary policy via the private sector’s optimality conditionrepresented by the DIS (Eq. 2.2). As a result, γ captures the impact of fiscal policy in the credibilityof forward guidance. Here, I start by showing that we can characterize credible forward guidancepolicies using Expression (2.50) for some γ > 0. Then I discuss how γ affects these policies.49Figure 2.4 plots the left- and right-hand sides of Eq. (2.50) for different announcements ofnominal interest rates to be implemented during the Recovery state. I use the parameterizationfrom Section 2.3.3 which satisfies that equation with strict inequality – for simplicity, subscript Ris omitted from these payoffs. The right-hand side of Eq. (2.50), labeled V CB,∗, is represented withan horizontal line because in the Recovery state the central bank solves its sequential problem ofchoosing its best response, which is an interior solution and not a function of the announced policieslying on the x-axis. In order to be consistent with the previous section, discretionary results areportrayed with blue solid lines. The left-hand side of the equation, labeled V CB,fg, is representedwith the red dashed curve, and plots the value of announcing a policy that the private sectorbelieves. Unlike V CB,∗, value V CB,fg varies with the announcements showing in the x-axis of thisgraph. This parabola is centered in the range of positive announcements because, in the Recoverystate, the natural interest rate is again positive and, hence, the central bank is not constrained toabsorb it with its nominal interest rate.In addition, Figure 2.4 shows two intersection points, E and E, where the central bank’s payofffrom sustaining nominal interest rates in state R (i.e., its credible FG) exactly matches the centralbank’s payoff under discretion. As a result, any forward guidance announcement iR in the interval[ifgR , ifgR ] defines a sustainable equilibrium, each of which is characterized by forward guidance levelschosen from that interval. At E, the nominal interest rate under discretion, i∗R, is equal to thatunder credible forward guidance, ifgR , meaning that the central bank is indifferent between eitherpromising a policy that convinces the PS, or choosing a policy that exactly matches its solution toits sequential problem. To the left of point E, a higher V CB,fg payoff in the open interval (ifgR , ifgR )indicates that the central bank can announce lower nominal interest rates in the Recovery state and,thus, exploit credibility gains – instead of implementing the solution to its sequential problem, i∗R.The intuition for this is the following. Forward-looking expectations of the PS regarding outcomeswhen the economy is away from the liquidity trap can affect outcomes during a liquidity trap. Butthe central bank needs to be credible to have an impact on PS’ expectations. If the central bankhas credibility, then Figure 2.4 shows that the central bank may prefer to promise monetary policiesmore expansionary relative to the discretionary policy (i.e., a nominal interest rate smaller thani∗R) that mitigates the drop in yZ and piZ (at the ZLB) at the cost of distorting yR and piR (in theRecovery state).At point E the payoff from implementing credible forward guidance is again equal to the payofffrom the policy under no forward guidance (point E). Here, the credibility gains from forwardguidance are exhausted at iR = ifgR – i.e., below ifgR further distortions in piR and yR in theRecovery state cannot compensate the improvements in piZ and yZ at the ZLB. To summarize,rates in [ifgR , ifgR ] are sustainable, but as we shall see in our welfare analysis in Section 4.3.3, whencredibility gains are exhausted at ifgR , the present discounted value of welfare will be larger for thecentral bank and the treasury.Finally, to the right of E or to the left of E, curve V CB,fg lies below V CB,∗. To the rightof E, any iR is more contractionary (i.e., higher) relative to ifgR . When nominal rates fall in this50range, the central bank is choosing a policy different to its optimal discretionary policy, which canonly be reasonable if the central bank is trying to attain some credibility gains. However, withiR > ifgR the central bank is increasing distortions in Z, which is the opposite that the central bankwould optimally seek to do in that state – recall that the central bank is also trying to eliminatethese distortions in Z but it cannot only because of the ZLB. Therefore, expanding distortions inZ more by choosing iR > ifgR goes counter the interest of the central bank, and the private sectorrationally anticipates this. In other words, the private sector understands that the central bankdoes not prefer such contractionary promise, and that the central bank would instead deviate tothe discretionary rate – analytically, V CB,fg < V CB,∗. This means that the central bank cannotguide private sector expectations with rates going in direction iR > ifgR . A simpler logic applies tothe left of E. In this case, it is clear that when iR < ifgR (= i∗R) the central bank is attempting tocollect gains from credible forward guidance that mitigate the distortions in Z. However, for tooexpansionary policies, iR < ifgR , the central bank has already exhausted the gains from credibleforward guidance because any Z-state improvements cannot offset the large distortions generated instate R. Because of this, such iR does not survive the rational expectations of the PS: Householdsanticipate that such a low rate gives the central bank incentives to deviate from iR to i∗R and,therefore, the equilibrium with too low rates falls apart.In summary, if for some parameterization with γ > 0 the sustainability constraint from Eq.(2.50) is satisfied with strict inequality, then the range of nominal interest rates that can be an-nounced, [ifgR , ifgR ], is nonempty. Furthermore, this means that we can find an interval [ifgR , ifgR ]where the model economy features multiple forward guidance policies that describe different equi-libria. Recall first from the No-FG equilibrium that fiscal policy becomes the only game in townduring liquidity traps. But if expansionary fiscal policy is costly (because debt has to be repaid inthe future), the treasury may not fully offset output drops with too aggressive stabilization policies.After internalizing the treasury’s response, the remaining portion of output gap can still give thecentral bank incentives to mitigate recessions with different levels of reputation-based forward guid-ance when it has access to it. It is worth pointing out that the previously described mechanismsoperating behind credible forward guidance matches expositions from related literature – see, forexample, Nakata (2018), Walsh (2018), and Dong and Young (2019).Before turning to the next section, I shall introduce the following definition that collects allthe nominal interest rates that constitute forward guidance announcements that are credible fordifferent parameters γ ≥ 0.Definition (Credibility region). The Credibility Region, I, is the collection of nominal interestrates defined as I ≡{ifgR : ifgR satisfies SC (Eq. 2.51) when γ ≥ 0}.This region captures all the nominal interest rates that can be sustained for those γ parametersthat satisfy the sustainability constraint at least with equality. Graphically, this region is formedwith all the interest rates that lie within intervals [ifgR , ifgR ] for some γ (as long as they are non-empty). An example of these intervals is displayed in Figure 2.4. Having introduced this definition,I now proceed to analyze how region I responds to changes in fiscal parameters.51Impact of fiscal policy on forward guidance credibilityIn this section, I analyze the sensitivity of credible FG to changes in parameter γ. To do so, I willstudy the effect of γ on region I.The two panels of Figure 2.5 show how the presence or absence of fiscal policy impacts on therange of credible forward guidance policies available to the central bank. The impact of fiscal policyin the economy is represented in the horizontal axis, and can be both inactive (γ = 0) or active(γ > 0). The forward guidance levels (i.e., the nominal interest rates to be implemented in theRecovery state) appear in the vertical axis. Panel A illustrates how the nominal interest rates ifgRthat lie in the credibility region, I, respond to different fiscal policy parameters (γ) when thereis no fiscal policy available in the economy. To remove fiscal policy from the model, observe thatsetting αTry = 0 returns a coefficient Φ = 0 in the treasury’s reaction function. Panel B, instead,represents the credibility region where there is fiscal policy available in the economy. The blue solidline plots the highest interest rate that satisfies the sustainability constraint on interest rates (SC)with equality; i.e., ifgR . This interest rate corresponds to the case where the central bank announcesa forward guidance level, ifgR , equal to its optimal discretionary policy level, i∗R – and recall thatifgR ≡ i∗R. The red dashed curve, instead, illustrates the lowest nominal interest rate that satisfiesthe SC with equality, ifgR . The resulting light-yellow shaded area that lies between the blue solidline and the red dashed line shows the region of sustainable (credible) forward guidance policiesthat the central bank can implement in state R. These are characterized by the vertical intervalsof nominal rates between these two contours, at every γ.Consider first the case of no fiscal policy (Panel A). When Φ = 0, fiscal policy becomes bZ = 0.This indicates that the treasury is not willing to exert fiscal effort in response to monetary policy,output, inflation, or the shock. The left panel of Figure 2.5 shows that, when debt does not matter,the levels of forward guidance the central bank can promise are unaffected by fiscal policy for everyparameter γ. Analytically, the DIS equation shows that changes in the γ parameter are irrelevantto all variables (and, in particular, to monetary policy) when bZ = 0. Then, if bZ = 0, changes inγ should not affect the nominal interest rate that the central bank is willing to sustain in recoverytimes, and this conjecture is graphically confirmed as the contours of the credibility region I areinvariant to different values of γ – the yellow region of the left panel of Figure 2.5 shows that allthe forward guidance levels that the central bank can sustain are the same for every γ. This resultcaptures credible policies similar to those characterized in the literature (see, for example, Nakata,2018).Panel B shows the case of an economy that has an additional government agency, a treasury,which implements fiscal policy to mitigate a recession. Observe first that, if fiscal policy is inactive(γ = 0), fiscal policy does not have any impact on inflation rates or output gaps. In particular, whenfiscal policy is inactive, the model environment resembles the case where Ricardian equivalence holds(i.e., agents have infinite planning horizons). If the credible forward guidance region is nonempty,we obtain the largest range of sustainable policies – observe the vertical difference between blueand red curves. The intuition is the following. Since γ = 0 allocates no role for fiscal policy52and therefore frees the treasury to act, a portion of the improvement in output gap (that wouldotherwise occur in each state if γ > 0) is forgone. As a result of this, the liquidity trap state is atits worst level. But this worsening of the recession actually provides the central bank more leewayto promise a more aggressive expansionary policy (i.e., lower nominal interest rates) and, hence,region I achieves the widest range.Panel B also allow us to capture how fiscal policy can erode the credibility of some forwardguidance policies. First note that, moving rightward on the horizontal axis, increases in γ indicatethat Ricardian equivalence breaks down as active fiscal policy carries a non-negligible effect: Whenγ increases, the treasury has incentives to implement expansionary fiscal policy (debt-financedlump-sum transfers to the private sector) that helps mitigate recessions. But this happens at thesame time forward guidance is also in place. Then, if forward guidance can be made crediblewithout fiscal policy (i.e., if I 6= ∅ when γ = 0), active fiscal policy can restrict the credibility offorward guidance depending on how large the wealth effect of debt-financed fiscal policy is. Theimplications of fiscal actions on I can be described by two changes in region I.The first change brought by fiscal policy is that region I shifts upward. In other words, theshape of I can be explained via the rising behaviour of its upper and lower contours of the regionas γ rises. I shall start by explaining the rise in the lower contour as γ increases. Abstracting, forthe moment, from forward guidance, we know that a positive γ provides incentives for the treasuryto use fiscal policy to mitigate a recession; i.e. to reduce the output gap during a liquidity trap.Therefore, as γ rises, then γbZ rises, which improves output gap in state Z, and also in the R-state– via the expectations of the Euler Equation. In summary, a rise in γ means that the discretionaryscenario (also known as No-FG) improves. Now, since we are explaining what happens underdiscretion, this description is absent from sustainable forward guidance policies. However, when weconsider forward guidance, we know that the central bank uses these discretionary outcomes (thatconstitute its payoff under discretion) to build reputation and conduct credible forward guidance.Hence, and as a result of the improvement in both Z and R states of the discretionary scenariobrought by fiscal policy when γ rises, the rising lower contour of region I reveals that a higher γnow makes it more difficult for the central bank to sustain too expansionary nominal interest ratespromises in R. The intuition is the following. Assume that îSR is the most expansionary forwardguidance policy that the central bank chooses when γ = 0. This policy is sustainable as longas it satisfies the sustainability constraint; i.e., if it is at least weakly preferred to rate i∗R of thediscretionary scenario – scenario the PS uses as a punishment. But with γ > 0 we just referred abovethat the discretionary scenario for the central bank improves relative to the γ = 0 case: If fiscalpolicy has an impact on the economy, the central bank already seizes gains stemming from activefiscal policy as (i) fiscal policy turns the recessionary Z-state into a milder episode, and (ii) it alsoimproves the R-state scenario via the expectations of such milder Z scenario. This means that theZ-state outcomes (now better because of fiscal policy) can only tolerate milder state-R distortionswith forward guidance and,35 therefore, any forward guidance level that remains credible (call it35In other words, any improvement of the discretionary scenario (and, specifically, of its ZLB outcomes when fiscal53i˜SR) must be more contractionary (i.e., i˜SR ∈ (̂iSR, i∗R)).36 In search of that i˜SR, the sustainabilityconstraint rules out rates of the type of îSR by moving upward in the y-axis and until the first onethat gives no incentives for the central bank to deviate is reached. This eventually shifts up thelower contour of region I.The upper contour of I also shifts up. Recall first that the nominal interest rate in the uppercontour of the region behaves in the exact same form as the nominal interest rate under discretion– formally, ifgR and i∗R match. As γ rises, we argued before that the discretionary scenario sees animprovement in both outcomes from states Z and R. And this improvement becomes particularlyrelevant in state R, as it allows the central bank to change its optimal response. Recall first thatfiscal policy mitigates the negative impact of the shock in every state (again, the shock affects theZ state directly, and state R indirectly via expectations). And as these better outcomes in Z andR are seized by the central bank, then the monetary authority can adjust its optimal choice in Rand absorb an even larger portion of the natural interest rise with the nominal interest rate; i.e.,rise i∗R. Analytically, it is straightforward to show that the response of the discretionary nominalinterest rate in state R is monotone increasing to γ, namely,di∗Rdγ={−[αCBy((1− (βPS)2 p)+ κ)+ αCBpi κ2 (1 + κ+ βPS)] pαTrτ βTr (1 + rR)2 rZφ2}dφdγ> 0(2.53)where the term in braces is positive, and also dφ/dγ > 0 – the positive sign of the derivativefollows from the negative term and a negative rZ . Derivative (2.53) shows that when fiscal policybecomes more active (γ rises), the nominal interest rate under discretion, i∗R, is further away fromthe zero-lower bound in state R. Intuitively, a larger γ releases part of the pressure that themonetary authority faces in the Recovery times since state-Z deflation affects state-R variables viaexpectations.Additionally, a larger impact of fiscal policy also contributes to a reduction of the region I, whichis reflected in Figure 2.5 via the shrinking in the credibility region I for each γ > 0. This behavioris explained by the upward shift in the contours that we described above, in combination with thesign of the derivative di∗R/dγ and an upper bound that the upper contour of I has. Specifically,as fiscal policy mitigates recession and deflation in state Z, it liberates the pressure on the centralbank’s nominal interest rate i∗R to absorb more of rate rt in state R. But, on the other hand, thispolicy has a positive impact) limits the credibility gains that a central bank using reputation-based forward guidancecan exploit in R.36Analytically, these gains can be seen by a rise in l∗Z (·) and l∗R (·) in Expression (2.51) since these losses becomenow closer to 0 for γ > 0. Specifically, we know that as γ rises, the satisfaction of Expression (2.51) is more difficultbecauseβp[lfgZ − l∗Z↑ ] ≥ l∗R↑− lfgR (2.52)Recall the numerical result in Figure 2.3 that shows the improvement of discretionary outcomes as γ increases. If forsuch γ > 0 the interest rate îSR (the rate the central bank sustained when γ = 0) turns out to be now too expansionaryrelative to those higher l∗Z (·) and l∗R (·) losses attained at some γ > 0 in the discretionary setup (i.e., where fiscalpolicy is already mitigating the losses in both states), then it means that îSR is no longer sustainable – Expression(2.51) would exhibit the opposite (and strict) inequality evaluated at îSR when γ > 0.54rise in i∗R is actually limited by r¯.37 Therefore, the combined effect of FG announcements droppingout of region I (which rise the lower contour of I), and the rise of i∗R (which rises the upper contourof I and has the distinctive feature of being limited by r¯), explains that region I shrinks as γ rises.A key finding that can be drawn from this experiment is that these two effects of fiscal policyon monetary policy (namely, the upward shift and the shrinking of the range of state-R nominalinterest rates) together imply that active fiscal policy can potentially crowd out some monetarypolicy promises, or in other words, erode the credibility of reputation-based forward guidance.Some comparative statics. Figure 2.6 plots the response of the nominal interest rate in stateR to changes in the treasury’s weight on tax distortions. The figure shows the lowest (i.e., moreexpansionary) and the highest (or discretionary) FG levels that satisfy the SC with equality using ared-dashed and a blue-solid line, respectively. The pink shaded region denotes the FG levels wherethe SC is satisfied with strict inequality. As can be seen in Figure 2.6, for large concerns over taxdistortions (large αTrτ ), the vertical distance between the lower-contour and upper-contour of theregion characterizes an interval of nominal interest rates that constitute credible FG policies for thecentral bank. Therefore, when the treasury’s concerns over tax distortions decrease (i.e., movingleftward on the x-axis), there is a reduction in the range of credible FG – which is captured by theshorter height of the pink region for each αTrτ .A similar reasoning to the one in Figure 2.5 can be used to interpret the response of FGobserved here. From the treasury’s F.O.C. we know that, given some monetary policy and privatesector expectations, the best response to a shock, bZ , is inversely related to concerns over taxdistortions – namely, a decrease αTrτ . The logic behind the rise in the lower contour of the regionis straightforward. First, a decrease in αTrτ making policy bZ more expansionary means that thetreasury is more actively implementing fiscal stabilization policies at the ZLB. But, then, thismakes some nominal interest rates not sustainable, with a same argument as before: A smaller αTrτimproves Z and R outcomes, and at the current FG level the PS rationally expects a deviationfrom the monetary authority. Therefore, the SC is negative, unless the central bank announcesless expansionary policies. This rises the lower contour of the region. When αTrτ decreases, alsothe upper contour also shifts up. As we argued above, this is because the solution under discretionfaces less pressure in state R when fiscal policy improves the Z state (here due to the decrease inαTrτ ). Finally, the upward shift of the contours, together with the existence of r as an upper boundto how far the state-R discretionary interest rate can rise explains the shrinking of the region inFigure 2.5.37From derivative (2.53) we note that i∗R increases as γ rises. Furthermore, it is easy to see from the expression ofi∗R in Proposition 2.3.1 that i∗R asymptotically approaches rR as γ rises (see Appendix). However, note that takingthis limit can make i∗Z = 0 become no longer a corner solution but an optimal solution for some positive γ (wherethe multiplier of the ZLB becomes 0, and the liquidity trap is no longer binding). The ZLB multiplier in state Z inour analysis, however, is positive for the γ values considered.55Welfare analysis and forward guidance-fiscal policy interactionThus far, we have assessed the sustainability of forward guidance while it interacts with fiscal policy.Also, we analyzed the effects of fiscal policy on forward guidance for different parameterizations ofγ. It would be reasonable to ask, then, how these policy responses affect welfare and equilibriumallocations. In this section, I analyze how welfare differs between the two different monetary policyregimes (one with no forward guidance, and one with forward guidance), in combination with fiscalpolicy.We first need to define a measure that captures the welfare impact of the equilibrium allocations.Note, however, that a direct measure of social welfare cannot be easily derived as we have twopolicymakers with differing objectives. To overcome this issue, I shall consider a convex combinationof the discounted losses standing at Z of both fiscal and monetary authorities and call it the social(i.e., private sector) losses. To that end, I will first call the present discounted value (PDV) oflosses for authority k = {CB, Tr} at time Z under regime j = {∗, fg} asV k,jZ ≡lk,jZ − βlk,jR1− β2pIt is straightforward to see that the gains from forward guidance are relatively larger than thoseunder discretion for authority k = {CB, Tr} if V k,fgZ ≥ V k,∗Z . Having defined these PDV of lossesstanding at Z for the treasury and the central bank, I now define the social (or private sector)losses under regime j = {∗, fg} asV PS,jZ = θVCB,jZ + (1− θ)V Tr,jZwhere θ ∈ [0, 1] captures the relevance of central bank’s policy objectives relative to those of thetreasury. When θ = 1, private sector losses exactly match those of the central bank. In thiscase, the private sector allocates no weight to tax distortions, and it instead values output gap andprice stability according to the central bank’s mandates over these variables. On the contrary, thepolar opposite case, θ = 0, means that the private sector has no concerns over price stability, andit only cares about close-to-zero output gaps and low taxation costs. Finally, for any θ ∈ (0, 1),social preferences exhibit some fraction of dislike for output gap distortions, inflation, and taxationcosts. Figure 2.9 shows the hump-shaped plot of both central bank and treasury value functions,which is a result of the quadratic structure of the losses in their response to forward guidanceannouncements from the central bank. Similarly to the description in Walsh (2018), these curveshighlight how credible forward guidance operates by affecting private sector expectations. In anutshell, the decreasing side of the curves shows that lowering nominal interest rates can enhancewelfare via state-Z outcomes that are improved at the expense of distortions in R – the oppositelogic applies if nominal interest rates are in the increasing side of the curves.For our current parameterization, the range of sustainable policies for every γ ≥ 0 is in thedecreasing side of the curves. This means that the central bank can be better-off by promisinglower nominal interest rates. I discuss the implications of this for the treasury and for different γ56parameters below.Table 2.2 presents central bank, treasury, and social welfare for two different parameterizationsof the fiscal parameter, γ, and under different specifications of monetary policy. Specifically, foreach parameterization γ, the table evaluates welfare when the central bank does not have access toforward guidance (row ‘No-FG’), and when it can access forward guidance (row ‘FG’). When thecentral bank can implement forward guidance, I set iR at it most expansionary (i.e., lowest) level,ifgR , which is the optimal sustainable forward guidance policy – as measured by the low welfarethat it yields the central bank. The first two columns of table 2.2 display the monetary and fiscalpolicies, with and without central bank access to forward guidance, and each evaluated at thecorresponding fiscal parameter. The last three columns report the welfare corresponding to thecentral bank, the treasury, and the private sector (labelled ‘Social’ for brevity). For simplicity,parameter θ has been set at 0.5.I shall start comparing the impact of γ in the No-FG scenarios beginning with the treasury– see column 4 in the No-FG row for both the top and the lower panels. When fiscal policy ismore effective to mitigate a recession during liquidity traps (i.e., fiscal policy has a higher wealtheffect due to a larger γ), then the treasury is willing to improve Z outcomes. This can be observedby the increase in bZ between No-FG levels when γ rises: As lump-sum transfers increase, theamount of debt issued to finance them rises – which rises output in the Euler Equation. Also, notethat a higher impact of fiscal policy in state Z affects R-outcomes. Specifically, when fiscal policyhas more power to prevent a recession, the improved variables at Z spill over R outcomes due toforward-looking inflation and output gap. As a result, R-outcomes improve too when γ rises, andall these effects combined rise treasury’s welfare.For the central bank, recall first that it always observes better outcomes in Z when fiscal policyis active, and these outcomes can be even better as fiscal policy has more effect; namely, γ rises.Likewise, even when the central bank has no access to forward guidance, it will still seize betteroutcomes in R because of the impact of state-Z fiscal policy on state-R variables via forward-looking variables. This result is magnified as γ rises, which shows in the larger central bank welfareunder No-FG as we switch from the top panel to the bottom panel. Moreover, the improvementin R outcomes explains the rise in the nominal interest rate (column 1) under discretion (No-FG).Specifically, better R-state outcomes allow the central bank to use its nominal interest rate toabsorb a higher portion of the natural interest rate, rt. This explains the rise in iR when wetransition between No-FG when γ rises. In summary, the general effect that a higher impact offiscal policy has in the economy is that welfare under No-FG for both the Treasury and the CentralBank (and, therefore, social welfare) can improve with credible forward guidance (columns 3 to 5).I now turn to analyze what happens when the central bank uses credible forward guidance,starting with the case of a fixed γ. When the monetary authority has access to forward guidance,it can lower the nominal interest rate to more expansionary levels when the economy is away froma liquidity trap. For example, when γ = 0.25 (top panel), iR drops from its discretionary level,0.0091, to a FG level of 0.0079 – a 13 percent decline. Now, let us focus on central bank’s welfare.57On the one hand, recall that the No-FG scenario represents the highest welfare the central bankachieves when it lacks access to reputation. But, on the other hand, this No-FG scenario alsoconstitutes the punishment whenever the CB deviates from an announced forward guidance policy.For this latter case, when the central bank builds reputation to lower iR from the discretionarylevel, it tolerates some distortions in y and pi in state R as long as these reduce the dispersion inoutput gap and inflation in Z. Eventually, these R-distortions are at their maximum level wheniR = ifgR , and that is where the central bank achieves lower losses and larger welfare. Finally, asγ rises, forward guidance can improve welfare further: While No-FG welfare rises as γ rises (i.e.,the punishment scenario), then if reputation-based forward guidance is still sustainable for somehigher γ, it must be that the gains associated to such forward guidance level yields at least a welfarehigher to the (now improved) punishment No-FG scenario.38I now consider fiscal tools. As we discussed above, a positive fiscal parameter γ makes thetreasury willing to increase fiscal effort to mitigate recessions. Now, for any given γ, the changein regime from No-FG to FG generates less fiscal effort (as observed in column 2). Recall that,regardless of the access to forward guidance, good or bad prospects about future y and pi affectthe response of bZ via private sector’s expectations, yeZ and pieZ . But when the central bank hasaccess to forward guidance, the monetary authority can affect these expectations. Specifically, ifthe central bank can convince the private sector about implementing a policy rate different than thediscretionary rate, then the Treasury issues lower lump sum transfers during Z – which have a lowerimpact in the economy as measured by γbt in the DIS. Hence, forward guidance makes the centralbank to be in charge of a higher share of the fight against a recession in Z, with lower fiscal actionfrom the treasury’s side. With respect to welfare, the treasury is better-off when the central bankuses forward guidance, and the intuition for this result is straightforward. With forward guidancein place, monetary policy affects PS’s expectations, and these impact on the reaction function ofthe treasury. The treasury is better-off because the central bank is now fighting the recession inZ: As forward guidance makes the treasury retire some of its fiscal assistance, the treasury suffersless distortions in R (distortions that arise for the treasury as it needs to collect less taxes to payits issued debt). To summarize, when the central bank can conduct forward guidance, not onlythe central bank does better (as compared between the FG rows of the central bank’s welfare), butalso the treasury can do even better (comparing FG rows of treasury’s welfare) because forwardguidance crowds out a portion of fiscal effort, thus lowering the amount of tax distortions thatit will generate in R for the treasury – and which are costly to the treasury. Finally, all theseeffects on treasury’s welfare are magnified when γ rises (and if some forward guidance levels arestill sustainable).To better explore how fiscal and monetary policy interact when the central bank seeks to im-plement credible forward guidance while the treasury conducts fiscal policy, I compare equilibriumoutcomes when the economy transitions between different levels of forward guidance.38However, as γ rises, it is more difficult to sustain forward guidance: Numerically, when γ = 0.5, the range ofpolicies is |0.0086− 0.0093| = 0.0007, while for γ = 0.25, the range of policies is |0.0079− 0.0091| = 0.0012.58Figure 2.7 depicts the equilibrium dynamics when the central bank cannot use forward guidance(blue solid lines) versus the equilibrium outcome where the central bank implements the lowestforward guidance policy (red dashed lines), which satisfies the SC and gives the highest welfare.For reference, the curves under no forward guidance are the same as those from Figure 2.3. Thefigure helps us to compare the different outcomes that can be attained when the central bankexhausts its credibility power to set the most expansionary level of FG. First of all, note thatthere are some similarities between the two curves. Again, the negative demand shock triggers animmediate response from fiscal and monetary authorities. Panel C shows that the nominal interestrate curves overlap in both equilibria at the time of the shock (state Z). This is because the nominalinterest rate strategy the central bank follows instructs the monetary authority to choose the samerate that the central bank would choose under discretion. Also, in state Z there is a drop in theoutput gap and a large deflation, which are mitigated with expansionary fiscal policy financed withdebt.The dynamics of the model economy in Figure 2.7 also evidences the mechanism that the centralbank uses to make FG credible (described in the previous section). Specifically, forward guidanceoffsets costs and gains between states Z and R, and this is channeled by the expectations of theprivate sector, which make the impact of credible monetary expansions in R states to cascade backinto Z states (and vice-versa). Credible FG therefore results in milder recessions (or even positiveoutput gaps), and mitigated deflation in both states. Panels A and B show that credible futurepromises of loose monetary policy during economic recoveries (i.e., lower interest rates in state R)can affect the economy’s outcomes in state Z. Overall, then, forward guidance alleviates deflationand recession during liquidity traps: The private sector anticipates that it will have less purchasingpower in state R (due to a looser monetary policy when the liquidity trap ends), and it thereforeprefers to consume more today – rising yZ and piZ , which lowers central bank losses. Therefore,when the central bank can use its reputation, it exploits credibility gains and it can impact theeconomy using FG via (i) a direct effect in state-R outcomes where piR and yR increase, and (ii) anindirect effect in state-Z outcomes, as the PS rationally expects that state-R monetary policy willbe delivered. To summarize, the first conclusion from Panels A and B in Figure 2.7 is that credibleFG can mitigate deflation (at Z and R) and recessions (at Z).I now turn to the analysis of fiscal policy under the presence of forward guidance. Under bothdiscretionary monetary policy and forward guidance, Panel D shows that the treasury implementsan expansionary fiscal policy to counter for part of the drop in output gap. Note, however, thatFigure 2.7 shows a change in the direction of the fiscal response compared to the scenario withoutforward guidance. To see analytically why this is the case, recall that the treasury’s response takesas given the actions from the central bank and PS expectations. Abstracting momentarily from therole of expectations, observe first that the treasury exerts the same positive level of effort upon anegative shock, −αTry γrZ/(αTry γ2 + βTrαTrτ (1 + rR)2). But, in fact, with credible FG, the centralbank can now affect future yR and piR outcomes, which are variables considered by the treasuryvia the (yeZ + pieZ) term. Therefore, since the treasury best responds not only to the central bank’s59nominal interest rate, but also to the expectation of future inflation and output gap (pieZ and yeZ),then the PS’ expectation of an improvement in future outcomes coming from credible forwardguidance eventually releases the pressure of the treasury to respond with the (large) expansionaryfiscal policy observed in Section 2.3. This implies that, during a recession, any sustainable forwardguidance level ifgR such that ifgR < i∗R can make fiscal responses to go in the opposite direction tomonetary expansions – i.e., the treasury can now exert a more contractionary fiscal policy whenthe central bank uses credible FG. This finding provides a new policy lesson, and it reveals thatreputation based forward guidance implemented during liquidity traps can yield a ‘loose monetary-tight fiscal’ policy mix.To further expand on why there can be responses of fiscal and monetary policy in oppositedirections during a recession, I turn to a graphical analysis that plots how the variables of theeconomy respond to credible forward guidance.Figure 2.8 illustrates the impact of different levels of credible forward guidance (nominal interestrate ifgR in the x-axis) on equilibrium outcomes (y-axis). In particular, Plots E and B show thatduring Recovery times (resp. ZLB times), more aggressive forward guidance can increase the outputgap (resp. mitigate a recession). In particular, note that (i) output in the R state collects boththe direct effect (coming from the contemporaneous rise in piR and yR), and indirect effects (as thePS anticipates an improvement of state-Z outcomes), and (ii) output in the liquidity trap state isbenefited by the indirect effect of PS expectations alone (the PS anticipates credible FG in stateR). Also, inflation in states Z and R (resp. Plots A and D) show that a more expansionaryforward guidance (i.e., as it moves leftward in the x-axis) can mitigate deflation in both states.Furthermore, since during liquidity traps the nominal interest rate cannot go below zero, theobserved improvement in deflation in state Z stems purely from the monetary policy’s effect onthe expectations of the private sector. This result coincides with what the literature calls thedeflation bias in expectations – see Nakov (2008). During the Recovery state, the possibility offuture deflation from state Z affects piR. Thus, inflation can be negative. However, Panel D inFigure 2.8 shows that forward guidance can counter deflation in R times via both its direct effect(the decrease in it), and its indirect effect (less deflation in state Z). In fact, in the R state levelsthat are closer to the lowest forward guidance equilibrium, ifgR , can actually revert what Eggertsson(2006) calls the deflation bias of discretionary policy.So far I have focused on all outcomes except from fiscal policy. In particular, Plot C showsthe response of debt and lump sum transfers during a liquidity trap. At the ZLB, a key feature inPlot C is that a more aggressive forward guidance can result in smaller lump-sum transfers. Thissmaller lump-sum transfers translate into small debt issuance, which carry a smaller impact in theDynamic IS curve. In other words, forward guidance may trigger a more austere fiscal policy as itcrowds out a portion of fiscal effort. To understand this, it is useful to recall that the central bankleads the treasury. Thus, after the monetary authority has set a policy, the treasury observes itand acts accordingly. The key intuition behind it is, again, the way the treasury best responds toforward guidance. In Section 2.3 we saw that the treasury issues debt in state Z based on private60sector’s expected inflation and output gap, as well as current monetary policy (set at 0 during theliquidity trap) and the magnitude of the shock. But now the central bank has access to forwardguidance that affects PS’ expectations. This makes the treasury follow a more conservative fiscalexpansion as forward guidance is now improving one of the variables the fiscal authority observes(the output gap). As a result of this, the treasury has now more incentives to implement lessexpansionary fiscal policies while the central bank attempts to implement FG.DiscussionIn search of linkages between fiscal policy effectiveness and the credibility of reputation-basedforward guidance, this last section additionally showed how the quantitative experiment performedin this chapter can give rise to a scenario where the policy choices of a central bank and a treasurycan generate a ‘loose monetary-tight fiscal’ mix in equilibrium during a recession. In other words,this last result implies that expansionary fiscal policy and reputation-based forward guidance canexhibit a substitutability component while a central bank and a treasury try to fight a crisis. Giventhat this chapter features a stylized model economy that analyzes existing links between one type ofunconventional monetary policy (forward guidance) and one type of impact of fiscal policy (wealtheffects of debt from fiscal expansions), this last conclusion regarding the substitutability of fiscal andmonetary policy during recessions should be treated with caution. However, this finding is relevantsince it accounts for the possibility of a lack of coordination between fiscal and monetary policies.In recent years, this has been empirically analyzed by Greenwood et al. (2014), who documentedfiscal and monetary policies going in opposite directions with regards to debt management in the USduring the financial crisis of 2008. Furthermore, such disconnect between fiscal and monetary policycan also be traced back to some narrative evidence – see, for instance, Draghi (2019) regardingwhether monetary and fiscal policy respond cooperatively to crises. With regards to the study offiscal-monetary interactions, this chapter contributes to this topic by capturing a policy interactionthat conflicts with the notion of coordinated monetary and fiscal responses during recessions.In summary, it is worth emphasizing that this result gives an example that opposes to theidea that fiscal and monetary responses be expansionary when fighting a crisis. In the light of thenumerical exercise conducted in this chapter, policies that may seem ‘misaligned’ can instead beinterpreted as optimal responses arising from a central bank and treasury interacting to preventrecessions and deflation in the context of liquidity traps.392.5 ConclusionThis chapter addresses the following two questions: (1) How does the credibility of forward guidancerespond to active fiscal policy? (2) What are key features arising from the interaction betweencredible forward guidance and active fiscal policy? To answer to these questions, I use a standard39This argument relates to Eggertsson (2006) who discusses that certain outcomes that arise during liquidity trapscan be the result of policy constraints and lack of commitment.61new Keynesian environment adjusted to capture only wealth effects from debt-financed fiscal policy.I outline two separate government agencies, a treasury and a central bank, that use fiscal andmonetary policy respectively to mitigate recessions and deflation during repeated liquidity trapstates triggered by negative shocks to the natural interest rate.When the central bank’s nominal interest rate hits the zero-lower-bound and this is the centralbank’s only instrument, I show that the treasury uses expansionary fiscal policy to mitigate aportion of the deflation and recession that the economy features during a liquidity trap. If, inturn, we allow a central bank to use the repeated structure of liquidity traps to implement forwardguidance, I find that multiple forward guidance promises can be made credible. When credible,forward guidance can make private sector’s expectations of states with improved outcomes to affectbad outcomes of the liquidity trap period. While this result is in line with the literature, a newfinding is that fiscal policy effectiveness can restrict the credibility of forward guidance by reducingthe range of announcements that the central bank can sustain. This is a novel result that adds to theliterature of credible forward guidance and highlights how this monetary tool might be restrictedwith active fiscal policy. Finally, the simple modelling approach taken in this chapter documentsthat fiscal and monetary policy can depict substitutability even during a crisis: When the economytransitions from an equilibrium without forward guidance to one with forward guidance, the use ofthis unconventional monetary policy tool can crowd out fiscal effort. In this context, this simplemodel provides an explanation to seemingly misaligned fiscal-monetary policies that emerge inequilibrium during recessions.The majority of the findings presented in this work correspond to a quantitative analysis per-formed on a simple model that is rich enough to represent credible forward guidance, and toillustrate how fiscal policy may limit the use of that monetary policy instrument by eroding itscredibility. However, some candidate extensions emerge as natural questions: For instance, therole of seigniorage revenues, the impact of the stochastic duration of ZLB episodes, to name a few.These are examples of questions not answered in this chapter, but which I plan to explore in futureresearch.622.6 TablesTable 2.1: Parameter valuesParameter Value Description Sourceβ 0.99 Discount factor (quarterly; r ≈ 0.01 D10, CD10, C11rZ −0.005 Natural interest rate shock (quarterly) EW03, C11, MNS17, ST18αCBpi 1.5 Central bank’s weight on inflation –αCBy 1 Central bank’s weight on output gap –αTrτ 0.5 Treasury’s weight on tax distortions –αTry 1 Treasury’s weight on output gap –γ {0, 0.5} Dynamic IS coefficient of bt D10κ 0.0329 Phillips Curve coefficient of yt D10, C11p 0.2 Probability of returning to ZLB from R –Note: Sources of values: Eggertsson and Woodford (2003) (EW03); Devereux (2010) (D10);Cook and Devereux (2011) (CD10); McKay et al. (2017) (MNS17); Christiano et al. (2011)(C11); Smets and Trabandt (2018) (ST18). Equal values assigned for {rS , rR, r}.Table 2.2: WelfareFiscal parameter: γ = 0.25Welfare†iR bZ Central bank Treasury Social‡No-FG 0.0091 0.0022 −1.27 −1.42 −1.35FG 0.0079 0.0015 −0.75 −0.83 −0.79Fiscal parameter: γ = 0.50Welfare†iR bZ Central bank Treasury Social‡No-FG 0.0093 0.0034 −0.70 −1.05 −0.87FG 0.0086 0.0028 −0.52 −0.76 −0.64Note: Nominal interest rate set at lowest forward guidancelevel. †: Welfare units multiplied by e5 to simplify comparison.‡: Parameter θ set at 0.5 (balanced losses).632.7 FiguresZZZRRRSS(1-p)(1-p)(p)(p)S SSS(1)(1)(1)(1)(1)(1)(1)𝑡 ൌ 0 𝑡 ൌ 1 𝑡 ൌ 2 𝑡 ൌ 3 𝑡 ൌ 4 𝑡 ൌ 5Figure 2.1: Histories of states of the world for t = 5 and transition probabilities (in parentheses)𝑡𝑡 + 1ShockCentral Bank chooses monetary policyPrivate Sector forms expectations of future inflation and output gapTreasury chooses fiscal policyEconomic variables realize𝑡𝑡Figure 2.2: Sequence of events640 1 2Time-2.5-2-1.5-1-0.50t10-4 A. Inflation0 1 2Time-6-5-4-3-2-101y t10-3 B. Output gap0 1 2Time-20246810i t10-3 C. Interest rate0 1 2Time-10123b t10-3 D. Debt0 1 2Time-10123t10-3 E. Taxes0 1 2Time-10123l t10-3 F. Lump-sum transfersFig. 9. Vbles vs. Time ($ = 0$ (blue lines), $ >0$ (green lines)) - Flag 17ZLB Recovery ZLB Recovery ZLB RecoveryZLB Recovery ZLB Recovery ZLB RecoveryFigure 2.3: Simulated time series without forward guidanceNotes: The lines represent the responses of equilibrium outcomes to a one-period shock to the natural interest ratewith inactive fiscal policy, γ = 0, (dashed green lines) and active fiscal policy, γ > 0 (solid blue lines). Variables inlog deviations (except it).0002 0004 0006 ifgR ifgR0010 0012 0014Interest rate, iR2015105010 6Value function, VCBE EFigure 2.4: Present discounted value of losses for the central bankNotes: The curves evaluate the central bank’s PDV of losses starting at period R and at different nominal interestrates, iR, when the central bank sustains a policy, VCB,S , (dashed red line), and when it implements the discretionaryoutcome, V CB,∗, (solid blue line).650.0 0.5 1.0Fiscal policy parameter, 0.00750.00800.00850.00900.00950.0100Interest rates, i R(a) FG only.0.0 0.5 1.0Fiscal policy parameter, 0.00750.00800.00850.00900.00950.0100Interest rates, i R(b) FG and fiscal policy.Figure 2.5: The credibility regionNotes: The plot illustrates the region of sustainable forward guidance policies (light-yellow shaded area) for everyfiscal policy parameter, γ. The region lies between the highest nominal interest rate that satisfies the sustainabilityconstraint (solid blue line) that coincides with the nominal interest rate at the discretionary equilibrium, and thelowest nominal interest rate that satisfies the SC (dashed red curve).0.001 0.003 0.005Fiscal policy parameter, Tr0.00750.00800.00850.00900.00950.0100Interest rates, i RFigure 2.6: The credibility region: Comparative staticsNotes: The plot illustrates the region of sustainable forward guidance policies (pink shaded area) for every fiscalpolicy parameter, αTrτ . The region lies between the highest nominal interest rate that satisfies the sustainabilityconstraint (solid blue line) that coincides with the nominal interest rate at the discretionary equilibrium, and thelowest nominal interest rate that satisfies the SC (dashed red curve).660 1 2Time-2.5-2-1.5-1-0.500.5t10-4 A. Inflation0 1 2Time-6-4-202y t10-3 B. Output gap0 1 2Time-50510i t10-3 C. Interest rate0 1 2Time-50510b t10-5 D. Debt0 1 2Time-50510t10-5 E. Taxes0 1 2Time-50510l t10-5 F. Lump-sum transfersZLB Recovery ZLB Recovery ZLB RecoveryZLB Recovery ZLB Recovery ZLB RecoveryFigure 2.7: Simulated time series with forward guidanceNotes: The lines represent the responses of equilibrium outcomes to a one-period shock to the natural interest rate,for the case with forward guidance (dashed red lines) and without forward guidance (solid blue lines) – in both casesfiscal policy is active, γ > 0. Variables in log deviations (except it).67ifgR Interestrates ifgRZA. Inflation (state Z)ifgR Interestrates ifgRy ZB. Output gap (state Z)ifgR Interestrates ifgRb Z, l ZC. Debt, Transfers (state Z)ifgR Interestrates ifgRRD. Inflation (state R)ifgR Interestrates ifgRy RE. Output gap (state R)ifgR Interestrates ifgRRF. Taxes (state R)Figure 2.8: Equilibrium outcomes and FGNotes: The lines show the response of equilibrium outcomes to a one-period shock to the natural interest rate fordifferent levels of sustainable forward guidance (solid lines). Variables in log deviations (except it).0.0080 0.0085 0.0090Interest rates, iRfg (when =0.25)-1.4-1.2-1-0.8-0.6Value function, VZCB,fg, V ZTr,fg10-5(a) Welfare (γ = 0.25).0.0086 0.0088 0.0090 0.0092Interest rates, iRfg (when =0.5)-11-10-9-8-7-6-5Value function, VZCB,fg, V ZTr,fg10-6(b) Welfare (γ = 0.50).Figure 2.9: Welfare for different levels of γ68Chapter 3Maturity Structure of Debt andSelf-fulfilling Debt Crises3.1 IntroductionRecent empirical evidence from the last European debt crisis indicates that sudden changes increditors’ beliefs about governments’ repayment capacity can influence the likelihood of defaults.During the crisis, the European Central Bank announced a liquidity support program (OutrightMonetary Transactions) to address the rise in bond yields volatility that was disconnected fromchanges in economic fundamentals. After the announcement, a large decline in yields occurredeven though the liquidity support was not used – see Figure 3.1. This evidence suggests that thepromise of liquidity itself could have brought calm to the markets, thus showing a link between thelikelihood of default and creditors’ beliefs about governments’ debt repayment capacity. Using aself-fulfilling debt crises model, several studies have explained this link with a focus on governments’repayment capacity of short-term debt (Cole and Kehoe, 1996, 2000; Conesa and Kehoe, 2017).However, the study of this link with a focus on governments’ repayment capacity of long-term debthas largely remained unexplored.In this chapter, we study the impact of long-term debt in a model of self-fulfilling debt crises.The main goal is to analyze how the maturity structure of debt affects governments’ default decisionsin a model of self-fulfilling debt crisis, and to understand its welfare implications. We show thatholding the ratio of debt-over-GDP constant, governments’ incentives to default decrease as thematurity of debt increases. This is because a higher debt maturity allows a government to roll overa lower fraction of debt in each period, thus making the government less exposed to negative shiftsin expectations. We also show that this decrease in governments’ default incentives coming fromlong-term debt can increase the welfare of the economy. Specifically, a government can decreasetotal debt to levels that are not vulnerable to creditors’ panic, in a shorter period of time comparedto the short-term debt case.The chapter consists of two parts. In the first part, we build on an earlier work of Cole andKehoe (2000) and consider a model of self-fulfilling debt crises featuring short-term debt in asmall open economy. A government which maximizes households’ welfare chooses spending on aconsumption good, issues one-period bonds, and has access to a competitive international creditmarket. In every period, given the bonds’ price, the government first chooses how much debtto sell, and then decides how much to spend and whether to default or not. Upon default, the69government is permanently excluded from the bond market, and seizes a share of the economy’sendowment thereafter. In this set-up, we first show that there can be an interval of debt levels inwhich repayment or default are rational expectations equilibria (the Crisis region). Within thisinterval, a sunspot can suddenly trigger creditors’ panic and force the government to default, thusconfirming creditors’ negative initial expectations. When initial debt lies in the Crisis region, weanalytically prove that the government tries to avoid self-fulfilling debt crises by lowering its debtto reach an interval of debt levels where government repayment is the only rational expectationsequilibrium (the No-Default region). We further show that the decrease in debt levels occurs in afixed number of periods that optimally depend on how large initial debt is within the Crisis region.In the second part of the chapter, we consider a model of self-fulfilling debt crises using debtof higher maturity in the form of coupon-paying bonds. This definition of long-term debt in linewith Hatchondo and Martinez (2009) assumes that creditors receive an infinite stream of couponpayments that geometrically decrease with time. The key aspect of long-term debt in the form ofa coupon-paying bond is that it lowers the amount of debt maturing in each period, and thereforeallocates the repayment of total debt more evenly across time. Also, this definition of long-term debtexplicitly introduces credit market freezes. In this set-up, we first study how the structure of theequilibrium price of debt changes compared to the short-term debt scenario, and analyze the changein the bounds of the Crisis region under long-term debt. In particular, we show that these boundsincrease as debt maturity rises, shifting upward the Crisis region relative to the one-period bondcase. The intuition is that coupon payments divide the repayment of total debt in many periods, andsince government incentives to default on less maturing debt decrease, creditors now expect defaultto occur only at higher levels of total debt. Second, we numerically investigate the implicationsof long-term debt in this model and compare it to the short-term debt scenario. Specifically, wedescribe how the presence of long-term debt shortens the time in which the government decreasesany initial debt starting in the Crisis region. Finally, we numerically show that total utility inthis model can be higher when the government issues long-term bonds. Intuitively, the fastestthe government exits the Crisis region, the less it discounts the payoff attained in the No-Defaultregion. These findings, therefore, suggest that lengthening debt maturity can improve an economy’swelfare.The chapter proceeds as follows. Section 3.2 outlines the set-up of the model under short-termdebt. Section 3.3 characterizes a Crisis region and analyzes an optimal debt policy to exit it. Themodel with long-term debt is introduced in Section 3.4. Section 3.5 presents the Crisis regionand illustrates government optimal decisions under this new type of debt. Section 3.6 providesa numerical example of the welfare changes under short-term and long-term debt. Concludingremarks are outlined in Section 3.7. Proofs are relegated to the Appendix.703.2 Baseline model3.2.1 SetupThe following model setup and notation follows Cole and Kehoe (2000) very closely. Consider agovernment in a small open economy inhabited by a continuum of identical households. Householdslive infinite periods and are uniformly distributed over the [0, 1] interval. The government maximizeshouseholds’ utilityE0∞∑t=0βtu (gt)where gt is government spending, β ∈ (0, 1) the discount factor, and u (·) satisfies u′ (·) > 0,u′′ (·) < 0 and u (0)→ −∞.The economy receives a deterministic sequence of endowments (see Conesa and Kehoe (2017)),Z1−zty =y, if zt = 1Zy, if zt = 0where y is a positive constant, Z ∈ (0, 1) is the share of output that is left after a default episode, andzt reflects the default decision of the government. When the government defaults on its obligations,zt = 0; otherwise, zt stays at 1. In the state space, z−1 reflects whether the government defaultedon its debt before; if that happened, z−1 remains at 0 forever. The government has access to theinternational credit market only if it did not default in the past. The international credit market iscompetitive, and has risk-neutral lenders who buy bonds and have finite wealth. Lenders and thegovernment take the price of a bond qt as given. If the government never defaulted before, it startseach period with a positive amount of maturing debt Bt. Thereafter, the government chooses gt,sells Bt+1 one-period bonds at price qt to finance its expenditures, and decides whether to repayor not its maturing debt Bt.The budget constraint of the government isgt + ztBt = Z1−zty + qtBt+1Agents in the model observe the realization of a sunspot ζt in every period. This randomvariable ζt is independent and identically distributed with uniform CDF on the unit interval [0, 1].The sunspot, together with the current maturing debt and the government past default decisions,define the aggregate state st in this economy, st = (Bt, z−1, ζt).The timing within a period is as follows. The government and the creditors observe the aggregatestate of the economy st. Taking the price qt as given and if no default occurred before, thegovernment issues Bt+1 bonds and the creditors purchase that quantity. At the end, the governmentmakes its default and spending decisions, zt and gt. It is worth noting that the default decisionoccurs after borrowing in this timing, in contrast to models of sovereign default where the borrowing71decision occurs after default and impede belief-driven crises (see Arellano, 2008; Arellano andRamanarayanan, 2012; Hatchondo and Martinez, 2009).3.2.2 An equilibrium with short-term debtThis section describes the government value function and the price of debt that define an equilibriumin the model.Government value function. Given the aggregate state st, the government chooses gt, Bt+1,and zt in two stages within each period. Conditional on no previous default, the value when thegovernment first decides how much to borrow from the international credit market is given byV (s) = maxB′u (g) + βEV(s′)(3.1)subject tog + zB =Z1−zy + q(s,B′)B′z =z(s,B′, q(s,B′))g =g(s,B′, q(s,B′))s′ =(B′, z, ζ ′)with optimal policy function B′ (s). When B′ (s) increases, default becomes more likely and theequilibrium price decreases. This drop in q will act as a no-Ponzi-game condition in problem (3.1),preventing large borrowings from the government.In the second stage, the government makes its default and spending decisions,maxg,zu (g) + βEV(s′)(3.2)subject tog + zB =Z1−zy + q(s,B′)B′z =1 or z = 0s′ =(B′, z, ζ ′)It is immediate to observe that policy functions z (s,B′, q (·)) and g (s,B′, q (·)) from problem(3.2) depend on the choice of B′. In the next sections, we will use this feature to solve thegovernment problem using a backward induction logic.Price of debt. The government borrows money from a competitive international credit market.Creditors in this market discount at the same rate as the government, and β = 1/ (1 + r), with rbeing a fixed interest rate. Hence, the equilibrium price satisfiesq(s,B′)= βE[z′((s′, B′(s′), q(s′, B′(s′))))](3.3)72The price of a one-period bond equals the inverse of the gross interest rate, adjusted by tomor-row’s expected probability of default, E [z′ (·)].Using eqs. (3.1)-(3.3), an equilibrium under short-term debt is defined as follows:Definition (Equilibrium with short-term debt). Given an initial stock of debt B0, and a givenrepayment decision from the past z−1, a recursive equilibrium with short-term debt is a governmentvalue function V (s); government policy functions B′ (s), g (s,B′, q (·)), and z (s,B′, q (·)); and aprice function q (s,B′) such that:1. Given g (s,B′, q (·)), z (s,B′, q (·)) and q (s,B′), the government policy function B′ (s) maxi-mizes the value function V (s) represented by the government problem (3.1).2. Given the value function V (s) and the price q (s,B′), policy functions g (s,B′, q (·)) andz (s,B′, q (·)) solve the second government’s problem represented by equation (3.2).3. The price q (s,B′) at which the government borrows from the international credit marketsatisfies (expected) zero profits of competitive risk-neutral creditors (equation (3.3)).3.3 The Crisis region under short-term debtWe now define different regions of debt related to the previous definition of a sunspot equilibrium.Then, we present two conditions defining bounds of these regions. Finally, we provide a numericalillustration of these regions and describe optimal debt policies for the government. Our exposition(including propositions, proofs and figures) follow Cole and Kehoe (2000) very closely.3.3.1 Debt regionsTwo thresholds B and B¯ characterize debt regions with different equilibrium outcomes for themodel economy.40 For debt levels B below B, the No-Default region defines an interval of debtwhere the government always chooses to repay its obligations. In contrast, debt levels B above B¯define a Default region where the government optimally chooses to default. Finally, starting froma debt level B ≤ B and as B increases, debt levels pass through an interval [B, B¯] called the Crisisregion where both equilibrium outcomes can occur with positive probability. When debt lies in thisregion, creditors expect default to occur with a fixed probability pi, and the sunspot ζ becomes acoordination device among them. Specifically, if ζ ≤ pi, the sunspot signals that the governmentwill default, and therefore creditors stop lending. As a result, q drops to 0, and the governmentdefaults because it cannot roll over its debt. Conversely, if ζ > pi, creditors expect the governmentto repay, and therefore q remains positive and provides liquidity to the government.40Bound B¯ will depend on pi, i.e. B¯ = B¯ (pi). For notational convenience, we omit this dependence wheneverpossible.733.3.2 The optimal price q under short-term debtIf the government always repaid (i.e., z−1 = 1), then the government and creditors trade bonds atthe equilibrium priceq((B, 1, ζ) , B′)=β, if B′ ≤ Bβˆ, if B < B′ ≤ B¯0, if B¯ < B′(3.4)with βˆ = β (1− pi). The equilibrium price (3.4) reflects the government choices of B′ (thefirst decision of the government). When B′ is above B¯, creditors expect default to occur withprobability 1, and therefore q = 0. When B′ is below B, creditors expect the government to defaultwith probability 0, and q = β. Otherwise, when tomorrow’s debt lies within[B, B¯], creditorsexpect default to occur with a fixed probability pi, implying that q = βˆ.413.3.3 Determining the region boundariesWe now present the conditions that characterize both thresholds B and B¯. Following the literature,we take t = 0 to be the first period.The No-Lending Condition (NLC) states that the government strictly prefers to default insteadof repaying maturing debt when the price of debt is 0. Analytically, this impliesV D (s, 0, 0) > V R (s, 0, 0) (3.5)where V R (·) is the value of repayment and V D (·) is the value function under default. Intuitively,the NLC suggests that the government defaults instead of repaying debt when it does not obtainany liquidity from the credit market. Therefore, when q = 0, the government choice of B′ becomestrivial and assumed to be 0. Finally, the equality of equation (3.5) characterizes the lower boundB of the Crisis region.The Participation Constraint (PC) states the condition for the government to always preferrepayment versus default,V R(s,B′, q) ≥ V D (s,B′, q) (3.6)This expression states that the government prefers to honour its debts when the value of repay-ment V R (·), is greater than defaulting and becoming excluded from the credit market, V D (·). Theequality of equation (3.6) characterizes the upper bound B¯ of the Crisis region, and the governmentprefers default at any debt level above B¯.41See Appendix section B.1 for the calculation of equation (3.4).743.3.4 Characterization of the lower bound of the Crisis regionIf the government chooses to default at t, the government budget constraint in that period isg = Zy + q(s,B′)B′After default, the government only finances spending with a share Z of total output y, and losesaccess to the international credit market. Therefore, the budget constraint from t + 1 onward issimply g = Zy. The value of default for any price q is therefore given byV D(s,B′, q)= u(Zy + q(s,B′)B′)+ βu (Zy)1− β (3.7)Setting q = 0 in the last expression leads to the No-Lending Condition (3.5) (left-hand side),V D (s, 0, 0) =u (Zy)1− β (3.8)To compute the right-hand side of the NLC, the budget constraint at t when the governmentstill repays but q = 0 is g = y−B. Thereafter, the government finances spending with total output(i.e., g = y) and V R (·) in the No-Lending Condition (3.5) rendersV R (s, 0, 0) = u (y −B) + β u (y)1− β (3.9)After replacing eqs. (3.8) and (3.9) in the NLC (3.5), the lower bound B of the Crisis region isthe debt level such that expressionu (Zy)1− β > u (y −B) + βu (y)1− β (3.10)holds with equality. Appendix section B.1.3 characterizes the existence and uniqueness of B.3.3.5 Characterization of the upper bound B¯ of the Crisis regionThe right-hand side of the Participation Constraint (3.6) follows immediately from equation (3.7).When government debt B′ lies in the Crisis region, the value of defaulting today isV D(s,B′, q)= u(Zy + βˆB′)+ βu (Zy)1− βwhere the equilibrium price of debt is now q = βˆ.To obtain the left-hand side of the Participation Constraint (3.6), the value of repaymentdepends on different paths of borrowing that the government can choose in the Crisis region. Inparticular, the government can decide to lower debt in T arbitrary periods, or also choose to neverrun down debt, i.e. T →∞. We will therefore compute V R (s,B′, q) describing the optimal choiceof {Bt+1} by dividing the government problem in two parts, first when government debt is in theCrisis region, and second when debt is in the No-Default region. Specifically, in the first part, initial75debt B0 starts in the Crisis region at t = 0, and the government exits the Crisis region in T periodsby choosing an optimal path {Bt+1}T−1t=0 . In the second part, the government begins already at theNo-Default region, and starting from t = T it will choose the optimal path of debt {Bt+1}∞t=T . Inwhat follows, we use a backward induction logic and start from the second problem.Assume that the government never defaulted in the past (z−1 = 1). The government problemafter reaching the No-Default region in T periods and starting with an initial debt level BT = B isgiven bymax{Bs+1,gs}∞s=T∞∑s=Tβsu (gs) (3.11)subject togs +Bs =y + qsBs+1 ∀s ≥ TBT =B givenBs ≤B ∀s ≥ T(Refer to Appendix section B.1 for the full solution.)The Euler Equationu′ (gt) = u′ (gt+1)states that the government smooths gt across periods and attains a constant level of spendingg¯ = y − (1− β)BSince the government’s optimal choice of debt remains at B for every period t = T, T+1, T+2, ...,the present value from smooth spending rendersu (y − (1− β)B)1− βand represents the government’s continuation payoff after reaching the No-Default region.Equipped with the previous result, we now turn to the first part of the government problem.Starting at t = 0 and with a given debt level B0 in the Crisis region, the government lowers debtin T periods to reach the No-Default region. Hence, the total payoff V T (B0) from lowering debt76in T ≥ 2 periods is42V T (B0) = max{Bt+1,gt}T−1t=0T−1∑t=0βˆtu (gt) + βˆT−1βu (y − (1− β)B)1− β +[1− βˆT−11− βˆ]βpiu (Zy)1− β (3.12)subject togt =y −Bt + qtBt+1 ∀t = 0, 1, ..., T − 2gt =y −Bt + qtB t = T − 1qt =βEt [zt+1]B0, B given, z−1 = 1The first two terms in equation (3.12) show the government’s payoff when the sunspot does nottrigger government default. With probability 1− pi, the (discounted) value of lowering Bt+1 yieldsa utility of u (gt) in each period t ≤ T − 1, and the government achieves the continuation payoffu (y − (1− β)B) from t ≥ T onward. Conversely, the last term in equation (3.12) captures thepayoff from following a policy T when the sunspot triggers government default. Specifically, thegovernment is subject to the realization of a self-fulfilling debt crisis with probability pi in everyperiod it remains in the Crisis region. When that is the case, it can only spend Zy.Maximization of expression (3.12) renders the same Euler Equation as before, u′ (gt) = u′ (gt+1),implying that government spending is constant during the transition. Following the literature,gt = gT (B0) denotes the optimal level of government spending given B0 and the chosen policy T .Replacing each gT (B0) in the T budget constraints, the optimal level gT (B0) is given bygT (B0) = y −[1− βˆ1− βˆT]B0 +[1− βˆ1− βˆT]βˆT−1βB (3.13)and the value function associated to problem (3.12) when the government runs down debt in Tperiods isV T (B0) =[1− βˆT1− βˆ]u(gT (B0))+ βˆT−1βu (g¯)1− β +[1− βˆT−11− βˆ]βpiu (Zy)1− β (3.14)This last expression (3.14) captures the value function for any policy T . Additionally, takingthe limit as T → ∞ in expressions (3.13) and (3.14) yields the value when the government neverlowers initial debt B0,42Refer to the Appendix section B.1 for the full calculation of V T (B0), the Euler Equation, and the calculationof gT (B0). Note that when T = 1, the government jumps from B0 to B and exits the Crisis region. Since nointermediate choice of {Bt+1} takes place during the transition, we omit writing this case here, and relegate itsdetails to the Appendix.77V∞ (B0) =u(y −(1− βˆ)B0)1− βˆ +βpiu (Zy)(1− β)(1− βˆ)Finally, we define the maximum across every payoff V T (B0) when T = 1, 2, ...,∞ as the left-hand side of the Participation Constraint (3.6), and obtain the final expression of the PC asmax{V 1 (B0) , V2 (B0) , ..., V∞ (B0)} ≥ u(Zy + βˆB0)+ βu (Zy)1− β (3.15)As before, the upper bound of the Crisis region is the debt level B¯ such that expression (3.15)holds with equality.3.3.6 Crisis equilibria under short-term debtFrom the last section, equations (3.10) and (3.15) characterized the lower and the upper boundsof the Crisis region respectively. In what follows, we numerically show Crisis regions for differentdefault probabilities. With a similar approach to Cole and Kehoe (2000), we first compute the Crisisregion in an equilibrium where the sunspot channel is switched off, i.e., pi = 0. This equilibrium iscalled a zero-probability crisis equilibrium. Then, we compute a Crisis region under strictly positivecreditors’ beliefs pi. Finally, we present analytical and numerical results on the government optimaldebt policies.43A zero-probability crisis equilibrium. When creditors’ belief of default pi becomes smaller,the government’s incentives to run down debt decrease. Intuitively, the government must preferto roll over its total initial debt when lenders never expect default, i.e. pi → 0. Analytically, thisimplies that V∞ should strictly dominate any other payoff V T (with T < ∞) in the LHS of theParticipation Constraint (3.15). (Appendix B.1 proves this argument.) Therefore, the ParticipationConstraint (3.15) under a zero-probability crisis equilibrium becomesu (y − (1− β)B0)1− β ≥ u (Zy + βB0) + βu (Zy)1− β (3.16)Finally, Appendix section B.1.3 establishes the existence and uniqueness of debt levels B andB¯ satisfying eqs. (3.10) and (3.16) at equality.Rewriting the No-Lending Condition (3.10) and the Participation Constraint (3.15) asUNLC (B) ≡ u (y −B) + β[u (y)1− β]− u (Zy)1− βUpi=0PC (B) ≡u (y − (1− β)B)1− β − u (Zy + βB)− βu (Zy)1− βwe observe that as debt levels increase, both curves monotonically decrease and cross the 0-line43For ease of exposition, we will focus on stationary debt policies when characterizing the Crisis region. For furtherdetails on the stationarity of debt policies, refer to the discussion in Cole and Kehoe (2000).78once. Appendix section B.1.3 analyzes functions UNLC (B) and Upi=0PC (B) on the domain of B.Figure 3.2 plots the Upi=0PC (B) and the UNLC (B)) as B increases. The government utility isu (gt) = log (gt). In addition, constant output y is set at 10, while the share of remaining outputafter default is Z = 0.9. The discount parameter is β = 0.96 , and the probability of default piis set to 0. The Crisis region is characterized by the curves Upi=0PC (B) and UNLC (B) where debtlevels B satisfy Upi=0PC (B) ≥ 0 ≥ UNLC (B). In the particular example of Figure 3.2, the intervalof debt levels where the sunspot ζt triggers self-fulfilling debt crises (with probability pi = 0) isB ∈ [9.28, 15.9].A positive-probability crisis equilibrium. We now compute a nonempty Crisis regionwhere self-fulfilling debt crises occur with positive probability, i.e. pi > 0. Using V∞ (B0), werewrite the Participation Constraint (3.15) asUpi>0PC (B) ≡u(y −(1− βˆ)B)1− βˆ +βpi1− βˆu (Zy)1− β − u(Zy + βˆB)− βu (Zy)1− βAppendix section B.1.3 shows that Upi>0PC (B) starts positive at B = 0 and monotonically de-creases crossing the 0 line as B rises. Since the No-Lending Condition (3.10) is independent frompi, Upi>0NLC (B) = UNLC (B) ∀pi ∈ [0, 1], and therefore has the same behavior as before.As creditor’s default belief pi increases, Figure 3.3 illustrates how the upper bound B¯ (pi) de-creases relative to B¯ (0). Appendix section B.1 analytically proves that dB¯ (pi) /dpi < 0. Thisresult implies that some debt levels below the upper bound B¯ (0) strictly violate the ParticipationConstraint, eq. (3.15), when pi > 0, thus making B¯ (pi) < B¯ (0). Intuitively, when pi rises, creditorsbelieve that some debt levels are now too large to be honoured, thus expecting the governmentto default for sure at these debt levels. Also, the government now prefers to default at large debtlevels B and collect βˆB, instead of facing creditors’ panic with probability pi and collect 0. Theseinteraction ultimately increases the interval where the government defaults with certainty, loweringthe upper bound B¯ of the Crisis region.Moreover, when pi > 0 and initial debt is not very large, it is intuitive that the government opti-mally evaluates policies to escape the Crisis region. As the next Proposition shows, the governmentwill choose different optimal policies for {Bt+1}, depending on the initial debt when pi > 0. Thisresult has been established by Cole and Kehoe (2000) and is one of the most relevant findings intheir paper. In the Appendix section B.1, we analytically prove the most important parts of theirProposition applied to our version of the model.Proposition 3.3.1. Let V T (B0) be the government’s payoff when its policy is to lower its debt toB in T periods. Then for any B0 < B (pi) there exists a T∗ (B0) ∈ {1, 2, ...,∞} that maximizes{V 1 (B0) , V2 (B0) , ..., V∞ (B0)}. Moreover, for pi close to 0, there are necessary regions of B0 withthe full range of possibilities T ∗ (B0) = 1, 2, ...,∞.Proof. See Appendix.When pi > 0 and B0 lies in the Crisis region, government’s default and repayment decisions79ultimately depend on the realization of the random variable ζt. When this is the case, the result inProposition 3.3.1 states that the government optimally chooses to exit the Crisis region in T ∗ (B0)periods. Furthermore, this result also states that for a small positive pi, the Crisis region can bedivided into different intervals of debt where the government optimally sets a different T ∗ (B0),with T ∗ (B0) = 1, 2, ...,∞. To understand the intuition behind, recall that the government knowsthat the sunspot triggers default when B0 ∈[B, B¯ (pi)]. Proposition 3.3.1, therefore, states thatthe government can optimally choose to exit the Crisis region to avoid being subject to creditors’beliefs.At a different initial B0 in the Crisis region, Figure 3.4 depicts three optimal debt policies whenself-fulfilling debt crises occur with positive probability. When B0 is close to B, the governmentlowers debt faster than when B0 approaches B¯ – see, for example, the lower optimal path for {Bt+1}when T ∗ (B0) = 5. As B0 increases, government debt is further from the No-Default region, andProposition 3.3.1 establishes that the optimal T ∗ (B0) does not decrease. The government thenfaces the following trade-off: on the one hand, the fastest it reaches the No-Default region, thelower the probability that the sunspot triggers default. On the other hand, a sharp decline in debtBt+1 leads to lower government spending. Furthermore, initial per-period utility levels u (·) aremore valuable because of discounting. To balance these gains and losses when B0 rises, Proposition3.3.1 states that the government can only delay its exit from the Crisis region, i.e. increase T ∗ (B0).Examples of these optimal choices are depicted by the middle and uppermost paths of {Bt+1}, withT ∗ (B0) = 10 and T ∗ (B0) = 15 respectively.To better illustrate these effects, we take the difference between two value functions at T andT + 1 for a given B0 ∈(B, B¯ (pi)),V T (B0)− V T+1 (B0) =1− βˆT1− βˆ[u(gT (B0))− u (gT+1 (B0))]+ βˆT [u (g¯)− u (gT+1)]+βˆT−1βpi1− β [u (g¯)− u (Zy)]The first term in brackets represents the loss associated to plan T versus T+1. Appendix sectionB.1.3 shows that gT (B0) < gT+1 (B0). This means that government spending is higher when thegovernment smooths g across more periods. Intuitively, when the government delays exiting theCrisis region from T to T + 1, it allocates the decrease of initial B0 into more periods. This impliesthat government spending is relatively higher at T + 1 than at T . In contrast, the second and thirdterms represent the government gains from choosing T . The appendix also shows that spending g¯is always bigger than gT for any finite T . In particular, u (g¯)−u(gT+1) is the benefit from attainingthe No-Default region earlier given policy T (which is faster than T + 1). Finally, the differenceu (g¯) − u (Zy) captures the relative gain once government spending reaches the constant level g¯.The higher is u (g¯) relative to u (Zy), the higher are the government’s incentives to exit the Crisisregion soon. Not surprisingly, this term increases as the transition T becomes shorter: the earlier80the government arrives at B, the lower will be the impact of discounting on this term.Some comparative statics. We analyze how the amount of output left after a default, Z,affects the bound of the Crisis region and the government value function.Proposition 3.3.2. An increase in Z (i) strictly decreases both bounds B and B¯ (pi) of the Crisisregion, (ii) strictly decreases the value of running down debt in one period V 1 (B0), and (iii) strictlyincreases the value of debt roll over V∞ (B0).Proof. See Appendix.Result (i) states that the lower and upper bounds of the Crisis region, B and B¯, are decreasingin Z,∂B∂Z= −u′(Zy)1−β yu′ (y −B) < 0,∂B¯ (pi)∂Z= −β u′(Zy)y1−β[1− pi1−βˆ]+ u′(Zy + βˆB¯)yu′(y −(1− βˆ)B¯)+ u′(Zy + βˆB¯)βˆ< 0To understand the intuition behind the first expression, assume that Z0 < Z1. When Z risesfrom Z0 to Z1, a default event reduces the economy’s output by less. This, in turn, improves thedefault outcome for the government for every debt level. In response to that, rational creditorshave to attribute a default probability pi to more debt levels below the initial B (Z0). Intuitively,creditors must expect that the government has less incentives to repay after the government’sdefault scenario becomes better. Therefore, the lower bound of the Crisis region needs to decreaseto define a new threshold B (Z1), such that B (Z1) < B (Z0) – below which creditors again expectrepayment with probability 1 again.A similar logic applies to the upper bound of the Crisis region. A rise in Z would only improvethe default scenario for the government. Since this increases the value V D, the government has nowgreater incentives to default for any debt level. But if B¯ (pi) remained unchanged, creditors wouldstill expect repayment of some levels of debt with probability 1 − pi, where the government nowstrictly prefers default. Therefore, rational creditors must adjust their beliefs and assign probability1− pi of repayment to lower debt levels, thus decreasing B¯ (pi) after Z rises.For part (ii), observe thatdV 1dZ= β∂B∂Z[u′ (g0)− u′ (g¯)]< 0By (i), recall that ∂B/∂Z < 0. Hence, an increase in the output left post-default also increasesthe distance between the initial stock of debt B0 and the lower bound of the No-Default region,B (Z1). Because of this, debt levels Bt+1 have to be reduced by a larger amount in order to attainthe (now further) bound B (Z1). As a result, this effect lowers total payoff V1 (B0).Part (iii) of the Proposition argues that when the optimal plan entails never running down debt,the government’s payoff responds positively to an increase in the productivity parameter Z.81dV∞ (B0)dZ=βpiu (Zy)(1− βˆ)(1− β)y > 0The last term of expression V∞ (B) indicates that if the government is entitled to permanentlykeep its initial debt level, then it will always be vulnerable to self-fulfilling debt crises. If thegovernment is likely to default in every period, then the government’s payoff can only rise if thebad outcome Zy increases.3.4 A model with long-term debtIn this Section, we present the law of motion of long-term debt issuances. We then show thegovernment problem that enters into the new definition of a sunspot equilibrium under long-termdebt. In Section 6, we show the associated Crisis region and government optimal choices of debtwhen bonds have long maturity.3.4.1 Law of motion of long-term debt and new budget constraintIn the previous sections, we presented Crisis regions for pi = 0 and pi > 0, and described optimalgovernment policies under short-term debt. Here, we study how the above results change afterswitching to long-term debt defined as in Hatchondo and Martinez (2009) – we also follow theexposition from Arellano and Ramanarayanan (2012).In this new set-up, it represents the issuance of long-term bonds at any period t. At period 0,the government now starts with a given amount of outstanding bonds issued from the past, i−1,and in every period t the government collects qtit after issuing it bonds. Also, recall that undershort-term debt, total outstanding debt was repaid in t + 1. In contrast to this case, under long-term debt the government now repays coupons worth δn−1 in every period t+ n and n ≥ 1, whereδ ∈ (0, 1) denotes the rate of geometric decrease of coupon payments across time.Conditional on no default, the government budget constraints for periods 0, 1, ..., t areg0 + i−1 = y + q0i0g1 + δi−1 + i0 = y + q1i1...gt + δti−1 + δt−1i0 + ...+ δit−2 + it−1 = y + qtitCalling Bt =∑t−1j=0 δt−j−1ij + δti−1 the stock of maturing debt (i.e., the total coupons due at tfrom past issuances), B0 equals i−1 and the budget constraint becomesgt + ztBt = Z1−zty + qt (st,Bt+1) it∀t (3.17)82In addition, the law of motion describing it isit = Bt+1 − ztBtδ (3.18)What equation (3.18) states is that (i) debt issuances depend on the default choice zt, and (ii)it units will become part of the stock of coupon payments Bt+1 that are due tomorrow. Finally,note that when δ equals 0, government issuance is it = Bt+1 – and we return to the short-termdebt set-up.3.4.2 Government value function and new price of debtUnder long-term debt, the choice of it occurs in stage 2 of the model, and equation (3.18) becomesan additional constraint to the government problem (3.1). Given state s, the first governmentdecision involves choosing B′ and i according toV (s) = maxi,B′u (g) + βEV(s′)(3.19)subject tog + zB =Z1−zy + q(s,B′)ii =B′ − zBδz =z(s,B′, q(s,B′))g =g(s,B′, q(s,B′))s′ =(B′, z, ζ ′)Since the second government decision remains unchanged, problem (3.2) depicts the choice ofz and g.Government and creditors trade long-term debt in a competitive international credit market atthe equilibrium priceq(s,B′)= βE[[1 + q(s′, B′(s′))δ]z′(s′, B′(s′), q(s′, B′(s′)))](3.20)Similar to the one-period bond case, the first term of the sum represents the value of a couponreceived tomorrow. The difference is now that a bondholder receives a stream of coupons in everyperiod t. Therefore, the second term can be interpreted as the value of reselling the long-term bondafter it paid its first coupon.We now define the equilibrium under long-term debt.Definition (Long-term debt). Given an initial stock of debt B0, and a given repayment decisionfrom the past z−1, a recursive equilibrium with long-term debt is a government value function V (s);policy functions B′ (s), i (s), g (s,B′, q (·)), and z (s,B′, q (·)); and a price function q (s,B′) suchthat:831. Given g (s,B′, q (·)), z (s,B′, q (·)) and the price q (s,B′), the government policy functionsB′ (s) and i (s) maximize the value function V (s) represented by the government problem(3.19).2. Given the value function V (s) and the price q (s,B′), policy functions g (s,B′, q (·)) andz (s,B′, q (·)) solve the second government’s problem represented by eq. (3.2).3. The price at which the government borrows from the international credit market (3.20) sat-isfies (expected) zero profits of competitive risk-neutral creditors.3.5 The Crisis region under long-term debtThe definition of debt regions outlined in Section 3 holds for long-term debt. In the next parts, wepresent new features of the price of debt, as well as the value functions that define the bounds ofthe Crisis region.3.5.1 Optimal price under long-term debt and determination of the regionboundsIn what follows, we particularize the equilibrium price of debt (3.20) when debt B′ lies in differentdebt regions. When B′ ≤ B, the international credit market expects no default and, hence, thegovernment always repays. The equilibrium price is then stationary and equal toq(s,B′)=β1− βδWhen debt lies in the Crisis region, the international credit market expects the government todefault with probability pi. Under debt roll over, the equilibrium price is again stationary but nowyieldsq(s,B′)=βˆ1− βˆδUsing a backward induction argument, we now illustrate the logic when the government runsdown debt in T = 2 periods – we relegate to the Appendix the general case when T is anyarbitrary number. Be qTt (s,B′) ≡ qTt the current price at t when exiting the Crisis region takes Tperiods. Starting at t = 0, in t = 1, 2 the optimal price does not depend on the realization of thesunspot ζt since creditors know that tomorrow’s choice of debt lies in the No-Default region. Hence,q21 = q22 = β/ (1− βδ). However, given that the government planned to exit in T = 2 periods, att = 0 creditors know that the government debt will lie in the Crisis region in t = 1. Thus, at t = 0the optimal price of debt isq20 = βE[(1 + q21δ)z′]= βˆ + βˆδβ1− βδ84Following this logic, the equilibrium price when the government exits the Crisis region in Tarbitrary periods is44q (·) =β1−βδ , if B′ ≤ B(1− (βˆδ)T−1−t) βˆ1−βˆδ + (βˆδ)T−1−t β1−βδ , if B < B′ ≤ B¯and t < T − 10, if B¯ < B′(3.21)The expression shows that price qTt is now the convex combination of the optimal pricesβ/ (1− βδ) and βˆ/(1 − βˆδ). In particular, the weights depend on both the government choiceof T , and the time left to exit the Crisis region. Finally, expression (3.21) is consistent with theprice being stationary when the government always rolls over debt that lies in the Crisis region.When T →∞, the second term vanishes since the international credit market expects the govern-ment to never lower debt, and hence q∞t = βˆ/(1− βˆδ) for any t.In what follows, we denote B and B¯ as B (δ) and B¯ (δ) in order to highlight the role of debtmaturity in the bounds of the Crisis region. Under long-term debt, the No-Lending Condition (3.5)and the Participation Constraint (3.6) from Section 3 will still characterize the lower and upperbounds of the Crisis region.Starting first with the No-Lending Condition, note that expression V D in equation (3.5) is thesame as equation (3.8): the value of default when nobody lends does not depend on the type ofbond. We now compute V R in the NLC (3.5). Recall that under δ = 0 (the short-term debt case),the government paid its total outstanding obligations B in the current period and financed g withy thereafter. This was equivalent to a situation where lenders froze credit for one period. Thedifference when δ > 0 is that now the government only repays δB in the current period, and whathappens thereafter depends on the duration of the credit market freeze.If we consider the extreme case where the credit market freezes loans for infinitely many periods,the government would never be able to issue debt again. Appendix section B.2.2 shows that thevalue function V R that results after iterating the law of motion for it (Eq. 3.18) isV R (B0, δ) =∞∑t=0βtu (gt) (3.22)with gt = y − δtB0.Expression (3.22) implicitly states that creditors never lend even when the government honoursits obligations in every period. Since this is a strong requirement, we implement the more realisticassumption that lenders freeze loans only for an arbitrary number of periods. In this set-up, we fixthe credit freeze to one period, what allows us to compare our results with Cole and Kehoe (2000).44From the previous example for T = 2, q = β/ (1− βδ) also holds when B′ ∈ [B, B¯] and t = 1. Therefore,q = β/ (1− βδ) is valid for the general T case when t = T − 1.85Hence, the government value function yieldsV R (B0, δ) = u (y −B0) + βu(y − 1−β1−βδ δB0)1− β (3.23)and the final expression for the No-Lending Condition under a one-period market freeze isu (Zy)1− β > u (y −B0) + βu(y − 1−β1−βδ δB0)1− β (3.24)Expression V D in the Participation Constraint (3.6) only needs substituting the stationary priceq by βˆ/(1− βˆδ)in eq. (3.7). For V R, we proceed with a backward induction logic similar to theone-period bond case, but using budget constraint (3.17) and adding the law of motion of issuances(3.18). Therefore, if no default occurred after the T periods of transition, the government problemat the No-Default region ismax{Bs+1,gs,is}∞s=T∞∑s=Tβsu (gs)subject togs +Bs =y + qsis ∀s ≥ Tis =Bs+1 − δBs ∀s ≥ Tqs =β/ (1− βδ) ∀s ≥ TBT =B givenBs ≤B ∀s ≥ T(Refer to Appendix section B.2 for the full solution.) The optimality condition u′ (gt) = u′ (gt+1)characterizes a constant government spending given byg¯ = y −(1− β1− βδ)BThis last expression leads to the continuation payoff u (g¯) / (1− β) once the government reachesthe No-Default region. Using this result when the government problem starts with an initial debtB0 > B (δ) and debt is lowered in T periods, givesV T (B0) = max{Bt+1,gt,it}T−1t=0T−1∑t=0βˆtu (gt) + βˆT−1βu(y −(1−β1−βδ)B)1− β +[1− βˆT−11− βˆ]βpiu (Zy)1− β (3.25)subject to86gt +Bt =y + qTt it ∀t = 0, 1, ..., T − 1it =Bt+1 −Btδ ∀t = 0, 1, ..., T − 2it =B −Btδ t = T − 1qTt =βEt[zt+1(1 + qTt+1δ)]B0, B given, z−1 = 1with associated Euler Equationu′ (gt) = u′ (gt+1)[βˆ(1 + δqTt+1)qTt]Given the assumptions on the international credit market, it is immediate to see that the termin square brackets in the Euler Equation is equal to 1. Particularly, creditors’ indifference conditionrequires balancing today’s cost of lending with tomorrow’s benefit of being paid a coupon plus thepossibility of reselling the bond. As a result, government spending will be constant during the firstt = 0, 1, ..., T − 1 periods as in the short-term debt case.Using again the T budget constraints, we can obtain the expression for the optimal gT (B0, δ)when the government lowers debt in T periods to exit the Crisis region – for a full derivation, seeAppendix section B.2.gT (B0, δ) = y −(1 + qT0 δ)B0 −B T−1∏j=0qTj1 + qTj δ1 +T−2∑k=0T−2−k∏j=0qTjT−1−k∏j=1(1 + qTj δ)−1With these results, we define the Participation Constraint (3.6) asmax{V 1 (B, δ) , V 2 (B, δ) , ..., V∞ (B, δ)} ≥ u(Zy + βˆ1− βˆδB)+ βu (Zy)1− β (3.26)whereV∞ (B, δ) =u(y −[1−βˆ1−βˆδ]B)1− βˆ +βpiu (Zy)(1− βˆ)(1− β)Finally, the equality of (3.26) solves B¯ (pi, δ) – which we denote as B¯ (δ).3.5.2 Zero- and positive-probability crisis equilibriumWe numerically show that a nonempty Crisis region can exist using long-term debt for pi ≥ 0.Following this, we illustrate different debt trajectories {Bt+1} under alternative plans of loweringdebt in T periods. All these proofs are relegated to the Appendix.87Appendix B.2 shows that there is one B (δ) defining a lower bound to the Crisis region whenδ > 0. For the upper bound B¯ (pi, δ), we also argue that V∞ (B, δ) dominates any other policyT < ∞ when pi goes to zero in the PC. Intuitively, when creditors expect default to occur withzero probability in the Crisis region, the government has no incentive to lower debt. Thus, theParticipation Constraint (3.26) becomesu(y −[1−β1−βδ]B¯ (0, δ))1− β = u(Zy +β1− βδ B¯ (0, δ))+ βu (Zy)1− βAppendix B.2 proves that there exists a unique level of debt B¯ (0, δ) that equates the previousexpression – and also for B¯ (pi, δ) at strictly positive probabilities of default.Before proceeding to the characterization of a nonempty Crisis region, it is worth noting thedifference between the amount of maturing debt versus the total amount of debt. To understandthis, we denote Bˆ ≡ B/ (1− δ) the total outstanding obligations for a given debt level B. In themodel with one-period bonds of the previous sections, any bond issued in period t matured at t+1.As a result, the amount of debt maturing at t+ 1 equated total outstanding debt, namely, B = Bˆ.With long-term debt, however, the amount of maturing debt in period t+ 1 only pays a fraction oftotal outstanding debt Bˆ. This is because coupons divide the payment of Bˆ into more periods, thusdecreasing maturing debt for every t. The plot below highlights the different repayment scheduleswhen the government repays the same stock of debt Bˆ using either one-period bonds (Figure 3.5,left panel) or long-term bonds with infinite coupon payments (Figure 3.5, right panel).The right panel of Figure 3.5 shows that coupon payments generate the same total debt as inthe left panel by allocating portions of debt more evenly across periods. This lowers the amountof maturing debt at every t, thus allowing the government to stand a higher total amount of debtBˆ. As a result of this, the government’s default incentives should intuitively decrease as δ rises,thus increasing thresholds B (δ) and B¯ (δ) where the government is indifferent between repaymentor default.Using the previous choice of parameters, Figure 3.6 characterizes three nonempty Crisis regions.To obtain this graph, we first replace the level of debt in the No-Lending Condition, eq. (3.24), andin the Participation Constraint, eq. (3.26), by Bˆ (1− δ). We do this to derive conclusions on thethresholds B (δ) and B¯ (δ) based on total amount of debt, when comparing them under short-term(δ = 0) and long-term debt (δ > 0). Then, we defineUNLC (B; δ) ≡ u (Zy)1− β − u (y − (1− δ)B)− βu(y − 1−β1−βδ δ (1− δ)B)1− βUpi>0PC (B; δ) ≡u(y − 1−βˆ1−βˆδ (1− δ)B)1− βˆ +βpi1− βˆu (Zy)1− β − u(Zy +βˆ1− βˆδ (1− δ)B)− βu (Zy)1− βand plot these curves as δ varies – the Appendix analytically describes their behavior as B increases.Debt levels B satisfying Upi>0PC (B; δ) ≥ 0 ≥ UNLC (B; δ) when δ = {0, 0.1, 0.2} characterize intervals88where self-fulfilling debt crises can occur with positive probability. When δ = 0, the governmentrepays its total outstanding obligations in one period. Instead, when δ > 0, total debt is spreadacross infinitely many periods, and from our previous intuition this should allow Bˆ to be higher.In particular, some debt levels that initially violated eqs. (3.24) and (3.26) when δ = 0 will nowsatisfy them since a higher δ decreases default incentives. Ultimately, this implies that the boundsof the Crisis region can only increase in order to keep the government indifferent between defaultingand repaying. The next Proposition formalizes this result.Proposition 3.5.1. When δ rises, the bounds B (δ) and B¯ (pi, δ) of the Crisis region increase.Proof. See Appendix.Figure 3.5 illustrated how an increase in debt maturity made future coupon payments a largershare of total debt. In line with this, the result in Proposition 3.5.1 analytically states that higherdebt maturity decreases government’s incentives to default. When δ increases, the governmentfaces lower obligations at early periods, therefore increasing government’s incentives to (i) repaywhen nobody lends (i.e., satisfy the NLC, eq. (3.24)), and (ii) remain in the international creditmarket (i.e., satisfy the PC, eq. (3.26)).Figure 3.7 plots trajectories of debt when δ > 0 and the government chooses the optimal timeto exit the Crisis region by lowering the total amount of debt. It is straightforward to observe thatthe optimal government policies {Bt+1} imply trajectories of debt that increase T as B0 rises. Thisis similar to the short-term debt case. The bounds of the Crisis region when δ > 0, however, areabove those of short-term debt – what coincides with the finding in Proposition 3.5.1. When δ > 0,the government can tolerate a higher total amount of debt since coupon payments allocate a largershare of obligations in future periods.3.6 Welfare analysisIn this section, we analyze the welfare implications of lengthening debt maturity. We first explorethe impact of short-term and long-term debt in the government’s optimal debt choices. We thenturn to the numerical comparison of welfare under bonds of different maturity.Starting at the same level of total outstanding debt, Figure 3.8 illustrates the optimal path{Bt+1} under the two classes of bonds. In particular, we set pi = 0.0001, and parameters δ =0 and δ = 0.1 denote one-period and long-term bonds, respectively. The plot shows that aninitial debt level Bˆ ≈ 11.15 is closer to the lower bound of the Crisis region under long-term debt(B (0.1) ≈ 10.23) than under short-term debt (B (0) ≈ 9.28). The reason why B (0.1) > B (0)is that creditors now expect default to occur at larger debt levels when δ is higher. Recall thatwhen debt maturity increases, some debt levels above threshold B (0) will now be independent fromthe sunspot variable ζt. At those levels of debt, not only the government’s incentives to defaultdecreased, but also creditors know that the government can tolerate a larger stock of total debt.89As a result, the government now repays these debt levels with probability 1, until it reaches thenew upper bound B (0.1).45Moreover, the number of periods that the government takes to escape the Crisis region T ∗ (B0)also varies as δ increases. The plot shows that when δ = 0, the government avoids self-fulfilling debtcrises by exiting the Crisis region in T (Bˆ) = 10 periods. When δ = 0.1, however, the governmentchooses a faster policy T (Bˆ) = 5. Specifically, when δ increases, any total debt is now closer to thenew lower bound of the No-Default region. This proximity of Bˆ to B (0.1) provides incentives todecrease T (Bˆ), and the government therefore becomes less vulnerable to self-fulfilling debt crises.Table 3.1 summarizes some important results from this example. Furthermore, the secondcolumn reports a greater welfare for bonds of longer maturity. Motivated by this observation, wenumerically explore the welfare implications of bond maturity for a wider range of initial debt levels.First, we build the value functions for short-term and long-term debt at different T ’s. Second, weevaluate each value function for every total outstanding debt Bˆ that lies in the Crisis region. Third,for each Bˆ, we identify which policy T achieves the highest value V T (·). We repeat this processboth for short-term and long-term debt. Fourth, we build the envelope function that collects thehighest V T (·), for every Bˆ and for each separate case, δ = 0 and δ > 0. Specifically, for eachδ = {0, 0.1}, the envelopes will show the value V T (·), for each Bˆ in the Crisis region and at theoptimal T (Bˆ). Finally, we use these envelopes to perform welfare comparisons.Figure 3.9 shows the envelope functions under short-term (red line) and long-term debt, forδ = 0.1 and 0.2 (blue line and dark line, respectively).46 It is immediate to observe that as δincreases, the envelopes are strictly above the one under short-term debt, for any initial total debtin the Crisis region. In other words, the government attains a higher total utility at any initial Bˆwhen debt maturity increases. To understand this, recall that when δ = 0 the government paidback total debt in one period. When δ > 0, however, the government repays the same total amountof debt using coupons in each period. Precisely, a higher δ decreases the value of the first couponsby transferring a portion of total debt to the future. Since this decreases both default incentivesand T (B0), the likelihood of self-fulfilling debt crises declines. This is a first gain coming from ahigher δ. But now, discounting also generates an additional government gain coming from a shortertransition T (B0). In particular, when the optimal T (B0) decreases, the government reaches theNo-Default region in a smaller number of periods, and therefore it discounts less the payoff u (g¯)attained when the government arrives to that region. These gains finally explain why welfareincreases compared to the one-period bonds case.45Proposition 3.5.1 of the previous section analytically proved this result.46This figure illustrates the government payoffs for a portion of total debt levels B. Refer to Appendix B.2 for adetailed description of the government’s payoff for debt levels starting at B = 0, which also illustrates how welfareincreases when pi rises.903.7 ConclusionThis chapter explored the impact of debt maturity in a model of self-fulfilling debt crises, featuringa small open economy where a benevolent government maximized household’s utility and coulddefault on its debt. The main finding showed that switching from short-term debt to long-termdebt can improve the welfare of the economy. Specifically, the first part of the chapter character-ized an interval of debt levels for short-term debt, where creditors’ expectations of default couldtrigger government’s default with a fixed probability. The second part of the chapter modified theinitial set-up by switching to long-term debt in the form of a coupon-paying bond. This type ofbond changed the structure of the equilibrium price of debt. Furthermore, long-term debt shiftedupward the bounds of the interval of debt levels where crises are belief-driven. Finally, the chapternumerically showed that long-term bonds can improve the economy’s welfare by decreasing defaultincentives of the government and the likelihood of self-fulfilling debt crises.This chapter suggested that lengthening debt maturity can increase welfare using an economicenvironment as in Cole and Kehoe (2000). Other works in the literature, however, have documentedtrade-offs coming from the use of both short-term and long-term debt in models of sovereign default.For example, Arellano and Ramanarayanan (2012) explore hedging and incentive benefits for thegovernment that are linked to the issuance of debt of different maturities. Whether there is anoptimal combination of short-term and long-term bonds within a model of self-fulfilling debt crisesremains an open question. We plan to explore this question by incorporating the possibility tochoose between different debt instruments in our analysis in future work.913.8 TablesT (Bˆ) V (Bˆ, T (Bˆ), δ)Short-term debt (δ = 0) 10 56.4204Long-term debt (δ = 0.1) 5 56.4272Table 3.1: Welfare under short-term and long-term debt. Utility and parameter values as reportedin text.923.9 FiguresFigure 3.1: Outright monetary transactions (OMT) announcement and long-term governmentbond yields (10-yr).Source: OECD and authors’ calculations. Units: percentages, not seasonally adj. Freq.: Monthly.93Figure 3.2: Crisis region when pi = 0, short-term debt (Parameters and functional forms: β = 0.96,Z = 0.9, y = 10, u()˙ = ln()˙.)Figure 3.3: Crisis region when pi = 0 and pi > 0, short-term debt (Parameters and functionalforms: pi = {0, 0.01}, β = 0.96, Z = 0.9, y = 10, u()˙ = ln()˙.)94Figure 3.4: Optimal {Bt+1} in the crisis region at different levels of initial debt. Short-term debt.(Parameters and functional forms: pi = 0.0001, β = 0.96, Z = 0.9, y = 10, u()˙ = ln()˙.)(a) Short-term debt (δ = 0). (b) Long-term debt (δ = 0.5).Figure 3.5: Maturing debt accumulated over periods, short-term vs. long-term debt. (Totaloutstanding debt ' 9.28.)95Figure 3.6: Crisis region when pi = 0 and pi > 0, long-term debt (Parameters and functional forms:pi = {0, 0.01}, β = 0.96, δ = {0, 0.1, 0.2}, Z = 0.9, y = 10, u()˙ = ln()˙.)Figure 3.7: Optimal {Bt+1} in the crisis region at different levels of initial debt. Long-term debt.(Parameters and functional forms: pi = 0.0001, β = 0.96, δ = 0.1, Z = 0.9, y = 10, u()˙ = ln()˙.)96Figure 3.8: Optimal {Bt+1} in the crisis region of short-term and long-term debt at the same levelof initial debt. (Parameters and functional forms: pi = 0.0001, β = 0.96, δ = {0, 0.1}, Z = 0.9,y = 10, u()˙ = ln()˙.)Figure 3.9: Envelope of the value functions for short-term and long-term debt for varying levels ofthe total amount of debt. (Parameters and functional forms: pi = 0.0001, β = 0.96, δ = {0, 0.1},Z = 0.9, y = 10, u()˙ = ln()˙.)97Chapter 4Fiscal Devaluations and Lags in FiscalPolicy4.1 IntroductionA well-known result in textbook models (e.g., Krugman and Obstfeld, 2009) states that countrieswith access to monetary policy can introduce nominal exchange rate devaluations to affect theircompetitiveness. However, when a country belongs to a currency union it cannot resort to exchangerate devaluations. Such constraint is experienced by each of the 20 economies of the European Unionthat have adopted the euro as their national currency, and where each country member follows themonetary policy that the European Central Bank sets.47When a country that is subject to shocks lacks access to monetary policy, fiscal devaluationshave been analyzed as an alternative tool for macro stabilization (e.g., Schmitt-Grohe´ and Uribe(2012)). Farhi et al. (2014) (henceforth FGI) took this insight one step further and theoreticallyshowed an equivalence between nominal and fiscal devaluations. In other words, a country withoutaccess to monetary policy can use a menu of tax adjustments to mitigate cyclical fluctuations justlike nominal exchange rate devaluations do. However, it has been argued that fiscal policy reformstake time to be implemented. For instance, Friedman (1948) long ago pointed out that limitationsto fiscal stabilization policies embodied in the form of lags could affect how such policies respond tobusiness cycles. Thus, and contrary to relatively faster adjustments in monetary policy, fiscal policychanges usually need time to be implemented. Delays in fiscal policy can often be rationalized bythe need of parliamentary discussions prior to a tax change implementation. In other cases, thesedelays can be explained due to characteristics inherent to a tax bill – for instance, when tax reformsexplicitly have a future implementation date. In this context, FGI’s theoretical finding that fiscaland nominal devaluations can equivalently yield the same real outcomes seems to necessarily relyon the assumption that there are no delays in the implementation of fiscal policies. But if delays arepresent in fiscal policy, the discussion about fiscal devaluations’ effectiveness is relevant for macropolicymaking, and in particular for countries belonging to currency unions.Using a new dataset on tax policy measures recently made available by Amaglobeli et al. (2018),I replicate part of their analysis to explore the existence of implementation lags associated to fiscalpolicy with a particular focus on countries that do not have access to monetary policy. The data47These countries are Austria, Belgium, Croatia, Cyprus, Estonia, Finland, France, Germany, Greece, Ireland,Italy, Latvia, Lithuania, Luxembourg, Malta, the Netherlands, Portugal, Slovakia, Slovenia and Spain.98comes from the Tax Policy Reform Database (hereafter TPRD), where the implementation lag ofa tax change refers to the time in days that this fiscal policy takes to be implemented after itsannouncement. Based on data from nine European countries that became part of the Eurozonein 1999, I document the distribution of the implementation lag in these countries consideringdifferent summary statistics for the 1999-2018 period. Specifically, I analyze what the distributionof implementation lags of fiscal policy is when we (1) identify all tax changes as well as theirdirection of change (i.e., tax increases and decreases), (2) categorize fiscal policies as anticipatedor unanticipated, and (3) analyze fiscal policy lags from a business cycle perspective.This chapter presents an empirical and theoretical study, where the empirical analysis consistsof two parts. In the first part, I work with the whole sample of European countries whose monetarypolicy is set by the ECB, and I look for the presence of lags in fiscal policy. The dataset includesapproximately 500 observations from nine countries belonging to the Euro Area for the 1999-2018 period. During that period, tax policy changes report an average and median delay in theirimplementation of approximately 6 and 3 months, respectively. Restricting the length of lags totwo years preserves the median and lowers the average although it still remains positive and above4 months. Interestingly, these patterns are robust to considering the direction of change of fiscalpolicies, namely tax decreases and increases, which exhibit the same statistics like those of thefull sample. This is the first evidence that suggests the existence of fiscal policy lags. FollowingMertens and Ravn (2012), I use a 90-day cutoff metric to catalogue tax changes as anticipated orunanticipated. I find that more than 50 percent of the subsample from the TPRD can be cataloguedas anticipated fiscal policies. Finally, I explore whether fiscal policy exhibits different lags duringthe business cycle. To that end, I focus on three pairs of periods of economic downturn and upturnsince the creation of the Euro Area. I find that in all these six periods implementation lags averageusually more than 3 months and often have medians of more than 2 months. Also, I identify thatin two out of three of these pairs of periods the averages and medians are larger during economicexpansions. While this finding is in line with Amaglobeli et al. (2018), my results strictly apply tocountries belonging to a currency union. In addition, I document one case where the recovery thatfollowed the recession featured shorter implementation lags. Lastly, I find that fiscal policies fellunder the category of anticipated fiscal policies approximately 50 percent of the times in each ofthese six economic phases. Overall, the empirical evidence suggests the existence of implementationlags of fiscal policy, and in particular that lags are present also in the different phases of businesscycles.In their work, FGI do a numerical illustration and calibrate an open economy with parametersto the values for Spain during the 2008 great financial crisis. The authors then study the abilitythat different types of fiscal devaluations have to mitigate negative shocks to the economy. In thesecond part of the empirical analysis I follow a similar procedure to the one outlined above to assessthe incidence of fiscal policy lags in Spain. Overall, key statistics for Spain exhibited lags of atleast one-quarter and up to two quarters. In particular, and based on 79 tax changes that tookplace within 1999-2018, the data shows that fiscal policy delays averaged approximately 5 months99and exhibited a median of 3 months. A similar average value can be found for the tax-decreaseand tax-increase categories. While these observations are in line with those of the full sample, thedelays tend to be lower than those European countries reported in the TPRD that do not use theeuro. In addition, the persistence of lags in fiscal policy makes a large share of it to be anticipated(approximately 65 percent). From a business cycle perspective, I compare characteristics fromfiscal policy lags across the 2001 economic downturn versus the 2012 European debt crisis (andtheir subsequent recoveries). During recessions, I find that fiscal lags in the 2001 recession averagedroughly 6 months, while in 2012 lags fell to less than 2 months – the same is true for the medians.Interestingly, these results could have been driven by the the 2008 great financial crisis, which tookplace in the middle of these recessions and largely affected economic conditions in several countries– see, for example, Mishkin (2017). While this conjecture would deserve further explorations,this finding a priori constitutes preliminary evidence suggesting that the 2008 great financial crisiscould have affected implementation lags of fiscal policy in the Spanish economy. Finally, I analyzewhether fiscal policy lags affect differently the type of tax change. FGI consider a fiscal devaluationconsisting of a menu of tax changes (specifically, variations in value-added, capital and payrolltaxes). In my analysis, and although the sample size is not too large, I provide evidence of thepresence of lags in those tax measures that are similar to the ones used in FGI’s fiscal devaluationexercise. Specifically, I find substantially large averages and medians for corporate income taxes(more than 3 months) and for personal income taxes (nearly 6 months). Likewise, the share offiscal policies that can be anticipated represent more than half of the realizations for both cases. Insummary, this empirical evidence shows that a large share of fiscal policy changes exhibit delays.From the perspective of FGI’s quantitative experiment, this means that fiscal devaluations mightoften be anticipated and thus yield smaller welfare gains. Moreover, while FGI also analyze the caseof a one-quarter anticipated fiscal devaluation and find non negligible welfare gains, our empiricalfindings suggest that further combinations of fiscal devaluations and lags should be considered sincefiscal instruments may exhibit larger delays as well as asymmetric delays.In the second portion of the chapter I present a model to analyze the impact of implementationdelays in fiscal devaluations. In particular, I present a two-period tradable-nontradable modelwith nominal rigidities based on the textbook exposition of Schmitt-Grohe´ and Uribe (2012) (seeSchmitt-Grohe´ et al. (2022)). This model is a simple yet rich enough presentation that relates toFarhi et al. (2014) and also to their static version of the model – see Farhi, Gopinath, and Itskhoki(Farhi et al.). This model is exploratory, and the purpose of it is to illustrate the large welfareimplications of devaluations that seek to replicate a nominal devaluation but that are subject tofiscal policy lags. I present this model following Schmitt-Grohe´ and Uribe (2012) very closely(included their notation), and I analyze the case of a negative shock to the economy. In thiscontext, I numerically present what the economy can achieve under a nominal devaluation and afiscal devaluation that adjust at the time of the shock. Next, I depart from their presentation inthat I consider an additional case of a fiscal devaluation with lags, which is performed under theassumption that a fiscal devaluation does not adjust at the business cycle frequency, but instead100with a one-period delay. When this happens, fiscal devaluations cannot mitigate output collapse,and therefore this policy cannot achieve the real allocations as a nominal devaluation that adjustsat the time of the shock.The findings documented in this chapter are relevant for two reasons. From an empirical view-point, the empirical facts exhibit the likely existence of delays in the implementation of tax changeswithin the context of countries belonging to a currency union. Hence, for those countries that lackunilateral access to monetary policy, the existence of delays in fiscal policy could be incorporated aspart of fiscal policy discussions, and in particular during recessions. From a theoretical perspective,the existence of lags in fiscal policy can transform a zero-delay fiscal policy into a too restrictiveassumption. This can be particularly relevant when a model assumes that fiscal policy can adjustat the business cycle frequency – which is typically assumed to be a characteristic of monetarypolicy. As exemplified in the simple theoretical exercise performed in this model, this assumptiondeserves special consideration as its relaxation may compromise the extent to which the equivalencebetween nominal and fiscal devaluations holds.The rest of the chapter is organized in three sections. Section 4.2 presents the empirical evi-dence about implementation lags in tax policy changes, both for the whole sample of Euro Areacountries, as well as for Spain. Section 4.3 presents a model and quantitatively measures the im-pact of monetary policy, as well as fiscal devaluations under different assumptions regarding theirimplementation. Section 4.4 concludes and discusses some possible avenues for future research.4.2 Data and empirical analysisIn this section I provide a description of the data used. Next, I document the empirical evidenceto explore the relevance of the research question.4.2.1 DataThe data used in this study comes from the Tax Policy Reform Database (hereafter TPRD). TheTPRD is a collaborative work between the International Monetary Fund and the InternationalBureau of Fiscal Documentation (IBFD).48 The database is built based on information on taxmeasures collected from the Organisation for Economic Co-operation and Development and theIBFD. Amaglobeli et al. (2018) provide a full description of the construction of the database,which is publicly available and is continuously updated.The entire dataset documents tax policy reforms for 23 economies: Australia, Austria, Brazil,Canada, China, Czech Republic, Denmark, France, Greece, Germany, India, Ireland, Italy, Japan,Korea, Luxembourg, Mexico, Poland, Portugal, Spain, Turkey, United Kingdom, and the UnitedStates. In this work I focus on fiscal policy lags from countries that cannot unilaterally conduct48This dataset was downloaded from https://www.tprdportal.org/#/ (accessed on February 12, 2023) and it isbased on D. Amaglobeli, V. Crispolti, E. Dabla-Norris, P. Karnane, and F. Misch, 2018, Tax Policy Measures inAdvanced and Emerging Economies: A Novel Database, IMF Working Paper, 18/110. This reference will be alsoadded to the corresponding following the citation convention from that section – see Amaglobeli et al. (2018).101stabilization policies using monetary tools and that might need to resort to fiscal policy to mitigatenegative shocks to the economy. To that end, I only consider the nine countries that adopted theEuro, both in year 1999 (Austria, France, Germany, Ireland, Italy, Luxembourg, Portugal, Spain)and in year 2001 (Greece).49For the purpose of this chapter, the first set of variables of interest are: (i) the announcementdate, (ii) the implementation date, and (iii) the implementation lag.50 The announcement datedocuments the exact date when the tax policy change was announced by government officials.Likewise, the implementation date states the exact date of effective implementation of the taxchange as per official government documents. The announcement and the implementation datesallow the construction of the implementation lag of fiscal policy variable (or, simply, the fiscal policylag), which captures the difference in days between the announcement date and the implementationdate. This variable is positive when the tax reform is implemented at a date posterior to itsannouncement date, and negative when the tax reform is retroactive. According to Amaglobeliet al. (2018), the existence of lags in these reforms can come from interactions between differentgroups affected by a new tax regulation or even parliamentary discussions occurring prior to theimplementation of the tax measure. These are the type of empirical facts relevant to this work,which renders a total sample of 514 observations with nonnegative lags for the nine countries inthe Euro Area for the period 1999-2018 – where some of the tax changes are implemented up toyear 2020.4.2.2 Empirical analysisIn this section I present key descriptive statistics to document the relevance of fiscal policy lags.Throughout the analysis, I shall only consider tax policy reforms with nonnegative lags and focuson countries belonging to the Euro area. However, the reader is referred to Amaglobeli et al. (2018)for a comprehensive analysis of tax policy reforms for a larger set of countries. In addition, partof the exposition in this section was motivated by the aforementioned paper, from where I shallborrow terminology too.Summary statisticsAfter the creation of the Euro Area in 1999, European countries signing the currency union agree-ment started to adopt the euro as the common unit of exchange. Country members can use fiscalpolicies against business cycle fluctuations, while monetary policy is in charge of the EuropeanCentral Bank (commonly referred to as ECB).49Although not all countries adopted the euro at the same time, I shall refer to 1999-2018 as the period underconsideration for all the countries in order to simplify the exposition.50It is important to mention that the TPRD includes more information on each tax policy reform. For instance,the database includes whether tax reforms include base or rate changes. Also, it documents six different categoriesof taxes. While considering these additional dimensions could result in interesting extensions, in this study I onlyfocus on the subset of them referred in the text.102From a total of 514 tax changes identified in the TPRD, Table 4.1 summarizes descriptivestatistics from their announcement and implementation dates. The first column shows that the an-nouncement of tax changes is condensed in a smaller span of years compared to the implementationof tax changes. Specifically, the 514 tax changes were announced within 20 years (from 1999-2018),while their implementation spanned over 22 years (from 1999-2020). Since this difference meansthat in some years there must have been only taxes implemented but not announced, this suggeststhe existence of lags in fiscal policy implementation. In particular, one implication of this discrep-ancy is that the average number of implemented tax reforms per year (second column) is smallerthan the number of announced tax changes per year – approximately 23 versus 26, respectively.Finally, columns 6, 7 and 8 in Table 4.1 show that tax changes are announced slightly earliercompared to their respective implementations. This indicates that implementations take longer tomatch the total number of announced taxes.Figure 4.1 plots the tax changes from our total sample according to their announcement andimplementation dates. Each bin represents a year, and the red bars with solid outline indicate theannouncement date of these tax changes, while the blue bars with dashed outline indicate theirimplementation date. It is important to mention that this plot shows all the tax changes that wereannounced on 1999 and implemented on 1999 (or later).51 Figure 4.1 shows that both distributionsof announcements and implementations of tax changes do not exactly coincide. Specifically, visualinspection of the plot reveals a slightly more dispersed distribution of tax implementations relativeto tax announcements, and also that the announcement of tax changes tend to accumulate moreobservations in earlier years (relative to their implementations). In turn, implementation lags takeplace more often at later dates. Also Figure 4.1 further illustrates the results from Table 4.1,showing that tax implementations span over more years, with part of the observations occurring inyears 2019 and 2020 (when there were no tax changes documented). All these observations wouldfavour the conjecture that Euro Area countries face delayed implementation of tax changes to someextent.Table 4.2 provides summary statistics for Euro Area countries for fiscal policy lags over theperiod 1999-2018. The first column reports statistics for the full sample. Following Amaglobeliet al. (2018), I also use the classification provided in the TPRD to identify whether there aredifferent delays conditional on the direction of the tax change. Therefore, I include two categoriesthat capture whether the tax change involved a tax decrease or a tax increase (columns 2 and 3,respectively). Tax decreases and increases are typically labelled as expansionary and contractionaryfiscal policies. As a result of this, reporting the direction of tax changes seems a sensible strategysince tax decreases may have different implementation dates relative to tax increases – for example,due to political economy aspects from different government policies.Table 4.2 shows that, in our specific sample of nine countries, the total aggregate tax measures inthe Euro Area went in both directions in roughly even shares (300 and 214 out of 514 observations).51The figure for non-Euro Area countries shows a similar pattern like that of Figure 4.1. For completeness, I plotthat figure – see Figure 4.2. The sample has 135 observations in total and includes four countries: Czech Republic,Denmark, Poland and United Kingdom.103In addition, the two categories of tax changes depict a similar behavior in terms of their summarystatistics. This also holds when we only keep observations with strictly positive lags and dropzero-lag tax changes. Because of the similar patterns, and since the more comprehensive sample oftax changes includes zero lags (which only account for 19 observations that represent less than 4percent of the total observations), I shall provide interpretations focusing only on tax changes withnonnegative lags instead of strictly positive lags.The full sample (first column) shows that, across 514 observations, tax policy changes for the1999-2018 period averaged 175 days of delay, and the median delay was of 94 days. These valuesaccounted for an average and median lag in tax policy implementation of approximately six andthree months, respectively. Tax reductions and increases (second and third column) do not differfrom the full sample, with nearly 6 months of delay on average. Also, tax increases and decreasesare in the proximity of the median of 3 months for the total sample – 87 and 97 days, respectively.Interestingly, this shows that tax decreases and increases exhibit similar lag characteristics likethose of the full sample, with the only difference arising in terms of the value of the maximumlag observed. Given that the table shows that roughly 95 percent of the sample would fall below2-year lags, I impose 730 days as an arbitrary cut-off to disregard too long tax implementationsand repeat the previous analysis.Table 4.3 is an adjusted version of Table 4.2 when we only consider fiscal policy with lags notexceeding 2 years. After dropping these extreme observations, we find that tax measures still exhibitconsiderable delays for the period 1999-2018. In particular, the full sample (first column) still showstax policy changes with an average delay of 144 days (approximately 5 months), and a median delayof 93 days (approximately 3 months). In addition, tax reductions and increases (second and thirdcolumn) are also similar along these descriptive statistics, with averages and medians roughly equalto 5 and 3 months each. Finally, Figure 4.3 shows the corresponding histogram of tax changesconsidering nonnegative lags and regardless of their direction of change. The plot’s illustrationprovides supplementary insights about characteristics of the sample reported in Table 4.3. Oneof the most salient features that the visual inspection of the plot reveals is an apparent bunchingof lags between 0 to 150 days. With relatively less density but still detectable in the histogram,another bunching occurs around lags between 150 and 350 days – i.e., almost half a year and a fullyear.As a final exercise, it is interesting to contrast these empirical findings with the fiscal policylags documented in the TPRD for four European countries that do not belong to the Euro Area.These account for a total of 129 observations: 46 for the Czech Republic, 22 for Denmark, 37 forPoland, and 24 for the United Kingdom. In particular, during the 1999-2018 period, Table 4.8shows that (1) the distribution of lags for all tax changes and for tax decreases and increases hasbeen the same across categories but, at the same time, that (2) the delays of fiscal policy have beenconsiderably larger relative to Euro Area countries. Specifically, the mean and median delay for allas well as for decreasing and increasing tax changes was around 210 days (i.e., 7 months). Thesevalues are well above those reported for Euro Area countries: non-Euro Area country members face104fiscal policy lags of 2 more months on average (i.e., 40 percent more), and a median of more thantwice that of Eurozone countries. An analysis of the reasons behind these differences is beyondthe scope of this chapter. However, one potential explanation for this evidence is that Euro Areamembers’ recognition of the lack of access to monetary policy might contribute to speed up theimplementation of fiscal policies. On the contrary, a second potential explanation is that Europeancountries that do not use the euro have access to monetary policy adjustments and might rely lessin instantaneous fiscal policy changes. In any case, these explanations are neither exhaustive norcomplete, and therefore a comprehensive analysis of the drivers of the observed empirical differencesis left for future research.Anticipated tax changesNext, I use the TPRD to explore the existence of tax changes that can be anticipated due to delaysin their implementation. To that end, I follow Mertens and Ravn (2012) and Amaglobeli et al.(2018) and apply a cut-off of 90 days to distinguish anticipated and unanticipated tax reforms– assuming that 90 days fall in the ‘unanticipated’ category. For the remainder of the chapter,our subsample from the TPRD shall preserve the constraint on maximum lags (i.e., I focus ontax changes with lags smaller or equal to 730 days). Also, I perform the analysis only includingobservations with nonnegative lags for European countries that became part of the Euro Area, andconsidering both tax increases and decreases. As a result, the sample consists of 497 observationsin total for the 1999-2018 period.Figure 4.4 shows the distribution of fiscal policy lags for our sample of nine Euro Area countriesdocumented in the TPRD for the period 1999 - 2018. This graph replicates Mertens and Ravn(2012) analysis, who used US data based on a similar narrative approach for the period 1947-2006.The plot has been constructed to capture the number of tax changes that fall within six bins, whereeach bin collects the number of tax changes that feature lags according to 6 intervals of time. Thefirst five bins show the number of tax changes with implementation lags of 30, 60, 90, 120 and 150days, and the sixth bin groups the quantity of tax changes that were implemented after 5 months.The most salient feature of Figure 4.4 is that the distribution of lags for Euro Area membersexhibits a similar twin peaked shape as the one documented by Mertens and Ravn (2012) for theUnited States. As portrayed in the figure, 97 tax reforms (approximately 20 percent of the sample)exhibit a lag in their implementation of up to 1 month (the first bin). However, the sixth bincaptures that a much bigger proportion of tax changes (accounting for approximately 35 percentof the sample) is implemented within 5 months and two years. The remaining 45 percent of thesample is roughly evenly spread around the second and fifth bins, where lags account for up to 3months.It is important to note that the frequency distribution of lags is not driven by only one ofthe nine Euro Area countries. In this sample, 80 percent of fiscal lags featuring more than 5months correspond to five countries and are distributed as follows: Two countries with more than20 percent each (Germany, 24 percent; Spain, 22 percent), one country at 16 percent (Austria),105and two countries (roughly) close to 10 percent (France, 9 percent; Greece 9 percent). Therefore,applying the 90-day cut-off to identify the proportion of anticipated tax changes in our sampleshows that 260 out of 497 observations, took place after the 90-day cutoff.52 This evidence thereforesuggests the sample of fiscal policies exhibits lags in their implementation in Euro Area countries,and that more than 50 percent of the policies can be labelled as anticipated.Tax changes and business cyclesNext, I describe the behavior of fiscal policy both during and away from recessionary periods. Fiscaldevaluations like the ones proposed by FGI are a policy used to fight recessions. Therefore, we nowanalyze whether tax reforms evidence different delays during crises. To identify business cycles inthe Euro Area countries considered, I follow the dates of the turning points listed by the OECDcalendar.53I consider six different time intervals since the creation of the Euro Area, which I group in threepairs of economic downturns and upturns. The first pair with a recessionary and an expansionaryphase embodies the economic downturn of 2001 until 2003, followed by the recovery from 2003 to2008. The second pair of economic downturn and upturn is represented by the Great FinancialCrisis, which as per the OECD dates affected Euro Area countries between 2008 and 2009, andafter which followed an economic expansion during the interval 2009-2011. Finally, the third paircontaining periods five and six represents the European debt crisis and its aftermath, which tookplace in 2011-2013 and 2013-2017, respectively. For simplicity, I only keep the data from theserecession-expansion pairs, and I disregard observations from the periods prior to the first pair andposterior to the third pair. Our sample for these six episodes therefore consists of 386 tax policychanges.Table 4.4 summarizes all this information. The first column describes how many tax changesare in each of the six periods. In addition, columns 2 and 3 show the total number of days andmonths for each period, where it is straightforward to observe that economic expansions roughlyduplicate the length of the recessions. Columns 4, 5 and 6 describe the average, the median andthe maximum lag in days. Finally, columns 7 and 8 report the total and the share of tax changesthat can be considered to be anticipated under the 90-day cutoff metric.Every period shows an average lag well above 3 months. The result for the median lag is moreconservative but always above 2 months and, except from Period 5, closer to 3 months. The recentwork of Amaglobeli et al. (2018) documented a decline in the median delay of some tax changesduring the global financial crisis for the sample of countries in the TPRD. At the same time, sometax measures reported a median lag increase in the recovery period following the crisis. In linewith Amaglobeli et al. (2018), the statistics reported in Table 4.4 show that a similar patterncan be observed in two out of three recession-recovery pairs from our Euro Area sample. During52In their paper,Mertens and Ravn (2012) find that approximately half of its sample (36 out of 70 observations)qualify as anticipated tax measures for the US economy.53https://www.oecd.org/sdd/leading-indicators/CLI-components-and-turning-points.pdf accessed onMarch 16, 2023.106the 2001 downturn and the 2012 European debt crises, median lags were 92 and 63 days, whichwere considerably smaller than the medians during the subsequent recoveries – 173 and 79 days,respectively. Interestingly, and differently to the results reported by Amaglobeli et al. (2018), whenthe sample is limited to Euro Area countries (and given our additional restrictions), fiscal policy lagsduring the 2008 great financial crisis and its subsequent recovery behaved in a different direction –with medians of 79 days (for the recession) and 75 days (for the following expansion). Finally, thelast two columns highlight the number and proportion of the total tax changes that can be labelledas anticipated. There are three interesting features. First, that except from period 5, the share ofpolicies that can be labelled as anticipated ranges from values close to 50 to above 60 percent. Thisindicates a large incidence of lags in fiscal policies. Second, that in the first two pairs of phases(periods 1-2 and 3-4) the share of anticipated fiscal policies during the recession is similar to thatof the recovery - respectively, 63 and 65 percent, and 48 and 46 percent. And third, that betweenthe three pairs there appears to be a decline in the share of anticipated fiscal policies. Althoughinteresting, the analysis of the source of these stylized facts for Euro Area countries is beyond thescope of this chapter at this point, and it is therefore left for future research.Figure 4.5 shows the distribution of lags for each of the tax changes announced within March2001 to December 2017. The histogram bins are similar to those in the previous figures, with lagsof the x-axis measured in days. The sample period has been divided in the six periods summarizedin Table 4.4. The top left and right panels show the histogram for tax changes during the 2001economic downturn (Panel A) and the posterior recovery which lasted until early 2008 (Panel B).The middle left and right panels report lag distributions for the 2008 Great Financial Crisis period(Panel C) and the post-crisis period (Panel D). Finally, the left and right panels show the 2012recession period, often called the ‘European debt crisis’, and the subsequent expansion (Panels Eand F, respectively). It is important to highlight that I zoom in the y-axis scale of the crisis panels(Panels A, C and E), which is smaller than the one from the non-crisis panels (Panels B, D and F).Figure 4.5 first reveals the existence of fiscal policy lags in all the subperiods. Moreover, andin line with Table 4.4, five out of the six subperiods under consideration portray their largest peakin the last bin. This implies that a large share of the tax policy implementation lags extend wellbeyond their announcement dates. Abstracting from the last bin, a second aspect in the plot is thatPanels A and C show, on closer inspection, a more uneven distribution compared to Panels B andD. In other words, the 2001 economic downturn and the 2008 great financial crisis evidenced morevarying shares of tax policy changes being implemented at different time horizons. For example,during the 2008 Great Financial Crisis, more tax changes were implemented within one and threemonths than with two months of delay. A similar behavior occurred during the 2001 economicdownturn.54 These two recessions contrast with the economic recoveries that followed after theseepisodes, where the return to normal times appears to be more stable in terms of the number of taxchanges expected to occur in different months after their announcement. This, hence, highlightsthe possibility that fiscal policy lags exist, and that these lags may correlate differently to business54The histograms in Panels A and B have a different scale.107cycles. However, this conclusion is not persistent as it harder to detect when comparing Panels Eand F.Overall, the main lesson from this section can be summarized as follows. For Euro Area coun-tries, there is evidence of fiscal policy lags during economic cycles. Most importantly, these fiscalpolicy delays are still observed regardless of the business cycle; there is a nondegenerate distributionof lags both during recessions and recoveries. This, therefore, points toward the consideration thatfiscal devaluations used during recessions can still be subject to delays that may affect the potencyin achieving the same real outcomes and welfare improvements than those of nominal devaluations.Case study: Fiscal policy lags in SpainIn their work, Farhi et al. (2014) calibrate their model for Spain and analyze the impulse response ofmacro variables and welfare implications of a fiscal devaluation policy during the 2008 crisis. First,FGI consider the welfare gains from an unanticipated fiscal devaluation. In addition, they comparetheir results for an unexpected fiscal devaluation against a one-quarter anticipated fiscal devaluation– which the authors label as an ‘incomplete fiscal devaluation’. While they still document theexistence of welfare improvements with incomplete fiscal devaluations, such improvement declinesrelative to the one achieved under an unanticipated fiscal devaluation. In this section I use theTPRD to illustrate how fiscal policy lags distribute in Spain, and provide evidence that there isa considerable share of fiscal adjustments that take time to be implemented. As we did in theanalysis of the whole sample, it is worth to point out that also for Spain I disregard observationswith implementation delays exceeding 2 years (i.e., above 730 days).The sample for Spain for the 1999-2018 period has 79 tax changes in total that exhibit nonneg-ative lags, and only 8 negative tax changes – for consistency with our previous analysis, we onlykeep tax changes that are nonnegative. Figure 4.6 shows the distribution of the announcementsand implementations with bins representing different years. The distribution reveals that imple-mentation of tax changes (dashed blue bars) do not exactly overlap with their announcements (redsolid bars). A priori, this hints at the possibility of the existence of lags in the implementation offiscal announcements, characteristic I discuss in further detail below.Table 4.5 describes key statistics for the observations from Spain, and it shows that the averagedelay of all 79 taxes since 1999 was of approximately 5 months, with a median delay slightlyabove 3 months. The subgroup of categories consisting of tax decreases (51 lags) and increases(28 lags) represent approximately two thirds and one third of the sample respectively, both withaverages of roughly 5 months. The median of fiscal policy lags, however, is considerably different.Tax decreases take approximately two quarters to be implemented, which almost doubles the rateof implementation of tax increases (slightly above one quarter).55 In addition, it is important tomention that the Spanish economy still features lower delays than those of non Euro Area economies– see Table 4.8. Overall, Table 4.5 reveals that fiscal policy exhibited lags key statistics of at least55An analysis of this observed difference is beyond the scope of this chapter but proposed as a subject for futureresearch.108one-quarter and up to two quarters.Figure 4.7 displays the distribution of tax changes in Spain as documented by the TPRD. Theleft panel of the figure shows the distribution of the total sample, and the middle and right panelsdo the same for tax decreases and increases, respectively. As portrayed for the whole sample’scase in the previous section, this plot shows a twin-peaked distribution for Spain. Specifically,the distribution of the observations for all the tax changes in the Spanish economy (left panel) istwin-peaked, with approximately 51 out of 79 tax policy changes taking place with more than 90days of delay. Furthermore, we find that roughly two thirds of the tax changes from the increasingand decreasing subgroups can be categorized as anticipated tax changes based on the 90-day-cutoffmetric. Considering FGI’s numerical experiment, the fact that a sizeable share of fiscal policychanges can be catalogued as anticipated reveals that incomplete fiscal devaluations can be a likelyscenario that a policymaker may face, which, in turn, can reduce the welfare gains from fiscalpolicies.In order to identify how fiscal policy lags have responded to different crisis phases, I shall nowturn to a business cycle analysis similar to that presented in the previous section.56 To that end,I present the recessions and economic recoveries for the Spanish economy in Figure 4.8 and theassociated descriptive statistics in Table 4.6. It is important to note that since we are consideringonly observations for the Spanish economy, some time periods have just a few (or no) observations.Because of that, the figure plots only the histograms for two recessionary periods coupled with theirsubsequent recoveries: the 2001 recession and the 2012 European debt crisis – for concreteness, Iwill mostly discuss the evidence from the recessionary periods. First of all, it is important to notethat in both recessions fiscal policy exhibited positive implementation lags. However, their averagesare very different: In the economic downturn of 2001, fiscal policy lags averaged roughly 6 months,while in the 2012 European debt crisis lags fell to less than 2 months. Moreover, the same is truefor the medians. Also, note that the sample has 10 and 11 observations for the recessions (left panelof Figure 4.8, top and bottom plots), but fiscal policy lags accumulate at different extremes of thedistribution. Using the definition of anticipated versus unanticipated fiscal policies we observe thatduring the 2001 economic downturn 9 out of 10 tax changes were anticipated, which is in starkcontrast with the proportion of 3 out of 11 anticipated tax changes during the 2012 economic crisis.In this chapter I do not discuss the reasons behind these empirical findings. However, twoobservations are in order. First, it is important to mention that the two recessions were one decadeapart. In this context, the 2008 great financial crisis took place between them, which changed theeconomic conditions not only in Spain but in many economies – for example, see Mishkin (2017).In favor of this conjecture, it is straightforward to observe that the frequency of lags of each pair ofperiods depicts a similar distribution in its own recession and expansion, but a different distributionrelative to the recession and expansion of the other pair. Specifically, during the first decade ofthe 2000’s, tax changes show longer implementation delays (both during the recession and the56For Spain I preserve the same business cycle dates as those from the Euro Area since matching the dates allowsan easier comparison with the crisis phases identified in our previous analysis for the whole sample.109recovery, as portrayed in the top plots) relative to the second decade (second plot). Second, it isworth to point out that the 2001 recession happened in the first years after the Euro Area was born,and where country members may have had less experience about how to respond to the recessionas members of a currency union. In any case, we cannot conclude whether the faster responsethat fiscal policy portrayed during (and after) the 2012 European debt crisis will become the ‘newnormal’ for future recessions. But overall, the pattern observed in Figure 4.8 does not in itselfexclude the possibility that fiscal policy can respond with delays during and after recessions.Finally, I will focus the last part of the empirical analysis to the type of taxes used. In theirpaper, Farhi et al. (2014) conduct a numerical exercise to understand the economy’s response andwelfare implications of a fiscal devaluation in Spain. The country is calibrated as an open economythat is subject to a negative shock to borrowing costs. Also, and as part of a currency union,the Spanish economy only has access to fiscal instruments. Upon the occurrence of the shock,the authors describe how an unanticipated fiscal devaluation consisting of a change in the VAT,a payroll tax and a capital tax can mitigate the output collapse. Then, they proceed to analyzethis policy when it is anticipated one quarter. With this in mind, I exploit one of the categoriesincluded in the TPRD which identifies the type of tax used in each fiscal policy implemented. In thisprocedure I seek to identify the delays that each type of tax presents in order to analyze whetherthey are differently affected by implementation lags, and also discuss whether fiscal devaluationscan be anticipated.The TPRD compiles VAT and payroll taxes (or ‘SSC’; i.e., Social Contributions). In addition,it separately collects information of fiscal policy affecting capital gains, both via personal incometaxes (labelled PIT) and corporate income taxes (labelled CIT) – for completeness, I consider bothof them. Our subsample now consists of 73 observations, and Figure 4.9 and Table 4.7 describerelevant characteristics of each type of tax. First of all, note that over the sample period the tableshows that the SSC and the VAT exhibit smaller delays. Specifically, fiscal implementation lags ofSSC and VAT averaged at 66 and 88 days respectively – i.e., roughly 2 and 3 months. Furthermore,the median delay in the implementation of these taxes also lies close to 2 months. On the contrary,the fraction of time that its takes for CIT and PIT to be implemented is substantially larger. Table4.7 reports that the average for the CIT during 1999-2018 was 128 days (slightly more than 4months), and the median lag was above 3 months. For the case of the PIT, the table shows anaverage and median lag above the 2 quarters. As portrayed in Figure 4.9, part of the summarystatistics of the CIT and PIT is explained by the twin peaks of their distributions. Using again the90-day cutoff measure we find that a large proportion of personal and corporate income taxes canbe anticipated by the private sector to occur beyond the first quarter of the respective tax changeannouncements. For the CIT, that proportion is more than half of the total CIT observations. Forthe PIT, the share amounts to 83 percent of the realizations. Although these numbers are smallerfor VAT and SSC, this empirical evidence highlights the possibility that there can be at least anasymmetric implementation of fiscal policy instruments.In summary, the descriptive analysis carried in this section for Spain points to two key takeaways110to understand fiscal policy characteristics in general, and fiscal devaluations in particular. The firstone is that an open economy belonging to a currency union can experience delays when it seeks toimplement fiscal stabilization policies. For Spain, a share of these policies can show delays exceedingthe first quarter, and in some occasions approximate to the second quarter. Although the latterproportion is smaller, the presence of lags in fiscal policy is nonnegligible, especially consideringthat fiscal policies with implementation lags beyond two years have been dropped from the sample.The second takeaway is that not only fiscal policy is subject to implementation lags, but also thatthese lags can be asymmetric within different categories of taxes. This can potentially affect furtherthe assessment of the final impact of fiscal policies when they are aimed at mitigating recessions.Extrapolation of these facts to one of FGI’s results (that welfare improves relatively less whenfiscal devaluations are anticipated) may potentially affect fiscal devaluations’ welfare impact whenthe time-horizon of fiscal implementation lags become longer. One model example of the implica-tions of such delay is presented and discussed in the next section.4.2.3 Discussion of empirical evidenceThis section provided evidence on the existence of lags across a sample of fiscal policies implementedin a subset of Euro Area countries. Before turning to the formal analysis in the next section, thereare aspects from the empirical findings that deserve attention: One relates to the effectiveness offiscal devaluations in the presence of lags, and the other one involves the relevance of the macrostabilization goal of fiscal policies. Since these aspects can be relevant for the interpretation of ourempirical findings, in this section I shall discuss them in further detail.Fiscal policy effectiveness and nominal rigiditiesProvided that a key empirical finding is that fiscal policy for a restricted set of countries from theTax Policy Reform Database exhibits nonnegative lags during recessions, a natural question thatarises is whether implementation lags matter for the ability of fiscal policy to effectively respondto economic downturns. In practice, although there can be a lag between a negative shock and thetime a specific stabilization policy is implemented, this does not necessarily imply that the policy isnot effective. Furthermore, the knowledge of a policy that will be implemented in the future likelytriggers an economy’s response today – see, for instance, Yang (2005) and Leeper et al. (2013). Asa result, this means that the future adjustment of fiscal instruments can affect variables’ currentbehaviour due to forward-looking agents. This adjustment, overall, can still make the final impactof fiscal policy non-negligible.To see why fiscal policy delays might not render fiscal devaluations ineffective (e.g., in welfareterms), it is important to recall that monetary policy can counter a negative shock in the presenceof nominal frictions. In particular, if monetary policy reacts instantaneously to a negative shockand it adjusts at a faster rate than the adjustment of prices, this policy can affect real outcomes.Likewise, if fiscal policy can be implemented at the business cycle frequency, then fiscal policy canbe effective against recessions. Now, when there are implementation lags, fiscal policy might still111be effective, but this depends on the relationship between fiscal policy lags and the rate of priceadjustment. Specifically, fiscal policy with moderate implementation lags might be still effectiveif this policy adjusts before prices do. But fiscal policy can also feature long lags such that pricesadjust even before fiscal changes take place.57 Since the direction of the response of real variablesin the presence of lags is not entirely clear, these cases highlight how delayed implementation offiscal policy can matter for its effectiveness.It is straightforward to exemplify with a slightly more formal argument some of the implicationsthat different lengths of fiscal policy lags can have on the effectiveness of fiscal devaluations in thecontext of nominal rigidities. Fiscal and monetary policy can stabilize an economy that featuresnominal rigidities, where nominal rigidities can take several forms – for example, staggered pricesetting (Calvo, 1983), price adjustments with quadratic costs (Rotemberg, 1982), frictions affectingthe adjustment of wages (via Calvo-type rigidities as discussed in Gal´ı, 2008), downward wagerigidity (Schmitt-Grohe´ and Uribe, 2016), to name a few. In the context of downward wage rigidity,Schmitt-Grohe´ and Uribe (2012) use a tradable-nontradable model to illustrate how a negativeshock to an economy that features a currency peg triggers unemployment. Upon a shock, firmslay off workers as firms observe higher real costs if wages cannot downwardly adjust. However, theadjustment in real variables stemming from the nominal friction can still be countered with a fiscaldevaluation when the country does not have access to monetary policy. Specifically, the authorsshow that a fiscal devaluation in the form of a wage subsidy can stabilize a currency-peg economyby decreasing hiring costs of firms and therefore restoring employment levels.Countries in a currency union (like those from the Euro Area analyzed in this section) presentsimilarities to the model described above in that these countries lack access to monetary policy.As a result, fiscal devaluations emerge as a candidate stabilization policy in the context of nominalrigidities that translate negative shocks to an economy’s real variables. In this context, Farhi et al.(2014) (thereafter FGI) showed how fiscal devaluations can replicate the same real outcomes asa nominal devaluation. But if fiscal policy does not adjust at the business cycle frequency, thenchanges to its effectiveness are likely to be non-trivial. Since lags are present in tax changes fromEuro Area countries, in the next section I formally discuss one way in which fiscal action can beless effective in its ability to overcome nominal rigidities (and, therefore, result in lower welfare)when fiscal policy is slow to respond to shocks.58Macro stabilizationThe previous empirical analysis also discussed the incidence of fiscal implementation lags in theEurozone during different business cycle phases. A priori, taxes enacted during a recession can likely57Even in this case, private sector’s rational expectations can carry effects to the present before fiscal policy changesoccur. Although the analysis of this mechanism is beyond the scope of this chapter, it is an interesting topic for fiscalpolicy design as it may result in policy prescriptions that have to entirely focus on the effect of current responses tofuture tax changes – rather than on the contemporary effect of current tax changes.58In this context, FGI also show that fiscal devaluations can replicate the real outcomes of nominal devaluations,but at the same time document that there are welfare losses when fiscal devaluations can be anticipated.112be thought to be related to macro stabilization objectives. Nevertheless, the actual backgroundat the time of the announcement of a tax change can more accurately indicate the actual purposeand objective of the policy. For example, some tax changes can be aimed at affecting rates ofeconomic growth. Also, some fiscal policies can be used for macroeconomic stabilization. In thecontext of this chapter, fiscal devaluations can be thought of an example of the latter type of policywhen it is aimed at absorbing negative shocks that distort variables such as output gap, inflation,and employment rates.59 Provided that the dataset consists of approximately 500 fiscal policiesfrom different countries, identifying the goal of each policy would require an in-depth analysis oftheir backgrounds as documented by different sources. While such analysis is not carried out inthe current chapter, we can still argue the existence of fiscal policies aimed at macro stabilizationby identifying some clear goals on a case-by-case basis. Specifically, the content of some taxchange announcements often refer to business cycles concepts – e.g., ‘recession’, ‘crisis’, ‘slowdown’,‘downturns’, to name a few. As long as these concepts are identifiable in the announcement of thepolicy, these policies can be considered likely to be fiscal stabilization policies. Since in this sectionwe also carried our analysis for the Spanish economy to focus on fiscal policies like those in FGI’snumerical illustration, I briefly exemplify below the existence of fiscal policies aimed at macrostabilization in Spain. After that, I shall briefly motivate the use of wage subsidies as in Schmitt-Grohe´ and Uribe (2012) (which will be the type of policy used in the model in the next section).In their model, FGI presented changes in value added taxes (VAT) as one instrument of the menuof taxes in their fiscal devaluation example. Looking for some similar tax changes in Spain usingthe Tax Policy Reform Database, we can find in 2011 a VAT policy where one of its salient aspectsconsisted in reducing VAT rates for the purchase of housing. According to the announcementreported in the dataset, this policy change was related to the banking crisis, which would makethis tax change to be aligned with macro stabilization goals. In line with this, another tax measureimplemented in 2012 involved a VAT change with explicit reference to the macro context, and wherethe country was trying to adjust the ratio of headline deficit to GDP. In this scenario, this policyconsisted of a tax increase. Although these two policies involve tax changes in different directions,they still serve the purpose of capturing VAT changes with stabilization objectives - like those inFGI’s numerical illustration.In addition to these examples, the analysis of currency pegs from Schmitt-Grohe´ and Uribe(2012) that will be used in the next section presents a model with wage subsidies. From anempirical point of view, the Tax Policy Reform Database reports some personal income tax policiesfor Spain involving wage subsidies as one of their elements, and which were used in the context ofan economic downturn. These policies were announced in 2005 and they provide evidence of theuse wage subsidies as a fiscal stabilization policy. When we turn to the formal analysis from thenext section, the model that I use will depict similar wage subsidies and it will show the impactthat fiscal lags in this policy can have on the effectiveness of fiscal devaluations.59Another example of the use of fiscal policy relates to electoral periods. See, for example, the analysis in Amaglobeliet al. (2018).1134.3 ModelIn this section I illustrate the impact that implementation lags can have on a fiscal devaluationwhen fiscal policy seeks to replicate the same economic outcomes of a nominal devaluation. Tothat end, I use the model of tradable-nontradable goods with nominal rigidities that studies fiscaldevaluations from Schmitt-Grohe´ et al. (2022) – I refer to that textbook as SUW.60 Here, I analyzehow a slight variation to their model, namely, delays in their implementation of fiscal policy, canbecome a relevant assumption for the equivalence between fiscal and nominal devaluations – similarto that presented in FGI. For consistency, and in order to facilitate the connection between thetextbook’s presentation and the model in this chapter, I use SUW’s concepts and also borrow theirnotation.I start by presenting the key economic relations from SUW that are needed to characterizethe equilibrium of the model economy. I simplify the exposition of this section by skipping thederivation of the intermediate set of solutions – the reader is referred to SUW for the completepresentation. Next, I solve the model closely following SUW’s exposition in their book chapter.In contrast to the authors, in the model presented here the economy is hit by a large negativeshock (similar to the 2008 Great Financial Crisis) and then it vanishes to let the economy revertto normal times. Finally, I use the parameters from SUW and perform a numerical exercise of themodel to carry a brief welfare analysis.4.3.1 Structure of the economy and equilibriumI now introduce the key elements of the model of SUW. Consider a two-period economy withtradable and nontradable goods. This economy has access to international capital markets whereit can lend and borrow units of the tradable good at an international (real) interest rate rt. Theeconomy receives a fixed endowment of a tradable good, QTt , with PTt being the price of thetradable good – superscript T refers to tradable goods. In addition, the economy has a productiontechnology for nontradable goods given byQNt = hαt , (4.1)for periods t = 1, 2, where α ∈ (0, 1) denotes the productivity of labor and ht the amount of hoursemployed by the firm to produce the nontradable good. Likewise, PNt is the price of the nontradablegood – superscript N denotes nontradable units. For future reference, we define the relative priceof nontradable goods in terms of tradable goods as pt ≡ PNt /P Tt .In each period the government of this economy has access to fiscal and monetary policies thatit uses against recessions. Following SUW, I use εt to denote the nominal exchange rate betweendomestic currency and foreign currency. Moreover, be P T,∗t the price of the tradable good in its60The model presented here follows the presentation from the manuscript version of this book. Chapter slides arestill available at http://www.columbia.edu/~mu2166/UIM/ (accessed on April 25, 2023). The corresponding chapterin that manuscript was based on Schmitt-Grohe´ and Uribe (2012).114corresponding foreign currency. Since the tradable good can be traded internationally, we assumethat the law-of-one-price for tradables holds and determines the price of P Tt , i.e.,εt =P TtP T,∗t.The government also has access to fiscal policy. For simplicity, fiscal policy will come in theform of wage subsidies, st ∈ (0, 1). To see the margin impacted by this wage subsidy, we need toresort to the determination of the price of the good in the nontradable sector, PNt . Unlike the caseof the tradable good, in this economy PNt depends on the supply and demand of the nontradablegood. Next, I shall begin with the supply side.In every period t = 1, 2, firms with access to technology given by Eq. (4.1) hire ht units of laborat a wage rate Wt to produce QNt units of the nontradable good at the given price PNt . Hence,firms profits are given byΠt ≡ PNt QNt −WthtThe competitive firm maximizes profits and chooses the optimal amount of labor ht needed toproduce the nontradable good in every t = 1, 2, which results in the supply of the nontradable goodPNt = (1− st)Wtαhα−1tThe expression shows that the firm hires labour as instructed by its marginal productivity (termαhα−1t ) and given the nominal wage net of the subsidy (term (1− st)Wt). To leave the previousequation in terms of the relative price of nontradables, pt, we first assume for simplicity that theforeign nominal price of tradables is P T,∗t = 1 for every t = 1, 2, which renders the law-of-one-priceP Tt = εt.This equation indicates that the variation in monetary policy cascades back to the nominal priceof tradable goods, P Tt . Hence, the optimizing behavior of the firm can be manipulated to yield thesupply schedulept = (1− st)Wtεtαhα−1t. (4.2)Now I turn to the demand side. The economy has a large number of identical households thatlive two periods. Households consume tradable and nontradable goods, and preferences over thegoods take the formU = lnC1 + β lnC2where β represents the discount factor with β ∈ (0, 1). Variable Ct denotes household consumptionof a good composite of tradable and nontradable goods. These goods are aggregated according tothe consumption aggregatorCt =(CTt)γ (CNt)1−γ115where CTt and CNt represent the tradable and the nontradable good, respectively, and γ ∈ (0, 1) isthe household’s preference over tradables.In every period, the household’s sources of income include the deterministic endowment of thetradable good QTt , profits from the firm Πt and wage wt (from a total of ht hours worked), whichare adjusted by an income tax τyt that will finance the wage subsidy from the government. Also,the household has access to international capital markets to manage the shortage or excess of thetraded good via the purchase of Bt foreign-denominated bonds at the real interest rate rt. Thismeans that the household receives proceeds from bond savings between periods – for simplicity, weassume initial bond holdings B0 to be fixed at zero. The household uses this income to consumetradable and nontradables goods, and also to save in bonds. As a result, the household’s budgetconstraint in period 1 in units of domestic currency isPN1 CN1 + PT1 CT1 + ε1B1 = (1− τy1 )(P T1 QT1 + Π1 +W1h1)(4.3)and in period 2 the budget constraint becomesPN2 CN2 + PT2 CT2 = (1− τy2 )(P T2 QT2 + Π2 +W2h2)+ ε2 (1 + r1)B1 (4.4)The demand schedule of the nontradable good comes from the solution to the utility maximizationproblem of a household subject to the budget constraints described above, and becomespt =1− γγCTtCNt(4.5)which represents the relative consumption of tradables and nontradables for each period t = 1, 2 asa function of the relative price of nontradables. Using the law of one price and after rewriting eqs.(4.3) and (4.4) in terms of the relative price of nontradable goods to combine both expressions (viabonds), it can be shown that the resulting intertemporal budget constraint and the Euler equationfor tradable goods yields the optimal choice of CTt for t = 1, 2,CT1 =11+β(QT1 +QT21+r1)and CT2 = β (1 + r1)11+β(QT1 +QT21+r1). (4.6)Finally, after imposing market clearing of non-traded goods,CNt = hαt , (4.7)the demand schedule of nontraded goods for every t = 1, 2 becomespt =1− γγCTthαt(4.8)(given that CT1 and CT2 satisfy Eq. (4.6)). Hence, Eq. (4.2) (the supply schedule), and Eqs. (4.6),(4.7) and (4.8) (the demand schedule) will help determine the equilibrium in this economy.116The economy exhibits nominal rigidities. In particular, we assume that nominal wages aredownard rigid. Following SUW, we impose that current wages cannot go below the level thatwages attained in the previous period,Wt ≥Wt−1, (4.9)and we also assume that there is a limited amount of labor that can be supplied in this economy,ht ≤ h¯. (4.10)Finally, we impose what SUW call the Labor Market Slackness Condition (LMSC),(Wt −Wt−1)(ht − h¯)= 0, (4.11)which guarantees that wages cannot adjust downwardly to clear the labor market whenever thereis unemployment (i.e., when ht < h¯, Wt must equate Wt−1). On the contrary, when there is alarge demand for labor by firms that exhausts the total endowment of labor in this economy (i.e.,ht = h¯), only then can wages rise to clear any excess in labor demand.From the previous description, households face income taxes, τyt that provide government rev-enues that finance wage subsidies, st. As per SUW, the government must choose the tax andsubsidy in order to satisfy the budget constraint of the given bystWtht = τyt(εtQTt +Wtht + Πt), (4.12)where Πt are the firm’s profits. These profits come from selling the non-traded good (producedwith technology from the expression in Eq. (4.1)) at price pt, minus total costs portrayed by thefirm’s labor demand at the real wage net from subsidies ((1− st)htWt/εt).Given our model assumptions, and given a process for the interest rate, the optimal behaviorof firms and households (Eq. (4.2) and Eqs. (4.8), (4.6) and (4.7), respectively), the conditions forwage stickiness (Eqs. (4.9), (4.10) and (4.11)), and the satisfaction of the budget constraint of thegovernment Eq. (4.12)), represents an equilibrium in this economy.4.3.2 Non-stochastic steady stateI now compute the equilibrium relations of this economy in the absence of shocks, and solve itfollowing the analysis of the pre-shock equilibrium that is presented in SUW. Following the authors,we use the full-employment steady state equilibrium as our benchmark.Assume that the economy starts with no initial bond holdings, and that there is an equilibriumworld interest rate r1 = r¯ > 0 – we use upper bars to denote steady state values. Moreover,conjecture that the equilibrium level of labor hired by firms (and offered by households) in the117full-employment steady state ish1 = h2 = h¯ and W1 = W2 = W0 = W¯ ,with W¯ > 0. (In a numerical exercise, this conjecture must be verified to ensure equilibriumconsistency.)The government follows the following policies. For its monetary policy, it pegs the nominalexchange rate at ε1 = ε2 = ε¯, with ε¯ > 0. For fiscal policy, I assume that in the steady state thegovernment implements no subsidy and sets s1 = s2 = s¯ = 0. Also, since the government does notprovide subsidies, then there is no need for collect income taxes (τyt = τ¯y = 0).For simplicity, we assume that the sequence of endowments of this economy is the same andequal to QT1 = QT2 = Q¯T > 0. Implementing this in Eq. (4.6) and setting β (1 + r1) = 1 yields thetotal consumption of tradablesCT1 =11 + β(2 + r11 + r1)Q¯T = Q¯T ⇒ C¯T = Q¯T > 0andCT2 = β (1 + r1)11 + β(2 + r11 + r1)Q¯T = Q¯T ⇒ C¯T = Q¯T > 0.Likewise, the equilibrium consumption of non-traded goods is atCN1 = h¯α > 0⇒ C¯N = h¯α > 0andCN2 = h¯α ⇒ C¯N = h¯α > 0Finally, to pin down the equilibrium prices associated to these previous quantities, the demandschedule formula yields a relative price of nontradables equal topt =1− γγCTtF (ht)=1− γγQ¯Th¯α⇒ p¯ = 1− γγQ¯Th¯α> 0,for every t = 1, 2. (For consistency, this price has to clear the goods market also using the supplyschedule formula at Wt = W¯ > 0, st = s¯ = 0, εt = ε¯ and ht = h¯ for every t = 1, 2.). Furthermore,for every t = 1, 2, and given that εt = ε¯ (and assuming PT,∗1 = PT,∗2 = P¯T,∗), the nominal price oftradables isP Tt = ε¯P¯T,∗ ⇒ P¯ T = ε¯P¯ T,∗ > 0,while the nominal price of nontradable goods isPNtP Tt=1− γγCTtF (ht)⇒ PNt =1− γγQTh¯αε¯P¯ T,∗ ⇒ P¯N = 1− γγQTh¯αε¯P¯ T,∗ > 0.118Finally, it is immediate to show that the constraints on nominal wages, labor supply and the LaborMarket Slackness Condition (LMSC) are valid under our conjecture – the government budgetconstraint is satisfied too. With the satisfaction of all these requirements, all the above expressionscharacterize the steady state variables of the economy in terms of the model parameters, hencedescribing the equilibrium outcome when the economy is at full employment.4.3.3 A negative shock to the economyIn this section I characterize the impact of a sudden unexpected shock to the world interest ratein period 1. Farhi et al. (2014) explore the role of fiscal devaluations using a numerical illustrationof a negative shock in a small open economy calibrated with the parameter values of Spain. Inparticular, the authors replicate the 2008-crisis output collapse triggered by a negative shock in theform of a sudden rise in borrowing costs. In the context of SUW’s model, this shock is equivalent toa rise from r1 = r¯ to r1 = r′1 > r¯. Following SUW, I use this example to replicate an approximationof Farhi et al. (2014) analysis under various cases with different access to monetary and fiscalstabilization policies.Case 1: No monetary-fiscal stabilization policiesStart assuming an economy without access to monetary or fiscal policies. For simplicity, this shallbe interpreted as a government still using a fixed nominal exchange rate, εt = ε¯ > 0, and issuingno subsidies, st = 0, in t = 1, 2. After the increase in r1 to r′1, it is immediate to observe from Eq.(4.6) thatdCT1dr1< 0, anddCT2dr1> 0 (4.13)Given this direction of change of CT1 and CT2 , we conjecture that this economy reaches anequilibrium with unemployment. Specifically, we guess (and later check for consistency) thath1 < h¯ and h2 = h¯.Contrary to the adjustment experienced in the consumption of tradable goods (where units in excesscan be exported), this guess suggests that the non-traded sector will have to adjust its productionas it is not possible to trade units that are not consumed. It is important to emphasize that herewe conjecture unemployment in period 1. This is because, given that dCT1 /dr1 < 0 and for a givenfixed relative price of nontradables, then the demand schedule (as an expression of the MRS ofhouseholds) instructs a decline in the demand for nontradable goods. This decline in CN1 entails adecline in labor since W1 cannot adjust downwards. Since the conjecture must satisfy the LMSCand W1 cannot decline, it is straightforward to pin down W1 = W0 = W¯ . It remains to calculatethe total labor demanded in this economy.In our guess we argued that the economy features a decline in the demand of labor, h1. To119confirm this guess the demand schedule and supply in the goods market yields that1− γγCT1 (r1)F (h1)= (1− s1)W1ε¯F ′ (h1)F (h1)F ′ (h1)=1− γγ1W1ε¯1− s1CT1 (r1) ,where we explicitly wrote tradable consumption as a function of r1. Total differentiation of theprevious equality yieldsdr1 = Γdh1 + ΘdW1 (4.14)where Γ,Θ < 0.61 Eq. (4.14) includes useful information about the response of the demand oflabor. If there is a shock to r1 in period 1, dr1 > 0. It is straightforward to observe then that Eq.(4.14) requires either that households work reduced number of hours (dh1 < 0) or an adjustmentin salaries (dW1 < 0). But when wages are downward rigid, dW1 = 0, and only labor can respondvia h1 < h¯. This level of unemployment eventually means that total production of nontradables,F (h1), must decline, and this pins down the equilibrium level of consumption of nontradables inperiod 1, CN1 . Likewise, using the supply schedule and provided that h1 < h¯ rises the marginalproductivity of labor, the equilibrium relative price of nontradables must decline.In period 2, households collect the gains in terms of tradable goods from their period-1 savings(when interest rates were high), and as dCT2 /dr1 > 0, then consumption of traded goods rises.Now, recall that: (1) For any given price p2, the marginal rate of substitution implicit in thedemand schedule indicates that there has to be an expansion in the consumption of nontradablegoods, CN2 , and (2) the nontradable good has to be domestically produced. Intuitivelly, this meansthat there has to be an increase in labor demanded by firms. And if such demand for labor risesabove the total supply of labor from this economy, h¯, only then wages will start to rise to clearany excessive demand from the labor market. Given the equilibrium level of h (determined via thedemand schedule and supply as we did in period 1), nominal wages and the equilibrium quantityof nontradable goods can be pinned down, as well as the relative price of nontradable goods.Case 2: Nominal devaluationsIn their exposition, SUW show that the full employment equilibrium can be restored when monetarypolicy can react to the negative shock; i.e., when the government uses the nominal exchange rateto stabilize the economy. Analytically, this means that the following equilibrium can be sustained,h1 = h¯ and h2 = h¯.61Coefficients are defined by Γ ≡ −γ/∆, Θ ≡ θ/∆, γ ≡ 1 − F (h1)F ′′ (h1)F ′ (h1)−2 > 0, θ ≡ − (W1)−2 < 0,∆ ≡ QT2 r1 (1 + β)−1 (1 + r1)−2 > 0. Terms for the nominal exchange rate and the wage subsidy do not appear sincethey were assumed to be fixed.120In the case analyzed by the authors (a drop in the interest rate), the authors highlight the existenceof many exchange rates that the government can use to guarantee full employment.62 However,they focus on a specific subset of these exchange rates that are aligned with conventional macropolicymaking, and which involves government interested in preserving price stability. In terms ofSUW’s model, this goal is achieved via controlling the volatility of PNt , the price of non-tradedgoods, for every t. Analytically, this means thatPNt = P¯N ,for every t = 1, 2, which ensures full employment and also that Wt = W¯ for every t. In this sectionI summarize the impact of the nominal devaluation using the total differentiation approach that Ipresented in the previous section – the reader is referred to the full solution of the model in SUWfor a more detailed exposition.When exchange rates can freely adjust to absorb the shock, total differentiation of the equalitybetween demand schedule and supply of goods becomesdr1 = Γdh1 + ΘdW1 + Φdε1 (4.15)where Γ,Θ < 0 and Φ > 0.63 (Now note that the nominal exchange rate is allowed to change, butwe still assume that there is no fiscal policy in place).In period 1 we know that interest rates rise (dr1 > 0), and by conjecture we want to maintain fullemployment (i.e., dh1 = 0). Also, by price stability, it is easy to see via Eq. (4.15) that dW1 = 0.64Therefore, any increase in dr1 will be replicated with increases in the nominal exchange rate, dε1.The reason is behind this change is explained by Eq. (4.13). Intuitivelly, a rise in the interestrate increases households’ willingness to save more now and spend more once the shock vanishes.This means a decline in the consumption of tradable and nontradables goods – as instructed bythe demand schedule. Moreover, by wage stickiness, firms will adjust the number of workers hired,thus resulting in unemployment. That was the final picture from the previous section. But nowthe government has access to monetary policy, and affect the nominal exchange rate to introducea nominal devaluation. Analytically, this entails rising ε1. The reason for conducting a nominaldevaluation is to then overcome the constraints imposed by wage stickiness, which preventingnominal wages prevents W1 from going down impedes the restoration of full employment in thelabor market. In turn, a nominal devaluation here can help reduce the real wage, which is equal to62The discussion about all the candidate equilibria relates to those discussed in SUW. For a similar argument fromthe perspective of nominal interest rates, see Gabaix (2020).63Coefficients are defined by Γ ≡ −γ/∆, Θ ≡ θ/∆, Φ ≡ φ/∆, γ ≡ 1 − F (h1)F ′′ (h1)F ′ (h1)−2 > 0, θ ≡− (W1)−2 < 0, ∆ ≡ QT2 r1 (1 + β)−1 (1 + r1)−2 > 0, φ ≡ 1 > 0.64Total differentiation of the first-order condition of the firm (which is equivalent to the relative supply multipliedby PTt ) givesdPN = d (1− s) + dW − (F ′ (h))2 dh.Since d (1− s) = 0, and dh = 0 (due to full employment), and given that the government cares about price stability(i.e., dPN = 0), then dW = 0.121W1/ε1 due to the law-of-one-price. Formally, and assuming that dPT,∗t = 0 for every t = 1, 2, thelaw-of-one-price implies that dP T1 = dε1. Therefore, a nominal devaluation, can alleviate the largecosts of that the firm would otherwise face. Analytically, and under our conjecture, Eq. (4.15)requires that dε1 > 0 when dr1 > 0.Conversely, in period 2 households consume the gains from their savings (in terms of tradedgoods), CT2 . Since by conjecture monetary policy will adjust to maintain full employment in thisperiod too, then the rise in CT2 and the fixed CN2 = F(h¯)will rise relative prices, p2. But fromhere it is straightforward to observe that the rise in the relative price of nontradables would affectprice stability if nominal exchange rates were not adjusted. To achieve that, the government canuse ε2 to transfer the rise in the relative price into a decline in the nominal exchange rate (i.e.,an appreciation). This way, the government can channel the rise in p2 into the price of the tradedgood and, therefore, preserve price stability – i.e., an unchanged PN2 .Case 3: Fiscal devaluationsIn this section, I explore what a government can do when it belongs to a currency union and itonly has access to fiscal policy. In a nutshell, I discuss whether fiscal devaluations can be useful fora country when fiscal devaluations adjust at the business cycle frequency, and also when this toolis subject to a delay in its implementation.One of the main results in Farhi et al. (2014) is that a country can replicate the real outcomesfrom a nominal devaluation using fiscal instruments. Furthermore, if the government faces someconstraints in the application of fiscal instruments (e.g., there is no access to the full set of fiscalinstruments, or the fiscal devaluation is anticipated), the authors show that fiscal devaluations canstill considerably increase welfare relative to the case of no fiscal devaluations. In the simple modelused in this chapter (which is a very succinct simplification of the DSGE model presented in Farhiet al., 2014), I shall first illustrate a scenario where access to fiscal devaluations can be beneficialand preserve full employment when fiscal devaluations can be used at the time of the shock. Thisexercise is straightforward, and it will closely follow the exposition from SUW. However, based onthe empirical evidence presented in the first section of this chapter, fiscal policy can be subject tolags. Hence, secondly I present a second scenario highlight how the polar opposite case in whichfiscal devaluations cannot be implemented at the business cycle frequency may yield no benefits tothe country. Specifically, this case will mean that fiscal policy can only be used in period 2.Let us first consider the first scenario where the country has access to fiscal devaluations at thetime of the negative shock. Following SUW we conjecture that, although the nominal exchange rateis pegged at εt = ε¯ > 0 for every t, the government can still preserve full employment. To that end,the government has access to fiscal policy in the form of wage subsidies, st, that firms receive as anaid to pay wages. In turn, subsidies are financed with taxes on household income. One importantcaveat to the use of fiscal policy pertains to its relation with price stability. As the country willseek to mitigate a shock with the subsidy and preserve full-employment, the price stability goalcalls for transmitting changes in the relative price pt to the price of tradable goods. However, and122unlike the previous section, the government cannot carry that transmission since fiscal policy doesnot affect the nominal exchange rate. This detail will become relevant when we analyze the secondscenario when fiscal devaluations can only be used in period 2.When exchange rates cannot adjust to absorb the shock but the government has access to fiscaldevaluations at the business cycle frequency, total differentiation of the equality between demandschedule and supply of goods yieldsdr1 = Γdh1 + ΘdW1 + Φdε1 + Ψds1 (4.16)where Γ,Θ < 0 and Φ,Ψ > 0.65 Using that in the first period our conjecture stated that therewould be full employment (i.e., dh1 = 0), and given that the government pegs the currency (i.e.,dε1 = 0), then Eq. (4.16) becomesdr1 = ΘdW1 + Ψds1.Upon a shock to the economy, dr1 > 0, the last expression shows that either wages should godown (dW1 < 0), or subsidies up (ds1 > 0). But the former response cannot occur since wages aredownward rigid by assumption. Eq. (4.16) therefore shows that a negative shock to the economycan be countered with change the latter change. Intuitively, when the government has access towage subsidies, it can use these to help firms mitigate the costs of hiring labor. As a result, thisfiscal policy can be interpreted as fiscal devaluation because it replicates the same real allocation asa nominal devaluation: The subsidy lets the nominal price of nontradables to go down, PN1 , whileit keeps the level of labor hired in this economy unchanged at h1 = h¯.In t = 2, the standard adjustment is as follows. The rise in r1 creates a larger consumptionof the tradable good, CT2 – see Eq. (4.13). This implies that the demand schedule of nontradablegoods also increases for any given price, meaning that more nontradables will be demanded atthat given price. But a higher demand for nontradables requires more labor. And, therefore, firms(the supply side) will face a larger cost of production after hiring more, thus rising the relativeprice of nontradables. More importantly, if the labor demanded seeks to exceed the limit of laboravailable h¯, only then wages start will adjust, thus also rising the price of nontradables – but nowvia increments in the supply schedule.In our case, and provided that we are characterizing an equilibrium with full employment (i.e.,h2 = h¯), then that means that after the negative shock the consumption of nontradables canincrease until CN2 = F(h¯). Now, the increase in the tradable good, CT2 , can trigger an increase inthe demand schedule for nontradables that translates into a demand of nontradable goods that islarge enough to require hiring more labor than h¯. When that is the case, then the adjustment fromthe firm’s side will take place in the two stages described in the previous paragraph. Specifically,in the first stage, the relative price p2 will rise driven by the rise in the amount of labor hired, h2,65Coefficients are defined by Γ ≡ −γ/∆, Θ ≡ θ/∆, Φ ≡ φ/∆, Ψ ≡ (1− st)−2, γ ≡ 1−F (h1)F ′′ (h1)F ′ (h1)−2 > 0,θ ≡ − (W1)−2 < 0, ∆ ≡ QT2 r1 (1 + β)−1 (1 + r1)−2 > 0, φ ≡ 1 > 0.123as long as labor has not reached the total employment available, h¯ – in this case W2 still remainsequal to W1. But if h2 is pushed beyond h¯, then the second stage starts where the relative pricewill rise driven by the rise in nominal wages (h2 cannot rise beyond h¯ and stays at that level). Thekey takeaway (and which will matter for price stability), is that in the latter step of adjusment thesupply schedule will now shift due to the rise in W2, and being responsible for the larger relativeprice of nontradables in equilibrium, p2. This rise in p2 matters to price stability. First, note thatp2 ≡ PN2P T2=PN2ε2=PN2ε¯2.The first equivalence is due to the definition of p2, the middle equality is given by the law-of-one-price, and the rightmost equality is because the nominal exchange rate is fixed – this economybelongs to a currency union. From this expression, the rise in p2 can only come from changes inPN2 , but this means that there is no price stability. The reason is that there is no way for thegovernment to use subsidies s2 to channel the change in the relative price of nontradables, p2, tothe nominal price of tradables: Due to the law of one price, the price of tradables, P T2 = ε2, andε2 is fixed. Hence, price stability cannot be secured in this case.The main lesson from the case where fiscal devaluations can adjust at the business cycle fre-quency is that the government can use this tool to replicate the real outcomes of nominal devalua-tions. This is in accordance with the equivalence result presented in the full fledged model in FGI.Moving to the other case, namely, when fiscal devaluations operate with delays, yields a slightlydifferent result which I show with a verbal argument.Section 2 provided evidence in favor of the conjecture that fiscal policy exhibits implementationlags in recessions and recoveries. Because of this, assume that this delay translates in this modelinto a government that only has the ability to use a nonnegative subsidy in t = 2. But as we justsaw from the previous results for fiscal devaluations, in period 2 there is no leeway for fiscal policyto operate. Specifically, in that period full employment has already been restored, and the mostthat the government would attempt to do would be to stabilize prices in the nontradable sector.As a result, fiscal policy becomes redundant if it does not adjust at the business cycle frequency.More importantly, in period 1 the economy would suffer from unemployment, but if the governmentcan only operate in period 2, then fiscal devaluations have no role in mitigating bad allocationsduring recessions. In other words, the equilibrium would be the same as the one analyzed when theeconomy suffers a shock and there is neither fiscal nor monetary devaluations available. Lastly, andbased on our previous discussion, if it is irrelevant for the government to collect taxes that financea policy that does not mitigate economic fluctuations, then it is reasonable that the governmentsets zero subsidies.I now perform a simple quantitative assessment to illustrate how fiscal devaluations might notaffect welfare if fiscal devaluations operate at a different frequency compared to the shock. Tomeasure welfare under different stabilization policies I follow the values assumed in the numericalexercise from SUW (see Table 4.9). The reader is referred to their book chapter for further details.124Table 4.10 shows the response of the key variables in the goods market (consumption and pricesof tradable and nontradable goods, together with the relative price of nontradables) and the labormarket (employment and nominal wage), as well as monetary and fiscal policy, for periods t = 1, 2.These variables are characterized for the four different regimes described in our analysis. The firstone is a fixed exchange rate regime (column 1), where the nominal exchange rate is fixed in bothperiods and does not respond to the negative shock – by definition, in this regime fiscal policycannot be adjusted. The second one (column 2) is a flexible exchange rate regime, which is theonly regime where the government will have access to monetary policy. The third one is a fiscaldevaluation regime (column 3). In this regime fiscal devaluations can be conducted at the businesscycle frequency. Finally, column 4 depicts a regime with fiscal devaluations that are conductedwith one period of delay.At time 1, we assume that the economy is hit by a sudden and unexpected shock to the interestrate that rises the cost of borrowing. The first thing to note in the four regimes is that the shocktriggers a decline in consumption of tradables in period 1 as households save a portion of theirendowment in order to seize the higher interest rate. This situation reverts in period 2, where theconsumption of tradables rises.In the first regime (column 1) domestic production of nontradables falls in period 1 and risesin period 2. This decline in consumption is matched with a contraction of output in the firstperiod, QN1 , as nontradable goods cannot be traded and depend instead on domestic production.Since wages are downward rigid, they remain at their original level (0.75), and the adjustment isevidenced instead in total labor hired. In particular, the level of employment in this economy islower than the available supply – reflected by h1 = 0.983, which is less than h¯ = 1. The reasonfor this outcome is that the price of nontradables is too high at the current level of the nominalwage. Furthermore, given that the nominal exchange rate cannot adjust, the government cannotintroduce a nominal devaluation that reduces the real wage and, therefore, firms hire less workers.The situation reverts in the second period, where the economy recovers and demand for nontradablerises, which makes firms hire labor up to h2 = 1(= h¯)and nominal wages rise in order to absorbthe additional demand for nontradable goods. Finally, it is worth to point out that the governmentcannot guarantee price stability as PNt varies with the business cycle in t = 1, 2.When there is access to monetary policy, the second regime (column 2) reveals that the gov-ernment can introduce a nominal devaluation in period 1 (ε1 = 1.017) in order to reduce the realwage. This, in turn, reduces the cost of hiring labor that the firm faces, and hence the policyfosters production of the nontradable good. Overall, the nominal devaluation prevents a declinein the consumption of the nontradable good during the recession, which in addition contributes torestoring full employment and, at the same time, preserves price stability of nontradable goods.Once the economy recovers in period 2, the economy seizes full employment as the consumptionof nontradable goods is restored. At that point, the government also introduces an appreciation ofits currency (ε2 = 0.978) in order to preserve price stability after the output collapse.The third regime (column 3) reveals a similar situation to the second regime, but with the125adjustment occuring from the fiscal side. In particular, in period 1 and during the recession, firmsare forced to layoff workers due to wages not adjusting downwardly. To prevent unemployment,the government instruments a positive wage subsidy of s1 = 0.017 (financed with income taxes,τy1 = 0.006), which alleviates the cost of producing units of the nontradable good at the given wage.As a result, the government is able to restore full employment, with the only caveat of fiscal policybeing that the government cannot maintain price stability of the nontradable good. Because of that(and for simplicity), we assume that the government sets subsidies to 0 once the shock vanishes –see SUW for a discussion about wage subsidies being removed after a recession.Finally, in the fourth regime (column 4), Table 4.10 illustrates the scenario of a governmentthat has access to fiscal tools that allow it to pursue fiscal devaluations. However, since fiscaldevaluations have a deferred response, they can only be adjusted in period 2. As a result of thisconstraint on fiscal policy, there is nothing that fiscal policy can prevent in period 1: The sameoutput collapse as in the case of the currency peg occurs at the time of the shock, and once theeconomy is away from the recession, fiscal policy becomes redundant to control fluctuations – theeconomy is already at full employment in period 2, and there is no room for fiscal policy to introduceprice stability. (For simplicity, we assume that if fiscal policy is not operative in period 2, thensubsidies are set at 0.)Table 4.11 illustrates how welfare responds across the four different regimes. Our measure ofwelfare comes given by the utility of households as per SUW specification,ln((CT1)γ (CN1)1−γ)+ β ln((CT2)γ (CN2)1−γ).Column 1 of the table shows that welfare is low when the economy suffers a negative shock and thegovernment has no access to monetary or fiscal tools to prevent the contraction of output. On thecontrary, the situation is reversed when the government can use nominal devaluations (column 2).Likewise, fiscal devaluations can also be effective at improving households’ welfare. However, thetable illustrates that this is only the case when fiscal devaluations can adjust at the business cyclefrequency – column 3. On the contrary, when fiscal policy fails to adjust instantaneously with theoccurrence of the shock, column 4 shows that there is no welfare improvement that can be derivedfrom the government intervention in the second period.This particular example depicted a scenario with fiscal policy adjusting very late to a recession– in this case, when the recession is already over. Although this is an extreme case, this modelis merely illustrative and it is only used to present a simple environment where delays can affectfiscal policy. In particular, the model shows that fiscal devaluations can fail to achieve the samereal allocations as nominal devaluations when fiscal devaluations to implementation lags.4.3.4 DiscussionThe model presented in this section described one way in which fiscal devaluations featuring lagscan potentially be ineffective to fight recessions. This result largely depends on the assumption126that tax policies are implemented in the second period: When fiscal policy acts late, it might notbe able to prevent real variables from absorbing the negative shock. However, these conclusionsalso depend on other assumptions and simplifications. Next, I shall briefly discuss some of theseaspects.Separable preferencesOne assumption related to the specification of preferences is that the model features both a time-separable and good-separable utility function. This assumption simplifies the analysis as it allowsconsumption decisions of tradable and non-tradable goods to be made independently. Althoughin this study I do not address how the results would respond to non-separable preferences, it isstraightforward to identify some ways in which our analysis would change using different prefer-ences. Suppose instead a utility specification that is non-separable with respect to consumption oftradables and non-tradables. As a result, the allocations arising from the decision problem of thehousehold will now likely depend on future fiscal policy since this policy will affect intertemporalchoices as tradable and nontradable consumption are linked. Technically, this means that the firstorder conditions of the household’s optimization problem would depict nonzero cross derivativesbetween goods. The intuition is that now there is substitutability between consumption of trad-ables and nontradables, and changes in one type of good can affect the marginal utility of the othergood. As wage subsidies will affect the relative price of nontradables (and since now the decisionover both goods will be linked), fiscal policy that is anticipated can likely affect the consumption ofboth goods before it is implemented. In summary, a non-separable utility would mean that fiscalpolicy implementation lags may affect consumption on non-tradables even during a recession sincethe Euler equation will prescribe a reaction of forward-looking households at the time of the shock.Role of capital controlsIn this chapter, I presented evidence of fiscal policy lags that may affect the applicability andeffectiveness of fiscal devaluations. In particular, in this section I formally discussed how fiscal policylags may affect that effectiveness when fiscal devaluations feature delays in their implementation.Before concluding, it is important to mention that the macro literature has also investigated theimpact of alternative stabilization instruments. While this chapter only analyzed fiscal devaluations,it is worth pointing out that these alternative policies are also part of policy toolkits. Briefly, oneexample of these instruments is the use of capital controls. Schmitt-Grohe´ and Uribe (2016) describehow capital controls can mitigate the welfare costs faced by an economy subject to a currency pegand which cannot implement the optimal exchange rate.66 In our context, addressing shocks to theeconomy using capital controls would mean that a government faces a trade-off between mitigatingthe collapse in consumption of tradables (and the resulting unemployment in the non-tradablesector) during the shock, at the cost of a moderate expansion in tradables once the shock vanishes.66The reader is referred to that paper for a survey on capital controls and their use for macro stabilization.127To explore this possibility, one candidate strategy would be to allow a government to tax the returnson net foreign assets. As analyzed before, a rise in the interest rate pushes households to increasesavings in the first period, so as to seize more tradable goods after the shock. Because of this, andabsent any capital controls, the fluctuation in consumption (and, therefore, unemployment) couldbe larger across periods. However, when the government can tax returns on net foreign assets, itmay provide a disincentive to the volatility in consumption and unemployment since the decline inconsumption at the time of the shock is anchored to the rise in tradable consumption after a shock.A priori, this mechanism might indicate that capital controls could theoretically improve welfare– for a further discussion on capital controls and their performance relative to fiscal devaluations,see for example Schmitt-Grohe´ and Uribe (2012).4.4 Concluding remarksIn this chapter I explored the incidence of fiscal policy lags on fiscal devaluations. In the first part,I investigated the existence of fiscal policy lags in the Euro Area since its creation, as well as inthe Spanish economy. To that end, I used the Tax Policy Reform Database which documents lagsin fiscal policy for different economies – see Amaglobeli et al. (2018). In the second part, I useSchmitt-Grohe´ and Uribe (2012) two-period model to illustrate the impact that implementationlags can have in a country’s fiscal policy from a welfare perspective. The goal of this chapter is toapproach some evidence and economic intuitions that can potentially affect the equivalence resultbetween fiscal and nominal devaluations analyzed by Farhi et al. (2014) under the presence of lags.The results from these two parts can be easily summarized. From the empirical perspective,and for both the Euro Area as a whole and for the Spanish economy in particular, there is evidencethat fiscal policy is subject to implementation lags across several dimensions. Apart from featuringpositive averages and medians, considerable shares of tax policy changes can be catalogued asanticipated under a metric used in the literature – see Mertens and Ravn (2012). Furthermore,over the business cycle, recessions feature shorter implementation lags more often. This result is,however, not consistently observed: In some cases, the economic expansions following an outputcollapse may feature shorter implementation lags relative to their preceding recession. Overall,these empirical facts suggest that fiscal policy might not adjust at the business cycle frequency.From the theoretical perspective, I take this observation to a model and argue that welfare mightnot be improved with fiscal devaluations when this policy does not respond at the time of a negativeshock. Delays in fiscal policy implementation can, therefore, affect the ability of fiscal devaluationsto replicate nominal devaluation outcomes.It is worth to point out that there are some important caveats to the analysis carried out.From the empirical point of view, for example, this chapter only focuses on the aggregate EuroArea level and in the Spanish experience. The focus on Spain is driven by the analysis conductedby Farhi et al. (2014). However, this investigation is silent about the role of fiscal policy lagsin other Euro Area countries. For instance, of particular interest could be the analysis of other128countries that were largely affected during the 2012 European debt crisis as measured by theincrease in the yields of long-term government bonds – e.g., Greece, Italy, and Slovenia. Fromthe theoretical perspective, this chapter presents a two-period model that seeks to rationalize ourempirical findings to illustrate how fiscal implementation lags can reduce welfare gains from fiscaldevaluations. However, the model presented here does so under the extreme assumption that fiscalpolicy cannot respond at the time of the shock. In addition, the model does not address the role ofprivate sector’s expectations. This limits the analysis of how households and firms respond to fiscalpolicies in the presence of fiscal implementation lags, which can impact the relationship betweenfiscal and nominal devaluations from Farhi et al. (2014). Finally, a calibration of Farhi et al. (2014)where fiscal policy features lags of different lengths is worth pursuing to understand how welfaregains respond to delayed fiscal devaluations – relative to the case already discussed in that paper.In the future I plan to study these issues with alternative modelling approaches.1294.5 TablesTable 4.1: Summary statistics (Euro Area countries – Period 1999-2018)TimespanAvg. taxchangesper yr.Std.Dev.Min. taxchangesin a yr.Max. taxchangesin a yr.Perc.25Perc.50Perc.75Announced 20 25.7 12.8 9 59 16.5 22.5 31Implemented 22 23.4 11.5 3 43 16 21.5 30Note: Total observations: 514 tax changes. Observations per country: Austria (42 obs.),France (73 obs.), Germany (73 obs.), Greece (47 obs.), Ireland (54 obs.), Italy (82 obs.),Luxembourg (20 obs.), Portugal (44 obs.), Spain (79 obs.). Lags: Nonnegative. Units: Totalannounced/implemented measures. Source: TPRD (Amaglobeli et al., 2018) and author’scalculations.Table 4.2: Summary statistics: Lags (EuroArea countries – Period 1999-2018)AlltaxesTaxdecreasesTaxincreasesObs. 514 300 214Avg. 175 174 177Std. Dev. 249 206 300Min 0 0 0Max 3765 1486 3765Perc. 25 50 48 51Perc. 50 94 97 87Perc. 75 240 240 240Perc. 90 377 377 349Perc. 95 614 555 622Perc. 99 999 999 758Note: Units in days. Lags: Nonnegative.Source: TPRD (Amaglobeli et al., 2018)and author’s calculations.130Table 4.3: Summary statistics: Restrictedlags (Euro Area – Period 1999-2018)AlltaxesTaxdecreasesTaxincreasesObs. 497 289 208Avg. 144 144 145Std. Dev. 139 135 143Min 0 0 0Max 714 714 696Perc. 25 50 43 51Perc. 50 93 96 83Perc. 75 220 212 233Perc. 90 330 342 327Perc. 95 403 403 415Perc. 99 643 622 666Note: Units in days. Lags: Nonnegativeand less than 2 yrs. Source: TPRD (Ama-globeli et al., 2018) and author’s calcula-tions.131Table 4.4: Summary statistics: Lags and business cycles (Euro Area – Period 1999-2018)TaxchangesDays MonthsAvg.lagMedianlagMax.lagAnticipatedtax changesShare ofanticipatedtax changesPeriod 1Recession49 822 27 134 92 530 31 63%Period 2Expansion106 1735 55 191 173 696 69 65%Period 3Recession27 457 15 133 79 338 13 48%Period 4Expansion37 730 22 108 75 299 17 46%Period 5Recession52 639 21 99 63 666 13 25%Period 6Expansion115 1736 56 127 79 535 52 45%Total obs. 386 195 51%Note: Period 1: 2001m3 to 2003m6. Period 2: 2003m7 to 2008m2. Period 3: 2008m3 to2009m6. Period 4: 2009m7 to 2011m5. Period 5: 2011m6 to 2013m3. Period 6: 2013m4 to2017m12. Lags: Nonnegative and less than 2 yrs. Source: TPRD (Amaglobeli et al., 2018)and author’s calculations.Table 4.5: Summary statistics: Restricted Lags (Spain– Period 1999-2018)All taxes Tax decreases Tax increasesObs. 79 51 28Avg. 154 160 144Std. Dev. 141 144 138Min 0 0 5Max 622 622 622Perc. 25 29 22 34Perc. 50 107 187 103Perc. 75 195 195 222Perc. 90 310 310 278Perc. 95 398 398 310Perc. 99 622 622 622Note: Units in days. Lags: Nonnegative and less than2 yrs. Source: TPRD (Amaglobeli et al., 2018) andauthor’s calculations.132Table 4.6: Summary statistics: Lags, Business cycles (Spain – Period 1999-2018)TaxchangesDays MonthsAvg.lagMedianlagMax.lagAnticipatedtax changesShare ofanticipatedtax changesPeriod 1Recession10 822 27 170.5 187 187 9 90%Period 2Expansion12 1735 55 350.7 277 622 11 92%Period 5Recession11 639 21 42.3 50 107 3 27%Period 6Expansion25 1736 56 156.4 195 310 17 68%Total obs. 58 40 69Note: Period 1: 2001m3 to 2003m6. Period 2: 2003m7 to 2008m2. Period 3 (skipped): 2008m3to 2009m6. Period 4 (skipped): 2009m7 to 2011m5. Period 5: 2011m6 to 2013m3. Period 6:2013m4 to 2017m12. Lags: Nonnegative and less than 2 yrs. Source: TPRD (Amaglobeli et al.,2018) and author’s calculations.Table 4.7: Summary statistics: Taxmeasures (Spain – Period 1999-2018)VAT SSC CIT PITObs. 7 5 20 41Avg. 88 66 128 205Std. Dev. 90 40 119 154Min 1 0 1 1Max 278 97 398 622Perc. 25 50 68 2 99Perc. 50 50 68 98 187Perc. 75 94 97 195 277Perc. 90 278 97 277 310Perc. 95 278 97 338 622Perc. 99 278 97 398 622Note: Units in days. Lags: Nonnega-tive and less than 2 yrs. Source: TPRD(Amaglobeli et al., 2018) and author’scalculations.133Table 4.8: Summary statistics: Direction of tax changes(Spain – Period 1999-2018)All taxes Tax decreases Tax increasesObs. 129 76 53Avg. 218 221 213Std. Dev. 161 144 185Min 0 0 0Max 655 601 655Perc. 25 96 122 50Perc. 50 214 216 201Perc. 75 287 287 287Perc. 90 480 386 489Perc. 95 584 561 587Perc. 99 655 601 655Note: Units in days. Lags: Nonnegative and less than2 yrs. Source: TPRD (Amaglobeli et al., 2018) andauthor’s calculations.Table 4.9: Numerical exercise: ParametersParameter Value Descriptionβ 0.8 Discount factorQT1 Endowment of tradablesh 1 Time endowmentr∗ 0.25 Nominal interest rate (Shock: 0.3)P T,∗t 1 Foreign nominal price of tradablesW0 0.75 Initial nominal wageγ 0.5 Tradables shareα 0.75 Productivity of laborAt 1 Firm productivityE 1 Exchange rate (pre-shock, fixed E.R.)B0 0 Initial bond holdingss 0 Subsidy (pre-shock)134Table 4.10: Numerical exercise: Equilibrium outcomesFixedexchange rateFlexibleexchange rateFiscaldevaluations(no lags)Fiscaldevaluations(with lags)t=1 t=2 t=1 t=2 t=1 t=2 t=1 t=2Goods marketCTt 0.983 1.022 0.983 1.022 0.983 1.022 0.983 1.022CNt 0.987 1 1 1 1 1 0.987 1QTt 1 1 1 1 1 1 1 1QNt 0.987 1 1 1 1 1 0.987 1P Tt 1 1 1.017 0.978 1 1 1 1PNt 0.996 1.022 1 1 0.983 1.022 0.996 1.022pt 0.996 1.022 0.983 1.022 0.983 1.022 0.996 1.022Labor marketht 0.983 1 1 1 1 1 0.983 1Wt 0.750 0.767 0.750 0.750 0.750 0.767 0.750 0.767Monetary policyEt 1 1 1.017 0.978 1 1 1 1Fiscal policyst 0 0 0 0 0.017 0 0 0τt 0 0 0 0 0.006 0 0 0Table 4.11: Present discounted value of total welfare.Fixedexchange rateFlexibleexchange rateFiscaldevaluation(no lags)Fiscaldevaluation(with lags)Welfare −0.0063 0.0002 0.0002 −0.00631354.6 Figures0102030405060Number of tax changes1999 2003 2007 2011 2015 2019YearsAnnounced tax changes (per yr.) Implemented tax changes (per yr.)Figure 4.1: Distribution of tax changes (Euro Area countries – Period 1999-2018)Notes: Total observations: 514 tax changes. Observations per country: Austria (42 obs.), France (73 obs.), Germany(73 obs.), Greece (47 obs.), Ireland (54 obs.), Italy (82 obs.), Luxembourg (20 obs.), Portugal (44 obs.), Spain (79obs.). Lags: Nonnegative. Each bin represents a year. Source: TPRD (Amaglobeli et al., 2018) and author’scalculations.0102030405060Number of tax changes1999 2003 2007 2011 2015 2019YearsAnnounced tax changes (per yr.) Implemented tax changes (per yr.)Figure 4.2: Distribution of tax changes (Non Euro Area countries – Period 1999-2018)Notes: Total observations: 135 tax changes. Observations per country: Czech Republic (47 obs.), Denmark (23obs.), Poland (40 obs.), United Kingdom (25 obs.). Lags: Nonnegative. Each bin represents a year. Source: TPRD(Amaglobeli et al., 2018) and author’s calculations.1360510152025Frequency0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750Lags of tax changes (in days)Lags of tax changes (in days) implemented within 2 yearsSource: Author's calculations. Data: Tax Policy Reform DatabaseFigure 4.3: Distribution of lags for the final sample (1999-2018)Notes: Frequency of lags in tax changes announced during 1999-2018. Lags: Nonnegative and less than 2 yrs. Totalobservations: 497 tax changes. Observations per country: Austria (42 obs.), France (69 obs.), Germany (71 obs.),Greece (42 obs.), Ireland (53 obs.), Italy (77 obs.), Luxembourg (20 obs.), Portugal (44 obs.), Spain (79 obs.). Figurelinked to Table 4.3. Source: TPRD (Amaglobeli et al., 2018) and author’s calculations.050100150200Frequency0-30 31-60 61-90 91-120 121-150 >151Implementation lagsHistogram of tax changes. Countries: All. Lags: Nonnegative and less than 2 yrs. Taxdirection: All (increases and decreases). Bins measured in days.Source: Author's calculations. Data: IMF - Tax Policy Reform DatabaseFigure 4.4: Distribution of lags (Euro Area – Period 1999-2018)Notes: Histogram of tax changes. Countries: All. Lags: Nonnegative and less than 2 yrs. Tax direction: All (increasesand decreases). Total observations: 497. tax changes. Observations per country: Austria (42 obs.), France (69 obs.),Germany (71 obs.), Greece (42 obs.), Ireland (53 obs.), Italy (77 obs.), Luxembourg (20 obs.), Portugal (44 obs.),Spain (79 obs.). Source: TPRD (Amaglobeli et al., 2018) and author’s calculations – the figure follows the graphicalrepresentation in Mertens and Ravn (2012).13705101520Freq.0-30 31-60 61-90 91-120121-150 >151Implementation lagsPanel A. Period 2001-2003.01020304050Freq.0-30 31-60 61-90 91-120121-150 >151Implementation lagsPanel B. Period 2003-2008.05101520Freq.0-30 31-60 61-90 91-120121-150 >151Implementation lagsPanel C. Period 2008-2009.01020304050Freq.0-30 31-60 61-90 91-120121-150 >151Implementation lagsPanel D. Period 2009-2011.05101520Freq.0-30 31-60 61-90 91-120121-150 >151Implementation lagsPanel E. Period 2011-2013.01020304050Freq.0-30 31-60 61-90 91-120121-150 >151Implementation lagsPanel F. Period 2013-2017.Figure 4.5: Distribution of lags: Business cycles. (Euro Area – Period 1999-2018)Notes: Bin units: days. Total obs: 386 tax changes. Observations per country: Austria (29 obs.), France (44 obs.),Germany (41 obs.), Greece (32 obs.), Ireland (43 obs.), Italy (73 obs.), Luxembourg (19 obs.), Portugal (40 obs.),Spain (65 obs.). Lags: Nonnegative and less than 2 yrs. Source: TPRD (Amaglobeli et al., 2018) and author’scalculations – the figure follows the graphical representation in Mertens and Ravn (2012).13805101520Number of tax changes1999 2003 2007 2011 2015 2019YearsAnnounced tax changes (per yr.) Implemented tax changes (per yr.)Figure 4.6: Distribution of tax changes (Spain – Period 1999-2018)Notes: Total observations: 79 tax changes. Each bin represents a year. Lags: Nonnegative and less than 2 yrs.Source: TPRD (Amaglobeli et al., 2018) and author’s calculations.139010203040Freq.0-30 31-60 61-90 91-120 121-150 >151Implementation lag in daysHistogram of tax changes(tax increases and decreases)010203040Freq.0-30 31-60 61-90 91-120 121-150 >151Implementation lag in daysHistogram of tax changes(tax decreases)010203040Freq.0-30 31-60 61-90 91-120 121-150 >151Implementation lag in daysHistogram of tax changes(tax increases)Figure 4.7: Direction of tax changes (Spain – Period 1999-2018)Notes: Histogram of tax changes. Country: Spain. Lags: Nonnegative and less than 2 yrs. Tax direction: All taxes(top figure), tax decreases (middle figure), tax increases (bottom figure). Source: IMF, TPRD (Amaglobeli et al.,2018) and author’s calculations – the figure follows the graphical representation in Mertens and Ravn (2012).14005101520Freq.0-30 31-60 61-90 91-120 121-150 >151Implementation lagsPanel A. Period 2001-2003.05101520Freq.0-30 31-60 61-90 91-120 121-150 >151Implementation lagsPanel B. Period 2003-2008.05101520Freq.0-30 31-60 61-90 91-120 121-150 >151Implementation lagsPanel C. Period 2011-2013.05101520Freq.0-30 31-60 61-90 91-120 121-150 >151Implementation lagsPanel D. Period 2013-2017.Figure 4.8: Distribution of lags: Business cycles. (Spain – Period 1999-2018)Notes: Bin units: days. Total obs: 58 tax changes. Period 1: 10 obs. Period 2: 12 obs. Period 5: 11 obs. Period 6:25 obs. Lags: Nonnegative and less than 2 yrs. Source: TPRD (Amaglobeli et al., 2018) and author’s calculations –the figure follows the graphical representation in Mertens and Ravn (2012).141012345Freq.0-30 31-60 61-90 91-120 121-150 >151Implementation lagsTax type: VAT.012345Freq.0-30 31-60 61-90 91-120 121-150 >151Implementation lagsTax type: SSC.0246810Freq.0-30 31-60 61-90 91-120 121-150 >151Implementation lagsTax type: CIT.0102030Freq.0-30 31-60 61-90 91-120 121-150 >151Implementation lagsTax type: PIT.Figure 4.9: Tax measures and business cycles (Spain – Period 1999-2018)Notes: Bins measured in days. Total obs: 55 tax changes. Observations per tax measure: VAT (7 obs.), SSC (5obs.), CIT (20 obs.), PIT (41 obs.). Lags: Nonnegative and less than 2 yrs. Source: TPRD (Amaglobeli et al., 2018)and author’s calculations – the figure follows the graphical representation in Mertens and Ravn (2012).142Chapter 5ConclusionThe first chapter (chapter 2) explores whether and how fiscal stabilization policies implemented bya treasury affects the credibility of a central bank’s forward guidance. To answer to this question, Ipresent a model in line with recent literature analyzing credible forward guidance based on centralbank reputation. Specifically, I use a stylized New Keynesian setup with repeated liquidity trapsand a central bank, and I include a treasury that chooses fiscal policy to fight recessions accordingto its own policy objectives. When fiscal policy becomes more effective against recessions, I findthat fiscal policy can limit the ability of a central bank to maintain the credibility of its forwardguidance announcements. The intuition behind this result is the following. On the one hand, whenfiscal policy becomes more effective to mitigate output drops, then recessions during liquidity trapsbecome milder. But on the other hand, milder recessions leave forward guidance with less leewayto improve economic crises. In other words, the central bank can promise less forward guidancelevels based on its reputation. As a result, more effective fiscal policy can potentially leave a centralbank with access to a set of forward guidance announcements that is smaller relative to the set ofpolicies it would observe in the absence of fiscal policy.This chapter contributes to the growing body of work analyzing unconventional monetary poli-cies, and in particular to the strand focused on the credibility of forward guidance. When a centralbank cannot commit and its promises are time-inconsistent, recent literature showed that reputa-tional concerns can restore the credibility of central bank’s forward guidance. However, monetarypolicy is not necessarily implemented in vacuum: Recent crises provided evidence of fiscal policyactively fighting recessions. Therefore, this chapter uses the reputational mechanism from previousworks to explore how forward guidance credibility responds to fiscal stabilization policies.While the model presented in this chapter makes some abstractions, it also suggests somepossible avenues for future research. For instance, in this setup I abstract from modelling seignioragerevenues that could arise from the promise of expansionary monetary policy – aspect that has beenstudied in previous literature; see for example Eggertsson (2006). In our context, this might expandthe possibility of strategic interactions that can be thought of between the central bank and thetreasury. Also, this chapter’s model abstracts from discussing the impact of liquidity trap episodesthat last more than one period. As this can affect the central bank’s incentives to keep its reputationas well as the treasury’s debt paths, a candidate first step would involve exploring how the resultsrespond to liquidity trap episodes of longer (but finite) duration. This type of questions are worthpursuing and I plan to analyze them in future research.The previous chapter 2 presented a scenario where there is a liquidity trap constraint on con-143ventional monetary policy, and where the literature has shown how this constraint could be solvedusing credible forward guidance. The study in chapter 2 was then focused on how this policysolution interacted with active fiscal policy. In chapter 3, instead, we explore another type of inter-action. Specifically, chapter 3 starts presenting the scenario where self-fulfilling crises constraint agovernment’s ability to finance its fiscal policy. While a theoretical solution to deal with this type ofcrises suggested some optimal paths of short-term debt, in this chapter we analyze the interactionbetween this solution and the lengthening of the maturity of debt. Switching to a long-term debtinstrument widely used in the sovereign debt literature, chapter 3 analyzes how a longer maturityof debt can help prevent self-fulfilling debt crises. We depart from the standard belief-driven setupwith one-period bonds, and when we implement bonds that mature in a longer horizon we findthat debt levels once affected by default fears from creditors are no longer subject to them. Theintuition is that access to long-term debt requires a government to repay only a portion of its totaloutstanding debt in each period. This, in turn, reduces the fear of sovereign default from creditors.In line with this result, we compute numerically optimal debt policies and argue that governmentsmay require less time to reach debt levels that eliminate the possibility of self-fulfilling debt criseswhen governments use long-term debt. For all these cases, we also find that lengthening debtmaturity can improve the welfare of consumers in the model economy.An important motivation of this study was the scenario experienced by some European countriesduring the 2012 debt crises. During that episode, the observed volatility of bond spreads and itspotential linkage with creditors’ pessimistic expectations could be theoretically explained by self-fulfilling debt crises frameworks like the one presented in chapter 3. However, although part of thatcrisis seemed to be controlled after the European Central Bank announced its liquidity program, thecurrent model is silent with respect to the possibility of having liquidity shocks affecting creditors.Exploring this dimension would allow us to investigate how liquidity shocks can reshape the crisisregion, or also to what extent varying the access to liquidity can affect optimal debt policies. Forinstance, finance literature has explored the impact of market liquidity and rollover risk providedthat creditors are exposed to liquidity shocks. He and Xiong (2012) introduce risk neutral creditorsthat are subject to a shock that forces them to exit the market and sell their bond holdings insecondary markets at a fraction of their market value. This assumption decreases the availableliquidity in credit markets, and could be implemented to explore existing trade-offs between long-term and short-term debt.Finally, in chapter 4 I study how fiscal policy lags can affect fiscal devaluations, which were aproposed policy for countries belonging to a currency union. I first conduct an empirical analysisthat focuses on the Euro Area and Spain using the Tax Policy Reform Database – see Amaglobeliet al. (2018). The focus on the Spanish economy is motivated by the analysis conducted by Farhiet al. (2014) who numerically assess the welfare gains from fiscal devaluations for that country. Forthe whole sample of Eurozone country members, I first documented the existence of implementationlags focusing the analysis on a set of descriptive statistics. While lags exhibited an average and amedian of four and three months, respectively, a considerable portion of the tax changes fell under144the category of anticipated fiscal policies. Similar but slightly larger delays can be found for Spain.In addition, these positive average and median lags, as well as the share of anticipated tax changes,are also present during business cycles for the aggregate Euro Area sample. Specifically, while fiscalpolicy delays are still present during recessions, these lags can exhibit faster implementation datesrelative to fiscal policy occurring during recovery periods. However, this fact does not appear to beconstant for the Euro Area as a whole since one recession features longer lags. While there are lessobservations available, a similar finding was identified for the Spanish economy too. The secondpart of this chapter presents a simple model to illustrate how lags can affect fiscal devaluationsfrom a welfare perspective. 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Monetary theory and policy, third edition. MIT Press Books 1.Werning, I. (2012). Managing a liquidity trap: Monetary and fiscal policy, manuscript.Woodford, M. (2003). Interest and prices: Foundations of a theory of monetary policy. Princetonuniversity press.Woodford, M. (2011). Simple analytics of the government expenditure multiplier. American Eco-nomic Journal: Macroeconomics 3 (1), 1–35.Woodford, M. (2012). Methods of policy accommodation at the interest-rate lower bound.151Woodford, M. and Y. Xie (2022). Fiscal and monetary stabilization policy at the zero lower bound:Consequences of limited foresight. Journal of Monetary Economics 125, 18–35.Yang, S.-C. S. (2005). Quantifying tax effects under policy foresight. Journal of Monetary Eco-nomics 52 (8), 1557–1568.152Appendix AAppendix to Chapter 2A.1 Scenario without forward guidanceDerivations of Optimality Conditions without forward guidanceThe following solutions closely follow Clarida et al. (1999), Jung et al. (2005), Nakata (2018)and Walsh (2018). Other references cited in text.TreasuryClaim. In state R, the treasury’s problem (2.13) can be compactly re-expressed asmax{bR,`R,τR}−12(αTry (yR)2 + αTrτ (τR)2)subject toyR = yeR − (iR − pieR − rR) + γbRbR = (1 + r) bZ − τR + `R`R = 0, bR = 0, 0 ≤ bR ≤ b¯, `R ≥ 0, τR ≥ 0where {iR, bZ , yeR, pieR, rR, r} are given.Proof. The general problem for the treasury ismax{τt,bt,`t}∞t−12Et∞∑k=0(βTr)k [αTry y2t+k + αTrτ τ2t+k]subject toyt = Etyt+1 − (it − Etyt+1 − rt) + γbt, ∀t;bt = (1 + r) bt−1 − τt + `t, ∀t;0 ≤ bt ≤ b¯ <∞, ∀t;`t ≥ 0, ∀t;and also taking initial debt level bt−1, current central bank’s choice it and future paths {yt+k, pit+k,it+k, τt+k, bt+k, `t+k}k≥1 as given.Note first that in state R, the period-t problem of the treasury can be written asmax{bt,`t,τt}−12(αTry (yR)2 + αTrτ (τR)2)+ Ψt153subject toyR = yeR − (iR − pieR − rR) + γbRbR = (1 + r) bZ − τR + `R0 ≤ bR ≤ b¯, and `R ≥ 0andΨt ≡ −12Et∞∑k=1(βTr)k (αTry (yt+k)2 + αTrτ (τt+k)2)and also taking as given initial debt level bZ , the central bank’s choice iZ , and variables piet ≡ Etpit+1and yet ≡ Etyt+1 representing PS’s expectations. For the expectation term in Ψt, note first thatwhen the state is R in period t, the state of the world in t + 1 will be a liquidity trap state Zwith probability p, or the absorbing steady state S with probability (1− p). However, note that inthe R state of the world, the treasury makes no actual choice due to our assumptions. Hence, wecan compute its losses in state R by just dropping term Ψt and combining the previous constraintswhile we make use of our assumptions that the treasury (i) sets `R = 0, and (ii) retires past debtwith taxation, bR = 0. The previous problem in state R therefore becomes (again, abstracting fromΨt since it is independent of today’s choices),max{bR,`R,τR}−12(αTry (yR)2 + αTrτ (τR)2)subject toyR = yeR − (iR − pieR − rR) + γbRbR = (1 + r) bZ − τR + `R`R = 0, bR = 0, 0 ≤ bR ≤ b¯, and `R ≥ 0where {iR, bZ , yeR, pieR, rR, r} are given.Claim. The solution to the treasury’s problem, Eq. (2.13), in state R yieldslTrR ≡ −12(αTry (yeR − iR + pieR + rR)2 + αTrτ ((1 + rR) bZ)2)Proof. By the previous result,max{bR,`R,τR}−12(αTry (yR)2 + αTrτ (τR)2)subject toyR = yeR − (iR − pieR − rR) + γbRbR = (1 + r) bZ − τR + `R`R = 0, bR = 0, 0 ≤ bR ≤ b¯, `R ≥ 0, τR ≥ 0where {iR, bZ , yeR, pieR, rR, rS} are given.154Plugging bR = 0, `R = 0 and given bZ and rR in the treasury’s BC yieldsτR = (1 + r) bZ (A.1)Now, plugging bR = 0 and (iR, rR) in the DIS rendersyR = yeR − iR + pieR + rR, (A.2)meaning that {yR} is given to the treasury from the point of view of a problem at period R.Therefore, using Eqs. (A.1) and (A.2) in the loss function in periods where state is R (lTrt (st = R) ≡lTrR ) yieldslTrR ≡ −12(αTry (yeR − iR + pieR + rR)2 + αTrτ ((1 + r) bZ)2). (A.3)Claim. The treasury’s problem can be compactly re-expressed in state Z asmax{bZ}−12αTry (yeZ − (iZ − pieZ − rZ) + γbZ)2 + βTrEtlTrt+1subject topiZ = κyZ + βPSpieZ0 ≤ bZ ≤ b¯,rZ givenlTrt+1 ≡ lTrt+1 (yt+1, τt+1) ≡ lTrt+1(it+1, bt, bt+1, `t+1, yet+1, piet+1, rt+1)and given {iZ , yeZ , pieZ} (for the first term of the objective) and{it+1, bt+1, `t+1, τt+1, yet+1, piet+1, rt+1}(for the second term of the objective) – to make the current state explicit, state-Z variables are onlywritten as xt = xZ .Proof. Note first that in state Z, the period t problem of the treasury can be written asmax{bt,`t,τt}−12(αTry (yt)2 + αTrτ (τt)2)+ βTrEtlTrt+1 (yt+1, τt+1) + Ψ′tsubject toyt = yet − (it − piet − rt) + γbtbt = (1 + r) bt−1 − τt + `t0 ≤ bt ≤ b¯, `t ≥ 0, τt ≥ 0{bt−1, rt, it, yet , piet } givenandlTrt+1 ≡ lTrt+1 (yt+1, τt+1) ≡ lTrt+1(it+1, bt, bt+1, `t+1, yet+1, piet+1, rt+1, r),155with{it+1, bt+1, `t+1, τt+1, yet+1, piet+1, rt+1}given (and where we made explicit the second term ofthe summation since it depends on bt+1), and also subject toΨ′t ≡ −12Et∞∑k=2βjlTrt+k, with lTrt+k ≡ αTry (yt+k)2 + αTrτ (τt+k)2 .where again {it+k, bt+k, `t+k, τt+k, rt+k} for k ≥ 2. Plugging yt = yet −(it − piet − rt)+γbt and τt = 0(by assumption of period st = Z) in the objective, linking the remaining constrains (together withthe fact that bt−1 = 0 in Z states), and dropping the Ψ′t term (the treasury’s choice at t will notaffect any variable at t+ 2), then if the current state is Z (and we rename period-t variables usingthe state subscript), then the treasury’s problem becomesmax{bZ}−12αTry (yeZ − (iZ − pieZ − rZ) + γbZ)2 + βTrEtlTrt+1subject topiZ = κyZ + βPSpieZ0 ≤ bZ ≤ b¯,lTrt+1 ≡ lTrt+1 (yt+1, τt+1) ≡ lTrt+1(it+1, bt, bt+1, `t+1, yet+1, piet+1, rt+1)and given rZ and {iZ , yeZ , pieZ} (for the first term of the objective) and{it+1, bt+1, `t+1, τt+1, yet+1, piet+1, rt+1}(for the second term of the objective).Lemma. Without forward guidance, the optimal level of debt that solves the treasury’s problem is,bZ =αTry γαTry γ2 + βTrαTrτ (1 + rR)2 (iZ − rZ − yeZ − pieZ)Proof. I build this problem using a backward induction logic. In period t + 1 the state is R, andtherefore after the natural interest rate is realized at rR, the payoff of a treasury in state R is givenbylTrR ≡ −12(αTry (yeR − iR + pieR + rR)2 + αTrτ ((1 + rR) bZ)2).Variable bZ is going to be determined by the fiscal authority from state Z (iR is determined bythe monetary authority making decisions in state R, and the same applies for PS’s decisions over(yeR, pieR)).In period t, the state is Z and the fiscal authority making decisions at Z solves the problemgiven by out previous resultsmax{bZ ,`Z ,τZ}−12αTry (yeZ − (iZ − pieZ − rZ) + γbZ)2 + βTrEtlTrt+1156subject to rZ given,0 ≤ bZ ≤ b¯,lTrt+1 ≡ lTrt+1 (yt+1, τt+1) ≡ lTrt+1(it+1, bt, bt+1, `t+1, yet+1, piet+1, rt+1)Given that the structure of the shock indicates the fiscal authority to expect state R with certainty,the expectation term from βTrEtlTrt+1 is equal to βTrlTrR – with the variables inside lTrR at state R.Plugging in this continuation value the expression for lTrR from above, the discretionary treasury atstate Z solves a two-period problem of the formmax{bZ ,`Z ,τZ}−12αTry (yeZ − iZ + pieZ + rZ + γbZ)2−βTr12(αTry (yeR − iR + pieR + rR)2 + αTrτ ((1 + rR) bZ)2)subject to0 ≤ bZ ≤ b¯,and {iz, rZ , yeZ , pieZ , iR, rR, yeR, pieR} given. The Lagrangean of the treasury’s problem at Z isL = −12αTry (yeZ − iZ + pieZ + rZ + γbZ)2 − βTr12(αTry (yeR − iR + pieR + rR)2 + αTrτ ((1 + rR) bZ)2)− λ1 (−bZ)− λ2(bZ − b¯)FOC w.r.t. bZ yields−122αy (yeZ − iZ + pieZ + rZ + γbZ)(dyeZdbZ+dpieZdbZ+ γ)−βTr 12αTrτ 2 ((1 + rR) bZ) (1 + rR)−λ1 (−1)−λ2 (1) = 0Throughout the analysis we look for an interior solution for fiscal policy. Hence, (λ1 = 0 andλ2 = 0) – the latter condition also implies nonnegative lump-sum transfers. Rearranging termsyields−αTry (yeZ − iZ + pieZ + rZ + γbZ)(dyeZdbZ+dpieZdbZ+ γ)− βTrαTrτ (1 + rR)2 bZ = 0Terms dyeZ/dbZ and dpieZ/dbZ capture the fact that the treasury is choosing debt which canpotentially affect future output gap and inflation and, therefore, have an impact in state Z via cur-rent PS expectations – for a discussion see, for example, Debortoli and Nunes (2013), Gnocchi andLambertini (2016), and Leeper et al. (2021). To compute the equilibrium, conjecture that yeZ andpieZ depend on next period variables (bt, rt+1); namely, yet ≡ EtY (bt, rt+1) and piet ≡ EtΠ (bt, rt+1).Y (bt, rt+1) is an auxiliary function denoting the expected output gap in the next period as a func-tion of the debt level chosen in period t (and outstanding, in period t + 1) for every exogenousstate rt+1 – the same idea applies to Π (bt, rt+1). Moreover, the functions are differentiable. Thus,157rewriting terms in the FOC yield0 = −αTry (EtY (bZ , rt+1)− iZ + EtΠ (bZ , rt+1) + rZ + γbZ)(dEtY (bZ , rt+1)dbZ+dEtΠ (bZ , rt+1)dbZ+ γ)− βTrαTrτ (1 + rR)2 bZ0 = −αTry (EtY (bZ , rt+1)− iZ + EtΠ (bZ , rt+1) + rZ + γbZ)(EtdY (bZ , rt+1)dbZ+ EtdΠ (bZ , rt+1)dbZ+ γ)− βTrαTrτ (1 + rR)2 bZIn equilibrium, a closed-form solution to the model can still be obtained given our assumptionswhen the economy is in state R. Specifically, recall that for whatever debt level bZ inherited inperiods where the state is R, we argued that (i) the fiscal authority will retire outstanding debt withtaxation (i.e., τR ≥ 0) , and (ii) the treasury does not issue any additional debt in state R (bR = 0)or give any extra lump-sum transfers (`R = 0). The combination of (i) and (ii) implies that anyexisting debt bZ inherited in the R state is absorbed via taxes, and not via debt financing. But,furthermore, since bR enters in the private sector’s DIS (but it is fixed at 0), then any outstandinglevel of debt bZ does not affect PS’s decisions via bR since fiscal policy only retires debt in Recoverytimes with taxes. As a result of this, dY (bZ ,rt+1)dbZ = 0 anddΠ(bZ ,rt+1)dbZ= 0, and plugging this in thelast F.O.C. expression yields(αTry γ2 + βTrαTrτ (1 + rR)2)bZ = −αTry γEtY (rt+1) + αTry γiZ − αTry γEtΠ (rt+1)− αTry γrZ ,where bZ yieldsbZ = Φ (iZ − yeZ − pieZ − rZ)with Φ ≡ αTry γ/(αTry γ2 + βTrαTrτ (1 + rR))– and recall that (1 + rS) = (1 + rR) = (1 + r).Central BankThe solution to the central bank’s problem for each state is conducted in two steps. First, wewill solve the optimization problem using a generic fiscal policy rule with given coefficients. Second,we will cast the results from the previous lemma using the fiscal policy obtained for the problemof the treasury. All these results are calculated using a version of the ZLB given by it ≥ ι – but, inthe chapter, ι is set at 0.Lemma. During the Recovery state, the discretionary Central bank sets the ‘targeting rule’yR +αCBpi καCBypiR = 0,158with associated inflation, output gap, and nominal interest rate given bypiR =αCByαCBy +αCBpi κ2βPSpieR, yR = − αCBpi καCBy +αCBpi κ2βPSpieR,andiR = yeR +(1 +αCBpi κβPSαCBy + αCBpi κ2)pieR + rR + γbR ≥ 0.During the ZLB state, the CB sets the ‘targeting rule’(1− γΦiZ) (αCBy + αCBpi κ2)(yt +αCBpi κβPSαCBy + αCBpi κ2piet)< 0with associated inflation, output gap, and nominal interest rate given bypiZ =((1 + γΦyZ)κ)yeZ +((1 + γΦpiZ)κ+ βPS)pieZ −(κ(1− γΦiZ))ι+ ((1 + γΦrZ)κ) rZyZ =(1 + γΦyZ)yeZ + (1 + γΦpiZ)pieZ −(1− γΦiZ)ι+ (1 + γΦrZ) rZand iZ = 0, with implicit iZ satisfying(iZ =) ι >1 + γΦyZ1− γΦiZyeZ +1 + γΦpiZ +A1− γΦiZpieZ +1 + γΦrZ1− γΦiZrZwhich is equivalent to the satisfaction of the following condition:ι(1− γΦiZ1 + γΦrZ)− 1 + γΦyZ1− γΦiZ(1− γΦiZ1 + γΦrZ)yeZ −1 + γΦpiZ +A1− γΦiZ(1− γΦiZ1 + γΦrZ)pieZ > rZ .Proof. Variables piet ≡ Etpit+1 and yet ≡ Etyt+1 are fixed. Consider then a treasury’s choice bt givenbybt = ΦrZrt + ΦiZit + ΦyZyet + ΦpiZpietwhen the ZLB binds, and bt = 0 away from the ZLB. Variable rt is the exogenous shock. First,note that the ZLB can be re-written asit ≥ ι⇒ −it ≤ −ι⇒ −it + ι ≤ 0At the ZLB, expression for bt in the DIS yieldsyt = yet − (it − piet − rt) + γ(ΦrZrt + ΦiZit + ΦyZyet + ΦpiZpiet)Following an exposition similar to Jung et al. (2005), I calculate the two different cases for states159Z and R. The Lagrangian at Z yieldsL = −12(αCBy (yt)2 + αCBpi (pit)2)− λ1t[pit − κyt − βPSpiet]− λ2t[yt − yet + (it − piet − rt)− γ(ΦrZrt + ΦiZit + ΦyZyet + ΦpiZpiet)]− λ3t [−it + ι]Combining the F.O.C.’s with respect to pit, yt and it renderyt = −αCBpi καCBypit − 1αCBy(1− γΦiZ)λ3t (A.4)For the equality constraints, it must be that∂L∂λ1t= 0⇒ pit = κyt + βPSpiet , (A.5)∂L∂λ2t= 0⇒ yt = yet − (it − piet − rt) + γ(ΦrZrt + ΦiZit + ΦyZyet + ΦpiZpiet). (A.6)For the inequality constraint,λ3t (−it + ι) = 0 (A.7)withλ3t ≥ 0 (A.8)andit ≥ ι (A.9)In the Z state, we look at the case where λ3t > 0 and the ZLB constraint binds. By thecomplementary slackness condition (Eq. (A.7)), it must be thatit = ι (A.10)Then, Eq. (A.4) becomes(1− γΦiZ) (αCBy yt + αCBpi κpit)= − λ3t︸︷︷︸>0< 0 (A.11)and using the NKPC and DIS (Eqs. (A.5) and (A.6)), we arrive at(1− γΦiZ) (αCBy + αCBpi κ2)(yt +αCBpi κβPSαCBy + αCBpi κ2piet)< 0Disregarding(αCBy + αCBpi κ2)> 0, denoting A ≡ αCBpi κβPSαCBy +αCBpi κ2 , and assuming that(1− γΦiZ)> 0160(we will later in the endogenous fiscal policy case that this can be the case), then1 + γΦyZ1− γΦiZyeZ +1 + γΦrZ1− γΦiZrz +1 + γΦpiZ +A1− γΦiZpieZ < iZand Eq. (A.10) implies that(iZ =) ι >1 + γΦyZ1− γΦiZyeZ +1 + γΦpiZ +A1− γΦiZpieZ +1 + γΦrZ1− γΦiZrz (A.12)So, the ZLB binds for some values of rZ : The conditions under which rt is negative enough to makeit = ι are:ι >1 + γΦyZ1− γΦiZyeZ +1 + γΦpiZ +A1− γΦiZpieZ +1 + γΦrZ1− γΦiZrz ⇒ ι− 1 + γΦyZ1− γΦiZyeZ −1 + γΦpiZ +A1− γΦiZpieZ >1 + γΦrZ1− γΦiZrZSince1+γΦrZ1−γΦiZ> 0, and if γΦiZ ∈ (0, 1), thenι(1− γΦiZ1 + γΦrZ)− 1 + γΦyZ1− γΦiZ(1− γΦiZ1 + γΦrZ)yeZ −1 + γΦpiZ +A1− γΦiZ(1− γΦiZ1 + γΦrZ)pieZ > rZ (A.13)(Note that if ΦiZ = ΦrZ = ΦyZ = ΦpiZ then the previous expression becomes ι(1−γΦiZ1+γΦrZ)− yeZ −1+γΦpiZ+A1+γΦrZpieZ > rZ . We will show that this can be the case when we use the result from the treasuryoptimization problem.) Finally, plugging it = ι and optimal choice of Tr, bZ = ΦrZrZ + ΦiZι +ΦyZyeZ + ΦpiZpieZ in the NKPC and DIS, and using subscripts to denote the state yieldspiZ =((1 + γΦyZ)κ)yeZ +((1 + γΦpiZ)κ+ βPS)pieZ −(κ(1− γΦiZ))ι+ ((1 + γΦrZ)κ) rZ (A.14)yZ =(1 + γΦyZ)yeZ + (1 + γΦpiZ)pieZ −(1− γΦiZ)ι+ (1 + γΦrZ) rZ (A.15)iZ = 0 (A.16)and implicit iZ satisfying(iZ =) ι >1 + γΦyZ1− γΦiZyeZ +1 + γΦpiZ +A1− γΦiZpieZ +1 + γΦrZ1− γΦiZrz (A.17)which is equivalent to the satisfaction of the following condition:ι(1− γΦiZ1 + γΦrZ)− 1 + γΦyZ1− γΦiZ(1− γΦiZ1 + γΦrZ)yeZ −1 + γΦpiZ +A1− γΦiZ(1− γΦiZ1 + γΦrZ)pieZ > rZ (A.18)161This finalizes our relevant results for state Z. At R, the Lagrangian for the central bank isL = −12(αCBy (yt)2 + αCBpi (pit)2)−λ1t[pit − κyt − βPSpiet]−λ2t [yt−yet+(it − piet − rt)−γ bt︸︷︷︸=0]−λ3t [−it + ι]Combining the F.O.C.’s with respect to pit, yt and it renderyt = −αCBpi καCBypit − 1αCByλ3t (A.19)For the equality constraints, it must be that∂L∂λ1t= 0⇒ pit = κyt + βPSpiet , (A.20)∂L∂λ2t= 0⇒ yt = yet − (it − piet − rt) + γbt. (A.21)For the inequality constraint,λ3t (−it + ι) = 0 (A.22)withλ3t ≥ 0 (A.23)andit ≥ ι (A.24)In the R state, the ZLB constraint is not binding. Hence, λ3t = 0. Therefore, combiningequations following the same steps as before yields the solution in the R state (the output targetingrule),yR = −αCBpi καCBypiR (A.25)and Eqs. (A.26), (A.27), (A.28)piR =αCByαCBy + αCBpi κ2βPSpieR (A.26)yR = − αCBpi καCBy + αCBpi κ2βPSpieR (A.27)iR = yeR +(1 +αCBpi κβPSαCBy + αCBpi κ2)pieR + rR + γbR (A.28)This completes the proof.162Lemma. During the Recovery state, the discretionary Central bank setsyR +αCBpi καCBypiR = 0,with associated inflation, output gap, and nominal interest rate given bypiR =αCByαCBy +αCBpi κ2βPSpieR, yR = − αCBpi καCBy +αCBpi κ2βPSpieR,andiR = yeR +(1 +αCBpi κβPSαCBy + αCBpi κ2)pieR + rR + γbR ≥ 0.During the ZLB state, the CB sets(1− γΦ) (αCBy + αCBpi κ2)(yZ + αCBpi κβPSαCBy + αCBpi κ2pieZ)< 0with associated inflation, output gap, and nominal interest rate given bypiZ = (1− γΦ)κyeZ +((1− γΦ)κ+ βPS)pieZ − (1− γΦ)κι+ (1− γΦ)κrZyZ = (1− γΦ) (yeZ + pieZ − ι+ rZ)and iZ = 0,and implicit iZ satisfying(iZ =) ι > yeZ +(1− γΦ +A1− γΦ)pieZ + rZwhich is equivalent to the satisfaction of the following condition:ι− yeZ −(1− γΦ +A1− γΦ)pieZ > rZ .Proof. For the endogenous fiscal policy case, the treasury’s choice bt in state Z is given bybt = Φ (it − (yet + piet )− rt) , (A.29)with Φ ≡ αTry γαTry γ2+βTrαTrτ (1+rR)2 . This is equivalent to the exogenous fiscal policy case where bt =ΦrZrt + ΦiZit + ΦyZyet + ΦpiZpiet and coefficients are given byΦrZ ≡ −Φ, ΦyZ ≡ −Φ, ΦpiZ ≡ −Φ, and ΦiZ ≡ Φ. (A.30)To prove this, note that if we impose the previous conditions on the exogenous fiscal policy case,163then we obtain:bt = ΦrZrt + ΦiZit + ΦyZyet + ΦpiZpiet ⇒ bt = Φ (it − (yet + piet )− rt)which is equal to Eq. (A.29). Therefore, the problems for the CB in states Z and R can becast without calculating again the optimization problems, but using instead our relations givenby expressions (A.30). Therefore, by only replacing results from the previous Lemma with theequivalences from (A.30), we obtain the following expressions.The optimality condition (A.4) becomesyt = −αCBpi καCBypit − 1αCBy (1− γΦ)λ3t (A.31)Then, imposing the condition on the multiplier, Eq. (A.31) becomes(1− γΦ) (αCBy yt + αCBpi κpit) < 0 (A.32)Using the NKPC, Eq. (A.32) yields(1− γΦ) (αCBy + αCBpi κ2)(yt + αCBpi κβPSαCBy + αCBpi κ2piet)< 0and from here it is easy to see thatγΦ = γ(αTry γαTry γ2 + βTrαTrτ (1 + rR)2)=αTry γ2αTry γ2 + βTrαTrτ (1 + rR)2 ∈ (0, 1)implying that 1− γΦ ∈ (0, 1). Thus, we only need thatyt +αCBpi κβPSαCBy + αCBpi κ2piet < 0.(disregarding(αCBy + αCBpi κ2)> 0). Now, plugging in the DIS, solving for it and calling A ≡αCBpi κβPSαCBy +αCBpi κ2 ,yet + rt +(1− γΦ +A1− γΦ)piet < it (A.33)Imposing condition (A.30) on expression (A.33), we can build the equivalent version of Eq. (A.12),(it =) ι > yet +(1− γΦ +A1− γΦ)piet + rt (A.34)Lastly, we can build the equivalent version of Eq. (A.13) by solving Eq. (A.34) for rt,ι− yet −(1− γΦ +A1− γΦ)piet > rt (A.35)164This expression is what we identify as condition C1 in text (imposing that ι = 0).Finally, using the DIS and NKPC under the endogenous fiscal choice, and labelling variableswith Z, then solution in the liquidity trap state is defined by Eqs.(1− γΦ) (αCBy + αCBpi κ2)(yZ + αCBpi κβPSαCBy + αCBpi κ2pieZ)< 0 (A.36)piZ = (1− γΦ)κyeZ +((1− γΦ)κ+ βPS)pieZ − (1− γΦ)κι+ (1− γΦ)κrZ (A.37)yZ = (1− γΦ) (yeZ + pieZ − ι+ rZ) (A.38)iZ = 0 (A.39)and implicit iZ from(iZ =) ι > yeZ +(1− γΦ +A1− γΦ)pieZ + rZ (A.40)which is equivalent to the satisfaction of the following condition (from Eq. (A.35)ι− yeZ −(1− γΦ +A1− γΦ)pieZ > rZ (A.41)This finalizes our relevant results for state Z. Finally, note that away from the ZLB (state R)the problem is the same as before, so it is not necessary to make any adjustments to the expressionspreviously obtained.Claim. Be T ∗ a random period where the natural interest rate returns to its steady state, rt = rS.The economy then exhibits two candidate steady states:1. A zero-inflation steady state characterized bypiS = 0, yS = 0, iS = rS, τS = 0, `S = 0, bS = 0, λ1S = 0, λ2S = 0, λ3S = 0(and the PS rationally expects (piet , yet ) = (piS , yS) = (0, 0)),2. A steady state with deflationary expectations characterized bypiS = −rS, yS = −1− βPSκrS, iS = 0, τS = 0, `S = 0, bS = 0,λ1S = αCBpi rS, λ2S =(αCBy1− βPSκ+ αCBpi κ)rS, λ3S =(αCBy1− βPSκ+ αCBpi κ)rS(and the PS rationally expects (piet , yet ) = (piS , yS) =(−rS ,−1−βPSκ rS)< (0, 0)).165Proof. The following exposition closely follows Jung et al. (2005). Assume that the economy is att = T ∗. First recall that we assume that fiscal instruments are perfectly correlated with the shock.Hence, for the treasury this implies that (τt, `t, bt) = (0, 0, 0). Likewise, combining the FOC of thecentral bank’s problem (and after plugging private sector expectations, piet ≡ Etpit+1 = pit+1 andyet ≡ Etyt+1 = yt+1, and given that uncertainty is resolved in state S) yield the following system ofequations,yS = −αCBpi καCBypiS − 1αCBy λ3SpiS =κ1−βPS ySiS = piS + rSλ3SiS = 0, λ3S ≥ 0, iS ≥ 0τS = 0, `S = 0, bS = 0(Steady-state variables denoted as piS , yS , τS , `S , bS , λ1S , λ2S and λ3S have already been replaced inthe treasury and central bank’s policy functions, and private sector expectations. Also note thatwe present a system where we plugged bS = 0 in DIS, simplified the NKPC, and set ι = 0.)We can construct and obtain an equilibrium with zero-inflation (which will generate zero-outputgap in steady state, and positive nominal interest rates). This equilibrium outcome is obtained whenthe ZLB condition is not binding: If λ3S = 0 and iS > 0, condition λ3SiS = 0 is satisfied, the firstequation isyS = −αCBpi καCBypiSPlugging it in the NKPC, it is easy to see that the equality holds if piS = 0. Taking this result tothe NKPC yields a zero-inflation steady state characterized bypiS = 0, yS = 0, iS = rS , τS = 0, `S = 0, bS = 0, λ1S = 0, λ2S = 0, λ3S = 0(and the PS rationally expects (piet , yet ) = (piS , yS) = (0, 0)).I now proceed to build an equilibrium with deflationary expectations (a self-fulfilling deflationaryspiral as in Jung et al., 2005). Using our system of equations, conjecture that pit+1 < 0. In steadystate, pit = pit+1 = piS , and since pit+1 < 0, then piS < 0. Also, yt = yS , and from the NKPC it canbe rewritten as yS =(1− βPS)piS/κ. Provided that piS < 0, then yS < 0, and this implies thatthe CB’s FOC in steady state is − (αCBy yS + αCBpi κpiS) = λ3S . As yS < 0 and piS < 0, this impliesthat λ3S > 0, which needs that iS = 0. Hence, the DIS in steady state is iS = piS + rS and providedthat iS = 0 and rS > 0, then piS = −rS < 0 – this proves the initial conjecture that there exists anequilibrium with deflationary expectations. Finally, replacing in variables yS , λ3S , λ2S , and λ1S , thesteady state with deflationary expectations is characterized bypiS = −rS , yS = −1− βPSκrS , iS = 0, τS = 0, `S = 0, bS = 0,λ1S = αCBpi rS , λ2S =(αCBy1− βPSκ+ αCBpi κ)rS , λ3S =(αCBy1− βPSκ+ αCBpi κ)rS166(and the PS rationally expects (piet , yet ) = (piS , yS) =(−rS ,−1−βPSκ rS)< (0, 0)).Corollary. The equilibrium payoffs are 0 when the economy reverts to its steady state at some t.Proof. The equilibrium payoffs when the economy reverts to its steady state at some date t arecalculated as follows. As in Walsh (2018), be ljk the value of losses in state k, with k = {Z,R, S}and j = {CB, Tr}. Then, given the zero steady state values of inflation, output gap, and taxes,the central bank’s and treasury’s per-period losses at the zero-inflation steady state becomelCBS ≡ lCBt(ySt , piSt)= 0, (A.42)andlTrS ≡ lTrt(ySt , τSt)= 0. (A.43)Proof of Proposition 2.3.1. Imposing Private sector’s rational expectations from states Z and R((yeZ , pieZ) = (piR, yR) and (yeR, pieR) = (ppiZ , pyZ)) on DIS and NKPC in the ZLB state and in theRecovery state yields the system of equationspiZ = κyZ + βpiR,yZ = yR − (iZ − yR − rZ) + γbZ ,piR = κyR + βppiZ ,yR = pyZ − (iR − ppiZ − rR) + γbR,(Note that here and in the next results we are already making use of the zero inflation equilibriumoutcomes under steady state.) Using the previous equations together with the monetary and fiscaloptimal policies in the ZLB state (Eqs. (2.27), (2.24), (2.25), (2.26)),iZ = 0, bZ =αTry γαTry γ2+βTrαTrτ (1+rR)2 (iZ − (yeZ + pieZ)− rZ), τZ = 0, `Z = bZ ,and in the Recovery state (Eqs. (2.22), (2.15), (2.17) (2.16)),iR = γbR + yeR +(αCBy +αCBpi κ2+αCBpi κβαCBy +αCBpi κ2)pieR + rR, bR = 0, τR = (1 + rR) bZ , `R = 0,I can build a system of equations with solution (denoted with asterisks)pi∗Z =βκαTrτ (αCBy +κ2αCBpi )(1+rR)2φ rZ , y∗Z =βαTrτ ((1−pβ2)αCBy +κ2αCBpi )(1+rR)2φ rZ , i∗Z = 0,b∗Z = −γαTry ((1−pβ2)αCBy +κ2αCBpi )φ rZ `∗Z = −γαTry ((1−pβ2)αCBy +κ2αCBpi )φ rZ τ∗Z = 0167pi∗R =βpκαCBy αTrτ β(1+rR)2φ rZ , y∗R = −β2pκ2αCBpi αTrτ (1+rR)2φ rZ ,i∗R =βp((1−pβ2+κ)αCBy +(1+κ+β)κ2αCBpi )αTrτ (1+rR)2φ rZ + rR,b∗R = 0 `∗R = 0τ∗R = −γαTry ((1−pβ2)αCBy +κ2αCBpi )(1+rR)φ rZand pi∗S = 0, y∗S = 0, i∗S = rS , b∗S = 0, `∗S = 0, and τ∗S = 0, with φ ≡ αCBy [(1 − pβ2)γ2αTry + (1 −pβ (β + κ))αTrτ β (1 + rR)2] + αCBpi κ2(γ2αTry + (1 + βp)αTrτ β (1 + rR)2).A.2 Scenario with forward guidanceProof of PropositionsThe proofs of propositions and notation of this section closely follow Kurozumi (2008) andNakata (2018) (with underlying logic based on Chari and Kehoe, 1990), except that here I adjustthese to incorporate another policymaker (the treasury). Part of the notation also uses Dongand Young (2019). In addition, I also follow Jung et al. (2005) to discuss uniqueness, and therevert-to-discretion strategy from Nakata (2018). Other references cited in text.Proof of Proposition 2.4.1. Call the strategy profile{σCB,∗, σTr,∗, σPS,∗}the solution under dis-cretion. This strategy profile can be built as follows. Recall the discretionary monetary policyresult,{i∗j}, the fiscal policy results,{b∗j , τ∗j , `∗j}, and the private sector results,{pi∗j , y∗j}, de-fined by Proposition (2.3.1) for each state j = {Z,R, S}. Be the strategy for the central bankσCB,∗ ={σCB,∗t(hCBt)}∞t=0. I define each element of strategy σCB,∗, σCB,∗t(hCBt), using the resultsfor monetary policy from Proposition (2.3.1) ∀t ≥ 0 and ∀ht asσCB,∗t(hCBt)= i∗k if rt = rk, with k = {Z,R, S} . (A.44)Similarly, for the treasury define σTr,∗ ={σTr,∗t(hTrt)}∞t=0, with each element of following fiscalpolicy results from Proposition (2.3.1) ∀t ≥ 0 and ∀ht asσTr,∗t(hTrt)= (b∗k, τ∗k , `∗k) if rt = rk, with k = {Z,R, S} . (A.45)Finally, be the strategy for the Private Sector σPS,∗ ={σPS∗t(hPSt)}∞t=0. Then, for each elementof strategy σPS,∗ it follows that, ∀t ≥ 0 and ∀ht asσPS,∗t(hPSt)= (pi∗k, y∗k) if rt = rk, with k = {Z,R, S} . (A.46)I seek to argue that the equilibrium outcome under discretion is a SE; i.e., that{σCB,∗, σTr,∗, σPS,∗}satisfies items 1-3 from the SE definition. Formally, I want to show that, for any current history, hktwith k = {CB, Tr, PS}, each continuation strategy σk,∗t for k = {CB, Tr, PS} is a best response tostrategy σ−k,∗. Let us show first that the central bank’s continuation strategy that can be formedwith strategy σCB,∗ with element specification in Eq. (A.44) satisfies the SE definition (item 2).168To show this, it is sufficient to apply the One-Shot Deviation Principle and rule out any profitabledeviations. For every t and history ht, I want to show that there is no welfare improving deviationcompared to the discretionary outcome; this means showing thatlCBt (yt (i∗t ) , pit (i∗t )) + βEtLCBt+1 ≥ lCBt (yt (it) , pit (it)) + βEtLCBt+1for every it such that it 6= i∗t . Due to the nature of the discretionary problem, we know that thecontinuation values βEtLCBt+1 at both sides of the inequality do not depend on the current choiceof it. Since the discretionary outcome of the left-hand side comes defined by the nominal interestrate that maximizes the stage payoff lCBt (·), i.e.,maxitlCBt (yt (it) , pit (it))+βEtLCBt+1 ≡ lCBt (yt (i∗t ) , pit (i∗t ))+βEtLCBt+1 ≥ lCBt (yt (it) , pit (it))+βEtLCBt+1then the inequality shown above holds. Therefore, σCB,∗t(hCBt)= i∗k if rt = rk with k = {Z,R, S},and since this holds for every element of the continuation strategy σCB,∗t , then the discretionaryspecification satisfies item 2 of the SE definition.For the treasury’s case, we want to show that its continuation strategy formed with strategyσTr,∗ with element specification in Eq. (A.45) satisfies the SE definition (item 3). The prooffollows a similar logic as that of the CB, with the only difference being that the treasury solvesproblem (2.13). Note that, since the central bank leads the treasury, then i∗t is such that itinternalizes the treasury’s response. As such, it rules out any incentives for the treasury to deviate.Therefore, after i∗t is chosen, the treasury must be better-off when it plays on the equilibrium path;i.e., σTr,∗t(hTrt)= (b∗R, τ∗R, `∗R) if rt = rk with k = {Z,R, S}. This holds for every element of thecontinuation strategy σTr,∗t , and it implies that the discretionary specification in Eq. (A.45) satisfiesthe SE definition (item 3).For the Private Sector, we need to show that its continuation strategy formed with strategyσPS,∗ with element specification in Eq. (A.46) satisfies the SE definition (item 1). We thereforewant to show that(σPS ,pi,t (rk) , σPS,yt (rk))yields (pi∗k, y∗k) for every k-state. Build the system ofequations using the DIS and NKPC at Z and R,σPS ,pit (rZ) = κσPS,yt (rZ) + βEt[σPS ,pit+1 (rZ)],σPS,yt (rZ) = Et[σPS ,yt+1 (rZ)]−(i∗Z − Et[σPS ,pit+1 (rZ)]− rZ)+ γb∗Z ,σPS ,pit (rR) = κσPS,yt (rR) + βEt[σPS ,pit+1 (rR)],σPS,yt (rR) = Et[σPS ,yt+1 (rR)]−(i∗R − Et[σPS ,pit+1 (rR)]− rR)+ γb∗R,and invoke rational expectations,Et[σPS ,pit+1 (rZ)]= σPS ,pit+1 (rR) , Et[σPS ,yt+1 (rZ)]= σPS ,yt+1 (rR) ,169Et[σPS ,pit+1 (rR)]= pσPS ,pit (rZ) , Et[σPS ,yt+1 (rR)]= pσPS ,yt (rZ) .This yieldsσPS ,pit (rZ) = κσPS,yt (rZ) + βσPS ,pit+1 (rR)σPS,yt (rZ) = σPS ,yt+1 (rR)−(i∗Z − σPS ,pit+1 (rR)− rZ)+ γb∗Z ,σPS ,pit (rR) = κσPS,yt (rR) + βpσPS ,pit (rZ) ,σPS,yt (rR) = pσPS ,yt (rZ)−(i∗R − pσPS ,pit (rZ)− rR)+ γb∗R.Solving further the system of equations and plugging in the expressions for (i∗Z , b∗Z , τ∗Z , `∗Z) and(i∗R, b∗R, τ∗R, `∗R) from Proposition (2.3.1) yieldsσPS ,pit (rZ) = θpiZrZ σPS,yt (rZ) = θyZrZ σPS ,pit (rR) = θpiRrZ σPS,yt (rR) = θyRrZwhich are the exact expressions for (pi∗Z , y∗Z) and (pi∗R, y∗R) from the same Proposition (and, therefore,from Eq. (A.46)).The solutions for the central bank and the treasury are unique. In addition, it is also easilyverified that the solution for the Private sector is unique. If we impose{σCB,∗, σTr,∗}, we wantto show for the PS that the solution yields the previous expressions σPS ,pit(hPSt)= θpik rk andσPS,yt(hPSt)= θykrk (for k = {Z,R}) in strategies σPS ,pis and σPS,ys for any s > t. Iterating forwardequations DIS and NKPC for any s > t and hPSs (and abstracting from arguments below), yieldsσPS,ys = −Es{ ∞∑k=0is+j − σPS,pis+1+j − rs+j + γbs+j}andσPS,pis = κEs{ ∞∑k=0βjσPS,ys+j}If the economy starts at Z,σPS,yZ = −Es{ ∞∑k=0is+j − σPS,pis+1+j − rs+j + γbs+j}=11− p[σPS,piR + rZ + γbZ − iR + pσPS,piZ + rR]σPS,piZ = κEs{ ∞∑k=0βjσPS,ys+j}=κ1− β2p(σPS,yZ + βσPS,yR)and if the economy starts at R,σPS,yR = −Es{ ∞∑k=0is+j − σPS,pis+1+j − rs+j + γbs+j}=11− p[p(σPS,piR + rZ + γbZ)−(iR − pσPS,piZ − rR)]170σPS,piR = κEs{ ∞∑k=0βjσPS,ys+j}=κ1− β2p(σPS,yR + βpσPS,yZ)and this yields the same system of equations outlined for the period-t strategies from before.Hence, plugging in the expressions for (i∗Z , b∗Z , τ∗Z , `∗Z) and (i∗R, b∗R, τ∗R, `∗R) from Proposition (2.3.1)yields the same exact expressions for (pi∗Z , y∗Z) and (pi∗R, y∗R) from Proposition (2.3.1). Therefore,σPS,yt(hPSt)= σPS,ys(hPSs) ∀s > t, and the solution is unique.We now want to prove the statement that σ∗ ={σCB,∗, σTr,∗, σPS,∗}is the worst SE. We dothis by contradiction as in Kurozumi (2008) but for our specific model. Assuming instead thatthere exists a σˆ ={σˆCB, σˆTr, σˆPS}worse than σ∗. If that is the case, then CB losses at any t,history hCBt , areV CB,∗(ht, σCB,∗, σTr,∗, σPS,∗)> Vˆ CB(ht, σˆCB, σˆTr, σˆPS)I.e.,−12Et{∑∞k=t βk−t[αCBy[σPS ,y,∗k(hk,(σCB,∗k(hCBk), σTr,∗k(hk, σCB,∗k(hCBk))))]2+αCBpi[σPS ,pi,∗k(hk,(σCB,∗k(hCBk), σTr,∗k(hk, σCB,∗k(hCBk))))]2]}>−12Et{∑∞k=t βk−t[αCBy[σˆPS ,yk(hk,(σˆCBk(hCBk), σˆTrk(hk, σˆCBk(hCBk))))]2+αCBpi[σˆPS ,pik(hk,(σˆCBk(hCBk), σˆTrk(hk, σˆCBk(hCBk))))]2]}(A.47)(where DIS, NKPC and ZLB and treasury’s policy function are skipped but these are satisfied inevery period).Recall that, if σˆ ={σˆCB, σˆTr, σˆPS}is the worst SE, it means that there is no profitabledeviation from it for every history ht – and, in particular, it means that σ∗ is not a deviation too.Since strategy σˆ is a very generic element, we use σ∗ to build how any candidate description of σˆ(but different from σ∗) may look. There are two groups formed by different candidate descriptionsfor σˆ, where σˆCB is the strategy that will allow deviations. Also note that all strategies below forσˆ will be defined in terms of σˆCB only, leaving σˆTr and σˆPS unspecified. The first group has onestrategy description for σˆ given byσˆCBt(hCBt)= σCB,∗t(hCBt)(A.48)for every t, given history ht and every future history induced by σˆ. The second group (with infinitestrategy descriptions except from the strategy described in the first group) is defined by some t˜, ht˜171and future history induced by σˆ such thatσˆCBt(hCBt)= σCB,∗t(hCBt), 0 ≤ t ≤ t˜− 16= σCB,∗t(hCBt), t = t˜free , t > t˜(A.49)If we can show that inequality from Eq. (A.47) does not hold for any σˆ prescribed in the first group(A.48) and the second group (A.49) of candidate descriptions, then we will have proven that σ∗ isthe worst SE by contradiction.I shall start with the second group first. Specifically, I want to show that there is a profitabledeviation from strategy profiles σˆ that follow Eq. (A.49). I focus on a deviation of the Central bankstrategy to σCB,dev – yet to be determined. Since σˆ is a Sustainable Equilibrium by assumption,we know that it must be such that ∀t, ht,Vˆ CB(ht, σˆCB, σˆTr, σˆPS) ≥ Vˆ CB (ht, σCB,dev, σˆTr, σˆPS) , (A.50)with Vˆ CB(ht, σCB,dev, σˆTr, σˆPS)given by−12Et{∑∞k=t βk−t[αCBy[σˆPS ,yk(hk,(σCB,devk(hCBk), σˆTrk(hk, σˆCBk(hCBk))))]2+αCBpi[σˆPS ,pik(hk,(σCB,devk(hCBk), σˆTrk(hk, σˆCBk(hCBk))))]2]}(where DIS, NKPC and ZLB and treasury’s policy function are satisfied in every period). Con-sider now a deviation from Eq. (A.49) to σCB,devt(hCBt)= σCB,∗t(hCBt). But under this specificdeviation, we know thatσˆTrk(hk,σˆCBk (·))= σˆTrk(hk,σCB,devk (·))= σˆTrk(hk,σCB,∗k (·))= σTr,∗k(hk,σCB,∗k (·)).Likewise, PS will observe both actions, and its unique best response is σPS,∗. I.e., formally we wouldobtain that σˆPSk(hk,σˆCBk (·) , σˆTrk(hk,σˆCBk (·)))is given by σPS,∗k(hk,σCB,∗k (·) , σTr,∗k(hk,σCB,∗k (·))).After plugging in the RHS of Eq. (A.50) the result for the PS inside the history observed by thetreasury, then taking this new object inside the history of the central bank yields the same presentdiscounted value for the central bank without forward guidance, V CB,∗(ht, σCB,∗, σTr,∗, σPS,∗).Finally, taking this into Eq. (A.50) yieldsVˆ CB(ht, σˆCB, σˆTr, σˆPS) ≥ Vˆ CB (ht, σCB,dev, σˆTr, σˆPS) = V CB,∗ (ht, σCB,∗, σTr,∗, σPS,∗) .But linking this chain of inequalities in our initial inequality (Eq. (A.47)), rendersV CB,∗(ht, σCB,∗, σTr,∗, σPS,∗)> V CB,∗(ht, σCB,∗, σTr,∗, σPS,∗)which is a contradiction. Hence, any candidate strategy σ̂ described in expression (A.49) cannot172do worse than σ∗. Following the same procedure with expression (A.48), it is easy to see that weobtain the same inequality as the one above – this is immediate because the candidate strategyσ̂ is exactly equal to σ∗. Therefore, there is no alternative strategy profile σˆ ={σˆCB, σˆTr, σˆPS}that does worse than the discretionary equilibrium strategy profile, σ∗ ={σCB,∗, σTr,∗, σPS,∗}and,hence, strategy profile σ∗ is the worst SE.Proof of Proposition 2.4.2. (⇒) We want to show that if σS = {σCB,S , σTr,S , σPS,S} is a SE withoutcome(iSt , bSt , ySt , piSt), then it satisfies items 1,2,3 of the Proposition 2.4.2.By assumption, if σS ={σCB,S , σTr,S , σPS,S}is a SE, then as per the definition of SE thecontinuation strategy of the private sector, σPS,St , satisfies rational expectations and the NKPCand the DIS for every current history hPSt . Hence, the outcome(iSt , bSt , ySt , piSt)satisfies a PSCE(and, therefore, item 1 of the proposition is satisfied). Following a similar logic, we can prove item2 too. To prove item 3, we follow Chari et al. (1998) and Kurozumi (2008), and assume, on thecontrary, that there exists a t˜ such that the Sustainability Constraint does not hold; i.e.,V CB,St˜(ht˜, σCB,S , σTr,S , σPS,S)< V CB,∗t˜(ht˜, σCB,∗, σTr,∗, σPS,∗)(A.51)with V CB,∗t˜(ht˜, σCB,∗, σTr,∗, σPS,∗)given by− 12Et{ ∞∑k=tβk−t[αCBy[σPS,∗k(hk,(σCB,∗k(hCBk), σTr,∗k(hk, σCB,∗k(hCBk))))]2+αCBpi[σPS,∗k(hk,(σCB,∗k(hCBk), σTr,∗k(hk, σCB,∗k(hCBk))))]2]}and V CB,St˜(ht˜, σCB,S , σTr,S , σPS,S)defined by− 12Et{ ∞∑k=tβk−t[αCBy[σPS,y,Sk(hk,(σCB,Sk(hCBk), σTr,Sk(hk, σCB,Sk(hCBk))))]2+αCBpi[σPS,pi,Sk(hk,(σCB,Sk(hCBk), σTr,Sk(hk, σCB,∗k(hCBk))))]2]}By definition of SE, since σS ={σCB,S , σTr,S , σPS,S}is a SE, then it has to satisfy that σCB,Srenders a payoff weakly better than any deviation to σCB,dev ∀t, ht – provided that the otherplayers are following the SE prescription,{σTr,S , σPS,S}. Formally, for t ≥ t˜, this means that forany deviation to σCB,dev,V CB,St˜(ht˜, σCB,S , σTr,S , σPS,S) ≥ V CB,St˜(ht˜, σCB,dev, σTr,S , σPS,S)(A.52)Consider the particular case of a deviation σCB,devt˜= σCB,∗t˜– for which the value of V is known.From the proof of Proposition 1, we know that the best response from the treasury to that devi-ation instructs the treasury to do σTr,St(ht,σCB,devt (·))= σTr,∗t(ht,σCB,∗t (·)), and for the Private173sector, σPS,St(hCBt , σCB,devt (·) , σTr,St(ht,σCB,devt (·)))= σPS,∗t(ht,σCB,∗t (·) , σTr,∗t(hk,σCB,∗t (·))).Plugging these in the RHS of Eq. (A.52) yields V CB,∗t˜(ht˜, σCB,∗, σTr,∗, σPS,∗). Taking this resultto Eq. (A.52) it means thatV CB,St˜(ht˜, σCB,S , σTr,S , σPS,S) ≥ V CB,St˜(ht˜, σCB,dev, σTr,S , σPS,S)= V CB,∗t˜(ht˜, σCB,∗, σTr,∗, σPS,∗)But plugging back this chain of inequalities in our initial statement again (Eq. (A.51)), rendersV CB,∗t˜(ht˜, σCB,∗, σTr,∗, σPS,∗)< V CB,∗t˜(ht˜, σCB,∗, σTr,∗, σPS,∗)(a contradiction). Hence, it can only be that Eq. (A.51) holds asV CB,St˜(ht˜, σCB,S , σTr,S , σPS,S) ≥ V CB,∗t˜(ht˜, σCB,∗, σTr,∗, σPS,∗)which proves item 3 of the proposition (the Sustainability constraint).(⇐) If (iSt , bSt , τSt , `St , ySt , piSt ) satisfy items 1,2,3 from Proposition 2.4.2, then (iSt , bSt , τSt , `St , ySt , piSt )is an outcome of a SE. We prove this by construction. Specifically, following Kurozumi (2008), we re-call the grim trigger strategy σS ={σCB,S , σTr,S , σPS,S}that supports outcomes(iSt , bSt , τSt , `St , ySt , piSt)like the one shown in text – see also Basso (2009) for a similar proof.Let us analyze first what happens if the central bank did not deviate. Since item 1 of theproposition is satisfied by(iSt , bSt , τSt , `St , ySt , piSt), then it is a PSCE, and this satisfies the PrivateSector part of the SE definition. The same applies for item 2 of the proposition. Finally, since thecentral bank did not deviate, we know that(iSt , bSt , τSt , `St , ySt , piSt)is realized and it satisfies item3 of the proposition (the sustainability constraint). So, the Central bank must be maximizing itspayoff, which matches the central bank part of the SE definition.Now let us analyze what happens if the central bank deviated at some point in the history ofpast or current events. If there existed a deviation, then the trigger strategy instructs (y∗t , pi∗t ) tobe chosen by the PS, and the CB and the treasury’s best responses are i∗t and (b∗t , τ∗t , `∗t ), whichis a PSCE. For the treasury, upon central bank’s deviation, b∗t will be chosen, which also satisfiestreasury’s policy (the PS’s and the treasury’s best responses are (y∗t , pi∗t ) and b∗t ). Finally, since theCB deviated, then it has to set i∗t , and since (i∗t , b∗t , τ∗t , `∗t , y∗t , pi∗t ) is realized, the CB attains V CB,∗.Since V CB,∗ is a value such that it emerges when all central bank, treasury and PS optimize (absentforward guidance), it is an equilibrium. Furthermore, since before we showed that the discretionaryequilibrium is a SE, then we also prove that this grim trigger strategy yields outcomes that belongto a SE. So, we proved by construction that this strategy sustains an outcome(iSt , bSt , τSt , `St , ySt , piSt)that is a SE.Lemma. Under discretion, and given {lCBZ (y∗Z , pi∗Z) , lCBR (y∗R, pi∗R) , lCBS (y∗S , pi∗S)} (the central bank’stime-invariant losses), the present discounted value of losses in periods where the state of the world174is Z, V CB,∗Z , is determined byV CB,∗Z =11− β2p(lCBZ (y∗Z , pi∗Z) + βlCBR (y∗R, pi∗R))+1− p1− β2pβ21− β lCBS (y∗S , pi∗S) .Proof. The proof consists on writing the recursive structure of the value function for the centralbank standing at state Z. Considering the first periods and grouping by states (these steps areskipped), the recursion yieldsV CB,∗Z =11− β2p(lCBZ (y∗Z , pi∗Z) + βlCBR (y∗R, pi∗R))+1− p1− β2pβ21− β lCBS (y∗S , pi∗S) .Corollary. The present discounted value of losses for the central bank under no forward guidancein state R isV CB,∗R ≡lCBR (y∗R, pi∗R) + βplCBZ (y∗Z , pi∗Z)1− β2pProof. To start at R, subtract lCBZ (y∗Z , pi∗Z) and divide by β expression VCB,∗Z from the previouslemma to stand from the central bank’s perspective at R (the result follows immediately).Corollary. The discounted value of the CB’s losses in the Recovery state can be reexpressed asV CB,∗R ≡ lCBR (y∗R, pi∗R) + βplCBZ (y∗Z , pi∗Z) + βlCBR (y∗R, pi∗R)1− β2pProof. The result follows from algebraic manipulation of the previous corollary’s expression.Sustainability constraint (SC)Simplified SC (proof). Algebraic manipulation of the sustainability constraint from the main textyieldsβp[LcbZ (y˜Z , ˜Z)− LcbZ (y∗Z , pi∗Z)] ≥ LcbR (y∗R, pi∗R)− LcbR (y˜R, ˜R)Corollary. If SC is satisfied with strict inequality, there exist multiple equilibria.Proof. For simplicity, consider an alternative version of the sustainability constraint (from previousproof),LcbR (˜iR) + βpLcbZ (˜iR) ≥ LcbR (i∗R) + βpLcbZ (i∗R)where arguments are functions of state-R nominal interest rates. Rearranging we obtainH (˜iR) ≡ LcbR (˜iR) + βpLcbZ (˜iR)− LcbR (i∗R)− βpLcbZ (i∗R) ≥ 0175Due to concavity and continuity of H (˜iR), if H > 0 at some i˜R, then there exists an i˜′R in theneighbourhood of i˜R such that H > 0.Proof (Asymptote of i∗R). From Proposition 2.3.1, recall that i∗R = θiRrZ + rR. Then,θiR ≡ (αCBy ((1−(βPS)2p) + κ) + αCBpi κ2(1 + κ+ βPS))pυ/φwhereφ ≡ αCBy ((1− p(βPS)2)γ2αTry + (1− pβPS(βPS + κ))υ) + αCBpi κ2(γ2αTry + (1 + βPSp)υ)and υ ≡ αTrτ βTr(1 + rR)2. Note that φ can be rewritten asφ ≡[αCBy(1− p (βPS)2)+ αCBpi κ2] γ2αTry +Hwith H terms independent from γ. Then, limγ→∞ φ→∞, and variable i∗R = θiRrZ + rR → rR.176Appendix BAppendix to Chapter 3The exposition and proofs of the appendix of this chapter closely follow Cole and Kehoe (2000).Other references cited in text.B.1 Short-term debtB.1.1 Determination of the equilibrium priceWhen B′ lies in the Crisis region, creditors expect the government to default with probability pi.Thus, the equilibrium price equation (3.3) leads toq(s,B′)= β [pi (0) + (1− pi)]⇒ q (s,B′) = β (1− pi)If B′ lies in the Default region, the international credit market expects the government to defaultwith probability 1, and therefore q (s,B′) = 0. In contrast, if B′ lies in the No-Default region, thegovernment never defaults and q (s,B′) = β.B.1.2 Determination of the lower bound of the Crisis regionTo show that there exists a unique B = B solution to equation (3.10) at equality, the next resultfollows.Lemma. There exists a unique B that is solution tou (Zy)1− β = u (y −B) + β[u (y)1− β].Proof. Using the No-Lending Condition, letF (B) =u (Zy)1− β − u (y −B)− βu (y)1− β = 0.Since (i) F (0) = [u (Zy)− u (y)] / (1− β) < 0, (ii) F (B → y) = u (Zy) / (1− β)−u (0)−βu (y) / (1− β)→∞, and (iii) dF (B) /dB = u′ (y −B) > 0, then there exists a unique B > 0 solution to F (B) =0.177B.1.3 Determination of the upper bound of the Crisis regionOptimal policy in the No-Default region under short-term debtThe claim below solves the government maximization problem under short-term debt. In particular,it shows that the optimal paths for government spending {gt} and bonds {Bt+1} are constant whenthere is no risk of default (i.e., pi = 0). maximization problem under short-term debt. In particular,it shows that the optimal paths for government spending {gt} and bonds {Bt+1} are constant whenthere is no risk of default (i.e., pi = 0).Claim (Short-term debt). If pi = 0 and given an initial debt Bt′ = B ≤ B, then governmentspending is constant and equal tog = y −B (1− β) ,and government debt Bs+1 equals the initial debt Bt′ for s ≥ t′.Proof. Starting at t′ and for any initial Bt′ = B, a general version of the government problem(equation (3.11) in the main text) can be written asmax{Bs+1}∞s=t′{gs}∞s=t′∞∑s=t′βsu (gs) (B.1)s.t. gs +Bs = y + qsBs+1 ∀s ≥ t′Bt′ = B given, and B ≤ BBs ≤ B ∀s ≥ t′.When pi = 0, the price q of a riskless bond equals β and the government budget constraint isgs +Bs = y + βBs+1 ∀s ≥ t′.The optimal paths {gs}∞s=t′ , {Bs+1}∞s=t′ of the government maximization problem (B.1) solveL =∞∑s=t′{βsu (gs)− λs [gs + y + βBs+1 −Bs]}The first-order and the transversality conditions are{gs} : βsu′ (gs)− λs = 0 ⇒ βsu′ (gs) = λs{Bs+1} : −βλs − λs+1 (−1) = 0 ⇒ βλs = λs+1lims→∞βsλsBs+1 = 0.From the first-order conditions,u′ (gs) = u′ (gs+1) ,178and this implies a constant government spending at gs = gs+1 = g.We now show that Bs equals the initial Bt′ = B for every s ≥ t′. Plugging g in two consecutivebudget constraints, the difference yieldsBs+2 −Bs+1 = 1β(Bs+1 −Bs) .Let ∆s+1 = Bs+1−Bs and notice that β−1 > 1. When ∆s+1 < 0, the optimal path of debt divergesto negative infinity. In this case, the government can always do better by increasing gt but thisviolates the sign of the transversality condition. When ∆s+1 > 0, government debt diverges topositive infinity. But this now violates constraint Bs ≤ B since the optimal path {Bs+1}∞s=t′ willhit the lower bound of the Crisis region for some s. Moreover, this violates the assumption thatlenders have a finite wealth. Thus, the only ∆s+1 consistent with the transversality condition andwith optimality of government spending is ∆s+1 = 0, namely, Bs+1 = Bs. Since this holds for everys ≥ t′, the optimal path {Bs+1}∞s=t′ is constant and equal to Bt′ = B, also satisfying constraintBs ≤ B for every s ≥ t′.Finally, optimal government spending in terms of the initial debt level Bt′ = B yieldsg +B = y + βB ⇒ g = y −B (1− β)General problem when debt is lowered in T periodsBe g¯ = y − (1− β)B. When T = {1, 2, 3}, the government’s objective functions areu (g0) + βu (g¯)1− βu (g0) + βˆu (g1) + βˆβu (g¯)1− β + βpiu (Zy)1− βu (g0) + βˆu (g1) + βˆ2u (g2) + βˆ2βu (g¯)1− β + βpiu (Zy)1− β(1 + βˆ)respectively. Therefore, the objective function of the government over T periods isT−1∑k=0βˆku (gk) + βˆT−1βu (g¯)1− β + βpiu (Zy)1− βT−2∑k=0βˆk,or, alternatively,T−1∑k=0βˆku (gk) + βˆT−1βu (g¯)1− β +[1− βˆT−11− βˆ]βpiu (Zy)1− β .179Solution of the general problem for any T ≥ 2The general problem isV T (B0) = max{Bt+1}T−1t=0 ,{gt}T−1t=0u (g0)+βˆu (g1)+βˆ2u (g2)+...+βˆT−1u (gT−1)+[1− βˆT−11− βˆ]βpiu (Zy)1− β +βˆT−1βu (g¯)1− βs.t. gt = y −Bt + qtBt+1 ∀t = 0, 1, .., T − 2gt = y −Bt + qtB t = T − 1qt = βE [zt+1] , zt = 1B0, B given; g¯ = y − (1− β)B.If B0 lies in the Crisis region, then qTt = βˆ. After plugging the constraints, the first-orderconditions for t yield{Bt} : βˆt−1u′ (gt−1) βˆ + βˆtu (gt) (−1) = 0⇒ u′ (gt−1) = u′ (gt)⇒ gt−1 = gtFrom the first-order condition, gt = gt+1 for every t = 0, 1, ..., T − 1. Furthermore, since g0depends on the initial debt level B0, then we label constant government spending as gt = gT (B0)for every t = 0, 1, ..., T − 1.Expression for gT (B0)To derive an expression for gT (B0), we first replace government spending by gT (B0) in the T budgetconstraints of the previous problem. Then, we multiply each expression by βˆt with t = 0, 1, ..., T−1,to obtainβˆ0gT (B0) + βˆ0B0 = βˆ0y + βˆ0βˆB1βˆ1gT (B0) + βˆ1B1 = βˆ1y + βˆ1βˆB1...βˆT−2gT (B0) + βˆT−2BT−2 = βˆT−2y + βˆT−2βˆBT−1βˆT−1gT (B0) + βˆT−1BT−1 = βˆT−1y + βˆT−1βB.Adding up these terms leads togT (B0)T−1∑k=0βˆk +B0 = yT−1∑k=0βˆk + βˆT−1βB,180and solving for gT (B0) rendersgT (B0) = y −[1− βˆ1− βˆT] [B0 − βˆT−1βB].The upper bound of the Crisis region when pi = 0Before characterizing the existence and uniqueness of the upper bound B¯ of the Crisis region,we introduce the next result that holds when pi = 0. When the probability of defaults is null, thegovernment should always choose to roll-over its current level of debt instead of adopting any policyof lowering debt in T periods.Lemma. Suppose that pi → 0 and consider V∞ (B0)− V T (B0). Then,V∞ (B0)− V T (B0) > 0for every T <∞.Proof. When pi → 0, the value function at T = 1 is defined asV 1 (B0) = u (y −B0 + βB) + βu (y − (1− β)B)1− β ,while for T ≥ 2,V T (B0) =[1− βT1− β]u(y −[1− β1− βT]B0 +[1− β1− βT]βTB)+ βTu (y − (1− β)B)1− β .Computing V∞ (B0)− V 1 (B0) renders11− β [u (y − (1− β)B0)− (1− β)u (y −B0 + βB)− βu (y − (1− β)B)] ,and since(1− β) (y −B0 + βB) + β (y − (1− β)B) = y − (1− β)B0,then by concavity of u (·) and Jensen’s inequality, u (y − (1− β)B0) − (1− β)u (y −B0 + βB) −βu (y − (1− β)B) > 0. Thus, V∞ (B0)− V 1 (B0) > 0.With a similar argument for V∞ (B0)− V T (B0) when T ≥ 2,V∞ (B0)− V T (B0) = 11− β[u (y − (1− β)B0)−(1− βT )u (gT (B0))− βTu (y − (1− β)B)] ,and since(1− βT ) [y − [ 1− β1− βT]B0 +[1− β1− βT]βTB]+ βT [y − (1− β)B] = y − (1− β)B0,181then, V∞ (B0)− V T (B0) > 0 – using concavity of u (·) and Jensen’s inequality.Having proved that V∞ > V T for pi > 0 and T <∞, we replace the left-hand side of Eq. (3.15)by V∞ (B) and proceed to characterize B¯ (0) (which for simplicity we call B¯).Lemma. There exists a unique B¯ that solvesu(y − (1− β) B¯)1− β = u(Zy + βB¯)+ βu (Zy)1− βProof. Using the Participation Constraint with V∞ (B) in the right-hand-side, we can define thefollowing functionH (B;β, y, Z) =u (y − (1− β)B)1− β − u (Zy + βB)− βu (Zy)1− β = 0Given that (i) H (0; ·) = [u (y)− u (Zy)] / (1− β) > 0, (ii) H (B → y/ (1− β) ; ·) = u (0) / (1− β)−u (Zy + βy/ (1− β)) − βu (Zy) / (1− β) → −∞, and (iii) ∂H (B; ·) /∂B = −u′ (y − (1− β)B) −u′ (Zy + βB)β < 0, then there exists a unique B¯ > 0 such that the Participation Constraint issatisfied with equality. Moreover, for any B < B¯ (0), then V∞ (B) > V D (B), and viceversa.Derivation of Figure 3.2Following a logic similar to Cole and Kehoe (2000), we describe the behavior of equations (3.10)and (3.15). Rewriting firstUpi=0PC (B) ≡u (y − (1− β)B)1− β − u (Zy + βB)− βu (Zy)1− βUNLC (B) ≡ u (y −B) + β[u (y)1− β]− u (Zy)1− β ,then it is immediate to observe that they decrease monotonically asB increases, since (i) Upi=0PC (B = 0) =UNLC (B = 0) = [u (y)− u (Zy)] / (1− β) > 0, (ii) limB→y/(1−β) Upi=0PC (B) = u (0) / (1− β) −u (Zy + yβ/ (1− β))− βu (Zy) / (1− β)→ −∞ and limB→y UNLC (B) = u (0) + u (y)β/ (1− β)−u (Zy) / (1− β) → −∞, and (iii) dUpi=0PC /dB = −u′ (y − (1− β)B) − u (Zy + βB)β < 0 anddUNLC/dB = −u′ (y −B) < 0.The upper bound of the Crisis region when pi > 0Lemma. For some pi > 0, there exists a unique B¯ that solvesu(y −(1− βˆ)B0)1− βˆ +βpiu (Zy)(1− β)(1− βˆ) = u(Zy + βˆB¯)+ βu (Zy)1− β .182Proof. As performed in the lemma of Appendix section B.1, we use equation (3.15) to defineH˜ (B;β, y, Z, pi) =u(y − (1− βˆ)B0)1− βˆ +βpiu (Zy)(1− β) (1− βˆ) − u(Zy + βˆB¯)− βu (Zy)1− β = 0.Given that (i) H˜ (0; ·) = u (y) /(1 − βˆ) − u (Zy) / (1− β) + βpiu (Zy) /[(1− β) (1 − βˆ)] > 0,67(ii) H˜ (B → y/ (1− β) ; ·) = u (0) / (1− β) + βpiu (Zy) /[(1− β) (1 − βˆ)] − u(Zy + βˆy/(1 − βˆ)) −βu (Zy) / (1− β)→ −∞, and (iii) dH˜ (B; ·) /dB = −u′(y − (1− βˆ)B)− u′(Zy + βBˆ)βˆ < 0, then apositive B¯ (·) satisfies equation (3.15) and is unique.Derivation of Figure 3.3Since function UNLC (B) is independent from pi, the same result in Section A.3.6 applies. Therefore,definingUpi>0PC (B) ≡u(y − (1− βˆ)B)1− βˆ +βpiu (Zy)(1− β) (1− βˆ) − u(Zy + βˆB)− βu (Zy)1− β ,it is easy to observe that Upi>0PC (B) monotonically decreases as B increases, since (i) Upi>0PC (B = 0) =[u (y)− u (Zy)] /(1−βˆ)+βpiu (Zy) / (1− β) (1−βˆ) > 0, (ii) limB→y/(1−β) Upi>0PC (B) = u (0) /(1−βˆ)+βpiu (Zy) / (1− β) (1−βˆ)−u(Zy+yβˆ/(1−β))−βu (Zy) / (1− β)→ −∞, and (iii) dUpi>0PC (B) /dB =−u(y − (1− βˆ)B)− u(Zy + βˆB)βˆ < 0.Lemma. When pi increases, the stationary upper bound B¯ (pi) decreases.Proof. By the previous lemma, there exists a unique B¯ (pi) that is solution to equation (3.15) atequality. Rewriting that equation asH =u(y −(1− βˆ)B)1− βˆ +pi(1− βˆ) βu (Zy)(1− β) − u(Zy +Bβˆ)− βu (Zy)1− β = 0,we can apply the implicit function theorem to get ∂B (pi) /∂pi = − (∂H/∂pi) / (∂H/∂B). Expression∂H/∂B and ∂H/∂pi yield∂H∂B= −[u′(y −(1− βˆ)B)+ u′(Zy +Bβˆ)βˆ]< 0,∂H∂pi= −βBu′(y −(1− βˆ)B)1− βˆ − u′(Zy +Bβˆ)+ β(1− βˆ)2 [u(y − (1− βˆ)B)− u (Zy)] .The term in brackets in the last expression is positive. Also, the No-Lending Condition impliesthat y − B < Zy. Adding Bβˆ to the last inequality and invoking concavity of u (·) renders67Since u (y) > u (Zy) and limpi→0 H˜ (0;β, y, Z, pi) > 0, then by continuity of pi in [0, 1], there is some strictlypositive pi such that H˜ (0;β, y, Z, pi) is still greater than 0.183u′(y −(1− βˆ)B)> u′(Zy +Bβˆ). Therefore, the first term is positive and ∂H/∂pi < 0, thusimplying that ∂B (pi) /∂pi < 0.We now introduce the next lemma which will be useful for proving Proposition 3.3.1.Lemma. Suppose that the government wants to run down debt in T periods (i.e., have BT = B).Denote the government spending at time t for t ≤ T − 1 along this path as gTt . Then,1. optimal gTt is constant for all t ∈ {0, ..., T − 1}, and2. gTt < g¯, for all t ∈ {0, ..., T − 1}.Proof. Throughout the proof, we assume that B0 starts in the Crisis region (B0 > B), and thatthe government optimally decides to lower debt in T periods. The government problem becomesV T (B0) = max{Bt+1}T−1t=0T−1∑t=0βˆtu(y −Bt + βˆBt+1)+ βˆT−1u (y −BT−1 + βB)+[1− βˆT−11− βˆ]βpiu (Zy)1− β + βˆT−1βu (y − (1− β)B)1− β .To prove part 1 of the Lemma, observe that optimality conditions for Bt+1 in t = 0, ..., T − 1yieldu′(y −B0 + βˆB1)= u′(y −B1 + βˆB2)⇒ g0 = g1......u′(y −BT−2 + βˆBT−1)= u′ (y −BT−1 + βB) ⇒ gT−2 = gT−1.Therefore, gt−1 = gt for every t = 1, ..., T − 1. In other words, the government smooths spendingwhile it decreases debt to reach at the No-Default region.To prove part 2, recall that if the government exits the Crisis region in T periods, then it mustbe that Bt > B for every t ≤ T − 1. If this was not the case, then the government would havealready exited the Crisis region at an earlier T˜ < T , thus contradicting the assumption that runningdebt in T periods was optimal. Hence, at t = T − 1, BT−1 > B implies gT−1 < g¯. But, since thegovernment smooths spending for T periods, gt = gT−1 impliesgt < g¯ ∀t = 0, 1, ..., T − 1.Therefore, gt is smaller than the optimal level of spending that the government attains at theNo-Default region, g¯.We now turn to the proof of Proposition 3.3.1, that follows a similar logic to the result establishedby Cole and Kehoe (2000).Proof of Proposition 3.3.1. The idea of the proof is to show that for some pi > 0 there is an initialrange of debt levels in the Crisis region[B, B¯ (pi)]where the government strictly prefers to run184down debt in one period. As B0 increases, we then show that T∗ (B0) monotonically increases;namely, that the government delays exiting the Crisis region as its outstanding obligations rise.Therefore, for a higher B0 in[B, B¯ (pi)], this result implies that the government optimally lowersinitial debt following policies T ∗ (B0) ∈ {1, 2, ...,∞}. Throughout the proof, we assume that pi > 0and that the government starts in the Crisis region (B0 > B). We also assume that the governmentchooses to run down debt in T periods (BT = B).68To prove that T ∗ (B0) = 1 strictly dominates any other plan T > T ∗ (B) for some interval in[B, B¯ (pi)], we use the value function when T = 1,V 1 (B0) = u (y −B0 + βB) + β u (g¯)1− β .The value function when T > 1 (evaluated at the optimal path {Bt+1} isV T (B0) =(1 + βˆ + ...+ βˆT−1)u(gT (B0))+ βˆT−1βu (g¯)1− β + [Value if default occurs] ,withgT (B0) = y − 1− βˆ1− βˆT B0 +1− βˆ1− βˆT βˆT−1βB.Starting at t = 0 and if T = 1, the government attains g¯ when t = 1. For any other policyT > 1, however, government spending at t = 1 is still at gT (B0). Using our previous result thatgT (B0) < g¯, this means that the continuation value of V1 (B0) is greater than VT (B0) for anyT > 1. If we then compare government spendings at t = 0, we know that g0 under policy T = 1will be strictly greater than any gT (B0) ify −B0 + βB > y − 1− βˆ1− βˆT B0 +1− βˆ1− βˆT βˆT−1βB,from where it follows thatB0 <B1− pi .Therefore, if B0 ∈ (B,B/ (1− pi)), V 1 (·) dominates any other value function V T (·), for T > 1.It is easy to show that V 1 (B0) > V2 (B0) > ... > V∞ (B0) when B0 = B/ (1− pi). First, notethatV 1 (B0) = u(y −B0 + βB)︸ ︷︷ ︸g1+ βu (y − (1− β)B)1− β ,V 2 (B0) =(1 + βˆ)u(y − B01 + βˆ+11 + βˆβˆβB︸ ︷︷ ︸)g2+ βpiu (Zy)1− β + βˆβu (y − (1− β)B)1− β .68Here, we skip some intermediate results from Cole and Kehoe (2000). In their proof, the authors analyticallyshow that (i) T ∗ (B0) increases by one period as B0 increases, and (ii) there cannot be sudden jumps from a finiteT ∗ (B0) to T ∗ (B0)→∞.185By comparing g1 (B0) and g2 (B0) at B0 = B/ (1− pi), we obtain that government spending in thefirst period are the same,g1 (B0) = y −B[1− βˆ1− pi]= g2 (B0) .Using the result from the previous lemma, it is easy to observe that g1 (B0) and g2 (B0) are strictlybelow g¯ for any pi > 0,g1 (B0) = g2 (B0) = y −B[1− βˆ1− pi]< y − (1− β)B = g¯,But, from period t = 1 onward, V 1 (B0) outperforms V2 (B0) since V1 (B0) reaches g¯ one periodbefore V 2 (B0) – the second part of the previous lemma guarantees that g2 (B0) < g¯, thus makingV 1 (B0) to strictly dominate V2 (B0). By the same token, we can extend this logic to argue thatV 1 (B0) dominates VT (B0) for any T > 1. Hence, T∗ (B0) = 1 is preferred to any other policy Twhen B0 = B/ (1− pi).To prove that every payoff V T (B0) is ranked in decreasing order at B0 = B/ (1− pi), takeV T (B0) and VT+1 (B0), for T > 1, and note that gT (B0) equalsgT (B0) = y −B[1− βˆ1− pi].Since gT = gT+1 in periods 0, 1, ..., T , value V T (B0) reaches g¯ before any VT+1 (B0), i.e., VT (B0) >V T+1 (B0). Thus, the government strictly prefers to run down debt in one period when B0 ∈[B,B/ (1− pi)] ⊂ [B, B¯].We now show that schedules T ∗ (B0) > 1 can also be optimal for debt levels B0 > B/ (1− pi)in the Crisis region. Specifically, as B0 increases, we need to prove that (i) every value functionV T (B0) strictly decreases, (ii) value functions VT (B0) with higher T decrease by less, and (iii)for small pi > 0, there exists a full range of possibilities T ∗ (B0) such that a higher T increases thegovernment payoff V T (B0), i.e. ∂VT (B0) /∂T > 0. As a result, (i) - (iii) imply that the optimalT ∗ (B0) rises as B0 increases in[B/ (1− pi) , B¯]. The idea behind this part of the proof is thata higher B0 increases the government’s debt burden when it has to run down debt. Therefore,an increase in B0 should intuitively make delay (namely, a higher T ) preferable. In particular,discounting makes the government prefer to divide the decrease of initial debt B0 into more periodsin order to avoid higher initial sacrifices of gt.To show (i), note thatddB0V T (B0) = −u′(gT)< 0 andddB0V∞ (B0) = −u′ (g∞) < 0implies dV T (B0) /dB0 < 0 for any T = 1, 2, ...,∞.186To show (ii), compare first any dV T (B0) /dB0 and dV∞ (B0) /dB0, and propose thatddB0V T (B0) <ddB0V∞ (B0)⇒ u′(gT)> u′ (g¯) .It can be easily shown that gT < g¯ when B0 > B/ (1− pi). With a similar logic, dV T−1 (B0) /dB0 <dV T (B0) /dB0, since u′ (gT−1) > u′ (gT ) when B0 > B/ (1− pi). Therefore,0 >ddB0V∞ (B0) > ... >ddB0V 2 (B0) >ddB0V 1 (B0) . (B.2)Intuitively, since increasing B0 worsens any scenario for the government, delaying more is preferablein order to sacrifice less spending in the initial periods (and gain more because of discounting).To prove (iii), we follow Cole and Kehoe (2000) and assume that T is continuous. Thus, we cantake the derivative of V T (B0) with respect to T and obtain,∂V T (B0)∂T= βˆT−1 ln βˆ×{ugT(gT)1− βˆT[βB − βˆB0]+ βu (g¯)1− β −β1− βˆ[(1− pi)u (gT (B0))+ (pi) u (Zy)1− β]}.(B.3)When B0 ∈ (B/ (1− pi) , B¯], the first term (multiplied by ln βˆ) is positive. This term shows thegovernment gain from remaining in the Crisis region in every period. The price in the Crisis regionβˆ, is always lower than the price of the No-Default region, β. However, for some large levels B0in the Crisis region, the government gain βˆB0 is greater than its gain βB when it arrives to theNo-Default region. As a result, this provides an incentive to delay exit. In contrast, the last terms(multiplied by ln βˆ) are the cost of staying in the Crisis region. Intuitively, the opportunity costof remaining in the Crisis region is βu (g¯) / (1− β), namely, the value of not attaining the saferg¯ level in the No-Default region. Finally, the last term shows that the government risks payoffu (Zy) / (1− β) with probability pi when it follows policy T .If we can find that ∂V T (B0) /∂T is positive for some small pi and B0 ∈ (B/ (1− pi) , B¯], thenthe benefit of delaying exit from the Crisis region is greater than the cost. When this is the case,there exist other possibilities T > 1 that will increase the government’s payoff when debt lies inthe Crisis region.Taking the limit on the last expression simplifies tolimpi→0∂V T (B0)∂T= βT−1 ln (β){βugT(gT)1− βT (B −B0) +β1− β[u (g¯)− u (gT (B0))]} .By the previous lemma, g¯ > gT (B0). Using property of strictly concave functions,ugT(gT) (g¯ − gT (B0))> u (g¯)− u (gT (B0)) ,and sinceg¯ − gT (B0) = 1− β1− βT [B0 −B] ,187thenugT(gT)( 1− β1− βT)[B0 −B] > u (g¯)− u(gT (B0)).Multiplying the last expression by ln (β) changes its sign. Rearranging terms in ∂V T (B0) /∂T andusing the last inequality impleslimpi→0∂V T (B0)∂T= βT−1 ln (β){βugT(gT)1− βT (B −B0) +β1− β[u (g¯)− u (gT (B0))]}> βT−1{ln (β)β1− βT ugT(gT)(B −B0) + β1− β ln (β)ugT(gT)( 1− β1− βT)[B0 −B]}= 0.Notice then that, for some B0 and pi very small, ∂VT (B0) /∂T > 0. Finally, the sign of equations(B.2) and (B.3) implicitly define a positive relation of T in terms of B0, thus proving that T∗ (B0)can take values {1, 2, ...,∞} for small values of pi, as B0 increases in[B, B¯ (pi)].Relation between gT (B0) and gT+1 (B0)Claim. If B0 > B/ (1− pi), then gT (B0) < gT+1 (B0).Proof. Be gT (B0) < gT+1 (B0). Theny −[1− βˆ1− βˆT]B0 +[1− βˆ1− βˆT]βˆT−1βB < y −[1− βˆ1− βˆT+1]B0 +[1− βˆ1− βˆT+1]βˆTβB.After rearranging terms,0 < βˆT−1(B0βˆ − βB),and this inequality holds only if the term in parenthesis is positive, what, in turn, implies thatB0 > B/ (1− pi).Comparative statics (region boundaries with respect to Z)Proof of Proposition 3.3.2. From equation 3.10 we can writeG (y, Z, β,B) =u (Zy)1− β − u (y −B)− βu (y)1− β = 0,and after applying the implicit function theorem, we get∂B∂Z= −u′(Zy)1−β yu′ (y −B) < 0.Assuming that V∞ dominates any other V T when pi is positive, we can write equation (3.15)188asF(y, Z, β, piB¯)=u(y −(1− βˆ)B¯)1− βˆ +βpiu (Zy)(1− βˆ)(1− β)− u(Zy + βˆB¯)− βu (Zy)1− β = 0.Using the implicit function theorem we get∂B¯∂Z= −−(β u′(Zy)y1−β[1− pi1−βˆ]+ u′(Zy + βˆB¯)y)−(u′(y −(1− βˆ)B¯)+ u′(Zy + βˆB¯)βˆ) ,and since the term in square brackets in the numerator is positive for any pi ∈ (0, 1), then ∂B¯/∂Z <0.If the government lowers debt in T = 1 period,V 1 (B0) = u (y −B0 + βB) + βu (y − (1− β)B)1− β ,and∂V 1∂Z= β∂B∂Z[u′ (g0)− u′ (g¯)].Given that ∂B/∂Z < 0, and since g0 < g¯ by the previous lemma, this implies that u′ (g0) > u′ (g¯),then ∂V 1/∂Z < 0.Finally, it follows immediately from V∞ that∂V∞ (B0)∂Z=βpiu (Zy)(1− βˆ)(1− β)y > 0.B.2 Long-term debtB.2.1 Determination of the equilibrium priceIf the government is in the No-Default zone, then it never defaults (z′ = 1) and price is stationary(q = q′). Henceq(s,B′)= βE[z′(1 + q′δ)]⇒ q (s,B′) = β1− βδIf price were stationary in the Crisis region, then government defaults with probability pi,and otherwise it repays with probability 1 − pi. Therefore, the equilibrium price is now q =β [(1− pi) (1 + qδ) + pi (0)] and yieldsq =β (1− pi)1− β (1− pi) δ =βˆ1− βˆδ189where we defined βˆ = β (1− pi).Finally, if debt is lowered in T periods, the equilibrium price can be built with the followingprocedure. We start with T = 1 and use a backward induction logic. Conditional on no previousdefault, the government will reach the No-Default region in t = 1, and the stationary price isq11 = β/ (1− βδ). At the initial period t = 0, the q10 comes defined asq10 = βE[z′(1 + q11δ)].and, after plugging the previous result, we obtain that q10 = q11 = β/ (1− βδ). Assume that T = 2;then prices in each period areq20 = βE[z′(1 + q21δ)]⇒ q20 = β [(1− pi) (1 + q21δ)]q21 = βE[z′(1 + q22δ)]⇒ β (1 + q22δ)q22 =β1− βδ .Then, q22 = q21 = β/ (1− βδ), andq20 = βˆ + βˆδβ1− βδ .Since q22 = q21 also occurs when T = 1 (namely, that q11 = q10), we can already identify that qTt−1 = qTt ;i.e., that qTt = β/ (1− βδ) when t ≥ T − 1.We do the same analysis when T = 3, with the only difference that we now disregard q33 sinceit will be equal to q32. Therefore,q30 = βE[z′(1 + q31δ)]= β[(1− pi) (1 + q31δ)]q31 = βE[z′(1 + q32δ)]= β[(1− pi) (1 + q32δ)]q32 = βE[z′(1 + q33δ)]= β(1 + q33δ).After some algebra, this results in q32 = β/ (1− βδ), q31 = βˆ + βˆδβ/ (1− βδ), and q30 = βˆ(1 + βˆδ) +(βˆδ)2β/ (1− βδ). Finally, this allows us to identify thatqT0 = βˆ(T−2∑k=0(βˆδ)k)+(βˆδ)T−1( β1− βδ),and, therefore, the equilibrium price isqTt =(1−(βˆδ)T−1−t)(βˆ1−βˆδ)+(βˆδ)T−1−t (β1−βδ)t < T − 1β1−βδ t ≥ T − 1190B.2.2 Value of repayment and determination of the lower bound of the Crisisregion under long-term debtValue of repayment when the credit market freezes for a finite number of periodsWe need two results to derive V R under long-term debt.69 The claims below state that the gov-ernment chooses a constant level of spending when there is no probability of default (i.e., pi = 0),and that if government spending is constant, the path {Bt+1} is constant.Claim. If pi = 0, then government spending is constant.Proof. When pi = 0, the equilibrium price of a riskless bond is fixed and equal to q = β/ (1− βδ).Hence, the per-period budget constraint isgt +Bt = y +β1− βδ (Bt+1 − δBt)⇒ gt = y +β1− βδBt+1 −11− βδBt,and the government chooses the sequence {Bt+1}∞t=0 that solvesmax∞∑t=0βtu (gt)s.t. gt = y +β1−βδBt+1 − 11−βδBt ∀t ≥ 0Bt = B given, and B ≤ BBt+1 ≤ B ∀t ≥ 0.First-order and transversality conditions yieldβtu′ (gt)[β1− βδ]+ βt+1u′ (gt+1)[− 11− βδ]= 0lims→∞βsλsBs+1 = 0.From the first expression we obtain that u′ (gt) = u′ (gt+1), implying that government spending isconstant.Claim. If gt is constant and initial debt B0 is riskless, then Bt = B0 for every t.Proof. Takegt = y +β1− βδBt+1 −11− βδBtgt+1 = y +β1− βδBt+2 −11− βδBt+1.69These results particularize intermediate steps from Appendix section B.1.3 for long-term debt.191Using that gt = gt+1 from Appendix section B.2.2 leads toBt+2 −Bt+1 = 1β(Bt+1 −Bt) .By a logic similar to the one displayed in the Appendix section B.1.3, we can argue that Bt+1 = Btfor every t.We can now build the value of repayment when the credit market freezes for T periods.Claim. The value of repayment when the market freezes for T periods is given byV R (B0, δ) =∞∑t=0βtu (gt) s.t. gt ={y − δTB0 t = Ty − 1−β1−βδ δTB0 t > TProof. In this problem, we are looking for the highest B0 such that the government does notdefault in the No-Default region (i.e., when pi = 0). Hence, we divide the problem in two parts.The first part is the government problem when the market freezes lending for T periods, namely,for t = 0, 1, ..., T − 1. The second part is when the market resumes lending for t = T, T + 1, ....We use a backward induction logic, starting from the second problem and then moving to the firstproblem.Starting at t ≥ T , Appendix section B.2.2 showed that the government spending is constant inevery period, and that debt is constant and equal to the debt level at the initial period. Since theinitial period in this problem is the date when the credit market resumed lending (t = T ), thendebt will be constant at Bt = BT . Therefore, government spending will be equal tog = y − 1− β1− βδBT ,and the value after the market resumes lending isu(y − 1−β1−βδBT)1− β .In the first part of the problem, there is no lending for t = 0, 1, ..., T − 1 periods. Since thegovernment collects nothing from its issuances, it = 0 implies Bt+1 = δBt t = 0, ..., T − 1. Theiteration of the last expression yields Bt = δtB0. Finally, the government’s payoff in the firstproblem isT−1∑t=0βtu(y − δtB0),and the government value of repayment when there is a T -period market freeze isV R (B0, δ) =T−1∑t=0βtu(y − δtB0)+ βTu(y − 1−β1−βδ δTB0)1− β .192It is worth noting that after a market freeze, lending resumes. But the government has alreadyachieved a lower level of debt, and as per Appendix section B.2.2 it will keep that level constant.Since the level of debt was the highest at the initial period, then if the government has not defaultedbefore, it will surely not default after lending resumes. Therefore, we can use this value of repaymentto compare it later against the value of default at equality, in order to characterize the highest B.Determination of the lower bound of the crisis regionHaving identified the No-Lending Condition under long-term debt, we now prove that there existsa unique solution such that the equation holds with equality under a one-period market freeze.Lemma. When T = 1, there exists a unique B that is solution tou (Zy)1− β = u (y −B) + βu(y − 1−β1−βδ δB)1− βProof. We start buildingF (B;Z, y, β, δ) =u (Zy)1− β − u (y −B)− βu(y − 1−β1−βδ δB)1− β = 0Provided that (i) F (0; ·) = 11−β [u (Zy)− u (y)] < 0, (ii) F (B → y; ·) = u (Zy) / (1− β) − u (0) −βu (y (1− δ) / (1− βδ)) / (1− β)→∞, and (iii) F ′ (B) = u′ (y −B)+δβu′ (y − δB (1− β) / (1− βδ))/ (1− βδ) > 0, then we are guaranteed that a maximum B (Z, y, β, δ) > 0 exists and is unique.B.2.3 Determination of the upper bound of the Crisis regionOptimal policy in the No-Default region under long-term debtClaim (Long-term debt). If pi = 0 and given an initial debt Bt′ = B ≤ B, then governmentspending is constant and equal tog = y −B(1− β1− βδ),and government debt Bs+1 equals the initial debt Bt′ for s ≥ t′.Proof. This problem is identical to the government problem from Appendix section B.2.2 (whichwe can particularize for any t = t′ instead of t = 0 as in Appendix section B.1.3). Moreover, wecan use the result from that section to establish that debt is constant and equal to the initial debt.After setting a constant g and B in the budget constraint, the optimal government spending isg = y −(1− β1− βδ)B0.193General problem when debt is lowered in T periods (long-term debt)The government problem can be written asV T (B0, δ) = max{Bt+1}T−1t=0T−1∑t=0βˆtu(y −Bt(1 + δqTt)+ qTt Bt+1)+ βˆT−1u(y −BT−1(1 + δqTT−1)+ qTT−1B)+[1− βˆT−11− βˆ]βpiu (Zy)1− β + βˆT−1βu(y −(1−β1−βδ)B)1− βsubject toqTt = βE[zt+1(1 + qTt+1δ)], z−1 = 1B0, B given.The first-order condition yieldsu′ (gt) = u′ (gt+1)βˆ(1 + δqTt+1)qTtGiven that B0 is in the Crisis region, and since every debt level {Bt+1} lies on it while the govern-ment plans to exit the region in T periods,70 the equilibrium price relationship qTt = βˆ(1 + δqTt+1)holds, and as a result we haveu′ (gt) = u′ (gt+1)⇒ gt = gt+1Expression for gT (B0, δ)We assume that the government is in the crisis region (B > B), and use every budget constraintwhen z = 1. Writing gT (B0 , δ) ≡ gT yieldsgT +Bt = y + qTt (Bt+1 −Btδ) .We initially solve for B0 for different T schemes in order to identify a pattern. After that, wecalculate a general formula for gT .When T = 2, the budget constraints during the transition areg2 +B0 = y + q20 (B1 −B0δ)⇒ B0 =y − g21 + q20δ+q201 + q20δB170If not, exit from the Crisis region would have occured for some Bt+1 in the optimal sequence, contradicting thefact that the government exits in T .194g2 +B1 = y + q21 (B −B1δ)⇒ B1 =y − g21 + q21δ+q211 + q21δB.And after plugging B1 into B0,B0 =y − g21 + q20δ[1 +q201 + q21δ]+q201 + q20δq211 + q21δB.Doing the similar proces for T = 3 and T = 4, we obtain thatB0 =y − g31 + q30δ[1 +q301 + q31δ+q301 + q31δq311 + q32δ]+q301 + q30δq311 + q31δq321 + q32δBB0 =y − g41 + q40δ[1 +q401 + q41δ+q401 + q41δq411 + q42δ+q401 + q41δq411 + q42δq421 + q43δ]+q401 + q40δq411 + q41δq421 + q42δq431 + q43δB.Hence, for any T we have thatB0 =y − gT1 + qT0 δ1 +T−2∑k=0T−2−k∏j=0qTjT−1−k∏j=1(1 + qTj δ)+T−1∏j=0qTjT−1∏j=0(1 + qTj δ)B,and after solving for gT ,gT (B0, δ) = y −(1 + qT0 δ)B0 −T−1∏j=0qTjT−1∏j=0(1 + qTj δ)B1 +T−2∑k=0T−2−k∏j=0qTjT−1−k∏j=1(1 + qTj δ)−1.The upper bound of the Crisis region when pi = 0Lemma. Suppose that pi → 0, and consider V∞ (B0, δ)− V T (B0, δ), where δ > 0. Then,V∞ (B0, δ)− V T (B0, δ) > 0for every T <∞.Proof. We use the formula for the prices under long-term debt, qTt . As pi tends to 0, each βˆ term isequal to β, and therefore, qTt = β/ (1− βδ). Labelling qTt ≡ q and plugging it into our expression195for gT (B, δ), we get thatgT (B0, δ) = y − (1 + qδ)[B0 −(q1 + qδ)TB] 1− q1+qδ1−(q1+qδ)T .Since 1 + qδ = 1/ (1− βδ) and q = β (1 + qδ), thengT (B0, δ) = y −[1− β1− βT] [B0 − βTB]( 11− βδ),(and it is straightforward to observe that δ = 0 leads to the same expression as our formergT (B, 0) ≡ gT (B)).Equation V∞ (B0, δ)− V 1 (B0, δ) is equal to11− β[u(y − 1− β1− βδB0)− (1− β)u(y − B01− βδ +βB1− βδ)+ βu(y − 1− β1− βδB)].Since the argument of the first term is equal to the (convex combination) of the second andthird terms’ arguments, we use strict concavity in u (·) with Jensen’s inequality to obtain thatV∞ (B0, δ)− V 1 (B0, δ) > 0. Equation V∞ (B0, δ)− V T (B0, δ) when T ≥ 2 yieldsV∞ (B0, δ)− V T (B0, δ) = 11− β [u(y − 1− β1− βδB0)− [1− βT ]u (gT (B0, δ))− βTu(y −(1− β1− βδ)B)],and provided that(1− βT ) gT (B0, δ)+βT [y − ((1− β) / (1− βδ))B] equals y−[(1− β) / (1− βδ)]B0,then strict concavity and Jensen’s inequality proves that the difference is strictly positive for anydebt level.Lemma. There exists a unique B¯ that solvesu(y − 1−β1−βδ B¯)1− β = u(Zy +β1− βδ B¯)+ βu (Zy)1− βProof. We define the following functionH (B;β, y, Z, δ) =u(y − 1−β1−βδ B¯)1− β − u(Zy +β1− βδ B¯)− βu (Zy)1− β = 0Given that (i) H (0; ·) = 11−β [u (y)− u (Zy)] > 0, (ii) u (0) / (1− β) − u (Zy + yβ/ (1− β)) −βu (Zy) /(1− β) → −∞, and (iii) ∂H (B; ·) /∂B = −u′ (y − B¯ (1− β) / (1− βδ)) / (1− βδ)−u′(Zy + B¯β/ (1− βδ)) (β/ (1− βδ)) < 0, then there exists a unique B¯ such that the PC is satisfied196with equality at pi → 0.Derivation of Figure 3.6Using the PC and the NLC, define the Upi=0PC (B; δ) ≡ UPC (B; δ) and Upi=0NLC (B; δ) ≡ UNLC (B; δ)curves asUPC (B; δ) =u(y − 1−β1−βδ B¯)1− β − u(Zy +β1− βδ B¯)− βu (Zy)1− βUNLC (B; δ) = u (y −B) + βu(y − 1−β1−βδ δB)1− β −u (Zy)1− β .Using our results from Appendix section B.2, then it is straightforward that (i) UPC (B; δ) =UNLC (B) > 0 at B = 0, (ii) UPC (B; δ) → −∞ as B → y (1− βδ) / (1− β) and UNLC (B; δ) →−∞ as B → y, and (iii) both U ′PC (B; δ) < 0 and U ′NLC (B) < 0. This describes the entirebehavior of the curves in Figure 6. The behavior of Upi>0PC (B; δ) is the only relevant for this graph –UNLC (B; δ) is independent of pi. However, we can again invoke the previous lemma to argue that(i) Upi>0PC (0; δ) > 0, (ii) Upi>0PC (B; δ) < 0 as B → y(1− βˆδ)/ (1− βδ), and (iii) U ′pi>0PC (B; δ) < 0.The upper bound of the Crisis region when pi = 0Lemma. There exists a unique B¯ (δ) that solvesu(y −B[1−βˆ1−βˆδ])1− βˆ +βpiu (Zy)(1− βˆ)(1− β)= u(Zy +βˆ1− βˆδB)+ βu (Zy)1− βProof. With a similar logic as before, we defineH˜ (B;β, y, Z, pi, δ) =u(y −B[1−βˆ1−βˆδ])1− βˆ +βpiu (Zy)(1− βˆ)(1− β)− u(Zy +βˆ1− βˆδB)− βu (Zy)1− β .Given that (i) H˜ (0; ·) = [u (y)− u (Zy)] /(1 − βˆ) > 0, (ii) H˜(B → y(1 − βˆδ)/(1 − βˆ); ·) =u (0) /(1− βˆ) +βpiu (Zy) /[(1− βˆ) (1− β)]−u(Zy+ yβˆ/(1− βˆ))−βu (Zy) (1−β)→ −∞, and (iii)dH˜ (B; ·) /dB = −u(y −B(1− βˆ)/(1− βˆδ))/(1− βˆδ)− u(Zy +Bβˆ/(1− βˆδ))βˆ/(1− βˆδ) < 0, thenthere exists a positive and unique B¯ (β, y, Z, pi, δ) that satisfies the PC with equality.B.2.4 Welfare analysis resultsProof of Proposition 3.5.1. We rewrite equation (3.24) in terms of total outstanding debt and defineH =u (Zy)1− β − u (y − (1− δ)B)− βu(y − 1−β1−βδ δ (1− δ)B)1− β ,197to apply the implicit function theorem. Therefore,dHdB= (1− δ)u′ (y − (1− δ)B) + βu′(y − 1− β1− βδ δ (1− δ)B)(δ (1− δ)1− βδ),anddHdδ= −B[u′ (y − (1− δ)B)− βu′(y − 1− β1− βδ δ (1− δ)B)[1− 2δ + βδ2(1− βδ)2]].The first derivative is positive. Also, expression y − (1− δ)B < y − (1− β) δ (1− δ)B/ (1− βδ)for every δ ∈ (0, 1), and then u′ (y − (1− δ)B) > u′ (y − (1− β) δ (1− δ)B/ (1− βδ)). Since theinner-most term in brackets is strictly less than 1, the outer-most expression in brackets is positive.Therefore, dH/dδ < 0. As a result, dB/dδ = − (dH/dδ) / (dH/dB) > 0.Using equation (3.26), we also rewrite it in terms of total outstanding debt to getG =u(y − 1−βˆ1−βˆδ (1− δ) B¯)1− βˆ +βpi1− βˆu (Zy)1− β − u(Zy +βˆ1− βˆδ (1− δ) B¯)− βu (Zy)1− β = 0.Applying the implicit function theorem,dGdB= −u′(y − 1− βˆ1− βˆδ (1− δ) B¯)(1− δ1− βˆδ)− u′(Zy +βˆ1− βˆδ (1− δ) B¯)(βˆ (1− δ)1− βˆδ)< 0anddGdδ=u′(y − 1−βˆ1−βˆδ (1− δ) B¯)1− βˆ B¯(1− βˆ1− βˆδ)2+ u′(Zy +βˆ1− βˆδ (1− δ) B¯) βˆB¯(1− βˆ)(1− βˆδ)2 > 0result in dB¯/dδ = − (dG/dδ) / (dG/dB¯) > 0.Further analysisUnder short-term debt, Figure B.1 graphically shows the envelope function Vˆ (solid line) that resultsfrom the value function V 5 (B0) (dotted line). Specifically, the government chooses T (B0) = 5approximately when Bˆ ∈ (10.23, 10.38). In this interval of total debt levels Bˆ, the envelope functionVˆ (B0) matches V5 (B0); after that interval the government switches to policy T (B0) = 6, makingVˆ (B0) lie strictly above V5 (B0).The same mechanics applies for the envelope function under long-term debt. The next figureillustrates the maximum payoff that the government attains when it follows the optimal strategiesprescribed under equilibrium. Specifically, we concentrate in the value that the government obtainsin the No-Default and the Crisis region. Notice first that we can disregard the value that thegovernment obtains in the Default region because when the government starts with initial debtB > B¯, the definition of a sunspot equilibrium prescribes that creditors will not lend money. As198Figure B.1: Value function V 5 and envelope curve as total amount of debt varies for short-termdebt. (Parameters and functional forms: pi = 0.0001, β = 0.96, δ = 0, Z = 0.9, y = 10, u()˙ = ln()˙.)a result, the price q drops to 0, and the government defaults and gets V D = u (Zy) / (1− β) forevery debt level B ≥ B¯, both under short-term and long-term debt. If we can show that in theCrisis region the envelope under long-term debt is above the envelope under short-term debt, thenwe can rule out any possibility that the economy attains a higher utility in the Default region whenδ = 0.With this logic, we can numerically prove that welfare under long-term debt is greater comparedto short-term debt by showing that (i) in the No-Default region, the government attains a higherutility under long-term debt when it smooths spending, and (ii), in the Crisis region, the envelopeof the family of value functions under long-term debt is always above the envelope under short-termdebt.Figure B.2 proves that welfare under long-term debt is greater than under short-term debt. Inparticular, points (1) and (2) indicate a change in the optimal government policy coming from aswitch between regions. For the short-term debt case, as B increases, (1) shows the change in thegovernment’s payoff when it switches from a policy of keeping a constant level of total debt to apolicy of lowering debt in T periods when pi > 0. Similarly, (2) has the same interpretation forlong-term debt bonds.In the No-Default region, both government value functions start at the same value when B = 0.As B increases, the optimal government spending starts to decrease, making V (B; δ = 0) declinemore than V (B; δ > 0). Intuitively, when total outstanding debt increases, a higher δ allocatesmaturing debt in several periods versus one-period bonds. As a result, when the maturing debt199Figure B.2: Debt regions and maximum V as total amount of debt varies, for short-term andlong-term debt. (Parameters and functional forms: pi = 0.0001, β = 0.96, δ = {0, 0.1}, Z = 0.9,y = 10, u()˙ = ln()˙.)that the government rolls over is a portion of the total amount of debt (i.e., δ > 0), the constantspending attained g¯ is greater compared to the case where maturing debt is the total amount ofdebt in every period (i.e., δ = 0).In the Crisis region, we numerically obtain that the envelope of the family of V T curves underlong-term debt (built with the logic of Figure 10) is strictly greater than the envelope of curvesV T under short-term debt. In the main text, we describe that when δ > 0, the government gainscome from (i) the faster decline of debt that decreases the likelihood of self-fulfilling debt crises,and (ii) the lower discount of the payoff under smooth spending u (g¯) (conditional on the sunspotnot triggering default), associated to the faster policy in (i).Figure B.3 and Figure B.4 illustrate other examples of welfare improvement under a highermaturity of debt, when creditors’ belief of government default increases from pi = 0.0001 to pi =0.001 and pi = 0.01.71 In order to have a Crisis region where the optimal choice of T varies whenpi = 0.01, we changed the default penalty from Z = 0.9 to Z = 0.86. Notice also that these figuresillustrate the envelopes of the value functions for a portion of the Crisis region, specifically whenT ∗ (B0) = 1. However, when considering a greater range of B, the envelope of δ = 0 switches toT ∗ (B0) = 2 before δ = 0.1 and δ = 0.2 – and the same happens for δ = 0.1 compared to δ = 0.2.As Figure B.3 and Figure B.4 show, there is a welfare improvement when δ rises, showing thatthe findings hold when beliefs rise.71In our numerical example, pi = 0.01 is half of the probability of default specified in Cole and Kehoe (1996). Foranother specifications on pi, see also Conesa and Kehoe (2017).200Figure B.3: Envelope functions as total amount of debt varies, for short-term and long-term debtand for a lower probability of crises (Parameters and functional forms: pi = 0.001, β = 0.96,δ = {0, 0.1, 0.2}, Z = 0.9, y = 10, u()˙ = ln()˙.)Figure B.4: Envelope functions as total amount of debt varies, for short-term and long-term debtand for a higher probability of crises (Parameters and functional forms: pi = 0.01, β = 0.96,δ = {0, 0.1, 0.2}, Z = 0.9, y = 10, u()˙ = ln()˙.)201
Thesis/Dissertation
eng
Economics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Devereux, Michael B.
Szkup, Michal
Essays on fiscal and monetary policy during economic crises
Text