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Three essays on heterogeneity in sectoral price flexibility Tugan, Mustafa 2014

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Three Essays On Heterogeneity inSectoral Price FlexibilitybyMustafa TuganB.Sc., Middle East Technical University, 2004M.Sc., Middle East Technical University, 2006A 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)February 2014? Mustafa Tugan 2014AbstractThis thesis focuses on the heterogeneity of price flexibility among sectors. For instance, doesa multi-sector model in which the frequency of price changes differs among sectors predictradically different dynamics for aggregate variables following a monetary policy shock thana one-sector model in which sectors are assumed to be homogenous in their frequencies ofprice changes? Is there any relative price effect of the shocks to monetary policy in theUnited States? If there is, can this relative price effect be related to the heterogeneity ofthe frequency of price changes among sectors? What insights can be gained if dynamicstochastic open-economy models are elaborated by including the heterogeneity of priceflexibility among sectors? These questions are among the types of the questions that Iaddress in this thesis.This thesis consists of three papers. The first paper studies the effects of a monetarypolicy shock on output, inflation and the real wage in the United States. Next, the dynamicsof these aggregate variables as predicted by the one- and multi-sector dynamic stochasticgeneral equilibrium (DSGE) models are compared. The main finding is that the dynamicspredicted by the multi-sector model are quite similar to those predicted by the one-sectormodel.The second paper focuses mostly on the effects of shocks to the federal funds rate ondisaggregated sectors? prices in the United States. The two main empirical findings in thischapter are the substantial heterogeneity in sectoral price responses to these shocks andiithat the price responses in sectors are only weakly associated with their frequency of pricechanges.The third paper, jointly written with Emek Karaca, is concerned with the effects ofpositive monetary shocks on output, the real exchange rate and the price level in developingcountries which have adopted an inflation targeting regime. We find that such shocks areassociated with a temporary rise in output; a temporary depreciation in the real exchangerate and a sizable contemporaneous increase in the price level in those economies.iiiPrefaceThe third paper of this dissertation is co-authored with Emek Karaca. The empiricalmodels in this paper were primarily developed by Mustafa Tugan. Emek Karaca andMustafa Tugan worked together to give a clear exposition of the findings in the empiricalsection. Both Emek Karaca and Mustafa Tugan were involved in developing the theoreticalmodels in the paper and in the calibration of the theoretical models? parameters. Theeconometric analysis for assessing the performance of the theoretical models considered inthe paper were mainly performed by Mustafa Tugan.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 How Important is Sectoral Heterogeneity in Price Flexibility in Explain-ing the Effects of Monetary Shocks in a DSGE Framework? . . . . . . 32.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 The Household . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.3 Equilibrium in the Frictionless Economy . . . . . . . . . . . . . . . 17v2.2.4 The Economy with Nominal Frictions . . . . . . . . . . . . . . . . . 232.2.5 The IS Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.6 Monetary Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Econometric Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.1 Calibrated Parameters of the Model . . . . . . . . . . . . . . . . . . 312.3.2 Estimated and Implied Parameters . . . . . . . . . . . . . . . . . . 362.3.3 Impulse Responses Predicted by the Models . . . . . . . . . . . . . 392.3.4 A Comparison of Model-Based Impulse Responses under AlternativeTaylor Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3.5 Explaining the Contrasting Findings . . . . . . . . . . . . . . . . . 502.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 Heterogeneity in Price Flexibility and Monetary Policy Shocks . . . . 583.1 The Empirical Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.1.1 Aggregate Dynamics after an Exogenous Shock in the Federal FundsRate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.1.2 Sectoral Price Responses after Interest Shocks . . . . . . . . . . . . 633.2 Theoretical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.2.1 The Structural Equations in the Models . . . . . . . . . . . . . . . 703.2.2 The Econometric Method . . . . . . . . . . . . . . . . . . . . . . . . 723.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.2.4 The Multi-Sector Model with Asymmetric Cost Structure . . . . . . 873.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944 Which Type of Model Best Captures the Effects of Monetary Shocks inDeveloping Countries? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97vi4.1 Empirical Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.1.1 Empirical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.1.2 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.2 Theoretical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.2.1 The Problem of Home and Foreign Households . . . . . . . . . . . . 1184.2.2 The Objective of Firms in the Home and Foreign Countries . . . . . 1274.2.3 Closing the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.3 Calibration and Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374.4 Quantitative Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1404.4.1 Output and Price Level Dynamics . . . . . . . . . . . . . . . . . . 1404.4.2 Real and Nominal Exchange Rate Dynamics . . . . . . . . . . . . . 1434.4.3 One- and Multi-Sector Models without Investment . . . . . . . . . . 1434.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149AppendicesA Appendix to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155A.1 Sectoral Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155A.1.1 Optimized Prices and Sectoral Inflation . . . . . . . . . . . . . . . . 155A.2 Sectoral Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157A.3 Percentile Group Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . 161A.4 Wages Set for an Hour of Differentiated Labor . . . . . . . . . . . . . . . . 162A.4.1 Composite Labor Demand Equation . . . . . . . . . . . . . . . . . . 164viiA.5 Model-Based Impulse Responses with the EfficientWeighting Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167B Appendix to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169B.1 The Bils, Klenow & Kryvtsov (2003) Model Reconsidered . . . . . . . . . . 169B.1.1 The Bils, Klenow & Kryvtsov (2003) Model . . . . . . . . . . . . . 169B.1.2 Findings from the Bils, Klenow & Kryvtsov (2003) Model . . . . . 172B.1.3 Testing a Critical Assumption in the Bils, Klenow & Kryvtsov (2003)Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174B.2 Estimation of Confidence Intervals for Figure 3.3 Using a Block-BootstrapMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179C Appendix to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181C.1 The Empirical Strategy in Clarida & Gali (1994) . . . . . . . . . . . . . . . 181C.2 Aggregate Dynamics after Monetary Shocks . . . . . . . . . . . . . . . . . 186C.2.1 Aggregate Dynamics in Empirical Model I after Monetary Shocks . 186C.2.2 Aggregate Dynamics in Empirical Model II after Monetary Shocksin the United States . . . . . . . . . . . . . . . . . . . . . . . . . . . 186C.3 Calibration of Models? Parameters . . . . . . . . . . . . . . . . . . . . . . . 189C.3.1 The Weak Link between the Level of Inflation and the Frequency ofPrice Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190C.3.2 Asymmetry in Currency Invoicing in International Trade betweenDeveloping and Advanced Economies . . . . . . . . . . . . . . . . . 192C.4 The One- and Multi-Sector Models? Dynamics with a Taylor-Type Rule . . 194viiiList of Tables2.1 Estimated Monetary Policy Rule: 1959Q1-2013Q1 . . . . . . . . . . . . . . 302.2 The Quarterly Frequencies of Price Adjustment over Different Percentiles ofthe Price Flexibility and Their Implied Durations . . . . . . . . . . . . . . . 322.3 The Calibrated Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4 Estimates of Structural Parameters . . . . . . . . . . . . . . . . . . . . . . 372.5 Estimates of Implied Parameters . . . . . . . . . . . . . . . . . . . . . . . . 382.6 Calibration for Explaining Contrasting Findings in Carvalho (2006) . . . . . 502.7 Cumulative Real Effects of Monetary Shocks and Their Persistence . . . . 523.1 Estimates of Structural Parameters . . . . . . . . . . . . . . . . . . . . . . 743.2 Calibrated Parameters (The Multi-Sector Model with Asymmetric CostStructure) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.3 Estimates of Structural Parameters (Multi-Sector Model with AsymmetricCost Structure) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.1 Adoption Dates of Inflation Targeting in Developing Economies . . . . . . . 112B.1 Sectors with a Significant Response from the Federal Reserve . . . . . . . . 176C.1 Calibration and Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 189C.2 Estimated Parameters of the Taylor Rule . . . . . . . . . . . . . . . . . . . 196ixList of Figures2.1 The Histogram of Quarterly Frequencies in Entry Level Item Categories . . 52.2 Impulse Responses to an Unanticipated 1% Fall in Rt . . . . . . . . . . . . 102.3 Impulse Responses to an Unanticipated 1% Fall in Rt (One-Sector Model) . 402.4 Impulse Responses to an Unanticipated 1% Fall in Rt (Multi-Sector Model) 412.5 The Model-Based Impulse Responses in the One- and Multi-Sector Modelsunder Alternative Taylor Rules after an Unanticipated 1% Fall in Rt . . . 432.5 The Model-Based Impulse Responses in the One- and Multi-Sector Modelsunder Alternative Taylor Rules after an Unanticipated 1% Fall in Rt (cont.) 442.5 The Model-Based Impulse Responses in the One- and Multi-Sector Modelsunder Alternative Taylor Rules after an Unanticipated 1% Fall in Rt (cont.) 452.5 The Model-Based Impulse Responses in the One- and Multi-Sector Modelsunder Alternative Taylor Rules after an Unanticipated 1% Fall in Rt (cont.) 462.5 The Model-Based Impulse Responses in the One- and Multi-Sector Modelsunder Alternative Taylor Rules after an Unanticipated 1% Fall in Rt (cont.) 472.5 The Model-Based Impulse Responses in the One- and Multi-Sector Modelsunder Alternative Taylor Rules after an Unanticipated 1% Fall in Rt (cont.) 482.6 The One- and Multi-Sector Model-Based Impulse Responses of Output Gapafter a Negative 1% m Shock in (2.3.7) . . . . . . . . . . . . . . . . . . . . 53x3.1 The VAR-Based Impulse Responses of Aggregate Variables to MonetaryShocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2 The Impulse Responses of the Price Levels of PCE Categories to an Unan-ticipated 1% Increase in the Federal Funds Rate Shocks . . . . . . . . . . . 643.3 Correlations of ?i with the Impulse Responses of Pi to an Unanticipated 1%Increase in Rt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.4 Impulse Responses to an Unanticipated 1% Rise in Rt (One-Sector Model) . 763.5 Impulse Responses to an Unanticipated 1% Rise in Rt (Multi-Sector Modelwith Symmetrical Cost Structure) . . . . . . . . . . . . . . . . . . . . . . . 773.6 Model- and VAR-Based Correlations of ?i with Clnpit1,20 . . . . . . . . . . . . 783.7 The Front-Loading Argument (The Multi-Sector Model with SymmetricCost Structure) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.8 Model-Based Clnpit2,3 (The Multi-Sector Model with Symmetric Cost Struc-ture) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.9 Impulse Responses to an Unanticipated 1% Rise in Rt (Multi-Sector Modelwith Asymmetric Cost Structure) . . . . . . . . . . . . . . . . . . . . . . . . 933.10 Model- and VAR-Based Correlations of ?i with Clnpit1,20 . . . . . . . . . . . . 943.11 Inflation Dynamics in the Low and High Labor-Share Industries (The Multi-Sector Model with Asymmetric Cost Structure) . . . . . . . . . . . . . . . . 954.1 Impulse Responses to Monetary Shocks in Developing Economies (EmpiricalModel II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.2 Impulse Responses in Each Country to Monetary Shocks in DevelopingEconomies (The VAR Model with Monthly Data) . . . . . . . . . . . . . . . 1154.3 Conditional Movements of the Real and Nominal Exchange Rates (EmpiricalModel II with Monthly Data) . . . . . . . . . . . . . . . . . . . . . . . . . . 116xi4.4 The Unconditional Co-movements of the Log-Changes in the Nominal andReal Exchange Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.5 Model- and Panel VAR-Based Impulse Responses of P and Y to z . . . . . 1414.6 Model- and Panel VAR-Based Impulse Responses of E and Q to z . . . . . 1444.7 Model- and Panel VAR-Based Impulse Responses of P , Y, E and Q to z(Without Investment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145A.1 Impulse Responses to an Unanticipated 1% Fall in Rt (Efficient CMD Esti-mator) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168B.1 The Bils, Klenow & Kryvtsov (2003) Model-Based Impulse Responses of theRelative Price to Monetary Shocks . . . . . . . . . . . . . . . . . . . . . . . 173B.2 Testing for the Significance of the Federal Reserve?s Response for SectoralPrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177C.1 Impulse Responses to Monetary Shocks in Empirical Model I . . . . . . . . 187C.2 Impulse Responses to Monetary Shocks in the United States (EmpiricalModel II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188C.3 Median Inflation Rates in Developing Economies (1999M1-2012M9) . . . . . 191C.4 Consumer Prices Inflation and the Turkish Lira Share in External Trade inTurkey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192C.5 One- and Multi-Sector Models with a Taylor-Type Rule . . . . . . . . . . . 197xiiAcknowledgmentsI am particularly indebted to my adviser, Paul Beaudry, for his excellent guidance through-out my thesis. Without such guidance, this thesis would not have been completed.My experience during my doctoral studies can be divided into two different periods: theperiod before I met Paul Beaudry and the period after I met him. Before I met him, myexperience in the Ph.D. program was not much different from a nightmare. I felt isolatedfrom society and worked all the time without ever knowing where to go and mostly I waslost and lacking guidance at that time.After I met him, on the other hand, I had a clear vision of where the research projectwas going. I had the relief that I could consult someone who could guide me when I faced aseemingly unsolvable problem. I am sure my view about his guidance is shared by all whohave had the privilege of having him as his/her supervisor. His attentive behavior towardshis students deserves much respect and gratitude.I also specially thank Viktoria Hnatkovska and Yaniv Yedid-Levi for their careful read-ing of the draft of the thesis and their helpful suggestions. I would also like to thank VadimMarmer as I benefited a lot from his comments and his long and insightful answers to thequestions I asked via e-mail.Last, but not least, I am thankful to Okan Yilankaya and Kadir Mercul. Their intimateand genuine attitude towards me in Canada reminded me of the beautiful people of mycountry, Turkey, when I was away a great distance from my home.xiiiDedicationTo my mother and father, Zehra and AlixivChapter 1IntroductionUntil recently, it has been standard practice in monetary economics to model the UnitedStates economy as consisting of identical firms which have the same frequency of priceadjustment. After a proliferation of studies investigating the price frequencies at the disag-gregated level, this practice is under question as sectors are found to differ largely regardingthe frequencies of price changes in the United States.Is this finding generalizable to the frequency of price changes among sectors in devel-oping economies? It is true that higher inflation in developing economies is likely to lowerthe standard deviation of the frequency of price changes among sectors by causing a largerincrease in the frequency of price changes in sectors where prices change infrequently com-pared with sectors where they change quite often. It is still, however, safe to argue thatheterogeneity in price flexibility among sectors matters, to some degree, for developingcountries. In my thesis, the implications of sectoral heterogeneity in price flexibility areinvestigated in terms of both the closed- and open-economy contexts.In the first paper, I depart from the identical-firms economy by adding heterogeneityin price flexibility to the model and investigate the consequences. I find that the aggregatedynamics in the heterogeneous-firms economy do not deviate sharply from those in theidentical-firms economy. This result contrasts with the finding in Carvalho (2006) andNakamura & Steinsson (2008b) that monetary shocks induce larger and more persistentreal effects in a heterogeneous-firms economy. These contrasting findings can be reconciled1with the fact that Carvalho (2006) and Nakamura & Steinsson (2008b) ignore the degreeof wage rigidity in the United States. When this is taken into account, the dynamics ofoutput after monetary shocks in the heterogeneous-firms economy and the identical-firmseconomy are similar.The second paper studies sectoral price responses to an exogenous interest rate shockin the United States. It has two main findings. First, an interest rate shock causes strongrelative price effects as price responses to such a shock differ largely among sectors. Second,sectoral asymmetries in the frequency of price changes are only weakly associated withsectoral price responses. I show that a multi-sector model where sectors differ not only inthe frequency of price changes but also in the structure of production costs is capable ofexplaining these two findings.The third paper investigates the movements of output, the price level, the bilateralnominal and real exchange rates with the United States following a monetary shock indeveloping economies with an inflation targeting regime. By means of an empirical model,we show that an expansionary domestic monetary shock in these economies causes a tem-porary increase in output, a temporary depreciation in the real exchange rate with theUnited States and a sizable contemporaneous increase in the price level. The multi-sectormodel where wage setting is staggered and there are asymmetries in the frequency of pricechanges among sectors proves equal to explaining these aggregate dynamics after such ashock. Contrasting with other staggered-wages models of Erceg, Henderson & Levin (2000)and Huang & Liu (2002), staggered wage setting in our model does not require completefinancial markets. Since financial markets in developing economies are underdeveloped andlack sophistication, the incomplete financial market assumption adds realism to our model.2Chapter 2How Important is SectoralHeterogeneity in Price Flexibilityin Explaining the Effects ofMonetary Shocks in a DSGEFramework?Until recently, it has been standard practice in the New-Keynesian tradition to model theUnited States economy as consisting of only one sector where firms are assumed to behomogeneous in regards to their ability to change their prices in each period. There are atleast two justifications for this assumption. First, when frequencies of price changes do notdiffer significantly between sectors, one-sector models are easy to justify. Second, even whenthey differ substantially, if estimated structural parameters of the model are, in essence,invariant to allowing heterogeneous sectors with varying degrees of price stickiness, one-sector economy models can again be justified. As a matter of fact, if any of these reasons isstrong and the gain from introducing sectoral heterogeneity into the model is limited, one-sector models should be preferred over multi-sector models as the latter models significantly3complicate the analysis. An analysis of the sectoral distribution of price change reveals littlesupport for the first justification as the sectoral distribution of price changes in the UnitedStates economy is quite wide. Figure 2.1 illustrates this point with the help of a histogramof quarterly mean frequencies of price change, including sales in Entry Level Items (ELIs)in the CPI using the frequency of price changes in Nakamura & Steinsson (2008a).1 Asshown in the figure, the United States economy features sectors with significantly differingprice flexibility. The frequencies range from as low as 0.02% to 100%. More importantly,the distribution of sectoral frequencies of price change more closely resembles a uniformdistribution than a tight normal distribution. Consequently, one-sector models are difficultto justify based on the fact that sectors have similar frequencies.This leaves only one channel through which one-sector models can be justified, namelythat inference from the structural DSGE models is insensitive to heterogeneity in frequen-cies of price change. Carvalho (2006) attempts to address this question within the contextof the simple Calvo model. His main conclusion is that interest shocks have substantiallylarger and more persistent non-neutral effects on output in the multi-sector economy thanin a one-sector economy when there is strategic complementarity between firms. Strate-gic complementarity refers to the phenomenon that firms have an incentive to raise theirprices when other firms do the same. In order to explain the findings in Carvalho (2006),note that prices are by definition slow to adjust in low-frequency sectors. With strategiccomplementarity in price setting, the limited response of prices in these sectors gives adisincentive for firms in high-frequency sectors to change their prices when the shock hap-pens. As a result, the behavior of prices in low-frequency sectors has a dominant effecton the behavior of overall prices. This dominant effect results in longer and larger effectsof monetary shocks on output in the multi-sector economy compared to those in the one-1Entry Level Items are product categories that the Bureau of Labor Statistics uses to meausure the CPI.Some examples of ELIs are ?Girls? Outwear? and ?Parking Fees?.4Figure 2.1: The Histogram of Quarterly Frequencies in Entry Level Item Categories0 10 20 30 40 50 60 70 80 90 1000246810121416% SharesQuarterly Frequencies (%)Source: Own estimates based on data in Nakamura & Steinsson (2008a)Note: To meausure the CPI, the Bureau of Labor Statistics divides products into Entry Level Item(ELI).Some examples of ELIs are ?Girls? Outwear?, and ?Parking Fees?.sector economy with the same average frequency of price change. Later, Schwartzman &Carvalho (2008) extend these findings for Taylor and sticky-information models. Lastly,Nakamura & Steinsson (2008b) find that stronger non-neutral effects of monetary shocksin multi-sector models can be generalized to menu-cost models. Indeed, they also find thisis the case when the frequency of price changes in the one-sector model is calibrated as themean frequency of price changes in the multi-sector economy.In this paper, I also study the one- and multi-sector economies and compare the aggre-gate dynamics after monetary shocks between the former and the latter. However, unlike5Carvalho (2006) and Nakamura & Steinsson (2008b) who isolate the effect of heterogene-ity in a simple model which performs unsatisfactorily in explaining what happens aftera monetary policy shock2, I work with a more complicated model which better explainsthese dynamics. I find that the dynamic properties of the model are largely insensitive toadding heterogeneity in the frequency of price changes to the model. This finding holdsirrespective of whether the model is closed with the estimated or hypothetical interest raterules.How can the contrasting findings regarding the real effects of monetary shocks in theone- and multi-sector economies in this paper and in Carvalho (2006) and Nakamura &Steinsson (2008b) be reconciled? I show that the finding in Carvalho (2006) and Nakamura& Steinsson (2008b), that monetary shocks have larger and more persistent real effects inthe multi-sector economy than those in the one-sector economy, is driven mainly by theirflexible-wages assumption which is at odds with the data. For example, Barattieri, Basu& Gottschalk (2010) investigate the sluggishness of wage adjustment in the United Stateseconomy and find that the expected average duration of wage contracts in the United Statesis 5.6 quarters. When accounting for such a degree of wage rigidity, I find the dynamics ofoutput in the one- and the multi-sector economies are alike for a given monetary shock.I follow two steps for the systematic exposition of this finding. First, I do a replicationexercise of Carvalho (2006) who considers a simple model that features Calvo-type nominalprice contracts and flexible wages. In line with the finding in Carvalho (2006), I findthat monetary shocks in the multi-sector economy have larger and more persistent realeffects than those in the one-sector economy. Next, the model is elaborated by introducing2For example, Woodford (2003) convincingly notes the simple Calvo (1983) model which features firmswith solely forward looking behavior fails to account for the delayed effects of nominal disturbances oninflation. Indeed, when an expansionary monetary policy shock happens, the simple Calvo model predictsinflation to peak before real GDP peaks. Yet, evidence from structural VAR models indicates that the peakin inflation actually occurs much later than the strongest effect on real GDP.6staggered wage setting in the form of Erceg, Henderson & Levin (2000) type nominal wagecontracts. I show that when costs are slow to respond to such shocks, the larger and morepersistent real effects of monetary shocks in the multi-sector economy relative to those inthe one-sector economy disappear.The organization of the paper is as follows: Section 2.1 describes the data and specifiesthe VAR estimation. Section 2.2 develops one- and multi-sector economy models whichexplain the economy?s response following the shock. Section 2.3 compares the one- andmulti-sector models using impulse response functions under estimated and hypotheticalinterest rate rules and discusses why the finding in this paper and in Carvalho (2006) andNakamura & Steinsson (2008b) differ. The last section concludes the paper.2.1 DataI study the aggregate dynamics following an unanticipated 1% fall in the federal funds ratewith the following vector autoregression (VAR) model,?t = B0 +kmax?k=1Bk?t?k +A0Et (2.1.1)where Et and kmax represent the vector of structural shocks that occur in the periodand the number of lags included, respectively. A contemporaneous response matrix ofthe variables to these shocks is shown by A0. Lastly, ?t denotes the vector of variablescontained in the VAR and is given as:?t =[Yt ? Y nt , pit, Wreal,t, Rt](2.1.2)where Yt?Y nt denotes the output gap. I follow Giordani (2004) and use capacity utiliza-tion in manufacturing as a measure of Yt ? Y nt . The next variable, pit, denotes annualized7inflation, which is measured as the annual percentage change in the GDP deflator. Realwage is represented by Wreal,t, which is defined as the hourly earnings in manufacturingdivided by the GDP deflator. Lastly, Rt shows the quarterly federal funds rate. I use quar-terly data that spans the period from 1959Q1 to 2013Q1 and the VAR contains four lagsof each variable. The ordering of the variables in (2.1.2) implies that the Federal Reserveis assumed to observe disturbances in the output gap, inflation and the real wage (k = 0)and respond to them contemporaneously. For the least square estimates of the coefficientsto be unbiased, the output gap, inflation and the real wage must be assumed to respond tomonetary shocks only with a quarter lag. This is in line with the recursive identificationof monetary shocks in Christiano, Eichenbaum & Evans (2005).It is notable that it is conventional to use real GDP in place of the output gap in(2.1.2). However, if real GDP is used, ?the price puzzle?, which refers to the counter-intuitive finding that an expansionary monetary policy shock lowers inflation, is observed.3 Giordani (2004) criticizes the practice of using real GDP in place of a measure of theoutput gap since using real GDP results in monetary shocks not being orthogonal to otherstructural shocks, which may be the cause of the puzzle. He shows no such problem existswhen the VAR contains a measure of the output gap. My results support this conjecture.Indeed, when I use a measure of the output gap as opposed to real GDP, the puzzle3Sims (1992) conjectures that the puzzle results from misspecification due to the failure of identifyingan exogenous monetary policy shock. To explain this misspecification problem, it must first be noted thatthe overall prices in an economy respond only sluggishly to commodity price shocks. Now, consider a fall incommodity prices. Since a fall in future inflation is anticipated, it is natural for a central bank to react tothis shock by lowering its interest rate. Despite the fall in interest rate, a fall in inflation may be observedsince the effect of the fall in commodity prices on prices may outweigh the effect of lower interest rates onprices. Note that no such control for commodity prices are contained in the VAR system above. Hence, itis possible the finding that expansionary monetary policy causes a fall in inflation may reflect the effectsof commodity prices on inflation and interest rates. In other words, what is regarded as an exogenousmonetary shock in the VAR system above might in fact reflect the endogenous policy response of monetaryauthorities to the commodity price shock. In this case, an economic interpretation of impulse responses isdifficult. To alleviate this problem, Sims (1992) includes a commodity price index in his VAR model. Incontradiction to this conjecture, adding a commodity price index in (2.1.2) before the interest rate does nothelp alleviate the puzzle in our results when real GDP is used in place of a measure of the output gap.8is eliminated to a large extent. Figure 2.2 shows impulse responses for the aggregatevariables in the VAR in (2.1.2) to a 1% expansionary interest rate shock. After this shock,the output gap shows a persistent increase. While the point estimates of impulse responsesfor the real wage indicate that the real wage rises, their confidence bands are not tightenough to conclude that the real wage shows a significant rise. After an insignificant fall inperiods immediately following the shock, inflation rises. It is notable that while inflationresponses to the expansionary shock are negative over some quarters, there is no puzzle inour results as these responses are not significant. Lastly, the federal funds rate stays belowthe pre-shock level for about three years after the shock.It is notable that aggregate dynamics reported in Figure 2.2 are in line with those inChristiano, Eichenbaum & Evans (2005). In this regard, the aggregate dynamics I aimto match with those in dynamic stochastic models in Section 2.3.3 are similar to thosereported in the literature.2.2 ModelThe model I employ for the analysis builds largely on the model in Giannoni & Woodford(2003). The model in Giannoni & Woodford (2003) departs from the simple textbook Calvomodel in four ways. First, the model features staggered wage setting along the lines ofErceg et al. (2000). Second, when not optimized, prices and wages are set according to thebackward-looking indexation rule. Third, consumer preferences exhibit habit persistence.Thus, an increase in today?s consumption increases today?s marginal utility, while it leadsto a fall in tomorrow?s marginal utility. Fourth, while wages and prices are set one periodin advance, the decision regarding real expenditure is made two periods in advance. Mystructural model is a modified version of this model. The model in Giannoni & Woodford(2003) include three modifications. First, firms have to pay their wage bill in advance,9Figure 2.2: Impulse Responses to an Unanticipated 1% Fall in Rt(a) Yt ? Y nt0 4 8 12 16 20?1?0.500.511.5QuartersPercent(b) pit0 4 8 12 16 20?0.4?0.200.20.40.60.81QuartersPercent(c) Wreal,t0 4 8 12 16 20?0.3?0.2?0.100.10.20.30.40.5QuartersPercent(d) Rt0 4 8 12 16 20?1.5?1?0.500.51QuartersPercentNote: In the figure, the solid line indicates the estimated point-wise impulse responses. The area betweenthe dashed lines shows the 95% confidence interval estimated with the method suggested by Sims & Zha(1999).implying an increase in the opportunity cost of holding money leads to an increase in thecost of production, all things being equal. Second, the model is extended to allow sectoralheterogeneity in price flexibility. Third, as opposed to two periods in advance, the decisionon real expenditure is made one period in advance. Such an assumption is consistent with10the VAR model presented in the last section where the output gap responds with a lag. Yet,the assumption in Giannoni & Woodford (2003), that real expenditure is predeterminedfor two periods, is debatable since such an assumption contradicts with their VAR modelwhere the output gap response is delayed by only one period.2.2.1 The HouseholdThe objective of the infinitely lived household is to maximize its lifetime utility as specifiedby the following utility function:4Ut = Et?1{??s=0?s[U(Ct+s ? bCt?1+s)?H(ht+s(i))]}(2.2.1)The household makes the decision for period t consumption one period in advance.The presence of habit formation in preferences implies that while an increase in today?sconsumption increases today?s marginal utility, it leads to a fall in the marginal utility ofthe next period due to the presence of the ?bCt?1+s term in the utility function. Theparameter b measures the degree of habit formation. In the standard utility function, bis taken as zero. When b is positive, the household is more intolerant to fluctuations inconsumption, and thus, maintains a smoother consumption profile. It is well known thatthe presence of habit formation in a model results in hump-shaped dynamics for outputafter monetary shocks. Such dynamics are present in the VAR-based impulse responsedisplayed in Figure 2.2.The consumption aggregator, Ct, is defined as;Ct =??J?j=1n1/?pj C(?p?1)/?pjt???p/(?p?1)(2.2.2)4Since each household supplies a differentiated type of labor, the hours of work supplied by each worker(ht+s(i)) is indexed with i in the utility function.11where J , nj and ?p denote the number of sectors in the economy, the number of firmsin the sector j and the elasticity of substitution between any two sectors in the economy,respectively. The aggregator consumption function for a sector, Cjt is defined as:Cjt =??n?1/?pjnj?0cjt(j?)(?p?1)/?pdj????p/(?p?1)(2.2.3)Here, cjt(j?) denotes the consumption of the differentiated good j? in sector j. Theelasticity of substitution of differentiated goods within sectors is ?p.5 It is notable that theonly source of asymmetry among sectors is the frequency of price changes in sectors. Whenthe frequency of price changes is taken as equal among all sectors, the aggregator functionin the economy reduces to the standard aggregator function for the one-sector model.6The representative household supplies a differentiated type of hours of work for firms.Let ht(i) in (2.2.1) be the hours of work supplied by type i. The function H is assumed tobe convex and increasing with ht(i).The Household?s Budget ConstraintLet wt(i) denote the nominal wage demanded by the owner of the differentiated labor typei for an hour work. Then, the budget constraint of the household supplying that type oflabor can be written as,5The purpose of this paper is to investigate the effects of introducing heterogeneity in the frequencyof price changes among sectors on the deep parameter estimates. Thus, it is important that the onlymodification to the one-sector model is to drop the assumption of the same frequency of price changes inall sectors. If elasticities of substitution for the goods between the sectors in (2.2.2) and within the sectorsin (2.2.3) are allowed to differ, this second modification to the one-sector model would make it impossibleto isolate the effects of allowing different frequencies of price changes among sectors on the estimates ofdeep parameters.6It is notable that Ct is defined as the CES function of a finite number of sectors whereas Cjt is definedas the CES function of the consumption of a continuum of differentiated types of goods in a sector. In thecase where I assumed a finite number of differentiated types of goods in each sector, the overall price in asector would be affected by the price set by each firm in that sector. The continuum of differentiated typesassumption is required to circumvent this complication.12PtCt +Bt+1 ? W$t + wt(i)ht(i) +J?j=1?jt (2.2.4)The household starts the period with a given wealth of W$t . It has labor income from thesupply of the hours of its differentiated labor type, wt(i)ht(i). In addition, the householdhas profit income from sectors in the economy shown by?Jj=1 ?jt which will be specifiedlater. The household allocates its total resources between holding a portfolio of assets andconsumption. The price of the consumption is given by Pt. The portfolio of assets that isacquired in period t and adds to the household?s wealth in period t+ 1 is denoted by Bt+1.Since both safe and risky assets are available in the economy, a typical portfolio may bewritten as:Bt+1 =???st+1|stq(st+1|st)BRt+1(st+1|st)??+BSt+1 (2.2.5)where st+1 and st denote the realization of the state of the world in period t+1 and thehistory of the states realized until period t, respectively. The risky and safe asset holdingsof the household are represented by BRt+1 and BSt+1, respectively. Regardless of the realizedstate in t + 1, the safe asset pays a nominal interest of Rt. Any risky asset pays one unitin nominal terms if the state is such that the risky asset is traded and pays zero otherwise.When the state in period t+ 1 is realized as st+1, the total?st+1|st q(st+1|st)BRt+1(st+1|st)adds to the wealth in period t + 1 by an amount of only BRt+1(st+1|st). Accordingly, thewealth of the household for the next period can be written as:W$t+1(st+1|st) = BRt+1(st+1|st) +RtBSt+1 (2.2.6)It must be emphasized that the presence of complete capital markets provides full insur-13ance for workers in an environment where they face idiosyncratic shocks. This enables thewriting of a budget constraint for a representative agent in an economy with differentiatedlabor as noted in Woodford (2003, ch.3).Optimal Consumption and Asset Holdings for the HouseholdThe first-order conditions for consumption and the holding of safe and risky assets aregiven by (2.2.7), (2.2.8) and (2.2.9), respectively.Et?1(?U(Ct ? bCt?1)?Ct? b? ?U(Ct+1 ? bCt)?Ct)= Et?1?tPt (2.2.7)Et?1?t = Et?1?t+1Rt (2.2.8)Et?1?t+1 = Et?1?tQt,t+1 (2.2.9)In (2.2.9), Qt,t+1 denotes the stochastic discount factor between the periods t and t+ 1when the state in period t + 1 is realized as st+1 and is given by the ratio of the utilityvalue of having an extra unit of money in the two periods for the given realization of thestate in the period t+ 1.By definition, it must hold that:PtCt =J?j=1PjtCjtPjtCjt =nj?0pjt(j?)cjt(j?)dj?where Pjt and pjt(j?) denote the price index of sector j and the price of the differentiated14good j? in the sector, respectively. Using these definitions and those in (2.2.2) and (2.2.3),the optimality conditions for Cjt and cjt(j?) can be written as:nj(PtPjt)?pCt = Cjt (2.2.10)cjt(j?) = n?1j(Pjtpjt(j?))?pCjt (2.2.11)Substituting these optimal conditions back into (2.2.2) and (2.2.3) gives the aggregateand sectoral price indices:Pt =??J?j=1njP1??pjt??1/(1??p)(2.2.12)Pjt =??n?1jnj?0pjt(j?)1??pdj???1/(1??p)(2.2.13)Note that each sectoral price index has a weight equal to the number of firms in a sector(nj) in the aggregate price index. In the next section, when the structural parameters areestimated, njs are calibrated as the weights of the sectors in the CPI.The optimality condition for hours of work will be discussed when the workers? decisionproblem is considered below.2.2.2 FirmsThe production function of the firm producing the differentiated good j? in the sector j isgiven as:yjt(j?) = ZtHjt(j?)? 0 < ? < 1 (2.2.14)15where Zt is the technology level which is assumed to be common among all firms.Hjt(j?) denotes the firm?s demand for the hours of composite labor which will be specifiedbelow.Each firm hires a continuum of differentiated types of labor with a mass of one. Thehours worked by the continuum of differentiated types of labor is combined with the firmproducing the differentiated good j? in the sector j to form the hours of composite labor,Hjt(j?), with the following technology:Hjt(j?) =??1?0hjt(j?, i)(?w?1)/?wdi???w/(?w?1)(2.2.15)where hjt(j?, i) denotes the demand for hours of work of the differentiated labor i bythe firm producing the differentiated good j? in the sector j. Since each differentiated laborsupplies hours of work for a continuum of firms, each firm has a negligible effect on thewage paid to the differentiated labor for hours of work, wt(i), and that to the compositelabor, Wt, and takes wt(i) and Wt as given. From the firm?s cost minimization, one canshow the hours of work of each differentiated labor demanded by the firm producing thedifferentiated good j? in the sector j can be written as:hjt(j?, i) =(Wtwt(i))?wHjt(j?) (2.2.16)Substituting (2.2.16) in (2.2.15) gives the aggregate wage index for the composite hours:Wt =??1?0wt(i)1??wdj??11??w(2.2.17)Let ht(i) denote the total demand for the differentiated labor i. One can write ht(i) as:16ht(i) =(Wtwt(i))?wHt, Ht =?Jj=1? nj0 Hjt(j?)dj? (2.2.18)Before concluding this section, it is convenient to present the cost function of theproducer of the j?th good in sector j, TCjt(j?). Since the wage bill must be paid prior toproduction, TCjt(j?) is given by:TCjt(j?) = RtWtHjt(j?) =(yjt(j?)Zt) 1?RtWt (2.2.19)2.2.3 Equilibrium in the Frictionless EconomyIn this section, it is assumed in each period that households, firms and workers optimallydecide on consumption, prices and wages, respectively. When prices are perfectly flexible,firms are able to set prices optimally in each period. The objective function of a firm canthen be written as:maxpjt(j?)pjt(i)yjt(j?)? TCjt(j?) (2.2.20)If we denote the potential outcome values of each variable with superscript n, then,one can write the optimality condition for the price set by the firm under the frictionlessmarket assumption as:ynjt(j?) + pnjt(j?)?ynjt(j?)?pnjt(j?)=?TCnjt(j?)?ynjt(j?)?ynjt(j?)?pnjt(j?)(2.2.21)Using (2.2.11) and the fact that output is demand determined, one can show that:pjt(j?)n = ?p?TCnjt(j?)?ynjt(j?) , ?p =?p?p?1? 1 (2.2.22)17where ?p shows the markup over marginal output at the potential output due to productdifferentiation. Furthermore, using (2.2.19), one can write the marginal cost of the firm(Sjt+s(j?)) as:Sjt+s(j?) =?TCnjt(j?)?ynjt(j?)= 1?ynjt(j?)1??? Z? 1?t RntWnt (2.2.23)This implies that one can write (2.2.22) as:(pjt(j?)nPnjt)(PnjtPnt)= ?p1?ynjt(j?)1??? Z? 1?t Rnt(WntPnt)(2.2.24)Or equivalently,(Y ntyjt(j?)n)1/?p= ?p1?ynjt(j?)1??? Z? 1?t Rnt(WntPnt)(2.2.25)Since the firms are assumed to have only negligible influence on aggregate variables,they take all the terms in (2.2.25) as given except yjt(j?)n.7 It can now be shown that thevalue of yjt(j?)n which satisfies (2.2.25) is unique. Since the marginal cost is increasing inyjt(j?)n, the right-hand side is an increasing function of yjt(j?)n, but the left hand side of(2.2.25) is decreasing in yjt(j?)n. It can, thus, be concluded that yjt(j?)n should be uniquelydetermined for given Y nt , Rnt , Wnt , Pnt , and Zt. Consequently, the amount supplied by firmsin the same sector must be equal to each other. This implies one can alternatively writeyjt(j?)n as ynjt. Since there is a one-to-one relationship between the amount of the goodsupplied and the price set by firms when aggregate variables are given, the same supplyof goods implies the same price set by the firms in the same sector. Thus, one can easilyshow from (2.2.11) that7It is notable that the small-firms assumption is also used in writing (2.2.22).18njynjt = Y njt (2.2.26)Furthermore, in a frictionless economy, sectors are homogeneous in every aspect. Thus,the quantity supplied by firms in different sectors must be the same since they face thesame optimality condition given by (2.2.25). That is,ynjt = ynj?t for j 6= j? (2.2.27)Hence, one can also drop the subscript j from ynjt and simply write ynt Lastly, using(2.2.2), (2.2.26) and (2.2.27), it is easy to show that:Y nt = ynt(2.2.25) can, thus, be rewritten as:1 = ?p1?Y nt1??? Z? 1?t RntWnreal,t, Wnreal,t =WntPnt(2.2.28)where Wnreal,t stands for the real wage paid to the composite hours of work. Let thelog-deviations of a variable from its corresponding steady state be denoted with a hat overthis variable. One can then write (2.2.28) in log-linearized form as:0 = ?pY? nt ? (1 + ?p)Z?t + R?nt + W?nreal,t, ?p =1? ??(2.2.29)where ?p is the elasticity of prices with respect to the supply of goods when the interestrate and wages paid for composite hours of work remain unchanged.In order to show that monetary policy shocks are irrelevant in the determination ofpotential output, it is necessary to write each endogenous variable in (2.2.29) as a function19of potential output. First, R?nt can be written as the log-linear form of equation (2.2.8):R?nt = ???1x?nt + ??1Etx?nt+1 + Etpint+1 (2.2.30)where x?t is given by:x?t =((Y?t ? bY?t?1)? b?Et(Y?t+1 ? bY?t))(2.2.31)and ? represents the intertemporal elasticity of substitution given by:? = ?U(1? ?b), ? > 0 (2.2.32)In the absence of habit formation, the intertemporal elasticity of substitution would beequal to ?U where8?U = ?UcUccY?(2.2.33)When there is habit formation in preference, the intertemporal elasticity of substitutionis modified and given by ? in (2.2.32).In writing (2.2.30), steady state prices are normalized to one and price inflation isdefined as pit+1 = logPt+1 ? logPt. In the frictionless-economy case, there is a Nash8(2.2.33) can be rewritten as:?U = ??YY?UcUcConsider preferences without habit formation. It can be shown that?UcUc=Uc(Ct+1)? Uc(Ct)Uc(Ct)=Uc(Ct+1)Uc(Ct)? 1 =Pt+1?RtPt? 1Thus, ?UcUc shows the percentage change in the price of next period?s consumption relative to that of today?sconsumption. Hence, ?U indicates the percentage fall in next period?s consumption when the relative priceof next period?s consumption increases by 1%. Hence, it must hold that ?U > 0.20equilibrium in which all prices respond fully and instantaneously to an aggregate shock.In other words, following an aggregate shock to the economy, prices reach their new levelsimmediately and remain there (Etpint+1 = 0). This suggests that the last term in (2.2.30)vanishes.The last variable in (2.2.29), whose log-deviation from its steady state needs to beexpressed as a function of the log-linearized output gap, is Wnreal,t. To do this, I firstspecify the wage-setting environment. Workers have monopsonistic power over the hoursthey work. Hence, once they set their hourly wage, they are required to supply labor tosatisfy all the demand from the aggregate output-producing firm. When workers are ableto optimally set wages each period, the optimality condition for the wage set by the ownerof the differentiated labor type i for hourly work (wt(i)n) is given by (2.2.34):Hnh?ht(i)n?wt(i)n= ?nt(ht(i)n + wt(i)n?ht(i)n?wt(i)n)(2.2.34)Using (2.2.34) and (2.2.18), one can show that:wt(i)n = ?wHnh?nt, ?w =?w?w ? 1(2.2.35)where ?w shows the markup over the marginal cost of supplying more hours imposedby the differentiated worker when setting his wage. To explain (2.2.35), note that Hh isthe utility cost of working an extra hour for the differentiated labor type i. Dividing thisby ?t, the utility gain of having an extra unit of money in period t, results in a measureof the worker?s marginal cost in money terms. Hence, a worker?s wage is set by imposinga markup over the marginal cost of working an extra hour, where the marginal cost ofworking is expressed in money terms. When labor is not differentiated, workers demandthe perfectly-competitive wage which is equal to the marginal cost of working an extra21hour without any markup over this cost.Using (2.2.7) and (2.2.18), one can alternatively write (2.2.35) as:Wnreal,t(Hntht(i)n)1/?w= ?wHnhEt(?U(Y nt ?bYnt?1)?Y nt? b? ?U(Ynt+1?bYnt )?Y nt) (2.2.36)For given values of Wnreal,t, hnt and Y nt , it can be shown that the left-hand side ofthe above equation is a decreasing function of ht(i)n, whereas the right-hand side is anincreasing function of of the same variable.9 Thus, ht(i)n must be uniquely determined fromthis equation. Under the flexible-wages assumption, ht(i)n is the same for all differentiatedlabor types since all labor suppliers are alike. This implies hnt (i) = Hnt and wnt (i) = Wnt .Taking the log-linear approximation of (2.2.36) yields that:W?nreal,t = ??1H H?nt + ??1x?nt (2.2.37)where ?H denotes the elasticity of the number of hours with respect to real wagechanges when the marginal utility of real income is constant (i.e., the Frisch-elasticity oflabor supply). It is given as:?H =HhHhhH?> 0 (2.2.38)Using (2.2.14) and (2.2.37), it is easy to show that:W?nreal,t = ?w(Y? nt ? Z?t)+ ??1Etx?nt , ?w =??1H?(2.2.39)where ?w denotes the output elasticity of the real wage for a constant marginal utilityof real income.9The right-hand side is increasing in ht(i)n as H is assumed to be increasing and convex in ht(i)n.22When (2.2.30) and (2.2.39) are substituted into (2.2.29), one obtains:0 =(?w + ?p)Y? nt ?(?w + 1 + ?p)Z?t + ??1Etx?nt+1 (2.2.40)Thus, monetary policy shocks are irrelevant as they do not appear in the equilibriumvalue of Y? nt in the frictionless economy which is determined by equation 2.2.40.2.2.4 The Economy with Nominal FrictionsSticky PricesIt is assumed that firms set prices one period in advance. In each sector, firms optimizetheir prices only when a price-change signal is received. The probability of receiving sucha signal is different in each sector and is given by 1 ? ?pj for sector j. The fraction offirms in sector j which receive a price change signal in each period is also given by 1??pj .When no such a signal is received, firms are assumed to set their prices according to thefollowing partial adjustment backward-looking indexation rule:p?jt(j?) = pjt?1(j?)(Pt?1Pt?2)?p(2.2.41)The tilde over p denotes the price set according to the backward-looking indexationrule. As noted by Woodford (2003), such a rule helps to explain that the peak effect of aninterest rate shock on inflation occurs later than the peak effect of the shock on the outputgap in the VAR.When a firm is capable of setting an optimal price, it sets pjt(j?)? to maximizeEt?1(??s=0?spjQt,t+s?jt+s(j?))(2.2.42)where Qt,t+s is the stochastic discount factor between period t and t + s and is given23by:Et?1(?t+s?s)= Et?1(Qt,t+s) (2.2.43)(2.2.42) implies that firms and households have the same stochastic discount factor.The profit of firm i in sector j is given by:10?jt+s(j?) = p?jt+s,t(i)yjt+s(j?)? TCjt+s(j?)where TCjt(j?) is the total cost of the firm, as given in (2.2.19), and p?jt+s,t(j?) is theprice set in period t + s by the firm that received a price-change signal in period t andwhich does not have the opportunity to set an optimal price between t and t+ s. Due tothe backward-indexation rule, one can write p?jt+s,t(j?) as:p?jt+s,t(j?) = p?jt(j?)?pt,t+s (2.2.44)where?pt,t+s =????????sk=1(Pt+k?1Pt+k?2)?pif s ? 11 if s = 0(2.2.45)The optimality condition in (2.2.42) for p?jt(j?) can be expressed as:Et?1(??s=0?spjQt,t+sd?jt+s(j?)dp?jt(j?))= 0 (2.2.46)From this, one can show that (2.2.47) is true:10It is notable that since firms are assumed to respond to monetary shocks with a one period delay,they have to condition their optimum price based on information until period t ? 1 rather than period t.Correspondingly, Et?1 appears in the objective function in (2.2.42), rather than Et.24Et?1(??s=0 ?spjQt,t+s?pt,t+s(Pt+sp?jt+s,t(j?))1+?pYt+s?(p?jt+s,t(j?)Pjt+sPjt+sPt+s? ?p Sjt+s(j?)Pjt+sPjt+sPt+s))= 0(2.2.47)and log-linearizing this equation yields:Et?1??s=0(??pj)s[p??jt(j?)? P?jt+s + P?jt+s ? P?t+s + ??pt,t+s?(R?t+s + W?real,t+s +(?py?jt+s(j?)? (1 + ?p)Z?t+s))]= 0(2.2.48)where ??pt,t+s is the log-deviation of ?pt,t+s from its steady state given by:??pt,t+s =????????ppit + ?ppit+1 + ? ? ?+ ?ppit+s?1 if s ? 10 if s = 0(2.2.49)In Appendix A.1, the dynamic equation for sectoral inflation (pijt = P?jt ? P?jt?1) isshown to evolve according topijt ? ?ppit?1 = ??pj(1 + ?p?p)Et?1(P?jt ? P?t)+?pjEt?1(R?t + W?real,t +(?pY?t ? (1 + ?p)Z?t))+ ?Et?1(pijt+1 ? ?ppit) (2.2.50)where ?pj is given by:?pj =1? ?pj?pj1? ??pj1 + ?p?p(2.2.51)Hence, sectoral inflation is a decreasing function of relative sectoral prices11, an increas-11This can be explained as follows, if the relative price of a sector was low in one period, the demand forits good would be high. This would induce firms in that sector to raise prices since their marginal cost is25ing function of the log-deviation of real marginal cost from its potential output level andan increasing function of expected sectoral inflation in the next period.It is notable that ?pj is decreasing and convex in ?pj , implying other things beingequal, the relative sectoral price today matters most for the sector where prices changemost frequently. This can be explained as follows: firms in sectors with a low frequencyof price changes expect not to reoptimize their prices for a considerable period of time.One may argue that this leads them to more often consider relative prices in subsequentperiods compared to firms in sectors where they are able to optimize their prices frequently.This leads the less-frequently optimizing firms to place greater weight on relative prices insubsequent periods and to place lower weight on relative prices today.It should be noted that the one-sector model differs from the multi-sector model onlyin terms of the equation governing the aggregate inflation. When sectors are homogeneousin their price flexibility, (2.2.50) reduces to:pit??ppit?1 = ?pEt?1(R?t+W?real,t+(?pY?t? (1+?p)Z?t))+?Et?1(pit+1??ppit)(2.2.52)where ?p is given by:12?p =1? ?p?p1? ??p1 + ?p?pSticky Wages for Hours of WorkThe hours of work are differentiated among suppliers. There is a continuum of differentiatedsuppliers of hours of work in the economy with a mass of one. The owner of each typeconvex and increasing in the output they supply.12In (2.2.51), in place of 1??jp, the weighted average of the frequency of price changes in sectors, whichI denote with (1? ?p), is used when writing ?p above.26sets an hourly wage in the monopsonistically competitive market and is prepared to supplythe number of hours demanded at this wage. Each supplier has a chance to optimize hiswages only when a wage-change signal is received. The probability of receiving such signalis given by 1? ?w. The fraction of the types receiving the signal is also given by 1? ?w.When such a signal is not received, workers set their wages by taking into account thechange in inflation in the last period:w?t(i) = wt?1(i)(Pt?1Pt?2)?w(2.2.53)When the wage-change signal is received, the objective of the owner of the differentiatedlabor type i is to set an hourly wage, wt(i)?, which lasts until the new wage-change signalis received. Indeed, the problem can be expressed as follows: 13maxwt(i)? Et?1{??s=0 ?sw(??sH(ht+s(i)) + ?t+sw?t+s,t(i)ht+s(i))} (2.2.54)where w?t+s,t(i) shows the wage set in period t+s by the supplier of hours of differentiatedlabor type i who optimized his wage at period t and is unable to reoptimize between periodt and t+ s, respectively. From (2.2.53), w?t+s,t(i) is given as:w?t+s,t(i) = w?t (i)?wt,t+s (2.2.55)where13Since the nominal wages are set one period in advance, the expectation operator is taken as Et?1 in(2.2.54).27?wt,t+s =????????sk?=1(Pt+k??1Pt+k??2)?wif s ? 11 if s = 0(2.2.56)In Appendix A.4, the wage inflation for the composite hours of work is shown to evolveaccording to (2.2.57):piwt ??wpit?1 = ?wEt?1(??1H?(Y?t ? Z?t)+??1x?t?W?real,t)+?Et?1(piwt+1??wpit)(2.2.57)where piwt and ?w are defined aspiwt = logWt ? logWt?1?w =(1? ?w)?w1? ?w?(1 + ?w??1H )Hence, the wage inflation of hours worked is an increasing function of output andexpected wage inflation and a decreasing function of the marginal utility from real income.Lastly, it needs to be noted that the nominal wage for hours worked falls with the realwage paid per hour of composite labor.2.2.5 The IS EquationLog-linearizing (2.2.7) and (2.2.8) yields that:Et?1x?t = Et?1x?t+1 ? ?Et?1(R?t ? pit+1)(2.2.58)where x?t is defined in (2.2.31). This equation, (2.2.58), implies the presence of the28working capital channel produces a higher response of output to an expansionary shockduring the periods directly following the shock. To see this, consider first the simple casewithout habit formation so that equation (2.2.58) reduces to a dynamic equation for realGDP. Since the working capital channel leads firms to set lower prices after an unanticipatedfall in the interest rate, it can be seen from (2.2.58) that lower inflation with the workingcapital channel produces a larger increase in output due to the expansionary shock forgiven values of the model?s parameters and expected output in the next period. Withhabit formation in preferences, on the other hand, as a household avoids large changes inits consumption pattern, the effect of the working capital channel on output is likely tolessen. However, the model would still produce a larger output increase with the workingcapital channel.2.2.6 Monetary PolicyThe recursive assumption that is used to identify monetary policy shocks in the VAR aboveimplies the current values of the output gap, inflation and real wage are predetermined.With this assumption, the OLS estimates become consistent. Using this and the fact thatfour lags of each variable are included in identifying the monetary policy shocks in theVAR model, I follow the standard practice and characterize the monetary policy with thefollowing Taylor rule:Rt = c+4?j=0?jy(Yt?j ? Y nt?j)+4?j=0?jpipit?j +4?j=0?jwWreal,t?j +4?j=1?jRRt?j + mt (2.2.59)where Rt denotes the federal funds rate and c is a constant term which denotes thesteady-state interest rate. ?y, ?jpi and ?jw show the response of the federal funds rate to thelagged j values of the output gap, price inflation and real wage, respectively. The response29Table 2.1: Estimated Monetary Policy Rule: 1959Q1-2013Q1?jy ?jpi ?jw ?jRj=0 0.36 0.06 -0.11j=1 -0.32 0.04 0.27 1.05j=2 0.01 0.16 -0.11 -0.31j=3 -0.01 -0.07 0.05 0.32j=4 0.02 -0.10 -0.08 -0.11Note: For the definitions of ?jy, ?jpi, ?jw and ?jR, see (2.2.59).of the federal funds rate to its own lags are denoted by ?jR. Lastly, mt is the monetaryshock in the period which is introduced by the Federal Reserve and is uncorrelated acrosstime and orthogonal to the explanatory variables of the policy rule. In Table 2.1, I reportthe estimates of ?jy, ?jpi, ?jw and ?jR.2.3 Econometric EstimationLet g(P) and f(P) be the model-based impulse responses for a given vector of modelparameters (P) in the one- and multi-sector models, respectively. The estimated vectorof model parameters in the multi-sector model (PM (A?n)) is given as the minimizer of thefollowing classical minimum distance measure:P?M (A?n) = arg minP(h?n ? f(P))?A??nA?n(h?n ? f(P)) (2.3.1)where A?n and h?n are the weighting matrix used and the estimated VAR-based im-30pulse response of the output gap, inflation, the real wage and the federal funds rate to anunanticipated 1% reduction in the federal funds rate between the 1st and 20th quarters,respectively. Lastly, n stands for the sample size of the data used to estimate the VAR-based impulse responses. Since using different weighting matrices would yield a differentestimator, P?M is written as a function of A?n. Similarly, the estimated vector of parame-ters in the one-sector model (PO(A?n)) are given as the minimizer of the following classicalminimum distance measure:P?O(A?n) = arg minP(h?n ? g(P))?A??nA?n(h?n ? g(P)) (2.3.2)As a weighting matrix, I use the diagonal matrix A?n where the diagonal elements aregiven by the inverse of the standard deviations of the VAR-based impulse responses.14This weighting matrix ensures that the deep parameters are estimated such that the more-precisely estimated impulse responses are given more importance.2.3.1 Calibrated Parameters of the ModelThere is information on price flexibility for approximately 270 ELI categories in Nakamura& Steinsson (2008a). Even if it is ideal to use all information about frequencies of pricechange for ELI in the CPI documented in Nakamura & Steinsson (2008a), adding sucha large number of sectors to the model results in an excessive number of variables in thedynamic system in the DSGE model which is beyond the computation limit of today?stechnology. As a practical solution, the number of sectors to be included in the multi-14Assume that n12 (h?n?h0) ? N(0,?0) where h0 denotes true impulse responses and ?0 is the asymptoticvariance-covariance matrix of the VAR-based impulse responses. In terms of efficiency, in the case ofcorrectly specified models, one might suggest using ???1n in place of A??nA?n. However, this hinders thestability of the minimization algorithm used in estimation. Giannoni & Woodford (2003) also note the sameproblem in their paper. To visualize how stability of the minimization algorithm is affected, I illustrate themodel-based impulse responses by using the efficient weighting matrix ???1 in place of A??nA?n in AppendixA.5.31Table 2.2: The Quarterly Frequencies of Price Adjustment over DifferentPercentiles of the Price Flexibility and Their Implied DurationsPercentiles Frequency Duration Weights(%) (Quarters) (%)One-Sector Economy 40.0 1.95 100.0Multi-Sector Economy0-10 10.0 9.49 14.910-20 19.8 4.53 18.420-30 28.8 2.94 7.030-40 37.5 2.13 6.140-50 45.6 1.64 4.250-60 53.5 1.30 4.660-70 59.4 1.11 5.370-80 67.3 0.89 11.680-90 74.9 0.72 9.090-100 90.9 0.42 19.0Note: Weights indicates the weights (nj) in the 2000 CPI expenditures of Entry Level Items (ELIs)given in Nakamura & Steinsson (2008a). Frequency refers to the percentage of firms which adjusttheir prices in a quarter. The frequencies in the multi-sector economy above show the median of thequarterly frequencies for different percentiles of price flexibility. The median implied duration inquarters for percentiles of price flexibility in the multi-sector economy are computed from the medianfrequencies with the formula ? 1ln(1?fq) where fq refers to the median quarterly frequencies in eachpercentile. The frequencies in the one-sector economy (fqOne) above show the weighted averagequarterly frequencies defined as (fqOne =?Jj=1 njfqj ) where fqj denotes the estimated quarterlyfrequency of ELIs. Lastly, the duration of price contracts in the one-sector economy is estimatedas ? 1ln(1?fqOne) .sector model is reduced to 10 in the following way. First, sectors are ordered according totheir quarterly frequency of price changes from lowest to highest using the data given inNakamura & Steinsson (2008a). Secondly, 10 percentile groups of price flexibility in the ELIcategories are formed and all ELI categories are included in one of these ten groups. Table322.2 gives some descriptive statistics about the price flexibility of the percentile groups.It must be noted that the frequencies of price changes in Nakamura & Steinsson (2008a)are expressed as monthly percentages for the ELIs. Estimating quarterly frequencies in-volves two steps. First, the expected length of a price quotation in a sector (dk) in Naka-mura & Steinsson (2008a) is estimated as 1?ln(1?fmk ) where fmk denotes the monthly fre-quency of price changes in sector k. Put differently, fmk can be rewritten as:fmk = 1? e? 1dkTo relate fmk to the Calvo (1983) model, let ?k be the constant hazard rate in sectork in the Calvo (1983) model. Note that of the total number of firms which set prices inperiod s < t, a share,1? e??k(t?s) (2.3.3)will again have received a price change signal between s and t in the Calvo (1983) model.If the unit of time is taken as a month (t-s=1) and ?k is set equal to 1dk , one can relate thefrequencies of price changes in the Calvo (1983) model to those reported in Nakamura &Steinsson (2008a).Second, I estimate the quarterly frequency of price changes in the Calvo (1983) model(f qj ) byf qj = 1? e?3??k (2.3.4)where t?s in (2.3.3) is assumed to be three since unlike Nakamura & Steinsson (2008a)who estimate monthly frequencies, I estimate three-month (quarterly) frequencies.In the one-sector economy, the weighted average of the quarterly frequency of priceadjustment in the overall economy is estimated as?Jj=1 njfqj where njs are the shares in33Table 2.3: The Calibrated ParametersOne-Sector Model Multi-Sector Model? = 0.99 ? = 0.99?w = 0.83 ?w = 0.83?p = 60.0 Sectors Stickiness Weight in PtSector 1 ?p1 = 0.90 n1 = 0.15Sector 2 ?p2 = 0.80 n2 = 0.18Sector 3 ?p3 = 0.71 n3 = 0.07Sector 4 ?p4 = 0.62 n4 = 0.06Sector 5 ?p5 = 0.54 n5 = 0.04Sector 6 ?p6 = 0.46 n6 = 0.05Sector 7 ?p7 = 0.40 n7 = 0.05Sector 8 ?p8 = 0.33 n8 = 0.12Sector 9 ?p9 = 0.25 n9 = 0.09Sector 10 ?p10 = 0.09 n10 = 0.19the 2000 CPI expenditures of ELIs reported in Nakamura & Steinsson (2008a) and f qj ismeasured using (2.3.4). The estimated value is about 40%. The implied duration indicatesa typical price contract in the United States lasts less than 1.95 quarters. Such duration islower than the value of the calibrated duration of price contracts in Giannoni & Woodford(2003). The calibrated duration in their paper is based on previous survey studies, such34as Blinder, Canetti, Lebow & Rudd (1998), which are not in line with the recent findingson price changes of ELI categories. Weights in the table correspond to the weights of thepercentile groups computed using the weights given in Nakamura & Steinsson (2008a).15The calibrated values for sectoral price rigidities (?pjs) in Table (2.3) correspond to 1?f q,where f q shows the frequency of quarterly price adjustments for the percentiles in Table2.2. It is evident in this table that the frequencies of price adjustment differ significantlyamong the percentile groups. More interestingly, the weights of the middle percentiles aregenerally less than those of the top and bottom percentiles. Thus, for the evolution ofoverall prices, it is more relevant to study the price dynamics in the top and the bottompercentiles than those in the middle percentiles.Other calibrated parameters for the one- and multi-sector models in Table (2.3) are ?and ?w. The calibrated value for ? is 0.99, which implies an annualized net real interestrate of 1%. The calibrated value for ?w is 0.83, which indicates an expected duration of5.6 quarters for a typical wage contract as found in Barattieri et al. (2010).16The sectors? frequencies still differ from their original frequencies in the multi-sectormodel since the frequencies in the same percentile group are equated to the median fre-quency of the group. However, the disparities in the sectoral frequencies are much higherin the one-sector economy where the frequencies of price changes for all sectors are equatedto the weighted average of the frequency of price changes in the aggregate economy. In this15It is notable that were the correlation between the weights and the frequencies of price changes insectors a strongly positive (negative), heterogeneity in price flexibility across sectors would only play asmall effect on the aggregate price dynamics as it would result in such dynamics being largely determinedby fast-adjusting (slow-adjusting) sectors and that slow-adjusting (fast-adjusting) sectors would have smalleffects on such dynamics. However, the correlation between sectoral weights and sectoral frequencies inTable 2.2 is almost zero. Hence, there is no a priori reason to expect that sectoral heterogeneity in priceflexibility may only have trivial effects on the aggregate dynamics.16Assuming wage rigidity is the same across sectors in the multi-sector model may be of some concernsince price dynamics across sectors following a monetary shock under this assumption may significantlydiffer from those under different wage rigidities across sectors. However, this concern is unwarranted sinceBarattieri et al. (2010) find little evidence of heterogeneity in the frequency of wage adjustment acrossindustries in the United States.35respect, the multi-sector economy is much closer to reality than the one-sector economy.2.3.2 Estimated and Implied ParametersThe calibrated parameters above reduce the parameters of the model to be estimated. Inaddition to these, one can further reduce the number of parameters to be estimated in thefollowing way. Labor has a share of 0.66 at the steady state, where the price set by thefirm producing the aggregate composite good must satisfy (2.2.22). Using this fact and(2.2.19), it is easy to show that the steady-state share of labor is given by:11?R??p= ??p??1= 0.66 (2.3.5)Thus, it must hold that:? = 0.66 ?p?p ? 1??1 (2.3.6)Hence, with the calibrated value of ?, ? can be obtained from (2.3.6) when ?p isestimated, and thus, it can be omitted from the list of variables which should be estimated.The list of parameters (P) which need to be estimated includes P = [?, ?H, ?p, ?w, b, ?p, ?w].Table 2.4 gives the estimated structural parameters for the one- and multi-sector mod-els.17 The estimates of the intertemporal elasticity of substitution (IES) (?) is quite closeto zero. Although, there is contrasting findings regarding IES, the estimated values in themodels are largely consistent with the empirical estimates of IESs in the literature.18 The17Apart from theoretical restrictions, the parameters are not restricted: ? > 0, ?H > 0, ?p > 1, ?w > 1,1?b?0, 1??p?0 and 1??w?0.18Hall (1988) estimates IES for the post-war period in the United States and finds that IES can be aslow as 0.1 and may well be equal to zero. Hall (1988) attributes the substantial IES estimate reportedin Summers (1984) for the post-war period to the invalid instrument used in the latter study. When alist of valid instruments is used for expected real returns, IES is estimated to be barely positive and notsignificantly different from zero. Similarly, the low values of IES given in Table 2.4 are also consistent withthe empirical estimate of IES in Boldrin, Christiano & Fisher (2001) (0.09) for the sample period of 1964-I36Table 2.4: Estimates of Structural Parameters(a) (b)One-Sector Model Multi-Sector Model? 0.011 0.011(0.009) (0.009)?H 0.066 0.001(0.645?) (0.762?)?p 67.232 19.621(82.695?) (20.948?)?w 5.695 5.192(6.453?) (11.755?)b 0.867 0.886(0.249?) (0.280?)?w 0.999 0.999(0.226?) (0.184?)?p 0.946 0.925(0.120?) (0.120?)Obj. Func. 14.267 11.795Note: The numbers in parentheses are standard errors. The asterisks next to the reported standard errorsindicate the estimated parameter values are close to their theoretical limits, and thus, their standard errorsmay be unreliable.estimated values of the Frisch-elasticity of labor supply (?H) in Table 2.4 are also close tozero. These values lie in the range of the estimates of the Frisch-elasticity of labor supplyand 1988-II as it lies within one standard deviation of the point estimate. However, the estimated valuesfor IES in Table 2.4 are not in line with those in Beaudry & van Wincoop (1996). Indeed, they estimate thevalue of IES to be close to one for non-durable consumption by using a panel of state-level data. However,the estimates for the entire United States economy are substantially lower even though they are found tobe positive and statistically different from zero.37Table 2.5: Estimates of Implied ParametersOne-Sector Model Multi-Sector Model?w = 1.131 ?w = 1.239 ?p1 = 0.001?p = 1.037 ?p = 1.054 ?p2 = 0.005?U = 0.075 ?U = 0.086 ?p3 = 0.013? = 0.677 ? = 0.702 ?p4 = 0.025?p = 0.478 ?p = 0.424 ?p5 = 0.042?w = 22.408 ?w = 1423.550 ?p6 = 0.067?w = 0.000 ?w = 0.000 ?p7 = 0.094?p = 0.008 ?p8 = 0.149?p9 = 0.241?p10 = 0.971in the published literature reported in Browning, Hansen & Heckman (1999). Regardingthe estimates of b, ?w and ?p, it can be noted that habit persistence in consumption andbackward-looking indexation in both wages and prices are large, which are also in line withthe estimated values in previous studies (See, for example, Giannoni & Woodford (2003)).Table 2.4 also shows that the one- and multi-sector models differ to some extent in theircapacity to account for the dynamics after the shock. The lower objective function valuefor the multi-sector model indicates that this model is, to some extent, more successfulthan the one-sector model in approximating the dynamic behavior of the variables after38the shock.Three points should be noted about the estimates of implied parameters in Table 2.5.First, the values for ?w and ?p indicate the markups in wage and price settings are low.Second, ?w = ??1H? has a very large value. This suggests that when wages and prices canbe set optimally in each period, an increase in output results in an unduly large increasein real wages because of a very low estimated value of the Frisch-elasticity of labor supply.However, given that wages are staggered and workers must supply any amount of hoursdemanded by firms at the set wages, the dynamic behavior of wages in the model is withinthe 95% confidence bands, despite a rise of a 0.5% in output following the shock. Third,the arithmetic mean of the slope coefficients in the inflation equations in the multi-sectormodel (?pj) is much higher than the slope coefficient in the one-sector model (?p) due tothe fact that ?pj is convex in ?pj .2.3.3 Impulse Responses Predicted by the ModelsFigure 2.3 and 2.4 show the predicted responses of the output gap, inflation, the real wageand the federal funds rate to an unanticipated 1% fall in interest rates over a five yearperiod in the one- and multi-sector models, respectively. Assuming that the monetarypolicy shocks can be recovered correctly with the recursive assumption in the VAR, boththe former and the latter models can be considered successful in explaining the reactionsof the variables included in the VAR model above. As in all periods, the responses staywithin the 95% confidence intervals.Regarding aggregate dynamics, by construction, the contemporaneous impulse re-sponses of the output gap, inflation and the real wage to the shock are zero. Startingwith the first quarter, the output gap increases due to unexpectedly low real interest ratesfollowing the expansionary interest shock. Initial responses of inflation are negative due39Figure 2.3: Impulse Responses to an Unanticipated 1% Fall in Rt(One-Sector Model)(a) Yt ? Y nt0 4 8 12 16 20?0.500.511.52QuartersPercent(b) pit0 4 8 12 16 20?0.4?0.200.20.40.60.81QuartersPercent(c) Wreal,t0 4 8 12 16 20?0.3?0.2?0.100.10.20.30.40.5QuartersPercent(d) Rt0 4 8 12 16 20?1.5?1?0.500.51QuartersPercentNote: The solid lines represent the VAR-based impulse responses and the area between dashed linesindicate the 95% confidence intervals estimated using the method suggested by Sims & Zha (1999). Thesolid lines marked with circles show the dynamic responses of the variables as predicted by the model.to the working-capital channel. Over time, inflation rises above its undistorted path dueto an increase in output, interest rates and nominal wages which cause marginal costs to40Figure 2.4: Impulse Responses to an Unanticipated 1% Fall in Rt(Multi-Sector Model)(a) Yt ? Y nt0 4 8 12 16 20?0.500.511.52QuartersPercent(b) pit0 4 8 12 16 20?0.4?0.200.20.40.60.81QuartersPercent(c) Wreal,t0 4 8 12 16 20?0.3?0.2?0.100.10.20.30.40.5QuartersPercent(d) Rt0 4 8 12 16 20?1.5?1?0.500.51QuartersPercentNote: The solid lines represent the VAR-based impulse responses and the area between dashed linesindicate the 95% confidence intervals estimated using the method suggested by Sims & Zha (1999). Thesolid lines marked with circles show the dynamic responses of the variables as predicted by the model.rise. The initial positive responses of the real wage result from the decrease in inflationand marginal utility of real income after the shock.41How does the one-sector model differ from the multi-sector model? In comparing Figure2.3 and Figure 2.4, it is notable that even if the multi-sector model outperforms the one-sector model in matching the estimated VAR-based impulse responses, when uncertainty inthe VAR-based impulse responses are considered, the success of the former model over thelatter model fades. At the very least, it is difficult to distinguish the aggregate dynamicsin the one-sector model from those in the multi-sector model in Figure 2.3 and Figure 2.4.It is therefore not possible to conclude the improvement in the multi-sector model relativethe one-sector model is important.2.3.4 A Comparison of Model-Based Impulse Responses underAlternative Taylor RulesVAR-based impulse responses may provide a good explanation of the outcome of a mone-tary shock for a given Taylor rule assuming the estimated VAR-based policy shocks cor-rectly identify the true shocks. VAR models, however, are unable to predict the dynamicbehavior of variables after a shock to the interest rate when a new monetary regime isconsidered. This results from the fact that inference in VAR models is susceptible to theLucas Critique.DSGE models are useful as they may help in predicting the dynamic behavior of vari-ables after a monetary policy shock when a new policy regime is implemented. Workingwith such models, however, necessitates making simplifying assumptions. In the contextof this paper, for example, the assumption that all sectors have the same price flexibilityrepresents a simplification assumption made in the one-sector model. Should the model-based impulse responses sharply differ between the models under different Taylor rules, thiswould call into question the inference drawn in the one-sector model since the simplifyingassumption in this model is unrealistic. In this section, I investigate this by consider-42Figure 2.5: The Model-Based Impulse Responses in the One- and Multi-SectorModels under Alternative Taylor Rules after an Unanticipated 1% Fall in Rt?=0.68 , ?y=0.27 , ?pi=1.05(a) Yt ? Y nt0 4 8 12 16 20?0.1?0.0500.050.10.150.20.25QuartersPercent(b) pit0 4 8 12 16 20?0.2?0.15?0.1?0.0500.050.10.15QuartersPercent(c) Wreal,t0 4 8 12 16 20?0.1?0.0500.050.10.150.20.25QuartersPercent(d) Rt0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.4QuartersPercent?=0.68 , ?y=0.27 , ?pi=1.60(e) Yt ? Y nt0 4 8 12 16 20?0.0500.050.10.150.2QuartersPercent(f) pit0 4 8 12 16 20?0.2?0.15?0.1?0.0500.050.10.15QuartersPercent(g) Wreal,t0 4 8 12 16 20?0.0500.050.10.150.2QuartersPercent(h) Rt0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.4QuartersPercent?=0.68 , ?y=0.27 , ?pi=2.15(i) Yt ? Y nt0 4 8 12 16 20?0.1?0.0500.050.10.150.20.25QuartersPercent(j) pit0 4 8 12 16 20?0.2?0.15?0.1?0.0500.050.10.15QuartersPercent(k) Wreal,t0 4 8 12 16 20?0.1?0.0500.050.10.150.20.25QuartersPercent(l) Rt0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.4QuartersPercentNote: The solid lines show the multi-sector model-based impulse responses and the area between dashedlines indicate the 95% model-based confidence intervals for the multi-sector model. The dotted lines markedwith asterisks display the dynamic responses of the variables as predicted by the one-sector model.ing different sets of coefficients for the Taylor rule equation and document the resultingmodel-based impulse responses for the one- and multi-sector models. In this theoreticalexperiment, the interest rate rule corresponds to the following Taylor rule considered byClarida, Gali & Gertler (1999):43Figure 2.5: The Model-Based Impulse Responses in the One- and Multi-SectorModels under Alternative Taylor Rules after an Unanticipated 1% Fall in Rt(cont.)?=0.68 , ?y=0.60 , ?pi=1.05(m) Yt ? Y nt0 4 8 12 16 20?0.0500.050.10.15QuartersPercent(n) pit0 4 8 12 16 20?0.2?0.15?0.1?0.0500.050.10.15QuartersPercent(o) Wreal,t0 4 8 12 16 20?0.1?0.0500.050.10.150.20.25QuartersPercent(p) Rt0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.4QuartersPercent?=0.68 , ?y=0.60 , ?pi=1.60(q) Yt ? Y nt0 4 8 12 16 20?0.1?0.0500.050.10.15QuartersPercent(r) pit0 4 8 12 16 20?0.2?0.15?0.1?0.0500.050.10.15QuartersPercent(s) Wreal,t0 4 8 12 16 20?0.0500.050.10.15QuartersPercent(t) Rt0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.4QuartersPercent?=0.68 , ?y=0.60 , ?pi=2.15(u) Yt ? Y nt0 4 8 12 16 20?0.0500.050.10.150.2QuartersPercent(v) pit0 4 8 12 16 20?0.2?0.15?0.1?0.0500.050.10.15QuartersPercent(w) Wreal,t0 4 8 12 16 20?0.0500.050.10.150.2QuartersPercent(x) Rt0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.4QuartersPercentNote: The solid lines show the multi-sector model-based impulse responses and the area betweendashed lines indicate the 95% model-based confidence intervals for the multi-sector model. Thedotted lines marked with asterisks display the dynamic responses of the variables as predicted bythe one-sector model.R?t = ?+ ?pi (Etpit+1 ? p?i) + ?y(Yt?j ? Y nt?j)44Figure 2.5: The Model-Based Impulse Responses in the One- and Multi-SectorModels under Alternative Taylor Rules after an Unanticipated 1% Fall in Rt(cont.)?=0.68 , ?y=0.93 , ?pi=1.05(y) Yt ? Y nt0 4 8 12 16 20?0.0500.050.10.15QuartersPercent(z) pit0 4 8 12 16 20?0.2?0.100.10.20.3QuartersPercent(aa) Wreal,t0 4 8 12 16 20?0.2?0.100.10.2QuartersPercent(ab) Rt0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.4QuartersPercent?=0.68 , ?y=0.93 , ?pi=1.60(ac) Yt ? Y nt0 4 8 12 16 20?0.0500.050.10.15QuartersPercent(ad) pit0 4 8 12 16 20?0.15?0.1?0.0500.050.1QuartersPercent(ae) Wreal,t0 4 8 12 16 20?0.0500.050.10.15QuartersPercent(af) Rt0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.4QuartersPercent?=0.68 , ?y=0.93 , ?pi=2.15(ag) Yt ? Y nt0 4 8 12 16 20?0.0500.050.10.15QuartersPercent(ah) pit0 4 8 12 16 20?0.2?0.15?0.1?0.0500.050.10.15QuartersPercent(ai) Wreal,t0 4 8 12 16 20?0.0500.050.10.150.2QuartersPercent(aj) Rt0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.4QuartersPercentNote: The solid lines show the multi-sector model-based impulse responses and the area betweendashed lines indicate the 95% model-based confidence intervals for the multi-sector model. Thedotted lines marked with asterisks display the dynamic responses of the variables as predicted bythe one-sector model.Rt = ?Rt?1 + (1? ?)R?twhere ? is the steady-state nominal interest rate and the bars over variables refer45Figure 2.5: The Model-Based Impulse Responses in the One- and Multi-SectorModels under Alternative Taylor Rules after an Unanticipated 1% Fall in Rt(cont.)?=0.79 , ?y=0.27 , ?pi=1.05(ak) Yt ? Y nt0 4 8 12 16 20?0.2?0.100.10.20.30.40.5QuartersPercent(al) pit0 4 8 12 16 20?0.2?0.100.10.20.3QuartersPercent(am) Wreal,t0 4 8 12 16 20?0.2?0.100.10.20.3QuartersPercent(an) Rt0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.4QuartersPercent?=0.79 , ?y=0.27 , ?pi=1.60(ao) Yt ? Y nt0 4 8 12 16 20?0.2?0.100.10.20.30.40.5QuartersPercent(ap) pit0 4 8 12 16 20?0.2?0.100.10.2QuartersPercent(aq) Wreal,t0 4 8 12 16 20?0.0500.050.10.150.2QuartersPercent(ar) Rt0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.4QuartersPercent?=0.79 , ?y=0.27 , ?pi=2.15(as) Yt ? Y nt0 4 8 12 16 20?0.2?0.100.10.20.30.40.5QuartersPercent(at) pit0 4 8 12 16 20?0.3?0.2?0.100.10.2QuartersPercent(au) Wreal,t0 4 8 12 16 20?0.100.10.20.3QuartersPercent(av) Rt0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.4QuartersPercentNote: The solid lines show the multi-sector model-based impulse responses and the area betweendashed lines indicate the 95% model-based confidence intervals for the multi-sector model. Thedotted lines marked with asterisks display the dynamic responses of the variables as predicted bythe one-sector model.to their steady-state values. ? is the interest rate smoothing parameter determined bythe monetary authority. Under this policy rule, the target interest rate, R?t , is not setinstantaneously. The sets of values for ?, ?pi and ?y are as follows:46Figure 2.5: The Model-Based Impulse Responses in the One- and Multi-SectorModels under Alternative Taylor Rules after an Unanticipated 1% Fall in Rt(cont.)?=0.79 , ?y=0.60 , ?pi=1.05(aw) Yt ? Y nt0 4 8 12 16 20?0.100.10.20.30.4QuartersPercent(ax) pit0 4 8 12 16 20?0.4?0.200.20.40.6QuartersPercent(ay) Wreal,t0 4 8 12 16 20?0.4?0.3?0.2?0.100.10.20.3QuartersPercent(az) Rt0 4 8 12 16 20?1?0.500.5QuartersPercent?=0.79 , ?y=0.60 , ?pi=1.60(ba) Yt ? Y nt0 4 8 12 16 20?0.2?0.100.10.20.30.40.5QuartersPercent(bb) pit0 4 8 12 16 20?0.2?0.100.10.2QuartersPercent(bc) Wreal,t0 4 8 12 16 20?0.1?0.0500.050.10.150.20.25QuartersPercent(bd) Rt0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.4QuartersPercent?=0.79 , ?y=0.60 , ?pi=2.15(be) Yt ? Y nt0 4 8 12 16 20?0.2?0.100.10.20.30.40.5QuartersPercent(bf) pit0 4 8 12 16 20?0.3?0.2?0.100.10.2QuartersPercent(bg) Wreal,t0 4 8 12 16 20?0.1?0.0500.050.10.150.20.25QuartersPercent(bh) Rt0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.4QuartersPercentNote: The solid lines show the multi-sector model-based impulse responses and the area betweendashed lines indicate the 95% model-based confidence intervals for the multi-sector model. Thedotted lines marked with asterisks display the dynamic responses of the variables as predicted bythe one-sector model.? ? {0.68, 0.79}47Figure 2.5: The Model-Based Impulse Responses in the One- and Multi-SectorModels under Alternative Taylor Rules after an Unanticipated 1% Fall in Rt(cont.)?=0.79 , ?y=0.93 , ?pi=1.05(bi) Yt ? Y nt0 4 8 12 16 20?0.100.10.20.3QuartersPercent(bj) pit0 4 8 12 16 20?0.3?0.2?0.100.10.20.30.4QuartersPercent(bk) Wreal,t0 4 8 12 16 20?0.3?0.2?0.100.10.20.30.4QuartersPercent(bl) Rt0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.4QuartersPercent?=0.79 , ?y=0.93 , ?pi=1.60(bm) Yt ? Y nt0 4 8 12 16 20?0.100.10.20.3QuartersPercent(bn) pit0 4 8 12 16 20?0.3?0.2?0.100.10.2QuartersPercent(bo) Wreal,t0 4 8 12 16 20?0.1?0.0500.050.10.150.20.25QuartersPercent(bp) Rt0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.4QuartersPercent?=0.79 , ?y=0.93 , ?pi=2.15(bq) Yt ? Y nt0 4 8 12 16 20?0.100.10.20.3QuartersPercent(br) pit0 4 8 12 16 20?0.3?0.2?0.100.10.2QuartersPercent(bs) Wreal,t0 4 8 12 16 20?0.1?0.0500.050.10.150.20.25QuartersPercent(bt) Rt0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.4QuartersPercentNote: The solid lines show the multi-sector model-based impulse responses and the area betweendashed lines indicate the 95% model-based confidence intervals for the multi-sector model. Thedotted lines marked with asterisks display the dynamic responses of the variables as predicted bythe one-sector model.?y ? {0.27, 0.60, 0.93}48?pi ? {1.05, 1.60, 2.15}These values for the coefficients reflect the estimates of the Taylor rule coefficients inClarida, Gali & Gertler (1999) for the Pre-Volcker and the Volcker-Greenspan periods, aswell as their means. 19Figure 2.5 shows the model-based impulse responses for the one- and multi-sector mod-els as well as the 95% confidence bands for the impulse responses of the variables in themulti-sector model under alternative Taylor rule coefficients. 20It is notable that the estimated one-sector model-based impulse responses are alwayscontained in the multi-sector model-based 95% confidence bands. This suggests when themonetary authority considers implementing a new policy rule, and the evolution of theeconomy in this new regime is a matter of interest, it would not be misleading to use the19The estimate of ?pi for the Pre-Volcker period is 0.83. As argued in Clarida, Gali & Gertler (1999),this makes the system of dynamic equations indeterminate. For this reason, I set the lowest value for ?pias 1.05.20In estimating standard deviations for the multi-sector model-based impulse responses, the delta methodis used. Indeed, by using the mean-value theorem, one can writefA(P?M (A?n)) = fA(PM0 ) +?fA(P?M )??P(P?M (A?n)? PM0)where P?M ?(P?M (A?n),PM0)and fA(.) denotes the multi-sector impulse responses obtained with alternativeTaylor-rule specifications.It can be shown that?n(P?Mn (A?n)? PM0 ) ? N(0, Vf )whereV f =(?f(PM0 )??PA?A?f(PM0 )?P?)?1?f(PM0 )??PA?A?0A?A?f(PM0 )?P?(?f(PM0 )??PA?A?f(PM0 )?P?)?1This suggests that:?n(fA(P?M (A?n))? fA(PM0 ))? N(0,?fA(PM0 )??PV f?fA(PM0 )?P)The plug-in method has been employed to obtain a consistent estimate of ?fA(PM0 )??P Vf ?fA(PM0 )?P .49Table 2.6: Calibration for Explaining Contrasting Findings in Carvalho (2006)(a): The Replication Exercise (Flexible Wages)? ?H ?p ?w b ?p ?w ?Y ?pi ? ?w1 0.5 11 21 0 0 0 0.33/4 1.24 0.92 0.001(b): With Estimated Wage Rigidity? ?H ?p ?w b ?p ?w ?Y ?pi ? ?w1 0.5 11 21 0 0 0 0.33/4 1.24 0.92 0.83one-sector model since the dynamic behavior of the variables as predicted by this modelunder the new regime does not differ sharply from the dynamic behavior of the variablesas predicted by the multi-sector model where the degree of heterogeneity in price flexibilityis close to the true degree of heterogeneity in price flexibility.2.3.5 Explaining the Contrasting FindingsIn order to isolate the effects of sectoral asymmetries in the frequency of price changes,both Carvalho (2006) and Nakamura & Steinsson (2008b) compare the response of theheterogeneous-firm multi-sector economy to that of the identical-firm one-sector economy.They find when parameters are realistically calibrated, monetary shocks have larger andmore persistent real effects, which sharply contrast with the findings in this paper. Indeed,irrespective of the policy rule specified, a comparison between the dynamics of the outputgap after the same shock in the one-sector and multi-sector economies in Figure 2.3, 2.4and 2.5 reveals such dynamics in the one- and multi-sector economies are very similar.50In order to reconcile the findings in Carvalho (2006) and Nakamura & Steinsson (2008b)with my findings, I start with a replication of the findings in Carvalho (2006). The modelin Carvalho (2006) deviates from the simple Calvo (1983) model in two ways. First, thereis heterogeneity in the frequency of price changes among sectors. Second, rather thanemploying homogeneous labor, firm-specific labor is used. To replicate his finding, I retainthe former but drop the latter feature of the model. Dropping the latter is inevitable sinceextending the analysis to staggered wage setting requires firm-specific labor to be replacedwith homogeneous labor.21 The model presented in Section 2.2 lends itself to do such areplication. Indeed, in this replication exercise, I consider a simplified version of the modeldiscussed in Section 2.2 where there is no habit formation(b = 0), no backward-lookingindexation rule in non-optimized prices and wages (?p = ?w = 0), and the working capitalchannel is not operational as firms pay the wage bill at the period that production takesplace. Carvalho (2006) calibrates ?, ?H , ?p as 1, 0.5 and 11, respectively. Carvalho (2006)closes the model with a policy rule given byRt = c+ ?y(Yt ? Y nt)+ ?pipit + mt where mt = ?mt?1 + ?t (2.3.7)To calibrate ?y, ?pi and ?, Carvalho (2006) relies on the estimates reported in Rude-21As extensively discussed in Woodford (2003), replacing specific labor markets with common labormarkets results in a fall in the degree of strategic complementarity. However, in both the one- and multi-sector models, each firm owns a fixed amount of capital that never depreciates and may not be reallocatedamong firms, preventing the rental rate of capital from being equalized economy widely. In addition, theproduction function is concave in labor input in both of the models. Under these assumptions, Woodford(2003) shows that there is strategic complementarity if? =?w + ?p + ?U1 + ?p?p< 1The calibrated parameters in the replication exercise gives ? = 0.81, which still implies strategic comple-mentarity in price setting so that fast-adjusting firms have a disincentive to strongly change prices whenprice responses of slow-adjusting firms are muted. It is notable that as a robustness check, I also calibratedthe parameters in such a way that they imply a larger degree of strategic complementarity such as ? = 0.10,the main result from the replication exercise, that heterogeneity in price flexibility plays only a small roleunder staggered wages, remains largely unchanged.51Table 2.7: Cumulative Real Effects of Monetary Shocks and Their Persistence(a): The Replication Exercise (Flexible Wages)Half-life(Quarters) Cumulative EffectThe One-Sector Economy 10 12.99The Multi-Sector Economy 20 35.53Ratio 2.74(b): With Estimated Wage RigidityHalf-life(Quarters) Cumulative EffectThe One-Sector Economy 11 71.74The Multi-Sector Economy 11 71.72Ratio 1.00Note: The normalized cumulative real effects of monetary shocks are measured as |m?1t??s=0 Y?t+s|. Thehalf-life refers to the first period in which the response of the output gap falls below the midpoint of itsimpact response.busch (2002). Similarly, I use the same estimates and calibrate ?y, ?pi and ?R as equalto 0.33/4, 1.24 and 0.92, respectively. I again use the same calibrated values for ?j andnj reported in Table 2.3. In addition to such parameters, ?w and ?w require calibration.I calibrate ?w to 21 based on the value in Christiano, Eichenbaum & Evans (2005). Ac-cordingly, wage rigidities in these models can be adjusted solely by changing the fractionof workers who do not obtain a wage-change signal in the period (?w). Since wages areflexible in Carvalho (2006), to replicate his finding, I choose ?w = 0.001. The calibratedparameters for the replication exercise is summarized in Panel (a) of Table 2.6.Carvalho (2006) measures the size of the real effects of monetary shocks in the one- andmulti-sector economies with normalized cumulative effects on the output gap defined as,52Figure 2.6: The One- and Multi-Sector Model-Based Impulse Responses of OutputGap after a Negative 1% m Shock in (2.3.7)(a) With Flexible Wages0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 6000.20.40.60.811.21.41.61.82QuartersPercent(b) With Estimated Wage Rigidity0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 600123456QuartersPercentNote: The dotted lines marked with asterisks and the solid lines show the model-based responses of theoutput gap to a negative 1% m shock in the one- and multi-sector economy, respectively.|m?1t??s=0Y?t+s| (2.3.8)where m?1t denotes the inverse of the size of the shock in (2.3.7). Carvalho (2006)measures the degree of persistence of monetary shocks on the output gap using the half-lifeof the output gap. This is defined as the first period in which the absolute value of theresponse of the output gap falls below that of the response of the output gap on impact.As evident in Panel (a) of Table 2.7, my finding under flexible wages conforms with themain finding in Carvalho (2006) since the cumulative real effects of monetary shocks in themulti-sector economy are larger than those in the one-sector economy by a factor of 2.74(35.53 in the multi-sector economy compared to 12.99 in the one-sector economy). Thehalf-life in the multi-sector economy is twice that in the one-sector economy, suggesting53that monetary shocks have more persistent real effects in the former.22 Panel (a) of Figure2.6 helps visualize these findings.Next, I consider that wage setting is staggered. Panel (b) of Table 2.6 reports calibratedparameters for the staggered wage-setting model with the calibrated wage rigidity param-eter (?w = 0.83) in the United States. Panel (b) of Table 2.7, on the other hand, showsthe degree of persistence and the size of the cumulative effects in the one- and multi-sectoreconomies under this calibrated parameter for wage rigidity. Three findings are notewor-thy. First, monetary shocks no longer have larger real effects in the multi-sector economythan the one-sector economy. This finding suggests that heterogeneity in the frequencyof price changes in a staggered wage-setting model adds much less to the real effects ofmonetary shocks when compared with a flexible wage-setting model. Second, monetaryshocks induce larger real effects in the identical-firms one-sector economy model with stag-gered wage setting than in the heterogeneous-firms multi-sector model under flexible wagesetting (71.74 in the former and 35.53 in the latter). For this reason, if one aims to inducemonetary shocks to have larger real effects, departing from the simple Calvo (1983) modelby adding nominal wage contracts, as per Erceg et al. (2000), is more effective than addingheterogeneity in the frequency of price changes among firms. Third, when wage setting isstaggered, it is no longer true that monetary shocks have more persistent real effects inthe multi-sector economy than the one-sector economy.23 Panel (b) of Figure 2.6 displaysthe response of the output gap after a 1% m shock for the staggered wage-setting modelswith the estimated degree of wage rigidity.2422The half-lives in the former and the latter are 20 and 10, respectively23The half-lives in the former and the latter are 11 and 11, respectively.24As previously discussed, the degree of wage rigidity in the models is calibrated using the estimateddegree of wage rigidity in Barattieri et al. (2010) who note their estimated degree of wage rigidity in theUnited States differs from the degree of wage rigidity chosen by some key papers which estimate DSGEmodels using macro data. For example, differently from the degree of wage rigidity in this paper (0.83),Christiano et al. (2005) and Giannoni & Woodford (2003) assume two thirds of wages are not optimizedeach quarter. As a robustness check, the one- and multi-sector models in this section are also estimated54Next, I explain why a higher degree of wage rigidity leads monetary shocks to havesimilar output dynamics in the one- and multi-sector economies. Firstly, consider theexplanation for the finding in Carvalho (2006) that monetary shocks have larger and morepersistent real effects in the multi-sector economy than in the one-sector economy whenwages are free to quickly respond to such shocks. In the multi-sector economy, firms witha broad spectrum of the frequency of price changes are present. That is, while some firmsrespond rapidly to monetary shocks, others are slower. In addition, the response of wagesto monetary shocks is strong as wages adjust quickly in Carvalho (2006). Faced withsuch profound changes in their marginal costs, fast-adjusting firms tend to show strongresponses. However, the existence of slow-adjusting firms in the multi-sector economygives a disincentive for the fast-adjusting firms to respond strongly to monetary shocks inCarvalho (2006) since the calibrated parameters imply there is strategic complementarityin price setting. This results in an initial muted price response to monetary shocks. In thefollowing periods, while a large fraction of firms obtain a chance to change prices at leastonce, the initial muted price responses restrain them to raise prices. In effect, the pricelevel responses under strategic complementarity stay muted compared to the price levelresponses that would prevail if there were no such strategic interaction in price setting.Since the output gap responses are the direct opposite of the price responses, the outputgap strongly responds to monetary shocks in the multi-sector economy. Nakamura &Steinsson (2013) note that if it were possible to transfer some price change signals fromthe fast-adjusting firms to the slow-adjusting firms, price adjustment would be much fasterand the real effects of monetary shocks would be much smaller. Such transfers are madehypothetically in the one-sector economy. Indeed, by calibrating the frequency of priceusing 0.66 as the degree of wage rigidity. None of the results in this section is affected largely from thischange. Indeed, I find that the ratio of the cumulative real effect of a monetary shock in the multi-sectormodel to that in the one-sector model is only 1.06 and that the half-life of output responses in both theone- and multi-sector models is 10 quarters.55changes in the aggregate economy as the weighted average of the frequency of price changesand treating all firms as identical, the price change signals are transferred implicitly fromthe fast-adjusting firms to slow-adjusting firms. Consequently, the one-sector economyinduces much faster price responses and much smaller real effects from monetary shocks.Why does a higher degree of wage rigidity lower the effects of heterogeneity in thefrequency of price changes on output dynamics in the model? Note that for strategiccomplementarity to play a non-trivial role, fast-adjusting firms should at least have aninclination to give a strong price response in the first place. Such an inclination existswhen wages are free to respond as fast-adjusting firms face profound changes in marginalcosts. In the case of staggered wage setting, since monetary shocks initially only cause amuted response in marginal costs, fast-adjusting firms are not likely to be inclined towardsgiving a strong price response. As a result, strategic complementarity plays a small roleunder staggered wage setting which induces output to display similar post-shock dynamicsin the one- and multi-sector models.Do the findings in Table 2.7 contrast with the finding in Chari, Kehoe & McGrattan(2000) that sticky wages quantitatively have small effects on the extent to which monetaryshocks induce persistent effects on consumption in their one-sector model? The findingof small effects from sticky wages in Chari et al. (2000) is similar to the finding in Table2.7 that the half-life of shocks to consumption after monetary shocks in the one-sectormodel under flexible wages is virtually the same as that in the same model under staggeredwages.25 However, such small effects from sticky wages do not hold if one considers themulti-sector model. Indeed, as reported in Table 2.7, I find the persistence of shocks toconsumption after monetary shocks in the multi-sector model under staggered wages is25The half-lives in the former and the latter are 10 and 11 quarters, respectively. It is notable that outputis simply given as consumption in the one- and multi-sector models since investment is excluded from bothof the models. Consequently, the half-lives of shocks to consumption and output must be identical in themodels.56drastically lower compared to that in the same model under flexible wages.26 This findingcontrasts sharply with the finding in Chari et al. (2000).2.4 ConclusionIn this paper, I have investigated the consequences of introducing heterogeneity in thefrequency of price changes into a dynamic stochastic model which satisfactorily providesan explanation of the dynamics of the output gap, inflation, the real wage and the federalfunds rate after an unanticipated interest shock. The main findings of this paper can besummarized as follows: first, the dynamic behavior of the variables after a monetary policyshock are similar in both models. This finding is robust to the alternative interest rulesspecified to close the model. Second, the finding that monetary shocks have similar realeffects in the one- and multi-sector economies contrasts sharply with the finding in Carvalho(2006). I show that the staggered wage setting plays an important role in bringing thedynamics of output in the multi-sector economy close to those in the one-sector economy.26The half-lives in the former and the latter are 11 and 20 quarters, respectively.57Chapter 3Heterogeneity in Price Flexibilityand Monetary Policy ShocksThis paper seeks to answer two questions: First, do shocks to monetary policy in theUnited States induce sectoral prices to exhibit common or divergent dynamics? Second,are such shocks the cause of different price dynamics in fast-adjusting sectors where priceschange often, compared to the slow-adjusting sectors where prices change infrequently?In regards to the first question, I find that monetary policy shocks in the United Stateslead to divergent sectoral price dynamics, which suggests there are relative price effectsof the monetary shocks. Indeed, I find that while the price responses in some sectors tomonetary shocks are muted, they are strongly positive or negative in others. This finding isin conformity with the finding in Balke & Wynne (2007). However, they surprisingly findthat a contractionary shock to monetary policy preponderantly results in positive initialprice responses. In contrast, I find that the initial price responses in sectors to such ashock are equally divided between negative and positive responses, resulting in the initialaggregate price responses staying muted.27 The difference in the distribution of negativeand positive price responses between Balke & Wynne (2007) and this paper can result fromthe fact that a measure of the output gap is missing in Balke & Wynne (2007). As arguedin Giordani (2004), the absence of an output gap measure in a VAR model may result27This finding is similar to the finding in Boivin, Giannoni & Mihov (2009).58in ?the price puzzle?, which refers to the counter-intuitive finding that an unanticipatedmonetary tightening causes an increase in the price level. I use the capacity utilizationrate as a measure of the output gap and find no evidence of the puzzle at the disaggregatedlevel in the VAR model.Next, I analyze the correlation between the frequency of price changes in a sectorand its impulse response functions to a contractionary monetary policy shock over fiveyears. I find the frequency of price changes in a sector does not play a decisive role inits price responses to monetary policy shocks as the correlations between the frequencyof price changes and the price responses are weak and never significantly different fromzero. This finding contrasts sharply with that in Bils, Klenow & Kryvtsov (2003) whofind that a higher frequency of price change for a given sector is associated with a higherprice response when a contractionary monetary shock occurs. I show the assumption inBils, Klenow & Kryvtsov (2003), that the isolated monetary shocks and sector-specificprice shocks are orthogonal, is violated for a considerable number of sectors. The violationof such a critical assumption may drive the counter-intuitive finding in Bils, Klenow &Kryvtsov (2003).Lastly, I attempt to develop a DSGE model to explain two important findings in thispaper: the interest rate shock causes strong relative price effects, and there is a weakassociation between the impulse response functions of sectoral prices and the frequency ofprice changes in sectors. Three DSGE models are considered: The first model is the one-sector model, in which it is assumed that the frequency of price changes is the same amongall sectors. The second model is the multi-sector model with symmetric cost structurein which sectors differ only in regards to the frequency of price changes. Lastly, thethird model is the multi-sector model with asymmetric cost structure in which sectorsdiffer not only in regards to the frequency of price changes, but also in their production59costs? structure. I show that while the one-sector model can explain the second findingsuccessfully, this model may not explain the strong relative price effects of the interestrate shock at the disaggregated level. Quite the opposite, the multi-sector model withsymmetric cost structure is successful in explaining relative price effects. Yet, this modelfails to account for the low correlations of the frequency of price changes with sectoral priceresponses over five years following an interest rate shock. The last model, the multi-sectormodel with asymmetric cost structure, on the other hand, successfully explains both of theaforementioned two findings of the empirical section. Therefore, I conclude this modeloutperforms the other two models.The organization of the paper is as follows. Section 3.1 presents the emprical strategy forisolating monetary shocks in the United States and studies the impulse response functionsof sectoral prices to such shocks. Section 3.2 develops three theoretical models and evaluatesthe success of these models in explaining the strong relative price effects of the interestrate shock and the weak correlations between the frequency of price changes and sectoralprice responses. The last section concludes the discussion.3.1 The Empirical SectionThis section develops my empirical strategy for analyzing sectoral price responses followinga contractionary shock to the federal funds rate. Before investigating how sectoral priceschange following an exogenous interest shock, it is first useful to study the aggregatedynamics.3.1.1 Aggregate Dynamics after an Exogenous Shock in the FederalFunds RateIn Tugan (2013), the following VAR model is considered,60?t = B0 +kmax?k=1Bk?t?k +A0Et (3.1.1)where structural shocks and the number of lags included are denoted by Et and kmax,respectively. A0 stands for the contemporaneous response matrix of the variables to theseshocks. Lastly, ?t denotes the vector of variables contained in the VAR and is given as:?t = [yt ? ynt , pit, wt, Rt] (3.1.2)I again consider the VAR model in (3.1.1) to study the aggregate dynamics in thispaper. The variables included in ?t are the capacity utilization rate in manufacturing asa measure of the output gap (yt ? y?nt ), annualized inflation (pit), the real wage (wt) andthe federal funds rate (Rt) (See Tugan (2013) for a detailed explanation of the variables).The VAR is quarterly and contains four lags of each variable. The sample spans the periodof 1959Q1-2013Q1. The ordering of the variables in the VAR system implies that thecapacity utilization, inflation and the real wage respond to the monetary policy shock withone quarter lag. This assumption is standard in the literature.Figure 3.1 displays the impulse responses of the variables contained in the VAR systemto an unanticipated 1% rise in the federal funds rate together with the 95% error bandsestimated with the method proposed by Sims & Zha (1999). Regarding the effect of theexpansionary policy shock, as shown in the figure:? The point estimates suggest the output gap stays below its pre-shock level for fouryears after the shock. In addition, in the first two years, the 95% confidence bandsindicate, the fall in the output gap is statistically significant with a trough occurringafter about one and half years following the shock.? Inflation is above its pre-shock level in the early periods following the shock but61Figure 3.1: The VAR-Based Impulse Responses of Aggregate Variables to MonetaryShocks(a) yt ? ynt0 4 8 12 16 20?2?1.5?1?0.500.51QuartersPercent(b) pit0 4 8 12 16 20?0.8?0.6?0.4?0.200.20.40.6QuartersPercent(c) wt0 4 8 12 16 20?0.5?0.4?0.3?0.2?0.100.10.20.3QuartersPercent(d) Rt0 4 8 12 16 20?1?0.500.511.5QuartersPercentNote: In the figure, the solid line indicates the estimated point-wise impulse responses.The area between the dashed lines shows the 95% confidence interval estimated with themethod suggested by Sims & Zha (1999).rises over time below the pre-shock level. The trough realizes after about three yearsaccording to the estimated impulse responses. It is notable that my finding of a62delayed effect from the interest shock on inflation is a general finding for the UnitedStates economy as noted by Woodford (2003).? The real wage falls after the shock, suggesting nominal wages fall relative to nominalprices following the expansionary shock.? Lastly, the federal funds rate remains above its pre-shock level for about three yearsfollowing the shock according to its point impulse response estimates.3.1.2 Sectoral Price Responses after Interest ShocksTo study sectoral price responses following an unanticipated 1% increase in the federalfunds rate, I consider again the same VAR in (3.1.1) but add the annualized percentagechange in a sectoral price to the vector of variables(?t) in (3.1.2). This variable is denotedby piit and is measured as 4? (lnpit ? lnpit?1).?t =[yt ? ynt , pit, piit, wt, Rt](3.1.3)Two identifying assumptions for isolating exogenous interest shocks in (3.1.3) are worthmentioning. First, the Federal Reserve observes sectoral price movements before setting thefederal funds rate. Second, there is at least a quarter lag in the response of sectoral pricesto the federal funds rate shocks. The latter is consistent with with the aforementionedassumption that there is a quarter delay in the aggregate price level?s response to thefederal funds rate shock. Were the sectoral prices assumed to respond contemporaneouslyto the shock while the aggregate price level was not, the analysis would be internallyinconsistent.Figure 3.2 shows the impulse responses of the sectoral and aggregate price levels to63Figure 3.2: The Impulse Responses of the Price Levels of PCE Categories to anUnanticipated 1% Increase in the Federal Funds Rate Shocks0 4 8 12 16 20?6?5?4?3?2?1012QuartersPercent  GDP DeflatorUpper 95% Conf. BandLower 95% Conf. BandPCE Sectoral PricesNote: In the figure, the thick solid line marked with circles shows the aggregate priceresponses following an unanticipated 1% increase in the federal funds rate whose 95%confidence intervals are marked by the thick dashed lines. The thin solid lines, on theother hand, display the sectoral impulse responses to the same shock.an unanticipated 1% increase in the federal funds rate.28 Sectoral price level impulseresponses represent the price level responses for 124 Personal Consumption Expenditure(PCE) categories for which I have an estimate of the frequency of price changes.A crucial finding in Figure 3.2 is that an unanticipated change in the federal funds rate28Since both pit and piit are measured as four times the difference in the log of price levels between twoperiods, to obtain impulse responses for sectoral and aggregate price levels ( denoted with lnpit and lnPt,respectively), cumulative impulse responses for pit and piit are obtained, which are then scaled down byfour.64produces relative price effects in the United States. The existence of such effects requiresonly the lowest and highest sectoral price level responses to differ significantly. A strongercondition is met in Figure 3.2. Indeed, not only the highest sectoral price level responsesdiffer radically from the lowest ones, but they are also outside the 95% confidence bandsfor the impulse responses of the aggregate price level.It is also notable that initially, positive and negative sectoral responses are distributedevenly. This results in an initial muted response of the aggregate price level. Following thisphase, the sectoral prices? responses are predominantly negative. As a consequence, theaggregate price level shows a decline following the initial phase. It is worth mentioning thatthese findings are in conformity with the findings in Boivin, Giannoni & Mihov (2009) whouse the factor augmented vector autoregression (FAVAR) approach to study the sectoralprice responses to a federal funds rate shock. They advocate their method by showing thata contractionary interest rate shock results in a fall in most sectoral prices and that thereis no evidence of a ?price puzzle? when the FAVAR approach is used. However, Hanson(2004) finds that the ?price puzzle? is mainly associated with the 1959-1979 sample period,and that evidence of a ?price puzzle? is weak during the 1976-2005 period which Boivin,Giannoni & Mihov (2009) consider. Whether the FAVAR approach alleviates the puzzleor not is uncertain when the data sample is extended back to 1959. It is notable that evenif my data sample covers the period in which Hanson (2004) finds strong evidence of thepuzzle, my results do not indicate a ?price puzzle?.How strong is the association between sectoral price level responses in Figure 3.2 andsectoral frequency of price changes? Is a higher frequency of price change in a sectoris associated with a higher or a lower price level response in periods? To answer thesequestions, I first define the frequency of price changes in sectors and describe my data onthe frequency of price changes. The frequency refers to the percentage of firms that adjust65their prices in a quarter. Our monthly frequency of price changes data for the UnitedStates comes from Nakamura & Steinsson (2008a). In contrast to the frequency of pricechanges in Bils & Klenow (2004), where only the frequency of price changes including salesare reported, using the frequency of price changes data in Nakamura & Steinsson (2008a)has the advantage that the frequencies in sectors are reported for both non-sales pricechanges and price changes including sales. Since Nakamura & Steinsson (2008a) find thatthe frequency of price changes including sales in some sectors differs radically from that ofnon-sales price changes to a great margin, it is important to check the robustness of myresults in this paper for non-sales price changes and price changes including sales.The frequency of price changes is estimated for entry level items (ELIs) of CPI inNakamura & Steinsson (2008a). However, for many components of CPI, price series forthe disaggregated sectors are not available for earlier periods, prior to 1970s. In contrast,sectoral price indexes can be obtained from 1959 for the bulk of the Bureau of EconomicAnalysis? personal consumption expenditure (PCE) categories. For this reason, I use thePCE categories for estimation. However, since the frequency of price changes are notreadily available for the PCE categories, the categories are mapped to the components ofthe CPI index in my analysis. To map ELIs in the CPI with the PCE categories, I use themapping that Andrea Tambalotti made available.29 If a PCE category is matched withonly one component of CPI, the frequency of price changes in that PCE category is takenas that of the CPI component. If there are multiple ELIs that map with a single PCEcategory, the frequency of price changes in this PCE category is measured as the weightedaverage of the frequency of price changes in these CPI components, with the weights givenas the 2000 CPI expenditures of the ELIs reported by Nakamura & Steinsson (2008a).Lastly, the frequencies of price adjustment within ELIs in Nakamura & Steinsson29I am grateful to Andrea Tambalotti for sharing his mapping with me.66(2008a) are reported as monthly percentages. The quarterly frequency of price changesin ELIs are estimated by using the method described in Tugan (2013). It is notable thatthe quarterly frequencies are found to differ substantially among the mapped PCE cate-gories. The frequencies range from as low as 6.7% in intracity mass transit to 100% in netpurchases of used motor vehicles.Now, I study the association of the frequency of price changes with the impulse re-sponses of sectoral prices to interest rate shocks. First, the impulse responses of sectoralprices to an unanticipated 1% federal funds rate shock have been obtained as in Figure3.2. Next, for each period, the correlation between the frequency of price changes in sec-tors and their impulse responses is estimated. Figure 3.3 demonstrates these correlationsand the corresponding 95% confidence bands. The correlations reveal that if prices changemore frequently in a sector, it is more likely that prices in that sector increase after acontractionary interest rate shock during the first year following the shock. This findingholds whether or not the frequency of price changes includes non-sales price changes (SeePanel (a) and Panel (b) of Figure 3.3). After one year following a contractionary monetaryshock, on the other hand, a higher frequency of price change in a sector is associated witha lower impulse response. It is notable that the correlations of the impulse responses withthe frequency of non-sales price changes and with the frequency of price changes includingsales are rather similar. Lastly, the fact that the value of zero is contained in the confidenceintervals for the correlations indicates that one may not reject the hypothesis of no corre-lation between the frequency of price changes and the impulse responses of sectoral pricesfollowing a contractionary federal funds rate shock. In other words, the price responsesafter a contractionary shock in sectors are only weakly associated with the fraction of firmsin the sector that change their prices in a quarter.These findings contrast with those in Bils, Klenow & Kryvtsov (2003) who find that67Figure 3.3: Correlations of ?i with the Impulse Responses of Pi to an Unanticipated1% Increase in Rt(a) With the Frequency of Non-Sales PriceChanges0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.40.60.81CorrelationQuarter(b) With the Frequency of Price ChangesIncluding Sales0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.40.60.81CorrelationQuarterNote: The solid lines display the correlations of frequency of price changes in a sector withthe impulse responses of sectoral prices to an unanticipated 1% increase in the federal fundsrate at each quarter. The 95% confidence intervals for these correlations are shown withdotted lines and are estimated using the block-bootstrap method explained in AppendixB.2.there is an anomaly in the relative price movements following an unanticipated change in thefederal funds rate. Indeed, the price of the flexible-price category rises significantly relativeto that of the sticky-price category in the first eight months following a contractionaryinterest shock. They reason there are two possible explanations for this finding: eitherthe sticky price models are incapable of explaining relative price movements followingthe exogenous monetary shocks or else inferred monetary shocks are not orthogonal topersistent price shocks in the flexible- and sticky-price categories. I discuss the Bils, Klenow& Kryvtsov (2003) model in detail in Appendix B.1 and offer an explanation for thecontrasting findings in this paper and theirs.68In the next section, I aim to explain the empirical findings in this section with thethree DSGE models. In the first model (one-sector model), the fraction of firms that maychange their prices in a quarter in all sectors after an interest rate shock are assumed tobe the same in all sectors. In this model, the frequency of price changes in the economy isapproximated by the median frequency of price changes in all sectors. It is notable that theassumption of the same frequency of price changes among sectors in the one-sector modeldoes not necessarily contradict with the finding in Bils & Klenow (2004) and Nakamura &Steinsson (2008a) that the distribution of frequency of price changes among sectors is widein the United States. Indeed, while the frequency of price changes may differ largely amongsectors for sector-specific and other types of shocks, they are the same for an interest rateshock. In the second model (multi-sector model with a symmetric cost structure), sectorsare allowed to differ only in the frequency of price changes after an interest rate shock. Inthe last model, (the multi-sector model with an asymmetric cost structure), sectors differnot only in the frequency of price changes but also in the cost structure. The performanceof these models in explaining the weak association of the frequency of price changes withimpulse responses of sectoral prices after an interest rate shock in the economy is thenevaluated. The findings are in favor of the multi-sector model with an asymmetric coststructure.3.2 Theoretical ModelsIn this section, I consider a variant of the theoretical model in Tugan (2013). Since themodel environment is discussed in detail in Tugan (2013), only a summary of main featuresof the model and the dynamic equations needed to solve the DSGE models are stated here:? Price setting is staggered along the lines of Calvo (1983).69? Wage setting is staggered along the lines of Erceg, Henderson & Levin (2000).? When the optimization signal is not received by firms and workers, wages and pricesare set according to the backward-looking indexation rule.? There is habit persistence in consumption.? It is assumed that consumption decisions are made and prices and wages are set oneperiod before observing the interest rate shocks.? Firms are obliged to pay their wage bill in advance. As a consequence, when themonetary authority decides to introduce an unanticipated increase in the interest rate,real marginal costs may rise despite a fall in output accompanying the contractionaryshock.? The one- and multi-sector models differ only in the assumption regarding the fre-quency of price changes after an interest rate shock. In the one-sector model, thefrequency of price changes in all sectors is assumed to be homogenous. In the multi-sector model, on the other hand, there is a heterogeneity in price setting amongsectors. As a matter of fact, in some sectors, prices change more frequently than inothers.3.2.1 The Structural Equations in the ModelsNow, I state the main equations of the model. Let the hat over variables, R?t, pit+1 and? denote the log-deviation of the variables from their corresponding steady states; thenominal interest rate; inflation in prices; and, the intertemporal elasticity of substitution,respectively. The IS equation is given by:Et?1x?t = Et?1x?t+1 ? ?Et?1(R?t ? pit+1)(3.2.1)70where x?t is defined as:x?t =((y?t ? by?t?1)? b?Et(y?t+1 ? by?t))(3.2.2)In (3.2.2), y?t, b and ? denote the log-change in output; the habit formation parameter;and, the discount factor, respectively.The second equation in the models is the wage inflation equation (piwt ):piwt ? ?wpit?1 = ?wEt?1(?wy?t + ??1x?t ? w?t)+ ?Et?1(piwt+1 ? ?wpit)(3.2.3)In (3.2.3), ?w and ?w denote the elasticity of real wages paid for the number of hoursworked with respect to output changes for a constant marginal utility of real income andthe backward-looking indexation parameter in wages, respectively. Lastly, letting 1? ?w,?H and ?w denote the probability of receiving a wage change signal by a differentiatedlabor type; the Frisch-elasticity of labor; and, the wage elasticity of substitution amongdifferentiated labor types, respectively, the wage stickiness parameter ?w in (3.2.3) can bewritten as:?w =(1? ?w)?w1? ?w?(1 + ?w??1H )The third equation in the models is the monetary policy rule. The monetary authorityis assumed to control the interest rate and implements the following Taylor Rule to stabilizethe economy:Rt = ?R?Rt?1 + [apipit + ay(yt ? ynt )]? ?R?[apipit?1 + ay(yt?1 ? ynt?1)] + t (3.2.4)71where ynt and t denote the potential output and the shock in monetary policy, respec-tively. The calibrated values for ?R, api and ay are given as 0.92, 1.24 and 0.33, respectively.These calibrated values are based on Rudebusch (2002).3.2.2 The Econometric MethodSince the number of sectors for which the frequency of price changes data is available is quitelarge, it is impractical to solve the DSGE models by considering each individual sector. Tocircumvent this problem, as in Tugan (2013), I reduce the number of sectors in the modelto 10 by including each sector in one of the percentiles of the frequency of price changesand approximating the frequency of price changes in a sector with the median frequency inthe percentile group where that sector is contained. It is notable that only the frequencyof price changes including sales are considered in calibration. Since the calibration of fk,?pk and some other parameters of the models are extensively discussed in Tugan (2013), Iskip describing the calibration method here and only discuss the estimation method for thefree parameters of the models. Let P denote the vector of free parameters to be estimated.In the models, P contains 7 parameters:P = [?, ?H, ?p, ?w, b, ?p, ?w]where ?p and ?p denote the price elasticity of substitution for sectoral goods and thebackward-looking indexation parameter in prices, respectively.30P?(A?T ) = arg minP(h?T ? f(P))?A??T A?T (h?T ? f(P)) (3.2.5)where A?T and f(P) show the weighting matrix and the model-based impulse responses30See (3.2.1) for the definition of ?, (3.2.2) for the definition of b, and (3.2.3) for the definitions of ?H,?w and ?w.72and correlations for a given parameter vector P. Lastly, h?T stands for the vector of esti-mated VAR-based impulse responses and correlations and is given by:h?T =???? Cyt?ynt1,20 , Cpit1,20, Cwt1,20, CRt1,20 ?Clnpit1,20 ,?i?????(3.2.6)where CZ1,20 denotes the impulse responses of the variable Z to an unanticipated 1% risein the federal funds rate between the 1th and 20th quarters following the shock as shownin Figure 3.1, and ?Clnpit1,20 ,?irepresents the correlations of the frequency of price changes insectors(?i) with the impulse responses of sectoral prices between the 1th and 20th quarters(Clnpit1,20 ) as in Figure 3.3. Lastly, T stands for the sample size of the data used to estimatethe VAR-based impulse responses. As a weighting matrix, I use the diagonal matrix whosediagonal elements are given by the inverse of standard errors of each term in h?T . Thismatrix ensures more precisely estimated VAR-based correlations and impulse responseshave larger weights when choosing parameters in (3.2.5).3.2.3 ResultsBefore presening the aggregate and disaggregated model-based dynamics following an unan-ticipated 1% increase in the federal funds rate and comparing the outcomes in the modelswith the VAR-based dynamics, I first report the structural parameter estimates in themodels in Table 3.1. Woodford (2003) shows that the backward-looking indexation rulein prices (?p) and the habit persistence in consumption (b) induce hump-shaped dynamicsafter a monetary shock. Hence, high estimates for such parameters in Table 3.1 can berelated to the hump-shaped dynamics of consumption in Figure 3.1. Low estimates for theintertemporal elasticity of substitution (?) and the Frisch-elasticity of labor (?H) are con-sistent with the estimates reported in Hall (1988) and Boldrin, Christiano & Fisher (2001).The estimated value for the backward-looking indexation rule in wages (?w) is close to its73Table 3.1: Estimates of Structural Parameters(a) (b)Multi-Sector ModelOne-Sector Model With Symmetric Cost Structure? 0.012 0.025?H 0.740 0.001?p 147.215 8.723?w 19.970 6.030b 0.800 0.641?p 0.999 0.915?w 0.999 0.157Obj. Func. 117.27 246.02Note: Obj. Func. indicates the estimated value for the minimization problem discussed in(3.2.5). A lower value of Obj. Func. indicates a more successful model for accounting foraggregate dynamics and correlations between the frequency of price changes and sectoralprice dynamics after an unanticipated 1% increase in the federal funds rate.upper theoretical limit of one, as Christiano, Eichenbaum & Evans (2005) assume. Lastly,the calibrated values for the price elasticity of substitution for sectoral goods (?p) and thewage elasticity of substitution among differentiated labor types (?w) in the literature varyenormously. Our estimates lie within the range of those calibrated values.Aggregate DynamicsFigure 3.4 and Figure 3.5 show the impulse responses of the output gap, inflation, thereal wage and the federal funds rate over five years after a 1% contractionary shock inthe federal funds rate in the one-sector model and the multi-sector model with symmetric74cost structure. It is evident from these figures that the impulse responses of the aggregatevariables are similar in these models. Output is constant on impact following the shock,by construction. Starting with the first period, output falls. This emanates from the factthat a higher interest rate reduces consumption by making saving more desirable. Theone-sector model fails to account for increased inflation following the shock. The fall inreal wage in the models is the product of two factors. First, a fall in output lowers nominalwage demand. To explain this, note that less effort is needed when output falls, and sincethe disutility from working is a convex function of effort, workers lowers their nominal wageif firms demand less effort. Second, an increase in prices contributes to a fall in real wagein earlier periods. Excessive fall in real wage in the multi-sector model with symmetric coststructure can be accounted for by the second factor.Model- and VAR-Based CorrelationsThe dynamics displayed in Figure 3.4 and Figure 3.5 reveal the DSGE models have similarpredictions regarding the impulse responses of aggregate variables. In what they differquite substantially is their prediction of the correlations of the frequency of price changeswith the impulse responses of sectoral prices (?Clnpit1,20 ,?i).Figure 3.6 shows the correlations of the frequency of price changes (?i) with the impulseresponses of sectoral prices after an unanticipated 1% increase in the federal funds rate(Clnpit1,20 ). In the one-sector model, the correlations have to be zero by definition since allsectors have the same price response to the interest rate shock.31 In the multi-sector modelwith symmetric cost structure, since it is assumed that sectoral prices are unresponsive31It is notable that in the one-sector model, it is assumed that the measured frequency of price changesdiffers among sectors for sector-specific and aggregate shocks, except the federal funds rate shock. For thefederal funds rate shock, on the other hand, it is assumed that the fraction of firms in all sectors thatchange their prices are the same. This implies when a contractionary interest rate shock occurs, prices inall sectors respond in the same way. Consequently, by construction, the correlations between the measuredfrequency of price changes and Clnpit1,20 are equal to zero in the one-sector model.75Figure 3.4: Impulse Responses to an Unanticipated 1% Rise in Rt(One-Sector Model)(a) yt ? ynt0 4 8 12 16 20?2?1.5?1?0.500.51QuartersPercent(b) pit0 4 8 12 16 20?0.8?0.6?0.4?0.200.20.40.6QuartersPercent(c) wt0 4 8 12 16 20?0.5?0.4?0.3?0.2?0.100.10.20.3QuartersPercent(d) Rt0 4 8 12 16 20?1?0.500.511.5QuartersPercentNote: The solid lines in panels show the VAR-based impulse responses and the area between dotted linesindicate the 95% confidence intervals estimated with the method suggested by Sims & Zha (1999). Thesolid lines marked with circles represent the dynamic responses of the variables as predicted by the model.to the shock on impact, the impact correlation is zero as illustrated in Panel (b) of thefigure. Following the impact period, the correlations are positive for two quarters. This isunconventional as it implies that a tightening of monetary policy shock leads to a higher76Figure 3.5: Impulse Responses to an Unanticipated 1% Rise in Rt(Multi-Sector Model with Symmetrical Cost Structure)(a) yt ? ynt0 4 8 12 16 20?2?1.5?1?0.500.51QuartersPercent(b) pit0 4 8 12 16 20?0.8?0.6?0.4?0.200.20.40.6QuartersPercent(c) wt0 4 8 12 16 20?0.5?0.4?0.3?0.2?0.100.10.20.3QuartersPercent(d) Rt0 4 8 12 16 20?1?0.500.511.5QuartersPercentNote: The solid lines in panels show the VAR-based impulse responses and the area between dotted linesindicate the 95% confidence intervals estimated with the method suggested by Sims & Zha (1999). Thesolid lines marked with circles represent the dynamic responses of the variables as predicted by the model.price response in sectors where firms change prices frequently than sectors where firmschange prices infrequently. Starting in the third quarter following the contractionary shock,the correlations become negative. This suggests that a higher frequency of price change in77Figure 3.6: Model- and VAR-Based Correlations of ?i with Clnpit1,20(a) One-Sector Model0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.40.60.81QuartersCorrelation(b) The Multi-Sector Model withSymmetric Cost Structure0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.40.60.81QuartersCorrelationNote: The solid lines show the VAR-based correlations and the area between dotted lines indicate the 95%confidence intervals estimated with the method suggested by Sims & Zha (1999). The solid lines markedwith circles represent the dynamic responses of the variables as predicted by the model.a sector is associated with a lower price response in these periods.It is notable that compared to the multi-sector model with symmetric cost structure,the one-sector model performs much better in explaining the correlations. However, thismodel may not explain the rich set of sectoral price dynamics following the interest rateshock displayed in Figure 3.2. The multi-sector model with symmetric cost structure, onthe other hand, can explain the wide distribution of the responses of sectoral prices to sucha shock; yet, this model is unable to explain the correlations.Why does the multi-sector model with symmetric cost structure fail to explain ?Clnpit1,20 ,?i?To answer this question, I now detail the price-setting behavior of the firms in each model.It is assumed that firms set prices before observing shocks to the interest rate. In each78sector, firms optimize their prices only when a price-change signal is received. The fractionof firms which receive this signal is different in each sector and is given by ?i = 1??ip forthe sector i. It is well known that this fraction is equal to the probability of receiving aprice change signal in each period in the Calvo (1983) model.Let pit?1(i?) and Pt denote the last period price of the good produced by the firm i? inSector i in the last period and the price of the composite consumption good, respectively.When no such signal is received, firms are assumed to set their prices according to thefollowing partial adjustment backward-looking indexation rule:p?it(i?) = pit?1(i?)(Pt?1Pt?2)?p(3.2.7)where the tilde over p denotes the price set according to the backward-looking indexa-tion rule and ?p shows the backward-looking indexation parameter. If 0 < ?p < 1, there ispartial backward-looking indexation in the economy.When a firm is capable of setting an optimal price, it sets pit(i?)? that maximizes:Et?1(??s=0?sipQt,t+s?it+s(i?))(3.2.8)where Qt,t+s is the stochastic discount factor between the period t and t + s and isgiven by:?it+s(i?), on the other hand, shows the profit of the firm i? in the sector i and is givenby:32?it+s(i?) = p?it+s,t(i)yit+s(i?)? TCit+s(i?) (3.2.9)32It is notable that since the firms are assumed to respond the monetary shocks with a one period delay,they have to condition the optimum price based on the information till the period t ? 1 rather than theperiod t. Correspondingly, Et?1 appears in the objective function, rather than Et in (3.2.8).79where p?it+s,t(i?) shows the price set in period t + s by the firm that received a price-change signal at the period t and does not have an opportunity to set an optimal pricebetween t and t+ s. Due to the backward-indexation rule, one can write p?it+s,t(i?) as:p?it+s,t(i?) = p?it(i?)?pt,t+s (3.2.10)where?pt,t+s =????????sk=1(Pt+k?1Pt+k?2)?pif s ? 11 if s = 0(3.2.11)TCit(i?) in (3.2.9), on the other hand, denotes the total cost of the firm. In the one-sector model and the multi-sector model with symmetric cost structure, all firms have thesame total cost structure. Since firms are assumed to pay the wage bill in advance, TCit(i?)for these models can be written as:TCit(i?) = RtWtLit(i?) (3.2.12)where Wt and Lit(i?) denote the nominal aggregate wage and the labor demanded bythe firm i? in the sector i, respectively.The optimality condition in (3.2.8) for p?it(i?) can be expressed as:Et?1(??s=0?spiQt,t+sd?it+s(i?)dp?it(i?))= 0 (3.2.13)Using 3.2.13, one can show the sectoral price inflation (piit) in the multi-sector modelwith symmetric cost structure evolves according to the following equation:80piit ? ?ppit?1 = ??ip(1 + ?p?p)Et?1(P?it ? P?t)+ ?ipEt?1(R?t + w?t + ?py?t)+ ?Et?1(piit+1 ? ?ppit)(3.2.14)where ?ip is the stickiness parameter in each sector and is defined as:?ip =1? ?ip?ip1? ??ip1 + ?p?p, ?p =1? ??(3.2.15)In (3.2.15), ?p and ? denote the elasticity of prices with respect to the supply of goodswhen interest rate and wages paid for composite hours of work stay the same and thereciprocal of the output elasticity of labor demand, respectively. The aggregate inflationequation in the multi-sector model is a weighted average of sectoral inflation in the economy:pit =M?i=1fipiit (3.2.16)where fi is the sector?s share of aggregate consumption expenditure at the steady state.The equation for pit in the one-sector model, on the other hand, is given by:pit ? ?ppit?1 = ?pEt?1(R?t + wt + ?py?t)+ ?Et?1(pit+1 ? ?ppit)(3.2.17)?p =1? ?p?p1? ??p1 + ?p?p(3.2.18)where 1 ? ?p denotes the frequency of price changes in the model economy. It ismeasured as the median frequency of price changes in the United States.Here, I aim to explain the strong positive correlations of the frequency of price changeswith the sectoral price responses following the shock (?Clnpit1,20 ,?i) in earlier periods and strong81negative correlations in later periods. I explain this by studying how firms in each sectorset prices when they have a chance to optimize.First, one can show from (3.2.13) that the following equation holds:Et?1????s=0?sipQt,t+s?pt,t+s(Pt+sp?it+s,t(i?))1+?pyt+s(p?it+s,t(i?)Pit+sPit+sPt+s? ?pSit+s(i?)Pit+sPit+sPt+s)?? = 0(3.2.19)where ?p ? 1 (i.e. ?p ? 1) shows the steady-state markup. Sjt+s(j?) denotes themarginal cost of the firm. Letting yit(i?) be the output of the firms, Sjt+s(j?) is defined as:Sit+s(i?) =?TCit(i?)?yit(i?)(3.2.20)Log-linearizing (3.2.19) yields:Et?1??s=0(??ip)s[p??it(i?)? P?it+s+ P?it+s? P?t+s+ ??pt,t+s?(R?t+s+w?t+s+?py?it+s(i?))]= 0 (3.2.21)where ??pt,t+s is the log-deviation of ?pt,t+s from its steady state and is given by:??pt,t+s =????????ppit + ?ppit+1 + ? ? ?+ ?ppit+s?1 if s ? 10 if s = 0(3.2.22)Using the approximation that P?t+s = P?t + pit+s +?s?1k=0 pit+k ? pit for s ? 1, (3.2.21)can be restated as:82p??it(i?) = Et?1P?t + (1? ??ip)Et?1??s=1(??ip)s(pit+s + (1? ?p)?s?1k=0 pit+k ? pit)+ (1? ??ip)Et?1s?it + (1? ??ip)Et?1??s=1(??ip)ss?it+s(3.2.23)where s?it+s shows the log-deviation of the real marginal cost of the firm from its steadystate and is given by:s?it+s = R?t+s + w?t+s + ?py?it+s(i?) (3.2.24)(3.2.23) can be rewritten as:p??it(i?) = Et?1P?t ? ?p??ipEt?1pit + Et?1??s=1(??ip)s((1? ?p??ip)pit+s)+ Et?1s?it + Et?1??s=1(??ip)s(s?it+s ? s?it+s?1)(3.2.25)Two points must be emphasized regarding (3.2.25). First, as ?p increases, firms giveless importance to inflation in subsequent periods. When ?p = 1, they set prices such thatimportance given to inflation in subsequent periods is minimized. An intuitive explanationcan be given for this: The fact that the prices are optimized only if the Calvo signal isreceived leads firms to take preemptive measures against expected inflation in subsequentperiods. When the degree of backward-looking indexation is high in an economy, firms areable to change prices by taking into account inflation realized in the previous period evenif prices are not optimized. This results in a decrease in the degree of firms? preemptivemeasures against expected inflation in the subsequent periods.Second, when expected real costs are higher in subsequent periods than today, the per-centage increase in prices is higher than the percentage change in the current real marginal83cost, holding fixed expected inflation in subsequent periods.33 Christiano, Eichenbaum &Evans (2005) refer to this as firms ?front-load? for the expected real cost increases in subse-quent periods in which the chance to optimize their prices is uncertain. The ?front-loading?is most relevant for firms in a sector where the frequency of price changes is lower (?ip ishigher) since a higher frequency of price change discounts the importance of real marginalcosts in subsequent periods on prices set by a firm when a Calvo signal is received. Forexample, consider an unanticipated rise in interest rate. A rise in today?s marginal costsis likely because of the working capital channel in the model. Yet, as interest rate returnsto its undistorted level and output and wages decrease, marginal costs are bound to fallin subsequent periods. In the flexible-price sector where firms can optimize prices often,marginal costs today have a decisive effect on prices set. For firms in the sector where priceflexibility is low, on the other hand, the extent that marginal costs in subsequent periodsare taken into account in price setting is much larger. I illustrate the ?front loading? argu-ment in Figure 3.7. In this figure, the sticky- and flexible-price sectors are defined as thepercentile groups with the lowest and highest frequency of price changes among 10 groupsin the model, respectively. It is evident from this figure that a contractionary monetaryshock results in an initial fall in prices in the sticky-price sector and an initial rise in pricesin the flexible-price sector. This results in a positive correlation between the frequency ofprice changes and sectoral price responses in the early periods following the contractionarymonetary shock.In subsequent periods, firms? marginal costs fall markedly due to a persistent fall in thereal wage, output and the interest rate. Because of a higher price flexibility in the flexible-price sector, prices in this sector fall more pronouncedly than those in the sticky-price33It is notable that since the effect of p??it(i?) on Et?1P?t is negligible, when setting prices, the firms treatsEt?1P?t as constant. Hence, whether the percentage change in the optimized prices outweighs that of thereal marginal cost today depends entirely on the statement above.84Figure 3.7: The Front-Loading Argument(The Multi-Sector Model with Symmetric Cost Structure)(a) Inflation0 4 8 12 16 20?0.15?0.1?0.0500.050.10.150.2QuartersPercent  Sticky?Price SectorFlexible?Price Sector(b) Price Level0 4 8 12 16 20?1.4?1.2?1?0.8?0.6?0.4?0.200.2QuartersPercent  Sticky?Price SectorFlexible?Price SectorNote: The dot-dashed lines marked with a plus sign and the solid lines marked with circles show themodel-based impulse responses of inflation and the price level in the flexible- and sticky-price sectors to a1% contractionary shock to the federal funds rate, respectively.sector during these periods. This explains the finding in Figure 3.6 that a higher frequencyof price change is associated with a lower price response in the multi-sector model withsymmetric cost structure in the third period and onwards.As evident in Figure 3.6, while the multi-sector model with symmetric cost structureexplains the correlations successfully in qualitative terms, an undesirable feature of thismodel is that the correlations predicted by the model are too high compared to thosefound in the data. This is a natural consequence of the fact that sectors in the model areassumed to be identical apart from their frequency of price changes. With this assumption,price responses in sectors in any period are ordered to a large extent according to sectoralfrequency of price changes after a contractionary interest rate shock. Figure 3.8 illustrates85Figure 3.8: Model-Based Clnpit2,3(The Multi-Sector Model with Symmetric Cost Structure)0 1 2 300.050.10.15QuartersPercent10?6?2?8??4???9?5?1??7??39?6?2?8??4???10?5?1??7??3Note: The points with numbers inside the figures show the model-based price responses in each sector toa 1% contractionary shock in the federal funds rate in the first and second quarters. Sectors in the figureare ordered according to the frequency of price changes from highest to lowest. For example, 1 in the figuredenotes the price response of the sector with the highest frequency of price changes in the first and secondperiods.this. In Panel (a) of the figure, the model-based sectoral price responses in the first andsecond periods are shown for the multi-sector model with symmetric cost structure. Sectorsare indicated by numbers and are ordered according to their frequency of price changesfrom highest to lowest. For example, 1 in the figure denotes the price response of the sectorthat has the highest frequency of price changes among sectors. As evident from Panel (a)of the figure, price responses in sectors in the first and second periods are largely orderedaccording to sectoral frequency of price changes. This causes the predicted correlationsbetween the frequency of price changes and the sectoral price responses in the model to beexceedingly high compared to those in the data.86Briefly, the discussion in this section notes that the strong positive correlations betweenthe frequency of price changes and sectoral price responses in the initial periods and thestrong negative correlations in subsequent periods are a direct consequence of the front-loading argument. However, such strong correlations conflict with the low VAR-basedcorrelations shown in Figure 3.3.3.2.4 The Multi-Sector Model with Asymmetric Cost StructureIn this section, I show that when there is asymmetry in the cost structure of firms indifferent sectors, not only is it possible to account for the low correlations in the data, butone can also explain the wide distribution of sectoral price responses to the shock which isevident in Figure 3.2.I consider a multi-sector model with asymmetric output elasticity of labor demand wheresectors differ not only in terms of price flexibility but also in terms of their cost structuresince production functions used by firms differ among sectors in this model. This contrastssharply with the multi-sector model with symmetric cost structure where firms use thesame production function. To explain this model, it is useful to first write the productionfunction that firms use to produce their output (yit(i?)) in the multi-sector model withsymmetric cost structure,yit(i?) = ZtHit(i?)? 0 < ? < 1 (3.2.26)where Zt and Hit(i?) denote the technology level and the demand of the firm for thecomposite labor, respectively. Lastly, ? denotes the reciprocal of the output elasticity oflabor demand in sectors which is assumed to be identical among all sectors. In the multi-sector model with asymmetric cost structure, a differential ?i for each sector is considered:87yit(i?) = ZtHit(i?)?i (3.2.27)Except the sectoral inflation equation (piit) and the nominal wage inflation equation(piwt ), the structural equations are the same as in Section 3.2.1. The only change in thesectoral inflation equation is that ? in (3.2.15) should be replaced by ?i. piwt , on the otherhand, is now given as:piwt ? ?wpit?1 = ?wEt?1(??1H H?t + ??1x?t ? w?t)+ ?Et?1(piwt+1 ? ?wpit)(3.2.28)where H?t denotes the total composite labor demand. Let Y?it and ni stand for sectoraloutput and the sectoral weight in total output, respectively. Then, H?t can be written as:H?t =10?i=1ni?iY?it (3.2.29)where sectoral output is a function of sectoral relative price and total output:Y?it = ??p(P?it ? P?t)+ Y?t (3.2.30)Calibration for the Multi-Sector Model with Asymmetric Cost StructureIt is notable that in the multi-sector model with symmetric cost structure, the numberof sectors is reduced to 10 since it is impractical to solve the model if all disaggregatedsectors, for which the frequency of price changes is available, are included. When groupingdisaggregated sectors into 10 groups, sectors are ordered by their frequency of price changes,and they are included in one of the ten groups. The frequency of price changes in a sector88is then approximated by the median frequency of price changes in its group. Since sectorsin the multi-sector model with symmetric cost structure only differ in their frequency ofprice changes and the frequency of price changes in all sectors contained in a group isapproximated by the median frequency of price changes in that group, sectors within thesame group must have the same sectoral inflation equation.However, in the multi-sector model with asymmetric cost structure, grouping disaggre-gated sectors based only on the frequency of price changes may not be justified. Thisresults from the fact that even when such sectors have a similar frequency of price changes,sectoral price dynamics following a monetary shock may be markedly dissimilar if theiroutput elasticity of labor demand largely differs. Consequently, in the multi-sector modelwith asymmetric cost structure, both the frequency of price changes and labor shares insectors are needed to solve the model. To calibrate these parameters, we first match 124PCE categories, for which the frequency of price changes is available and whose price re-sponses are shown in Figure 3.2, with the industries reported by Close & Shulenburger(1971).34 If an industry is matched with only one PCE category, the frequency of pricechanges in that industry is taken as the one in the PCE category. If there are multiplePCE categories that match with a single industry, the frequency of price changes in thisindustry is measured as the weighted average of the frequency of price changes in thesePCE categories, with the weights given as the sum of the expenditure shares of the ELIs in2000 that are mapped with the PCE categories in Section 3.1.2. Labor shares in industriesare calibrated as those in 1948 reported by Close & Shulenburger (1971). Weights of eachindustry are calibrated as the sum of the weights of the PCE categories that match withthe industry. However, some PCE categories may not be matched with an industry, caus-34It may be useful here to exemplify our matching. For example, the PCE categories ?Tires? and?Accessories and parts? are matched with the industry of ?Motor vehicles and equipment? in Close &Shulenburger (1971).89ing the sum of the industries? weights to be less than one. Consequently, the weights ofthe industries need to be rescaled so that their sum is equal to one. Table 3.2 reports thecalibrated values for industries? labor share, the frequencies of price changes and weight.It is notable that while the petroleum and air-transportation industries have virtually thesame frequency of price changes,35 they markedly differ in their labor shares.36In Table 3.3, I report the structural parameter estimates in the multi-sector model withasymmetric cost structure. It is notable that the value of the objective function in themulti-sector model with asymmetric cost structure is lower compared the ones in the one-sector model and multi-sector model with symmetric cost structure as reported in Table3.1, suggesting that the multi-sector model with asymmetric cost structure is the mostsuccessful in accounting for the aggregate dynamics and the correlations.In Figure 3.9, I display the aggregate dynamics in the multi-sector model with asym-metric cost structure, after an unanticipated 1% increase in the federal funds rate, whichare largely in conformity with the aggregate dynamics in the previous two models as shownin Figure 3.4 and Figure 3.5.However, as evident in Figure 3.10, the correlations between the frequency of pricechanges and sectoral price responses in the multi-sector model with asymmetric cost struc-ture differ markedly from those in the multi-sector model with symmetric cost structure.As a matter of fact, contrasting with the latter, the former can more successfully explainthe low VAR-based correlations in the data.This can be attributed to the fact that when the asymmetric cost structure acrossindustries is introduced in the multi-sector model, sectoral price responses may delink fromsectoral frequency of price changes following the contractionary monetary shock. This point35The frequency of price changes in the petroleum and air-transportation industries are 0.97 and 0.94 inthe former and latter, respectively36The labor shares in the petroleum and air-transportation industries are 0.32 and 0.88 in the former andthe latter, respectively90Table 3.2: Calibrated Parameters(The Multi-Sector Model with Asymmetric Cost Structure)Industry Labor Share Frequency WeightFood 0.74 0.66 0.143Tobacco 0.52 0.69 0.019Textiles 0.80 0.64 0.002Apparel 0.87 0.66 0.064Paper 0.64 0.58 0.003Printing 0.79 0.20 0.013Chemicals 0.56 0.39 0.030Petroleum 0.32 0.97 0.060Furniture 0.82 0.51 0.023Fabricated metal 0.78 0.37 0.009Electrical machinery 0.77 0.50 0.019Transportation equipment and ordinance 0.89 0.21 0.035Motor vehicles and equipment 0.63 0.60 0.049Instruments 0.79 0.25 0.001Miscellaneous manufacturing industries 0.75 0.41 0.011Railroad trasportation 0.82 0.56 0.001Local, suburban, highway passanger transportation 0.87 0.14 0.005Water transportation 0.87 0.65 0.001Air transportation 0.88 0.94 0.014Trasporation services 0.80 0.29 0.002Telephone and telegraph 0.80 0.65 0.037Radio broadcasting and television 0.83 0.34 0.014Electric, gas, and sanitary services 0.55 0.71 0.067Wholesale trade 0.71 0.27 0.005Retail trade 0.6 0.38 0.008Hotels and other lodging places 0.69 0.75 0.040Personal services 0.64 0.12 0.028Miscellaneous business services 0.7 0.18 0.136Automobile repair 0.62 0.42 0.016Miscellaneous repair services 0.48 0.22 0.001Motion pictures 0.77 0.44 0.001Amusements 0.75 0.20 0.004Medical and other health services 0.4 0.13 0.061Educational Services 0.90 0.18 0.037Nonprofit membership organizations 0.98 0.25 0.025Miscellaneous professional services 0.58 0.15 0.013Note: Labor shares in industries are calibrated from the labor shares in industries in 1948 asreported by Close & Shulenburger (1971). See text for explanations related with the frequency ofprice changes in industries and the weight of industries.is illustrated in Figure 3.11 where the sectoral inflation dynamics in the petroleum and air-transportation industries. As suggested by the frequency of price changes in Table 3.2,91Table 3.3: Estimates of Structural Parameters(Multi-Sector Model with Asymmetric Cost Structure)? ?H ?p ?w b ?p ?w Obj. Func.0.014 0.001 81.4 12.68 0.78 0.97 0.87 53.59Note: Obj. Func. indicates the estimated value for the minimization problem discussedin (3.2.5).almost all firms in both industries optimize their prices each month. However, the formerhas a much lower labor share than the latter. For this reason, in Figure 3.11, the former andthe latter are labeled as the low and high labor-share industries, respectively. As evidentin the figure, inflation dynamics initially differ largely in the former and the latter despitehaving virtually the same frequency of price changes. Indeed, while inflation is almostunchanged in the former one period after the shock, inflation in the latter shows a strongincrease one period after the shock, suggesting sectoral price responses may substantiallydiffer across industries with a similar frequency of price changes in the multi-sector modelwith asymmetric cost structure, causing the association between the frequency of pricechanges and sectoral price responses in the sectors following a contractionary shock to below compared to that in the multi-sector model with symmetric cost structure. 3737This can be explained as follows: Since the frequency of price changes in both of the industries is almostone, it is reasonable to assume firms optimize their prices each period. Under this assumption, it can beshown that firms set prices relative to the aggregate price by imposing some constant mark up over realmarginal costs (sit+s(i?)) which can be written in its log-deviation as:s?it+s(i?) =1? ?i?iy?it(i?) + R?t + w?tA contractionary shock has two effects on prices which work in opposite directions. The first is that marginalcosts increase due to the working-capital channel in the model and an increase in R?t. The second is that afall in output results in a fall in marginal costs. The second effect is more decisive in the low labor-shareindustry since the real marginal costs faced by firms in the low labor-share industry would fall much moremarkedly compared to those in the high labor-share industry for a given fall in their output as the former92Figure 3.9: Impulse Responses to an Unanticipated 1% Rise in Rt(Multi-Sector Model with Asymmetric Cost Structure)(a) yt ? ynt0 4 8 12 16 20?2?1.5?1?0.500.51QuartersPercent(b) pit0 4 8 12 16 20?0.8?0.6?0.4?0.200.20.40.6QuartersPercent(c) wt0 4 8 12 16 20?0.5?0.4?0.3?0.2?0.100.10.20.3QuartersPercent(d) Rt0 4 8 12 16 20?1?0.500.511.5QuartersPercentNote: The solid lines show the VAR-based impulse responses and the area between dotted lines indicatethe 95% confidence intervals estimated with the method suggested by Sims & Zha (1999). The solid linesmarked with circles represent the dynamic responses of the variables as predicted by the model.In addition to bringing the correlations closer to those found in the data, the multi-has much lower ?i. This explains why prices in the low labor-share industry fall, while they increase stronglyin the high labor-share industry one period after the shock.93Figure 3.10: Model- and VAR-Based Correlations of ?i with Clnpit1,20(a) The Multi-Sector Model withSymmetric Cost Structure0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.40.60.81QuartersCorrelation(b) The Multi-Sector Model withAsymmetric Cost Structure0 4 8 12 16 20?1?0.8?0.6?0.4?0.200.20.40.60.81QuartersCorrelationNote: The solid lines show the VAR-based ?Clnpit1,20 ,?iand the area between dotted lines indicates the 95%confidence interval for ?Clnpit1,20 ,?ithat is estimated using the block-bootstrap method described in AppendixB.2. The solid lines marked with circles represent ?Clnpit1,20 ,?ipredicted by the model.sector model with asymmetric cost structure can also explain the wide distrubution ofsectoral price responses to a contractionary interest rate shock displayed in Figure 3.2.My findings in this section suggest adding asymmetries in the cost structure is crucial inexplaining the low correlations between the frequency of price changes and the sectoralprice responses to an interest rate shock.3.3 ConclusionIn this paper, the implications of heterogeneity in price flexibility at the disaggregated levelare studied. I have found that price responses to an unanticipated change in the interest94Figure 3.11: Inflation Dynamics in the Low and High Labor-Share Industries(The Multi-Sector Model with Asymmetric Cost Structure)0 4 8 12 16 20?0.3?0.25?0.2?0.15?0.1?0.0500.050.1QuartersPercent  Low?Labor?Share IndustryHigh?Labor?Share Industryrate differ substantially among sectors. Based on this finding, it is safe to claim thatinterest rate shocks have strong relative price effects at the disaggregate level. Next, I haveinvestigated whether this differential price response across sectors can be associated withthe wide distribution of the frequency of price changes in the United States. The findingsin this paper indicate that the association is weak. Lastly, the performances of three DSGEmodels are evaluated in explaining the aforementioned findings in the empirical section. Ithas been shown that the one-sector model may not explain the wide distribution of sectoralprice responses following the shock. It is possible to account for this finding by using amulti-sector model where sectors differ only in their frequency of price changes. However,contrary to the weak association of the frequency of price changes with sectoral priceresponses in the data, this model predicts a strong correlation between these variables.For this reason, an alternative multi-sector model has been considered. In this model,95sectors differ not only in the frequency of price changes but also in their cost structure.Such sectoral asymmetries in the cost structure and the frequency of price changes haveproved important in successfully explaining the strong relative price effects of the interestrate shock and the weak association of the frequency of price changes with sectoral priceresponses in the data.96Chapter 4Which Type of Model BestCaptures the Effects of MonetaryShocks in Developing Countries?There are many developing countries that have been exploring alternative monetary regimesafter years of high and variable inflation. However there remains considerable debateregarding the appropriate framework for analyzing monetary policy in such an environment.In particular, these economies are different on many fronts from those of more developedcountries, and therefore monetary models appropriate for the most developed countries areunlikely to be appropriate for developing countries.The goal of this essay is to develop a model which is appropriate for monetary policyanalysis in developing economies. Obviously, among many candidate models, the modelsin which key variables respond very differently than they do in actual economies to thesame type of shock studied are not appropriate for such an analysis. To reduce the numberof candidate models, Christiano, Eichenbaum & Evans (1998) suggest applying the Lucasprogram. Here, we follow this advice and apply the Lucas program using monetary shocks.This involves three steps. First, we attempt to isolate monetary shocks in developingeconomies which adopted an inflation targeting regime. In the second step, we study thedynamic behavior of output, the price level, the real and exchange rates in developing97economies following an expansionary monetary shock that results in a 1% increase in theprice level in the long-run.38 In the last step, the same experiment is conducted in twodifferent model environments and the outcomes in these models are compared with those inactual economies. The model in which the outcomes fit best with those in actual economiesis nominated as a viable candidate for monetary policy analysis in developing economies.Now, we elaborate on each of these steps. In the first step, for identifying monetaryshocks in developing economies under inflation targeting, we make two assumptions. Thefirst is that monetary shocks have no effect on the level of real variables in the long-term.This assumption is consistent with a broad class of models where monetary shocks haveno long-run effect on real variables. The second identifying assumption is that monetaryshocks in developing economies do not affect the aggregate price level in the United Statesin the long-term. This assumption is in conformity with the small-country assumption fordeveloping economies which is often made in the literature. With these assumptions, weshow monetary shocks can be isolated.Alternatively, the recursive assumption may be invoked to isolate monetary shocks.This method requires placing short-run restrictions on the contemporaneous response ofvariables as opposed to the long-term restrictions in our method. At this point, it is usefulto review the recursive assumption and discuss the reasons why adopting the recursive as-sumption may be unsuitable for isolating monetary shocks in developing countries. In therecursive assumption, a monetary authority is assumed to set its operating instrument byobserving movements in two different sets of variables. The first set of variables containsvariables that may respond only with a lag to monetary policy shocks and whose currentvalues are known to the monetary authority before a decision on its operating instrument38A 1% increase in the price level in the long-run is just a normalization. Indeed, it is by assumption inthe empirical model that monetary shocks must result in an increase in the price level. We simply normalizethe shock so that it induces a 1% increase in the price level in the long-run.98is made. The second set of variables, on the other hand, consists of variables that maycontemporaneously respond to monetary policy shocks and whose current values are un-known to the monetary authority before setting its operating instrument. The necessity ofincluding variables in one of these sets lies at the root of the controversy over the recursiveassumption for identifying shocks to monetary policy in developing economies. For exam-ple, in which set should the price level be included? Including it in the first set impliesprices are sluggish in responding to monetary policy shocks. Such an assumption wouldbe in conflict with the fact that a considerable share of prices change in a typical monthin developing economies. Additionally, because of the fast response of exchange rates tomonetary policy shocks and the strong pass-through of exchange rates into import pricesin developing countries, it is plausible to assume that monetary shocks affect prices con-temporaneously through their effect on exchange rates. Consequently, including the pricelevel in the first set of variables is questionable. Including it in the second set of variables,however, necessitates the assumption that central banks do not observe current values ofthe price level before setting their operating instrument. However, they collect data on alarge volume of prices and are likely to predict the general trend in prices over any period.For this reason, including the price level in the second set of variables is questionable, too.In our view, the price level in developing economies belongs to neither the first nor thesecond set of variables. Yet, that it has to be included in either of the two sets if therecursive assumption is adopted for isolating monetary shocks in developing countries leadus to abandon this strategy.In the next step, we characterize our experiment. We study how output, the bilat-eral real and nominal exchange rates with the United States and consumer prices move indeveloping countries under the inflation targeting regime after an expansionary domesticmonetary shock that results in a 1% long-run increase in the price level. We find this99shock is characterized by a temporary rise in output, a short-lived depreciation in the realexchange rate, a sizable overshooting of the nominal exchange rate and a 0.5% contempo-raneous increase in the consumer prices in these countries.Our findings of short-lived effects from monetary shocks on output and the real ex-change rates in developing economies contrast sharply with the long-lasting and persistenteffects of monetary shocks on such variables in advanced economies. For example, whileChristiano et al. (2005) find the effect of a monetary shock on output in the United Statesdissipates in about three years, we find the effect from a monetary shock on output be-comes negligible in less than one year in developing economies. Similarly, Rogoff (1996)finds shocks to the real exchange rates in advanced economies have a half-life of three tofive years, whereas we find in this study that shocks to the real exchange rate in developingeconomies have a half-life of less than a year. In addition, the speed of price adjustment isdifferent between advanced and developing economies. In fact, while the inertial characterof inflation results in a slow price adjustment in advanced countries, we find price adjust-ment is fast in developing economies. Moreover, the extent to which inflation has inertiais greatly limited in developing economies. As a matter of fact, prices adjust half-way, ormore, within the same period as the shock and the full price adjustment occurs in only oneyear. We show such short-lived real effects and faster price adjustment following a mone-tary shock in developing economies can be traced to the higher pass-through of exchangerates into import prices, the fact that import prices are largely denominated in the foreigncurrency and the fact that prices change more frequently in developing economies.In the last step, we turn to assess the ability of two dynamic stochastic general equilib-rium models to explain these findings. Before describing their differences, we discuss theirsix common features. First, there are Calvo-type nominal price contracts. Second, thefrequency of price changes differs between the home and foreign countries. Third, regard-100less of being domestic or foreign, if a firm sets prices in the home (foreign) currency, it issubject to the price rigidity in the home (foreign) country. Fourth, insurance is incompleteas households in both domestic and foreign countries only have access to the non-state con-tingent foreign asset. Fifth, in regards to the real side of the models, both of the modelsmaintain that acquiring new capital is subject to adjustment costs and capacity utilizationcan be variable. Sixth, they incorporate staggered wage setting.Next, we describe how these models differ. The first model is a one-sector model withidentical firms that have the same frequency of price changes. In contrast, the secondmodel is a multi-sector model with heterogeneous firms which have different frequencies ofprice changes. Indeed, while prices remain unchanged for long durations in some sectors,they change frequently in others. We then compare the outcomes in these models to thosein the actual economies after the monetary shock that causes a 1% long-run increase inthe price level. We find the latter is particularly accurate in accounting for the aggregatedynamics in the actual economies.The organization of the paper is as follows: Section 4.1 presents our empirical strategyfor isolating monetary shocks in developing economies and reports our findings on the con-sequences of monetary shocks in developing economies with the inflation targeting regime.Section 4.2 develops two dynamic stochastic sticky price small-open economy models. Sec-tion 4.3 describes the estimation and calibration of the models? parameters. Section 4.4evaluates the success of the models in accounting for the outcomes of a domestic monetaryshock in the actual economies that are reported in Section 4.1. The last section concludes.4.1 Empirical SectionIn this section, we develop an empirical model for studying the dynamics of output, the realexchange rate and the price level in developing countries under inflation targeting following101a positive monetary shock. In the next section, we consider an empirical model for isolatingmonetary shocks that closely follows the strategy in Clarida & Gali (1994). However, sincemonetary shocks in developing countries and the United States are not identified separately,we argue in our second empirical model that this strategy is questionable. Next, we developan empirical model which enables us to study monetary shocks in developing countries andthe United States separately.4.1.1 Empirical ModelsEmpirical Model IBy employing a Blanchard & Quah (1989) type decomposition, Clarida & Gali (1994)identify various structural shocks in four developed countries. In contrast to their concen-tration on developed countries, our focus is on developing economies. We first consider anempirical model based on the strategy in Clarida & Gali (1994). However, as opposed toestimating a VAR model for each country as in Clarida & Gali (1994), we estimate thefollowing panel VAR model for the group of developing countries under inflation targeting,Xi,t =pmax?p=1BpXi,t?p + ?i + ui,t (4.1.1)where ?i is the time-invariant country-specific fixed-effect term and pmax denotes thenumber of lags included in the panel VAR regression. We use both quarterly and monthlydata to estimate (4.1.1) with the lag lengths chosen to be four and twelve, respectively. Theendogenous variables in the panel VAR system of (4.1.1), Xit, consist of three variables,Xi,t =???????Yi,t ??Y?t?Qi,t?Pi,t ??P ?t??????(4.1.2)102where ?Yi,t??Y?t is the difference between the log-changes in economic activity in thecountry of interest and the United States. For the quarterly data, we measure ?Yi,t??Y?twith real GDP differences in Economy i and the United States as in Clarida & Gali (1994).For the monthly data, on the other hand, we measure it with the differences in industrialproduction indexes between Economy i and the United States.39 The second variable in(4.1.2), ?Qi,t, denotes the percentage change in the bilateral real exchange rate of thecountry of interest with the United States. Qi,t is defined as the cost of the consumptionbasket in the United States relative to that in the country of interest in the same currency.40Lastly, ?Pi,t ??P ?t denotes inflation differences in consumer prices between the countryof interest and the United States.Clarida & Gali (1994) assume there are three different structural shocks which accountfor the movements of the variables in Xi,t.41 These are: supply difference shocks in thecountry of interest and the United States (pi,t?p?t ); demand difference shocks in the UnitedStates and the country of interest (d?t ?di,t); and, money difference shocks in the country ofinterest and the United States (mi,t? m?t ). Demand shocks can be regarded as governmentspending shock or any other demand shock apart from money shocks.The identification of structural shocks is achieved by placing restrictions on the long-run response matrix. To explain the identification method, let ui,t ? N(0,?) where ? isthe non-diagonal variance-covariance matrix of ui,t. Also, suppose that ui,t is related tothe structural shocks in the following way,39Where data for seasonally adjusted series are available, we used these series. Otherwise, we obtainedseasonally adjusted series from non-seasonally adjusted series by using the Demetra + program from Eu-rostat.40Let Ei,t be the home currency price of the United States dollar in economy i. Also denote P ?t and Pi,tas indexes of the consumption basket in the United States and Economy i, respectively. We measure Qi,t asEi,tP?tPi,t. Hence, a rise in Qi,t is associated with a depreciation of the real exchange rate vis-a-vis the UnitedStates.41Since our empirical approach is related to the empirical strategy in Clarida & Gali (1994), we give areview of their method in Appendix C.1.103ui,t = C0i,t, i,t =??????????pi,t ? p?td?t ? di,tmi,t ? m?t??????????, i,t ? N(0, C?10 ?C?10?)(4.1.3)where C0 is a 3?3 matrix of the contemporaneous responses of the variables to shocks.It is notable that due to the assumption of independence among different type of structuralshocks, the variance-covariance matrix, C?10 ?C?10?, is diagonal. Furthermore, under thenormalization that the variance-covariance matrix of structural shocks is an identity matrix,the following equality has to hold:C0C0? = ? (4.1.4)Clarida & Gali (1994) identify structural shocks by imposing restrictions on the effectsof these shocks on the level of the output difference, the real exchange rate and the pricelevel difference in the long-run. Denoting the matrix of the long-run impulse responses byD, Clarida & Gali (1994) isolate structural shocks by assuming that D is lower triangular,D =??????????d11 0 0d21 d22 0d31 d32 d33??????????(4.1.5)The ordering of the variables in (4.1.2) implies only supply shocks influence the level ofthe output difference in the long-run. Neither demand nor money shocks have a permanenteffect on the level of the output difference. Regarding the real exchange rate, its level isaffected permanently by supply or demand shocks. Lastly, all three shocks have a long-runimpact on the level of the CPI difference.In order to uniquely recover structural shocks, in addition to the lower triangularity of104the long-run matrix, it is necessary to impose sign restrictions on D. A larger supply andmonetary shock in Economy i compared to the United States are assumed to increase thelong-run levels of GDP and CPI in Economy i relative to the United States, respectively(d11 > 0, d33 > 0). In addition, a larger demand shock in Economy i compared tothe United States is assumed to appreciate the long-run level of the real exchange rateof Economy i relative to the United States (d22 > 0). This can happen if governmentspending mostly fall on non-traded goods.Some restrictions on the long-run impact matrix in Clarida & Gali (1994) are debatable.For example, the sign restriction that an expansionary fiscal shock in Economy i appreciatesthe real exchange rate in the long-run should necessarily be taken with a grain of salt (Forexample, see Ravn, Schmitt-Groh & Uribe (2007) for counter evidence). Similarly, theexclusion restriction in Clarida & Gali (1994), that the fiscal shocks have no long-runeffect on the level of output, is subject to criticism as it is quite likely that fiscal shockssuch as spending shocks on education and infrastructure impact the long-run output level ina country. Based on these considerations, we slightly modify the long-run impact responsematrix. Indeed, as in Clarida & Gali (1994), we assume monetary shocks have a long-runimpact on neither output level nor the real exchange rate level. Yet, we do not place anyrestriction regarding the long-run impact of productivity and demand shocks on the levelof any of the variables. Let ~D denote the modified long-run impact matrix of structuralshocks with the above noted restrictions on the level of the variables. This matrix can thenbe written as~D =??????????~d11 ~d12 0~d21 ~d22 0~d31 ~d32 ~d33??????????(4.1.6)105In addition to the restrictions in (4.1.6), in Appendix C.1, we show in (C.1.13) that ~Dmust also satisfy~D~D? =(I ?pmax?p=1Bp)?1?(I ?pmax?p=1Bp)??1(4.1.7)The modified long-run impact matrix of structural shocks, ~D, has seven free parameterswhereas ~D~D? is symmetric so it has only six independent elements. Hence, it is not possibleto uniquely recover all the parameters of the ~D matrix. In particular, an analysis of thedynamic responses of the variables following productivity and demand shocks necessitatesknowing the elements in the first and second columns of (4.1.6), respectively. Yet, suchan analysis is not feasible as the elements in these columns are unidentifiable given thestructure of ~D. However, the third column can be uniquely recovered. This allows us toinvestigate dynamic responses of the variables to monetary shocks. To prove this, note firstthat since the model is not uniquely identified, there are many matrices satisfying (4.1.7).Letting ~D and ~DA be two of such matrices (i.e. both ~D and ~DA are block lower-triangularas stated in (4.1.6) and satisfy (4.1.7)), we can always find a square block lower-triangularorthonormal matrix ~? such that~DA = ~D~? (4.1.8)One can show the reason for ~? matrix to be block lower-triangular and orthonormal inthree steps. First, we show ~? is orthonormal. Since ~D and ~DA satisfy (4.1.7), the followingequation has to hold:~D~?~??~D? = ~D~D? (4.1.9)Multiplying both sides with ~D?1 from the left and with ~D?1?from the right yields106~?~?? = I where ~D is invertible by assumption. The implication being that ~? has to be anorthonormal matrix.Second, note that ~? = ~D?1~DA. Since the product of two block lower-triangular matriceshas to be block lower-triangular, ~? has to be block lower-triangular, as well. Hence, onecan write ~? as~? =??????????~?11 ~?12 0~?21 ~?22 0~?31 ~?32 ~?33??????????(4.1.10)Third, multiplying both sides of (4.1.8) with ~?? and using the fact that ~? is orthonormalyields ~DA~?? = ~D. Since ~DA?1 and ~D are block lower-triangular, ~?? = ~DA?1 ~D, ~?? must alsobe block lower-triangular. This implies ~?31 and ~?32 are equal to zero as well. Furthermore,since ~? is orthonormal, ~? has to be in the form of one of two matrices:~? =??????????~?11 ~?12 0~?21 ~?22 00 0 ?1??????????or ~? =??????????~?11 ~?12 0~?21 ~?22 00 0 1??????????(4.1.11)Lastly, the final step in uniquely identifying the monetary shock requires the assumptionthat an expansionary monetary shock results in a permanent rise in price level differencesbetween the developing economies and the United States.42 This sign restriction uniquelyidentifies the third column by ensuring ~?33 = 1. Therefore, even if there are many matricessatisfying both (4.1.7) and (4.1.9), their third column must be the same. Identifying theelements of the third column this way enables us to analyze dynamic responses of thevariables to monetary shocks.4342Therefore, ~d33 is positive in (4.1.6)43Here, it is natural to ask whether structural monetary shocks can be identified by placing restrictions107Empirical Model IIClarida & Gali (1994) employ their strategy for isolating structural shocks in developedeconomies. In comparison to developed countries, an analysis of the dynamic responses ofvariables to structural shocks in developing countries may require more demanding assump-tions. In particular, note that Clarida & Gali (1994) isolate differences in structural shocksbetween the country of interest and the United States, mi,t?m?t , rather than isolating themseparately, mi,t and m?t . When only differences in shocks are isolated, a 1% expansionarymonetary shock in the country of interest is implicitly assumed to induce the same dynamicsas a 1% contractionary monetary shock in the United States. Under the symmetric-countryassumption, this may be a plausible assumption if one studies the movements in Yi,t ? Y?t ,Qi,t and Pi,t ? P ?t between a developed economy and the United States. Yet, it is notrealistic to maintain the symmetric-country assumption for a developing economy and theUnited States. For example, the coefficients of exchange rate pass-through into import andconsumer prices in developing economies and the United States are markedly dissimilar.Furthermore, the frequencies of price changes among sectors in developing economies con-trast with those in the United States. These asymmetric features may cause the dynamicsof Yi,t?Y?t , Qi,t and Pi,t?P ?t between developing economies and the United States after a1% expansionary monetary shock in developing economies to differ significantly from thoseafter a 1% contractionary monetary shock in the United States.44only on the long-run responses matrix to monetary shocks. By writing the equation for the structural shockexplicitly in (4.1.12), we show that this is not possible:i,t = C?10 ui,t = ~D?1(I ?pmax?p=1Bp)ui,t (4.1.12)Since monetary shocks are ordered as the third element of i,t, recovering them requires the third rowof the inverse of ~D in (4.1.12). Yet, the third row cannot be identified by placing restrictions only in thelong-run effects of monetary shocks on the level of output differences and the real exchange rate betweenthe United States and the developing country. Consequently, structural monetary shocks are unidentifiablein Empirical Model I.44Apart from these asymmetric features, a difference in the monetary shock process between developing108For this reason, we believe it is more plausible to study the consequences of monetaryshocks in developing economies and the United States separately. To achieve this, weconsider the same panel VAR model in (4.1.1), yet the vector of variables, Xi,t, is nowgiven asXi,t =?????????????Y?t?Yi,t?Qi,t?P ?t?Pi,t????????????(4.1.13)Here, ?Y?t (?Yi,t) and ?P ?t (?Pi,t) denote the log-change in output and the consumerprice level in the United States (the country of interest), respectively.Fluctuations in the vector of variables in Empirical Model II are assumed to be drivenby five structural shocks in the following order:1. Supply shocks in the United States (p?t )2. Supply shocks in developing economies (pi,t)3. General preference shocks (di,t)4. Monetary shocks in the United States (m?t )5. Monetary shocks in developing economies (mi,t)Our goal is to analyze dynamic responses of the variables to monetary shocks in theUnited States and developing countries separately. This can be achieved if the followingeconomies and the United States may also result in the dynamics of Yi,t?Y?t , Qi,t and Pi,t?P?t between de-veloping economies and the United States after a 1% expansionary monetary shock in developing economiesdiffering significantly from those after a 1% contractionary monetary shock in the United States.109assumptions are made regarding the D? matrix which shows the long-run level responses ofthe variables in developing economies to each shock in Empirical Model II:D? =????????????????d?11 d?12 d?13 0 0d?21 d?22 d?23 0 0d?31 d?32 d?33 0 0d?41 d?42 d?43 d?44 0d?51 d?52 d?53 d?54 d?55????????????????(4.1.14)In the structure of (4.1.14), monetary shocks in the United States have been constrainedto have no impact on the long-run level of output in both economies and the real exchangerate. In addition to these constraints, monetary shocks in the developing economies arerestricted to have no permanent impact on the price level in the United States. This as-sumption is consistent with both the small-country assumption for developing economiesand the standard practice of modeling the United States as a closed economy in the lit-erature. In fact, our maintained assumption in Empirical Model II regarding the effectof domestic monetary shocks in developing economies is weaker than the small-countryassumption in our theoretical models presented in Section 4.2. Indeed, while the assump-tion in Empirical Model II constrains domestic monetary shocks in developing economiesto have no long-term impact on the price level in the United States, the small-countryassumption in our theoretical model imposes that they have a negligible impact on theprice level in the United States in the short- and long-terms.Now, we aim to separately analyze the dynamic responses of the variables to monetaryshocks in the United States and developing economies. This can be achieved if the elementsof the fourth and fifth columns of (4.1.14) are known. By following the same arguments inSection 4.1.1, it can be shown that Empirical Model II is unidentified and there are many110matrices satisfying (4.1.14) and (4.1.15),D?D?? =(I ?pmax?p=1Bp)?1?(I ?pmax?p=1Bp)??1(4.1.15)By following exactly the same arguments in Section 4.1.1, it is easy to show that anytwo such matrices D? and D?A have the same fourth and fifth columns. This results fromthe fact that the orthonormal square matrix, ??, linking these two matrices must be in thefollowing form:?? =??????????????????11 ??12 ??13 0 0??21 ??22 ??23 0 0??31 ??32 ??33 0 00 0 0 1 00 0 0 0 1????????????????(4.1.16)Having identified the fourth and fifth columns of (4.1.14) this way, an analysis of thedynamic responses of the variables to the monetary shocks in the United States and thedeveloping economies is straightforward.4.1.2 Empirical ResultsThis section presents our findings on the responses of domestic economic activity, thebilateral real exchange rate with the United States and prices after domestic monetaryshocks in developing countries under an inflation targeting regime. Since the adoptiondates of the inflation targeting regime were not the same among the countries in oursample, we have an unbalanced panel data. As stated in Arellano & Bond (1991), thisdoes not fundamentally change our analysis since we only require the assumption thatobservations are independently distributed in the initial cross-section and that subsequent111Table 4.1: Adoption Dates of Inflation Targeting in Developing EconomiesMonthly Data Quarterly DataCountry Effective IT adoption date Effective IT adoption datePoland 1998-M10 1998-Q4Brazil 1999-M6 1999-Q2Chile 1999-M9 1999-Q3Colombia 1999-M9 1999-Q3South Africa 2000-M2 2000-Q1Thailand 2000-M5 2000-Q2Mexico 2001-M1 2001-Q1Hungary 2001-M6 2001-Q2Peru 2002-M1 2002-Q1Philippines 2002-M1 2002-Q1Guatemala 2005-M1 2005-Q1Indonesia 2005-M7 2005-Q3Romania 2005-M8 2005-Q3Turkey 2006-M1 2006-Q1Serbia 2006-M9 2006-Q3Source: Roger (2009)additions and deletions occur randomly. Table 4.1 reports the adoption dates of inflationtargeting in the developing countries contained in our sample for which we have quarterlyor monthly data.Our source of data on the level of economic activity, bilateral nominal exchange rateswith the United States and consumer prices in our sample of countries is the IMF?s In-ternational Finance Statistics data. Our data spans the post-inflation targeting period foreach country until March, 2013. Due to data limitations on industrial production indexfor some developing countries at the monthly frequency, Chile, Colombia, Guatemala, In-donesia, Peru, Philippines and South Africa are dropped from the sample at the monthlyfrequency. Instead of the industrial production index, real GDP is used at the quarterly fre-112Figure 4.1: Impulse Responses to Monetary Shocks in Developing Economies(Empirical Model II)Quarterly(a) Y?t0 4 8 12?0.100.10.20.3QuartersPercent(b) Yi,t0 4 8 12?0.500.51QuartersPercent(c) Qi,t0 4 8 12?0.500.511.5QuartersPercent(d) P ?t0 4 8 12?0.100.10.20.3QuartersPercent(e) Pi,t0 4 8 1200.511.5QuartersPercentMonthly(f) Y?t0 12 24 36?0.200.20.40.6MonthsPercent(g) Yi,t0 12 24 36?1012MonthsPercent(h) Qi,t0 12 24 36?10123MonthsPercent(i) P ?t0 12 24 36?0.2?0.100.10.2MonthsPercent(j) Pi,t0 12 24 3600.511.5MonthsPercentNote: Our calculations are based on the IMF?s International Finance Statistics. The solidlines indicate the estimated point-wise impulse responses. The area between the dashedlines shows the 90% confidence interval estimated using the Bayesian method suggested bySims & Zha (1999).quency. Since the series of real GDP are available for most sample countries, our quarterlydata contains a larger sample of economies.4545Before presenting our results, it is essential that logged real exchange rates of developing economiescompared to the United States, Qi,t, the logged real GDP and CPI in Economy i and the United States(denoted by Yi,t, Y?t , Pi,t and P?t , respectively) all have unit roots. For the series pertaining to developingeconomies, we estimated a panel autoregressive model with country-specific fixed effects containing fourand twelve lags for the quarterly and monthly data, respectively. With the level specification, we performthe augmented Dickey-Fuller test. The unreported results indicate that one cannot reject the null that allfive series contains a unit-root at the 5% significance level. With the growth specification, on the otherhand, the null is rejected strongly at the 5% significance level. Hence, we conclude that all five series haveunit roots.113We now study the aggregate dynamics after an expansionary domestic monetary shockin developing economies using Empirical Model II. 46 These aggregate dynamics are dis-played in Figure 4.1. It is evident from this figure that an expansionary monetary shockin developing economies? causes a modest, short-lived impact on output in the United States;47? induces an increase in the level of output in developing countries relative to its undis-torted path which lasts for about one year;? depreciates the real exchange rate on impact, implying that the goods from thedeveloping economies is worth less in terms of the goods from the United States;48? leads to either a small, temporary increase or no change at all in the price level ofthe United States; and,? results in a permanent increase of the price level in the developing economies.Such findings only show the average impulse response functions for the group of de-veloping countries which adopted an inflation targeting regime. However, the impulseresponse functions of the variables to an expansionary domestic monetary shock in eachcountry in the group differ radically from the average impulse response functions. Figure4.2 illustrates this point. The impulse response functions of output, the real exchangerate and the price level to an expansionary monetary shock in each country is obtained46We study aggregate dynamics following monetary shocks in Empirical Model I and following monetaryshocks in the United States in Empirical Model II in Appendix C.2.47When the quarterly data is considered, output in the United States shows modest but significantresponses on impact and in the first period. This unusual finding may result from the fact that some of thecountries in our sample are large economies as they are included in the G20.48It is notable that the real exchange rate attains its highest level on impact, and falls afterwards. Thiscontrasts with the finding in Section C.2.2 that the real exchange rate exhibits hump-shaped dynamicsafter monetary shocks in the United States. Hump-shaped dynamics following monetary shocks in theUnited States are also found by Clarida & Gali (1994) and Eichenbaum & Evans (1995) for the bilateralreal exchange rates between the United States and other developed countries.114Figure 4.2: Impulse Responses in Each Country to Monetary Shocks in DevelopingEconomies(The VAR Model with Monthly Data)(a) Yi,t0 5 10 15 20 25 30 35?6?4?202468MonthsPercent(b) Qi,t0 5 10 15 20 25 30 35?10?50510MonthsPercent(c) Pi,t0 5 10 15 20 25 30 35?0.500.511.522.53MonthsPercentNote: Our calculations are based on the IMF?s International Finance Statistics. The solidline marked with circles indicates the median of the estimated point-wise impulse responsefunctions in the group in each period. The dot-dashed line shows the country-specificimpulse response functions separately estimated for each country in the group using theVAR version of Emprical Model II.separately by considering the country-specific VAR model version of Empirical Model IIwith monthly data. The size of the shock in each country is normalized to induce the samelong-run response in the price level. It is evident from this figure that the impulse responsefunctions of all three variables in the individual countries differ radically from the medianimpulse functions in the group.The Conditional and Unconditional Co-movements of the Real and NominalExchange RatesNext, we show the co-movements of the real and nominal exchange rates conditional onthe domestic monetary shock in the Empirical Model II with monthly data. The impulse115Figure 4.3: Conditional Movements of the Real and Nominal Exchange Rates(Empirical Model II with Monthly Data)(a) ?Qi and ?Ei0 12 24 36?0.4?0.200.20.40.60.811.21.4MonthsPercent(b) Qi and Ei0 12 24 36?0.200.20.40.60.811.21.41.6MonthsPercentNote: Our calculations are based on the IMF?s International Finance Statistics. Thedotted lines marked with circles in Panel (a) and Panel (b) indicate the log-change and thelevel impulse response functions of the real exchange rate to the domestic monetary shock,respectively. The dot-dashed lines marked with asterisks in Panel (a) and Panel (b) showthe log-change and the level impulse response functions of the nominal exchange rate tothe domestic monetary shock, respectively.response functions of the nominal exchange rates (E?) are obtained as Q? + P? ? P? ?. It isevident from Panel (a) of Figure 4.3 that conditional on the domestic monetary shock, thedeviation (in percent) of the log-change in the nominal and real exchange rates from theirundistorted path follow a similar pattern. Such co-movements are also noticeable from thecommon pattern of the impulse response functions of the level nominal and real exchangerates in Panel (b).Lastly, the co-movements of log-changes in the real and nominal exchange rates are an-alyzed unconditionally for each developing economy that has adopted an inflation targetingregime using monthly data in Figure 4.4. Again, we find movements in the real exchange116Figure 4.4: The Unconditional Co-movements of the Log-Changes in the Nominaland Real Exchange Rates(a) Brazil1999 Jul 2002 Dec 2006 Jun 2009 Nov?15?10?505101520DatePercent(b) Hungary2001 Jul 2004 Jun 2007 Jun 2010 May?10?5051015DatePercent(c) Mexico2001 Feb 2004 Mar 2007 Mar 2010 Apr?10?505101520DatePercent(d) Poland1998 Nov 2002 Jun 2006 Feb 2009 Sep?10?5051015DatePercent(e) Romania2005 Sep 2007 Jul 2009 Jun 2011 Apr?8?6?4?2024681012DatePercent(f) Serbia2006 Oct 2008 May 2010 Jan 2011 Aug?10?5051015DatePercent(g) Turkey2006 Feb 2007 Dec 2009 Sep 2011 Jul?10?505101520DatePercent  Real Exchange RateNominal Exchange RateNote: Our calculations are based the IMF?s International Finance Statistics. The dottedline marked with circles indicates the log-change in the real exchange rates for each country.The dot-dashed lines marked with asterisks show the log-change in the nominal exchangerates for each country.rates closely follow those in the nominal exchange rates.1174.2 Theoretical ModelsIn this section, we present two small-open economy DSGE models. In the next section, westudy the consequences of a monetary shock that causes a 1% increase in the price levelin the long-term for each model, and compare the outcomes in these models with those inthe actual economies to the same shock. We start by presenting models with the problemof Home and Foreign households.4.2.1 The Problem of Home and Foreign HouseholdsThere is a continuum of infinitely lived households in each country with a mass of one andindexed with h. Each household is comprised of two members. They aim to maximizetheir joint lifetime discounted utility with the discount factor given by ?. In period t, themembers of the hth household in the Home country have to make a sequence of decisions.First, they have to choose how much to consume from the home-country non-traded finalconsumption good (Ct). Second, they optimally choose how intensively they supply theircapital (ut) in each period. Third, they decide on the amount of investment (It), andtherefore, on the next period?s capital stock (Kt+1). Fourth, they have to decide on theamount of optimal holdings of a one-period risk-free foreign bond (Bt+1) which pays agross nominal return of RBt . Lastly, only one of the household members obtains a chanceto renegotiate its wage contract each period. The wage contract made in any period lastsfor two periods and has to be signed before observing the shock. The problem of the Homehousehold can be put more compactly as follows:maxCt,ut,It,Kt+1,Bt+1,xtEt??s=0?t+s(C1??ct+s ? 11? ?c?n?1+?nt+s,i1 + ?n?n1+?nt+s,i1 + ?n)(4.2.1)where n?t,i and nt,i are the hours worked by the members of the household whose wage118contracts are signed in period t and t ? 1, respectively. ?c and ?n stand for the recipro-cal of the intertemporal elasticity of substitution and the Frisch-elasticity of substitution,respectively. In solving 4.2.1, the household has the following budget constraint:Pt+s(Ct+s + It+s + a(ut+s)Kt+s)+ Et+sBt+1+s = xt+s,in?t+s,i + xt?1+s,int+s,i+Rkt+sut+sKt+s +RBH,t?1+sEt+sBt+s + ?t+s(4.2.2)In writing (4.2.2), we follow Christiano et al. (2005) and assume that increasing capacityutilization (ut) involves real costs in units of the final good denoted by a(ut).49. The priceof the home non-traded final good is denoted by Pt. Et stands for the nominal exchangerate between the currency of the home country (?) and the foreign country (??). Rkt denotesthe rental rate of capital paid to the owners of capital stock. The gross nominal return onthe holdings of last period?s foreign risk-free bonds is shown with RBH,t?1. xt,i and xt?1,iin (4.2.2) represent the hourly wage earnings of the household member who negotiates hiswage in period t and t?1, respectively. Lastly, ?t shows the profits of firms which belong tothe household. In sum, the representative household earns wage, capital, profit and interestincome. The household uses its resources to finance purchases of the final consumptiongood, investment, the cost associated with varying ut and purchases of foreign bonds.The law of motion for capital in the home country is given as:Kt+1 = (1? ?)Kt + ?(ItKt)Kt (4.2.3)49Let the bar symbol over the variables show the steady-state values of these variables. At the steadystate, capital is fully utilized, u? = 1. The function a(u) has the following properties: a(1) = 0, a?(u) > 0and a??(u) > 0.119where ?(ItKt)Kt shows the additional capital stock which new investment in the currentperiod makes available for the next period.50The problem of the foreign household is similar. Her optimization problem and flowbudget constraint can be written as:maxC?t ,u?t ,I?t ,B?t+1Et??s=0?t+s(C?1??ct+s ? 11? ?c?n??1+?nt+s,i1 + ?n?n?1+?nt+s,i1 + ?n)(4.2.4)P ?t+s(C?t+s + I?t+s + a(u?t+s)K?t+s)+B?t+1+s = x?t+s,in??t+s,i + x?t?1+s,in?t+s,i+ R?kt+su?t+sK?t+s +RBF,t?1+sB?t+s + ??t+s(4.2.5)where the variables denoted with the superscript ? represent the foreign counterpartsof the home variables. It is notable that the gross nominal return pertinent to the holdingsof the risk-free bond in the foreign country in (4.2.5), RBF,t?1, may differ from RBH,t?1 in(4.2.2). Following Devereux & Smith (2005), we assume that countries face a debt-elasticinterest rate. Let the net position of the home country in the risk-free bond be given asBt. The debtor country has to pay a higher interest rate than the lender country due toupward-sloping bond supply in international financial markets. The differential betweenRBF,t?1 and RBH,t?1 depends on the net bond holdings of the countries in the following way:RBH,t = ?(Bt+1 ? B?)RBF,t (4.2.6)where ?(Bt+1 ? B?)satisfies ? (0) = 1 and ?? (.) < 0. Since there is a continuum of50At the steady state, I? = ?K?. The function ?(ItKt)has the following properties. ? (?) = ?, ?? (?) = 1,?? (.) > 0 and ??? (.) < 0. The last assumption implies that ??? (.) is concave that emanates from the factthat new investment is subject to adjustment costs.120households in both countries, bond holdings of any individual household (Bt+1) has only anegligible effect on the net position of countries? bond holdings (Bt+1). Thus, householdsdo not internalize the interest rate country faces.51The optimality conditions for the Home household with respect to Ct, ut, It, Kt+1 andBt are given as:C??ct = ?tPt (4.2.7)a?(ut) = rkt , rkt = Rkt /Pt (4.2.8)?tPt = ?t??(ItKt)(4.2.9)?t = ?Et[??t+1Pt+1a(ut+1) + ?t+1Rkt+1ut+1 + ?t+1((1? ?)? ??(It+1Kt+1)It+1Kt+1+ ?(It+1Kt+1))](4.2.10)?tEt = ?Et?t+1RBH,tEt+1 (4.2.11)where ?t and rkt are the marginal utility of nominal income and the real rental priceof capital in the home country, respectively. ?t, on the other hand, stands for the shadowvalue of having one more unit of next period?s capital stock. In other words, it shows the51Assuming a debt-elastic differential in the two countries? interest rates is a standard way to circum-vent the problem of multiple steady states in imperfect financial markets. Without such an assumption,stationarity of the model would not be ensured as when a shock is introduced into the model, the modeloscillates between different steady states without ever reaching a stable equilibrium. For a more completedescription, see Schmitt-Grohe & Uribe (2003) and Boileau & Normandin (2008) who describe the problemof multiple steady states in the small- and large-open economy models with imperfect financial markets,respectively. They also evaluate different methods to circumvent this problem.121amount of the final good the household is willing to forgo in the current period to haveone more unit of capital stock in the next period. The condition (4.2.7) states that thehousehold equates the marginal utility of consumption with its marginal cost. As well, thecondition (4.2.8) implies that incremental variations in ut would cost a?(ut)Kt in resourcesbut since it allows the household to supply more capital services in the current period,the real income of the household rises by rktKt. At the optimal ut, these two should beequal. In (4.2.9), the left-hand side is the opportunity cost of investing an incrementalamount. At optimum, this is equated to the utility gained from making that incrementalinvestment as it allows the household to have ??(ItKt)more capital in the next period.The condition (4.2.10) indicates that the marginal utility of having an extra unit of capitalstock in the next period is the sum of three terms. ???t+1Pt+1a(ut+1) is the utility costassociated with the deviation of the capacity utilization rate in the next period from itssteady state. The second term, ??t+1Rkt+1ut+1, indicates that having an extra unit ofcapital stock in the next period would increase nominal income by Rkt+1ut+1. The thirdterm, ??t+1((1? ?)? ??(It+1Kt+1)It+1Kt+1+ ?(It+1Kt+1))denotes the utility gain of retainingthe extra unit of capital in period t + 2. Lastly, the optimal bond holdings equation inequation (4.2.11) states that purchasing an extra unit of foreign risk-free bonds would costEt in period t and would yield RBH,tEt+1 of nominal income in period t+ 1. Regarding theequivalent problem of households in the foreign country, all of the first-order conditions,except that of the bond holdings, are similar. The optimality condition for the foreign-household?s bond holdings, on the other hand, can be written as follows:??t = ?Et??t+1RBF,t (4.2.12)Using (4.2.7) and (4.2.11) along with their counterparts for the foreign household, the122equation for the real exchange rate between the home and foreign countries (Qt) can bewritten as:?c[Et(C?t+1 ? C?t)? Et(C??t+1 ? C??t)]= Et(Q?t+1 ? Q?t)+ ??(0)Y? B?t+1 (4.2.13)where Qt = EtP?tPt. In our paper, the bars and hats over the variables stand for thesteady-state values and the log-deviation of the variables from their steady states, respec-tively. The only exception is B?t+1 which is defined as Bt?B?Y? where Y? is the steady-statevalue of the aggregate final-good output. Defining B?t+1 this way makes it convenient totake a log-linear approximation of the domestic budget constraint.Aggregate Wage EquationBefore describing our wage setting environment, we discuss two main methods discussedin the literature for incorporating staggered wage setting into models. The first is theErceg, Henderson & Levin (2000) model where households supply differentiated hours ofwork and the chance to optimize their wages in each period is random with some givenprobability. The second is the Huang & Liu (2002) model where households are againassumed to supply differentiated hours of work, yet, the duration of wage contracts arenon-random as these contracts stay in effect for a specified duration of time.While these models are widely used, they are not particularly suitable for studying de-veloping countries since they assume complete financial markets. Yet, financial markets inthese economies are infant and lack sophistication. Based on this, we maintain householdsmay hold only non-state contingent bonds. However, writing a single budget constraintwhen a household consists of only one member is not possible due to the assumption ofincomplete financial markets in a staggered wage setting environment. A non-degenerate123income distribution of households due to incomplete insurance accounts for this fact asnoted by Huang & Liu (2002). Indeed, since workers renew their wage contracts in differ-ent periods under the staggered wage setting, their wage income differs with incompleteinsurance. Consequently, the problem of households in the economy with incomplete insur-ance might not be reduced to that of the ?representative? household as their incomes wouldnot be alike. Solving such a model involves the difficult task of following the non-degenerateincome distribution period-by-period which can be computationally demanding.Erceg et al. (2000) and Huang & Liu (2002) circumvent this problem by assumingcomplete financial markets. Under complete insurance, state-contingent assets are tradedto eliminate idiosyncratic risks among households. In staggered wage setting environments,these risks are associated with uncertainty about the timing of wage contract renewals. Forexample, when an expansionary monetary shock happens, in the absence of full insurance,workers whose contracts are renewed soon may be in an advantageous position comparedto workers whose contracts are renewed late. However, under complete financial markets,these idiosyncratic risks are eliminated since income transfers through state-contingentbonds among households exactly offset wage income differences among households so thathouseholds have the same income in all periods. In other words, income distribution isdegenerate under complete insurance.To the best of our knowledge, what is left unexplored in the literature is that id-iosyncratic risks under staggered wage setting can be eliminated even when insurance isincomplete. To achieve this, we modify the Huang & Liu (2002) model. We now explainthis. Note that households contain two members in our models, the wife and the husband,who negotiate their wages with employers in even and odd periods, respectively. Clearly,the wages of wives and husbands will be dissimilar after an expansionary monetary shockwith incomplete insurance. However, given that the wage income of households is given by124the sum of wives? and husbands? wages, even in the absence of income transfers throughfinancial assets, households? income will be alike after such a shock. Consequently, theincome distribution of households is degenerate as they all have the same income. Thus,we can consider the problem of a ?representative household? instead of household-specificmaximization problems. Achieving staggered wage setting without sacrificing the incom-plete financial market assumption in developing countries adds realism to our model.Now, we describe the home wage setting environment in detail. There is a continuumof employment offices with a mass of one in the home economy. They combine the dif-ferentiated hours of work supplied by the members of households (n?t,i and nt,i)52 into acomposite labor of (Nt) and sell it to the firms. The employment offices use the followingtechnology to form the composite of labor:Nt =??1?0n?(?w?1)/?wt,i di+1?0n(?w?1)/?wt,i di???w/(?w?1)(4.2.14)The optimization problem of employment offices can be written as:maxn?t,i,nt,iWtNt ?1?0xt,in?t,idi?1?0xt?1,int,idi (4.2.15)where, because of the assumption of a continuum of employment offices, individualoffices do not have an effect on the aggregate wage (Wt) and the wages set by the owners ofthe differentiated labors in period t and t?1 (xt,i and xt?1,i). Employment offices? demandfor differentiated labor of workers whose wages are set in period t and t? 1 are given by:n?t,i =(xt,iWt)??wNt ; nt,i =(xt?1,iWt)??wNt (4.2.16)52For definitions of n?t,i and nt,i, see (4.2.1).125From (4.2.16), it is clear that ?w is the wage elasticity of substitution among differen-tiated hours. In period t, one member of the households sets his wage before observing theshock that will remain fixed in period t and period t + 1. Hence, his optimality problemcan be written as:53maxxt,iEt?1[(?n?1+?nt,i1 + ?n+ ?txt,in?t,i)+ ?(?n1+?nt+1,i1 + ?n+ ?t+1xt,int+1,i)](4.2.17)Having renegotiated his wage in period t, the household member must supply differ-entiated hours of work as demanded by the employment offices due to the binding wagecontract in period t and period t+1. Due to the continuum of differentiated hours supplied,each individual worker has negligible effect on the aggregate wage. Using this and the factthat households? budget constraints are identical, the contracted wage in period t for allworkers is the same, allowing us to drop the subscript i in xt,i and write xt:x1+?w?nt =?w?w ? 1Et?1W?w+?w?nt N1+?nt + ?Et?1(W ?w+?w?nt+1 N1+?nt+1)Et?1(?tW?wt Nt)+ ?Et(?t+1W?wt+1Nt+1) (4.2.18)By using (4.2.14), (4.2.16) and the fact that all of the contracted wages are equal, onecan show that the aggregate wage equation is given by:Wt =(x1??wt + x1??wt?1) 11??w (4.2.19)53It is notable that in our notation, the hours supplied by the workers who do not renegotiate their wagesare shown without a tilde over n. Since it is not possible to renegotiate the wage in period t+ 1 once wageis set at period t, the hours supplied by the worker in the next period who set a wage at period t is shownwith nt+1,i not with n?t+1,i.126The wage-setting behavior of the owners of differentiated labor types in the foreigncountry is the same, yielding similar equations for the contracted and aggregate wages.4.2.2 The Objective of Firms in the Home and Foreign CountriesFirms Producing the Final Good in the Home and Foreign CountriesThe non-traded final goods in both of the countries are produced by a continuum of per-fectly competitive firms. Firms produce the final goods by using the following technologywhich involves combining goods from different sectors:Yt =(kmax?k=1f1/?k Y(??1)/?k,t) ???1(4.2.20)where Yt and Yk,t denote the amount of the final good produced by firms and the outputof Sector k, respectively. fk, ? and kmax denote the sectoral weight, constant elasticity ofsubstitution for sectoral goods in the final good production and the total number of sectorsin the home country, respectively. It is easy to show that the demand for sectoral goodsand the aggregate price index (Pt) are given by:Yk,t = fk(Pk,tPt)??Yt (4.2.21)Pt =(K?k=1fkP1??k,t) 11??(4.2.22)where Pk,t denotes the aggregate price index of sector k. Since the final-good firms inthe foreign country solve a similar problem, for the sake of brevity, we omit writing the127equations for the sector-specific foreign demand (Y ?k,t) and the foreign aggregate price (P ?t ).Firms Producing Sector k Output in the Home and Foreign CountriesIn both countries, sectoral goods are produced by an infinitely large number of perfectlycompetitive firms. The home firms producing sectoral goods combine domestic goods(YH,k,t) and import goods (YF,k,t) to produce sectoral output (Yk,t) with the followingtechnology:Yk,t =((1? ?) 1?Y (??1)/?H,k,t + ?1?Y (??1)/?F,k,t) ???1 (4.2.23)where ? and ? represent the steady-state weight of the import good in the home countryand the elasticity of substitution between the domestic and import goods, respectively. Itis straightforward to show the demands for the domestic goods and those imported by thehome country in sector k are given as:YH,k,t = (1? ?)(PH,k,tPk,t)??Yk,t ; YF,k,t = ?(PF,k,tPk,t)??Yk,t (4.2.24)where PH,k,t and PF,k,t denote domestic and import price indexes in sector k in thehome country, respectively. Using (4.2.23) and (4.2.24), one can write the sector k priceindex in the home country (Pk,t) as the weighted average of domestic and import priceindexes in sector k:Pk,t =((1? ?)P 1??H,k,t + ?P1??F,k,t) 11?? (4.2.25)Sector k?s good in the foreign country is again produced by perfectly competitive firms.128Yet, the technology combining home and foreign goods in sector k to produce its outputmay involve a lower steady-state share of imports in the foreign country than in the homecountry. Indeed, the foreign technology is given by:Y ?k,t =((1? ??) 1?Y ?(??1)/?F,k,t +(??) 1?Y ?(??1)/?H,k,t) ???1(4.2.26)It is clear from (4.2.26) that the steady-state import share in the foreign country is(??),which is smaller than the steady-state import share in the home country ? when ??1. Thisassumption is convenient since it allows us to study small- and large-open economies withinthe same model. Indeed, for a large economy, one can take ? = 1. For a small economy,on the other hand, one can assume ? is arbitrarily large as the size of its trading partnersis much larger compared to its size.We also give sector k?s price index and the demands for the home and foreign goods insector k in the foreign country as:P ?k,t =((1? ??)P ?1??F,k,t +(??)P ?1??H,k,t) 11??(4.2.27)Y ?F,k,t =(1? ??)(P?F,k,tP?k,t)??Y ?k,t ; Y ?H,k,t =(??)(P?H,k,tP?k,t)??Y ?k,t (4.2.28)where the variables denoted with asterisks (?) show the foreign counterparts of thehome variables.129The Invoice Currency and Pricing of Internationally Traded GoodsThe home-import good in sector k (YF,k,t) is produced by perfectly competitive home-import firms. Producing the home-import good involves combining intermediate foreigngoods which are invoiced in different currencies. Indeed, while some intermediate goodsare invoiced in the home currency (?), others are invoiced in the foreign currency (??). Inproducing the home-import good in Sector k, the home-import firm combines output fromthe foreign firms which set prices in the home and foreign currencies (denoted by YF,?,k,tand YF,??,k,t, respectively) with the following technology:YF,k,t =((1? ????)1?p Y (?p?1)/?pF,?,k,t + ??1?p?? Y(?p?1)/?pF,??,k,t) ?p?p?1(4.2.29)where ?p stands for the elasticity of substitution between intermediate foreign goodsinvoiced in different currencies and ???? denotes the steady-state weight of the foreign-currency-invoiced intermediate foreign goods in the home-import price index of sector k.It is easy to show that the price index for the home-import good (denoted by PF,k,t andexpressed in the home currency) and the demand for the intermediate foreign goods aregiven as:PF,k,t =((1? ????)P1??pF,?,k,t + ???? (EtPF,??,k,t)1??p) 11??p (4.2.30)YF,?,k,t = (1? ????)(PF,?,k,tPF,k,t)??pYF,k,t ; YF,??,k,t = ????(EtPF,??,k,tPF,k,t)??pYF,k,t (4.2.31)130where PF,?,k,t and PF,??,k,t represent the prices set for the intermediate foreign goodsthat are invoiced in the home and foreign currencies, respectively.The home-export good is produced similarly. Indeed, perfectly competitive foreignimporters in sector k combine output from the home firms which set prices in the homeand foreign currencies (denoted by Y ?H,?,k,t and Y ?H,??,k,t, respectively) with the followingtechnology:Y ?H,k,t =(??1?p Y ?H,?,k,t(?p?1)/?p + (1? ??)1?p Y ?H,??,k,t(?p?1)/?p) ?p?p?1 (4.2.32)where ?? is the steady-state share in sector k?s foreign-import price index of the home-currency-priced intermediate home-export goods. The foreign-import price index (denotedby P ?H,k,t and expressed in the foreign currency) and the demands for the intermediatehome-export goods can be written as:P ?H,k,t =(??( 1EtP ?H,?,k,t)1??p+ (1? ??)P ?H,??,k,t1??p) 11??p(4.2.33)Y ?H,?,k,t = ??(1EtP?H,?,k,tP?H,k,t)??pY ?H,k,t ; Y ?H,??,k,t = (1? ??)(P?H,??,k,tP?H,k,t)??pY ?H,k,t (4.2.34)where P ?H,?,k,t and P ?H,??,k,t denote the prices set for the intermediate home-export goodswhose prices are invoiced in the home and foreign currencies, respectively.131Home and Foreign Firms Producing Varieties for Intermediate GoodsThe intermediate domestic and import goods in both the home and foreign countries arecomposite goods composed of a variety of goods produced by firms engaging in monopolisticcompetition. The production technology used in the production of intermediate domesticgoods is given as:YH,k,t =??1?0YH,k,j,t(?p?1)/?pdj???p?p?1(4.2.35)where YH,k,j,t denotes demand for variety j of the firm producing the domestic inter-mediate good in the home country in sector k. One can show that YH,k,j,t and the priceindex for the domestic intermediate good in the home country in sector k (PH,k,t) can bewritten as:YH,k,j,t =(PH,k,j,tPH,k,t)??pYH,k,t (4.2.36)PH,k,t =??1?0PH,k,j,t1??p??11??p(4.2.37)where PH,k,j,t is the price set by the monopolistically competitive firm producing varietyj of the domestic intermediate good. When producing variety j, the firm employs thecomposite labor (NH,k,j,t) together with capital (KH,k,j,t) and uses the following productionfunction:YH,k,j,t = K1??H,k,j,tN?H,k,j,t (4.2.38)132where ? is the steady-state share of labor in the home country. In each period, only afraction of the firms producing different varieties in sector k obtains a price-change signal.When firms obtain such a signal, they set prices with their intermediate domestic-goodssuppliers. These prices remain constant until a new price-change signal is obtained. Duringthis time, firms are obliged to supply any quantity demanded of their varieties. In the one-sector model, it is assumed sectors have the same frequency of price changes which is givenby the weighted average of the frequencies of price changes in sectors. In the multi-sectormodel, on the other hand, the probability of receiving such a signal differs by sector. Forthe varieties of domestic sector k?s good in the home country, let 1 ? ?k indicate theprobability of receiving the price-change signal in each period. Then, the objective of thefirm producing variety j which obtains a price-change signal in period t can be written as:Et??s=0?s?sk(XH,k,j,tYH,k,j,t+s ?Wt+sNH,k,j,t+s ?Rkt+sKH,k,j,t+s)(4.2.39)where XH,k,j,t denotes the contracted price for the home variety j in sector k?s domesticgood in the home country. Let ?H,k,t be defined as:?H,k,t =( 1PH,k,t)??p (PH,k,tPk,t)??(Pk,tPt)??Yt (4.2.40)Then, from the first-order condition of (4.2.39), XH,k,j,t can be written as:XH,k,j,t =?p?p ? 1( 11? ?)1??( 1?)? Et??s=0 ?s?sk(W?t+sRkt+s1???H,k,t+s)Et??s=0 ?s?sk?H,k,t+s(4.2.41)133Since the objective function is identical across the firms that produce differentiatedgoods in sector k and obtain a price-change signal in the same period, their contractedprices are the same (XH,k,j,t = XH,k,t). This, together with the Calvo-type randomizationassumption, implies that PH,k,t can be rewritten as:PH,k,t = (1? ?H,k)XH,k,t + ?H,kPH,k,t?1 (4.2.42)Similar to the domestic intermediate good, the home-export goods are composite goodsmade up of a continuum of varieties produced by monopolistically competitive firms:Y ?H,?,k,t =(? 10 Y?H,?,k,j,t(?p?1)/?pdj) ?p?p?1 ;Y ?H,??,k,t =(? 10 Y?H,??,k,j,t(?p?1)/?pdj) ?p?p?1 (4.2.43)where the demand for the home-export variety of j priced in the home currency (theforeign currency) is denoted by Y ?H,?,k,j,t (Y ?H,??,k,j,t).The monopolistically competitive firm producing variety j and the aggregator firm de-manding this variety invoice in the same currency. It is also notable that while the varietiesproduced for the home-export firms are allowed to be invoiced in different currencies in themodel, the demand elasticity between any two home-export varieties is not affected by theinvoice currency. Indeed, the demand elasticity between any two home-export varieties isequal to ?p, regardless of whether they are priced in the same or different currencies.54Next, we write the maximization problem of the firm that produces variety j for thehome exporters and that set prices in the home currency (the foreign currency) as (4.2.44)54See Equation (4.2.29) and (4.2.43).134((4.2.45)):Et??s=0?s?sk(X?H,?,k,j,tY?H,?,k,j,t+s ?Wt+sN?H,?,k,j,t+s ?Rkt+sK?H,?,k,j,t+s)(4.2.44)Et??s=0?s??sk(Et+sX?H,??,k,j,tY?H,??,k,j,t+s ?Wt+sN?H,??,k,j,t+s ?Rkt+sK?H,??,k,j,t+s)(4.2.45)where 1-??k is the constant probability of receiving a price-change signal in the foreign-sector k, which is allowed to differ from that in the home-sector k (1-?k). Since inflationinfluences the frequency of price changes, the assumption of a constant probability of re-ceiving a price change signal may be considered strong due to variable inflation in oursample of developing economies. We discuss this in Appendix C.3.1 and conclude that overstable inflation periods, the frequency of price changes in sectors can be considered stable.In writing (4.2.44) and (4.2.45), we make an important assumption that the invoice cur-rency of monopolistically competitive home-export firms also determines the price rigiditywhich the firms face. Indeed, while the prices set in the home currency remain fixed withthe probability of ?k in each period, those set in the foreign currency are subject to theprice rigidity in the foreign sector k and remain fixed with the probability of ??k. We alsomake an analogous assumption for the monopolistically competitive home-import firms.One can show that the optimal prices set for the home-export varieties j which areinvoiced in the home currency (X?H,?,k,j,t) and the foreign currency (X?H,??,k,j,t) can bewritten as:135X?H,?,k,j,t =?p?p ? 1( 11? ?)1??( 1?)? Et??s=0 ?s?sk(W?t+sRkt+s1????H,?,k,t+s)Et??s=0 ?s?sk??H,?,k,t+s(4.2.46)X?H,??,k,j,t =?p?p ? 1( 11? ?)1??( 1?)? Et??s=0 ?s??sk(W?t+sRkt+s1????H,??,k,t+s)Et??s=0 ?s??sk Et+s??H,??,k,t+s(4.2.47)where ??H,?,k,t+s and ??H,??,k,t+s are defined as:??H,?,k,t+s =(1P ?H,?,k,t+s)??p ( 1EtP ?H,?,k,t+sP ?H,k,t+s)??p (P ?H,k,t+sP ?k,t+s)??(P ?k,t+sP ?t)??Y ?t (4.2.48)??H,??,k,t+s =(1P ?H,??,k,t+s)??p (P ?H,??,k,t+sP ?H,k,t+s)??p (P ?H,k,t+sP ?k,t+s)??(P ?k,t+sP ?t)??Y ?t (4.2.49)The maximization problem of foreign firms can analogously be written.4.2.3 Closing the ModelOur first approach to close the model is to assume the growth of nominal spending followsan exogenous process in both countries:136logZt ? logZt?1 = ?z(logZt?1 ? logZt?2) + zt zt ? N(0, ?z2 )logZ?t ? logZ?t?1 = ?z(logZ?t?1 ? logZ?t?2) + z?t t ? N(0, ?z?2 )(4.2.50)where Zt = PtYt and Z?t = P ?t Y ?t denote nominal spending in the home and foreigncountries, respectively.4.3 Calibration and EstimationThis section discusses calibration of the models? parameters. It should be noted that sincemonthly frequencies of price changes are readily available, whereas quarterly frequenciesare not, we assess the ability of the models by comparing the outcomes from the modelswith those in the actual economies using monthly data. In Table C.1 of Section C.3 ofthe appendix, we present calibrated parameter values along with a source on which webase our calibration for these parameters. We start with ?p. It is taken to be equal to11, implying an average markup of 10%, which is the estimated markup rate for the autoindustry of the United States in Bresnahan (1981). We set ? = 0.008, implying an annualrate of depreciation of 10%, which is the estimated annual rate of depreciation in theUnited States in Christiano & Eichenbaum (1992). We calibrate the values for ?c, ?n, ?a,??, ??Y? , ?, ? and ? directly from the sources outlined in Table C.1. ? is set to 1.03?112 ,which implies an annual real interest rate of 3%.Next, we calibrate the frequency of price changes in each sector. It is noteworthy thatsince the main trading partners of developing economies are advanced countries, the price-stickiness parameters and sectoral weights in the foreign country (denoted by ??k and fk)need to be calibrated as those in advanced countries when we study aggregate dynamicsfollowing monetary shocks in developing economies in our model. When calibrating these137parameters, we rely on the estimates reported in Carvalho & Nechio (2011).55 They es-timate the weighted average of the frequency of price adjustments (?67k=1 fk(1 ? ??k)) inthe United States as 0.21. Based on this, we take the foreign price stickiness, ??k, in theone-sector model as 0.79.The home frequency of price changes, 1 ? ?k, in the one-sector model is calibrated as27.2%. That is, on average, 27.2% of prices change in each month in developing economies,which is in line with the estimates of the mean frequency of price changes in Mexico inGagnon (2009) when inflation remained between 4% and 14%. We do not have estimatesof sectoral frequency of price adjustments in developing economies. In calibrating sec-toral price stickiness in developing economies for the multi-sector model, we ensure that?671 fk(1??k) = 0.272. We also assume that the expected duration of price contracts in ahome sector is shorter than that in its foreign counterpart by some factor, say by D. If D istaken as 1.45, we find that the aforementioned condition is met. That is, if sectoral pricesin these economies changes 1.45 times more frequently than those in the United States,the condition that?671 fk(1? ?k) = 0.272 is met. With such an assumption, the sectoralfrequency of price changes in the home country can be calibrated using the following steps.First, estimate the expected duration of price contracts in a sector in the United Stateswith the following formula,d?k = ?1ln??kSecond, estimate the expected duration of sectoral price contracts in the home country55It is notable that while Carvalho & Nechio (2011) use the data from Nakamura & Steinsson (2008a)who report the frequency of price changes and the expenditure share for 271 categories of goods and servicesin the United States, to make their model computationally manageable, Carvalho & Nechio (2011) onlyinclude 67 sectors in their model by aggregating some sectors.138by assuming that it is 1.45 times shorter than that in the United States,dk =d?k1.45In the last step, estimate sectoral price stickiness in developing economies with,?k = e?1dk 56Even if the frequency of price changes is calibrated for 67 sectors, we only include 3sectors in our multi-sector model. The reason is that we have to estimate some parametersusing minimum distance estimation in our paper and it is not computationally feasible todo estimation with 67 sectors. In reducing the number of sectors to three, we first order thesectors according to their frequencies of price changes. Next, we include the sectors whosefrequency of price changes lies in [0, 33],[34, 66] and [66,100] percentiles of frequencies ofprice changes in the first, second and third group, respectively. The frequency of pricechanges that represents each group is approximated by the median frequency of pricechanges in each group. The expenditure share of each group (fk), on the other hand, istaken as the sum of the expenditure shares of the sectors forming the group.In calibrating the shares of final consumption (sc), investment (sm) and home imports(?) in GDP, we use data for these series from the World Bank?s World DevelopmentIndicators in 2002. sc, sm and ? are taken as the median values in the group. ? whichdenotes the economic size of the foreign country relative to that of the home country istaken as 1000. ? is set to be very high for developing economies, in line with the commonsmall-country assumption for these countries in the literature. It is notable that setting ?to a large value for developing economies, together with the assumption of no international56This follows from dk = ?1ln?k139borrowing at the steady state, requires that the steady-state shares of exports and importsin the foreign country be only 1? as big as those in the home country. This is the essenceof the small-country assumption in our model. The share of the home exports priced inthe home currency (??) and the share of the home imports priced in the foreign currency(???) are calibrated based on the findings in Section C.3.2 for Turkey.Lastly, in order to calibrate ?z, which represents the persistence in the exogenousnominal spending growth process in (4.2.50), the Panel AR(12) model for log changesin the monetary aggregates M1 and M2 are estimated for our sample using monthly datawith country-specific fixed effects. The sum of AR coefficients for M1 and M2 are estimatedas 0.35 and 0.29, respectively. Based on this, we set ?z = 0.32.To study dynamics after nominal spending shocks, both models are log-linearizedaround the zero-inflation and zero-debt steady state.4.4 Quantitative ResultsIn this section, our aim is to evaluate the ability of the one- and multi-sector models toaccount for the dynamics of output, the price level, the real and nominal exchange ratesafter monetary shocks in developing economies which adopted an inflation targeting regime.4.4.1 Output and Price Level DynamicsFigure 4.5 displays the model- and panel VAR-based impulse response functions of output(Y?t) and the price level (Pt) in the home country.57 In this figure, the dashed lines with57It is notable that real spending (denoted by Yt) differs from domestic output. We denote domesticoutput in the home country as Y?t. Y?t can be written as:Y?t =K?k=1fk(1? ?)Y?H,k,t +K?k=1fk???Y??H,?,k,t +K?k=1fk?(1? ??)Y??H,??,k,t140Figure 4.5: Model- and Panel VAR-Based Impulse Responses of P and Y to z(a) Price Level (P )0 12 24 3600.511.5MonthsPercent(b) Output (Y)0 12 24 36?0.500.511.52MonthsPercentNote: Our calculations are based on the IMF?s International Finance Statistics. Thedotted lines with pentagrams and the dashed lines with squares indicate the model-basedimpulse response functions in the one- and multi-sector models, respectively. The solidlines show the estimated point-wise panel VAR-based impulse response functions. Thearea between the dotted lines shows the 90% confidence interval estimated with the methodsuggested by Sims & Zha (1999).pentagrams and dotted line with squares show the impulse response functions to a domesticexpansionary shock in the one- and multi-sector models, respectively. The panel VAR-basedimpulse responses of the variables in developing economies obtained in Empirical ModelII with the monthly data are displayed with the solid lines. Lastly, the area between thedotted lines show the 90% confidence interval of the panel VAR-based impulse responsefunctions estimated with the method suggested by Sims & Zha (1999). It is notable that forboth the model- and panel VAR-based impulse response functions, we consider a monetaryshock in developing economies that results in a 1% long-run increase in P .We first discuss the price level dynamics. A striking observation in Figure 4.5 is thatthe price level responses in the multi-sector model stays muted compared to those in the141one-sector model. This point is explained succinctly in Nakamura & Steinsson (2013) forthe case of no strategic interaction among firms. Suppose that an economy has two sectors.Let the first sector have a low frequency of price changes so that it takes quite a while forfirms in this sector to respond to an aggregate shock (the sticky-price sector). Let thesecond sector have high price flexibility so that prices may respond fast to an aggregateshock in this sector (the flexible-price sector). It can be argued that firms in the flexible-price sector might have a chance to change their prices several times before firms in thesticky-price sector do so for the first time. However, apart from the period in which firmsin the flexible-price sector obtain a chance to change their prices for the first time, theprice adjustment in this sector in accompanying periods adds little to the aggregate priceadjustment since firms adjust fully to the shock when they first obtain a chance to respond.In other words, apart from the first responses, all other price responses in the flexible-pricesector are ?wasted?. For the complete aggregate price adjustment, it is crucial that firms inthe sticky-price sector obtain a chance to change their prices at least once after the shock.Nakamura & Steinsson (2013) note that if it were possible to have a more even distributionof the frequency of price changes among sectors, the aggregate price adjustment would bemuch faster. This conjecture is supported by our findings. Indeed, in the one-sector model,by taking the weighted average of the frequencies of price changes among sectors as thefrequency of price changes in the economy, some price changes are implicitly reallocatedfrom the flexible-price sector to the sticky-price sector. As a result, it is not surprising toobserve a stronger contemporaneous response of the aggregate price level and faster priceadjustment in the one-sector model than in the multi-sector model.Regarding output, it is clear in Figure 4.5 that output shows less persistent dynamicsin the one-sector model than the multi-sector model. This can be accounted for by a fasterprice adjustment in the former.1424.4.2 Real and Nominal Exchange Rate DynamicsFigure 4.6 displays the dynamics of nominal and real exchange rate in the one- and multi-sector models along with their panel VAR-based dynamics.58 It is evident that the nominalexchange rate undershoots its new long-run level, which contrasts with a sizable overshoot-ing of the nominal exchange rate in the actual economies shown in this figure. This mainlyresults from the muted initial impulse response functions of the real exchange rate.Our findings regarding the models indicate that both the one- and multi-sector modelsare of limited ability in explaining the aggregate dynamics in developing economies follow-ing a monetary shock. Indeed, some impulse response functions stay out of 90% confidenceintervals. Particularly, nominal exchange dynamics in the actual economies are poorlypredicted by these models.How can the predictions of the one- and multi-sector models be improved? We show inthe next section that when adjustment costs of new capital are so large that they prohibitinvestment, the extent to which the exchange rate overshoots increases and the models?performance improves to a certain degree.4.4.3 One- and Multi-Sector Models without InvestmentTo understand the reason for the limited degree of exchange rate overshooting in themodels, it is useful to consider the real exchange rate equation in the model. It can beshown from (4.2.13) that the % deviation of the real exchange rate (Q?t) from its steady58To obtain 90% confidence intervals for the impulse response functions of the nominal exchange rate,we first obtain 1000 randomly generated impulse response functions of the nominal exchange rate over 36months (Ei) as Ei = Qi ? P i ? P ?iwhere Qi, P i, P ?idenote randomly generated impulse functions of thereal exchange rate, the price level in developing economies and the United States, respectively. The areathat stays within the 5th and 95th percentile of the distribution of randomly generated impulse responsefunctions of the nominal exchange rate is reported in Figure 4.5 as the 90% confidence interval for theimpulse response functions of the nominal exchange rate.143Figure 4.6: Model- and Panel VAR-Based Impulse Responses of E and Q to z(a) Nominal Exchange Rate (E)0 12 24 3600.511.522.533.5MonthsPercent(b) Real Exchange Rate (Q)0 12 24 36?1?0.500.511.522.53MonthsPercentNote: Our calculations are based on the IMF?s International Finance Statistics. Thedotted lines with pentagrams and the dashed lines with squares indicate the model-basedimpulse response functions in the one- and multi-sector models, respectively. The solidlines show the estimated point-wise panel VAR-based impulse response functions. Thearea between the dotted lines shows the 90% confidence interval estimated with the methodsuggested by Sims & Zha (1999).state in the models is given by,?c[Et(C?t+1 ? C?t)? Et(C??t+1 ? C??t)]= Et(Q?t+1 ? Q?t) + ??(0)Y? B?t+1 (4.4.1)Since we maintain the small-country assumption, the impulse response functions offoreign consumption should be negligible after a monetary shock in developing economies(C??t ? 0). This, together with the small value of calibrated interest elasticity of foreigndebt (??(0)Y? ), implies thatEt(Q?t ? Q?t+1) ? ?cEt(C?t ? C?t+1) (4.4.2)144Figure 4.7: Model- and Panel VAR-Based Impulse Responses of P , Y, E and Q to z(Without Investment)(a) Price Level (P )0 12 24 3600.511.5MonthsPercent(b) Output (Y)0 12 24 36?0.500.511.52MonthsPercent(c) Nominal Exchange Rate (E)0 12 24 3600.511.522.533.5MonthsPercent(d) Real Exchange Rate (Q)0 12 24 36?1?0.500.511.522.53MonthsPercentNote: Our calculations are based on the IMF?s International Finance Statistics. Thedotted lines with pentagrams and the dashed lines with squares indicate the model-basedimpulse response functions in the one- and multi-sector models, respectively. The solidlines show the estimated point-wise panel VAR-based impulse response functions. Thearea between the dotted lines shows the 90% confidence interval estimated with the methodsuggested by Sims & Zha (1999).5959Our models predict a strong correlation between relative consumption and the real exchange rate inconditional expectations. This does not necessarily contradict with the well-known weak unconditionalcorrelation in the data. As shown by Corsetti, Dedola & Leduc (2008), persistent productivity shocks may145From (4.4.2), we conjecture that the weak contemporaneous response of the real andnominal exchange rates in the models can be traced to a weak contemporaneous responseof consumption. Put differently, should the contemporaneous response of consumptionhave increased, the undesirable outcome of the nominal exchange rate undershooting inthe models would be avoided. To this end, it is useful to consider the resource constraintin the home country:sCC?t + sI I?t +sI?( 1?? (1? ?))u?t = Y?t (4.4.3)where sC and sI are the steady-state shares of consumption and investment in realspending in the home country, respectively. We conjecture that by increasing the contem-poraneous response of Ct for some given Yt, excluding investment in the models may resultin a more profound contemporaneous response of Qt, which may help the models to predictan overshooting of the exchange rates after monetary shocks.Figure 4.7 offers supporting evidence for our conjecture that when investment is ex-cluded from the models, Qt gives a stronger contemporaneous response. This helps themodels predict the nominal exchange rate overshoots its long-run level after the mone-tary shocks as found in the actual economies. Moreover, unlike the price dynamics in theone-sector model, the price dynamics in the multi-sector model never stay out of 90% con-fidence intervals of the impulse response functions of the aggregate variables in the actualeconomies when investment is too costly to make.Lastly, one may argue that instead of excluding investment, the one- and multi-sectormodels without a variable rate of capacity utilization (ut) would produce a higher exchangerate overshooting in the real and nominal exchange rates since the contemporaneous re-lower the correlation by causing consumption and the real exchange rate to move in opposite directions onimpact due to strong wealth effects under incomplete financial markets.146sponse of consumption would be stronger without a variable capacity utilization. However,we find excluding the variable ut has a negligible effect on the extent of overshooting. Thereason is that when capacity is fully utilized in all periods (u?t = 0), the rental rate of capitalincreases immediately when an expansionary monetary shock occurs, causing a strongercontemporaneous response of the price level and a weaker contemporaneous response ofreal spending. Consequently, when capital is assumed to be fully utilized in all periods,both u?t and Y?t fall, causing a small change in C?t. This results in the nominal and realexchange rate overshooting being limited after the monetary shock (see (4.4.2)).4.5 ConclusionIn this paper, we have studied what happens to output, the price level, the real and nominalexchange rates after a positive domestic monetary shock in developing economies under aninflation targeting regime. We have found such a shock causes a short-lived rise in output, atemporary real exchange rate depreciation, a sizable overshooting of the nominal exchangerate and an increase in the price level in the short- and long-terms in these countries.Then, we have compared these findings with the outcomes in the one- and multi-sectormodels under staggered wage setting. When adjustment costs of acquiring new capital islow, neither the former nor the latter can successfully account for the nominal exchangerate overshooting following domestic monetary shocks in the actual economies. Yet, whensuch costs are large, we have found that the multi-sector model successfully explains theaggregate dynamics following domestic monetary shocks in developing economies.147Chapter 5ConclusionThis thesis considers the implications of heterogeneity in price flexibility among sectors. Itconsists of three papers. 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(A.1.1)Each period, ?pj represents the fraction of firms not receiving a price-change signal.pjt,t?1(j?) denotes the period t?1 prices of firms in sector j which are not able to optimizein period t. Let kj be the number of firms with the price pjt?1(j?) in the period t?1. Due tothe randomization assumption in the Calvo model, there are exactly ?pjkj non-optimizingfirms in period t with price pjt,t?1(j?) = pjt?1(j?). Note also that the space of pjt?1 isequivalent to the space of pjt,t?1 due to randomization. In other words, for an arbitrarypjt?1, there exists pjt,t?1 such that these two are equal. Consequently, the following musthold:155n?1jnj?pj?0pjt,t?1(j?)1??pdj? = ?pjn?1jnj?0pjt?1(j?)1??pdj? = ?pjP 1??pjt?1Hence, (A.1.1) can be rewritten as:P 1??pjt = (1? ?pj)p?jt1??p + ?pj(Pt?1Pt?2)?p(1??p)P 1??pjt?1 (A.1.2)Log-linearizing this equation gives:P?jt = (1? ?pj)p??jt + ?pj(?ppit?1 + P?jt?1)or equivalently,p??jt ? P?jt =?pj1? ?pj(pijt ? ?ppit?1)(A.1.3)where inflation in sector j is defined as:pijt = P?jt ? P?jt?1156A.2 Sectoral InflationFor the sectoral inflation equation, I work with a more general model where the elasticityof substitution of goods in different sectors is allowed to differ from that of goods in thesame sector. Thus, while the sectoral aggregator consumption given in (2.2.3) is the same,the economy-wide consumption aggregator given in (2.2.2) is changed and given as:Ct =??J?j=1n1/?j C(??1)/?jt???/(??1)(A.2.1)Then it can be shown that y?t+s(j?) = ??(P?jt+s? P?t+s)??p(p??jt+s,t(j?)? P?jt+s)+ Y?t+s,and one can rewrite (2.2.48) as:Et?1??s=0(??pj)s[(1 + ?p?p)(p??jt(j?)? P?jt+s)+ (1 + ?p?)(P?jt+s ? P?t+s)+ (1 + ?p?p)??pt,t+s?(R?t+s + W?real,t+s +(?pY?t+s ? (1 + ?p)Z?t+s))]= 0(A.2.2)Rearranging terms yields:p??jt(j?) =(1???pj)(1+?p?p)Et?1??s=0(??pj)s[(1 + ?p?p)P?jt+s ? (1 + ?p?)(P?jt+s ? P?t+s)? (1 + ?p?p)??pt,t+s+(R?t+s + W?real,t+s +(?pY?t+s ? (1 + ?p)Z?t+s))] (A.2.3)(A.2.3) can be shown to be equal to:157p??jt(j?) = (1? ??pj)Et?1P?jt +(1???pj)(1+?p?p)?Et?1(? (1 + ?p?)(P?jt+s ? P?t+s)+ R?t + W?real,t +(?pY?t ? (1 + ?p)Z?t))+ ??pj(1???pj)(1+?p?p)?Et?1??s=0(??pj)s[((1 + ?p?p)P?jt+1+s ? (1 + ?p?)(P?jt+1+s ? P?t+1+s)? (1 + ?p?p)??pt,t+1+s)+(R?t+1+s + W?real,t+1+s +(?pY?t+1+s ? (1 + ?p)Z?t+1+s))](A.2.4)Using the fact that ??pt,t+1+s =(??pt+1,t+1+s+?ppit), (A.2.4) can alternatively be rewrittenas:p??jt(j?) = (1? ??pj)Et?1P?jt+(1???pj)(1+?p?p)Et?1(? (1 + ?p?)(P?jt+s ? P?t+s)+ R?t + W?real,t +(?pY?t ? (1 + ?p)Z?t))+ ??pj(1???pj)(1+?p?p)Et?1??s=0(??pj)s[(1 + ?p?p)P?jt+1+s ? (1 + ?p?)(P?jt+1+s ? P?t+1+s)? (1 + ?p?p)(??pt+1,t+1+s + ?ppit)(R?t+1+s + W?real,t+1+s +(?pY?t+1+s ? (1 + ?p)Z?t+1+s))](A.2.5)For each step in the following equations, I rewrite the equation which comes beforethem in an alternative form:p??jt(j?) = (1? ??pj)Et?1P?jt +(1???pj)(1+?p?p)?Et?1(? (1 + ?p?)(P?jt+s ? P?t+s)+ R?t + W?real,t +(?pY?t ? (1 + ?p)Z?t))???pj?pEt?1pit??pj(1???pj)(1+?p?p)Et?1??s=0(??pj)s ?[(1 + ?p?p)P?jt+1+s ? (1 + ?p?)(P?jt+1+s ? P?t+1+s)? (1 + ?p?p)(??pt+1,t+1+s + ?ppit)+(R?t+1+s + W?real,t+1+s +(?pY?t+1+s ? (1 + ?p)Z?t+1+s))](A.2.6)158From (A.2.3), it can be shown that the last term in (A.2.6) equals ??pjEt?1p??jt+1(j?).Hence, (A.2.6) is given by:p??jt(j?) = (1? ??pj)Et?1P?jt +(1???pj)(1+?p?p)Et?1(? (1 + ?p?)(P?jt+s ? P?t+s)+ R?t + W?real,t +(?pY?t ? (1 + ?p)Z?t))? ??pj?pEt?1pit + ??pjEt?1p??jt+1(j?)(A.2.7)p??jt(j?)? P?jt =(1???pj)(1+?p?p)Et?1(? (1 + ?p?)(P?jt+s ? P?t+s)+ R?t + W?real,t +(?pY?t ? (1 + ?p)Z?t))+??pjEt?1(p??jt+1(j?)? P?jt ? ?ppit)(A.2.8)p??jt(j?)? P?jt =(1???pj)(1+?p?p)Et?1(? (1 + ?p?)(P?jt+s ? P?t+s)+ R?t + W?real,t +(?pY?t ? (1 + ?p)Z?t))+??pjEt?1(p??jt+1(j?)? P?jt+1 + P?jt+1 ? P?jt ? ?ppit)(A.2.9)p??jt(j?)? P?jt =(1???pj)(1+?p?p)Et?1(? (1 + ?p?)(P?jt+s ? P?t+s)+ R?t + W?real,t +(?pY?t ? (1 + ?p)Z?t))+??pjEt?1(p??jt+1(j?)? P?jt+1 + pijt+1 ? ?ppit)(A.2.10)Using (A.1.3), one can show that:?pj(1??pj)(pijt ? ?ppit?1) =(1???pj)(1+?p?p)Et?1(? (1 + ?p?)(P?jt+s ? P?t+s)+ R?t + W?real,t +(?pY?t ? (1 + ?p)Z?t))+??pjEt?1(?pj(1??pj)(pijt+1 ? ?ppit) + pijt+1 ? ?ppit)(A.2.11)159It is easy to show from (A.2.11) that:pijt ? ?ppit?1 = ??pj(1 + ?p?)Et?1(P?jt ? P?t)+?pjEt?1(R?t + W?real,t +(?pY?t ? (1 + ?p)Z?t))+ ?Et?1(pijt+1 ? ?ppit) (A.2.12)where ?pj is given by:?pj =1? ?pj?pj1? ??pj1 + ?p?p(A.2.13)In the models considered above, the maintained assumption is that ?p = ?. Substitutingthis assumption into (A.2.12) gives (2.2.50), which was the equation I aimed to show.160A.3 Percentile Group InflationIn this appendix, the inflation equation for a percentile group is shown. First, it is no-table that the frequency of price changes in sectors included in the same percentile isapproximated by the median frequency in the percentile. Letting piFt denote inflation in apercentile group, I define piFt as the weighted average of inflation in the sectors includedin the percentile group (pift):piFt =?f nfpift?f nfwhere nf denotes the weight of the sector whose frequency is approximated by themedian frequency of price changes in the percentile group, respectively. Apart from theirfrequencies of price changes, sectors are identical. This, combined with the fact thatfrequencies of the sectors in the same percentile group are assumed to be the same, impliesthat all sectors within the same percentile group have the same inflation equation. Hence,it must hold thatpiFt = pift161A.4 Wages Set for an Hour of Differentiated LaborTaking the first-order condition in (2.2.54) by using (2.2.18) and (2.2.55) yields:Et?1{??s=0(?w?)s?wt,t+s(Wt+sw?t+s,t(i))?w?Ht+sWt+sw?t+s,t(i)(?w ? 1)?(Hh(ht+s(i))?w1Wt+s??t+s?sw?t+s,t(i)Wt+s)}= 0(A.4.1)(A.4.1) can easily be log-linearized since only log-deviations of the terms in the squarebrackets remain and those of all other terms outside the square brackets vanish. To seethis, note that the owner of the differentiated labor type i will set the wage by imposing amarkup of ?w over the marginal cost of working as given by (2.2.35) in the steady state.Then, it is easy to confirm the steady-state value of the term given in square bracketsis zero. Hence, the log-deviation of all terms outside the square brackets disappear. Toshow how overall wages evolve over time, it is convenient to define ??t+s as ??t+s = ?t+sPt+s?s .(A.4.1) can then be rewritten as:Et?1{??s=0(?w?)s?wt,t+s(Wt+sw?t+s,t(i))?w?Ht+sWt+sw?t+s,t(i)(?w ? 1)(Hh(ht+s(i))?w1Wt+s???t+sPt+sw?t+s,t(i)Wt+s)}= 0(A.4.2)A log-linearized version of (A.4.2) is given by:Et?1{??s=0(?w?)s(??1H h?t+s(i)????t+s ? W?real,t+s ? w??t+s,t(i) + W?t+s)}= 0 (A.4.3)Using (2.2.18), (A.4.3) can be rewritten as:Et?1{??s=0(?w?)s(??1H H?t+s ????t+s ? W?real,t+s ? (1 + ?w??1H )(w??t+s,t(i)? W?t+s))}= 0 (A.4.4)162By inserting w??t+s,t(i) = w??t (i)+ ??wt,t+s and arranging terms in (A.4.4), it is easy to showw??t (i) =1? ?w?(1 + ?w??1H )Et?1{??s=0(?w?)s(??1H H?t+s ????t+s ? W?real,t+s ? (1 + ?w??1H )(??wt,t+s ? W?t+s))}60(A.4.5)Then, (A.4.5) can be restated as:w??t (i) = (1? ?w?)W?t +1??w?(1+?w??1H )Et?1(??1H H?t ????t ? W?real,t)+ (1??w?)?w?(1+?w??1H )?Et?1{??s=0(?w?)s(??1H H?t+1+s ????t+1+s ? W?real,t+1+s ? (1 + ?w??1H )(??wt,t+1+s ? W?t+1+s))} (A.4.6)Using ??wt,t+1+s = ??wt+1,t+1+s + ?wpit, (A.4.6) can be rewritten as:w??t (i) = (1? ?w?)W?t +1??w?(1+?w??1H )Et?1(??1H H?t ????t ? W?real,t)? ?w??wpit+ ?w?1??w?(1+?w??1H )Et?1{??s=0(?w?)s(??1H H?t+1+s ????t+1+s? W?real,t+1+s ? (1 + ?w??1H )(??wt+1,t+1+s ? W?t+1+s))}(A.4.7)Hence, from (A.4.5), (A.4.7) can be restated as:w??t (i) = (1? ?w?)W?t +1? ?w?(1 + ?w??1H )Et?1(??1H H?t ????t ? W?real,t)? ?w??wpiwt + ?w?Et?1w??t+1(i) (A.4.8)60Let the percentage change in overall wages be given as piwt = logWt ? logWt?1. Then, ??wt,t+s is givenas:??wt,t+s ={?ppiwt + ?ppiwt+1 + ? ? ?+ ?ppit+s?1 if s ? 10 if s = 0163or as:w??t (i)? W?t =1? ?w?(1 + ?w??1H )Et?1(??1H H?t ????t ? W?real,t)+ ?w?Et?1(w??t+1(i)? W?t+1 + piwt+1 ? ?wpit)(A.4.9)By following the steps in Appendix A.1.1, one can easily show that(w??t (i)? W?t)=?w(1? ?w)(piwt ? ?wpit?1)Using this, it is easy to restate (A.4.9) as:piwt ? ?wpit?1 =(1??w)?w1??w?(1+?w??1H )Et?1(??1H H?t ????t ? W?real,t)+ ?Et?1(piwt+1 ? ?wpit)(A.4.10)I show in the following section that H?t = 1?(Y?t? Z?t). Further, it holds from (2.2.7) thatEt?1???t = ???1Et?1x?t. Hence, (A.4.10) can be written as:piwt ? ?wpit?1 = ?wEt?1(??1H?(Y?t ? Z?t)+ ??1x?t ? W?real,t)+ ?Et?1(piwt+1 ? ?wpit)(A.4.11)where ?w is given as:?w =(1? ?w)?w1? ?w?(1 + ?w??1H )A.4.1 Composite Labor Demand EquationI now aim to show that164H?t =1?(Y?t ? Z?t) (A.4.12)First, it is easy to see from (2.2.10) and (2.2.11) thatnjC? = nj c?j(j?) = C?j (A.4.13)This, together with (2.2.2) and (2.2.3), giveC?t =J?j=1njC?jt (A.4.14)njC?j =nj?0c?jt(j?)dj? (A.4.15)Since investment is absent in the model and supply is demand determined, the followingequalities must hold:c?jt(j?) = y?jt(j?), C?jt = Y?jt, C?t = Y?t (A.4.16)where Y?jt denotes sector j?s output. The total demand for the composite labor demand,Ht, is given byHt =J?j=1nj?0Hjt(j?)dj? (A.4.17)where the composite labor demand of the firm producing the j?th type good in sector j165is denoted by Hjt(j?). Using (2.2.14), Hjt(j?) can be written asHjt(j?) =(yjt(j?)Zt) 1?(A.4.18)Log-linearizing (A.4.18) yieldsH?jt(j?) =1?(y?jt(j?)? Z?t) (A.4.19)Furthermore, assuming all firms have access to the same technology Z?, one can showfrom (A.4.13), (A.4.17) and (A.4.18) that H?j(j?) = H? for all j and j?. Hence, the log-linearapproximation of (A.4.17) is given byH?t =J?j=1nj?0H?jt(j?)dj? (A.4.20)Lastly, using (A.4.14), (A.4.15), (A.4.16), (A.4.19) and (A.4.20), one can show thatH?t =1?(Y?t ? Z?t) (A.4.21)(A.4.21) is what I have intended to show in this section.166A.5 Model-Based Impulse Responses with the EfficientWeighting MatrixWith the assumption that both models are correctly specified, the efficient classical mini-mum distance (CMD) estimator corresponds to A?nAn = ???1.61 In Figure A.1, I illustratethe model-based impulse responses with this choice of the weighting matrix. Using theefficient weighting matrix seems to hinder the stability of the minimization algorithm asshown in the figure. Note how well the VAR- and model-based first period impulse re-sponses of the variables overlap with each other while they differ sharply in other periods.This may result from the fact that such a choice of the weighting matrix puts too muchweight on first period impulse responses and too little weight on impulse responses in otherperiods. This can be explained with the fact the VAR-based first-period impulse responsesare estimated much more precisely than those in other periods. It is true that the diagonalweighting matrix used in 2.3.1 and 2.3.2 also places more importance on the impulse re-sponses with lower variances. However, in the efficient weighting matrix case, first-periodimpulse responses are assigned more weight, not only since they have low variances, butalso since the covariances between any two first period impulse responses are much moreprecisely estimated. This leads the minimization algorithm to choose model parameters insuch a way that matching the first period impulse responses has an undue weight and allother period impulse responses have little weight.61For the definitions of An and ??, see (2.3.2).167Figure A.1: Impulse Responses to an Unanticipated 1% Fall in Rt(Efficient CMD Estimator)(I) One-Sector Model(a) Yt ? Y nt0 4 8 12 16 20?0.500.511.52QuartersPercent(b) pit0 4 8 12 16 20?0.4?0.200.20.40.60.81QuartersPercent(c) Wreal,t0 4 8 12 16 20?0.4?0.200.20.40.6QuartersPercent(d) Rt0 4 8 12 16 20?1.5?1?0.500.51QuartersPercent(II) Multi-Sector Model(e) Yt ? Y nt0 4 8 12 16 20?0.500.511.52QuartersPercent(f) pit0 4 8 12 16 20?0.4?0.200.20.40.60.81QuartersPercent(g) Wreal,t0 4 8 12 16 20?0.4?0.200.20.40.6QuartersPercent(h) Rt0 4 8 12 16 20?1.5?1?0.500.51QuartersPercentNote: The solid lines show the VAR-based impulse responses and the area between dashedlines indicates the 95% confidence interval estimated with the method suggested by Sims &Zha (1999). The lines marked with circles represent the dynamic responses of the variablesas predicted by the model.168Appendix BAppendix to Chapter 3B.1 The Bils, Klenow & Kryvtsov (2003) ModelReconsideredIn this section, I aim to explain why the empirical strategy for studying the relative priceeffects of monetary shocks in the United States in Bils, Klenow & Kryvtsov (2003) producesdifferent outcomes than those in Section 3.1.2. Before such an analysis, it is useful to reviewthe Bils, Klenow & Kryvtsov (2003) model.B.1.1 The Bils, Klenow & Kryvtsov (2003) ModelBils, Klenow & Kryvtsov (2003) investigate sectoral price responses to a monetary policyshock using the following empirical method:lnpit = ?ikmax?k=kmin?kt?k + ?i + ?it+ ?t + ?it (B.1.1)where lnpit denotes the logged price of sector i. ?i and ?t?k show the frequency of pricechanges in sector i and innovations in the monetary policy instrument, respectively. Theerror component in the panel data estimation of (B.1.1) is composed of sector- and time-specific terms. Sector-specific terms, which are included to allow each disaggregated serialto have a different intercept and a different trend, are denoted by ?i and ?i, respectively.169Time-specific terms, on the other hand, are meant to capture factors unobservable to theresearcher and are assumed to affect all prices by the same magnitude in period t and aredenoted by ?t. The maximum number of periods that a monetary policy innovation mayhave an impact on pit is denoted by kmax.I assume some delays may occur for prices in a sector when responding to monetaryshocks. kmin denotes the number of lags in sectoral price responses. In estimating (B.1.1), Imaintain there is a quarter delay in sectoral price responses to monetary shocks (kmin = 1).Differently, Bils, Klenow & Kryvtsov (2003) assume monetary shocks have a contempora-neous impact on sectoral prices (kmin = 0). However, such an assumption is at odds withthe empirical strategy for isolating monetary shocks in Bils, Klenow & Kryvtsov (2003),which requires aggregate price level to respond to monetary shocks with a lag.Lastly, it is notable that sector-specific errors (?it) in the Bils, Klenow & Kryvtsov(2003) model are assumed to follow an AR(2) process and is given by:?it = ?1?it?1 + %1?i?it?1 + ?2?it?2 + %2?i?it?2 + uit (B.1.2)The specification for sector-specific shocks in (B.1.2) implies that persistence in sector-specific shocks depends on the frequency of price changes in sectors. To explain why suchan assumption is made in the Bils, Klenow & Kryvtsov (2003) model, consider first thata sector-specific shock emerges in the fully flexible-price sector. As all prices in this sectorcan adjust instantaneously to any type of shock, it is expected to have transitory effectson this sector?s price, pit. Next, consider a sector-specific shock hits a sticky-price sectorwhere only a small fraction of firms can reset prices each period. Since it may take quitea while for firms in this sector to adjust fully to the shock, the shock is likely to havemore persistent effects on this sector?s price. Lower persistence of sector-specific shocks inflexible-price categories are reflected in the Bils, Klenow & Kryvtsov (2003) model in the170conjecture that %1 and %2 have negative signs in (B.1.2).The first step in Bils, Klenow & Kryvtsov (2003) is to obtain structural monetaryshocks (?t). To do so, I assume that the Federal Reserve uses the federal funds rate as itspolicy instrument and uses the following interest rate rule:Rt = ?0 +4?k=0?y?yn,k(yt?k?ynt?k) +4?k=0?pi,kpit?k +4?k=0?w,kwt?k +4?k=1?i,kRt?k + t (B.1.3)where Rt, yt ? ynt , pit and wt are defined in (3.1.2). After obtaining monetary shocksby using (B.1.3), the Cochrane-Orcutt procedure is used to estimate the coefficients in theBils, Klenow & Kryvtsov (2003) model.62 In this model, the relative price effects of acontractionary monetary policy shock are given as:?k(?90 ? ?10) (B.1.5)where ?90 and ?10 show the frequencies of the sectors which lie on the 90th and 10th62Since ?it in (B.1.1) is autoregressive, the OLS estimates of the coefficients in (B.1.1) are inefficient. Inaddition, OLS standard errors of those coefficients are incorrect. If the true values of the autoregressivecoefficients were known, (B.1.1) could easily be estimated by multiplying each side with the following term:1? (?1 + %1?i)L? (?2 + %2?i)L2 (B.1.4)where L is the lag operator. It needs to be emphasized that the transformation required for each sectoris different as ?i varies across sectors. It is intuitive to transform (B.1.1) this way since multiplying ?itwith (B.1.4) yields errors of the transformed model(?it) which are uncorrelated, and thus, the OLS with thetransformed model is efficient. Yet, autoregressive coefficients are unknown and need to be estimated. Inestimating these coefficients, Bils, Klenow & Kryvtsov (2003) employed the well-known Cochrane-Orkuttiterative procedure. In this procedure, parameters in (B.1.1) are first estimated with OLS. Then, estimatedOLS residuals in (B.1.1) are used to obtain the first round estimate of %1,?1, %2 and ?2 in (B.1.2). Then,both dependent and independent variables are transformed using these first round autoregressive estimatesinstead of the true autoregressive coefficients in (B.1.4). After both the dependent and independent variablesare transformed this way, the second round coefficient estimates are obtained as well as the second roundOLS residuals in (B.1.1). Using the second round OLS residuals, the second round autoregressive coefficientsin (B.1.2) are obtained and the variables are transformed using these second round autoregressive coefficientsonce again. This iteration continues until the autoregressive coefficient estimates in two consecutive roundsdiffer no more than some threshold. As a convergence criteria, I chose estimates of ?1 from two consecutiverounds with a change of less than 0.01.171percentile of price flexibility, respectively. I call the categories which lie at these percentilesof price flexibility as the flexible- and sticky-price categories, respectively. The intuitionfor (B.1.5) is as follows. Note that when the Federal Reserve introduces an unanticipated1% increase in the federal funds rate, the percentage change in pit of the flexible-pricesector (the sticky-price sector) is given by ?90?k (?10?k). Accordingly, (B.1.5) measuresthe percentage change in pit of the flexible-price sector relative to that of the sticky-pricesector in the kth period following the shock. Since the frequencies of price changes in thesticky- and flexible-price categories differ significantly, the dynamic behavior of pit in bothsectors may show substantial differences unless the estimate of ?ks are too small.B.1.2 Findings from the Bils, Klenow & Kryvtsov (2003) ModelNext, I study the relative price effects of monetary policy shocks in the Bils, Klenow &Kryvtsov (2003) model using my sample, which spans the period of 1959Q1-2013Q1. FigureB.1 shows the percentage change in pit of the flexible-price sector relative to that of thesticky-price sector following an unanticipated 1% increase in the federal funds rate in theBils, Klenow & Kryvtsov (2003) method ( ?k(?90 ? ?10) ) . In Panel (a) and Panel (b)of this figure, the dynamics of the relative price after the shock are estimated using thefrequency of non-sales price changes and the frequency of price changes including sales,respectively. Two points are noteworthy regarding Figure B.1. First, the inclusion ofprice changes during sales in measuring the frequency of price changes in sectors has onlya small effect on our results since the relative price dynamics after the shock are rathersimilar when the frequency of price changes includes or excludes price changes during sales.Second, our results are even more striking than the results in Bils, Klenow & Kryvtsov(2003). Indeed, following a contractionary monetary shock, I find the relative price staysabove its undistorted path for about three years compared to only three quarters as found172Figure B.1: The Bils, Klenow & Kryvtsov (2003) Model-Based Impulse Responsesof the Relative Price to Monetary Shocks(a) With the Frequency of Non-Sales PriceChanges0 4 8 12 16 20?0.4?0.200.20.40.60.811.21.4QuartersPercent(b) With the Frequency of Price ChangesIncluding Sales0 4 8 12 16 20?0.4?0.200.20.40.60.811.21.4QuartersPercentNote: In the figure, the solid line indicates the estimated point-wise impulse responses.The area between the dashed lines shows the two standard deviation confidence intervalsfor the estimate of ?k(?90 ? ?10) in (B.1.1). As in Bils, Klenow & Kryvtsov (2003), inestimating these confidence intervals, the uncertainty in estimating structural monetaryshocks in (B.1.3) is not taken into account.in Bils, Klenow & Kryvtsov (2003).Barth & Ramey (2001) claim a rise in inflation after a contractionary monetary shockcan be explained with the working-capital channel, which they show to be operative in theUnited States. Indeed, firms? requirement to pay input costs in advance raises factor costswhen the interest rate increases. Consequently, when there is an unanticipated rise in theinterest rate, firms raise their prices for some periods after the shock despite the downwardpressure on prices from a reduction in output following the shock. If the relative priceresponses in the flexible-price sector were positive only for a few periods, such a channel173could be invoked to explain the relative price puzzle. Yet, this channel may not explainthe positive responses of the relative price for three years as found in this paper.B.1.3 Testing a Critical Assumption in the Bils, Klenow & Kryvtsov(2003) ModelThe reliability of the findings in Figure B.1 depends on whether monetary shocks areorthogonal to sector-specific errors in period t in (B.1.1). To see this, note from (B.1.1)that the unbiasedness of coefficients requires E(?it|t) = 0. That is, sector-specific shocksshould be orthogonal to monetary shocks to have unbiased estimates. If this condition doesnot hold, the GLS estimates of the parameters in (B.1.1) will have bias and the resultsobtained with the Bils, Klenow & Kryvtsov (2003) model may be questionable. Next, I aimto test this hypothesis. Note, when obtaining monetary shocks, Bils, Klenow & Kryvtsov(2003) assume the Federal Reserve only responds to inflation in the general price level anddoes not take into account movements in any sectoral price. If this really holds, then, thesector specific shocks will be orthogonal to monetary shocks and the coefficients in the Bils,Klenow & Kryvtsov (2003) model can be estimated unbiasedly. One way to test whetherthe Federal Reserve responds to sectoral prices, apart from the general price level, is toincorporate a single sectoral price index in its policy reaction function in (B.1.3) and testwhether the coefficients pertinent to the Federal Reserve?s response to sectoral prices arejointly zero. The following regression is considered for this test:Rt = ?0+4?k=0?y?yn,k(yt?k?ynt?k)+4?k=0?pi,kpit?k+4?k=0?lnpi,klnpit?k+4?k=0?w,kwt?k+4?k=1?i,kRt?k+?t(B.1.6)Estimating (B.1.6) requires the assumption that the Federal Reserve observes sectoral174prices and may respond to them if this is desired. Also, it requires sectoral prices to respondto monetary shocks with at least a quarter lag. The only difference between (B.1.3) and(B.1.6) is the term?4k=0 ?lnpi,klnpit?k. The structural monetary shocks in (B.1.3) can beassociated with those in (B.1.6) in the following way:t = ?t +4?k=0?lnpi,klnpit?k (B.1.7)The maintained assumption in the Bils, Klenow & Kryvtsov (2003) model is that?lnpi,0 = ?lnpi,1 = ? ? ? = ?lnpi,4 = 0. If this assumption does not hold, E(t?it) 6= 0and the GLS estimators in the Bils, Klenow & Kryvtsov (2003) model will be inconsistent.One way to interpret the rejection of the null is that the response of the Federal Reserve toprice movements in these sectors is not confined to their marginal effects on the aggregateprice level. Apart from this, the Federal Reserve gives a statistically different response tosectoral shocks in these sectors. By using (B.1.6), I perform an F-test for the null hypoth-esis for each sector. I find among 125 sectors, the null hypothesis is rejected for 19 sectors.Table B.1 lists the sectors for which the null is rejected. For the remaining sectors, theFederal Reserve?s response is confined to its response to the change in the general pricelevel inflation caused by sector-specific price shocks in these sectors.Next, I study the association between the likelihood that the Federal Reserve providesa significant response to sectoral price shocks and the frequency of price changes. For thispurpose, a scatter plot of F-values and the frequency of non-sales price changes and ascatter plot of F-values and the frequency of price changes including sales are displayedwith circles in Panel (a) and Panel (b) of Figure B.2, respectively. The 90% critical value foran F (5, T ? 25) random variable, where T indicates sample size, is graphically representedby the thick solid line. If an F-value is greater than the critical value, the null hypothesis,175Table B.1: Sectors with a Significant Response from the Federal ReserveTires Dental ServicesWatches Nursing homesPoultry Other recreational servicesFish and Seafood Musical instrumentsVegetables (fresh) Commercial and vocational schoolsChildren?s and infants? clothing Professional and other servicesLubricants and fluid Clothing repair, rental, and alterationsMiscellaneous household products Carpets and other floor coveringsNewspapers and periodicals Telecommunication servicesElectric appliances for personal carethat the Federal Reserve is only concerned with the general price level inflation and doesnot respond to sectoral prices, should be rejected.In Figure B.2, I also show the fitted line for the following regression:Fi = a0 + a1?i (B.1.8)A positive (negative) slope of the fitted line suggests the higher (lower) the frequencyof price changes in a sector, the more (less) probable is the Federal Reserve to provide asignificant response to sectoral price shocks. I estimate (B.1.8) with both the frequency ofprice changes including sales (?salesi ) and the frequency of non-sales price changes (?regulari )and report the findings in (B.1.9).Fi = 1.19 + 0.18?regulari Fi = 1.44? 0.45?salesi (B.1.9)These findings indicate that while the likelihood of the Federal Reserve to provide asignificant response to sectoral price shocks is positively associated with the frequency ofnon-sales price changes, it is negatively associated with the frequency of price changes176Figure B.2: Testing for the Significance of the Federal Reserve?s Response forSectoral Prices(a) With the Frequency of Non-Sales PriceChanges0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 101234567FrequencyFtest  F?Test Values90% Rejection ValueFitted Line(b) With the Frequency of Price ChangesIncluding Sales0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 101234567FrequencyFtest  F?Test Values90% Rejection ValueFitted LineNote: In the figure, the circles show a scatter plot of the F-value for the null hypothesisand the frequency of price changes in sectors. The thick solid line indicates the 90% criticalvalue for the F-test. The dotted lines marked with asterisks show the fitted line for theregression in (B.1.8).including sales. However, both the positive and negative associations are weak. To seethis point, note that the fitted lines in both cases are flat and they always remain belowthe critical value within the admissible region of the frequency of price changes in sectors.Hence, the frequency of price changes in a sector does not seem to be an important factorin the decision of the Federal Reserve once the effect of these shocks on the general pricelevel inflation is controlled.To sum up, the findings in this section indicate that structural monetary shocks that areneeded to estimate the Bils, Klenow & Kryvtsov (2003) model (t in (B.1.1)) are correlatedwith sector-specific price shocks (?it in (B.1.1)). Since this assumption is critical in the177Bils, Klenow & Kryvtsov (2003) model and is shown to be violated for a non-negligiblenumber of sectors in our sample, it can be argued that the results in the Bils, Klenow &Kryvtsov (2003) model are questionable.178B.2 Estimation of Confidence Intervals for Figure 3.3Using a Block-Bootstrap MethodThe ?block-of-blocks? bootstrap method of Politis & Romano (1992) is used to estimatethe confidence interval for the correlation between the frequency of price changes in sectorsand sectoral price responses to a 1% increase in the federal funds rate. The following stepsare followed to construct the confidence intervals.1. Let Zt and Zt be defined as:Zt =???? yt ? ynt , pit, wt, Rt, pi1t, pi2t, . . . piit, . . . pi125t????Zt =???? Zt Zt?1 Zt?2 Zt?3 Zt?4????(B.2.1)where piit in Zt represents the annualized percentage change in the price of the sectori.2. Next, T ? b + 1 overlapping blocks are formed where T is the sample size and b isthe fixed length of blocks. The first contains observations | Z1 Z2 . . . Zb |?. Thesecond contains observations | Z2 Z3 . . . Zb+1 |?. The last contains observations| ZT?b+1 ZT?b+2 . . . ZT |?.3. For the block length, the following values are considered b ? {6, 9, 12}. Since theresults are similar, only the results for b = 12 are displayed in Figure 3.3.4. A random sample of size T is constructed by resampling from these blocks withreplacement. Since T/b is not an integer, the last block has been truncated.5. Next, for each random sample, the correlation between the frequency of price changes179and sectoral price responses to a 1% increase in the federal funds rate is estimatedby the method discussed in Section 3.1.2.6. These steps are repeated 500 times. The confidence intervals displayed in Figure 3.3give the area between the 2.5th and 97.5th percentiles of these randomly generatedcorrelations.180Appendix CAppendix to Chapter 4C.1 The Empirical Strategy in Clarida & Gali (1994)The following empirical model for quarterly data is used in Clarida & Gali (1994):Xi,t = Bi,0 +pmax?p=1Bi,pXi,t?p + ui,t (C.1.1)where the vector Xi,t is defined in (4.1.2). (C.1.1) in the companion form is given as:????????????Xi,tXi,t?1...Xi,t?p+2Xi,t?p+1????????????=????????????Bi,000...0????????????+????????????Bi,1 Bi,2 . . . Bi,p?1 Bi,pI 0 . . . 0 0... . . . ... ... ...... ... . . . ... ...0 0 . . . I 0????????????????????????Xi,t?1Xi,t?2...Xi,t?p+1Xi,t?p????????????+????????????ui,t00...0????????????(C.1.2)Or, rewrite (C.1.2) in a compact form as:Zi,t = Ai0 + Ai1Zi,t?1 + Ui,t (C.1.3)where Zi,t, Ai0, Ai1 and Uit are defined in the following way,181Zi,t =????????????Xi,tXi,t?1Xi,t?2...Xi,t?p????????????Ai0 =????????????Bi000...0????????????Ai1 =?????????????Bi1 Bi2 . . . Bipmax?1 BipmaxI 0 . . . 0 0.... . ................. . .......0 0 . . . I 0?????????????Ui,t =????????????ui,t00...0????????????=????????????Ci0i,t00...0????????????(C.1.4)In the last part, we use ui,t = Ci0i,t for writing Ui,t (For definitions of Ci0 and i,t, see(4.1.3)).Suppose that i,t = I. This corresponds to the study of dynamics of the variables inXit to an unanticipated unit change in each of the three structural shocks, separately. Inother words, each column of Ci0 shows contemporaneous impulse responses of the variableto a one unit increase in one of the three structural shocks.Let the impulse responses of Zi in the jth period be defined as:EtZi,t+j ? Et?1Zi,t+j (C.1.5)Now, our goal is to calculate the cumulative responses over j periods. First, note thatdue to the AR(1) structure in (C.1.3), it is easy to show that the jth period impulseresponse of Zi is given by:EtZi,t+j ? Et?1Zi,t+j = Aji1 Ui,t (C.1.6)Denoting the cumulative impulse response of Zi over j with (Zcum.i,t+j), one can write:182Zcum.i,t+j =j?m=0(EtZi,t+m ? Et?1Zi,t+m) = (I + Ai1 + A2i1 + ? ? ?+ Aji1)Ui,tWhen j ??, it is easy to verify that the cumulative impulse response is given by,Zcum.i,t+? = (I ?Ai,1)?1Ui,t ? (I ?Ai,1)Zcum.i,t+? = Ui,t (C.1.7)Using (C.1.4) and i,t = I, one can write (C.1.7) as:(I ?Bi1)Xcum.i,t+? ?Bi2Xcum.i,t?1+? ? ? ? ? ?BipmaxXcum.i,t?pmax+1+? = Ci0 (C.1.8)Note that Xcum.i,t+? = Xcum.i,t?1+? = ? ? ? = Xcum.i,t?pmax+1+? since for any p > 0,Xcum.i,t?p+? =?1?m=?p(EtXi,t+m ? Et?1Xi,t+m)? ?? ?0+??m=0(EtXi,t+m ? Et?1Xi,t+m) = Xcum.i,t+?(C.1.9)Thus, one can rewrite (C.1.8) as follows:(I ??pmaxp=1 Bip)Xcum.i,t+? = Ci0Xcum.i,t+? =(I ??pmaxp=1 Bip)?1Ci0(C.1.10)Note from (4.1.2) that the variables in Xi,t are in first differences. Then, the cumulativeimpulse responses of these differenced variables over j periods is the jth period level impulseresponses. To see this, consider a differenced variable y, one can show the equivalence ofthe cumulative impulse responses of ?y over j periods and the level impulse response of yin jth period in the following way:183?ycum.t+? =(Etyt ? Etyt?1)? (Et?1yt ? Et?1yt?1)+ (Etyt+1 ? Etyt)? (Et?1yt+1 ? Et?1yt)+ (Etyt+2 ? Etyt+1)? (Et?1yt+2 ? Et?1yt+1)...+ (Etyt+? ? Etyt?1+?)? (Et?1yt+? ? Et?1yt?1+?)= Etyt+? ? Et?1yt+?(C.1.11)Hence, (C.1.11) implies that the cumulative response of ?yt over the long-run is thelevel impulse response of y to the shock in the long-run. Denoting the long-run levelimpulse response matrix of the variables in Xi,t with Di, from (C.1.10) and (C.1.11), itfollows thatDi =(I ?pmax?p=1Bip)?1Ci0 (C.1.12)Then, it is easy to show thatDiD?i =(I ?pmax?p=1Bip)?1?i(I ?pmax?p=1Bip)??1(C.1.13)Clarida & Gali (1994) assume that Di satisfies the exclusion and sign restrictions in(4.1.5). With such identfiying assumptions, this matrix can be uniquely identified as theCholesky decomposition of (C.1.13). Further, it can be shown from (C.1.12) thatCi0 =(I ?pmax?p=1Bip)Di (C.1.14)Moreover, if Ci0 is assumed to be invertible, structural shocks can be recovered as184i,t = C?1i0 ui,t (C.1.15)185C.2 Aggregate Dynamics after Monetary ShocksC.2.1 Aggregate Dynamics in Empirical Model I after MonetaryShocksHere, we report our findings on aggregate dynamics in Empirical Model I after monetaryshocks.Aggregate dynamic responses are displayed in Figure C.1 over five years at the quarterlyand monthly frequencies. It is evident from this figure that a higher monetary shock indeveloping economies, relative to one in the United States, is associated with a short-livedincrease in the level of output in the former relative to that of the United States. It quicklyfalls again to the level of the undistorted path. Similarly, the real exchange rate exhibits atemporary upward movement after the shock, indicating a temporary depreciation in thereal exchange rate against developing economies. At the quarterly (monthly) frequency,our results for Empirical Model I suggest that the real exchange rate stays depreciatedrelative to its undistorted path for about six quarters (12 months) after the shock. It canalso be seen that the real exchange rate exhibits hump-shaped dynamics after the shock.Lastly, a positive monetary shock causes the price level in developing economies to riserelative to the price level in the United States on impact.C.2.2 Aggregate Dynamics in Empirical Model II after MonetaryShocks in the United StatesWe have discussed aggregate dynamics after an expansionary domestic monetary shock indeveloping countries in Empirical Model II in Section 4.1.2. This section, on the otherhand, discusses our findings on aggregate dynamics after an expansionary monetary shockin the United States in Empirical Model II. The results are presented in Figure C.2.186Figure C.1: Impulse Responses to Monetary Shocks in Empirical Model IQuarterly(a) Yi,t ? Y?t0 4 8 12?0.500.51QuartersPercent(b) Qi,t0 4 8 12?1012QuartersPercent(c) Pi,t ? P ?t0 4 8 1200.511.5QuartersPercentMonthly(d) Yi,t ? Y?t0 12 24 36?1012MonthsPercent(e) Qi,t0 12 24 36?10123MonthsPercent(f) Pi,t ? P ?t0 12 24 3600.511.5MonthsPercentNote: Our calculations are based on the IMF?s International Finance Statistics. The solidlines indicate the estimated point-wise impulse responses. The area between the dashedlines shows the 90% confidence interval estimated using the Bayesian method suggested bySims & Zha (1999).Following a monetary shock in the United States:? output in both the United States and developing economies stays above its undis-torted level for about a year;187Figure C.2: Impulse Responses to Monetary Shocks in the United States(Empirical Model II)Quarterly(a) Y?t0 4 8 12?0.500.511.5QuartersPercent(b) Yi,t0 4 8 12?10123QuartersPercent(c) Qi,t0 4 8 12?10?505QuartersPercent(d) P ?t0 4 8 120123QuartersPercent(e) Pi,t0 4 8 12?0.500.511.5QuartersPercentMonthly(f) Y?t0 12 24 36?10123MonthsPercent(g) Yi,t0 12 24 36?20246MonthsPercent(h) Qi,t0 12 24 36?10?505MonthsPercent(i) P ?t0 12 24 360123MonthsPercent(j) Pi,t0 12 24 36?1012MonthsPercentNote: Our calculations are based on the IMF?s International Finance Statistics. The solidlines indicate the estimated point-wise impulse responses. The area between the dashedlines shows the 90% confidence interval estimated using the Bayesian method suggested bySims & Zha (1999).? the real exchange rate appreciates on impact, and compared to the undistorted path,it stays appreciated for about 9 months; and,? the price level in both the United States and developing economies contemporane-ously rises.188C.3 Calibration of Models? ParametersTable C.1: Calibration and EstimationParameters Description Values Source?p Price elasticity of demand for varieties within the same sector 11 Bresnahan (1981)?w Wage elasticity of labor demand 4 Huang & Liu (2002)?c Inverse of elasticity of intertemporal substitution 5 Hall (1988)?n Inverse of Frisch-elasticity of labor supply 1 Carvalho & Nechio (2011)?a Inverse of the elasticity of capacity utilization with respect tothe rental rate of capital0.01 Christiano et al. (2005)?? =??? I?K??? The elasticity of the adjustment cost technology for investmentwith respect to ItKt-0.75 Devereux & Hnatkovska (2011)??Y? Elasticity of interest rate to net foreign assets -0.01 Devereux & Smith (2005)? Elasticity of substitution between the home and foreign goods 1.5 Carvalho & Nechio (2011)? Elasticity of substitution between different sector goods 1 Carvalho & Nechio (2011)? Labor share in GDP 0.66 Christiano et al. (2005)? Discount factor 1.03?112 Christiano et al. (2005)?k and ??k Price stickiness in sectors See text Carvalho & Nechio (2011)fk Expenditure share of sectors See text Carvalho & Nechio (2011)? Monthly rate of depreciation on capital 0.008 Christiano & Eichenbaum (1992)?z The persistence in nominal spending growth shocks 0.32 See text? Relative size of the foreign country 1000 See textsc % Share of final consumption expenditure in GDP 66 See textsi % Share of investment in GDP 20 See text?? Share of home exports invoiced in the home currency 0.05 See text???? Share of home imports priced in the foreign currency 0.95 See text? Share of home country imports in GDP 0.35 See text189C.3.1 The Weak Link between the Level of Inflation and the Frequencyof Price ChangesA seemingly odd assumption in our dynamic stochastic sticky-price small-open economymodels is that the frequency of price changes is independent of inflation. Indeed, regardlessof changes in inflation over the period in the economy, it is assumed that a constant fractionof firms change their prices in each period. While it is odd to make this assumption forunstable inflationary periods, there is some empirical evidence in favor of the constantfrequency of price changes in low and moderate inflation climates. For example, Nakamura& Steinsson (2008a) look at the relationship between the median frequency of price changesand inflation in the United States. During the low and stable inflation period of 1998-2005, even though their results indicate a positive association between inflation and thefrequency of price changes, the estimated coefficient is not statistically different from zero.Similarly, Gagnon (2009) studies the association of the frequency of price changes andinflation in Mexico for both high and low inflation periods. He finds during moderateinflation periods, between 5-15% annual change in the price level, the frequency of pricechanges is largely dissociated with inflation. Indeed, while a rise in inflation in these stableperiods results in a higher likelihood of observing price increases, it leads to a decline inthe probability of observing price decreases, leaving the frequency of price changes largelyunaffected with inflation in Mexico during low inflation periods. Higher inflation in theseperiods is accounted for through increases in the magnitude of average price adjustmentresulting largely from the change in the distribution of price changes in favor of priceincreases in these periods. However, when inflation rises to high levels, as one wouldexpect, Gagnon (2009) finds the frequency of price changes moves closely with inflation.Hence, whether constant frequency of price changes is a reasonable assumption or notdepends largely on the level of inflation which developing countries experienced in the190Figure C.3: Median Inflation Rates in Developing Economies(1999M1-2012M9)1999 2003 2007 20110246810PercentDateNote: Our calculations are based on the IMF?s International Finance Statistics.period. To evaluate this, we study the evolution of the median inflation rate in developingeconomies over our sample period in Figure C.3. The median inflation rate denotes themedian of the twelve-month percentage changes in the CPI of the countries in each period.It is noteworthy that inflation has been stable in the inflation targeting countries. Thisresult, together with the finding in Gagnon (2009) that the frequency of price changes isindependent of inflation over stable inflation periods, offers supporting evidence for ourassumption that the frequency of price changes is constant in developing economies for theperiod.191Figure C.4: Consumer Prices Inflation and the Turkish Lira Share in ExternalTrade in Turkey(a) TL Shares in External Trade1996 2000 2004 2008 2012012345678910PercentDate  TL Share in ExportsTL Share in Imports(b) Consumer Prices Inflation1996 2000 2004 2008 201201020304050607080PercentDateNote: Our calculations are based on the Turkish Statistical Institute data. In Panel A, thedotted lines with circles and the dashed lines with multiplication signs indicate the shareof TL-denominated exports in total Turkish exports and the share of TL-denominatedimports in total Turkish imports, respectively.C.3.2 Asymmetry in Currency Invoicing in International Tradebetween Developing and Advanced EconomiesIt is a well-known fact that there is an asymmetry between developing and advancedeconomies in regards to the currency in which exports and imports are denominated. In-deed, while exports and imports are largely denominated in home currencies in advancedeconomies, they are largely denominated in foreign currencies in developing economies.For example, in their study of pricing decision of the exports and imports in the UnitedStates, Gopinath & Rigobon (2008) report that 97% of exports and 90% of imports arepriced in the United States dollar. To exemplify the pricing practices of exporters andimporters in developing economies, we look at exports and imports by currency in Turkey.192In Figure C.4, we illustrate the share of exports (imports) priced in the Turkish Lira(TL)in total exports (imports) as well as the inflation in consumer prices between 1996 and2012. Inflation is measured as the percentage change in CPI over the last twelve months.It is notable that the remarkable success in bringing down inflation has produced only amodest rise in the shares of TL denominated exports and imports over the recent years.Indeed, the shares of TL-denominated exports and imports have stayed at very low levelsbelow 5% during this period. Our conjecture is that this finding holds generally for alldeveloping economies and currency invoicing in international trade happens largely withthe foreign currencies in this group.193C.4 The One- and Multi-Sector Models? Dynamics with aTaylor-Type RuleIn this section, we analyze aggregate dynamics in the one- and multi-sector models withoutinvestment by considering a Taylor-type interest rate rule in the home and foreign countriesinstead of considering exogenous nominal spending growth. In doing so, we assume that inaddition to the international foreign bond (Bt+1), there is a domestic bond (Dt+1) whichis traded only domestically, supplied in zero net supply and pays a gross nominal interestof Rt. The interest rates in the home country (Rt) and in the foreign country (RB?F,t) areset according to the following rules:R?t = ?pi ? pit + ?y ? Y?t + rtwhere rt = ?rrt?1 + ?t and ?t ? N(0, ?2?)R?BF,t = 0.79? R?BF,t?1 + (1-0.79)? 2.15? pi?t + (1-0.79)? 0.93? Y??t + r?twhere r?t ? N(0, ?2r? )(C.4.1)The coefficients for the foreign interest-rate rule reflect the estimates of the Taylor rulecoefficients in Clarida et al. (1999) for the Volcker-Greenspan periods. The coefficients inthe home interest-rate rule, on the other hand, have to be estimated since we do not havethe estimates of the reaction function of the monetary authorities under inflation targetingin developing economies. Two cases are considered when estimating the parameters. Thefirst is that ?t is a white noise (?r = 0). The second is that the shock to the home interestrate can be persistent (?r > 0). In the first case, the estimated vector of parameters (P)consists of P =[?pi ?y]. In the second case, it includes P =[?pi ?y ?r]. Let f(P)denote the impulse response functions of the price level, output, the real exchange rate andthe nominal exchange rate in developing economies for some P between the 0th and 12th194months. We estimate P as the classical minimum distance estimator and denote it withP?(A?n):P?(A?n) = arg minP(h?n ? f(P))?A??nA?n(h?n ? f(P)) (C.4.2)where A?n is the weighting matrix used. h?n shows the impulse response functions of theprice level, output, the nominal and real exchange rates in the actual economies betweenthe 0th and 12th months. Lastly, n stands for the sample size of the data used to estimatethe VAR-based impulse response functions. Since using different weighting matrices wouldyield different estimators, P? is written as a function of A?n. As a weighting matrix, wechoose the widely-used diagonal matrix whose diagonal elements are given as the inverseof standard deviations of empirical impulse responses.(See, for example, Christiano et al.(2005) and Giannoni & Woodford (2003)). This weighting matrix ensures more preciselyestimated impulse response functions are given more importance than the less preciselyestimated ones.63Table C.2 shows the estimated parameters of the Taylor rule specified in C.4.1. Allowingpersistence in the shocks to the interest rate in the home country significantly improvesboth the one- and multi-sector models? performance as it leads to a sharp fall in theweighted distance between the model- and VAR-based impulse response functions (SeeObj. Func. in the table). Figure C.5 visualizes this. In Panel A of this figure, the impulseresponse functions of the aggregate variables to an expansionary white-noise shock to thehome interest-rate rule in (C.4.1) are illustrated. Both the one- and multi-sector modelsare incapable of explaining the aggregate dynamics when shocks to the interest rate in thehome country are transitory.63To do the estimation, the lower and upper bounds for the parameters have to be entered in the com-puter program. For the parameters[?pi ?y ?r], we set the lower and upper bounds as [1.00,2.14],[0,2.00],[0,0.99], respectively.195Table C.2: Estimated Parameters of the Taylor RuleTransitory Shocks Persistent ShocksOne-Sector Multi-Sector One-Sector Multi-Sector?pi 1.00? 1.00? ?pi 1.12 1.64(0.004) (0.002)?y 0.24 2.00? ?y 0.15 1.52(0.14) (0.035) (0.990)?r 0 0 ?r 0.61 0.79(0.006) (0.004)Obj. Func. 619.36 1986.26 Obj. Func. 78.80 65.41Note: The numbers in parentheses denote estimated model-based standard errors. The numbers with anasterisk indicate that standard errors are not reported since the estimates of the parameters are close toeither its lower bound or its upper bound as discussed in Footnote 63. Obj. Func. indicates the value ofthe objective specified in (C.4.2).Next, we consider that the shocks to the interest rate in the home country are persistent.Panel B of Figure C.5 shows the model- and VAR-based impulse response functions witha persistent interest-rate rule. It is clear from this figure that with such a high persistencein the shocks, the dynamics of the price level, output and the nominal and real exchangerates after the monetary shock in both the one- and multi-sector models align quite wellwith those found in the data. While the model-based impulse response functions in boththe former and the latter stay within the 90% confidence intervals for the panel VAR-basedimpulse response functions, it is evident that the latter is more successful than the formerin explaining the movements of output, the real and nominal exchange rates in the actualeconomies following the monetary shock.196Figure C.5: One- and Multi-Sector Models with a Taylor-Type RulePanel A: Transitory Shocks (?r = 0)(a) P0 12 24 360123MonthsPercent(b) Y0 12 24 36?2024MonthsPercent(c) E0 12 24 3602468MonthsPercent(d) Q0 12 24 36?20246MonthsPercentPanel B: Persistent Shocks (?r > 0)(e) P0 12 24 3600.511.5MonthsPercent(f) Y0 12 24 36?1012MonthsPercent(g) E0 12 24 3601234MonthsPercent(h) Q0 12 24 36?10123MonthsPercentNote: Our calculations are based on the IMF?s International Finance Statistics. The dotted lines with pentagramsand the dashed lines with squares indicate the model-based impulse response functions in the one- and multi-sectormodels, respectively. The solid lines show the estimated point-wise panel VAR-based impulse response functions.The area between the dotted lines shows the 90% confidence interval estimated with the method suggested by Sims& Zha (1999).197

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