Essays in International Portfolio Choice and Monetary Policy by Changhua Yu B.A., Renmin University of China, 2003 M.A., Peking University, 2006 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Economics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) September 2012 c© Changhua Yu 2012 Abstract Empirical evidence shows that equity home bias is a prevailing fact for most countries, but standard international monetary business cycle models with nominal bonds hardly generate equity home bias for plausible prefer- ence parameter values. By incorporating inflation-indexed bonds, I show in chapter 2 that international monetary business cycle models can explain home bias in equities. Inflation-indexed bonds can hedge real exchange rate risk and domestic equities serve as a hedge against domestic labor income risk conditional on real exchange rates. Moreover, this chapter accounts for counter-cyclical movements of net foreign asset positions. These results are robust in environments either with complete markets or with incomplete markets. How does international financial market integration alter international risk sharing? Chapter 3 develops a tractable center-periphery model with portfolio choice to investigate this issue. I compare three stages of financial integration. The first stage is financial autarky. The second stage is the central country becomes financially integrated with peripheral countries, but there is no financial integration between peripheral countries. The third stage is all financial markets are integrated into each other. From finan- cial autarky to partial financial integration, volatility of consumption in all countries drops. From partial financial integration to full financial integra- tion, volatility of consumption in peripheral countries decreases; however, consumption volatility rises in the central country when the central country is relatively large. When peripheral countries are large, the degree of in- ternational risk-sharing for all countries increases in the process of financial integration. What’s the optimal monetary policy in an economy with financial fric- tions? Chapter 4 investigates optimal monetary policies in a dynamic stochas- tic general equilibrium model with sticky prices, sticky wages and credit- market imperfections. Credit frictions distort allocations and prices, and generate large volatility of aggregate variables via the financial accelera- tor. Policy-makers can take advantage of a debt deflation channel to push down volatility of endogenous variables through the financial decelerator. ii Abstract In the optimized linear interest rate rule, interest rate decreases in asset prices but the response is quantitatively small. This optimized rule exhibits high persistence. Within a class of simple linear interest rate rules, a strict inflation-targeting rule has a larger welfare loss. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . x Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Equity home bias in a monetary model . . . . . . . . . . . . 1 1.2 International financial markets and international risk sharing 8 1.3 Credit friction, asset price and optimal monetary policy . . . 16 2 Equity Home Bias in a Monetary Model. . . . . . . . . . . . 20 2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.1.1 Consumption composite . . . . . . . . . . . . . . . . . 21 2.1.2 Households . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.3 Wholesale firms . . . . . . . . . . . . . . . . . . . . . 26 2.1.4 Intermediate producers . . . . . . . . . . . . . . . . . 29 2.1.5 Government . . . . . . . . . . . . . . . . . . . . . . . 30 2.1.6 Market clearings and competitive equilibrium . . . . 31 2.2 Steady state equity portfolios . . . . . . . . . . . . . . . . . . 33 2.2.1 Steady states and log-linearizations . . . . . . . . . . 33 2.2.2 Empirics on international asset markets . . . . . . . . 39 2.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . 43 2.4.1 A model with nominal consols . . . . . . . . . . . . . 45 2.4.2 The benchmark model with inflation-indexed bonds . 46 iv Table of Contents 2.4.3 Alternative preference specification . . . . . . . . . . 54 2.4.4 Nominal rigidities in a model with nominal bonds . . 55 2.4.5 Home bias in consumption in a model with nominal bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.5 Incomplete asset markets . . . . . . . . . . . . . . . . . . . . 63 2.5.1 Investment efficiency shocks . . . . . . . . . . . . . . 64 2.5.2 Government expenditure shocks . . . . . . . . . . . . 66 2.5.3 Preference shocks . . . . . . . . . . . . . . . . . . . . 69 2.5.4 All shocks . . . . . . . . . . . . . . . . . . . . . . . . 70 2.6 International portfolio dynamics . . . . . . . . . . . . . . . . 72 2.6.1 Numerical solution method . . . . . . . . . . . . . . . 72 2.6.2 Measuring international portfolios . . . . . . . . . . . 72 2.6.3 Data moments v.s. model moments . . . . . . . . . . 78 3 International Financial Markets and International Risk Shar- ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.1 A one-good economy . . . . . . . . . . . . . . . . . . . . . . 81 3.1.1 Households . . . . . . . . . . . . . . . . . . . . . . . . 82 3.1.2 International financial markets . . . . . . . . . . . . . 83 3.1.3 Competitive equilibrium . . . . . . . . . . . . . . . . 86 3.1.4 Solving the model . . . . . . . . . . . . . . . . . . . . 86 3.2 A model with multiple tradable goods . . . . . . . . . . . . . 97 3.2.1 Consumption composite . . . . . . . . . . . . . . . . . 98 3.2.2 Households . . . . . . . . . . . . . . . . . . . . . . . . 99 3.2.3 Financial markets . . . . . . . . . . . . . . . . . . . . 100 3.2.4 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.2.5 Competitive equilibrium . . . . . . . . . . . . . . . . 102 3.2.6 Steady state . . . . . . . . . . . . . . . . . . . . . . . 102 3.2.7 Solving portfolios and the dynamic system . . . . . . 103 3.2.8 Impulse-responses . . . . . . . . . . . . . . . . . . . . 109 3.2.9 Calibration and numerical results . . . . . . . . . . . 110 4 Credit Friction, Asset Price and Optimal Monetary Policy 116 4.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.1.1 Households . . . . . . . . . . . . . . . . . . . . . . . . 116 4.1.2 Entrepreneurs . . . . . . . . . . . . . . . . . . . . . . 121 4.1.3 Retailers . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.1.4 Government . . . . . . . . . . . . . . . . . . . . . . . 126 4.1.5 Aggregation and market clearing conditions . . . . . 127 4.2 Characterizing the equilibrium . . . . . . . . . . . . . . . . . 128 v Table of Contents 4.2.1 Competitive equilibrium . . . . . . . . . . . . . . . . 128 4.2.2 The Ramsey equilibrium . . . . . . . . . . . . . . . . 129 4.3 Specification and calibration . . . . . . . . . . . . . . . . . . 129 4.4 Ramsey steady state and dynamics . . . . . . . . . . . . . . 132 4.4.1 Ramsey steady state . . . . . . . . . . . . . . . . . . 132 4.4.2 Ramsey dynamics . . . . . . . . . . . . . . . . . . . . 133 4.5 Ramsey and optimized impulse responses . . . . . . . . . . . 137 4.6 Welfare evaluation: optimized v.s. linear interest rate rules . 140 4.6.1 Comparing simple interest rate rules . . . . . . . . . 140 4.6.2 Welfare evaluation . . . . . . . . . . . . . . . . . . . . 143 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Appendices A Appendix to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . 160 A.1 Quarterly data on asset returns . . . . . . . . . . . . . . . . 160 A.2 Annual data on national accounts and net foreign assets . . . 160 A.3 Estimating returns on human capital . . . . . . . . . . . . . 161 A.4 Alternative currency pricing: pricing-to-market . . . . . . . . 164 A.4.1 Intermediate producers . . . . . . . . . . . . . . . . . 165 A.4.2 Market clearings and the competitive equilibrium . . 166 A.4.3 Steady state portfolios . . . . . . . . . . . . . . . . . 166 A.5 Foreign currency forward contract . . . . . . . . . . . . . . . 167 B Appendix to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . 169 B.1 Log-linearization . . . . . . . . . . . . . . . . . . . . . . . . . 169 B.1.1 Budget constraints . . . . . . . . . . . . . . . . . . . 169 B.1.2 First-order approximation to Euler equations . . . . . 170 B.1.3 Solving the dynamic system . . . . . . . . . . . . . . 171 B.2 Impulse responses . . . . . . . . . . . . . . . . . . . . . . . . 174 B.3 Financial autarky . . . . . . . . . . . . . . . . . . . . . . . . 174 C Appendix to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . 176 C.1 A complete set of equilibrium conditions . . . . . . . . . . . 176 C.2 The deterministic steady state . . . . . . . . . . . . . . . . . 178 vi List of Tables 1.1 Home bias in equity portfolios . . . . . . . . . . . . . . . . . . 5 1.2 Correlation of growth rate of real GDP per capita (Latin America) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Correlation of growth rate of real GDP per capita (Asia) . . . 13 2.1 Relative covariances of relative returns conditional on RER: Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2 Relative covariances of relative returns conditional on RER: Human capital . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3 Parameter values . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.4 Optimal home equities held by home households (nominal consols) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5 Optimal bond assets-GDP ratios (nominal consols) . . . . . . 44 2.6 Covariance ratios for hedging human capital income risk (nom- inal consols) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.7 Covariance ratios for hedging real exchange rate risk (nominal consols) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.8 Types of bonds . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.9 Optimal home equities held by home residents (inflation-indexed bonds) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.10 Covariance ratios for hedging real exchange rate risk (inflation- indexed bonds) . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.11 Data and model moments for asset returns . . . . . . . . . . 53 2.12 Optimal home equity shares of home equities: nominal con- sols and various nominal rigidities . . . . . . . . . . . . . . . 58 2.13 Optimal home equity shares held by home households: nom- inal consols and no home bias in consumption . . . . . . . . . 59 2.14 Optimal home equity shares held by home households: nom- inal consols and investment shocks . . . . . . . . . . . . . . . 65 2.15 Optimal home equity shares held by home households: nom- inal consols and government expenditure shocks . . . . . . . . 68 vii List of Tables 2.16 Optimal home equity shares held by home households: nom- inal consols and preference shocks . . . . . . . . . . . . . . . . 70 2.17 Optimal home equity shares held by home households: inflation- indexed bonds and all shocks . . . . . . . . . . . . . . . . . . 71 2.18 Optimal home equity shares held by home households: nom- inal consols and real shocks . . . . . . . . . . . . . . . . . . . 71 2.19 Second moments (1) . . . . . . . . . . . . . . . . . . . . . . . 74 2.20 Second moments (1) (continued) . . . . . . . . . . . . . . . . 75 2.21 Second moments (2) . . . . . . . . . . . . . . . . . . . . . . . 76 2.22 Second moments (2) (continued) . . . . . . . . . . . . . . . . 77 3.1 Optimal equity shares held by each country in partial financial integration (nA = 1 2 , nB = nC = 1 4) . . . . . . . . . . . . . . . 89 3.2 Optimal equity shares held by each country in global financial integration (nA = 1 2 , nB = nC = 1 4) . . . . . . . . . . . . . . . 89 3.3 Optimal equity shares held by each country in partial financial integration (nA = 1− 2n, nB = nC = n) . . . . . . . . . . . . 96 3.4 Optimal equity shares held by each country in global financial integration (nA = 1− 2n, nB = nC = n) . . . . . . . . . . . . 97 3.5 Parameter values . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.6 Optimal equity shares held by each country under different financial architectures . . . . . . . . . . . . . . . . . . . . . . 112 3.7 Conditional volatility (%): the benchmark . . . . . . . . . . . 113 3.8 Conditional volatility (%): an alternative calibration . . . . . 114 4.1 Structural parameters . . . . . . . . . . . . . . . . . . . . . . 131 4.2 Ramsey steady state . . . . . . . . . . . . . . . . . . . . . . . 132 4.3 Cyclical implications of optimal policy . . . . . . . . . . . . . 133 4.4 Degree of nominal rigidity and optimal policy . . . . . . . . . 135 4.5 Simple interest rate rules . . . . . . . . . . . . . . . . . . . . 141 4.6 Cyclical implications of simple interest rate rules . . . . . . . 142 4.7 Welfare results . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.1 Optimal bond assets-GDP ratio: inflation-indexed bonds . . . 146 5.2 Bond-GDP ratio . . . . . . . . . . . . . . . . . . . . . . . . . 146 A.1 Optimal home equity shares held by home households: nom- inal consols and local currency pricing . . . . . . . . . . . . . 167 viii List of Figures 1.1 Imports, exports and portfolio investments in Argentina, Brazil and Mexico . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2 Imports, exports and portfolio investments in Indonesia and Malaysia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Imports, exports and portfolio investments in Thailand . . . . 12 2.1 A simulation path for relative interest payments . . . . . . . 50 2.2 Optimal portfolios and nominal rigidities, σ = γ = 1 . . . . . 56 2.3 Optimal portfolios and nominal rigidities, σ = γ = 2 . . . . . 57 2.4 Optimal portfolios and home bias in consumption, σ = γ = 1 60 2.5 Optimal portfolios and home bias in consumption, σ = γ = 2 61 3.1 Financial architecture . . . . . . . . . . . . . . . . . . . . . . 84 4.1 Responses of variables to one percent positive technology shock139 4.2 Policy frontiers . . . . . . . . . . . . . . . . . . . . . . . . . . 144 ix Acknowledgements While working on this thesis, I received constructive suggestions and comments from many people. I would like to acknowledge the enormous help given to me in writing this dissertation. I wish to thank my supervisor Michael B. Devereux for his invaluable advice, unfailing encouragement and considerate support over my years of study at UBC. I am especially grateful to my thesis committee Viktoria Hnatkovska and Henry Siu, whose insight- ful comments and feedback greatly improved both the substantive coverage and the exposition. Particular thanks go to Paul Beaudry and Yaniv Yedid- Levi for many stimulating conversations and helpful comments. I specially thank Patrick Francois, Giovanni Gallipoli, Mathias Hoffmann, Hiroyuki Kasahara, Amartya Lahiri, Hao Li, Vadim Marmer, Ivan Pastine, Tuvana Pastine, Anji Redish and Martin Schneider for their beneficial suggestions. My special thanks are owned to my fellow students at UBC. I also thank participants at UBC macroeconomics lunch seminars and UBC macroeco- nomics seminars. I gratefully thank my parents, who have supported me throughout my years of education, both morally and financially. Finally, financial support from the China Scholarship Council is gratefully acknowl- edged. x Dedication To my parents xi Chapter 1 Introduction Over the past few decades, capital flows across countries have reached a historical high and financial linkage across borders has become stronger and stronger. On the one hand, we observe a large increase in capital flows between developed countries and developing countries, also in conjunction with enlarged capital flows among developed countries. On the other hand, households in most countries still hold most of domestic equities in their country level equity portfolios. This thesis mainly tries to rationalize inter- national portfolio choice in the context of financial globalization and explores how financial integration affects international risk-sharing. The Great Re- cession caused by 2007 − 2009 financial crisis recalls an investigation on financial frictions and policy studies. I then study optimal monetary poli- cies in an closed economy with financial frictions. In the following parts, I will illustrate these issues in three sections, which correspond to three fol- lowing chapters in this thesis. In the first section, I discuss how to explain equity home bias in an open economy monetary model. In the second sec- tion, I explore how financial integration changes international risk-sharing. In the last section, I present how monetary and fiscal authorities implement policies in an economy with credit frictions. 1.1 Equity home bias in a monetary model With the development of financial globalization in the past few decades, portfolio flows across countries have reached a historical high, and partic- ularly, trade volume of assets across borders has surged dramatically since the early of 1990’s. Nevertheless, countries still exhibit sizable home bias in equity portfolios. For instance, the share of country equity portfolios in domestic equities was 82% in the United States in 2005 (see Sercu and Vanpee (2007), summarized in table 1.1).1 International portfolio choice 1Sercu and Vanpee (2007) has recently documented the share of country equity port- folios in domestic equities for a large number of countries, whereas Tesar (1993) provide early evidence on home bias in equities for the group of seven countries. 1 1.1. Equity home bias in a monetary model theories (e.g., Lucas 1982; Baxter and Jermann 1997) fail to explain this phenomenon often referred to the puzzle of home bias in equity portfolios (Obstfeld and Rogoff 2001). A number of recent studies suggest potential resolutions to the equity home bias puzzle, relying on the notion that holding domestic equities pro- vides a hedge for non-diversifiable income risk. Coeurdacier, Kollmann and Martin (2010) use an international real business cycle model with real bonds, while Engel and Matsumoto (2009a) and Engel and Matsumoto (2009b) employ an international monetary model with foreign currency forward con- tracts. Coeurdacier and Gourinchas (2011) offer empirical evidence on using bonds to hedge real exchange rate risk and domestic equities to hedge do- mestic labor income risk.2 Nevertheless, in standard international monetary business cycle models with nominal bonds (e.g., Chari, Kehoe and McGrat- tan 2002), the equilibrium domestic equities held by domestic households are significantly sensitive to variations of preference parameter values. In contrast to existing theories, this chapter shows that international monetary business cycle models can be reconciled with equity home bias. The contributions of this chapter are two-fold. First, I show that, by incorporating inflation-indexed bonds, international monetary business cy- cle models deliver home bias in equities for plausible preference parameter values. Inflation-indexed bonds can hedge real exchange rate risk and do- mestic equities can hedge against domestic labor income risk conditional on real exchange rates. Moreover, equity home bias is robust in environments either with complete markets, or incomplete markets. This explanation is motivated by the data. I present empirical evidence on bonds hedging real exchange rate risk and domestic equities hedging domestic labor income risk, conditional on real exchange rates, for a sub-sample of OECD countries (the group of seven countries, plus Denmark and Spain). Second, I explore in- 2These arguments are in line both with the hedging motive and the diversification motive promoted by Baxter and Jermann (1997). The hypotheses in Baxter and Jermann (1997) are based on the assumption of one-good world economy and therefore they focused on the unconditional correlation between labor income and capital income. Recent works study multiple tradable goods in a world economy and they focus on conditional correlation between labor income and capital income. Coeurdacier, Kollmann and Martin (2010) address home bias in equities in an international real business cycle model with real bonds and technology and investment efficiency shocks. In addition, Coeurdacier and Gourinchas (2011) exploit equity home bias in an endowment economy with real bonds and provided relevant empirical evidence. On the other hand, equity home bias studies by Engel and Matsumoto (2009a) and Engel and Matsumoto (2009b) explain the phenomenon as a result of nominal stickiness being highly strong, given that foreign currency forward contracts or one-period nominal bonds are equivalently to one-period real bonds when the degree of price stickiness tends to unity. 2 1.1. Equity home bias in a monetary model ternational portfolio dynamics. In the data sample, net foreign asset posi- tions counter-cyclically move with output. I find that the model with either complete markets or incomplete markets can account for counter-cyclical movements of net foreign asset positions.3 Based on monetary business cycle models (see Chari, Kehoe and Mc- Grattan 2002; Schmitt-Grohe and Uribe 2006), this chapter investigates international portfolio choice in a two-country dynamic stochastic general equilibrium (DSGE, hereafter) monetary model with an asset menu of equity and debt portfolios, investment, sticky prices and sticky wages.4 I show that when asset markets are complete, the equilibrium equity portfolios can be decomposed into three components: market portfolio, portfolio for hedging labor income risk, and portfolio for hedging real exchange rate (or terms of trade) risk. Here, market portfolio is considered constant, one half, which reflects the traditional view on diversifying income risk across country bor- ders (see, Lucas 1982; Baxter and Jermann 1997). It is argued that home bias in equity portfolios is present when: (a) domestic equities hedge do- mestic labor income risk conditional on real exchange rates, and (b) bonds hedge real exchange rate risk. Results indicate that condition (a) is satisfied in various environments, either with complete asset markets, or with in- complete asset markets, whereas condition (b) holds when inflation-indexed bonds are traded in international bond markets. These two conditions are motivated by the data. Evidence shows that 6 out of 9 country-pairs in the data sample support condition (a) and all of 9 country-pairs support condition (b). In the benchmark model with producer currency pricing, monetary and technological shocks are considered, and hence, asset markets are complete 3The ratio of net foreign asset (NFA, hereafter, are the sum of net FDI outflows, net equity asset outflows, net debt asset outflows and reserves) to GDP in the United States declined steadily from 8% in 1970 to −23% in 2004. Moreover, net foreign assets counter-cyclically move with output for 8 out of 9 countries in the data sample, including the United States, Britain and Germany. The correlation between cyclical components of U.S. NFA-GDP ratio and U.S. GDP is −0.27 during 1970 − 2004. From the text book definition of net foreign asset positions and current account, it is known that the change of net foreign asset positions is the sum of current account and wealth effect from changes of asset prices. The wealth channel here has a large contribution to the net foreign asset dynamics. The correlation between the cyclical components of wealth effects and of GDP is −0.11 in the United States. Coeurdacier, Kollmann and Martin (2010) report second moments for portfolios in an international real business cycle model with complete asset markets. 4There isn’t investment in Engel and Matsumoto’s work and the equilibrium equities are very sensitive to preference parameter values. 3 1.1. Equity home bias in a monetary model from a first-order approximation perspective.5 Consequently, households across countries can achieve perfect risk sharing. However, international portfolio choice depends on the type of debt instruments available in asset markets. I find that in standard international monetary models with nomi- nal bonds, the component of equity portfolios for hedging labor income risk is positive and extremely stable, whereas the component for hedging real exchange rate risk is sensitive to variations of preference parameter values. Accordingly, equilibrium equities vary significantly under different values of preference parameters. I further find that a monetary model with real bonds can deliver robust home bias in equities, since returns on real bonds perfectly correlate with real exchange rates. Nonetheless, it is difficult to identify real bonds in the data. Conversely, inflation-indexed bonds seem to be a natural alternative to real bonds in hedging real exchange rate risk. A caveat here is that a significant difference between these two types of bonds remains because real bonds are state-contingent whereas inflation-indexed bonds are nominal, and therefore, non-state contingent. This chapter shows that inflation-indexed bonds in this model are quantitatively akin to real bonds in hedging real exchange rate risk. By incorporating inflation-indexed bonds, the model delivers robust equity home bias. To this effect, an intuitive explanation for inflation-indexed bonds hedg- ing real exchange rate risk is as follows. Monetary shocks have transitory effects on output, consumption and real exchange rates, but they have persis- tent effects on nominal prices. When nominal bonds, say, nominal consols or one-period nominal bonds, are traded internationally, relative interest payments on nominal bonds disconnect with real exchange rates because of high volatility of nominal prices. However, inflation-indexed bonds could bring this disconnection back, because the volatility of inflation rates is far smaller than that of nominal prices in a monetary model with monetary policy shocks. Thereby, inflation-indexed bonds quantitatively mimic real bonds in terms of hedging real exchange rate risk. 5Pricing-to-market is the other common pricing strategy for firms. Recent evidence from micro data (see Gopinath and Rigobon 2008 ; Gopinath, Itskhoki and Rigobon 2010) shows that 90% (97%) of U.S. imports (exports) are priced in dollars. When firms take advantage of pricing-to-market, real exchange rates disconnect with terms of trade, and consequently one debt instrument cannot hedge terms of trade and real exchange rate fluctuations simultaneously. However, it is shown that home equity holdings vary only slightly larger than those featuring producer currency pricing, responding to different values of preference parameters. 4 1.1. Equity home bias in a monetary model Table 1.1: Home bias in equity portfolios Country Share of portfolios in domestic equity (%, Year 2005) Austria 58.5 Canada 76.6 Denmark 62.7 France 68.8 Germany 57.5 Italy 57.1 Japan 91.9 Spain 86.3 UK 65.0 US 82.2 Average 76.61 Source: Open access publications from Katholieke Universiteit Leuven, Sercu and Vanpee (2007), adapted by permission. Equity home bias in 2005, CPIS data. Empirical data employed in this chapter takes into consideration infla- tion factors introduced by indexed bonds. The reason behind this rationale is that recent developments of bond markets have seen a burgeoning supply of inflation-index bonds in developed countries. For instance, Campbell, Shiller and Viceira (2009) document that the outstanding supply of Trea- sury Inflation-protected Securities (TIPS) accounted of 10% of marketable Treasury debt in the United States in 2008 and that inflation-indexed gov- ernment bonds increased to 30% of British government bonds in 2008.6 Besides the large trade of inflation-indexed bonds in the corresponding mar- kets, the other consideration behind this rationale depends on bond returns observed in the data sample. Returns on bonds and bond return differen- tials across countries in the data sample show that bond returns are around one-half as volatile as output is. The model with inflation-indexed bonds could produce similar volatility of bond returns. Consequently, this model 6Inflation-indexed bonds are usually long-term bonds (term-to-maturity in excess of one year). Long-term bonds are as important as equities in international asset markets. Survey data from U.S. Treasury International Capital System showed that the share of long-term securities in foreign holdings of U.S. securities was 92% and that this share in U.S. holdings of foreign securities was 93% at the end of 2008. Here, long-term debt securities accounted for both 70% of U.S. external long-term liabilities and 30% of U.S. external long-term assets. 5 1.1. Equity home bias in a monetary model not only explains equity home bias but also matches the second moments of bond returns, thus providing a clear advantage over existing ones. To address the issue of how different monetary policy specifications change the optimal portfolios, exogenous money supplies replace Taylor rules in the benchmark model. It reveals that equilibrium equity holdings remain fairly stable. An examination of whether preference specifications affect equilibrium equity portfolios follows next. The benchmark model uses a non-separable preference between consumption and real money balance, given that the latter affects marginal utility of consumption directly. To eliminate the direct effect of money balance on consumption, a separable preference between consumption and real money balance is employed. Re- sults show that equity home bias is robust to preference parameter values. When asset markets are complete, households in both countries achieve perfect risk sharing. However, when asset markets are incomplete, the de- gree of risk sharing is limited (see Corsetti, Dedola and Leduc 2008) and accordingly, portfolios are also adjusted correspondingly. This circumstance motivates further interest to understand how equity portfolios in incom- plete markets are different from those in complete markets. To this end, market incompleteness is introduced by incorporating additional indepen- dent shocks into the benchmark model. According to the recent estimation of DSGE models (Justiniano, Primiceri and Tambalotti 2010), several other shocks are considered to largely contribute to business cycle fluctuations, including investment efficiency shocks, government expenditure shocks and preference shocks. Two versions of market incompleteness are considered here. In the first version, one extra shock is added to the benchmark model separately, whereas the second version includes all shocks. The results show that equilibrium equity portfolios seem to be invariant to the concurrence of these additional shocks.7 To better understand how inflation matters for nominal bonds hedging real exchange rate risk, I then remove monetary shocks, keep only all real shocks, and use nominal bonds instead of inflation- indexed bonds in the model. The results show that nominal bonds can also hedge real exchange rate risk. Pioneering researchers have proposed numerous ways to solve the puz- 7 A question arises here as to equilibrium equity portfolios when only real shocks hit the economy. To answer this question, monetary shocks are disregarded and two versions of models with complete asset markets and real shocks are considered instead. When (1) only technology shocks and government expenditure shocks or (2) only technology shocks and preference shocks appear, the model generates super home bias in equities (home equity shares held by home households are larger than unity). However, incorporating monetary shocks can bring super home bias in equities down to home bias in equities. 6 1.1. Equity home bias in a monetary model zle of equity towards home.8 In Cole and Obstfeld (1991), terms of trade themselves can facilitate risk sharing across borders under special parame- terizations. Following this logic, Heathcote and Perri (2008) later argue that equity home bias is optimal in a two-country DSGE environment. However, their results are sensitive to preference parameter values. In contrast to these, efforts by a separate group of researchers took account of the influ- ence of non-tradable goods. For instance, Baxter, Jermann and King (1998) show that, even when allowing for non-traded goods and non-traded factors, equities of domestic tradable goods never optimally exhibit home bias. Sim- ilar implications of a richer model environment were examined by Pesenti and van Wincoop (2002) and Collard, Dellas, Diba and Stockman (2009).9 Hnatkovska (2010) study an economy with two-country, tradable goods, non-tradable goods sectors, and production. She shows that low diversifi- cation happens because variations in relative prices increase the riskiness of foreign assets and facilitate international risk-sharing. Yet another strand of research efforts emphasized hedging non-diversifiable human capital income (see Engel and Matsumoto 2009a; Engel and Matsumoto 2009b; Coeur- dacier, Kollmann and Martin 2010; Coeurdacier and Gourinchas 2011).10 Transaction cost is another important consideration in explaining equity home bias. Obstfeld (2007) Ohlin lecture provides an excellent summary on international asset trade and costs of trade.11 Nonetheless, the latest lit- erature on the subject reveals information asymmetry in explaining capital flows. For example, Tille and van Wincoop (2010) investigate the role of information dispersion in a general equilibrium environment.12 8An influential survey was originally conducted by Adler and Dumas (1983). Later, Lewis (1999) provide a summary of recent development of home bias in equities. The latest comprehensive survey work on open economy macro-finance can be found in Coeurdacier and Rey (2012). 9Tesar (1993) studies a two-period model with multiple non-tradable goods. In a partial equilibrium environment, Pavlova and Rigobon (2010) investigate portfolio choice in an endowment economy. 10Kollmann (2001) investigates the robustness of home bias in equities when these are the only international assets, whereas Collard, Dellas, Diba and Stockman (2009) illustrate that optimal equity shares are sensitive to trade elasticity. 11Coeurdacier (2009) shows that equities may not bias the home country for reasonable trade cost. 12Private information and the associated cost have been considered as important fac- tors determining international equity flows (see Nieuwerburgh and Veldkamp 2009; Al- buquerque, H. Bauer and Schneider 2009; Mondria and Wu 2010). However, from an empirical perspective, Didier, Rigobon and Schmukler (2011) present fund-level evidence showing that information cost might not be the key role in determining cross-country portfolios. 7 1.2. International financial markets and international risk sharing In the past few years, researchers have come up with various dynamic general equilibrium portfolio choice models. Devereux and Sutherland (2010a), Devereux and Sutherland (2011a) and Tille and van Wincoop (2010) inde- pendently and simultaneously develop a solution method advancing DSGE models with portfolio choice. Their proposed methods stay very close to the familiar first- and second-order approximation techniques.13 In this chap- ter, the solution method for general equilibrium rational expectations mod- els with higher order approximations are based on Schmitt-Grohe and Uribe (2004), and solutions to portfolio choice are based on the method introduced by Devereux and Sutherland (2010a) and Devereux and Sutherland (2011a). 1.2 International financial markets and international risk sharing Financial integration starting from 1980’s has significantly boosted inter- national capital flows among developed economies and between developed countries and developing countries. This phenomenon has motivated a large expanding volume of literature that try to evaluate gains from financial in- tegration (see recent studies, Bonfiglioli 2008; Kose, Prasad and Terrones 2009b; Nicolo and Juvenal 2012). Kose, Prasad, Rogoff and Wei (2009a) summarize that gains from financial integration are mixed in the literature. Although they are large between developing countries and developed countries, capital flows among developing countries remain still limited. Figure 1.1-1.3 show that portfolio investments across developing countries are far lower than the corresponding commodity trade.14 Developing coun- tries hold a dominant part of portfolios either in the world center, such as the United States, or regional centers such as Britain and Singapore. For instance, the share of exports or imports of goods and services between Argentina and other developing countries such as Brazil, Chile and Mex- ico, in Argentina’s total exports or imports is over 30% on average during 1997− 2009. However, in the same period, Argentina’s portfolio investment assets in these developing countries only account for less than 5% of its to- 13Devereux and Sutherland (2007), Devereux and Sutherland (2008) and Devereux and Sutherland (2010b) apply their method to monetary policy and portfolio choices. Further- more, Evans and Hnatkovska (2005) develop a similar solution method using perturbation and projection methods, whereas Stepanchuk and Tsyrennikov (2011) study international portfolios in a wealth-recursive equilibrium. Benigno and Nistico (2012) explore home bias under model uncertainty. 14Fratzscher and Imbs (2009) illustrate that portfolio investment exclusively increases risk-diversification across countries. 8 1.2. International financial markets and international risk sharing tal overseas portfolio investments.15 Eichengreen and Park (2003) show that bilateral bank claims in Asia are much more smaller than Europe. Based on high-frequency indicators for equity market integration, Yu, Fung and Tam (2010) find that Asian countries have experienced financial integration for the past decade but this process is quite different from mature markets. In other words, financial integration among developing countries stays at a low level. Naturally, two concerns are present about the observed low financial in- tegration among developing economies. One is that developing countries have already achieved optimal allocation of assets across borders according to their income-output processes. The other is that some prevalent barri- ers prevent capital flows across developing economies. Table 1.2 and 1.3 show the pairwise correlation of growth rate of real GDP per capita among a sample of developing economies. The output growth rate correlations among Argentina, Brazil, Mexico and Chile are essentially zero. Asian de- veloping economies except for China have a higher cross-correlation in the data sample, but it is almost the same as the correlation between these de- veloping countries and regional core countries, such as Singapore and Japan. The growth rate of output in China seems uncorrelated with both advanced economies and developing economies. In terms of these income processes, the point of view that developing economies have already fully diversified their income risk across countries seems not consistent with the data. Alter- natively, there might exist unexploited gains from financial integration for developing economies. These gains from diversification might be prevented by barriers such as capital account regulation, financial and economic under- development (see quantitative evaluations in Mendoza, Quadrini and Rios- Rull 2009b; Bai and Zhang 2010; Bai and Zhang 2012). In Callen, Imbs and Mauro (2011), they show based on a sample of 74 countries that welfare gains remain untapped because of non-diversifiable enforcement costs. An interesting question is whether further financial integration, say, fi- nancial integration among developing countries, could make all participat- ing countries better off. Nevertheless, the literature doesn’t say much about 15A country’s external assets consists of portfolio investment assets, foreign exchange reserves and foreign direct investment. Portfolio investment assets have two sub-categories, equities and bonds. For many developing countries, foreign exchange reserves account for a dominant part of external assets. Nevertheless, portfolio investment assets are also an equally significant part of external assets in these countries. In practice, a geographical break of foreign exchange reserves is not available publicly, so the use of portfolio assets can basically capture the distribution of trade in assets for developing countries given the assumption that the geographical composition of portfolio assets held by monetary authorities is the same as other sectors within a country. 9 1.2. International financial markets and international risk sharing 0.00 20.00 40.00 60.00 80.00 100.00 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 Argentina Exports to United States Imports from United States Portfolio assets in United States 0.00 10.00 20.00 30.00 40.00 50.00 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 Argentina Exports to Brazil, Chile and Mexico Imports from Brazil, Chile and Mexico Portfolio assets in Brazil, Chile and Mexico 0.00 10.00 20.00 30.00 40.00 50.00 60.00 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 Brazil Import from United States Portfolio assets in United States Exports to United States 0.00 5.00 10.00 15.00 20.00 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 Brazil Exports to Argentina, Chile and Mexico Import from Argentina, Chile and Mexico Portfolio assets in Argentina, Chile and Mexico 0.00 20.00 40.00 60.00 80.00 100.00 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 Mexico Exports to United States Imports from United States Portfolio assets in United States 0.00 1.00 2.00 3.00 4.00 5.00 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 Mexico Exports to Argentina and Brazil Imports from Argentina and Brazil Portfolio assets in Argentina and Brazil Figure 1.1: The graphs show the percentage share of imports to/exports from/portfolio assets in major foreign countries and regions when the home country is Argentina, Brazil and Mexico, respectively. The horizontal axis is year and the vertical axis is percentage. Data sources: Direction of Trade and CPIS datasets, IMF, 2011. 10 1.2. International financial markets and international risk sharing 0.00 10.00 20.00 30.00 40.00 50.00 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 Indonesia Exports to Japan, Korea and Singapore Imports from Japan, Korea and Singapore Portfolio assets in Japan, Korea and Singapore 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 Indonesia Exports to United States Imports from United States Portfolio assets in United States 0.00 5.00 10.00 15.00 20.00 25.00 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 Indonesia Exports to Malaysia and Mainland of China Imports from Malaysia and Mainland of China Portfolio assets in Malaysia and Mainland of China 0.00 10.00 20.00 30.00 40.00 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 Malaysia Exports to Japan, Korea and Singapore Imports to Japan, Korea and Singapore Portfolio assets in Japan, Korea and Singapore 0.00 5.00 10.00 15.00 20.00 25.00 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 Malaysia Exports to United States Imports to United States Portfolio assets in United States 0.00 10.00 20.00 30.00 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 Malaysia Exports to Mainland of China, Indonesia and Thailand Imports to Mainland of China, Indonesia and Thailand Portfolio assets in Mainland of China, Indonesia and Thailand Figure 1.2: The graphs show the percentage share of imports to/exports from/portfolio assets in major foreign countries and regions when the home country is Indonesia and Malaysia, respectively. The horizontal axis is year and the vertical axis is percentage. Data sources: Direction of Trade and CPIS datasets, IMF, 2011. 11 1.2. International financial markets and international risk sharing 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Thailand Exports to Japan, Korea and Singapore Imports from Japan, Korea and Singapore Portfolio assets in Japan, Korea and Singapore 0.00 20.00 40.00 60.00 80.00 100.00 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Thailand Exports to United States Imports from United States Portfolio assets in United States 0.00 5.00 10.00 15.00 20.00 25.00 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Thailand Exports to Mainland of China, Malaysia and Indonesia Imports from Mainland of China, Malaysia and Indonesia Portfolio assets in Mainland of China, Malaysia and Indonesia Figure 1.3: The graphs show the percentage share of imports to/exports from/portfolio assets in major foreign countries and regions when the home country is Thailand. The horizontal axis is year and the vertical axis is percentage. Data sources: Direction of Trade and CPIS data sets, IMF, 2011. 12 1.2. International financial markets and international risk sharing Table 1.2: Correlation of growth rate of real GDP per capita (Latin America) USA Argentina Brazil Mexico Chile USA 1 Argentina 0.0554 1 Brazil 0.0877 0.3377 1 Mexico 0.5294* 0.2636 -0.1491 1 Chile 0.4512* 0.1922 0.0468 0.1736 1 Note: The table shows pairwise correlation of growth rate of real GDP per capita. A star denotes a significance level of 0.05. Yearly data during 1985− 2009. Data source: Penn World Table 7.0. Table 1.3: Correlation of growth rate of real GDP per capita (Asia) USA Singapore Japan Indonesia Malaysia Thailand China USA 1 Singapore 0.4376* 1 Japan 0.4809* 0.6302* 1 Indonesia -0.2712 0.4384* 0.3509 1 Malaysia 0.2739 0.8789* 0.6261* 0.6811* 1 Thailand -0.0239 0.5397* 0.6286* 0.7639* 0.7515* 1 China -0.0928 0.0893 -0.3075 0.0877 0.0622 0.0628 1 Note: The table shows pairwise correlation of growth rate of real GDP per capita. A star denotes a significance level of 0.05. Yearly data during 1985−2009. Data source: Penn World Table 7.0. gains for developed countries when developing countries could fully diversify their portfolios worldly. I try to bridge this gap and investigate this issue in a center-periphery framework with international portfolio choice. This chapter develops a tractable center-periphery framework with in- ternational portfolio choice. International financial architectures are divided into three stages according to financial development. The first stage is fi- nancial autarky, in which no financial assets are traded across borders. The second stage is partial financial integration, in which the central country becomes financially integrated with peripheral countries, but there is no fi- nancial integration between peripheral countries. This case is closed to the current financial architecture. The third stage is global financial integration, in which all financial markets are integrated. I find that (a) from finan- cial autarky to partial financial integration, volatility of consumption in all 13 1.2. International financial markets and international risk sharing countries drops, and (b) from partial financial integration to global financial integration, volatility of consumption in peripheral countries decreases, but consumption volatility rises in the central country when the central country is large. In other words, peripheral countries monotonically increase inter- national risk-sharing in the process of financial integration. Nevertheless, the core country achieves its highest risk-sharing in the medium stage of financial integration when peripheral countries are small. When peripheral countries are large, consumption volatility in all countries decreases in the process of financial integration. From financial autarky to partial financial integration, each country has access to more financial assets and consequently achieves a better risk shar- ing than financial autarky. From partial financial integration to global fi- nancial integration, the mechanism lies in the demand effect induced by country size (see empirical evidence in Martin and Rey (2004)). Consider that the core country is far larger than other countries. From partial to global integration, the world demand for peripheral assets increases because of imperfect correlation of shocks, which implies peripheral asset prices go up and consequently their return rates decrease. However, the total demand for core assets reduces because of demand shifting towards peripheral assets, and then the asset price in the core country goes down and accordingly its re- turn rate is higher. Both accounts imply a narrower cross-country premium between assets in the core and peripheral countries in partial financial in- tegration than in global financial integration. The change of cross-country premium between different stages of financial integration can be treated as risk-sharing premium gained by peripheral countries. On the other hand, when the core country is large, income risk in this large country represents global income risk. In equilibrium, the response of consumption on impact can be decomposed into two parts. One is income effect, which comes from the change of output responding to shocks. The other is wealth effect, which represents the change of values of net foreign asset positions. When the core country is relatively large, it has a power over other assets in financial mar- kets due to size effect, and the wealth effect favors the core country most in partial financial integration. When it is not large, the core’s privilege disappears. Hence I obtain a monotonic reduction in consumption volatility for all countries with the proceeding of financial integration. In the baseline model, I abstract the effect of changes of intertemporal prices on international risk-sharing. As emphasized by Cole and Obstfeld (1991), terms of trade can move around responding to shocks and thereby income risk can be automatically shared through relative price changes under particular parameterizations. I then extend the baseline model to a world 14 1.2. International financial markets and international risk sharing economy in which each country produces one tradable good and consumes all types of tradable goods. Labor supply is endogenous and exogenous shocks are productivity shocks. I obtain similar results. A shortcoming of analyses in this chapter is that I don’t compute welfare for each country in each stage of financial integration. In order to calculate welfare, I need to take third-order approximations to Euler equations for portfolios in all countries to obtain first-order approximations to portfolios, and to take second-order approximations to the rest of equations in the competitive equilibrium. This exercise will be left in my work in progress (Yu 2012). This chapter contributes to two branches of literature. First, a strand of literature provides evaluation of gains from financial integration (Cole and Obstfeld 1991; Baxter and Crucini 1995; Brandt, Cochrane and Santa- Clara 2006; Martin 2010). Contrary to the international risk-sharing litera- ture, I find that financial integration is not necessarily reduce consumption volatility for all participants. This view is related to the imperfect credit market literature (say, Matsuyama 2008) but I focus on a quite different environment in which capital inflow/outflow restrictions among developing economies are the source of imperfect credit markets. Recently, a growing literature focus on the effect of financial development on international risk sharing in environments with heterogeneous agents (for instance, Mendoza, Quadrini and Rios-Rull 2009a; Angeletos and Panousi 2011). They only consider household-specific shocks and there are no country-specific shocks in their models. I focus on instead country-specific shocks here. Second, this chapter also contributes to the stream of literature on international portfo- lio choice. While most of this literature focus on a two-country framework (for instance, Devereux and Sutherland 2011b), I develop a center-periphery framework with different stages of financial integration, rather than creating new assets in the process of financial integration (Devereux and Sutherland 2009). In terms of framework, the center-periphery structure is similar to Pavlova and Rigobon (2008) and Goldberg and Tille (2009). Pavlova and Rigobon (2008) discuss portfolio choices and focus on international propagation of shocks when the central country is financially constrained. My work is quite different from theirs and I focus on financial integration and international risk-sharing. Goldberg and Tille (2009) explore monetary policy and invoice of international trade but without portfolio choice. The solution method used in this chapter is based on Devereux and Sutherland (2011a) who use higher order approximations to obtain optimal portfolios. 15 1.3. Credit friction, asset price and optimal monetary policy 1.3 Credit friction, asset price and optimal monetary policy The 2007 − 2009 financial turmoil revives a burgeoning interest in fi- nancial factors in the study of business cycles (i.e., Carlstrom and Fuerst 1997; Curdia and Woodford 2009; Curdia and Woodford 2010; Devereux and Yetman 2010; Gertler and Kiyotaki 2010; Kiyotaki and Moore 2012), a tradition starting from Bernanke and Gertler (1989).16 The major point lies in that financial shocks have important effects on macroeconomy (see Jermann and Quadrini 2012) and/or that financial factors play an impor- tant role in amplifying the propagation of shocks in a closed economy (i.e., Kiyotaki and Moore 2012) or across boarders (see Devereux and Yetman 2010; Faia and Iliopulos 2010). Recent data show that the financial frictions are important factors generating large consumption and output volatility, which in turn incur some welfare losses. The question I am interested in in this chapter is whether high volatility of consumption and output could be avoided if monetary authorities have the proper instruments. If a finan- cial accelerator can produce large responses of fundamentals to exogenous shocks, it is also true that policy makers could use a financial decelerator to stabilize the economy and improve welfare. Based on a medium scale closed economy model with sticky prices, sticky wages and financial frictions, this chapter shows that if the debt deflation channel is strong enough, monetary authorities should create deflation in booms and inflation in recessions in order to reduce the volatility of constrained borrowers’net worth. In this chapter, I investigate optimal monetary policies in an economy with staggered nominal wage setting in addition to staggered price setting and credit-market imperfections. The specific questions I ask are as follows. How do monetary policies respond to asset prices when credit frictions ex- ist in an economy? Does the response to financial factors make sense as a general policy? In other words, which kind of simple linear rules can im- plement the optimal monetary policy? As in recent contributions, volatility of price inflation prompts dispersion in prices across goods and henceforth inefficient output levels. Analogously, staggered wage contracts induce wage dispersion and hence inefficiencies in the distribution of employment across households (see Erceg, Henderson and Levin 2000). With credit market frictions, when nominal returns on debt are not state-contingent, varying the price level in response to shocks allows to transfer real wealth between 16Brunnermeier, Eisenbach and Sannikov (2012) provide an up-to-date survey on fi- nancial frictions in macroeconomics. 16 1.3. Credit friction, asset price and optimal monetary policy debtors and creditors, which in turn amplifies shocks through the financial accelerator (Bernanke, Gertler and Gilchrist 1999). Hence, achieving the optimal allocation would require to trade off costs of price dispersion and of wage dispersion, and the effect of debt deflation in the imperfect credit market as well. I model credit frictions following the collateral channel of Kiyotaki and Moore (1997). Specifically, financial imperfections originate from a problem of debt contract enforcement between debtors and creditors. When debts require to be secured by collateral, which is also an input in production, the amount of debts positively co-moves with values of collateral. At the same time, the financial costs between external borrowing and internal funds negatively vary with borrowers’ balance of sheet, which generate endogenous external finance premium and volatile asset price over the business cycle. The model studied here has several key features besides credit frictions. First, it embeds monopolistic competition both in goods market and labor market, and Calvo-style nominal rigidities both in wage contracts and price resetting. Second, I consider intertemporal nominal debt contracts, in which debt contracts are signed at current period and will be repaid next period. Movements in inflation affect not only labor and goods market allocations but also the real repayment of debts. For instance, when an unanticipated positive technology shock happens, monetary authority should restore labor market efficiency by creating deflation such that the labor allocation is close to the efficient level in the absence of nominal inertia over the business cycle. On the other side, price stickiness requires stabilizing inflation to reduce the cost of price dispersion. Furthermore, at the same time, the associated deflation will transfer wealth from debtors to creditors, which creates an endogenous counter-cyclical net worth shock. When this induced net worth shock appears unexpectedly in a credit-frictionless economy, borrowers can smooth consumption and adjust their capital holdings through borrowing as much as they can until there is no arbitrage in asset markets. As a result, the adjustment of productive inputs are ignorable. Notwithstanding, the effect of net worth shocks on reallocation of productive factors in an economy with credit frictions is not negligible any more, since constrained borrowers can expand or shrink large amount of their holdings of productive factors via leverage, which could produce large reallocation of resources between debtors and creditors. From the point of view of method, this chapter investigates monetary policies in terms of social welfare criterion. Unlike the output-inflation volatility frontier, a second-order approximation to an economy around its distorted steady state could provide correct rank of alternative monetary 17 1.3. Credit friction, asset price and optimal monetary policy policy rules (see Schmitt-Grohe and Uribe 2004). And thereby one can draw confident conclusion about the recommendable monetary policy. The main finding is that monetary policy has high persistence in interest rate and it negatively reacts to asset prices, although the response is quite small conditional on controlling for inflation rate and output gap. The basic intuition is as follows. The financial accelerator could generate large responses of output to shocks in an economy absent of any nominal rigidity. However, the large volatility of output may be not desirable in terms of social welfare. With credit frictions, policy-makers could take advantage of the financial decelerator to drive down the volatility of output and thus enhance social welfare. In the model economy, asset price is mainly driven by borrowers’ (also entrepreneurs’) demand for capital. When a positive shock appears, a higher return to capital induces an increase in demand for capital and therefore a boom in asset price. However, credit constrained borrowers have to resort to lenders (also households) to obtain external funds. A lower interest rate responding to an increase in asset price would induce a lower supply of private credit and hence only part of demand for capital is satisfied in equilibrium, which consequently stabilizes the volatility of output. At the same time, the associated external finance premium increases under the optimal policy because positive technology shocks bring about a higher return to capital while the deflation in nominal prices induces a higher ex post real debt repayment and the latter effect dominates the former. Hence, borrowers’ net worth decreases and thereby external finance premium goes up. However, the coefficient for asset price in the interest rate rule is low. After comparing several simple interest rate rules widely discussed in the literature, I find that an interest rate rule with a large response to the lagged interest rate has a relatively low welfare loss. Strict inflation targeting and traditional Taylor rules have relatively large welfare losses. Another finding of this chapter is that after incorporating fiscal policy into the model economy, at the long-run steady state, the optimal tax rate on capital is of −5.2%, and labor income tax rate is of 51%, for there is monopolistic power for final goods production, which in turn distorts the capital allocation in the entrepreneur sector. Subsidizing capital income could improve the efficiency of capital allocation. If all incomes are taxed at the same rate, then the income tax rate is 6.9%. In monetary models, most of literature considering financial factors have a sole form of nominal rigidity - staggered price setting, and monetary policy rules that keep inflation rate constant also achieve optimal welfare level (see Faia and Monacelli 2007; Carlstrom, Fuerst and Paustian 2010; Curdia and Woodford 2009; Curdia and Woodford 2010). Nevertheless, staggered wage 18 1.3. Credit friction, asset price and optimal monetary policy contracts seem to play an important role in explaining cyclical patterns of the U.S. economy (see Christiano, Eichenbaum and Evans 2005; Schmitt- Grohe and Uribe 2006). So I take into account of wage stickiness in this chapter. Taylor (2008) suggests that monetary policy should negatively respond to credit spread in the Taylor rules. However, after incorporating credit frictions in a dynamic stochastic general equilibrium model, Curdia and Woodford (2009) and Curdia and Woodford (2010) find that mere existence of spread matters little while spread adjustment can improve upon the Tay- lor rule but the size of response to credit is not robust to underlying nature of shocks. Carlstrom et al. (2010), De Fiore and Tristani (2009) and Be- nigno and Faia (2010) conclude that inflation stabilization is nearly optimal to technology shocks. The existing literature mainly focus on goods market frictions, i.e., price stickiness and credit market frictions (Monacelli 2008; Monacelli 2009). Either credit spread is produced in a reduced form, i.e., Curdia and Woodford (2009, 2010), or no role of credit spread and asset price is played in the model economy. Faia and Monacelli (2007) study an optimal linear interest rate rule in an economy of Carlstrom and Fuerst (1997); nevertheless, they only take into account of sticky prices and intra- period nominal debts, and therefore, stabilizing price inflation is apparently optimal. When inter-period nominal debts are introduced, ex post unantic- ipated variation in nominal prices associated with shocks could redistribute real wealth between debtors and lenders through the debt deflation channel, which has been discussed by Iacoviello (2002). However, monetary policy is not evaluated in terms of welfare criterion in his chapter. This chapter tries to bridge this gap and to investigate monetary policy based on welfare comparison in a richer environment. 19 Chapter 2 Equity Home Bias and International Portfolio Dynamics in a Monetary Open-Economy Model Can we reconcile the observed equity home bias and the workhorse mon- etary business cycle model? This chapter will investigate this issue in deep detail. In the remainder of this chapter, section 2.1 describes the household’s problem, the producer’s problem, market clearings and the corresponding equilibrium. Section 2.2 derives a solution for the equilibrium equity port- folios and provides relevant evidence. Section 2.3 reports data set and esti- mates of exogenous processes. Section 2.4 introduces numerical results for optimal portfolios whose robustness is correspondingly verified. Section 2.5 extends the model with complete markets to models with incomplete mar- kets, and section 2.6 discusses the dynamics of portfolios. Final remarks are in the last chapter. 2.1 The model In this section, I describe the model economy and display the problems faced by households and all types of firms. In addition, I describe the mone- tary and fiscal policies. The sources of uncertainty in the benchmark model are a shock to technology and a shock to monetary policy. The world economy consists of two equally large countries, denoted by H, the home country, and F, the foreign country. Goods and assets are interna- tionally traded without cost. The population in each country has a contin- uum of households with unit measure who live infinite horizons. Households choose labor supply, consumption, money balance and portfolio in each pe- riod. Labor union sets wage contracts in a staggered way for each type of labor services. Wholesale firms competitively produce wholesale goods us- 20 2.1. The model ing labor services and physical capital and sell them to intermediate goods producers. Physical capital is accumulated by wholesale firms. Intermediate producers purchase homogeneous wholesale goods in the competitive goods market and differentiate them into imperfectly substitutable goods through different brands. They adjust price for each variety in a staggered way. The local bundlers combine intermediates into local goods composite and sell it to domestic households and export to the foreign country. Assume that firms use producer currency pricing and the law of one price holds at each individual good level because of no trade costs. Government maintains a balanced budget each period. International assets consist of equities and bonds. Dividends from all domestic firms, including wholesale firms and intermediate producers, are distributed to holders of domestic equities at the end of each period. 2.1.1 Consumption composite Consumption composite in each country requires local intermediate goods and imported intermediate goods from the other country. It is aggregated via a CES form, Ct = [ a 1 γ CC γ−1 γ H,t + (1− aC) 1 γC γ−1 γ F,t ] γ γ−1 (2.1) where γ > 0 is the elasticity of substitution between imports and exports. CH,t (CF,t) is consumption demand for home (foreign) produced goods by home households. In the symmetric deterministic steady state, aC is the share of consumption spending devoted to local goods. A preference bias towards local goods implies, 1/2 < aC < 1. Expenditure minimization associated with the consumption composite (2.1) gives the consumer price index at the home country, Pt = [ aCP 1−γ H,t + (1− aC)P 1−γF,t ] 1 1−γ (2.2) where PH,t and PF,t respectively denote prices of home bundled goods and foreign bundled goods in terms of home currency. We assume that the law of one price holds at individual good level and thereby, agents in both countries face the same prices of intermedi- ate goods after adjusted by nominal exchange rates, i.e., PF,t = P ∗ F,tEt and P ∗H,t = PH,t/Et, with Et the price of foreign currency in terms of domestic currency. Foreign variables are denoted by ∗ at their superscripts. Demand 21 2.1. The model for home goods by home consumers is given by CH,t = aC(PH,t/Pt) −γCt and demand for foreign goods by home consumers has a form of CF,t = (1 − aC)(PF,t/Pt)−γCt. Foreign country has a similar expression for con- sumer price index, P ∗t = [ aCP ∗(1−γ) F,t + (1− aC)P ∗(1−γ)H,t ] 1 1−γ where P ∗t is the price of foreign consumption composite in terms of foreign currency. 2.1.2 Households Households in the home country have a life-time utility function of the form, Et [ ∞∑ τ=t βτU(Cτ , Hτ , Mτ Pτ ) ] (2.3) where the operator Et here represents the conditional expectation over all states of nature, Cτ is the home consumption, Hτ stands for labor supply and Mτ represents nominal money balance. Following the open economy literature, i.e., Schmitt-Grohe and Uribe (2003), I take use of endogenous discount factor, βt+1 = βtv(t), with β0 = 1, where v(t) ≡ ζ0C̄t−ζ1 , 0 < ζ1 < 1, ζ0 is a parameter which is chosen such that the discount factor at the steady state is equal to the inverse of risk-free rate. C̄t is the aggregate home consumption. When they made decisions, households treat aggregate consumption as given. As in Chari, Kehoe and McGrattan (2002), the period utility function takes a form of, U(Ct, Ht, Mt Pt ) = 1 1− σ (ωC θ−1θt + (1− ω)(MtPt ) θ−1 θ ) θ θ−1 1−σ + η0 (1−Ht)1−η 1− η (2.4) where σ is the coefficient of relative risk aversion or the inverse of elastic- ity of intertemporal substitution, η characterizes Frisch elasticity of labor 22 2.1. The model supply.17 ω describes the deterministic steady state share of consumption in the consumption and real money balance composite. θ characterizes the interest rate elasticity of money demand. Before exploring households’ budget constraints, let’s look at first the la- bor market. Following Erceg et al.(2000), I assume that the home household is a monopoly supplier of a differentiated labor service, Hjt, j ∈ [0, 1], as an imperfect substitute for other labor services. It’s convenient to further as- sume that there is a labor union (see, Schmitt-Grohe and Uribe 2006) which combines home households’ labor services and sells the service package to local firms. The aggregate labor input, Hdt , has the Dixit and Stiglitz (1977) form, Hdt = (∫ 1 0 H θw−1 θw j,t dj ) θw θw−1 , θw > 0 (2.5) where θw is elasticity of substitution between labor services. The labor union minimizes the cost of providing a given amount of the aggregate labor services, Hdt , given the wage rate Wj,t set by the home households, and sells the service package to firms at the wage rate Wt, Wt = (∫ 1 0 W 1−θwj,t dj ) 1 1−θw (2.6) The demand for labor service j is given by, Hj,t = ( Wj,t Wt )−θw Hdt (2.7) Total labor services supplied by an individual household are then given by, Ht = ∫ 1 0 Hj,tdj = H d t ∫ 1 0 ( Wj,t Wt )−θw dj = Hdt s w t+1 with swt+1 ≡ ∫ 1 0 ( Wj,t Wt )−θw dj describing wage dispersion in the labor market. The first equality above states that different labor services are indifferent to labor suppliers and therefore, the total labor supply is the sum of differen- tiated labor services. The second equality indicates that there is a wedge 17Christiano, Trabandt and Walentin (2010) propose another interpretation of labor supply Ht that it measures numbers of people working rather than hours of working. The value of η in the calibration is consistent with this interpretation. However, I still use Frisch labor supply elasticity for 1/η here for convenience. 23 2.1. The model between labor supply and labor demand. The higher the wage dispersion is in labor markets, the larger the wedge becomes. In the deterministic steady state, wage rates are the same cross labor services and thereby wage dispersion is one. Notice that households face the same labor demand curve, choose iden- tical consumption goods and hold the same financial assets. Therefore, the model is reduced to a representative agent model. Next I will use this ex- position convenience instead in the following analysis. The household’s budget constraint at period t states that consumption expenditure plus asset positions for the next period must equal the total income, Ct + qtψH,t+1 + stq ∗ tψF,t+1 + p b tBH,t+1 + stp b∗ t BF,t+1 +mt = (qt+dt)ψH,t+st(q ∗ t+d ∗ t )ψF,t+(p b t+r b t )BH,t+st(p b∗ t +r b∗ t )BF,t+wtH d t + mt−1 pit +trt (2.8) For saving notions, a uppercase letter Xt denotes nominal price in local currency if applicable and the corresponding lower case xt represents real price in terms of local real consumption, xt ≡ Xt/Pt. The real exchange rate is defined as, st ≡ EtP ∗t /Pt. Asset positions consist of money holdings, home and foreign equity hold- ings, and home and foreign debt portfolios. qt (q ∗ t ) represents the price of home (foreign) equity and the corresponding term ψH,t+1 (ψF,t+1) represents the share of home (foreign) equities purchased by the home household at the end of period t. Holders of equities will receive the corresponding share of dividends, dt (d ∗ t ), which are defined later. Similarly, p b t (p b∗ t ) indicates the price of home (foreign) bonds. BH,t+1 (BF,t+1) denotes the quantity of home (foreign) bonds held by the home resident at the end of period t. Each unit of home (foreign) bonds gives rbt (r b∗ t ) unit of real consumption composite to the claimant. Labor income is given by wtH d t . Finally, the household re- ceives a lump-sum government transfer trt. Foreign household has a similar exposition. Before deriving the optimality conditions for the household, let me de- scribe the bond market first. In the budget constraint (2.8) above, I present a generic form for bond returns. For instance, when nominal consols are traded across borders, they pay one unit of local currency per unit of bonds each period and then interest payments read rbt = 1 Pt , rb∗t = 1 P ∗t . Other types of bond instruments will be introduced later. In each period, the home household maximizes its life-time utility (2.3) 24 2.1. The model with respect to consumption, money balances, holdings of international as- sets, subject to the labor demand (2.7) and the budget constraint (2.8). The optimality conditions for consumption and holdings of financial assets are as follows, Uc,t = Et [v(t)Uc,t+1ri,t+1] , with i = a1, a2, b1, b2 (2.9) where Uc,t represents marginal utility of consumption at period t, and, ri,t+1 expresses real gross returns to asset i in terms of consumption composite at the home country, which are defined later in equation (2.10). The first-order conditions derived above illustrate the usual “consumption Euler equation” which links the marginal cost of a foregone unit of current consumption to the expected marginal benefit from international asset returns in the following period. The real gross return to asset i, ri,t+1, is defined as follows, ra1,t+1 ≡ qt+1 + dt+1 qt ra2,t+1 ≡ st+1 st q∗t+1 + d∗t+1 q∗t rb1,t+1 ≡ pbt+1 + r b t+1 pbt rb2,t+1 ≡ st+1 st pb∗t+1 + rb∗t+1 pb∗t (2.10) Analogously, the consumption Euler equations for the foreign household can be written as, Uc∗,t = Et [ v∗(t)Uc∗,t+1 st st+1 ri,t+1 ] , with i = a1, a2, b1, b2 The optimal condition for real balance holdings is given by, 1 = Um,t Uc,t + 1 Rt+1 (2.11) where Um,t denotes marginal utility of real money balance. Rt+1 is the risk- free nominal interest rate denominated in local currency, and it is determined by the following Euler equation, 1 Rt+1 = Et [ v(t) Uc,t+1 Uc,t Pt Pt+1 ] (2.12) Let Xw,t be the Lagrange multiplier for labor demand constraint (2.7), which is interpreted as wage markup. The household’s optimal labor supply is set to equalize the marginal value of labor income divided by wage mark- up, Xw,t, and marginal disutility of labor supply, given nominal wages preset 25 2.1. The model by the household, − Uh,t Uc,t = wt Xw,t (2.13) The household sets nominal wages according to the staggered contracts as in Calvo (1983). In each period a fraction 1 − ξw of wage contracts are renegotiated. When the household is able to reset wage contract j, it chooses W̃j,t to maximize its utility function, given the demand for labor service j and other wage contracts, while the remaining wage contracts, Wj,t, which cannot be re-optimized, are adjusted according to, Wj,t = pi χ t−1Wj,t−1 where pit = Pt/Pt−1 is home consumer price index inflation in period t and χ ∈ [0, 1] represents indexation of wage contract to CPI inflation. The first-order condition for optimal wage-setting w̃ can be written as, 0 = Et ∞∑ τ=0 βt+τξ τ wUc,t+τH d t+τ ( w̃t wt+τ )−θw τ∏ k=1 ( pit+k piχt+k−1 )θw θw − 1 θw w̃t∏τ k=1 pit+k piχt+k−1 − wt+τ Xw,t+τ (2.14) Here I have substituted the real wage rates, wt+τ ≡Wt+τ/Pt+τ and w̃t+τ ≡ W̃t+τ/Pt+τ , for nominal wages, where the optimal nominal wage rate W̃t is reset by the household in period t. According to the optimal wage-setting, the household prices its labor service j so that the sum of expected dis- counted marginal utility of the income from an additional unit of labor supply equals its expected sum of discounted marginal disutility from labor services. When ξw tends to zero, the household can reset its wages every period. 2.1.3 Wholesale firms At time t, the homogenous wholesale goods are produced by a perfectly competitive, representative firm. The firm produces goods by combining physical capital and labor aggregate according to the following technology, Yt = Za,tK α t (H d t ) 1−α (2.15) 26 2.1. The model where 0 < α < 1 and Kt denotes the time t home country’s physical capital services. Total factor productivity, Za,t, is an exogenous random variable. Physical capital stock, which is assumed to be owned by the wholesaler, and also finally owned by the household, evolves according to, Kt+1 = Φ ( It Kt ) Kt + (1− δ)Kt (2.16) Here, It denotes time t gross investment in the home country. The trans- formation function, Φ, characterizes the technology that transforms current investment into installed physical capital for use in the next period and it is continuously increasing and concave. 0 < δ < 1 is the physical rate of depreciation. In the following analysis, I use the specification as follows, Φ ( It Kt ) = It Kt − φ 2δ ( It Kt − δ )2 Parameter φ characterizes the elasticity of price of capital with respect to investment capital ratio. The wholesaler’s problem is to maximize the following expected dis- counted profits, Et +∞∑ τ=0 Qt,t+τ ( Pw,t+τYt+τ − PI,t+τIt+τ −Wt+τHdt+τ ) subjective to technology (2.15) and capital evolvement (2.16). Where Qt,t+τ is stochastic discount factor for nominal returns at period t + τ , Qt,t+τ ≡ βτ+t βt Uc,t+τPt Uc,tPt+τ . Pw,t, PI,t stand for the price of wholesales and the price of investment goods, respectively. The optimality condition for labor demand reads, wt = (1− α)YtpH,t HdtXp,t where Xp,t ≡ PH,tPw,t , the price mark-up for the wholesaler or the inverse of real marginal cost for intermediate producers. The shadow price for physical capital qk,t relates to marginal product of investment via, qk,t = [ Φ′(It/Kt) ]−1 pI,t Optimal investment decision is determined by the wholesaler’s Euler 27 2.1. The model equation, which reads, qk,t =Et { Λt,t+1 [ qk,t+1 ( (1− δ) + Φ ( It+1 Kt+1 ) − Φ′ ( It+1 Kt+1 ) It+1 Kt+1 ) + αYt+1pH,t+1 Kt+1Xp,t+1 ]} where Λt,t+1 is stochastic discount factor for real returns at period t + 1, Λt,t+1 ≡ βt+1βt Uc,t+1 Uc,t . In each period, the dividends from the wholesaler are retained revenues, which equal the firm’s total revenues minus wage bills and minus purchases of investment goods for use in the following period. Real dividends from the wholesale firm become, dw,t = pw,tYt − pI,tIt − wtHdt Notice that the profits from the wholesale-good sector may not always be positive. For instance, when the current shock is highly adverse while the expectation of future shocks is quite promising, then it’s optimal for the firm to issue debts or equities to finance its investment.18 However, in the steady state of the model, the profit, dw,t, is always positive and a small deviation from its steady state could rarely make profits negative when taking approximation around the steady state. Similarly to consumption composite, gross investment is also aggregated using local goods and imported goods, which has a form of, It = [ a 1 γI I I γI−1 γI H,t + (1− aI) 1 γI I γI−1 γI F,t ] γI γI−1 where aI represents the deterministic steady state share of investment ex- penditure on home goods, which has an analogous interpretation as aC . Minimizing the cost of a unit of investment, given the price of goods, implies that investment expenditure on home goods is, IH,t = aI(PH,t/PI,t) −γI It, and on foreign goods is, IF,t = (1− aI)(PF,t/PI,t)−γI It. Minimization investment cost implies the price of investment, PI,t = [ aIP 1−γI H,t + (1− aI)P 1−γIF,t ] 1 1−γI (2.17) 18Modigliani-Miller theorem holds in the current environment. 28 2.1. The model 2.1.4 Intermediate producers There exists a continuum measure of intermediate producers in both countries, each of whom employs wholesale goods to produce one type of variety j by differentiating the wholesale products into various brands. Each variety is not a perfect substitute for other varieties, and thereby, interme- diate producers have a monopoly power over consumers. The intermediate consumption bundle, YH,t, is aggregated from varieties, YH,j,t, j ∈ [0, 1], though the Dixit-Stiglitz technology, YH,t = (∫ 1 0 Y θp−1 θp H,j,t dj ) θp θp−1 (2.18) where θp > 1 is the elasticity of substitution between varieties. The higher the market competitiveness is, the larger θp becomes. Minimizing the cost of one unit of intermediate consumption bundle, the price reads, PH,t = (∫ 1 0 P 1−θp H,j,t dj ) 1 1−θp The demand for variety j becomes, YH,j,t = ( PH,j,t PH,t )−θp YH,t (2.19) Intermediate producer j obtains nominal profits in period t, Dp,j,t ≡ (PH,j,t − Pw,t) ( PH,j,t PH,t )−θp YH,t (2.20) Intermediate producers choose prices to optimize their discounted profits subject to the market demand for their varieties (2.19). Prices of interme- diates are set according to Calvo’s style. In each period, there is a fraction ξp of intermediate producers who can re-optimize their prices while the re- maining intermediate firms keep their prices unchanged. The intermediary j’s objective function follows as, max PH,j,t Et +∞∑ τ=0 [ ξτpQt,t+τDp,j,t+τ ] (2.21) The optimal price P̃H,j,t is set such that the sum of expected discounted 29 2.1. The model profits from selling intermediates equals the expected discounted cost from purchasing whole goods given the random duration of price readjustment, P̃H,j,t = θp θp − 1 Et {∑+∞ τ=0 ξ τ pQt,t+τYH,t+τP 1+θp H,t+τ/Xp,t+τ } Et {∑+∞ τ=0 ξ τ pQt,t+τYH,t+τP θp H,t+τ } (2.22) Summing up real dividends over all intermediate producers yields, dp,t ≡ ∫ 1 0 dp,j,tdj = pH,tYH,t(1− s p t Xp,t ) (2.23) with price dispersion in the intermediate variety market, spt ≡ ∫ 1 0 ( PH,j,t PH,t )−θp . 2.1.5 Government Following the literature and the practice in many central banks, mon- etary policy is represented by a Taylor rule, which is subject to stochastic financial shocks. Suppose that there are financial market shocks which affect the equilibrium nominal interest rate. According to the literature (McCal- lum (1999); Taylor and Williams (2010)), the Taylor rule takes a form of, log(Rt+1/R) = ρr log(Rt/R)+(1−ρr) [αpi log(pit/pi) + αy log(gdpt/gdp)]+r,t (2.24) where r,t denotes an innovation to the nominal interest rate Rt+1, which has an identical and independent normal distribution with mean zero. GDP in a country is defined as gdpt ≡ pH,tYH,t. ρr characterizes the persistence of nominal interest rate. αpi and αy represent the degree of response of interest rate to inflation and output gap. This simple monetary policy rule is designed to reflect the basic princi- ple of monetary policy of leaning against the wind of inflation and output movements. I exclude nominal exchange rates from the simple interest rate rule since exchange rates have a complete pass-through in the current envi- ronment; in other words, the CPI inflation has included all of information contained in the changes of nominal exchange rates. On the other hand, researchers find that interest rate rules reacting to nominal exchange rates are inferior to those excluding them in terms of welfare(Taylor and Williams 2010). When monetary authorities implement an interest rate rule, the quantity of money mt circulated in the financial market is endogenously determined 30 2.1. The model by the intertemporal substitution of money balance. Fiscal authorities al- ways maintain a balanced budget constraint. The seigniorage from issuing fiat money is redistributed in a lump-sum fashion to the domestic household in each period, trt = mt − mt−1 pit (2.25) 2.1.6 Market clearings and competitive equilibrium Market clearing conditions for home and foreign intermediate goods can be written as, CH,t + C ∗ H,t + IH,t + I ∗ H,t = YH,t (2.26) CF,t + C ∗ F,t + IF,t + I ∗ F,t = Y ∗ F,t (2.27) Market clearing conditions for international assets follow as, ψH,t+1 + ψ ∗ H,t+1 = 1 ψF,t+1 + ψ ∗ F,t+1 = 1 (2.28) BH,t+1 +B ∗ H,t+1 = 0 BF,t+1 +B ∗ F,t+1 = 0 (2.29) According to wage aggregation equation (2.6), real wage evolves as w1−θwt = ξww 1−θw t−1 (pi χ t−1/pit) 1−θw + (1− ξw)w̃1−θwt (2.30) Labor market clears, Ht = ∫ 1 0 Hj,tdj = H d t ∫ 1 0 ( wj,t wt )−θw dj = Hdt s w t (2.31) with wage dispersion swt in the labor market, which evolves as, swt = (1− ξw) ( w̃t wt )−θw + ξw ( wt−1 wt )−θw ( pit piχt−1 )θw swt−1 (2.32) The evolvement for the aggregate price has a form of, 1 = ξppi θp−1 H,t + (1− ξp)p̃1−θpH,t (2.33) with piH,t is PPI inflation and p̃H,t ≡ P̃H,j,tPH,t when firm j re-optimizes its price P̃H,j,t. 31 2.1. The model Price dispersion is measured by spt , which evolves as, spt = (1− ξp)p̃−θpH,t + ξppiθpH,tspt−1 (2.34) Supply of wholesale goods is equal to demand for wholesale goods, Yt = ∫ 1 0 YH,j,tdj = s p tYH,t (2.35) Total dividends collected from all producers within a country read, dt = pH,tYH,t − (1− α)pH,tYH,t s p t Xp,t − pI,tIt = pH,tYH,t ( 1− (1− α) s p t Xp,t − pI,tIt pH,tYH,t ) (2.36) Notice from the last equality in (2.36) that dividends are a time-varying fraction of GDP. When prices of goods are flexible, the labor income share, (1 − α) s p t Xp,t , is constant. If an economy doesn’t have capital accumulation, investment-GDP ratio, pI,tIt pH,tYH,t , is zero, and therefore, dividend-GDP ratio is constant. In the economy studied here, dividend-GDP ratio varies over time. CPI inflation relates PPI inflation through the following way, pit = pH,t−1 pH,t piH,t (2.37) Let κt ≡ {Kt, K∗t , Rt, R∗t , swt , sw∗t , spt , sp∗t , ZAt , ZEt , } denote the state of the world at time t, where a set of exogenous variables ZEt ≡{ Za,t, Z ∗ a,t, r,t, ∗ r,t } , and a set of portfolios, ZAt ≡ { ψH,t, ψF,t, ψ ∗ H,t, ψ ∗ F,t, BH,t, BF,t, B ∗ H,t, B ∗ F,t } A competitive equilibrium is a set of home household’s decision rules, C(κt), CH(κt), CF (κt), H(κt), m(κt), w̃(κt), portfolio choices ψH(κt), ψF (κt), BH(κt), BF (κt); a set of foreign household’s decision rules, C ∗(κt), C∗H(κt), C∗F (κt), H ∗(κt), m∗(κt), w̃∗(κt), portfolio choices ψ∗H(κt), ψ ∗ F (κt), B ∗ H(κt), B∗F (κt); a set of home firms’ decision rules, K(κt), H d(κt), YH(κt), I(κt), IH(κt), IF (κt), d(κt), p̃H(κt); a set of foreign firms’ decision rules, K ∗(κt), Hd∗(κt), Y ∗F (κt), I ∗(κt), I∗H(κt), I ∗ F (κt), d ∗(κt), p̃∗H(κt); a set of price func- tions q(κt), q ∗(κt), pb(κt), pb∗(κt), qk(κt), q∗k,t(κt), pH(κt), p ∗ F (κt), w(κt), 32 2.2. Steady state equity portfolios w∗(κt), R(κt), R∗(κt), s(κt), ri(κt), for i = a1, a2, b1, b2, sw(κt), sw∗(κt) , sp(κt), s p∗(κt), pi(κt), pi∗(κt), piH(κt) and pi∗F (κt); such that (1) households’ decision rules solve the households’ problems; (2) firms’ decision rules solve the firms’ problems; (3) the appropriate market clearing conditions hold. 2.2 Steady state equity portfolios 2.2.1 Steady states and log-linearizations In the deterministic steady state, all variables are unchanged in both countries. Let a variable without time subscript denote its deterministic steady state, and a variable with a hat stands for its log deviation from its steady sate, i.e., x̂t ≡ log(xt)− log(x). Since there are two types of shocks, a technological shock and a monetary shock, and two types of assets, equities and bonds, in the world economy, asset market is complete up to a first- order approximation.19 International risk sharing condition (Backus-Smith condition) can be written as,20 σ(Ĉt − Ĉ∗t ) = ŝt (2.38) The terms of trade, which is defined as tott ≡ PF,tPH,t , relates the real exchange rate via ŝt = (2aC − 1)t̂ott up to a first-order approximation.21 In the symmetric steady state, we have ψH = ψ ∗ F , ψF = ψ ∗ H = 1 − ψH , BH = B ∗ F , and BF = B ∗ H = −BH . Now let’s consider a static model in which the economy starts from its long-run steady state and households face uncertainty in the next period. Portfolio choice decisions are made in the current period. One can show that there exists a unique portfolio 19One can check that the spanning condition holds since the number of shocks is the same as the number of assets and that the rank condition holds as long as returns to assets are not perfectly correlated with each other. The assets traded in the international market here satisfy these two conditions. 20Notice that in the benchmark model with non-separate preference between consump- tion and real money balance, the log-linearized marginal utility of consumption contains real money balance and consumption. However, the share of consumption, ω, in the pref- erence is close to one, 0.9977 in the calibration. Therefore, for exposition convenience but without loss of accuracy, I get rid of real money balance here. When preference is separable between consumption and real money balance, equation (2.38) holds exactly. 21Normalizing home (foreign) prices for imports and exports by the home (foreign) aggregate price in equation (2.2) and its foreign counterpart, applying the law of one price, substituting terms of trade and real exchange rates into these two price equations, we have tott = [ aCs 1−γ t −(1−aC) aC−(1−aC)s1−γt ] 1 1−γ . 33 2.2. Steady state equity portfolios that satisfies the following static budget constraints for consumption and is consistent with the linearized risk sharing condition (2.38), 22 Ct = wtH d t + dtψH + d ∗ t (1− ψH)st + rbtBH − rb∗t BHst C∗t = w ∗ tH d∗ t + dt(1− ψH) st + d∗tψH − rbtBH st + rb∗t BH Taking a first-order log-linear approximation of static budget constraints around their steady state, then subtracting foreign consumption from home consumption, and finally replacing relative consumption with real exchange rate in the Backus-Smith condition (2.38), I obtain accordingly, 1 σ ŝt = d C (2ψH−1)d̂Rt + wHd C ŵH R t +2 [ (1− ψH) d C − BH C rb ] ŝt+2r bBH C r̂bRt (2.39) with relative dividends d̂Rt ≡ d̂t − d̂∗t , relative labor income ŵH R t ≡ (ŵt + Ĥdt )− (ŵ∗t + Ĥd∗t ), and relative interest payments r̂bRt ≡ r̂bt − r̂b∗t . Let labor income share be ls ≡ wHdC and equity income share dC = 1− ls. Projecting all relevant variables in the relative budget constraint (2.39) onto real exchange rates, and then subtracting the projection terms from the relative budget constraint, consequently, we have, (1−ls)(2ψH−1)(d̂Rt −βd̂R,ŝŝt)+ls(ŵH R−β ŵH R ,ŝ ŝt)+2 BHr b C (r̂bRt −βr̂bR,ŝŝt) = 0 (2.40) where βx̂,ŝ ≡ cov(x̂t,ŝt)var(ŝt) denotes the coefficient in the projection of x̂t onto ŝt. Notations cov and var stand for covariance and variance, respectively. Multiplying both sides of equation (2.40) by (d̂Rt − βd̂R,ŝŝt) and taking ex- pectations conditional on information up to period t− 1, yields, ψH = 1 2︸︷︷︸ Market portfolio + (−1)1 2 ls 1− lsβ s w,d︸ ︷︷ ︸ Human capital hedge + (−1)r bBH d βsr,d︸ ︷︷ ︸ Exchange rate hedge (2.41) with βsw,d ≡ cov(ŵH R − β ŵH R ,ŝ ŝt, d̂ R t − βd̂R,ŝŝt) var(d̂Rt − βd̂R,ŝŝt) 22Notice that the portfolio choice is constant in a linearized dynamic system. One can also apply this result directly to obtain equation (2.39). 34 2.2. Steady state equity portfolios βsr,d ≡ cov(r̂bRt − βr̂bR,ŝŝt, d̂Rt − βd̂R,ŝŝt) var(d̂Rt − βd̂R,ŝŝt) In the log-linearized model, optimal shares of home equities held by the home resident are determined by three terms. The first one is the market portfolio, in which one half comes from the home equities and the other half from foreign equities since both countries are symmetric. The market port- folio reflects the diversification motive emphasized by Baxter and Jermann (1997). The second term, human capital hedge, captures how well the home equity could hedge home human capital income risk. This term is expressed as a relative covariance between the component of human capital income orthogonal to real exchange rates and the component of dividends of equi- ties orthogonal to real exchange rates. The last term, a relative covariance between the component of interest payments on bonds orthogonal to real exchange rates and the component of dividends orthogonal to real exchange rates, captures the relationship between bond assets and equity assets. No- tice that bond position BH shows up in home equity holdings, but this is not the trouble here.23 The reason I use this expression is that we could see clearly the roles of equity assets in hedging human capital income risk and of bond assets in determining home equity bias. Following the convention in the literature (Engel and Matsumoto (2009a)), I label the last term in the home equity holding (2.41) as exchange rate hedge since this term disap- pears when relative bond returns are perfectly correlated with real exchange rates. There are several points worth of making in equation (2.41). When there’s no non-diversifiable income, i.e., labor income or income from non- tradable goods, and relative bond returns are uncorrelated with relative equity returns conditional on real exchange rates, households in both coun- tries then perfectly diversify their income risk and hold only the market portfolio (see, Lucas (1982), which makes the share of home equities equal to one half.24 When non-diversifiable income in an economy becomes unig- norable, households in both countries have to take it into account and hedge 23We can obtain ψH andBH by projecting both sides of equation (2.39) onto the relative dividends from equities d̂Rt and onto the relative interest payments r̂ bR t respectively and then use the same procedure in the main text to get expressions for ψH and BH . See also, Coeurdacier and Gourinchas (2011). 24In a two Lucas trees world economy, in which both trees produce the same kind of fruit, households hold exactly the market portfolio since risk sharing implies households fully diversify idiosyncratic income shocks via choosing the market portfolio. This result is independent of preference parameters and of correlation between productivity shocks to trees. 35 2.2. Steady state equity portfolios this income risk using assets traded in the international market. If the rel- ative returns on human capital is negatively correlated with relative equity returns conditional on real exchange rates, the motive for hedging human capital income risk leads the home household to hold more home equities since home equities pay more when labor income in the home country is low, which implies the term of human capital hedge in equation (2.41) is positive. When the correlation between relative returns on bonds and relative returns on equities is negligible, the motivation for hedging human capital income risk completely pins down how many home equities the home household holds. Home bias is present when the term human capital hedge is positive. Nevertheless, even if human capital hedge term in equation (2.41) is positive, the types of bonds available in the international asset market are equally important in determining home equity holdings. Consider two simple model economies, in which both markets are complete, and the only differ- ence comes from the types of bonds, for instance, one is with real consols whose payoffs are in local consumption goods and the other with nominal consols whose payoffs are denominated in national currency. Obviously, the motives for holding market portfolio and for hedging labor income risk are the same in both countries since asset market is complete, but the motive for hedging exchange rate risk is different. In the country with real consols, relative bond returns are perfectly correlated with real exchange rate and thereby the exchange rate hedge term is zero under any type of shocks while this term in the economy with nominal consols might not be zero and it depends on types of shocks. 36 2.2 . S tead y state eq u ity p ortfolios Table 2.1: Relative covariances of relative returns conditional on RER: Bonds Variable USA UK Japan Germany France Canada Italy Denmark Spain βsr,d 0.00342 0.0471 0.0250 0.00561 0.0132 0.0344** 0.00417 0.0458*** -0.0152 (0.0310) (0.0288) (0.0164) (0.00717) (0.00903) (0.0153) (0.00958) (0.0105) (0.0101) Observations 61 61 61 61 61 61 61 61 61 Adj. R2 0.0002 0.069 0.042 -0.009 0.012 0.095 -0.015 0.152 0.015 Note: Robust standard errors are in parentheses. *** stands for a significance p < 0.01, ** for p < 0.05 and * for p < 0.10. Foreign returns are constructed through a weighted average of home currency real returns on foreign assets. The weight is bilateral trade volume. The return differential is defined as home real return minus the weighted foreign returns. The effective real exchange rate is defined as a weighted average of bilateral real exchange rates using bilateral trade volume as a weight. Then for each type of assets, regressing the return differentials on changes of effective real exchange rates and obtain the residuals, which are orthogonal to the effective real exchange rates. Next regressing the residuals from the bond return differential regression on the residuals from the stock return differential regression yields the coefficient reported above. Returns on bond are the yields to 5-year government bonds. The appendix provides detailed information on data source and estimation method. 37 2.2 . S tead y state eq u ity p ortfolios Table 2.2: Relative covariances of relative returns conditional on RER: Human capital Variable USA UK Japan Germany France Canada Italy Denmark Spain βsw,d -0.247*** 0.00814 -0.0848 0.0979 -0.0540 0.123 -0.297* -0.170 -0.331*** (0.088) (0.150) (0.0995) (0.0804) (0.111) (0.0778) (0.160) (0.123) (0.104) Observations 61 61 61 61 61 61 61 61 61 Adj. R2 0.094 -0.017 -0.004 0.009 -0.011 0.022 0.044 0.019 0.130 Note: Robust standard errors are in parentheses. *** stands for a significance p < 0.01, ** for p < 0.05 and * for p < 0.10. Foreign returns are constructed through a weighted average of home currency real returns on foreign assets. The weight is bilateral trade volume. The return differential is defined as home real return minus the weighted foreign returns. The effective real exchange rate is defined as a weighted average of bilateral real exchange rates using bilateral trade volume as a weight. Then for each type of assets, regressing the return differentials on changes of effective real exchange rates and obtain the residuals, which are orthogonal to the effective real exchange rates. Next regressing the residuals from the bond return differential regression on the residuals from the stock return differential regression yields the coefficient reported above. Returns on equity are from stock index returns which include dividends and capital gains. Returns on human capital are estimated from data. The appendix provides detailed information on data source and estimation method. 38 2.2. Steady state equity portfolios 2.2.2 Empirics on international asset markets As emphasized by van Wincoop and Warnock (2006), “any explanation for portfolio home bias in the context of GE[general equilibrium] models only makes sense if the resulting covariance ratio is consistent with the data”. Be- fore going through the calibration of each term in equation (2.41), let’s take a look at international asset markets first. Notice that the covariance ratios βsw,d and β s r,d don’t depend on model specifications and can be obtained di- rectly from the data. To obtain these covariance ratios, we need to know returns on human capital and returns on bonds and equities. The data used in this subsection is quarterly data. Data sources and the estimation method are described in the appendix. Since long-term bonds are the most widely traded bond assets in the bond asset category, I report the covariance ratio for bonds, βsr,d, for 5-year government bonds. 25 Table 2.1 shows that in a sub-sample of OECD countries, seven out of nine country pairs (home v.s. foreign country pair) have zero covariance ratio βsr,d. This covariance ratio in the remaining two country pairs, Canada and Denmark, is statistically significantly different from zero, but it is close to zero economically.26 Now let’s move to the covariance ratio for human capital, βsw,d. Table 2.2 presents the estimated results for βsw,d. For 6 out of 9 country pairs, the human capi- tal covariance ratio, βsw,d, is negative. Particularly, this ratio is significantly negative for the USA, Italy and Spain, with −24.7%, −29.7% and −33.1%, respectively. 25The covariance ratio βsr,d for other types of bonds, i.e., 3-month treasury bill, 2-year government bond, and 10-year government bond, is quite similar to 5-year government bond. The results are available upon request. 26This covariance ratio might not be accurately mapped from model to data since the volatilities of real exchange rates and of returns on stocks are far larger than the volatilities of returns on bonds in the data, while in general equilibrium models like the one in this chapter, volatilities of real exchange rates and of equity returns are similar to the volatility of bonds. Thereby, a close to zero covariance ratio βsr,d in the data might be implied from extremely high volatility of real exchange rates that dominates the relative returns on bonds denominated in the same currency while relative returns on bonds denominated in national currency don’t necessarily hedge real exchange rate fluctuations as long as their volatility is far lower than real exchange rates. However, following the literature, i.e., Coeurdacier (2009) and others, I adopt the interpretation of bonds hedging real exchange rate risk well when the covariance ratio βsr,d is close to zero, ignoring the volatility wedge between models and data. However, I present second moments for bond return differentials (local return differentials) across borders in the data in the following section to get rid of volatile real exchange rates in the data. 39 2.3. Calibration 2.3 Calibration Since this chapter focuses not only on the steady state portfolios but also on first-order portfolios, while data on first-order portfolios is available at yearly frequency, thereby, I take use of annual data instead of quarterly. Variables and their sources are given in the appendix. The parameters of the model are chosen to be close to the standard choice for the values of parameters in literature. Parameters and their values are listed in table 2.3. Relative risk aversion parameter in the benchmark model is chosen to be σ = 2, which is adopted by Corsetti, Dedola and Leduc (2008) and Coeur- dacier, Kollmann and Martin (2010). Following Obstfeld and Rogoff (2001) and Christiano, Eichenbaum and Evans (2005), the labor supply elasticity is set to be unity, which implies η = 7/3. The estimation of elasticity of sub- stitution between imports and exports is ambiguous in literature. Corsetti, Dedola and Leduc (2008), who calibrate their model to the U.S. relative to a set of OECD countries on annual data, estimated that γ = 0.85, while Bernard et al. (2003) find that the elasticity equals 4 based on trade data. Here I assume that γ = γI = 2 in the benchmark model, as in Coeurdacier et al. (2010). I also consider a large range of trade elasticity and relative risk aversion to check the robustness of the results when necessary. The interest rate elasticity of money demand is chosen to be θ = 0.5, which is taken from Hoffman, Rasche and Tieslau (1995), whose estimation is based on a cross-country study. Empirical evidence suggests that the elasticity of substitution between differentiated labor services, θw, is between 2 and 6, which implies the wage markup rate is between 20% and 100%. Erceg, Henderson and Levin (2000) use the value of θ = 4. I take θ = 6 in the benchmark model. Elasticity of substitution between goods varieties is set at θp = 11, implying a 10% price markup. The wage contract duration parameter ξw = 0.82 4 implies an average contract duration of 1/(1 − 0.824) = 1.8 years, around 7 quarters, which is slightly higher than the estimate in Christiano, Eichenbaum and Evans (2005).27 Recently, Barattieri, Basu and Gottschalk (2010) find that the probability that an individual will experience a nominal wage change is at most 18% in the average quarter (a minimal duration of 5.6 quarters), which is based on Survey of Income and Program Participation for the period 1996 − 1999 in the United States. Price duration parameter is set at ξp = 0.754, which implies average duration of price is around 1.5 years, which is 27Wage contract duration is 5 quarters when there is no habit formation in their model. 40 2.3. Calibration Table 2.3: Parameter values Parameter Description Value Parameters in utility function β Discount factor 0.96 σ Risk aversion 2 η Frisch labor elasticity parameter 7/3 θ Interest rate elasticity of money demand 0.5 γ = γI Elasticity of substitution between home and foreign goods 2 aC = aI Share of home-traded goods 0.85 θw Elasticity of substitution between labor services 6 θp Elasticity of substitution between interme- diate goods 11 ξw Wage contract duration parameter 0.82 4 ξp Price duration parameter 0.75 4 ω Share of consumption 0.9977 η0 Coefficient for leisure 3.6405 χ Wage indexation 0 Parameters in production δ Depreciation rate 0.1 α Capital share 0.30 φ Curvature of adjustment cost function 0.1 Parameters in shock processes ρa Persistence of TFP shocks 0.905 ρr Persistence of money supply shocks 0.80 σa Standard deviation of TFP shocks 1.36% σr Standard deviation of interest rate shocks 1.36% 41 2.3. Calibration in the reasonable range of price stickiness. Both consumption and investment expenditure exhibit home bias in the U.S. and developed economies, and the share of investment expenditure on imports is higher than that of consumption. Following the international business cycle literature, i.e., Backus, Kehoe and Kydland (1995), I choose consumption and investment home bias to be aC = aI = 0.85, which implies imports-GDP ratio is 15% in the steady state. Corsetti, Dedola and Leduc (2008) assume that the share of home-traded goods in tradable consumption basket is 0.72 and the share of non-traded goods is 0.45, which implies imports-GDP ratio is (1− 0.72)× (1− 0.45) = 15.4%. Since the data used in this chapter is yearly data, the discount factor, summarized by function β, is set at 0.96 in the steady state, which implies that an annualized risk-free rate of 4%. The data in the sample shows that average fixed investment-GDP ratio is 20%. Then the share of capital in production function is calibrated as α = 0.3. As is well known, without capital adjustment cost, the volatility of investment in the business cycle model is quite large compared to that of GDP. Thereby, the elasticity of price of capital with respect to investment- capital ratio is set such that the investment volatility relative to GDP volatil- ity in the model matches that in the data. Finally, I set the depreciation rate of capital equal to 10% annually. There are two other free parameters to be pinned down. One is η0, which is chosen such that the labor supply at the steady state is H = 0.3, implying that the household will supply 30% of total hours to the labor market. The second is ω, which is set to match the yearly data on M1-GDP ratio, 15.2%, in the United States. I collect a data set covering a group of 10 OECD countries, including U.S., Japan, U.K., Germany, France, Italy, Canada, Spain, Austria and Den- mark. Once a home country is chosen, the foreign country is constructed through aggregating the rest of countries in the data sample using bilateral trade as weight. The appendix shows the data sources as well as the con- struction method for the foreign country. I estimate a VAR process of TFP using maximum likelihood for the U.S. and the constructed foreign country aggregates,[ log(Za,t) log(Z∗a,t) ] = [ 0.905 0 0 0.905 ] [ log(Za,t−1) log(Z∗a,t−1) ] + [ a,t ∗a,t ] with standard deviation of innovations σa = σ∗a = 1.36%, and cross corre- lation corr(a, ∗ a) = 0.26. 42 2.4. Numerical results According to the Taylor principle in literature (say, Taylor and Williams, 2011), the coefficient αpi in the Taylor rule (2.24) should be larger than unity to avoid indeterminacy in equilibrium. I set αpi = 1.5 and αy = 0.1, which are common used values in the Taylor rule. Assume that innovations in the Taylor for each country have a normal distribution with mean zero and constant standard deviation. Given these two parameter values, I estimate the persistence parameter, ρr, in the Taylor rule (2.24) for the Home and Foreign countries respectively based on the U.S.-foreign country pair and obtain ρr = 0.80 and the standard deviation σr = 1.36% with cross-country correlation corr(r, ∗ r) = 0.5. 28 2.4 Numerical results The usual solution method for rational expectations models can be ap- plicable here for a linearized dynamic system. As is well-known, portfolio decisions in the linearized model are indeterminate and therefore, we need to take a second-order approximation to the Euler conditions for portfolios to pin down the zeroth order portfolios. Devereux and Sutherland (2011a) provide an algorithm to calculate the zeroth-order optimal portfolios. Table 2.4: Optimal home equities held by home households (nominal con- sols) γ = 0.50 γ = 0.85 γ = 1 γ = 1.1 γ = 1.5 γ = 2 γ = 3 σ = 0.5 -4.38 -12.64 42.53 8.68 0.96 -0.25 -0.89 σ = 1 -0.13 0.42 0.80 1.15 5.04 -10.16 -3.12 σ = 1.5 0.30 0.85 1.20 1.50 3.74 54.89 -4.71 σ = 2 0.46 1.00 1.33 1.60 3.47 17.02 -5.91 σ = 5 0.70 1.21 1.51 1.74 3.19 8.81 -9.49 σ = 10 0.76 1.27 1.55 1.78 3.12 7.76 -11.49 Note: This table reports the optimal share of home equities held by home households. σ is the coefficient of relative risk aversion and γ is the elasticity of substitution between imports and exports. Each unit of consol bond pays one unit of local currency each period. When γ = 2, σ = 2, home residents hold 1702% of foreign equities. 28The coefficients in the interest rate rule are consistent with the rule studied in Chari et al. (2002) and Christiano, Trabandt and Walentin (2010). The latter authors obtain the interest rate rule via a Bayesian estimation based on quarterly data in U.S., in which ρr = 0.87, αpi = 1.43 and αy = 0.07. 43 2.4. Numerical results Table 2.5: Optimal bond assets-GDP ratios (nominal consols) γ = 0.50 γ = 0.85 γ = 1 γ = 1.1 γ = 1.5 γ = 2 γ = 3 σ = 0.5 1.76 4.35 13.07 2.40 0.01 0.37 0.53 σ = 1 0.40 0.19 0.06 0.06 1.31 3.45 1.19 σ = 1.5 0.25 0.04 0.08 0.18 0.92 17.06 1.68 σ = 2 0.19 0.02 0.13 0.23 0.84 5.14 2.05 σ = 5 0.10 0.10 0.20 0.29 0.77 2.56 3.16 σ = 10 0.07 0.12 0.23 0.30 0.76 2.24 3.78 Note: This table reports the ratio of bond assets to its GDP. σ is the coeffi- cient of relative risk aversion and γ is the elasticity of substitution between imports and exports. Each unit of consol bonds pays one unit of local cur- rency each period. Table 2.6: Covariance ratios for hedging human capital income risk (nominal consols) γ = 0.50 γ = 0.85 γ = 1 γ = 1.1 γ = 1.5 γ = 2 γ = 3 σ = 0.5 -0.23 -0.23 -0.23 -0.23 -0.23 -0.23 -0.24 σ = 1 -0.22 -0.23 -0.23 -0.23 -0.23 -0.23 -0.24 σ = 1.5 -0.22 -0.22 -0.22 -0.23 -0.24 -0.23 -0.24 σ = 2 -0.22 -0.22 -0.22 -0.23 -0.27 -0.23 -0.23 σ = 5 -0.22 -0.22 -0.22 -0.22 -0.23 -0.23 -0.23 σ = 10 -0.22 -0.22 -0.22 -0.22 -0.22 -0.23 -0.23 Note: This table reports the ratio of covariance between human capital re- turns orthogonal to real exchange rates and equity returns orthogonal to real exchange rates to the variance of equity returns orthogonal to real exchange rates, βsw,d, in the main text. σ is the coefficient of relative risk aversion and γ is the elasticity of substitution between imports and exports. Nominal consols are traded in the international bond markets. 44 2.4. Numerical results Table 2.7: Covariance ratios for hedging real exchange rate risk (nominal consols) γ = 0.50 γ = 0.85 γ = 1 γ = 1.1 γ = 1.5 γ = 2 γ = 3 σ = 0.5 12.48 12.99 13.25 13.46 -14.87 13.23 14.22 σ = 1 10.92 10.74 7.90 16.61 13.06 13.36 14.11 σ = 1.5 10.19 5.81 14.65 13.29 12.72 13.17 13.98 σ = 2 9.69 24.67 13.33 12.79 12.28 13.05 13.87 σ = 5 8.38 13.11 12.29 12.16 12.33 12.80 13.73 σ = 10 7.53 12.51 12.04 11.98 12.22 12.72 13.67 Note: This table reports the ratio of covariance between human capital re- turns orthogonal to real exchange rates and equity returns orthogonal to real exchange rates to the variance of equity returns orthogonal to real exchange rates, βsr,d, in the main text. σ is the coefficient of relative risk aversion and γ is the elasticity of substitution between imports and exports.Nominal consols are traded in the international bond markets. 2.4.1 A model with nominal consols Can a standard two-country monetary business cycle model with equities and nominal consols generate reasonable home bias in equity? The answer is it depends. Particularly, it depends critically on two preference parameters: the coefficient of relative risk aversion and trade elasticity. In a model with nominal consols, interest payments on bond assets are denominated in national currency (i.e., one dollar interest payment by one unit of home bonds and one euro by one unit of foreign bonds). Express the interest payment in terms of national real consumption and obtain, rbt = 1 Pt and rb∗t = 1 P ∗t . Table 2.4 presents optimal home equities held by the home resident at steady state. The number in each cell in the table denotes the optimal shares of home equities for different combination of the coefficient of relative risk aversion and trade elasticity. Taking σ = 2 and γ = 2 for example, which is wildly used in the literature, the home resident holds 1702% of home equities, an extremely unreasonable value. As many authors have pointed out, there are no consensus on the values for relative risk aversion and trade elasticity in the open-macroeconomics literature on the one hand (see, Corsetti, Dedola and Leduc 2008), and on the other hand, these two parameters seem to be important in determining asset holdings in the international macro-finance literature. For robustness check, table 2.4 lists home equity shares under a large range of relative risk 45 2.4. Numerical results aversion, σ ∈ [0.5, 10], and trade elasticity, γ ∈ [0.5, 3]. The numerical experiment illustrates that the share of home equities is extremely sensitive to relative risk aversion σ and trade elasticity γ. There’s no clear increase or decrease pattern with the changes of these two parameters. On the side of bonds, table (2.5) illustrates the ratio of external bond assets to GDP. The bond assets-GDP ratio also swings a lot across different preference parameters. A following question is how human capital hedge motive and exchange rate hedge motive contribute to the optimal shares of home equities. I then decompose the equity portfolio according to equation (2.41) and calcu- late each component. Combining steady state levels of labor income share, dividend-interest payment ratio and optimal bond holdings, with uncondi- tional covariance ratios βsw,d and β s r,d delivers all terms in (2.41). Table 2.6 reports the covariance ratio βsw,d. For a wide range of parameter values, this ratio remains almost the same, around −0.22, which is consistent with the data, implying that the home household holds about 40% of home equity shares for hedging human capital income risk. Clearly, the large swings of optimal shares of home equities come from the exchange rate hedge motive. Table 2.7 shows the covariance ratio for hedging real exchange rate risk. We can see that nominal consols don’t hedge real exchange rate risk well since βsr,d has a very large value and vary largely with the preference parameters. Do monetary or technological shocks change the responses of optimal equity shares hedging human capital income risk? The results (not reported here) show that either monetary shocks or technological shocks doesn’t change the covariance ratio βsw,d much. 2.4.2 The benchmark model with inflation-indexed bonds Inflation-indexed bonds Until now, I have analyzed different components of optimal shares of home equities held by the home household when nominal consols are traded in the international bond markets, and find that the incentive for hedging human capital income risk is quite stable and is also consistent with what we have observed in the data. Bonds seem to be a key to obtain a ro- bust home equity shares across different preference parameters and across diverse shocks. Recently, inflation-indexed government bonds have become available in many countries, including Canada, France, Japan, U.K. and U.S., and the stock of these bonds grows rapidly in bond markets. Britain first issued inflation-indexed government bonds in the 1980s and the U.S. 46 2.4. Numerical results government first introduced Treasury Inflation-protected Securities (TIPS) in 1997. Up to 2008, the outstanding supply of inflation-indexed bond in the U.K. grew to around 30% of British public debts and TIPS accounted for 10% of marketable U.S. Treasury debt (Campbell, Shiller and Viceira 2009). Inflation-indexed bonds in these countries usually have long matu- rities and naturally, they are corresponding to the inflation-indexed consols in the model studied here. Beyond the evidence above, a natural concern appears. It seems that equity home bias has existed for a long time, even before the first issuance of inflation-linked bonds. However, notice that, in the early years, international capital flows were significantly restricted and consequently we observed eq- uity home bias in decades ago because of lack of financial integration. Equity home bias becomes a puzzle only recently when international financial mar- kets are integrated. The time of issuance of inflation linked bonds was quite consistent with the start of massive financial market integration, which im- plies that inflation linked bonds could be an endogenous response to financial market integration.29 Accordingly, Inflation linked bond seem a good and reasonable candidate hedging against real exchange rate risk. Before going through how well inflation-indexed bonds hedge real ex- change rate risk, let’s begin with real consol bonds, which pay a fixed amount of consumption composite to claimants each period. The key in- sight for hedging real exchange rate risk comes from the relative interest payments. The relative interest payment on the real bond is rrRt ≡ 1st and its log-linearization version reads, r̂rRt ≡ −ŝt (2.42) which implies that the relative interest payment r̂rRt perfectly moves with real exchange rate ŝt; in another word, real consols could perfectly hedge real exchange rate risk. This point has confirmed by Coeurdacier et al. (2010). Table 2.8 reports gross returns and relative interest payments on bonds in detail. When nominal consols are traded in the international asset markets, the relative interest payment rcRt by nominal consols in period t is r cR t ≡ P ∗ t Pt 1 st . Log-linearizing this relative interest payment yields, r̂cRt ≡ (P̂ ∗t − P̂t)− ŝt (2.43) Notice that relative prices show up in the relative interest payment above 29I thank the external examiner, Mathias Hoffmann, brought up this concern. 47 2.4. Numerical results besides real exchange rates. We know that relative prices varies a lot with monetary shocks (non-stationary nominal prices) and relative interest pay- ment r̂cRt disconnects with real exchange rate ŝt. Thereby nominal consols are not good for hedging real exchange rate risk, which is the case in the model with nominal consols. Nevertheless, when monetary shocks are not important, or there are not monetary shocks, relative prices change mildly and nominal consols could be helpful to hedge real exchange rate risk. This point will be confirmed in the next section of incomplete asset markets when I shut down nominal rigidities or monetary shocks. Now let me describe inflation-indexed consols. Suppose that the economy starts from its steady state, say, period 0. Assume that the face value for one unit of consols is set at P b and the preset interest rate is R − 1 per period. Without inflation indexation, interest payment is P b(R − 1) dollars per period. Now suppose that interest rate is fully indexed to past inflation. Then one unit of such consols delivers, P b(R − 1)Πt−1s=0pis, with pi0 = 1, dollars to claimants in period t. This payment can be simplified as P b(R − 1)Pt−1P0 = (R − 1)Pt−1, where I have used the assumption of P b = P0 = 1 for simplicity. In many cases, bond issuers negotiate interest rates on long-term bonds with bond holders. Thereby instead of paying fixed interest rate R − 1 per period, bond issuers promise to pay time- varying interest rates. Assume that the interest rate on consols varies with short-term interest rate, say Rt − 1, in period t, determined in the previous period t − 1. Thereby, each unit of this type of inflation-indexed bond delivers (Rt − 1)Pt−1 dollars in period t. The relative real interest payment on inflation-indexed bond can then be written as, rdRt ≡ Rt − 1 R∗t − 1 pi∗t pit 1 st (2.44) Take a log-linearization of rdRt in the equation above and read, r̂dRt ≡ ( p̂i∗t − 1 1− β R̂ ∗ t ) − ( p̂it − 1 1− β R̂t ) − ŝt (2.45) Notice that, besides real exchange rates, there are two extra terms, in- flation rate adjusted by short-term interest rate in each country, show up in the equation above. It is this relative inflation rates that reduce the volatility of relative interest payment r̂dRt when the elasticity of intertem- poral substitution is low. With low EIS (or high risk aversion), households have higher incentive to smooth consumption, which implies nominal in- 48 2.4. Numerical results Table 2.8: Types of bonds Bond types Gross real return (Home) Relative interest payment ( HomeForeign ) Real consol P b(R−1)Pt+P bt P bt−1 1 pit P b(R−1) P b(R−1) 1 st = 1st One-period nominal bond Rt pit Rt−1 R∗t−1 1 Et Nominal consol P b(R−1)+P bt P bt−1 1 pit P b(R−1) P b(R−1) P∗t Pt 1 st = P∗t Pt 1 st Inflation- indexed bond P b(Rt−1)Pt−1+P bt P bt−1 1 pit P b(Rt−1) P b(R∗t−1) pi∗t pit 1 st = Rt−1R∗t−1 pi∗t pit 1 st terest rates respond positively with expected inflation rates, which in turn reduces the volatility of the terms in the brackets in equation (2.45). Conse- quently, inflation-indexed bonds could hedge real exchange rate risk pretty well. In the following analysis of portfolio dynamics, I make use of this type of inflation-indexed bond as the benchmark instead of nominal consols to study international portfolio choices. In order to get a sense of how bonds are different in terms of relative interest payments, I plot a simulation path of 125 periods based on the benchmark calibration. Figure 2.1 shows relative interest payments to dif- ferent bonds, which are expressed in units of local consumption goods r̂iRt +ŝt with i = r, c, d. Relative interest payments on inflation-indexed bonds move very tightly around those on real bonds. Nevertheless, relative interest pay- ments on nominal bonds, including nominal consols and one-period nominal bonds, have a large volatility compared with inflation-indexed bonds since monetary shocks have persistent effects on nominal prices. Given the benchmark calibration, I calculate the steady state home equity holdings by home households. Table 2.9 show that with inflation- indexed consols traded internationally, households in either country opti- mally hold around 90% of their domestic equity shares. Compared with the model with nominal bonds, the equilibrium equity shares in the model with inflation-indexed bonds are quite robust across different values of trade elasticity and elasticity of intertemporal substitution. How well does domestic equity hedge domestic labor income risk, con- ditional on real exchange rates? Does the inflation-indexed bond hedge real exchange rate risk? Following the decomposition of the steady state equity portfolio in equation (2.41), it shows that the covariance ratio for hedging la- 49 2.4. Numerical results 20 40 60 80 100 120 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 A simulation path Relative real interest payments Benchmark Real Consol One-period Figure 2.1: A simulation path for relative interest payments on different types of bonds based on the benchmark calibration. The red solid line is for the benchmark model with inflation-indexed bonds. The blue dotted line represents a model with real bonds indexed to local produced goods. The pink dashed line denotes a model with one-period nominal bonds. The grey dashed line is for a model with nominal consols. 50 2.4. Numerical results Table 2.9: Optimal home equities held by home residents (inflation-indexed bonds) γ = 0.50 γ = 0.85 γ = 1 γ = 1.1 γ = 1.5 γ = 2 γ = 3 σ = 0.5 1.11 1.04 1.03 1.02 1.00 1.00 1.05 σ = 1 1.12 1.01 0.98 0.97 0.94 0.93 0.99 σ = 1.5 1.11 0.97 0.94 0.93 0.89 0.90 0.96 σ = 2 1.10 0.95 0.92 0.90 0.87 0.87 0.94 σ = 5 1.06 0.89 0.85 0.84 0.81 0.82 0.91 σ = 10 1.04 0.86 0.82 0.81 0.78 0.80 0.89 Note: This table reports the optimal share of home equities held by home households in a model with TFP shocks and monetary shocks. Inflation- indexed consol derivatives are used. The model specification is same as the benchmark model. σ is the coefficient of relative risk aversion and γ is the elasticity of substitution between imports and exports. Table 2.10: Covariance ratios for hedging real exchange rate risk (inflation- indexed bonds) γ = 0.50 γ = 0.85 γ = 1 γ = 1.1 γ = 1.5 γ = 2 γ = 3 σ = 0.5 0.37 0.32 0.30 0.29 0.24 0.19 0.09 σ = 1 0.51 0.44 0.42 0.40 0.34 0.28 0.17 σ = 1.5 0.55 0.49 0.46 0.44 0.38 0.31 0.20 σ = 2 0.58 0.51 0.48 0.47 0.40 0.33 0.21 σ = 5 0.61 0.55 0.52 0.51 0.44 0.37 0.24 σ = 10 0.64 0.57 0.54 0.52 0.45 0.38 0.25 Note: This table reports the ratio of covariance between human capital re- turns orthogonal to real exchange rates and equity returns orthogonal to real exchange rates to the variance of equity returns orthogonal to real exchange rates, βsr,d, in the main text. σ is the coefficient of relative risk aversion and γ is the elasticity of substitution between imports and exports. Inflation- indexed bonds are traded in the international bond markets. 51 2.4. Numerical results bor income risk, βsw,d, is around −0.23 (similar to table 2.6 and not reported here) and remains quite stable across preference parameters, which matches the data pretty well. The covariance ratio for hedging real exchange rate risk, βsr,d, shown in table 2.10, is highly small and close to zero. For instance, in the benchmark calibration, σ = γ = 2, βsr,d = 0.33 for inflation-indexed bonds, while βsr,d = 13.05 for nominal bonds. 30 To wrap up, a model with inflation-indexed bonds can deliver quite robust equity home bias, and fur- thermore, the covariance ratios for hedging real exchange rate risk and labor income risk are also consistent with the data. Data and model moments for asset returns Inflation-indexed bonds could effectively hedge real exchange rate risk and hence a model with these bonds could deliver robust home bias in equity portfolios. A following question is then how well the second moments for bonds generated by the model are consistent with those in the data. I find that the model studied here not only explains home bias in equity portfolios, but also produces second moments for bond returns which are consistent with the data. Table 2.11 presents second moments for CPI inflation rate, equity return rate and bond return rate both in the model and in the data. Because of data availability on equity returns and bond returns, I only report the second moments for the quarterly data. The first column labeled data shows that inflation rate, bond real return rate and bond real return rate differential across borders (home real return rate net of foreign real return rate) are one half as volatile as output when the U.S. is the home country.31 Inflation rate is pro-cyclical, bond return rate is weakly counter-cyclical and bond return rate differential is acyclical in the data. The column labeled model (1) shows the corresponding model moments in the benchmark model with inflation-indexed bonds. The volatilities and cyclicality for inflation rate, bond return rate and bond return rate differential in the model are quite closed to the data. As in the literature, the model studied here hardly captures equity returns in the stock markets. For comparison convenience, I also report second moments in a model with nominal bonds. The column labeled model (2) shows that the volatilities of bond return rate and bond return rate differential are almost two times as large as those in the data. 30When I use exogenous money supplies (in the following section) rather than Taylor rules , βsr,d is close to zero in terms of magnitude, say, around 0.1. 31In calculating bond return rate differentials, both return rates are denominated in local real consumption composite, rather than in the same consumption composite. 52 2.4. Numerical results Table 2.11: Data and model moments for asset returns Data Model (1) Model (2) Model (3) Model (4) Standard deviation relative to GDP Inflation rate 0.57 0.66 0.66 0.70 0.70 Bond real return rate 0.60 0.54 0.89 0.42 0.71 Bond real return rate differential 0.48 0.48 0.92 0.41 1.01 Equity real return rate 5.55 0.41 0.41 0.35 0.35 Cross correlation with GDP Inflation rate 0.36 0.47 0.47 0.81 0.81 Bond real return rate -0.15 -0.17 -0.14 0.49 -0.80 Bond real return rate differential -0.04 -0.09 -0.13 0.28 -0.50 Equity real return rate 0.06 0.39 0.39 0.34 0.34 Note: Model specifications are model (1) with Taylor rules and inflation- indexed bonds, model (2) with Taylor rules and nominal consols, model (3) with exogenous money supplies and inflation-indexed bonds, and model (4) with exogenous money supplies and nominal consols. Bond return rate refers to the return rate on 5-year government bonds. Bond return rate differential in the data is the average of 5-year government bond return rate differentials between the U.S. and other countries in the data sample. Because of data availability, the moments in the data column are calculated based on quarterly data starting from 1980s and data periods vary with countries. The numbers in the model columns show second moments generated by various models. The model period is one-year. 53 2.4. Numerical results Alternative monetary policy: exogenous money supply An alternative way of conducting monetary policy for monetary author- ities is to control the money supply circulated in the economy. Suppose that monetary policy is represented as a growth rate schedule, gm,t, which is subject to stochastic financial shocks. Assume that there are financial market shocks which affect the quantity of money circulated in the financial market. The quantity of real money balance mt in the economy is assumed to evolve as, mt = mt−1 pit gm,t (2.46) Based on the U.S. and foreign country pair, I estimate a first-order VAR process for monetary growth, which reads,[ gm,t g∗m,t ] = [ 0.357 0 0 0.357 ] [ gm,t−1 g∗m,t−1 ] + [ m,t ∗m,t ] with standard deviation of innovations σm = σ∗m = 4.15%, and cross cor- relation corr(m, ∗ m) = 0.20, which is slight larger than 0.1 in Kollmann (2001) but lower than 0.5 in Chari, Kehoe and McGrattan (2002). I redo the experiments above and find that a model with inflation- indexed consols could deliver quite robust home bias in equity portfolios. The decomposition of steady state equity holdings are similar to the case with Taylor rules. Particularly, the covariance ratio βsr,d for hedging real exchange rate risk is highly close to zero, around 0.1. The second moments for bond return rates are reported in table 2.11. The column labeled model (3) shows that bond return rates and bond return rate differentials are less volatile than GDP, as in the data. The last column labeled model (4) shows that a model with nominal bonds generates quite high volatilities for bond return rates and return rate differentials. 2.4.3 Alternative preference specification When preference is nonseparable in consumption and real money balance demand, marginal utility of consumption depends on real money balance and thus portfolio Euler equations also in turn contain real money balance. In order to compare the results in the new Keynesian model to those in real business cycle model without money balance, it’s better to make the 54 2.4. Numerical results preference separable in consumption and money balance, U(Ct, Ht, Mt Pt ) = C1−σt − 1 1− σ − η0 H1+ηt 1 + η + ν0 1− ν ( Mt Pt )1−ν (2.47) Based on this alternative utility function, I recalibrate the model given the targets described in section 2.3 and resolve for steady state portfolios. Similarly to the model with nominal bonds and non-separable preference, the numerical result shows that optimal shares of home equities held by the home household remain sizably sensitive to trade elasticity and relative risk aversion. Contrastingly, the motive for hedging human capital income risk remains stable and consistent with data. Inflation-indexed bonds could bring robust equity home bias back as in the benchmark model. 2.4.4 Nominal rigidities in a model with nominal bonds How does optimal shares of home equities held by home households change with nominal rigidities in a model with nominal bonds? Engel and Matsumoto (2009a) firstly investigate the effect of nominal rigidities on port- folio choices and find that home equity shares are increasing with price stick- iness given the trade elasticity. However, this monotonic relationship doesn’t sustain in a model with investment. Table 2.12 displays optimal shares of home equities under different combination of nominal rigidities ξ ≡ ξw = ξp (price stickiness is set as the same as wage stickiness) and trade elasticity γ when money supply shocks and TFP shocks are present.32 The smaller ξ is, the more flexible prices and wages are. When nominal rigidities are remark- ably high, the household holds most of home equities. For instance, the price and wage duration must be longer than 11−0.7 = 3.3 years in order to make a household with log-preference bias toward home equities. Nevertheless, the motive for hedging human capital income risk seems not changing much with nominal rigidities. The bottom graphs in figure 2.2 show the covariance ratio βsw,d and the implied equity shares for hedging human capital income risk in an economy with log-preference and unitary trade elasticity. Even deviating from log preference and unit trade elasticity, say σ = 2 and γ = 2, the optimal equity shares for hedging human capital income risk remains quite similar to the case with log-preference. The bottom graphs in figure 2.3 illustrate this point. Based on these analyses, that a model with high nominal rigidities can deliver robust home bias in equity lies in a fact that 32Engel and Matsumoto (2009a) treat money supplies as exogenous shocks. To facilitate comparison, I also use money supply shocks in this and the following subsection. 55 2.4. Numerical results 0 0.2 0.4 0.6 0.8 1 −5 −1 0 1 5 Price and wage stickiness H om e eq ui ty s ha re Optimal Home Equity Share 0 0.2 0.4 0.6 0.8 1 −2 −1 0 1 2 Price and wage stickiness B on d− G D P r at io Optimal Home Bond−GDP Ratio 0 0.2 0.4 0.6 0.8 1 −0.4 −0.3 −0.2 −0.1 Price and wage stickiness C ov ar ia nc e ra tio Human Capital Covariance Ratio 0 0.2 0.4 0.6 0.8 1 0.8 0.9 1 1.1 Price and wage stickiness H om e eq ui ty s ha re Optimal Home Equity for Hedging Human Capital Figure 2.2: Optimal portfolios and nominal rigidities. σ = γ = 1. The last graph reports the sum of equity portfolios for hedging human capital income risk and the market portfolio. 56 2.4. Numerical results 0 0.2 0.4 0.6 0.8 1 −30 −25 −20 −15 −10 −5 0 5 Price and wage stickiness H om e eq ui ty s ha re Optimal Home Equity Share 0 0.2 0.4 0.6 0.8 1 −2 −1 0 1 2 Price and wage stickiness B on d− G D P r at io Optimal Home Bond−GDP Ratio 0 0.2 0.4 0.6 0.8 1 −0.4 −0.3 −0.2 −0.1 Price and wage stickiness C ov ar ia nc e ra tio Human Capital Covariance Ratio 0 0.2 0.4 0.6 0.8 1 0.8 0.9 1 1.1 Price and wage stickiness H om e eq ui ty s ha re Optimal Home Equity for Hedging Human Capital Figure 2.3: Optimal portfolios and nominal rigidities. σ = γ = 2. The last graph reports the sum of equity portfolios for hedging human capital income risk and the market portfolio. 57 2.4. Numerical results nominal bond becomes actually equivalent to real bond when the degree of price stickiness tends to unit. Table 2.12: Optimal home equity shares of home equities: nominal consols and various nominal rigidities γ = 0.50 γ = 0.85 γ = 1 γ = 1.1 γ = 1.5 γ = 2 γ = 3 ξ = 0.05 0.45 1.01 1.40 1.74 4.86 -22.12 -3.46 ξ = 0.10 0.43 1.02 1.41 1.77 5.20 -17.37 -3.35 ξ = 0.15 0.42 1.02 1.44 1.81 5.69 -13.71 -3.22 ξ = 0.20 0.39 1.02 1.46 1.87 6.46 -10.85 -3.08 ξ = 0.25 0.36 1.02 1.50 1.95 7.81 -8.59 -2.91 ξ = 0.30 0.33 1.03 1.55 2.07 10.70 -6.81 -2.73 ξ = 0.35 0.27 1.03 1.64 2.25 20.82 -5.35 -2.53 ξ = 0.40 0.20 1.04 1.77 2.59 -74.45 -4.17 -2.30 ξ = 0.45 0.09 1.05 2.04 3.34 -9.90 -3.21 -2.06 ξ = 0.50 -0.10 1.09 2.80 6.55 -4.32 -2.42 -1.81 ξ = 0.55 -0.49 1.22 17.68 -7.36 -2.24 -1.77 -1.54 ξ = 0.60 -1.70 0.34 -0.82 -0.97 -1.16 -1.22 -1.26 ξ = 0.65 31.94 0.86 0.23 -0.01 -0.51 -0.76 -0.97 ξ = 0.70 2.68 0.92 0.56 0.38 -0.07 -0.38 -0.68 ξ = 0.75 1.71 0.94 0.72 0.60 0.24 -0.05 -0.38 ξ = 0.80 1.36 0.95 0.81 0.73 0.46 0.22 -0.10 ξ = 0.85 1.19 0.95 0.87 0.82 0.64 0.46 0.20 ξ = 0.90 1.08 0.96 0.91 0.88 0.77 0.65 0.47 ξ = 0.95 1.01 0.96 0.94 0.93 0.88 0.82 0.72 ξ = 1.00 0.97 0.96 0.96 0.96 0.95 0.94 0.92 Note: This table reports the optimal home equities held by home households. ξ is the probability of price and wage non-adjustment each period as of Calvo’s style and γ is the elasticity of substitution between imports and exports. Each unit of consol bond pays one unit of local currency each period. 2.4.5 Home bias in consumption in a model with nominal bonds Is home bias in equity portfolios related to home bias in consumption? Based on the assumption of unitary trade elasticity and of log preference in consumption, Heathcote and Perri (2008) show that home bias in equity is positively correlated with home bias in consumption in a two-country real business cycle model. Figure 2.4-2.5 confirms their results in a model with 58 2.4. Numerical results Table 2.13: Optimal home equity shares held by home households: nominal consols and no home bias in consumption γ = 0.50 γ = 0.85 γ = 1 γ = 1.1 γ = 1.5 γ = 2 γ = 3 ψH -3.30 1.32 1.00 0.33 -0.53 -0.84 -1.05 Bond-GDP ratio 1.32 -0.14 -0.01 0.18 0.42 0.49 0.52 ψH,w 0.84 0.92 0.94 0.96 1.00 1.03 1.04 ψH,r 4.14 -0.40 -0.06 0.62 1.53 1.87 2.09 βsw,d -0.19 -0.23 -0.24 -0.25 -0.28 -0.29 -0.30 Note: This table reports optimal portfolios held by home households. γ is the elas- ticity of substitution between imports and exports. The first row labeled ψH shows the optimal portfolio share of home equities by home households. The following row is the bond-GDP ratio. The third row labeled ψH,w is the optimal equity share used to hedge human capital risk and ψH,r is the optimal share of home equities for the exchange rate hedge motive. The covariance ratio of human capital and equity re- turns βsw,d is shown in bottom line of the table. The model specification is as the benchmark except that there’s no home bias in consumption, aC = 0.5. nominal bonds, TFP shocks and money supply shocks. The top graph of figure 2.4 shows that with log preference and unitary trade elasticity, optimal shares of home equities increase with the share of home consumption and investment expenditure on home goods when this share is relatively large, say, 70%. Nevertheless, this relationship collapses when we deviate from the unitary trade elasticity. The top graph of figure 2.5 shows that with a coefficient of relative risk aversion σ = 2 and trade elasticity γ = 2, we obtain a reverse relationship between the share of home equities and the share of home consumption expenditure on home produced goods. Home equity shares exhibit either supper home bias (with a share of home equities higher than one) when home bias in consumption is large or super anti- home bias (with a share of home equities lower than zero) when the share of consumption expenditure on local goods is relatively low. Next we turn to the component of optimal shares of home equities for hedging human capital income risk. The bottom two rows of graphs in figure 2.4-2.5 show that the covariance ratio βsw,d of human capital income with dividends from equities conditional on real exchange rates doesn’t vary much with the consumption expenditure share on local goods, with a range of −0.2 − −0.3, which is consistent with the data and the implied home equity shares for hedging human capital income risk lie within 0.35− 0.53. 59 2.4. Numerical results 0.5 0.6 0.7 0.8 0.9 1 −5 −1 0 1 5 Home bias in consumption H om e eq ui ty s ha re Optimal Home Equity Share 0.5 0.6 0.7 0.8 0.9 1 −2 −1 0 1 2 Home bias in consumption B on d− G D P r at io Optimal Home Bond−GDP Ratio 0.5 0.6 0.7 0.8 0.9 1 −0.4 −0.3 −0.2 −0.1 Home bias in consumption C ov ar ia nc e ra tio Human Capital Covariance Ratio 0.5 0.6 0.7 0.8 0.9 1 0.8 0.9 1 1.1 Home bias in consumption H om e eq ui ty s ha re Optimal Home Equity for Hedging Human Capital Figure 2.4: Optimal portfolios and home bias in consumption. σ = γ = 1. The last graph reports the sum equity portfolios for hedging human capital income risk and the market portfolio. 60 2.4. Numerical results 0.5 0.6 0.7 0.8 0.9 1 −5 −1 0 1 5 Home bias in consumption H om e eq ui ty s ha re Optimal Home Equity Share 0.5 0.6 0.7 0.8 0.9 1 −2 −1 0 1 2 Home bias in consumption B on d− G D P r at io Optimal Home Bond−GDP Ratio 0.5 0.6 0.7 0.8 0.9 1 −0.4 −0.3 −0.2 −0.1 Home bias in consumption C ov ar ia nc e ra tio Human Capital Covariance Ratio 0.5 0.6 0.7 0.8 0.9 1 0.8 0.9 1 1.1 Home bias in consumption H om e eq ui ty s ha re Optimal Home Equity for Hedging Human Capital Figure 2.5: Optimal portfolios and home bias in consumption. σ = γ = 2. The last graph reports the sum equity portfolios for hedging human capital income risk and the market portfolio. 61 2.4. Numerical results In the literature, unit trade elasticity plays an important role in deter- mining optimal portfolios. Cole and Obstfeld (1991) firstly investigate the effect of portfolio diversification on welfare and find that complete home bias in equity is optimal when trade elasticity is unitary. Engel and Mat- sumoto (2009a) (EM, hereafter) and Devereux and Sutherland (2008) also confirm this finding.33 In their models, they both assume that there isn’t home bias in consumption, that is, aI = aC = 0.5. 34 In order to compare with their results, I produce optimal portfolios when there isn’t home bias in consumption in table 2.13. The first row labeled ψH is the optimal shares of home equities, following which bond assets-GDP ratio is reported. The next two rows show the decomposition of home equity holdings. ψH,w is the sum of market portfolio 1/2 and the optimal portfolio for hedging human capital income risk. ψH,r is the remaining hedge motive for equities con- ditional on real exchange rates. When trade elasticity is unity, the home household completely hold home equities and zero bond assets, financial autarky is optimal even when international financial markets are available to all households. However, once deviating from unitary trade elasticity, I get either super home bias or super foreign bias in equities. The large shift of optimal equity shares across preference parameters is mainly induced by bond holdings since the motive for hedging human capital income risk is quite stable across trade elasticity. To facilitate further comparison with EM’ model, I develop a model with pricing-to-market. The appendix describes the model structure in detail. In the model with pricing-to-market, I use nominal consols. I further assume that there isn’t physical capital and home bias in consumption. Other pa- 33Engel and Matsumoto (2009b) develop further a model including home bias in con- sumption based on EM (2009a). They solve for the foreign equity holdings by home residents and compare how the equity holdings vary with home bias in consumption and price and wage stickiness. 34They share many features in their models. First, they assume no home bias in con- sumption. Second, labor is the only input in production (no capital). Third, they use price stickiness as nominal rigidities. Forth, technological and monetary shocks are the exogenous driving forces. In terms of the asset menu, Devereux and Sutherland (2008) ex- ploit different types of asset menu, including real bond indexed to locally produced goods, nominal bonds denominated in each currency and equities, while Engel and Matsumoto (2009a) assume that equities and forward currency forward contracts are traded. Their results for home equity holdings are based on a key assumption in their models: no home bias in consumption. One can show that in a first-order approximated version of this type of model (even with investment and wage stickiness), asset holdings don’t depend on (a) relative risk aversion, (b) variance and persistence of shocks, (c) monetary policy, either monetary growth rates or coefficients in the Taylor rule. Home bias in equity then mainly depends on the trade elasticity and the extent of price stickiness. 62 2.5. Incomplete asset markets rameters are the same as the benchmark model. The appendix also provides a version of model with foreign currency forward contracts. Since both of models generate similar results, I report the results of the model with nom- inal consols. Table A.1 presents the optimal home equities, which are very similar to table 1 in Engel and Matsumoto (2009a), since both of models have similar structures.35 When deviating from no home bias in consump- tion, I get a similar portfolio shares of home equities as in table 2.4. In another experiment, I introduce investment into the model, and find that, even though there is no home bias in consumption, the model could still generate relatively robust home bias in equities. The home equity holdings are similar to an economy with ac = 0.68 and without investment. However, this investment channel becomes weaker when home bias in consumption is higher or the degree of nominal rigidity is larger. To sum up, when nominal consols are the mainly type of bond assets in the international financial market, home bias in consumption and investment seems to play a quite limited role in solving the puzzle of home bias in equity portfolios while price stickiness seems to be a promising candidate of solving this puzzle but we need incredibly high price stickiness to deliver robust home bias in equities! The reason for the latter case is that the exchange rate hedge term (the last term) in the optimal equity portfolio (2.41) tends to disappears as prices become more stickier because nominal consols are asymptotically equivalent to real consols. 2.5 Incomplete asset markets This section extends the benchmark model with complete asset markets to one with incomplete markets. The way I introduce market incompleteness is to incorporate additional shocks into the baseline model. In the following subsections, three types of extra shocks will be considered separately. The first subsection takes into account of investment specific shocks (see Fisher 2006; Greenwood et al. 1988; Greenwood et al. 1997; Greenwood et al. 2000), since empirical evidence shows that investment specific shocks have a large contribution to business cycle fluctuations (Justiniano, Primiceri and Tam- balotti 2010). In the second subsection I add government expenditure shocks to the benchmark model because government spending and taxes have im- portant effects on consumption and output (see Blanchard and Perotti 2002; Mertens and Ravn 2010; Fisher and Peters 2010; Lez-Salido and Vall 2007; 35The parameters used here are different from EM’s but the patterns for shares of home equities are exactly the same. 63 2.5. Incomplete asset markets Ramey 2011; Romer and Romer 2010). In the third subsection, I will in- vestigate optimal shares of home equities in a model with preference shocks since many researchers find that preference shifts play a non-ignorable role in explaining business cycle movements (see Ireland 2001; Justiniano, Prim- iceri and Tambalotti 2010). At the end of this section, I include all shocks in the model in order to compare quantitative effects of shocks on optimal equity portfolios. Monetary policy is conducted through exogenous money supplies in this section. 2.5.1 Investment efficiency shocks Let the investment efficiency shock be Zi,t, which effects the transforming efficiency between investment and capital goods. Physical capital accumu- lation now becomes, Kt+1 = Φ ( It Kt ) KtZi,t + (1− δ)Kt (2.48) Shadow price for physical capital yields, qk,t = [ Φ′(It/Kt) ]−1 pI,t/Zi,t (2.49) Optimal investment decision then follows, qk,t =EtΛt,t+1 { qk,t+1 [ (1− δ) + Zi,t+1 ( Φ( It+1 Kt+1 )− Φ′ ( It+1 Kt+1 ) It+1 Kt+1 )] + αYt+1pH,t+1 Kt+1Xp,t+1 } (2.50) Other optimality conditions are the same as in the benchmark model. As in Coeurdacier et al. (2010), investment efficiency shocks are an AR(1) process, [ log(Zi,t) log(Z∗i,t) ] = [ 0.79 0 0 0.79 ] [ log(Zi,t−1) log(Z∗i,t−1) ] + [ i,t ∗i,t ] where σi = σ∗i = 1.73%, corr(i, ∗ i ) = 0.19. Coeurdacier et al. (2010) explore a real model with technological and investment efficiency shocks and show that optimal equity shares don’t re- spond to trade elasticity and relative risk aversion. After shutting down monetary shocks and removing nominal rigidities, and keeping inflation- 64 2.5. Incomplete asset markets indexed consols rather than nominal consols, the model investigated here degenerates to a real model analogous to Coeurdacier et al. (2010) and the numerical results deliver quite similar robust results as theirs. I then add nominal rigidities and monetary shocks into the real model. Asset markets now become incomplete because there are six types of shocks but only four types of assets. The results show that the marginal contribution of nominal rigidities and the additional shocks to optimal home equity shares are mod- erate. However, when nominal consols, instead of inflation-indexed consols, are traded in international financial markets, optimal home equity shares be- come quite sensitive to preference parameters again. Table 2.14 displays the optimal equity shares. Analogously to the benchmark model, equity shares are varying much across different preference parameter values. In another experiment, which isn’t reported here, I get rid of monetary shocks and keep only real shocks in the monetary model with nominal consols. Quite contrasting to the results in the previous model with monetary shocks, the optimal equity shares bias the home countries and are robust to preference parameters. Table 2.14: Optimal home equity shares held by home households: nominal consols and investment shocks γ = 0.50 γ = 0.85 γ = 1 γ = 1.1 γ = 1.5 γ = 2 γ = 3 σ = 0.50 -4.29 -0.37 0.48 0.76 0.85 0.49 -0.05 σ = 1.00 -0.21 0.56 0.80 0.87 0.57 -0.04 -0.60 σ = 1.50 0.25 0.82 0.98 1.02 0.65 -0.09 -0.71 σ = 2.00 0.42 0.93 1.08 1.12 0.79 0.02 -0.68 σ = 5.00 0.67 1.11 1.25 1.32 1.30 0.80 -0.10 σ = 10.00 0.74 1.16 1.31 1.39 1.55 1.31 0.47 Note: This table reports the optimal share of home equities held by home households in a model with TFP shocks, investment efficiency shocks and monetary shocks. σ is the coefficient of relative risk aversion and γ is the trade elasticity. Nominal consols are traded. Each unit of bond asset pays one unit of local currency each period. 65 2.5. Incomplete asset markets 2.5.2 Government expenditure shocks A model with government expenditure shocks Assume that government consumption composite has a form of, Gt = [ a 1 γG G G γG−1 γG H,t + (1− aG) 1 γGG γG−1 γG F,t ] γG γG−1 (2.51) Minimizing the cost of unit government expenditure on goods implies that the price of government consumption composite is as follows, PG,t = [ aGP 1−γG H,t + (1− aG)P 1−γGF,t ] 1 1−γG (2.52) The government expenditure on home goods can be written as GH,t = aG(PH,t/PG,t) −γGGt and on foreign goods is, GF,t = (1−aG)(PF,t/PG,t)−γGGt. Government revenues consist of two parts. The first part comes from seignior- age revenues via printing fiat money and the second from tax revenues. As- sume that labor income tax rate and profit tax rate to be constant at τh and τf , respectively. Assume further that no tax occurs on financial assets. Tax revenues through levying on labor services are,∫ 1 0 Wj,tHj,tτhdj = WtH d t τh (2.53) and profit taxes are Dtτf . Assume government balances its budget every period, Dtτf +WtH d t τh +Mt −Mt−1 = GtPG,t + Trt (2.54) Notice that government expenditures are exogenous, tax revenues are en- dogenously determined by households and firms. The sum of seigniorage income and tax revenues may not completely cover government expendi- ture each period. For simplicity, I introduce lump-sum transfers between government and households to balance government budget constraint each period. The household’s budget constraint will now take into account of taxes, Ct + qtψH,t+1 + stq ∗ tψF,t+1 + p b tBH,t+1 + stp b∗ t BF,t+1 +mt = (qt + dt(1− τf ))ψH,t + st(q∗t + d∗t (1− τf ))ψF,t + (pbt + rbt )BH,t+ st(p b∗ t + r b∗ t )BF,t + wtH d t (1− τh) + mt−1 pit + trt (2.55) 66 2.5. Incomplete asset markets After-tax gross returns can then be written as follows, ra1,t+1 ≡ qt+1 + dt+1(1− τf ) qt ra2,t+1 ≡ st+1 st q∗t+1 + d∗t+1(1− τf ) q∗t rb1,t+1 ≡ pbt+1 + r b t+1 pbt rb2,t+1 ≡ st+1 st pb∗t+1 + rb∗t+1 pb∗t The optimal intratemporal substitution between consumption and leisure yields, − Uh,t Uc,t = wt(1− τh) Xw,t (2.56) which states that marginal rate of substitution between labor supply and consumption equals after-tax wage. Goods market clearing conditions now state that private consumption, government consumption and investment demands for each good are equal to the supply of each good, CH,t + C ∗ H,t + IH,t + I ∗ H,t +GH,t +G ∗ H,t = YH,t (2.57) CF,t + C ∗ F,t + IF,t + I ∗ F,t +GF,t +G ∗ F,t = Y ∗ F,t (2.58) Other optimal conditions are the same as in the benchmark model. Calibration and equity portfolios The new unknown parameters are γG, τh, and τf . Assume that tax rates on labor income and on profits are the same, τh = τf , and they are chosen such that the ratio of government consumption to GDP is 16.4%, which is taken from the data sample. In the calibration, tax rates are set at τh = τf = 20.1%, which lie in the range of tax rates discussed in the literature. Assume further that γG = γ in the calibration. Government expenditure shocks are estimated from the data sample and evolve as,[ log(Gt/G) log(G∗t /G) ] = [ 0.61 0 0 0.61 ] [ log(Gt−1/G) log(G∗t−1/G) ] + [ g,t ∗g,t ] where σg = σ∗g = 2.32%, corr(g, ∗ g) = 0.60. Now let’s turn to the numerical experiments. First, consider a model with technological shocks, monetary shocks and government expenditure shocks, in which nominal consols are traded internationally. Table 2.15 67 2.5. Incomplete asset markets shows that introducing government expenditure shocks doesn’t produce ro- bust optimal shares of home equities. For instance, at σ = 2 and γ = 2, the home household has an extremely large short position in home equities. How do the optimal shares of home equities hedging human capital in- come risk vary with shocks happening in an economy? In another experi- ment, I investigate optimal equity shares in a model with inflation-indexed consols and real shocks (technological and government expenditure shocks). The results (not reported here) show that equity portfolios exhibit super home bias for all combinations of trade elasticity and relative risk aversion. For instance, when σ = γ = 2, home residents hold 184% of home equity and short sell 84% of foreign equities. It seems that even with real bond to hedge real exchange rate risk, home bias in equity still depends on how the components of human capital income orthogonal to real exchange rates (or terms of trade) and returns to equities orthogonal to real exchange rates (or terms of trade) respond to shocks and how large the responses are. Although returns to local equities can still be used to hedge local labor income risk, it is quite hard to conclude that the shares of home equities lie exact between one half and one to all main types of shocks. Put it another way, the puzzle of home bias in equity remains in a real model with technological shocks and government expenditure shocks! Nevertheless, this trouble seems to disappear when I incorporate nominal rigidities and monetary shocks into the model with inflation-indexed bonds. Table 2.15: Optimal home equity shares held by home households: nominal consols and government expenditure shocks γ = 0.50 γ = 0.85 γ = 1 γ = 1.1 γ = 1.5 γ = 2 γ = 3 σ = 0.50 -12.14 8.53 3.90 2.23 0.26 -0.57 -1.15 σ = 1.00 -0.54 0.39 1.34 2.37 -8.99 -3.64 -2.60 σ = 1.50 0.07 0.99 1.86 2.40 28.26 -6.86 -3.39 σ = 2.00 0.29 1.17 1.81 2.39 12.67 -10.20 -3.89 σ = 5.00 0.59 1.40 1.93 2.38 4.64 -26.75 -4.99 σ = 10.00 0.67 1.45 1.95 2.37 5.37 -17.14 -5.36 Note: This table reports the optimal share of home equities held by home households in a model with TFP shocks, investment efficiency shocks and monetary shocks. σ is the coefficient of relative risk aversion and γ is the trade elasticity. Nominal consols are traded. Each unit of bond asset pays one unit of local currency each period. 68 2.5. Incomplete asset markets 2.5.3 Preference shocks Let a preference shift for the home household be Zb,t, which enters the household’s preference through subjective discount factor βt (as in Justini- ano et al. 2010). The preference of the home representative household now becomes, Et +∞∑ τ=0 { βt+τZb,t+τU(Ct+τ , Ht+τ , Mt+τ Pt+τ ) } Optimality conditions for portfolios yield, Uc,t = Et [ v(t) Zb,t+1 Zb,t Uc,t+1ri,t+1 ] , with i = a1, a2, b1, b2 Intertemporal preference shifts alter the household’s optimal consump- tion path. When Zb,t is higher than its expected value at period t, the house- hold is willing to consume more today and consume less in future periods. Observe first that preference shocks (or taste shifts) change the stochastic discount factor and act as demand shocks, which die out in the long run. Second, only temporary shocks have real effects on consumption and output. Justiniano et al. (2010) provide an estimation of preference shocks based on U.S. macroeconomic data and the following process for preference shocks is borrowed from theirs,[ log(Zb,t) log(Z∗b,t) ] = [ 0.674 0 0 0.674 ] [ log(Zb,t−1) log(Z∗b,t−1) ] + [ b,t ∗b,t ] where σb = σ∗b = 0.16%, corr(b, ∗ b) = 0.20. Notice that the volatility of preference shocks is quite small compared to technological and monetary shocks. It’s highly possible that in a model with technological, monetary and preference shocks, the first two shocks domi- nate the last one. Table 2.16 shows the numerical experiments. As in the previous subsection, I consider first a model with inflation-indexed consols and real shocks (technological and preference shocks). The numerical results display that home equity exhibits super home bias, as shown in the case with government spending shocks. Home residents hold 165% of home equities in the benchmark calibration! However, after adding monetary shocks and nominal rigidities into the model, I can get a reasonable and robust home bias in equities. 69 2.5. Incomplete asset markets Table 2.16: Optimal home equity shares held by home households: nominal consols and preference shocks γ = 0.50 γ = 0.85 γ = 1 γ = 1.1 γ = 1.5 γ = 2 γ = 3 σ = 0.50 16.51 3.84 2.64 2.10 0.90 0.22 -0.37 σ = 1.00 -0.78 -0.27 0.46 2.01 -2.57 -1.78 -1.52 σ = 1.50 0.05 0.79 1.42 2.12 -139.45 -3.93 -2.20 σ = 2.00 0.31 1.03 1.57 2.11 12.39 -6.27 -2.65 σ = 5.00 0.64 1.29 1.72 2.10 5.59 -25.90 -3.74 σ = 10.00 0.73 1.35 1.75 2.10 4.94 -129.96 -4.23 Note: This table reports the optimal share of home equities held by home households in a model with TFP shocks, preference shocks and monetary shocks. σ is the coefficient of relative risk aversion and γ is the trade elas- ticity. Nominal consols are traded. Each unit of bond asset pays one unit of local currency each period. 2.5.4 All shocks In this subsection, I examine how incomplete asset markets affect equity portfolios when all of shocks considered above are included in the model. This is a horse race among different shocks. When nominal bonds are used in the international market, as in other cases discussed above, home equity holdings are quite sensitive to trade elasticity and relative risk aversion. However, the component of equity holdings for hedging labor income risk remains quite stable, varying between −0.27 and −0.23. When inflation- indexed bonds are traded in the international market, robust equity home bias is restored (see table 2.17). Until now, we might conclude that monetary shocks and inflation matters for nominal bonds hedging real exchange rate risk. As a final remark in this section, I shut down monetary shocks and keep the remaining four types of real shocks and nominal rigidities in the model. The asset markets are still incomplete since there are four assets and eight shocks in the model. The numerical experiment shows that, even with nominal bonds, the model could produce quite robust home bias in equities (table 2.18). 70 2.5. Incomplete asset markets Table 2.17: Optimal home equity shares held by home households: inflation- indexed bonds and all shocks γ = 0.50 γ = 0.85 γ = 1 γ = 1.1 γ = 1.5 γ = 2 γ = 3 σ = 0.50 -0.74 -0.09 0.21 0.38 -0.69 -19.77 -16.00 σ = 1.00 0.94 0.93 0.93 0.93 0.91 0.89 0.83 σ = 1.50 0.95 0.92 0.91 0.90 0.87 0.84 0.78 σ = 2.00 0.95 0.91 0.89 0.88 0.85 0.81 0.75 σ = 5.00 0.93 0.88 0.86 0.85 0.81 0.77 0.71 σ = 10.00 0.92 0.87 0.85 0.84 0.80 0.75 0.69 Note: This table reports the optimal share of home equities held by home households in a model with TFP shocks, monetary shocks, investment efficiency shocks, government expenditure shocks and preference shocks. Inflation-indexed consol derivatives are used. The model include all shocks consider in the main text. σ is the coefficient of relative risk aversion and γ is the trade elasticity. Table 2.18: Optimal home equity shares held by home households: nominal consols and real shocks γ = 0.50 γ = 0.85 γ = 1 γ = 1.1 γ = 1.5 γ = 2 γ = 3 σ = 0.50 1.09 0.94 0.91 0.80 0.72 0.60 0.42 σ = 1.00 -0.25 0.63 0.86 0.80 -0.51 -1.33 -1.57 σ = 1.50 0.82 0.97 0.70 0.73 0.73 0.71 0.68 σ = 2.00 0.86 0.17 0.71 0.71 0.69 0.65 0.59 σ = 5.00 0.91 0.66 0.69 0.68 0.64 0.60 0.53 σ = 10.00 0.95 0.67 0.68 0.67 0.63 0.58 0.52 Note: This table reports the optimal share of home equities by home house- holds in a model with TFP shocks, investment efficiency shocks, government expenditure shocks and preference shocks. Nominal consols are used here. The model include all real shocks consider in the main text. Parameters are the same as cases with incomplete markets. σ is the coefficient of relative risk aversion and γ is the trade elasticity. 71 2.6. International portfolio dynamics 2.6 International portfolio dynamics 2.6.1 Numerical solution method Building on Devereux and Sutherland (2010a), a model has to be taken a third-order approximation around its steady state in order to find the first- order accurate equilibrium portfolios. The policy functions for equilibrium portfolios thus can be written as a function of state variables as usual, ψj,t = ψj +%j · κ̂t, Bj,t = Bj +%j · κ̂t, with j = H, or F . κ̂t denotes a vector for the percentage deviation of state variables, which are defined in section 2.1.6, and %j is the corresponding coefficient vector in the policy function. 36 The coefficient vector %j can be obtained using a third-order approximation of the household’s consumption Euler equations and a second-order approximation of the rest of equilibrium equations in the model. Here, I use Schmitt-Grohe and Uribe (2004) algorithm to calculate the second-order approximation and rewrite the formula for equilibrium portfolios as in Devereux and Sutherland (2010a) such that Schmitt-Grohe and Uribe’s method can be applied here directly. The Technical Appendix provides steps in detail on calculating first-order components of foreign asset dynamics. 2.6.2 Measuring international portfolios Before turning to the quantitative properties of the empirical and the- oretical economy, I start with summarizing international financial variables used in this chapter. Net foreign equity assets at the end of period t are measured as (ψH,t+1 − 1)qt + ψF,t+1q∗t st and net foreign bond assets can be written as BH,t+1p b t +BF,t+1p b∗ t st. The sum of net foreign equity assets and net foreign bond assets are net foreign assets. Substituting net foreign assets, dividends, consumption and investment demands into the home household’s budget constraint, yields, NFAt+1 = NXt +NFIt + Value-of-Assett = CAt +NFAt + V ALt (2.59) where NFAt+1 denotes net foreign asset positions at the end of period t. Net export (trade balance), NXt, is exports net imports, and net factor income from abroad, NFIt, is the factor income paid by foreigners (dividend and interest payment inflows, here) minus factor income paid to foreigners (dividend and interest payment outflows). From the textbook definition of 36Including newly defined variables besides the original variables in the state vector: net foreign asset positions at the end of period in both countries. 72 2.6. International portfolio dynamics current account, we know current account is given by CAt = NXt +NFIt. Value-of-Assett represents the period t value of external assets held at the end of period t− 1, which includes the net foreign asset positions at the end of period t− 1, NFAt, and wealth effect in period t, V ALt. Net equity purchases are the changes of equity portfolios, (ψH,t+1 − ψH,t)qt + (ψF,t+1 − ψF,t)q∗t st, which are equity asset outflows minus equity asset inflows. Net bond purchases are analogously defined as (BH,t+1 − BH,t)p b t +(BF,t+1−BF,t)pb∗t st, which are bond asset outflows minus bond as- set inflows. Changes of net foreign asset positions are given by ∆NFAt+1 ≡ NFAt+1 − NFAt. Up to a first-order approximation, current account is zero.37 Taking a first-order approximation around the steady state, changes of net foreign asset positions can be written as, ∆NFAt+1 = (ψH − 1)q∆q̂t + ψF q∗(∆q̂∗t + ∆ŝt) +BHpb∆p̂bt +BF p b∗(∆p̂b∗t + ∆ŝt) +O (‖ ‖) ≡ V ALt +O (‖ ‖) (2.60) where ∆xt+1 ≡ xt+1 − xt. Changes of net foreign assets mainly come from wealth effects, V ALt. Changes of net equity assets are the sum of net equity purchases and wealth effects from equity assets. Similarly, changes of net bond assets are equal to net bond purchase plus wealth effects from bond assets. 37See also the technical appendix in Coeurdacier et al. (2010). 73 2 .6. In tern ation al p ortfolio d y n am ics Table 2.19: Second moments (1) NSP NSP NSP SP Variables Data (US) Data (Mean) Money and TFP TFP Money Money and TFP Home equity share (%) 82 77(0.17) 84 84 84 87 Std. relative to GDP Investment 2.85 2.98(0.25) 2.80 1.87 3.41 2.75 Consumption 0.78 0.87(0.31) 0.50 0.65 0.36 0.54 Employment 0.68 0.70(0.46) 1.13 0.30 1.52 1.12 Real exchange rate 3.58 3.49(0.54) 0.66 0.78 0.55 0.61 Net foreign assets 0.96 1.34(0.30) 1.55 2.67 0.54 1.28 ∆(Net foreign assets) 0.98 1.24(0.32) 0.73 1.15 0.41 0.64 ∆(Net foreign equity assets) 0.90 1.10(0.39) 1.47 1.03 2.09 1.74 ∆(Net foreign bond assets) 0.45 0.83(0.60) 1.16 1.07 1.83 1.32 Series corr. GDP 0.54 0.56(0.24) 0.55 0.90 0.26 0.56 Investment 0.71 0.64(0.17) 0.30 0.86 0.16 0.30 Consumption 0.62 0.63(0.16) 0.81 0.93 0.54 0.84 Employment 0.55 0.63(0.18) 0.19 0.79 0.17 0.20 Real exchange rate 0.73 0.64(0.16) 0.71 0.93 0.39 0.67 Net foreign assets 0.44 0.41(0.49) 0.87 0.89 0.70 0.86 ∆(Net foreign assets) -0.03 -0.07(1.70) -0.02 0.02 -0.12 -0.02 ∆(Net foreign equity assets) -0.03 -0.04(3.41) -0.06 -0.07 -0.30 -0.10 ∆(Net foreign bond assets) 0.19 0.06(1.98) 0.13 0.45 -0.22 -0.0474 2 .6. In tern ation al p ortfolio d y n am ics Table 2.20: Second moments (1) (continued) NSP NSP NSP SP Variables Data (US) Data (Mean) Money and TFP TFP Money Money and TFP Cross corr. with home GDP Investment 0.85 0.88(0.08) 0.93 0.95 0.97 0.92 Consumption 0.89 0.80(0.13) 0.87 0.90 0.90 0.87 Employment 0.89 0.73(0.36) 0.68 -0.32 0.97 0.63 Real exchange rate 0.07 0.04(8.05) 0.60 0.58 0.59 0.57 Net foreign assets -0.20 -0.06(4.42) -0.29 -0.61 -0.47 -0.44 ∆(Net foreign assets) -0.27 -0.20(0.62) -0.29 -0.17 -0.53 -0.31 ∆(Net foreign equity assets) -0.21 -0.02(11.27) 0.16 0.02 -0.13 0.05 ∆(Net foreign bond assets) -0.17 -0.26(0.58) -0.39 -0.19 0.03 -0.21 Cross country corr. GDP 0.58 0.28(1.21) 0.23 0.19 0.28 0.30 Investment 0.20 0.19(1.57) 0.40 0.30 0.43 0.41 Consumption 0.46 0.11(3.43) 0.76 0.80 0.69 0.82 Employment 0.53 0.24(1.07) 0.13 0.57 0.11 0.13 Return on equity assets 0.53 0.44(0.30) 0.53 0.46 0.54 0.52 Return on bond assets 0.53 0.56(0.14) 0.53 0.73 0.52 0.52 Note: The column labelled Data (US) reports the data moments for the US and Data (Mean) reports the weighted average of data moments in the data sample. Numbers in parentheses in column Data (Mean) denote coefficient of variation. NSP stands for a model without TFP spillover across borders and SP represents a model with TFP spillover. The column labelled Money and TFP displays simulated moments in the model with TFP shocks and money supply shocks. The column labelled TFP reports model moments for the model with TFP shocks only and money for the model with money supply shocks only. 75 2 .6. In tern ation al p ortfolio d y n am ics Table 2.21: Second moments (2) NSP SP SP Variables Data (US) Data (Mean) Taylor rule Investment Gov. ex- pense Home equity share (%) 82 77(0.17) 87 88 83 Std. relative to GDP Investment 2.85 2.98(0.25) 2.44 3.05 2.89 Consumption 0.78 0.87(0.31) 0.58 0.54 0.63 Employment 0.68 0.70(0.46) 0.93 1.13 1.17 Real exchange rate 3.58 3.49(0.54) 0.70 0.62 0.65 Net foreign assets 0.96 1.34(0.30) 2.19 1.38 1.39 ∆(Net foreign assets) 0.98 1.24(0.32) 0.84 0.67 0.69 ∆(Net foreign equity assets) 0.90 1.10(0.39) 0.73 1.52 0.86 ∆(Net foreign bond assets) 0.45 0.83(0.60) 1.36 1.29 0.62 Series corr. GDP 0.54 0.56(0.24) 0.69 0.56 0.55 Investment 0.71 0.64(0.17) 0.38 0.35 0.30 Consumption 0.62 0.63(0.16) 0.90 0.84 0.85 Employment 0.55 0.63(0.18) 0.19 0.20 0.22 Real exchange rate 0.73 0.64(0.16) 0.80 0.69 0.68 Net foreign assets 0.44 0.41(0.49) 0.92 0.87 0.86 ∆(Net foreign assets) -0.03 -0.07(-1.70) -0.01 0.01 -0.02 ∆(Net foreign equity assets) -0.03 -0.04(-3.41) -0.04 -0.16 -0.14 ∆(Net foreign bond assets) 0.19 0.06(1.98) -0.04 -0.12 0.48 76 2 .6. In tern ation al p ortfolio d y n am ics Table 2.22: Second moments (2) (continued) NSP SP SP Variables Data (US) Data (Mean) Taylor rule Investment Gov. ex- pense Cross corr. with home GDP Investment 0.85 0.88(0.08) 0.91 0.88 0.92 Consumption 0.89 0.80(0.13) 0.88 0.83 0.84 Employment 0.89 0.73(0.36) 0.46 0.63 0.59 Real exchange rate 0.07 0.04(8.05) 0.61 0.54 -0.20 Net foreign assets -0.20 -0.06(-4.42) -0.52 -0.44 -0.44 ∆(Net foreign assets) -0.27 -0.20(-0.62) -0.15 -0.34 -0.32 ∆(Net foreign equity assets) -0.21 -0.02(-11.27) -0.10 -0.12 -0.07 ∆(Net foreign bond assets) -0.17 -0.26(-0.58) 0.11 0.00 -0.26 Cross country corr. GDP 0.58 0.28(1.21) 0.37 0.30 0.27 Investment 0.20 0.19(1.57) 0.61 0.29 0.41 Consumption 0.46 0.11(3.43) 0.84 0.81 0.84 Employment 0.53 0.24(1.07) 0.49 0.16 0.14 Return on equity assets 0.53 0.44(0.30) 0.69 0.53 0.53 Return on bond assets 0.53 0.56(0.14) 0.59 0.54 0.54 Note: The column labelled Data (US) reports the data moments for the US and Data (Mean) reports the weighted average of data moments in the data sample. Numbers in parentheses in column Data (Mean) denote coefficient of variation. NSP stands for a model without TFP spillover across borders and SP represents a model with TFP spillover. The column labelled Taylor rule displays simulated moments in the model with TFP shocks and interest rate shocks. The column labelled Investment reports model moments for the model with TFP shocks, investment efficiency shocks and money supply shocks and Gov. expense for the model with TFP shocks, government expenditure shocks and money supply shocks. 77 2.6. International portfolio dynamics 2.6.3 Data moments v.s. model moments First of all, I review some properties of international business cycles in major developed economies in my data set. These properties refer to the second moments of yearly time series detrended with Hodrik-Prescott filter and to cross correlations between these series. The first data columns la- belled “Data(US)” in table 2.19-2.20 and table 2.21-2.22 show the (relative) second moments and cross correlations for U.S. economy (the home country) and for the world economy which is aggregated from 9 other countries in the data using bilateral trade volume as weights. The next data columns labeled “Data(Mean)” in these two tables report the output-weighted average of sec- ond moments and the average of cross correlations when each country in the sample is treated as the home country and the rest of 9 countries as the for- eign country. Net foreign assets, changes of net foreign assets and changes of net foreign equity/bond assets are normalized by national GDP. GDP, investment, consumption are in real terms and logged. Real exchange rates are logged as well. Finally, all variables are detrended by Hodrick-Prescott filter (the smoothing parameter is 100). Instead of reporting the actual stan- dard deviations, I list the standard deviations of variables relative to that of GDP and I also use the relative moments in numerical experiments. The data shows that volatilities of net foreign assets, net foreign equity assets and net foreign bond assets are of the same order as output both in levels and first differences. Net foreign assets counter-cyclically move with output. Now we turn to the simulated second moments of the model. The column labeled “Money and TFP” displays the results for a model with inflation- indexed bonds and exogenous money supply shocks. The following three columns present results for the model with TFP shocks only, monetary sup- ply shocks only and a model with technology spill-over, respectively.38 All of these columns show that net foreign assets generate the same volatilities as output and that they have high persistence and are counter-cyclical. How does the model generate counter-cyclical movement of net foreign asset positions? Recall that the change of NFA is equal to the wealth effect (see equation (2.60)). In the steady state of a symmetric equilibrium, all real variables are the same across countries, q = q∗, pb = pb∗, and the share of home equity held by home consumer is identical to the share of foreign equity held by foreigners, that is, ψH + ψF = 1 and analogously, BH + BF = 0. 38When the number of shocks is less than the number of assets, at least some of assets are indeterminate in equilibrium. In the computation, I force the variance of one type shock close to zero such that I can calculate the steady state portfolios. 78 2.6. International portfolio dynamics Changes of net foreign assets can be rewritten as, ∆NFAt+1 =ψF q ∗ ( ∆q̂∗t + ∆ŝt −∆q̂t ) +BHp b ( ∆p̂bt −∆p̂b∗t −∆ŝt ) +O (‖ ‖) (2.61) Consider now a positive TFP shock happening at the home country. Outputs on impact and in future periods arise because of higher technolog- ical level on impact and higher investment in the following periods, which causes an increase of real dividends in the following periods and a rise of real interest payments because of expected deflation in the near future at home. Higher expected dividends and interest payments push up prices of home equities and home bonds.39 What happens in the foreign county is quite different from the home country. Investment at the foreign country doesn’t change much, and consequently, the price of foreign equities only goes up a little bit. However, the foreign real interest rate rises because of deflation spilled over from the home country, which implies foreign bond price has a similar increment as home bond price. On the other hand, home real exchange rates depreciate in response to the positive TFP shock at home. Thereby, the relative price of foreign bonds pb∗t st pbt increases. Relative price of foreign equities q∗t st qt depends on the relative increments in real exchange rates and in home equity price. In the model studied here, the latter has a larger increment than the former, and then the relative price of home eq- uities drops on impact. In the baseline calibration, home residents have a long position in home bond (BH > 0) and 0 < ψF < 1, which implies the total effect of a technological shock on net foreign assets is negative. Similar intuition holds for other types of shocks. Table 2.21 shows that either in a model with complete asset markets or with incomplete asset markets, net foreign asset positions counter-cyclically move with output. Besides portfolio dynamics, I also produce the second moments for other aggregate variables. The benchmark model with complete asset markets delivers similar second moments as in Chari et al. (2002). However, both of theirs and mine suffer common weaknesses. That is, cross country con- sumption correlation is two high compared to the cross correlation between output. Volatility of real exchange rates is much lower than that in data. Nevertheless, cross country correlations for investment and employment in 39Since households smooth their consumption paths, stochastic discount factors in both countries don’t move around much. Thereby, asset prices are mainly pinned down by dividend flows or interest payment flows in future periods. 79 2.6. International portfolio dynamics the current chapter are consistent with data both in a model with complete asset markets or incomplete asset markets. 80 Chapter 3 International Financial Markets and International Risk Sharing How does financial integration increase international risk sharing? Are all participating countries better off in the process of financial globaliza- tion? I’ll investigate this issue based on a center-periphery framework in this chapter. This chapter is organized as follows. Section 3.1 describes a simple one-good world economy and characterizes the optimal portfolios for each country in each stage of financial integration. Section 3.2 extends our results to the case of many goods. Final remarks are in chapter 5. 3.1 A one-good economy The world economy consists of three countries, country A, B, and C. Country A is the central country, which has a continuum of population with nA = 1 2 measure, and the index for population lies in [0, 1 2). Country B and C are peripheral countries and they have equal population size of nB = nC = 1 4 . The index of population in country B has a range of [ 1 2 , 3 4) and the population in country B lies in [34 , 1]. There is an infinitely lived representative agent in each country. In order to illustrate how portfolio choices affect international risk sharing, I first consider a world economy with only one tradable good. In other words, this section focuses on the aspect of common discount factor in international asset pricing and doesn’t take into account of the effect of terms of trade on portfolios and interna- tional risk-sharing. I’ll leave this part to the third section. Each country can issue equities which entitle the claimant a corresponding share of profits in that country. Whether equities can be traded across borders depends on the prevailing international financial architecture. I’ll describe it later. The household in each country receives an endowment and chooses consump- tion and a portfolio. There are no transaction cost for international trade 81 3.1. A one-good economy in goods and assets. The uncertainty in this world economy comes from endowment shocks in these three countries. There are three types of financial architecture in the world economy, which correspond to three stages of international financial integration. The first stage is financial autarky, in which only goods and services can be traded across borders. Trade balance is maintained period by period and there is no international capital flows. With the development of interna- tional financial markets, assets can be gradually traded across countries, but not for all countries. In the current financial architecture, capital flows between developing economies and developed economies are far larger than those among developing economies. Brazil, Argentina, Indonesia, Malaysia and Thailand are good examples of peripheral countries in this type of fi- nancial architecture. The United States and other regional countries, say, the Britain and Singapore are examples of the core country. I call this case as partial financial integration, in which capital can flow freely between the central and peripheral countries, but capital flows are restricted between peripheral countries. The last case is full financial integration, in which all countries can have access to domestic and all foreign financial markets. In this model economy, there are three types shocks, one endowment shock in each country. Full financial integration implies that asset markets are complete up to a first-order approximation. 3.1.1 Households Households in country i have a life-time utility function of the form, Et [ ∞∑ τ=t βi,τU(Ci,τ ) ] (3.1) where the operator Et here represents the conditional expectation over all states of nature, Ci,τ is the consumption in country i = A,B,C in period τ . Following the open macroeconomic literature, i.e. Schmitt-Grohe and Uribe (2003), I take use of an endogenous discount factor to make sure that the rational expectations equilibrium exists, βi,t+1 = βi,tv(i, t), with βi,0 = 1, where v(i, t) ≡ ζ0C̄−ζi,t , 0 < ζ < 1. ζ0 is a parameter which is chosen such that the discount factor at the steady state is equal to the inverse of risk-free rate. C̄i,t is the per capita consumption. When they made decisions, households 82 3.1. A one-good economy treat per-capita consumption as given. The household’s preference has a form of power utility, U(Ci,t) = (Ci,t) 1−σ − 1 1− σ (3.2) where σ is the coefficient of relative risk aversion or the inverse of elasticity of intertemporal substitution. 3.1.2 International financial markets With development of financial markets, asset markets in different coun- tries gradually integrate into one another. Without loss of generality, I consider three stages of international financial market development. The first one is financial autarky, in which there are no trade in assets across countries. the second stage is that financial markets are partially integrated across borders, in which central country A has access to all financial mar- kets in the world while peripheral countries can only hold domestic assets and assets in the core country (see figure 3.1). The last stage is the fully financially integrated world, in which each country has access to all financial markets in the world. Suppose that each country provides one type of asset, equity. Each equity share in country i gives the claimant a corresponding share of profits in that country. For tractability, assume that equity assets have a duration of one-period. Let the price of equity be Qi,t in country i, and the profits are endowments Yi,t in country i. The gross return on equities in country i can be written as, Ri,t = Yi,t Qi,t (3.3) Let Xji,t, i, j = A,B,C, be the share of equities issued by country j and held by country i at the end of period t − 1. I normalize the total share of equities in each country to be unity. The net foreign asset positions, NWi,t+1, then can be written as the sum of holdings of foreign equities by the home household net home equities held by foreigners, NWA,t+1 ≡ ( XAA,t+1 − 1 ) QA,t +X B A,t+1QB,t +X C A,t+1QC,t (3.4) NWi,t+1 ≡ ( Xii,t+1 − 1 ) Qi,t +X A i,t+1QA,t +X j i,t+1Qj,t (3.5) with i, j = B,C and i 6= j. The aggregate resource constraint faced by the central country has the 83 3.1. A one-good economy ff ff ff A B C A A A A AKA A A A AU ff ff ff A B C A A A A AKA A A A AU - ff Figure 3.1: Financial architecture. The left panel is for partial financial integration. The right panel is for global financial integration. A denotes for the central country. B and C stand for peripheral countries. following form, CA,t +NWA,t+1 = YA,t +NWA,tRA,t + α ′ A,tRx,t (3.6) The market value of equities in peripheral country B and C held by the central country is, αA,t ≡ [ XBA,tQB,t;X C A,tQC,t ] and excess returns Rx,t ≡ [RB,t −RA,t;RC,t −RA,t]. Total expenditures on the left hand side consist of consumption expenditure and a portfolio taken to the next period. The right hand side contains resources of total income, which include profits, YA,t, and gross returns on net foreign assets characterized by the last two terms in the equation above. Notice that the household’s budget constraint in country A can be obtained through dividing the population size on both sides of equation (3.6). I focus on aggregate resource constraint instead. Peripheral countries face similar resource constraints. Total income from output and returns on equity assets equals total expenses, Ci,t +NWi,t+1 = Yi,t +NWi,tRA,t + α ′ i,tRx,t (3.7) with i = B,C. The market value of equity holdings in peripheral country B is αB,t ≡ [ (XBB,t − 1)QB,t;XCB,tQC,t ] and it is αC,t ≡ [ XBC,tQB,t; (X C C,t − 1)QC,t ] in country C. I will focus on three stages of financial market development. In the first stage, financial markets are autarky, and the household in each coun- try only hold his own equities, Xii = 1, and X j i = 0, with i, j = A,B,C and i 6= j. In the second stage, the household in the central country has 84 3.1. A one-good economy access to global financial markets and can hold any types of assets in the world. The financial market in the central country is also available to house- holds in peripheral countries. Nevertheless, the household in one peripheral country is completely excluded from the other peripheral country because the under-development of financial markets in peripheral countries. For in- stance, capital account regulation in developing economies prevents foreign- ers from investing in the domestic market, or contract enforcement problems are so prevailing, or information cost is so high that no foreigners would like to invest in these countries. An extreme case considered here is that a pe- ripheral country is completely excluded from the other peripheral country, Xji = 0, i, j = B,C and i 6= j. In the third case, financial markets are glob- ally integrated, households in any country can participate trading assets in any financial market. In the financial autarky case, consumption equals output in each period and each country, Ci,t = Yi,t and X i i = 1, i = A,B,C. When households can choose foreign equities, the optimality conditions for portfolios are char- acterized by the consumption Euler equations in each country. I list these conditions in two separate sets. When the financial architecture is in stage two, only the household in the core country can choose three types of eq- uities and peripheral countries can have access to domestic and the core country’s equities. The optimality conditions for holding equities follow as, UAC,t = Et [ v(A, t)UAC,t+1Rj,t+1 ] , with j = A,B,C (3.8) UBC,t = Et [ v(B, t)UBC,t+1Rj,t+1 ] , with j = A,B (3.9) UCC,t = Et [ v(C, t)UCC,t+1Rj,t+1 ] , with j = A,C (3.10) where U iC,t denotes marginal utility of consumption in country i in period t. These conditions state that the forgone marginal utility of one dollar today equals the expected discounted marginal utility tomorrow from gross returns to such one foregone dollar. When the world economy is in global financial integration of stage three, households in each country can trade assets with the other two countries. The optimality conditions for portfolios read, U iC,t = Et [ v(i, t)U iC,t+1Rj,t+1 ] , with i, j = A,B,C (3.11) Endowments in each country follow an auto-regressive process of order one, log Yi,t+1 = (1− ρ) log Yi + ρ log Yi,t + i,t+1 (3.12) 85 3.1. A one-good economy where Yi stands for the steady state endowments in country i and 0 < ρ < 1 is the persistence of endowment shocks. The innovation i,t+1 has an i.i.d normal distribution with mean zero and standard deviation σ. For tractability of the solution, I assume that innovations across borders are uncorrelated. 3.1.3 Competitive equilibrium The asset market clearing condition requires that the sum of equity shares issued by each country is unity, XjA,t +X j B,t +X j C,t = 1 , with j = A,B,C (3.13) Goods market clearing condition implies that total absorption of goods is equal to total endowments, CA,t + CB,t + CC,t = YA,t + YB,t + YC,t (3.14) Notice also that the global net foreign asset positions should be zero, NWA,t +NWB,t +NWC,t = 0 and αA,t + αB,t + αC,t = 0. Now I have described the world economy. The rational expectations competitive equilibrium is the defined as follows, Definition 1. A competitive equilibrium consists of a sequence of asset prices Qi,t, i = A,B,C, t = 0, 1, 2, · · · and a sequence of allocations Ci,t and of portfolios Xji,t, i, j = A,B,C, t = 0, 1, 2, · · · , such that (a) given prices, consumption and portfolio decisions by the household in each coun- try solve the corresponding household’s problem, and (b) prices clear asset markets in each country. In the deterministic steady state, endowments in each country are con- stant. Let the steady state subjective discount factor be 0 < β ≡ v(i, ·) < 1, i = A,B,C. Gross returns to assets equal R = 1β . I consider a symmetric steady state in which consumption per capita are the same in all countries. Therefore, net foreign asset positions are equal to zero in the steady state and consumption equals output. 3.1.4 Solving the model As shown in Devereux and Sutherland (2011a), I need to take a second- order approximation to Euler equations (3.8)-(3.10) and a first-order approx- imation to other equilibrium conditions to obtain a solution for the steady 86 3.1. A one-good economy state portfolio since certainty equivalence implies that portfolio is indetermi- nate in a linearized model. In order to pin down the steady state portfolio, I need first to solve a linearized version of the rational expectations equi- librium and then substitute the corresponding terms into the second-order approximation to the Euler equations to obtain the steady state portfolios. After obtaining the portfolio, I substitute them back into the solution to the rational expectations equilibrium to get the final solution to the linearized dynamic system. Notice that up to a first-order approximation, expected excess returns on equities (say, the return of equity A relative to the return of equity B) are zero. Thereby, the linearized version of the dynamic sys- tem in partial financial integration is exactly the same as global financial integration. The only difference between these two financial regimes lies in the second-order approximation to consumption Euler equations. The linearized system Let a variable with a hat denote the log deviation from its determinis- tic steady state. Since population in each country is fixed, therefore, the log deviation for an aggregate variable is the same as a per capita vari- able. First, taking a log-linearization to resource constraints (3.6)-(3.7), the relative resource constraints read, 3 4 ÑWB,t+1 + 1 4 ÑWC,t+1 = 3 4β ÑWB,t + 1 4β ÑWC,t + 3 4 α̃′BR̂x,t + 1 4 α̃′CR̂x,t − 1 2 ∆ŶB,t + 1 2 ∆ĈB,t (3.15) 1 4 ÑWB,t+1 + 3 4 ÑWC,t+1 = 1 4β ÑWB,t + 3 4β ÑWC,t + 1 4 α̃′BR̂x,t + 3 4 α̃′CR̂x,t − 1 2 ∆ŶC,t + 1 2 ∆ĈC,t (3.16) where ÑWB,t ≡ 4NWB,t C̃ and ÑWC,t ≡ 4NWC,t C̃ are the ratios of net foreign assets to total output in country B and C, respectively. Notice that net foreign asset positions are zero at the steady state, and therefore, ÑW i,t, i = B,C is a first-order term. α̃i ≡ 4ᾱi βC̃ with i = B,C denotes the ratio of the steady state net equity holdings to total output in peripheral country i. ∆Ĉi,t ≡ ĈA,t − Ĉi,t expresses the consumption of central country A rela- tive to the consumption in peripheral country i, and similarly, the relative 87 3.1. A one-good economy endowment is given by ∆Ŷi,t ≡ ŶA,t − Ŷi,t. Taking a first-order approximation to the Euler equations (3.8)-(3.10), subtracting (3.9) from (3.8) and subtracting (3.10) from (3.8) respectively, yields, (σ − ζ)∆ĈB,t = Et [ σ∆ĈB,t+1 ] (3.17) (σ − ζ)∆ĈC,t = Et [ σ∆ĈC,t+1 ] (3.18) Four equations (3.15)-(3.18) form a dynamic system with two endogenous state variables ÑWB,t+1, ÑWC,t+1, and two endogenous control variables, ∆ĈB,t, ∆ĈC,t. Following Blanchard and Kahn (1980) and Devereux and Sutherland (2011a), the appendix provides steps for the solution in detail. I then substitute the solution to consumption, asset prices and excess returns into the second-order approximation to consumption Euler equations and obtain the steady state portfolios. In the second stage with partial financial integration, the second-order approximations to Euler equations (3.8)-(3.10) read, Et {[ σ∆ĈB,t+1 0 0 σ∆ĈC,t+1 ] R̂x,t+1 } = 0 In the third stage with global financial integration, the second-order approx- imations to Euler equations (3.11) yield, Et { σ∆ĈB,t+1R̂x,t+1 } = 0 Et { σ∆ĈC,t+1R̂x,t+1 } = 0 Observe that there are two restrictions on portfolio choices in partial financial integration while there are four restrictions on portfolio choices in global financial integration. Combining with three asset market clearing conditions (3.13) and two net foreign asset positions, the portfolio in each country can be completely pinned down. Results Table 3.1 and table 3.2 reports the steady state equity holdings by differ- ent countries in partial financial integration and global financial integration, respectively. In global financial integration, households achieve effective risk-sharing through fully diversifying portfolios in international financial markets. For instance, the core country A holds 12 1 1−βρ shares of equities 88 3.1. A one-good economy issued by country B and C, respectively, and it holds 1 − 12 11−βρ shares of domestic equities. Peripheral country B holds 14 1 1−βρ shares of equities of country A, 1− 34 11−βρ shares of domestic equities and 14 11−βρ shares of equities of country C. When international financial markets are partially integrated, peripheral countries don’t have the chance to fully diversify their income risk. The only channel of improving risk sharing is to hold equities issued by the core country. Table 3.1 shows that peripheral country B or C chooses to purchase 27 1 1−βρ shares of equities in the core country, which are higher than those in full financial integration. The core country would hold more equities in the peripheral countries in partial financial globalization than those in global financial integration. I summarize the solution for portfolios in the following proposition. Proposition 1. In a one-good endowment world economy with population size nA = 1 2 , nB = 1 4 and nC = 1 4 , the steady state portfolios don’t depend on the coefficient of relative risk aversion. Moreover, peripheral countries hold more equities issued by the core country in partial financial integration than those in global financial integration. Table 3.1: Optimal equity shares held by each country in partial financial integration (nA = 1 2 , nB = nC = 1 4) Equity A Equity B Equity C Country A 1− 47 11−βρ 47 11−βρ 47 11−βρ Country B 27 1 1−βρ 1− 47 11−βρ 0 Country C 27 1 1−βρ 0 1− 47 11−βρ Table 3.2: Optimal equity shares held by each country in global financial integration (nA = 1 2 , nB = nC = 1 4) Equity A Equity B Equity C Country A 1− 12 11−βρ 12 11−βρ 12 11−βρ Country B 14 1 1−βρ 1− 34 11−βρ 14 11−βρ Country C 14 1 1−βρ 1 4 1 1−βρ 1− 34 11−βρ Once obtaining the steady state portfolios, I substitute these portfolios into the solution to the linearized dynamic system and get the final solution. 89 3.1. A one-good economy Based on these solutions, I compare the volatility of consumption across financial architectures, which is reported in proposition 2. Proposition 2. In a one-good endowment world economy with population size nA = 1 2 and nB = nC = 1 4 , the conditional variance of consumption in all countries are the same in financial autarky and they are also the same in global financial integration. If 0 < 1−β+βζ σ 1−βρ < 7 3 , the conditional variance of consumption in each country in each financial regime is ranked as, vart−1(Ĉ p A,t) < vart−1(Ĉ g A,t) < vart−1(Ĉ a A,t) vart−1(Ĉ g B,t) < vart−1(Ĉ p B,t) < vart−1(Ĉ a B,t) vart−1(Ĉ g C,t) < vart−1(Ĉ p C,t) < vart−1(Ĉ a C,t) where vart−1 denotes variance conditional on information up to period t− 1 and superscript p, g and a stand for a variable in partial financial integra- tion, global financial integration and financial autarky, respectively. Proposition 2 states the main result in this chapter. The volatility of consumption in peripheral countries continually decreases in financial inte- gration while it is not necessary for the core country to experience mono- tonically reduction in consumption volatility. As we know, in the process of international financial integration, peripheral countries have access to more and more international financial markets and their income risk can be grad- ually diversified. Thereby, consumption in these countries becomes much smoother. Nevertheless, the core country in the second stage of financial integration has already got the opportunity to diversify its income risk in global financial markets. It can do better in this medium stage than in full financial integration since peripheral countries are mutually excluded from each other in the medium stage. The proof of proposition 2 can be directly obtained from the solution to the linearized system. In financial autarky, consumption equals out- put in each country and each period, vart−1(Ĉai,t) = σ 2 with i = A,B,C. In global financial integration, individual income risk is fully diversified through international financial markets, and the volatility of consumption is vart−1(Ĉ g i,t) = 3 8σ 2 with i = A,B,C. In partial financial integration, the core country can take advantage of holding global assets to achieve a better risk-sharing than the global financial integration. For the core country, its consumption variance is σ2 ( 1 2 − 114 1−β+βζ σ 1−βρ )2 + 2σ2 ( 1 4 + 1 28 1−β+βζ σ 1−βρ )2 . For 90 3.1. A one-good economy the peripheral countries, the variance of consumption in either country be- comes σ2 ( 1 4 − 14 1−β+βζ σ 1−βρ )2 + σ2 ( 1 2 + 1 14 1−β+βζ σ 1−βρ )2 + σ2 ( 1 4 + 5 28 1−β+βζ σ 1−βρ )2 . Comparing terms listed above, I can get the results in proposition 2. Notice that 0 < β < 1, 0 < ρ < 1, σ > 0 and 0 < ζ < 1. As long as ζ is not too large and σ is not too small, condition 0 < 1−β+βζ σ 1−βρ < 7 3 holds. For reasonable parameter values used in the literature, say, ζ is close to zero and σ > 0.5, this condition definitely holds. To better understand how portfolios alter the volatility of consumption across international financial regimes, let’s turn to the impact response of consumption to exogenous endowment shocks. Let ξ ≡ 114 1−β+βζ σ 1−βρ > 0. The response of consumption in country A to endowment shocks on impact reads, ∂ĈpA,t ∂t = [1, 0, 0]ᵀ︸ ︷︷ ︸ Income effect + [ −1 2 − ξ, 1 4 + 1 2 ξ, 1 4 + 1 2 ξ ]ᵀ ︸ ︷︷ ︸ Wealth effect ∂ĈgA,t ∂t = [1, 0, 0]ᵀ︸ ︷︷ ︸ Income effect + [ −1 2 , 1 4 , 1 4 ]ᵀ ︸ ︷︷ ︸ Wealth effect ∂ĈaA,t ∂t = [1, 0, 0]ᵀ︸ ︷︷ ︸ Income effect where superscript ᵀ denotes the transpose of a vector. The vector of endow- ment shocks is t = [A,t, B,t, C,t] ᵀ. When international financial markets are integrated, there are two channels affecting households’ consumption. The first channel is income effect, which is generated by endowment shocks. The household in country A increases his consumption one for one respond- ing to the domestic endowment shock, given returns on his portfolio un- changed, while the consumption doesn’t change with foreign endowment shocks through this income channel. The income effect is the same across financial regimes from the equations above. The second channel is wealth effect, which works through the changes of returns on portfolios. Since the household in country A has a negative net position in country A’s equities, in other words, foreigners have a long position in equities issued by country A, the household needs to pay dividends to foreigners. In partial integration, country A’s household is required to pay 12 +ξ dollars to households in other countries, while he only needs to pay 12 dollars to foreigners in global finan- 91 3.1. A one-good economy cial integration. On the other hand, the household receives dollar payments from other countries. The results show that country A receives more div- idends from abroad in partial financial integration than in global financial integration. The sum of wealth effect and income effect determines a country’s aggre- gate exposure to income risk. In financial autarky, the domestic household faces individual income risk. In global financial integration, the income risk is fully diversified through international financial markets. Since the core country accounts for one half population of the world, it can only diversify one half of domestic income risk in global financial integration. Neverthe- less, the core country could diversify more domestic income risk by holding more foreign equities and less domestic equities in partial global financial integration. Thereby, the degree of international risk-sharing is higher in partial financial integration when the core country is relatively large. For peripheral country B (similarly for country C, omitted here because of symmetry), the response of consumption to endowment shocks on impact is as follows, ∂ĈpB,t ∂t = [0, 1, 0]ᵀ︸ ︷︷ ︸ Income effect + [ 1 2 + ξ,−3 4 + 5 2 ξ, 1 4 − 7 2 ξ ]ᵀ ︸ ︷︷ ︸ Wealth effect ∂ĈgB,t ∂t = [0, 1, 0]ᵀ︸ ︷︷ ︸ Income effect + [ 1 2 ,−3 4 , 1 4 ]ᵀ ︸ ︷︷ ︸ Wealth effect ∂ĈaB,t ∂t = [0, 1, 0]ᵀ︸ ︷︷ ︸ Income effect The income effect is the same across financial regimes and also the same across countries responding to a domestic endowment shock. Compared to global financial integration, households in the peripheral countries in par- tial financial integration hold more domestic equities and equities issued by the core country. Therefore, risk exposure to domestic and the core country’s income risk is higher in the medium stage of financial integration for peripheral countries. Observe that when 1−β+βζ σ 1−βρ < 1, or equivalently, (1−ρ)σ > ζ, a rise in country C’s endowment can spill over to the other pe- ripheral country and consumption in country B increases, even though these two countries don’t have a direct financial and trade linkage between each 92 3.1. A one-good economy other. This kind of spill-over is propagated through the core country, which holds equity shares issued by country B. When the endowment in country C increases, wealth in both the core country and country C increases. Higher consumption induced by larger wealth leads to households’ more savings in order to smooth consumption overtime, which in turn drive up asset prices in all countries. Households then re-balance their portfolios responding to shocks. The net foreign asset position in country B drops and consumption goes up accordingly. A following question is if the core country achieves a higher risk-sharing in partial integration, why doesn’t the core country in global integration choose the same portfolio as in partial integration? The reason is that households face different asset prices in these two regimes and thereby the optimal portfolio for the core country in partial integration is no longer feasible in global integration. Country size and risk-sharing In the baseline model, I consider that the core country has a dominant population in the world. In order to explore whether the results in the base- line model are robust to country size, I investigate the optimal portfolios and consumption volatility under different country sizes. Peripheral coun- tries are still assumed to be symmetric, nB = nC = n, while the size of the central country becomes nA = 1 − 2n. Let the core country be the larger country, 0 < n ≤ 13 . Following the same procedure of solving the rational expectations equi- librium in the previous section, the optimal portfolios for each country are listed in table 3.3 and table 3.4. Similar to the benchmark case, peripheral countries would hold more shares of equities issued by the core country in partial financial integration than in global financial integration. Proposi- tion 3 shows that global financial integration lowers consumption volatility for all countries compared to financial autarky, regardless of country sizes. Moreover, when the size of the core country is relatively small (it is still the largest country), for instance, nA = 1 3 , consumption volatility in the core country monotonically decreases in the process of financial integration. Nevertheless, when it is large enough, the core country achieves the best risk-sharing in partial financial integration, as stated in proposition 2. Proposition 3. In a one-good endowment world economy with population size nA = 1 − 2n and nB = nC = n, 0 < n ≤ 13 , the conditional variance of consumption in all countries are the same in financial autarky and they 93 3.1. A one-good economy are also the same in global financial integration; volatility of consumption is lower in global financial integration, that is, vart−1(Ĉ g i,t) < vart−1(Ĉ a i,t), with i = A,B,C. If n(1 + χ) > 23 > nχ, with χ ≡ 22−n − βn(1−ρ− ζσ ) (2−n)(1−βρ) , the ranking of conditional variance of consumption for the core country reads, vart−1(Ĉ g A,t) < vart−1(Ĉ p A,t) < vart−1(Ĉ a A,t) If n(1+χ) < 23 , the conditional variance of consumption in the core country is ranked as follows, vart−1(Ĉ p A,t) < vart−1(Ĉ g A,t) < vart−1(Ĉ a A,t) The conditional variance of consumption in peripheral countries is ranked as follows, vart−1(Ĉ g j,t) < vart−1(Ĉ p j,t) < vart−1(Ĉ a j,t) ,j = B,C Risk-sharing premium Another way to explore the cost and benefit of international risk-sharing is to examine the cost of insurance in different stages of financial develop- ment. When households can pool individual income risk together, the cost of insurance would fall. Lower cost of insurance implies higher expected returns on insurance given the ex post payoffs by assets. As we know, ex- pected returns on assets are the same across asset types up to a first-order approximation. I need to use higher order approximations to pin down ex- pected asset returns. Based on the Euler conditions for country A’s house- hold (other Euler equations deliver the same results), the expected excess returns on the equity issued by country j, j = B,C, can be expressed as, Et (Rj,t+1 −RA,t+1) = RσEt ( ĈA,t+1(R̂j,t+1 − R̂A,t+1) ) = Rσcovt ( ĈA,t+1, R̂j,t+1 − R̂A,t+1 ) where covt denotes covariance conditional on information up to date t. This condition indicates that equity j will entitle its holders an excess return if it can not hedge against consumption risk. Once solving for the dynamic system, I obtain excess returns on equities in different stages of financial integration. In partial financial integration, 94 3.1. A one-good economy the excess returns read, Et ( Rpj,t+1 −RpA,t+1 ) = Rσ2Aσ(3nχ− 1) with, j = A,B (3.19) When the core country is large, that is, a small n such that 3nχ − 1 < 0, the expected return on equity A is higher than that on equity j since equity j can effectively hedge against the consumption income risk for country A’s household. Thereby, equity A should compensate its share holders more in order to make them hold this type of asset. When 3nχ − 1 > 0, equity j needs to pay its holders excess returns. Notice that ∂χ∂n > 0, when peripheral countries are large enough, the latter condition holds. In global financial integration, asset markets are complete up to a first- order condition. The excess returns on equities are as follows, Et ( Rgj,t+1 −RgA,t+1 ) = Rσ2Aσ(3n− 1) with, j = A,B (3.20) When three countries are equally large, n = 13 , returns on all equities are the same up to a second-order approximation. Nevertheless, equity A has an excess return over other equities in all other cases with 0 < n < 13 . When the core country is large, the equity shares in this country account for a dominant part of assets in international financial markets. In other words, equity A is backed by the major source of uncertainty in the world economy. The household in country A wants to get rid of his income risk by selling his own equity shares to foreigners. Nevertheless, on the demand side, households in peripheral countries can’t completely absorb the amount of equity shares sold by the core country because of their smaller country sizes. In equilibrium, the price of equity A must be driven down and the expected return on equity A must be pushed up to attract foreigners to hold this type of equity. With the liberalization of international financial markets, households in peripheral countries gradually have access to more financial assets in other countries. The expansion of the number of assets available to households could reduce the cost of insurance against country-specific shocks. In partial financial integration, the demand for equity A by peripheral households is higher than that in global integration, which in turn drives up the price of equity A and consequently lowers its expected excess return rate. Peripheral households gain a premium in asset markets from partial integration to global integration. Let the international risk-sharing premium be measured 95 3.1. A one-good economy by, RSPj ≡ Et ( RgA,t+1 −Rgj,t+1 ) − Et ( RpA,t+1 −Rpj,t+1 ) = 3nRσ2Aσ(χ− 1) > 0 with, j = A,B (3.21) which states that households in peripheral countries obtain a positive risk- sharing premium in the process of financial integration. This premium in- creases in the size of peripheral countries and in the coefficient of relative risk aversion. Some special shock processes In the previous section, I assume that shocks across countries are not correlated. The motivation for this simplified assumption is to introduce the need for risk-sharing across borders. If shocks in peripheral countries are perfectly positively correlated, of course, there is no need for the periph- eral countries to share individual income risk in global financial integration because of symmetry. The volatility of consumption will be the same both in partial integration and global integration. On the other hand, if these shocks are perfectly negatively correlated, peripheral countries can arrive at perfect risk-sharing by holding their equities only. The core country doesn’t have other asset markets to share its income risk. The volatility of consump- tion for the core country in global financial integration will be the same as the case of financial autarky. For general shocks processes, whether financial integration improves international risk-sharing depends on country sizes, as outlined in proposition 3. Table 3.3: Optimal equity shares held by each country in partial financial integration (nA = 1− 2n, nB = nC = n) Equity A Equity B Equity C Country A 1− 4n2−n 11−βρ 2(1−2n)2−n 11−βρ 2(1−2n)2−n 11−βρ Country B 2n2−n 1 1−βρ 1− 2(1−2n)2−n 11−βρ 0 Country C 2n2−n 1 1−βρ 0 1− 2(1−2n)2−n 11−βρ 96 3.2. A model with multiple tradable goods Table 3.4: Optimal equity shares held by each country in global financial integration (nA = 1− 2n, nB = nC = n) Equity A Equity B Equity C Country A 1− 2n1−βρ 1−2n1−βρ 1−2n1−βρ Country B n1−βρ 1− 1−n1−βρ n1−βρ Country C n1−βρ n 1−βρ 1− 1−n1−βρ 3.2 A model with multiple tradable goods In the previous section, I focus on the common discount factor and ab- stract from an important aspect of international risk-sharing, terms of trade. As emphasized by Cole and Obstfeld (1991), terms of trade could play an important role in international risk sharing. The basic logic is that terms of trade will appreciate responding to an adverse productivity shock in a country. Even though international financial markets may not be available to households in some countries, as long as goods and services can freely flow across borders, terms of trade could still buffer the adverse effect of a nega- tive productivity shock. A follow-up question is how international financial integration improves international risk-sharing when I take into account of both terms of trade and the common discount factor. Another new element in this section is that I consider a production economy in which labor ser- vices are the only input. With endogenous labor supply, households have an extra channel to smooth consumption overtime. The model structure is similar to the previous section. I present only the new model elements here. Country size is the same as baseline case, nA = 1 2 and nB = nC = 1 4 . Each country has two types of agents, households and firms. Households in each country choose a consumption composite, labor supply and an international portfolio in each period. Firms produce a unique type of tradable good. Therefore, firms in different countries produce dif- ferent kinds of tradable goods. The productivity suffers a random shock in each country. Final goods producers in each country combine three trad- able goods into domestic consumption composites, which is consumed by domestic households only. I will investigate three stages of financial devel- opment: financial autarky, partial financial integration and global financial integration. 97 3.2. A model with multiple tradable goods 3.2.1 Consumption composite Consumption composites in each country require locally produced goods and imported goods from the other two countries. The consumption com- posite in country A, CA,t, has a form of, CA,t = [ α 1 γ 1 (C A A,t) γ−1 γ + ( 1 2 ) 1 γ (1− α1) 1 γ (CBA,t) γ−1 γ + ( 1 2 ) 1 γ (1− α1) 1 γ (CCA,t) γ−1 γ ] γ γ−1 (3.22) where γ > 0 is the elasticity of substitution between imports and exports. CjA,t, i = A,B,C, is consumption demand for goods produced at country j by the household in country A. In the symmetric deterministic steady state, α1 is the share of consumption spending devoted to goods produced by country A. A preference bias towards local goods implies, 1/2 < α1 < 1. Expenditure minimization associated with the consumption composite (3.22) gives the consumer price index at country A as, PA,t = [ α1(P A A,t) 1−γ + 1− α1 2 (PBA,t) 1−γ + 1− α1 2 (PCA,t) 1−γ ] 1 1−γ (3.23) where P jA,t, j = A,B,C, denotes the price of good j in terms of currency in country A. Demand for goods produced at country A by country A consumers is given by CAA,t = α1(P A A,t/PA,t) −γCA,t. Demand for coun- try j, j = B,C, goods by country A consumers has a form of CjA,t = 1−α1 2 (P j A,t/PA,t) −γCA,t. Analogously, in the peripheral world, country B and C have a similar consumption composite defined over three types of goods, Ci,t = [ (1− α1) 1 γ (CAi,t) γ−1 γ + (α1α2) 1 γ (Cii,t) γ−1 γ + (α1(1− α2)) 1 γ (Cji,t) γ−1 γ ] γ γ−1 (3.24) where Ci,t, i = B,C, denotes consumption composite in country i. C j i,t, j = B,C and j 6= i, represents the consumption demand for goods produced at country j (source country) by agents in country i (destination country). α1 is the share of consumption expenditure on goods produced by peripheral countries. α2 is the share of local goods in the peripheral consumption composite at country B and C. Thereby, agents in country B or C spend a 98 3.2. A model with multiple tradable goods share of α1α2 on local products. Expenditure minimization implies that, Pi,t = [ (1− α1)(PAi,t)1−γ + α1α2(P ii,t)1−γ + α1(1− α1)(P ji,t)1−γ ] 1 1−γ (3.25) with i = B,C. Pi,t denotes the consumer price index in country i and P ji,t is the price of goods produced at country j in terms of currency at country i. Demand for goods produced in country j, j = B,C, by agents at country i, i = B,C can be written as, CAi,t = (1 − α1) ( PAi,t/Pi,t )−γ Ci,t, Cii = α1α2 ( P ii,t/Pi,t )−γ Ci,t and C j i = α1(1 − α2) ( P ji,t/Pi,t )−γ Ci,t with i 6= j. Since goods can be freely traded across borders without any cost, the law of one price holds for each good variety. Furthermore, suppose that all countries take use of international currency, say, dollar, as their account unit, thereby, PAA,t = P A B,t = P A B,t, P B A,t = P B B,t = P B C,t and P C A,t = P C B,t = P C C,t. For simplicity, let the goods produced in country A be the numeraire and set PAA,t = 1. 3.2.2 Households Households in country i have a life-time utility function of the form, Et [ ∞∑ τ=t βi,τU(Ci,τ , Ni,τ ) ] (3.26) where the operator Et here represents the conditional expectation over all states of nature, Ci,τ is the consumption, and Ni,τ stands for labor supply in country i = A,B,C. βi,τ is the endogenous subjective discount factor. The household’s preference has a form of, U(Ci,t, Ni,t) = (Ci,t) 1−σ − 1 1− σ − η0 (Ni,t) 1+η 1 + η (3.27) where σ is the coefficient of relative risk aversion or the inverse of elasticity of intertemporal substitution, η characterizes Frisch elasticity of labor supply.40 Labor market is competitive, and therefore labor supply is determined 40Christiano, Trabandt and Walentin (2010) propose another interpretation of labor supply Ni,t that it measures numbers of people working rather than hours of working. The value of η in the calibration is consistent with this interpretation. However, I still use Frisch labor supply elasticity for 1/η here for convenience. 99 3.2. A model with multiple tradable goods by real wage and marginal rate of substitution between consumption and leisure, − U i N,t U iC,t = Wi,t Pi,t (3.28) where U iC,t and U i N,t stand for marginal utility of consumption and marginal disutility of labor supply, respectively in country i at time t. Wi,t is wage rate. 3.2.3 Financial markets There are three types of equities in financial markets. The claimants of equities obtain a corresponding share of profits in the issuance country. Let the profits from production be Πi,t at country i, which is defined later. The gross return on equities at country i can be written as, Ri,t = Πi,t Qi,t The resource constraint faced by agents in the central country has the following form, PA,tCA,t +NWA,t+1 = NA,tWA,t + ΠA,t +NWA,tRA,t + α ′ A,tRx,t (3.29) Expenditures on the left hand side consist of consumption expenditure and a portfolio taken to the next period. The right hand side contains the resources of total income, which include labor income, NA,tWA,t, profits, ΠA,t, and gross returns on net foreign assets in the last two terms. Peripheral countries face similar resource constraints. The total income from sales of output and returns on equity assets equals total expenses, Pi,tCi,t +NWi,t+1 = Ni,tWi,t + Πi,t +NWi,tRA,t + α ′ i,tRx,t (3.30) with i = B,C. Net equities in peripheral country B is αB,t ≡ [(XBB,t − 1)QB,t;X C B,tQC,t] and in country C is αC,t ≡ [XBC,tQB,t; (XCC,t − 1)QC,t]. In partial financial integration, the core country A has access to global financial assets, the optimality condition for its portfolio choice is, UAC,t PA,t = Et [ v(A, t) UAC,t+1 PA,t+1 Rj,t+1 ] , with j = A,B,C (3.31) 100 3.2. A model with multiple tradable goods This condition is the usual consumption Euler equation for equities issued by country j, which states that the forgone marginal utility of one dollar today equals the expected discounted marginal utility tomorrow from gross returns to such one dollar asset j purchased in period t. In the peripheral sphere, households in country B (or C) have access to equities in the central country A and their own asset market. Their corresponding optimal conditions for portfolios yield, UBC,t PB,t = Et [ v(B, t) UBC,t+1 PB,t+1 Rj,t+1 ] , with j = A,B (3.32) UCC,t PC,t = Et [ v(C, t) UCC,t+1 PC,t+1 Rj,t+1 ] , with j = A,C (3.33) In global financial integration, households in any country can trade eq- uities with households in other countries. The optimality conditions are as follows, U iC,t Pi,t = Et [ v(i, t) U iC,t+1 Pi,t+1 Rj,t+1 ] , with i, j = A,B,C (3.34) 3.2.4 Firms At time t, the homogenous goods in each country are produced by a competitive, representative firm. The firm produces the good using labor services according to the following technology, Yi,t = Zi,tN 1−θ i,t (3.35) with 0 < θ < 1. 1−θ is labor income share and Ni,t denotes the time t labor demand. Total factor productivity, Zi,t, is an exogenous random variable, which has the following form of, logZi,t+1 = (1− ρ) logZi + ρ logZi,t + i,t+1 where Zi stands for the steady state technology in country i and ρ charac- terizes the persistence of technology shocks. Technology innovation i,t has an i.i.d. normal distribution with mean zero and standard deviation σ,i. In the competitive labor market, labor demand is determined by the 101 3.2. A model with multiple tradable goods marginal product of labor and wages, Wi,t = P ii,tYi,t(1− θ) Ni,t Subtracting payrolls from the firm’s total revenues, I obtain profits in coun- try i, Πi,t = P i i,tYi,t −Ni,tWi,t 3.2.5 Competitive equilibrium The asset market clearing condition requires that the sum of equity shares for each type of equity is unity, XjA,t +X j B,t +X j C,t = 1 , with j = A,B,C (3.36) Goods market clearing condition implies that total absorption of good j is equal to its total production in country j , CjA,t + C j B,t + C j C,t = Yj,t , with j = A,B,C (3.37) Notice also that the global net foreign asset positions should be zero, NWA,t +NWB,t +NWC,t = 0 and αA,t + αB,t + αC,t = 0. The rational expectations competitive equilibrium is the defined as fol- lows, Definition 2. The competitive equilibrium consists of a sequence of goods prices P ii,t, wages Wi,t and asset prices Qi,t, i = A,B,C, t = 0, 1, 2, · · · and a sequence of allocations Ci,t, Ni,t and of portfolios X j i,t, i, j = A,B,C, t = 0, 1, 2, · · · , such that (a) given prices, consumption, labor and portfolio decisions by the household in each country solve the household’s and firms’s problems, and (b) prices clear labor markets, goods markets and asset mar- kets in each country. 3.2.6 Steady state In the deterministic steady state, the technological level keeps constant in each country. I consider a symmetric steady state in which consumption per capita, output per capita and labor service per capita are the same in all countries. Thereby, net foreign asset positions are equal to zero in the steady state and consumption per capital equals output per capita. Relative prices in the steady state are equal to one. Let a variable without time subscript 102 3.2. A model with multiple tradable goods denote its deterministic level and a variable with a tilde on the head denote per capita variable. After some algebraic manipulation, I have the steady state technology ZB = ZC = 2 −θZA, and labor supply, Ñ = [ 1− θ η0 ( 2θZA )1−σ] 11+η−(1−θ)(1−σ) and output thereby can be written as, Ỹ = 2θZA [ 1− θ η0 ( 2θZA )1−σ] 1−θ1+η−(1−θ)(1−σ) Other variables in the steady state are as follows. YA = 1 2 Ỹ , YB = YC = 1 4 Ỹ , Πi = θYi, Qi = θβYi with the inverse of gross returns to assets, β, Ci = Yi. 3.2.7 Solving portfolios and the dynamic system First, wee need to work out the log-linearized version of the model. Let a variable with a hat denote the log deviation from its deterministic steady state. Since population in each country is fixed and thereby log derivation for an aggregate variable is the same as a per capita variable. Let the goods produced by the central country be numeraire, PAA,t = 1. Log-linearized consumption demand for each good in each country yields, ĈAA,t = γP̂A,t + ĈA,t, Ĉ B A,t = γ(P̂A,t − P̂BB,t) + ĈA,t, ĈCA,t = γ(P̂A,t − P̂CC,t) + ĈA,t ĈAB,t = γP̂B,t + ĈB,t, Ĉ B B,t = γ(P̂B,t − P̂BB,t) + ĈB,t, ĈCB,t = γ(P̂B,t − P̂CC,t) + ĈB,t ĈAC,t = γP̂C,t + ĈC,t, Ĉ B C,t = γ(P̂C,t − P̂BB,t) + ĈC,t, ĈCC,t = γ(P̂C,t − P̂CC,t) + ĈC,t The log-linearized versions of price indices (3.23) and (3.25) can be writ- ten as, P̂A,t = 1− α1 2 (P̂BB,t + P̂ C C,t) P̂B,t = α1α2P̂ B B,t + α1(1− α2)P̂CC,t P̂C,t = α1(1− α2)P̂BB,t + α1α2P̂CC,t (3.38) Combining optimal labor supply (3.28) and production (3.35), yields 103 3.2. A model with multiple tradable goods labor demand in country i = A,B,C, and the linearized equations read, N̂i,t = 1 θ + η ( P̂ ii,t − P̂i,t − σĈi,t + Ẑi,t ) (3.39) Ŷi,t = 1 θ + η Ẑi,t + 1− θ θ + η ( P̂ ii,t − P̂i,t − σĈi,t ) (3.40) Linearizing goods market clearing conditions (3.37) and get, 2ŶA,t = 2α1ĈAA,t + (1− α1)ĈAB,t + (1− α1)ĈAC,t (3.41) ŶB,t = (1− α1)ĈBA,t + α1α2ĈBB,t + α1(1− α2)ĈBC,t (3.42) ŶC,t = (1− α1)ĈCA,t + α1(1− α2)ĈCB,t + α1α2ĈCC,t (3.43) I then express output, labor, and prices as functions of consumption and exogenous shocks. Substitute these variables into the linearized version of budget constraints (3.29)-(3.30) and obtain, 3 4 ÑWB,t+1 + 1 4 ÑWC,t+1 = 3 4β ÑWB,t + 1 4β ÑWC,t + 3 4 α̃′BR̂x,t + 1 4 α̃′CR̂x,t + ΓA1∆ẐB,t + ΓA2∆ẐC,t + ΓA3∆ĈB,t + ΓA4∆ĈC,t (3.44) 1 4 ÑWB,t+1 + 3 4 ÑWC,t+1 = 1 4β ÑWB,t + 3 4β ÑWC,t + 1 4 α̃′BR̂x,t + 3 4 α̃′CR̂x,t + ΓA2∆ẐB,t + ΓA1∆ẐC,t + ΓA4∆ĈB,t + ΓA3∆ĈC,t (3.45) where ÑWB,t ≡ 4NWB,t C̃ and ÑWC,t ≡ 4NWC,t C̃ are the ratios of net foreign assets to GDP in country B and C, respectively. Notice that net foreign asset positions are zero at the steady state, and therefore, ÑW i,t, i = B,C is a first-order term. α̃i ≡ 4ᾱi βC̃ with i = B,C denotes the ratio of the zeroth order net equity holdings to GDP in the peripheral country i. ∆Ĉi,t ≡ ĈA,t− Ĉi,t expresses consumption of central country A relative to consumption at peripheral country i, and similarly, relative technological level is given by ∆Ẑi,t ≡ ẐA,t − Ẑi,t. ΓAs with s = 1, 2, 3, 4 are functions of structural parameters in the model, which are given in the appendix. Taking a first-order approximation to the Euler equations (3.31)-(3.33), subtracting (3.32) from (3.31) and subtracting (3.33) from (3.31) respec- 104 3.2. A model with multiple tradable goods tively, yields, (σ − ζ)∆ĈB,t + ∆P̂B,t = Et [ σ∆ĈB,t+1 + ∆P̂B,t+1 ] (σ − ζ)∆ĈC,t + ∆P̂C,t = Et [ σ∆ĈC,t+1 + ∆P̂C,t+1 ] with ∆P̂i,t ≡ P̂A,t − P̂i,t, i = B,C. Substituting relative prices out in equations above using relative consumption and technology, I obtain, Et [ (σ − ΓB3)∆ĈB,t+1 − ΓB4∆ĈC,t+1 ] = −Et [ ΓB1∆ẐB,t+1 + ΓB2∆ẐC,t+1 ] + (σ − ζ − ΓB3)∆ĈB,t − ΓB4∆ĈC,t + ΓB1∆ẐB,t + ΓB2∆ẐC,t (3.46) Et [ (σ − ΓB3)∆ĈC,t+1 − ΓB4∆ĈB,t+1 ] = −Et [ ΓB1∆ẐC,t+1 + ΓB2∆ẐB,t+1 ] + (σ − ζ − ΓB3)∆ĈC,t − ΓB4∆ĈB,t + ΓB1∆ẐC,t + ΓB2∆ẐB,t (3.47) where ΓBs, s = 1, 2, 3, 4 are functions of parameters in the model, which are defined in the appendix. Up to a first-order approximation, portfolio choices are indeterminate because of certainty equivalence, I need a second-order approximation to ob- tain the zeroth-order portfolio. Following Devereux and Sutherland (2011a), I take a second-order approximation to the Euler equations (3.31)-(3.33) in partial financial integration, and obtain, Et {[ −σ∆ĈB,t+1 −∆P̂B,t+1 0 0 −σ∆ĈC,t+1 −∆P̂C,t+1 ] R̂x,t+1 } = 0 (3.48) Notice that when financial markets are not fully accessible to all countries, the asset market is incomplete. There are two constraints on portfolio choices imposed by equations (3.48). Combining with three asset market clearing conditions (3.36) and another two symmetric asset holdings, say country B v.s C and central country v.s. peripheral countries, I have seven equations and seven unknowns for portfolio Xji . Once I solve the linearized rational expectations equilibrium and I can obtain the zeroth portfolio. When financial markets are globally integrated together, the second- 105 3.2. A model with multiple tradable goods order approximations to Euler equations (3.34) read, Et [ R̂x,t+1(−σ∆ĈB,t+1 −∆P̂B,t+1) ] = 0 Et [ R̂x,t+1(−σ∆ĈC,t+1 −∆P̂C,t+1) ] = 0 There are four constraints on portfolio choices imposed by Euler equations. Combining with three asset market clearing conditions and two symmet- ric conditions for asset holdings, say country B v.s C and central country v.s. peripheral countries, I have nine equations and nine unknowns for the portfolio Xji . Until now, I have reduced the linearized dynamic system to a system of four equations, that is, two resource constraints (3.44)-(3.45) and two Euler equations (3.46)-(3.47). There are four unknowns ∆ĈB,t, ∆ĈC,t, ÑWB,t+1 and ÑWC,t+1. ∆ẐB,t and ∆ẐC,t are exogenous variables. Since excess return vector R̂x,t+1 is unpredictable conditional on information up to date t and thereby is uncorrelated with relative consumption ∆Ĉi,t and relative technology ∆Ẑi,t, i = B,C. Let ξi,t+1 ≡ α̃′iR̂x,t+1, which is uncorrelated with variables at date t. Let a vector xt ≡ [ÑWB,t; ÑWC,t], which contains the predetermined variables, and a vector yt ≡ [∆C̃B,t; ∆C̃C,t], which includes control variables. The four-equation system can be rewritten as follows,[ xt+1 Et(yt+1) ] = Ã [ xt yt ] + f̃t (3.49) Coefficient matrix Ã and exogenous process f̃t are defined in the appendix. There are four eigenvalues for system (3.49), λ1 = 1− ζ σ − [ α1(2α2−1)−σ(θ−1)η+θ ] (α1−2α1α2) γ−α21γ(2α2−1)2+ (α1(2α2−1)−1)(θ−1)η+θ λ3 = 1 β λ2 = 1− ζ σ + (2α1−1) [ σ(θ−1) η+θ −2α1+1 ] 2(α1−1) [ 2α1γ− θ−1η+θ ] λ4 = 1 β Observe that λ2 < 1 when 1 2 ≤ α1 ≤ 1 and λ3 > 1, λ4 > 1. λ1 < 1 over a wide range of reasonable parameter values. Thereby, the dynamic system in this model has a unique solution according to Blanchard and Kahn (1980). 106 3.2. A model with multiple tradable goods The appendix provides details about how to find the solution. For the control vector, I have, yt+1 = D1ξt+1 +D2εt+1 +D3 [ xt+1 ∆Ẑt ] (3.50) where coefficient vectors (matrices) Ds, s = 1, 2, 3, are defined in the ap- pendix. Relative technology vector is ∆Ẑt ≡ [∆ẐB,t; ∆ẐC,t], exogenous shock vector is given by εt ≡ [A,t; B,t; C,t] and unexpected excess returns to assets ξt+1 ≡ [ξA,t+1; ξB,t+1]. The evolution of state vector xt+1 is deter- mined by the upper line of equation (3.49). Ex post excess returns on assets can be expressed as a function of ex- ogenous shocks, R̂B,t+1 − R̂A,t+1 = R11ξt+1 +R21εt+1 (3.51) R̂C,t+1 − R̂A,t+1 = R12ξt+1 +R22εt+1 (3.52) where Rsk, s, k = 1, 2, are constant and defined in the appendix. Recall that the expression for unexpected excess returns to assets ξt+1 = α̃R̂x,t with α̃ ≡ [α̃′B; α̃′C ]. In the benchmark model, international financial markets are partially integrated and there is no financial asset flows cross peripheral countries. Thereby α̃ = [ α̃11 0 0 α̃22 ] with some constant α̃ii, i = 1, 2. Next I write the marginal utility of peripheral countries relative to the central country as, −σ∆ĈB,t+1 −∆P̂B,t+1 = D11ξt+1 +D21εt+1 + otherst+1 (3.53) −σ∆ĈC,t+1 −∆P̂C,t+1 = D12ξt+1 +D22εt+1 + otherst+1 (3.54) where otherst+1 stands for terms which are predetermined at data t + 1. Similarly to Rsk, Dsk, s, k = 1, 2, are also constant and are defined in the appendix. I then substitute relative excess returns in equation (3.51)-(3.52) and relative consumption governed by equation (3.50) into the second-order ap- proximation to Euler equations (3.48) and achieve the following results. Result 1. When international financial markets are partially integrated, say, there isn’t asset trade within peripheral countries, optimal equities α̃ 107 3.2. A model with multiple tradable goods held by each country are determined by the following two equations, (D11H̃ +D21)Σ ˜̃ R1 = 0 (3.55) (D12H̃ +D22)Σ ˜̃ R2 = 0 (3.56) where H̃ = α̃ ˜̃ R, ˜̃ R = R̃−1 [ R21 R22 ] , and R̃ = I2− [ R21 R22 ] α̃. ˜̃ Ri is the trans- pose of the i-th row of matrix ˜̃ R. Σ is the variance-covariance of exogeneous shock vector εt. Result 1 shows that the optimal portfolio choices are determined by a nonlinear equation systems. When the financial market in a peripheral country is separated from the other peripheral country, the asset market is incomplete. Analogously, complete financial integration makes asset market complete up to a first-order approximation, in which agents in each country could achieve perfect risk sharing up to a first-order approximation. The optimal equities held by each country can be obtained by the following result. Result 2. When international financial markets are fully integrated, the optimal equities α̃ held by each country are determined by the following four equations, (D11H̃ +D21)Σ ˜̃ R ′ = 0 (3.57) (D12H̃ +D22)Σ ˜̃ R ′ = 0 (3.58) where H̃ = α̃ ˜̃ R, ˜̃ R = R̃−1 [ R21 R22 ] , and R̃ = I2 − [ R21 R22 ] α̃. Σ is the variance-covariance of exogenous shock vector εt. Once obtain the optimal zeroth-order portfolio α̃, I have the final so- lution to the linearized rational expectations equilibrium. The results are summarized in the following proposition. Proposition 4. The solution to the linearized rational expectations equilib- rium is as follows, yt = D3 [ xt ∆Ẑt−1 ] + (D1H̃ +D2)εt (3.59) 108 3.2. A model with multiple tradable goods xt+1 = 1 β xt + S1ρ∆Ẑt−1 + S2yt + (H̃ + S1Φ)εt (3.60) ∆Ẑt = ρ∆Ẑt−1 + Φεt (3.61) where S1 = [ 3 2 −12 −12 32 ] [ ΓA1 ΓA2 ΓA2 ΓA1 ] , S2 = [ 3 2 −12 −12 32 ] [ ΓA3 ΓA4 ΓA4 ΓA3 ] and Φ = [ 1 −1 0 1 0 −1 ] . The control variables are stacked in vector yt = [∆ĈB,t; ∆ĈC,t] and endogenous state variables are stacked in xt = [ÑWB,t; ÑWC,t]. Exogenous variables in (3.61) are included in ∆Ẑt = [∆ẐB,t; ∆ẐC,t]. 3.2.8 Impulse-responses Based on the proposition 4, consumption in each country can be explic- itly written as, ĈA,t = 1 2 θ + η θ + η + (1− θ)σ [1, 1] [ S1Φ + S2(D1H̃ +D2) ] εt − 1− α1 2 1 + η θ + η + (1− θ)σ [1, 1] [ ΓP1Φ− ΓP2(D1H̃ +D2) ] εt + 1 + η θ + η + (1− θ)σ [1, 0, 0]εt + otherst ≡fCAεt + otherst (3.62) There are three innovation terms affecting unexpected change in consump- tion conditional on predetermined variables at date t, otherst. The top line on the right hand of equation (3.62) shows the wealth effect, which indicates the effect of unanticipated excess returns to external wealth on consump- tion due to unanticipated shocks. The second line displays the terms of trade effect, which states that consumption moves with the change of rel- ative prices due to innovations. The third line reports the income effect, which comes from the change of output due to technological changes given all else constant. The income effect appears only when domestic innovation is present. Consumption in the peripheral countries has a form of, ĈB,t = fCAεt − [ (D1H̃ +D2) ] U εt + otherst (3.63) ĈC,t = fCAεt − [ (D1H̃ +D2) ] L εt + otherst (3.64) 109 3.2. A model with multiple tradable goods where [·]U and [·]L denote the upper row and lower row of matrix [·]. These two consumption response functions come directly from the policy function for relative consumption in proposition 4. Analogously, labor service in the central country has a similar exposition, N̂A,t =− 1 2 σ θ + η + (1− θ)σ [1, 1] [ S1Φ + S2(D1H̃ +D2) ] εt − 1− α1 2 1− σ θ + η + (1− θ)σ [1, 1] [ ΓP1Φ− ΓP2(D1H̃ +D2) ] εt + 1− σ θ + η + (1− θ)σ [1, 0, 0]εt + otherst ≡fNAεt + otherst (3.65) Notice that labor supply negatively responds to external wealth since higher external wealth implies higher consumption and then make leisure more desirable, which induces deduction of labor services in the labor market. The response of labor supply to price and technology changes depend on the elasticity of intertemporal substitution. When agents have log-preference σ = 1, income effect and terms of trade effect play no roles in determining labor services. Labor demands in the peripheral countries are characterized as follows, N̂B,t = fNAεt − fNBRεt + otherst (3.66) N̂C,t = fNAεt − fNCRεt + otherst (3.67) where fNBR and fNCR are constant vectors given in the appendix. 3.2.9 Calibration and numerical results As I have shown that the optimal portfolio held by each country is deter- mined by nonlinear equations in result (1) and (2). In order to calculate the business cycle properties as well as to compare volatility of consumption and output across different financial architectures, I resort to numerical results instead. The parameters in table 3.5 used here are very standard and taken from the literature. The model period is assumed to be yearly. Thereby, the discount rate in the steady state is set as β = 0.96, with a 4% annu- alized interest rate. There are two parameters in the endogenous discount factor, ζ and ζ0. ζ is set to be a small positive number, say, 0.001, and ζ0 is chosen such that the steady state discount rate is β = 0.96. The elasticity of intertemporal substitution is chosen at σ = 2. The Frisch labor elastic- 110 3.2. A model with multiple tradable goods Table 3.5: Parameter values Parameter Description Value β Steady state discount rate 0.96 σ Risk aversion 2 η Frisch labor elasticity parameter 1 η0 Coefficient for leisure 1 γ Elasticity of substitution between home and foreign goods 2 α1 = α2 Share of home-traded goods 0.85 ζ Parameter in the endogenous discount factor 0.001 ζ0 Parameter in the endogenous discount factor varying 1− θ Labor income share 23 ρ Persistence of TFP shocks 0.90 σ Standard deviation of TFP shocks 1.36% ity is η = 1. Elasticity of substitution between imports and local goods is assumed to be γ = 2 in the benchmark case. There is no consensus about this trade elasticity in the literature (see a discussion in chapter 2) and I take use of a wide range of values of trade elasticity in the following section as a sensitivity check. The expenditure share of home goods in the home consumption basket is set at α1 = α2 = 0.85, which implies that households in country A spend 85% of income on goods produced by country A and that household in country B and C spend 85% × 85% = 72.25% of income on locally produced goods. Labor income share is 1−θ = 2/3, which implies that dividend share is 1/3. Technologies are assumed to follow an AR(1) process with persistence ρ = 0.90 and a standard deviation of σ = 1.36%. First, let’s look at the zeroth order portfolio held by each country under the benchmark calibration. Table 3.6 reports the cross-country portfolios under the partially financial integration and the global integration. When financial markets are partially integrated, households in country A sell short their own equities and have a large long position in foreign equities. Country B and Country C also sell short their own equities and have a large long position in equities of Country A. These optimal shares of equities seem unreasonable compared to those in the data. Like other researches (see chapter 2 for details), equity holdings in each country exhibit super foreign bias and are quite sensitive to structural parameters. When all financial markets are integrated into a united world financial market, households in 111 3.2. A model with multiple tradable goods Table 3.6: Optimal equity shares held by each country under different fi- nancial architectures Partial integration Global integration Equity A Equity B Equity C Equity A Equity B Equity C Country A -28.22 29.22 29.22 -23.75 24.75 24.75 Country B 14.61 -28.22 0.00 12.38 -29.99 6.24 Country C 14.61 0.00 -28.22 12.38 6.24 -29.99 each country alter their portfolios to obtain extra gains from risk sharing across borders. The right columns in table 3.6 illustrates this point. As in proposition 1, households in peripheral countries would hold more equities in the core country in partial financial integration than those in global financial integration. Does financial integration reduce the volatilities of consumption and out- put? The answer is it depends. Table 3.7 shows the conditional volatility of consumption, output and real exchange rates. We can see that volatil- ity of consumption in all countries significantly drop when world economies parade from financial autarky to partial financial integration. The extra gains from partial financial integration to global financial integration are mixed. In peripheral country B and C, volatility of consumption decreases; nevertheless, in country A, the volatility of consumption increases a little bit. We know that country A has access to global financial markets both in partial integration and global integration. The effect of financial inte- gration in the third stage (global financial integration) on country A seems unclear since the market is incomplete in the second stage (partial finan- cial integration). Nevertheless, in the peripheral countries, household could have access to more financial markets from financial autarky to partial in- tegration, and then to global integration, which implies that households in these two countries can share their risk with the rest of world and thereby reduce the volatility of consumption. Labor supply has a similar pattern as consumption. Partial integration reduces labor volatility by a large ex- tent. Global financial integration seems moderately increase the volatilities of labor in all countries. 112 3.2. A model with multiple tradable goods Table 3.7: Conditional volatility (%): the benchmark Autarky Partial integration Global integration Consumption in A 0.897 0.739 0.745 Consumption in B 0.675 0.637 0.629 Labor in A 0.449 0.294 0.295 Labor in B 0.587 0.389 0.394 GDP in A 1.063 1.257 1.251 GDP in B 0.851 1.014 0.998 RER in A 0.969 1.010 0.991 RER in B 0.774 0.823 0.832 TOT in A 1.384 1.443 1.416 TOT in B 1.215 1.296 1.315 Note: This table reports the conditional standard deviation for variables of interest in different financial regimes. Parameter values are the benchmark calibration. RER is effective real exchange rate and TOT is effective terms of trade. 113 3.2. A model with multiple tradable goods Table 3.8: Conditional volatility (%): an alternative calibration Autarky Partial integration Global integration Consumption in A 0.833 0.833 0.833 Consumption in B 0.833 0.833 0.833 Labor in A 0 0 0 Labor in B 0 0 0 GDP in A 1.36 1.36 1.36 GDP in B 1.36 1.36 1.36 RER in A 0 0 0 RER in B 0 0 0 TOT in A 1.666 1.666 1.666 TOT in B 1.696 1.696 1.696 Note: This table reports the conditional standard deviation for variables of interest in different financial regimes. Parameter values are the benchmark calibration except for σ = γ = 1 and α1 = α2 = 0.5. RER is effective real exchange rate and TOT is effective terms of trade. Unitary trade elasticity As discussed in the literature, when the elasticity of substitution be- tween imports and exports, γ, is unity, terms of trade may effectively hedge against income risk across countries (Cole and Obstfeld 1991). Table 3.8 reports the second moments for variables of interest when γ = σ = 1. Not surprisingly, terms of trade can effectively hedge idiosyncratic income risk. Moreover, the effective real exchange rates are constant (when γ = 1) and labor supply is also unchanged (when γ = σ = 1). I can show that equi- ties are perfect substitutes up to a first order approximation, which implies that international portfolios are indeterminate. In other words, any feasible portfolio choice is optimal for households in all countries. Alternative shock processes In the baseline calibration, I assume that the correlation between pro- ductivity innovations is zero. This assumption is suitable for developing countries in Latin America (see, table 1.2). Obverse that output among Asian economies (see table 1.3) seems more synchronized than other de- veloping economies. In order to see whether the ranking of consumption volatility change with shock processes, I set the cross correlation to be corrt(A,t+1, B,t+1) = corrt(A,t+1, C,t+1) = 0.4 and corrt(B,t+1, C,t+1) = 0.7. Numerical results show that the ranking of consumption volatility is 114 3.2. A model with multiple tradable goods unchanged. Sensitivity analysis As in the literature, there is no consensus on the values of the trade elasticity and the elasticity of intertemporal substitution. I then vary these two values around their benchmark values. The results show that the rank of consumption volatility is the same as the benchmark case. Notice that the first-order approximation of the model can’t be used to analyze welfare changes across financial architectures, a complete welfare analysis is required to take a third order approximation to Euler equations for portfolios and a second approximation to other equations in the dynamic system. It is out of the scope of this chapter (see a work in progress, Yu 2012). However, volatility of consumption is still an important indicator for welfare analysis. 115 Chapter 4 Credit Friction, Asset Price and Optimal Monetary Policy Financial frictions play an important role in propagating shocks across sectors and countries, particularly at the time of financial crises, in which households and firms face binding collateral constraints. The constrained borrowers are then forced to cut their borrowing and consequently to reduce their demand for goods and assets. This in turn generates quite volatile out- put and consumption. In terms of the perspective of consumption smooth- ing, large consumption volatility isn’t a good thing. So there is room for policy makers to pursue optimal policies in order to stabilize the economy. This chapter will study how monetary and fiscal authorities quantitatively conduct optimal policies in the case of binding collateral constraints. I first outline the model in Section 2. After defining the equilibrium in Section 3, I specify functions used in the model and calibrate them in Section 4. In Sec- tion 5, I present the Ramsey steady state and discuss the optimal monetary and fiscal policy as well as second moments of several key variables. Section 6 displays the Ramsey optimal monetary policy and the optimized linear monetary policy rule. In Section 7, I investigate the welfare of alternative implementable interest rate rules. Conclusion is made in the final chapter. 4.1 The Model There are two types of agents in the economy, households, and en- trepreneurs, with the same population size of unit measure. Both of them are long-lived and risk averse. 4.1.1 Households The household supplies lt hours of labor, makes consumption decision ct, purchases risk-free bonds bt+1 and chooses holdings of capital good k s t+1 for 116 4.1. The Model the next period, which consists of, for instance, housing and other durable goods. The preference for a household is given by, E0 { +∞∑ t=0 βtU(ct, k s t , lt) } (4.1) Consumption composite ct is aggregated via a CES form, ct = [∫ 1 0 c 1− 1 φ it di ] 1 1− 1 φ (4.2) where cit is a consumption variety in the consumption composite. φ is the elasticity of substitution between consumption varieties. Nominal price for consumption is Pt, which is aggregated by individual prices Pit, Pt = [∫ 1 0 P 1−φit di ] 1 1−φ (4.3) There are continuous labor markets with unit measure. Following Schmitt- Grohe and Uribe (2006), nominal wage W jt in labor market j is set by households or labor union. Households face the same labor demand curve and choose identical consumer goods and capital goods. Given the wages prevailing in labor markets, households allocate hours across different la- bor markets. The household supplies ljt hours to labor market j to satisfy demand for that sort of labor service, ljt = ( wjt wt )−φ̃ ldt where wjt = W jt Pt , wt = Wt Pt . φ̃ is the elasticity of substitution between labor services. ldt is total labor demanded by entrepreneurs. Total hours supplied by household i are, lt = ∫ 1 0 ljtdj = l d t ∫ 1 0 ( wjt wt )−φ̃ dj (4.4) 117 4.1. The Model The household’s budget constraint reads, ct + qt [ kst+1 − (1− δ)kst ] + bt+1 = btRt pit + (1− τ ft )(Ft +Dt) + (1− τht )ldt ∫ 1 0 wjt ( wjt wt )−φ̃ dj (4.5) where qt is the price of capital goods, Rt nominal gross interest rate, Ft profits from retailers, Dt profits from capital goods producers, pit = Pt Pt−1 gross price inflation, δ depreciation rate for capital goods, τ ft profit tax and τ lt labor income tax. The left hand side of equation (4.5) denotes a house- hold’s expenditure, which consists of consumption of final goods, change of capital goods plus replacement cost, purchase of bonds. The income for the household contains debt repayments from entrepreneurs, after-tax profits and after-tax labor income. Profits come from labor union and retailers, which are postponed to later. Wages are set by labor union in terms of Calvo style. At each period a fraction of α̃ markets could reset their nominal wages, while the remaining markets adjust wages according to W jt = W j t−1pi χ̃ t−1 or w j t = w j t−1pi χ̃ t−1/pit, 0 ≤ χ̃ ≤ 1 denotes the degree of indexation to inflation. The household’s objective is to choose allocations { ct, lt, bt+1, wjt , kst+1 } to maximize preference (4.1), facing constraints (4.4) and (4.5). Let λt and λt(1−τ lt)wt µt be multipliers for constraints (4.4), (4.5), respectively. The Lagrangian functions is, L =E0{ +∞∑ t=0 βt{U(ct, kst , lt) + λt[ btRt pit + (1− τ ft )Ft + (1− τht )ldt ∫ 1 0 wjt ( wjt wt )−φ̃ dj − ct − qt ( kst+1 − (1− δ)kst )− τ lst − bt+1] + λt(1− τ lt )wt µt lt − ldt ∫ 1 0 ( wjt wt )−φ̃ dj }} (4.6) F.O.N.C.s are as follows, Uc(t) = λt (4.7) 118 4.1. The Model λt = βRt+1Et { λt+1 pit+1 } (4.8) − Ul(t) = λt(1− τ l t )wt µt (4.9) λtqt = βEt {λt+1qt+1(1− δ) + Uk(t+ 1)} (4.10) Equation (4.7) states that marginal utility of consumption equals the shadow price of real income. Equation (4.8) shows that a marginal incre- ment of savings in period t increases the quantity of available goods at time t + 1 by an amount of Rt+1/pit+1, which has a discounted marginal value βRt+1Et { λt+1 pit+1 } . The marginal benefit is equal to the marginal cost of the initial purchase of bond in period t, λt. The optimal condition for capital in equation (4.10) has a similar interpretation. For labor supply (4.9), marginal dis-utility of labor supply is equal to the marginal value of after-tax labor income divided by wage mark-up µt. Notice that in a symmetric equilibrium, all labor markets are the same. Consider a particular labor market j in which wage could be reset to w̃t at period t. The Lagrangian function is written as, Lw =Et{ +∞∑ s=0 (βα̃)s[λt+sw̃t+sl d t+s ( w̃t+s wt+s )−φ̃ (1− τ lt+s) − λt+s (1− τ l t+s)wt+s µt+s ldt+s ( w̃t+s wt+s )−φ̃ ]} =Et{ +∞∑ s=0 (βα̃)s[λt+sw̃ 1−φ̃ t l d t+s (∏s k=1 pi χ̃ t+k−1∏s k=1 pit+k )1−φ̃ wφ̃t+s(1− τ lt+s) − λt+s (1− τ l t+s)wt+s µt+s ldt+sw̃ −φ̃ t (∏s k=1 pi χ̃ t+k−1∏s k=1 pit+k )−φ̃ wφ̃t+s]} (4.11) Given prices {wt+s, ldt+s, τ lt+s, λt+s} for all s ≥ 0, the F.O.N.C.s can be 119 4.1. The Model simplified as, 0 =Et +∞∑ t=0 (βα̃)tλt+sl d t+s ( w̃t wt+s )−φ̃ s∏ k=1 ( pit+k piχ̃t+k−1 )φ̃ φ̃− 1 φ̃ (1− τ lt+s)w̃t∏s k=1 pit+k piχ̃t+k−1 − wt+s(1− τ l t+s) µt+s (4.12) The first term in the bracket stands for the marginal utility from labor income by increasing one unit of labor service at period t. The second term in the bracket represents the marginal cost from labor supply. The optimal wage resetting rule requires that, after taking account of the probability of not being able to reset wages in all future periods, the expected discounted value of labor income equals the expected discounted cost of labor service. Write the discounted expected utility from labor income as, f1t ≡ φ̃− 1 φ̃ Et +∞∑ t=0 (βα̃)tλt+sl d t+s ( w̃t wt+s )−φ̃ s∏ k=1 ( pit+k piχ̃t+k−1 )φ̃ (1− τ lt+s)w̃t∏s k=1 pit+k piχ̃t+k−1 and the discounted expected dis-utility from labor supply reads, f2t ≡ Et +∞∑ t=0 (βα̃)tλt+sl d t+s ( w̃t wt+s )−φ̃ s∏ k=1 ( pit+k piχ̃t+k−1 )φ̃ wt+s(1− τ lt+s) µt+s Write f1t and f 2 t in recursive forms, f1t = φ̃− 1 φ̃ w̃tλt(1− τ lt ) ( wt w̃t )φ̃ ldt + α̃βEt (pit+1 piχ̃t )φ̃−1( w̃t+1 w̃t )φ̃−1 f1t+1 (4.13) f2t = −Ult ( wt w̃t )φ̃ ldt + α̃βEt (pit+1 piχ̃t )φ̃( w̃t+1 w̃t )φ̃ f2t+1 (4.14) f1t = f 2 t (4.15) 120 4.1. The Model 4.1.2 Entrepreneurs Entrepreneurs own firms. They hire labor services ldt from labor markets and purchase capital goods ket+1 as productive inputs in the next period from capital market. Output produced by entrepreneurs is the whole-sale good, which cannot be directly consumed. Production is of constant return to scale for a given level of technology level at, Yt = F (k e t , l d t ; at) (4.16) The log of technology evolves as an AR(1) process with persistence of ρa, ln(at/ā) = ρaln(at−1/ā) + at (4.17) where at follows as i.i.d. normal distribution with mean zero and a standard deviation of σa. ā is the value of at at its long-run steady state, which is normalized to be one. Let the nominal price of the whole-sale good be Pwt . The price markup for the whole-sale good then can be written as Xt = Pt Pwt , a similarly expres- sion in Bernanke, Gertler and Gilchrist (1999). Entrepreneurs lack of sufficient internal funds to finance purchases of capital goods. An alternative way of obtaining external funds is to issue corporate bonds. Nevertheless, because of contract enforcement problems, entrepreneurs can only issue an amount of corporate bonds up to a fraction of their holdings of capital goods. In the spirit of Kiyotaki and Moore (1997), entrepreneurs face a following borrowing constraint in nominal terms, Bet+1 ≤ κEt ( Qt+1k e t+1 Rt+1 ) (4.18) where Bet+1 is the nominal purchase of bonds at the end of period t, whose payoff in the following period is Bet+1Rt+1. Qt+1 stands for nominal price of capital at period t+ 1. Write it in real terms, bet+1 ≤ κEt ( qt+1k e t+1pit+1 Rt+1 ) (4.19) where bet+1 ≡ Bet+1/Pt, qt+1 ≡ Qt+1/Pt+1. Entrepreneurs’ preference is defined, E0 { +∞∑ t=0 (βγ)tU(Cet ) } (4.20) 121 4.1. The Model with < 0γ < 1. They are less patient than households. In a symmetric equilibrium, all entrepreneurs are identical. A represen- tative entrepreneur faces a following budget constraint, cet + qt [ ket+1 − (1− δ)ket ] + betRt pit = (1− τkt ) ( Yt Xt − wtldt ) + bet+1 (4.21) The left-hand side is an entrepreneur’s expenditure in period t, which in- cludes consumption, installment of new capital plus replacement cost, and debt payments. The income side contains returns to physical capital (the first term on the right-hand side) and newly issued corporate bonds. The entrepreneur’s objective is to choose { cet , ldt , bet+1, ket+1 } to maximize (4.20) subject to (4.19) and (4.21). Let λ̃t and λ̃tϕt be multipliers for these two constraints respectively. The entrepreneurial Lagrangian function reads, Le =E0{ +∞∑ t=0 (βγ)t{U(Cet ) + λ̃t[(1− τkt ) ( Yt Xt − wtldt ) + bt+1 − cet − qt [ ket+1 − (1− δ)ket ] − τ let − btRt pit ] + λ̃tϕt ( κ qt+1k e t+1pit+1 Rt+1 − bet+1 ) }} (4.22) The optimality conditions include, Ue(t) = λ̃t (4.23) qtλ̃t = λ̃tϕtκEt ( qt+1pit+1 Rt+1 ) +Et { γβλ̃t+1 [ qt+1(1− δ) + (1− τkt+1) Fkt+1 Xt+1 ]} (4.24) λ̃t = λ̃tϕt + Et ( γβ λ̃t+1Rt+1 pit+1 ) (4.25) Flt Xt = wt (4.26) These equations have straightforward interpretations. In equation (4.23), marginal benefit of consumption is equal to the shadow price of real income, λ̃t. The right-hand side of equation (4.24) states that a marginal increase of net worth in capital investment in period t enlarges both the quantity of available returns to capital at time t + 1 by an amount of qt+1(1 − δ) + (1− τkt+1)Fkt+1Xt+1 , which has a marginal value of λ̃t+1, and the value of levered 122 4.1. The Model investment opportunity at time t by the amount of ϕtκEt ( qt+1pit+1 Rt+1 ) . The sum of these two effects equals the marginal value of wealth worth of one unit of capital in period t. Optimal debt issuance has a similar interpretation. Combining equation (4.24) and (4.25), yields an expression for external finance premium ϕt, ϕt = Et γβ λ̃t+1λ̃t [ qt+1(1−δ) qt + (1− τkt+1) Fkt+1qtXt+1 ] − Rt+1pit+1 1− κEt ( qt+1pit+1 qtRt+1 ) (4.27) which measures the value of relaxing the borrowing constraint or the value of levered investment opportunity. Notice that γβ λ̃t+1 λ̃t is a representative entrepreneur’s pricing kernel. The numerator in the last fraction denotes the excess return on holding capital goods relative to bond repayment at time t + 1, multiplied by leverage ratio 1/ ( 1− κEt ( qt+1pit+1 qtRt+1 )) , which is the marginal value of relaxing borrowing constraint. Capital goods are produced via using final consumption goods. The law of motion for aggregate capital is, it = kt+1 − (1− δ)kt (4.28) Investment in capital goods incurs adjustment costs, c(it). No arbitrage condition requires that price of capital goods should be the same as marginal cost of investment, qt = c ′(it) + 1 (4.29) and profits from capital goods production are redistributed to households, which are equal to, Dt = itqt − (c(it) + it) (4.30) Labor services used by entrepreneurs are aggregated via a constant elas- ticity of substitution (CES) technology, ldt = [∫ 1 0 (ldjt ) 1− 1 φ̃dj ] 1 1− 1 φ̃ (4.31) and nominal wage is accordingly given by, Wt = [∫ 1 0 (W jt ) 1−φ̃dj ] 1 1−φ̃ (4.32) 123 4.1. The Model 4.1.3 Retailers Retailers purchase whole-sale goods from entrepreneurs in the competi- tive whole-sale goods market, differentiate them by color or brands without additional costs, and then sell them in the final good market. Market de- mand for brand i is yit = ( Pit Pt )−φ yt, where yt is total absorption, which is defined as, yt = ct + c e t + it + c(it) + gt (4.33) where gt is government expenditure. Retailer of brand i obtains a profit of Fit, Fit = ( Pit Pt − 1 Xt )( Pit Pt )−φ yt When retailers get the chance to reset their prices, they choose P̃t; oth- erwise, price will be adjusted with past inflation, Pt = Pt−1pi χ t−1. Retails’ problem is to maximize the sum of expected discounted profits from period t onward and the Lagrangian function follows as Lr =Et +∞∑ s=0 αsΛt,t+sPt+s ( P̃t Pt )1−φ s∏ k=1 ( piχt+k−1 pit+k )1−φ yt+s − 1 Xt+s ( P̃t Pt )−φ s∏ k=1 ( piχt+k−1 pit+k )−φ yt+s (4.34) where Λt,t+s ≡ βs Uc(t+s)Uc(t) PtPt+s The F.O.N.C.s follow as, 0 =Et +∞∑ s=0 αsΛt,t+sPt+s ( P̃t Pt )1−φ 1 P̃t s∏ k=1 ( piχt+k−1 pit+k )1−φ yt+s − φ φ− 1 1 Xt+s 1 P̃t ( P̃t Pt )−φ s∏ k=1 ( piχt+k−1 pit+k )−φ yt+s (4.35) This equation has an analogous explanation as wage resetting. The expected discounted value of sale proceeds is equal to the expected discount cost of providing these final goods. 124 4.1. The Model Rewriting the equation above into two parts, x1t and x 2 t , yields, x1t = φ φ− 1Et +∞∑ s=0 αsΛt,t+s Pt+s P̃t 1 Xt+s ( P̃t Pt )−φ s∏ k=1 ( piχt+k−1 pit+k )−φ yt+s = φ φ− 1 1 Xt ytp̃ −φ−1 t + βαEt { Uc(t+ 1) Uc(t) ( p̃t p̃t+1 )−φ−1( piχt pit+1 )−φ x1t+1 } (4.36) where p̃t = P̃t Pt . x2t = Et +∞∑ s=0 αsΛt,t+s Pt+s P̃t ( P̃t Pt )1−φ s∏ k=1 ( piχt+k−1 pit+k )1−φ yt+s = ytp̃ −φ t + βαEt { Uc(t+ 1) Uc(t) ( p̃t p̃t+1 )−φ( piχt pit+1 )1−φ x2t+1 } (4.37) x1t = x 2 t (4.38) Total profits distributed to households are, Ft = ∫ 1 0 Fitdi = ∫ 1 0 ( Pit Pt − 1 Xt )( Pit Pt )−φ ytdi = yt ∫ 1 0 ( Pit Pt )1−φ di− yt Xt ∫ 1 0 ( Pit Pt )−φ di = yt − yt Xt spt (4.39) 125 4.1. The Model where spt is price dispersion and is defined as, spt ≡ ∫ 1 0 ( Pit Pt )−φ di = (1− α) ( P̃t Pt )−φ + α(1− α) ( P̃t−1pi χ t−1 Pt )−φ + α2(1− α) ( P̃t−2pi χ t−1pi χ t−2 Pt )−φ + · · · = (1− α)p̃−φt + α ( pit piχt−1 )φ spt−1 (4.40) 4.1.4 Government Government collects tax revenues τt and issues government bond b g t+1 to finance government expenditure gt. Tax revenues consist of labor income tax, income tax from the entrepreneurial sector and profit tax, τt = τ h t l d t ∫ 1 0 wjt ( wjt wt )−φ̃ dj + τ ft (Ft +Dt) + τ k t ( Yt Xt − wtldt ) + τ lht + τ le t = τht l d twt + τ f t Ft + τ k t ( Yt Xt − wtldt ) (4.41) Flow of funds for the government in a cashless economy is given by, bgt+1 − bgtRt pit = gt − τt (4.42) Let nt+1 ≡ Rt+1bgt+1 be government’s liability at the end of period t. Government’s flow of funds can be written as, nt+1 Rt+1 + τt = gt + nt pit (4.43) Government expenditure is driven by an exogenous process, ln(gt/g) = ρgln(gt−1/g) + g t (4.44) where g denotes government’s steady state expenditure. 126 4.1. The Model 4.1.5 Aggregation and market clearing conditions Law of large number implies that nominal price evolution follows as, P 1−φt = α(Pt−1pi χ t−1) 1−φ + (1− α)P̃ 1−φt (4.45) Writing it in real terms, yields, 1 = α(piχt−1/pit) 1−φ + (1− α)p̃1−φt (4.46) Whole-sale good is transformed to final goods via a one-for-one technol- ogy. Demand for whole-sale good is, Yt = yt ∫ 1 0 ( Pit Pt )−φ di = spt yt (4.47) Similarly, nominal wage evolves as, w1−φ̃t = α̃w 1−φ̃ t−1 (pi χ̃ t−1/pit) 1−φ̃ + (1− α̃)w̃1−φ̃t (4.48) Labor market clears, lt = ∫ 1 0 ljtdj = l d t ∫ 1 0 ( wjt wt )−φ̃ dj = ldt s w t (4.49) where wage dispersion is evolving as, swt = (1− α̃) ( w̃t wt )−φ̃ + α̃ ( wt−1 wt )−φ̃( pit piχ̃t−1 )φ̃ swt−1 (4.50) Capital market clearing condition reads, kt+1 = k e t+1 + k s t+1 (4.51) Bond market clearing condition is, bt+1 = b e t+1 + b g t+1 (4.52) 127 4.2. Characterizing the equilibrium In the case with nominal rigidities, an interest rate rule is applied here, ln ( Rt+1 R ) = αpiln (pit pi ) +αpiw ln ( piWt piW ) +αyln ( yt y ) +αqln ( qt q ) +ρrln ( Rt R ) (4.53) where αpi, αpiw , αy, αq and ρr are coefficients for the corresponding target variables. 4.2 Characterizing the equilibrium 4.2.1 Competitive equilibrium A symmetric equilibrium is defined in the usual way. Given the house- hold’s, entrepreneur’s and government’s initial debts, { b0, be0, bg0 }, initial capital holdings { ks0, ke0 }, initial price and wage dispersion { sp0, sw0 } and the stochastic processes, {gt, at}, a symmetric equilibrium is an allocation, { ct, kst+1, bt+1, lt, µt, λt, w̃t, f1t , f2t , p̃t, x1t , x2t , Ft, Dt, ket+1, ldt , Yt, bet+1, ϕt, cet , λ̃t, it, nt+1, τt, kt+1, yt }, price system, { pit, wt, qt, Xt }, evolvement of price and wage dispersion, { swt , spt }, and government policy, { τ lt , τkt , τ ft , Rt+1 }, such that: • { ct, kst+1, bt+1, lt, µt, λt, w̃t, f1t , f2t } solve the household’s prob- lem subjective to the sequence of household budget constraints and staggered wage setting; • { ket+1, ldt , Yt, bet+1, ϕt, cet , λ̃t, it } solve the entrepreneur’s problem sub- jective to the sequence of entrepreneur’s flow of funds and borrowing constraints; • { p̃t, x1t , x2t , Ft } solve retailer’s problem constrained by staggered price setting; • { nt+1, τt } satisfy government budget constraints, given government policy; • the whole-sale good market clears • the final goods markets clear; • the labor markets clear; • the capital market clears; 128 4.3. Specification and calibration • the bond market clear; • and Rt ≥ 1, and ϕt ≥ 0, for all t ≥ 0 The competitive equilibrium satisfies equations (4.7)-(4.10), (4.13)-(4.15), (4.16), (4.19), (4.21), (4.23)-(4.26), (4.28), (4.29), (4.30), (4.33), (4.36)- (4.40), (4.41), (4.43), (4.46), (4.47), (4.48), (4.49), (4.50), (4.51), (4.52), exogenous processes, (4.17), (4.44), and government policy. 4.2.2 The Ramsey equilibrium The Ramsey planner’s problem is to find the fiscal and monetary policies such that in the competitive equilibrium, the weighted sum of the house- hold’s and entrepreneur’s expected lifetime utility reaches the highest level. In choosing optimal policies, the government commits to honor its chosen policies in the past. In all periods, private agents maximize their objective functions taking the whole policy plan as given. Formally, the Ramsey equilibrium is a set of allocations { ct, kst+1, bt+1, lt, µt, λt, w̃t, f 1 t , f 2 t , p̃t, x 1 t , x 2 t , Ft, Dt, k e t+1, l d t , Yt, b e t+1, ϕt, c e t , λ̃t, it, nt+1, τt, kt+1, yt }, price system, { pit, wt, qt, Xt }, evolvement of price dispersion, { swt , spt } and government policy, { τ lt , τkt , τ ft , Rt+1 }, for all t ≥ 0 that maximize, W0 ≡ E0 { +∞∑ t=0 [ ωβtU(ct, k s t , lt) + (1− ω)(βγ)tU(Cet ) ]} (4.54) where ω is the weight on households in the planer’s objective function. The complete set of equilibrium conditions are listed in the appendix. 4.3 Specification and calibration Preferences for households and entrepreneurs have forms of, U(ct, k s t , lt) = c1−σt − 1 1− σ + jln(k s t )− l1+θt 1 + θ U(Cet ) = (cet ) 1−σ − 1 1− σ Production follows as, F (ket , l d t ; at) = at(k e t ) α0(ldt ) 1−α0 129 4.3. Specification and calibration Investment adjustment cost is set as, c(it) = ψ 2 ( it kt )2 kt Parameters in this system are { σ, β, γ, j, θ, δ, ψ, α0, φ, φ̃, κ, α, α̃, χ, χ̃, ā, g, ρa, σa, ρg, σg }. We need extra constraints to figure out the parameters. Parameters are divided into two groups. The first group of parameters are chosen from the literature. Preferences are set as log-preferences for both households and entrepreneurs, σ = 1. β = 1.04−1/4 such that the annual real interest rate is 4% when inflation is zero. Risk-spread is set to be 2% which induces γ = 0.98. Frisch labor elasticity is set at θ = 0.47. Depreciation rate is δ = 0.025. Share of capital in production is chosen to be α0 = 0.3. As in Christiano, Eichenbaum and Evans (2005), elasticity of substitution between final goods is φ = 6, and elasticity of substitution across labor services is set as φ̃ = 21. The probability of resetting price is α = 0.6, and α̃ = 0.64 for wage readjustment. Indexation to inflation for price χ = 0, and χ̃ = 0, in the benchmark model. Technology level is normalized to be ā = 1. Technology shocks and government expenditure shocks are taken from Schmitt-Grohe and Uribe (2006), ρa = 0.8556, σa = 0.0064, ρg = 0.87, σg = 0.016. Loan- to-value κ = 0.80, which is consistent with literature, i.e., Iacoviello (2002). The weight of households in the social welfare function is set be to 1/2 in the benchmark model. Following Mendoza, Razin and Tesar (1994), income tax rates are set as τk = 0.407 and τh = 0.258. I set τ f = τk, and lump-sump taxes are not applicable. The second group of parameters consists of the elasticity of utility with respect to capital goods consumption j, investment adjustment cost ψ, and government expenditure at the deterministic steady state g. I choose j and ψ such that investment-GDP ratio is si = 16.03%, which is the quarterly aver- age value of fixed investment-GDP ratio in USA over 1951−2009, and hold- ings of capital by households to holdings of capital by entrepreneurs ratio se is 1.32 in USA, which is defined as (residential fixed assets + consumer durables) nonresidential fixed assets . Government expenditure-GDP ratio is set to be st = 17%, based on which government expenditure is solved. Table 4.1 lists structural parameters used by this chapter. The appendix provides details of deterministic steady state. 130 4.3. Specification and calibration Table 4.1: Structural parameters Parameter Value Description β (1.04)−1/4 Households’ subjective discount factor (quarterly) σ 1 Elasticity of intertemporal substitution γ 0.98 Entrepreneur’s relative subjective discount factor (quarterly) θ 0.47 Frisch labor elasticity δ 0.025 Depreciation rate (quarterly) φ 6 Price-elasticity of final goods varieties φ̃ 21 Wage-elasticity of labor varieties κ 0.8 Loan-to-Value ratio α 0.6 Fraction of firms not setting prices optimally α̃ 0.64 Fraction of labor markets not setting wages optimally χ 0 Degree of price indexation χ̃ 0 Degree of wage indexation α0 0.3 Share of capital in production J 0.2652 Preference parameter ψ 40.5305 Capital adjustment cost parameter σa 0.0064 Std. dev. of technology innovation ρa 0.8556 Serial correlation of technology innovation σg 0.016 Std. dev. of government expenditure ρg 0.87 Serial correlation of government expenditure ω 0.5 Social welfare parameter 131 4.4. Ramsey steady state and dynamics 4.4 Ramsey steady state and dynamics 4.4.1 Ramsey steady state In the deterministic long-run steady state of the Ramsey equilibrium, all endogenous variables keep constant. Table 4.2 displays the Ramsey steady- state values of inflation, nominal interest rate, labor and capital income tax rates under a few environments of interests. Consider first a case in which profit tax rate is the same as capital in- come tax in an economy with credit frictions, shown in the first line of table 4.2. The steady-state inflation is zero because nominal rigidities such as price stickiness, wage stickiness and nominal interest rate are completely forecastable in the deterministic steady state. Capital income tax rate is negative, an order of −5.2% rather than zero. The intuition is that mo- nopolistic competition in the final goods markets provides also monopolistic power with entrepreneurs who use capital goods in their production. A Sub- sidy to capital-goods could improve the efficiency of capital goods market. This finding is consistent with Schmitt-Grohe and Uribe (2006) who get a subsidy of −6.3%. Tax rate on labor income is 52%, which is higher than that of Schmitt-Grohe and Uribe (2006). Furthermore, I investigate whether credit frictions could affect the opti- mal tax rate in the Ramsey steady state. The second line in table 4.2 reports that when there are no credit frictions, i.e., ϕ = 0, inflation is still zero, and subsidy to capital income becomes −5.8%, similar to the case with credit frictions, while labor income tax rate drops dramatically to 26%. The reason is that capital is efficiently allocated (in the sense of absence of credit con- straints) in an economy without credit frictions, thereby labor supply may be higher in that economy and so do output and consumption. Government can still finance its expenditure via a lower labor income tax rate. Consider an income tax regime, in which all of incomes are taxed by the same rates, τf = τk = τl. The last line of table 4.2 illustrates that income tax rate is around 7% in an economy with credit frictions. In the following analysis, I will focus on a regime without fiscal policy. Table 4.2: Ramsey steady state Economy Steady State (%) τf χ χ̃ ϕ pi R τh τk τk 0 0 1− γ 0.0 4.00 50.89 -5.16 τk 0 0 0 0.0 4.00 26.32 -5.78 τk = τl 0 0 1− γ 0.0 4.00 6.90 6.90 132 4.4. Ramsey steady state and dynamics 4.4.2 Ramsey dynamics In this section, I will study the cyclical implications under the Ramsey monetary policy. I shut down active fiscal policies in this section. Table 4.3 shows the steady state, standard deviation, serial correlation, and correlation with output for a number of variables of interest. The second moments are computed using Monte Carlo simulations. I perform 1000 simulations of 200 periods for each variable. The second moments are averages over the 1000 simulations. Note that the standard deviation of inflation and interest rate are at moderate levels. In the real business cycle literature with financial frictions, output usu- ally negatively correlates with external finance premium, in which positive (negative) technology shocks make entrepreneurs wealthier (poorer) and their associated external finance premia decrease (increase) with the rise (decrease) of net worth, which in turn enhances (drives down) output in the economy. When nominal debt is introduced into the financial mar- ket, another concern is the debt deflation. When there are time-varying nominal prices, real wealth will be redistributed between debtors and cred- itors. Accordingly, the credit constrained debtors have to accommodate asset holdings, i.e., capital here, which in turn propagates shocks through this collateral channel. Table 4.3: Cyclical implications of optimal policy Variable Steady State Standard Deviation Serial Corre- lation Correlation with Output Rt 4 0.49 -0.2 -0.17 pit 0 0.93 0.37 -0.28 yt 1.31 1.18 0.92 1 ct 0.97 0.96 0.92 1 cet 0.05 0.91 0.93 0.98 ϕt 0.02 2.14 0.5 0.46 qt 2.18 0.75 0.91 0.95 it 0.18 1.46 0.92 1 ket 3.85 0.47 0.99 0.29 lt 0.82 0.39 0.75 0.54 wt 0.93 0.91 0.95 0.98 Note: Rt and pit are expressed in percentage per year. The steady state values of other variables are expressed in levels. The second moments in the table corresponding to percent deviations from their steady state values. 133 4.4. Ramsey steady state and dynamics In a model with sticky price, nominal debt and borrowing constraint, policy-makers trade off price dispersion cost and debt inflation. Iacoviello (2002) analyzes an economy in this kind of environment. He finds that nom- inal debt could reduce the volatilities of output and inflation for a number of shocks. However, his results are not based on social welfare maximiza- tion. From the point of view of social welfare, volatilities of consumption and labor supply are not good things because welfare function is concave in consumption and labor supply. In the current environment, monetary pol- icy affects the economy through three channels. The first is price rigidity, which requires low volatility of inflation. The second is wage rigidity, which requires changes of inflation to make real wages reach their efficient lev- els. The last one works via the collateral channel, through which inflation volatility could accelerate or decelerate propagation of shocks. When the endogenous variables are stationary processes, large volatility maybe reduce ex ante welfare. If this happens, the Ramsey planner could use monetary policy to offset this large volatility through the collateral channel. Two variables from table 4.3, external finance premium and price of capital, il- lustrate this point. In the benchmark model, volatility of external finance premium is 2.1%, which is nearly twice as output while asset price is less volatile than output. Moreover, asset price is highly serially correlated but the serial correlation of external risk premium is moderate. The most sur- prising result is the positive correlation between output and external finance premium, which implies that entrepreneurs’ net worth decreases when there are positive technology shocks. The intuition is that, with the Ramsey pol- icy, inflation drops a lot when there are positive technology shocks hitting the economy, and hence debt deflation makes entrepreneurs pay more to creditors, although returns to capital increase associated with positive tech- nology shocks. The debt deflation channel dominates the higher returns to capital and thus entrepreneurs’ net worth is lower than their steady state level; hence it follows an rise in external finance premium. Consider then how nominal rigidities affect asset prices and external fi- nance premium. Panel A in table 4.4 shows the second moments of variables of interest under different price and wage rigidities when debts are issued in nominal terms. Under flexible price and wage (α = 0, α̃ = 0), inflation volatility is highly large while volatility of nominal interest rate is pretty small, which is consistent with the findings in literature, i.e., Schmitt-Grohe and Uribe (2006). However, nominal debts still play a role in buffering volatility of output and consumption. The standard deviation of external finance premium is ϕ = 2% and highly positively correlates with output. When there is only price stickiness in the economy, (α = 0.6, α̃ = 0), 134 4.4. Ramsey steady state and dynamics Table 4.4: Degree of nominal rigidity and optimal policy Panel A: Nominal Debt α α̃ Statistics Rt pit yt ϕt qt it 0.64 0.6 Mean 4 0 1.31 0.02 2.18 0.18 STD 0.49 0.93 1.18 2.14 0.75 1.46 Ser. Corr. -0.2 0.37 0.92 0.5 0.91 0.92 Corr. with yt -0.17 -0.28 1 0.46 0.95 1 0 0.64 STD 0.56 2.29 1.38 2.33 0.89 1.73 Ser. Corr. -0.2 0.01 0.81 0.72 0.78 0.8 Corr. with yt -0.26 -0.36 1 0.93 0.95 1 0.6 0 STD 1.42 0.62 1.3 0.98 0.84 1.63 Ser. Corr. -0.1 -0.24 0.91 0.07 0.89 0.9 Corr. with yt 0.17 -0.02 1 -0.4 0.95 1 0 0 STD 0.12 1.99 1.36 1.75 0.88 1.71 Ser. Corr. 0.83 -0.07 0.85 0.73 0.83 0.85 Corr. with yt -0.85 -0.3 1 0.93 0.95 1 Panel B: Indexed Debt 0.64 0.6 Mean 4 0 1.31 0.02 2.18 0.18 STD 1.84 1 1.22 2.74 0.79 1.54 Ser. Corr. -0.18 0.3 0.86 0.34 0.83 0.84 Corr. with yt 0.25 -0.05 1 -0.05 0.95 1 0 0.64 STD 2.61 4.21 1.48 5.97 0.98 1.89 Ser. Corr. -0.23 -0.33 0.81 0.68 0.77 0.79 Corr. with yt 0.3 0.26 1 -0.69 0.95 1 0.6 0 STD 2.15 0.8 1.39 2.82 0.92 1.77 Ser. Corr. -0.17 -0.36 0.85 0.34 0.82 0.84 Corr. withyt 0.26 0.33 1 -0.75 0.95 1 0 0 STD 2.55 - 2.34 50.92 1.63 3.12 Ser. Corr. -0.1 - 0.88 0.65 0.86 0.87 Corr. with yt 0.14 - 1 -0.84 0.96 1 Note: Rt and pit are expressed in percentage per year. The steady state values of other variables are expressed in levels. The second moments in the table corresponding to percent deviations from their steady state values. 135 4.4. Ramsey steady state and dynamics monetary policy should remedy the distortion of price dispersion in goods market. Inflation volatility drops dramatically and nominal interest rate is more volatile than the flexible case. On the other hand, lower inflation volatility makes debt inflation channel silent and external finance premium then negatively co-moves with output. When wage stickiness appears only in the economy, (α = 0, α̃ = 0.64), monetary policy needs to improve labor market efficiency through moving real wage close to the efficient level. Inflation volatility increases and so does external finance premium. In the benchmark model, α = 0.6, α̃ = 0.64, volatilities of inflation and nominal interest rate are between the case with flexible wages and the case with flexible prices. External finance premium also lie in that range but with a positive correlation with output. Across different cases, volatilities of asset price, output and net invest- ment don’t change too much and their correlation with output are quite similar across different environments. The reason is that asset price plays a role in allocating capital between households and entrepreneurs, and so does investment. A rise in asset price will decrease the demand for capital and a fall in asset price has an opposite effect. Collateral constraints make asset price more volatile in real business cycles (see the bottom line in Panel B of table 4.4). Nevertheless, debt deflation could dump this kind of volatility. That is why the correlation between inflation and output is negative and significant, and external finance premium positively moves with output. In panel B of table 4.4, I shut down debt deflation channel, that is, nominal debt is fully indexed to inflation and hence nominal interest rate is thereby the real interest rate. In the benchmark case, α = 0.6, α̃ = 0.64, the volatility of variables in the table increases because without debt deflation entrepreneurs needn’t repay as much as the case with debt deflation. The financial accelerator via an increase in entrepreneurs’ net worth drives the economy much volatile, which holds across regimes with different nominal rigidities. Notice that when there is only price stickiness in an economy, the volatility of inflation is still positive, 0.8%, rather than zero. The reason is that we take an approximation to the dynamic system around its distorted steady state, where monopolistic power exists both in retail sector and labor union, beyond credit frictions. Consequently, there is a tradeoff between inflation volatility and output volatility.41 When both prices and wages 41I also consider a case in which subsidies to monopolistic price-setters and wage-setters are applicable in the deterministic steady state. The results which are not reported here display that distortion of credit frictions dominates the distortion of monopolistic power both in goods and labor markets. The relative large positive volatility of inflation reflects 136 4.5. Ramsey and optimized impulse responses are flexible, the economy with fully inflation-indexed debts exhibits exactly the same second moments as in a real business cycle model. Volatilities of output, external finance premium, asset price and net investment are much higher than those with nominal rigidities. Therefore, the basic results in this section are that Ramsey planner is willing to bear some cost of price dispersion and/or wage dispersion to re- duce the volatility of endogenous variables since large volatility of variables is more undesirable comparing to the inefficiency of output from nominal rigidities. When nominal debt is introduced into the economy, debt defla- tion provides Ramsey policy-maker with another channel to push down the aggregate volatility through the financial decelerator. 4.5 Ramsey and optimized impulse responses In this section, I will find the optimized linear interest rate rule which contains only observable macroeconomic variables. I define the distance between the competitive equilibrium under a linear rule and the Ramsey equilibrium as follows. Let IRRT denotes the impulse-responses of variables associated with the Ramsey equilibrium of length T quarters. Analogously, let IRCET denotes the impulse-responses associated with the competitive equilibrium of length T quarters under a particular linear rule. Let the vec- torized difference be x ≡ vec(IRRT − IRCET ). The linear rule that minimizes the distance xTx is called the optimized linear rule. The optimized linear rule is given by,42 log ( Rt+1 R ) =0.154 log (pit pi ) + 0.0874 log ( piWt piW ) + 0.0048 log ( yt y ) − 0.0145 log ( qt q ) + 0.797 log ( Rt R ) (4.55) which exhibits a passive response to inflation. Notice that nominal interest rate negatively responds to the change of asset prices. The logic is similar to Faia and Monacelli (2007) but on an opposite argument. Asset price volatility in an economy with credit frictions is large because of collateral a large gap between the distorted steady state and the efficient steady state. 42Credit spread between external financial premium and risk free rate is quite small. Including the spread in the linearized interest rule wound’t change the coefficients for other variables. 137 4.5. Ramsey and optimized impulse responses constraints, which in turn increase the volatilities of capital allocations be- tween households and entrepreneurs, and so does investment. The increase in investment volatility in turn induces an increase in consumption volatil- ity and therefore reduces ex ante social welfare. The Ramsey policy aims to reduce the volatility of output and consumption. Associated with a de- crease of interest rate responding to ascending in asset price, households are less willing to lend entrepreneurs some funds, and therefore, demand for capital in intermediate goods production is restrained. The volatility of consumption is brought down. Notice that asset prices positively move with output. Consequently, the negative response of interest rate to asset price is not large, conditional on output gap. In other words, once we control for inflation rate and output gap, the optimal interest rate doesn’t necessarily respond to asset prices. Figure 4.1 displays the responses of variables of interest to an unexpected one-percent positive technology shock at at date t. The solid lines denote responses in the Ramsey equilibrium and the dashed line stands for responses in the competitive equilibrium associated with the optimized linear rule (4.55). The equilibrium dynamics of non-policy variables in the optimized linear rule mimic those associated with the Ramsey equilibrium quite well. Labor supply drops nearly −0.6% on impact in the Ramsey policy. The reason is that because of deflation, households obtain much real wealth transfer from their nominal debt payment by entrepreneurs. This kind of income effect dominates the substitution effect which comes from higher real wages. Accordingly, labor supply drops while consumption increases. The response of output jumps up to 0.6% on impact because of reduction of labor supply. The Ramsey planner responds to the positive technology shock by tightening monetary policy, while the optimized linear rule calls for an easing of monetary policy. A deflation immediately follows by the tightening monetary policy. With the easing of monetary conditions, inflation goes gradually back to its steady state level. At the same time, the associated external finance premium increases on impact because entrepreneurs’ net worth declines for the sake of debt deflation. In aggregate, entrepreneurs can still borrow 0.4% of their initial debts to finance their purchases of capital. 138 4.5. Ramsey and optimized impulse responses 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 Output Optimized Rule Ramsey Policy 0 5 10 15 20 -0.6 -0.4 -0.2 0 0.2 Hours 0 5 10 15 20 0 0.2 0.4 0.6 0.8 Real Wage Rate 0 5 10 15 20 -0.2 0 0.2 0.4 0.6 0.8 Price of Capital 0 5 10 15 20 0 0.5 1 1.5 Net Investment 0 5 10 15 20 0 0.2 0.4 0.6 0.8 Entrepreneurs Borrowing 0 5 10 15 20 -2 -1.5 -1 -0.5 0 0.5 Inflation 0 5 10 15 20 -1 -0.5 0 0.5 Nominal Interest Rate Figure 4.1: Responses of variables to one percent positive technology shock at in an economy with credit frictions. Red solid line denotes the Ramsey policy and blue dashed line is the optimized linear interest rule. 139 4.6. Welfare evaluation: optimized v.s. linear interest rate rules 4.6 Welfare evaluation: optimized v.s. linear interest rate rules This section I will assess alternative interest rate rules based on social welfare evaluation. I take second order approximations to the competi- tive equilibrium induced by a specified interest rule around its deterministic steady state. In particular, this analysis focuses on the conditional and unconditional expected discounted utility of agents in the economy. The conditional welfare is calculated conditional on the steady state levels of variables in the economy. Social welfare function is given by formula (4.54) and it can also be written in recursive forms. Households’ welfare can be written in a recursive form, Wh,t = U(ct, k s t , lt) + Et { +∞∑ s=1 βsU(ct+s, k s t+s, lt+s) } = U(ct, k s t , lt) + βEt {Wh,t+1} (4.56) Entrepreneurs’ welfare becomes, We,t = U(C e t ) + Et { +∞∑ s=1 βsU(cet+s) } = U(cet ) + βγEt {We,t+1} (4.57) Social welfare is of the form, Wt = ωWh,t + (1− ω)We,t (4.58) 4.6.1 Comparing simple interest rate rules Simple interest rules considered in this analysis consist of the following specifications: (a) a strict inflation targeting rule (IR, hereafter), (b) a sim- ple Taylor rule (TRI), (c) a Taylor rule (TR), (d) a simple Taylor rule with a positive response to asset price (TRI +αq > 0), (e) a simple Taylor rule with a negative response to asset price (TRI +αq < 0), (f) a Taylor rule with a positive response to asset price (TR +αq > 0), (g) a Taylor rule with a negative response to asset price (TR +αq < 0), (h) an alternative Taylor rule with a smoothing term (TR + inertia 1), (i) an alternative Taylor rule with a smoothing term (TR + inertia 2), (j) an alternative Taylor rule with a high response to output gap (TR + high output response), (k) an alter- 140 4.6. Welfare evaluation: optimized v.s. linear interest rate rules native Taylor rule with a low response to output but with high persistence (TR + low output response), (l) a Taylor rule with a positive response to wage inflation (TR + piw), (m) the optimized linear rule. Table 4.5 displays the coefficients for each specific simple interest rate rule. Table 4.5: Simple interest rate rules Rule Description αpi αpiw αy αq ρr (a) IT pit = 0 for all t (b) TRI 1.5 0 0 0 0 (c) TR 1.5 0 0.5 0 0 (d) TRI + αq > 0 1.5 0 0 0.2 0 (e) TRI +αq < 0 1.5 0 0 -0.2 0 (f) TR + αq > 0 1.5 0 0.5 0.2 0 (g) TR + αq < 0 1.5 0 0.5 -0.2 0 (h) TR + inertia 1 3 0 0.8 0 1 (i) TR + inertia 2 1.2 0 1 0 1 (j) TR + high output response 1.5 0 1 0 0 (k) TR + low output response 1.2 0 0.06 0 1.3 (l) TR + piw 1.5 1 0.5 0 0 (m) Optimized linear rule 0.154 0.0874 0.0048 -0.0145 0.797 Some remarks are made here. The coefficient of asset price in the linear rules (d)-(g) are taken from Faia and Monacelli (2007). Rules (h) and (i) react to the lagged interest rate with a coefficient of one. Rule (h) has a higher response to inflation. Thereby, these two rules have considerable inertia. Rule (j) has a high response to output gap, which is proposed by researchers in the literature. Rule (k) is suggested by Rotemberg and Woodford (1998). First, I explore the second moments of nominal interest rate, inflation, output, external finance premium, asset price and net investment in the competitive equilibrium induced by the specified linear interest rules (see table 4.6). An noticeable observation is that volatility of external finance premium varies a lot across linear interest rate rules. One way to measure a stabilizing policy rule is to investigate its policy frontier, the pair of standard deviation of inflation and standard deviation of output gap. Figure 4.2 plots this policy frontier under different policy rules. The Ramsey policy (point R) yields the smallest inflation-output-gap standard deviation combination (the closest point to the origin) and the optimized linear rule (point m) generates a larger but near optimal pair. The simple Taylor rule (Rule b) and Taylor rule with wage inflation (Rule l) have the largest output 141 4.6. Welfare evaluation: optimized v.s. linear interest rate rules Table 4.6: Cyclical implications of simple interest rate rules Rule Statistics Rt pit yt ϕt qt it IT Mean 4 0 1.31 0.02 2.18 0.18 STD 6.54 0 3.55 63.96 2.53 4.78 Ser. Corr. -0.24 -0.24 0.77 0.67 0.75 0.77 Corr. with yt 0.3 -0.31 1 -0.84 0.97 1 TRI STD 8.05 5.38 6.12 237.86 4.56 8.56 Ser. Corr. 0.76 0.76 0.75 0.61 0.73 0.74 Corr. with yt -0.67 -0.96 1 -0.97 0.97 1 TR STD 2.74 2.77 0.7 15.53 0.42 0.81 Ser. Corr. 0.78 0.78 0.77 0.68 0.74 0.76 Corr. with yt -0.77 -1 1 0.93 0.96 1 TRI + αq > 0 STD 2.95 2.6 1.73 36.94 1.19 2.27 Ser. Corr. 0.77 0.77 0.77 0.59 0.74 0.76 Corr. with yt -0.75 -0.98 1 -0.93 0.96 1 TRI + αq < 0 STD 3.93 1.9 2.79 156.47 2.24 4.14 Ser. Corr. 0.54 0.25 0.71 0.63 0.7 0.71 Corr. with yt -0.37 -0.48 1 -0.95 0.99 1 TR+ αq > 0 STD 2.69 2.77 0.6 20.67 0.34 0.66 Ser. Corr. 0.78 0.77 0.76 0.65 0.72 0.75 Corr. with yt -0.76 -1 1 0.91 0.95 1 TR+ αq < 0 STD 2.86 2.81 0.88 6.91 0.55 1.06 Ser. Corr. 0.78 0.78 0.77 0.76 0.74 0.76 Corr. with yt -0.77 -1 1 0.97 0.96 1 TR + inertia 1 STD 2.24 2.08 1.79 33.59 1.23 2.35 Ser. Corr. 0.87 0.74 0.83 0.57 0.81 0.82 Corr. with yt -0.8 -0.99 1 -0.89 0.96 1 TR + inertia 2 STD 1.9 2.16 0.58 26.13 0.32 0.63 Ser. Corr. 0.89 0.71 0.92 0.64 0.93 0.94 Corr. with yt -0.92 -0.94 1 0.77 0.95 1 TR + high STD 2.94 3.06 0.41 31.35 0.21 0.41 output response Ser. Corr. 0.82 0.79 0.74 0.64 0.68 0.72 Corr. with yt -0.76 -1 1 0.86 0.96 0.99 TR + low STD 0.51 1.06 1.14 2.59 0.73 1.42 output response Ser. Corr. 0.94 0.41 0.92 0.4 0.92 0.93 Corr. with yt -0.85 -0.56 1 0.44 0.95 1 TR +piw STD 8.05 5.26 6.63 258.73 4.93 9.28 Ser. Corr. 0.75 0.74 0.76 0.61 0.74 0.75 Corr. with yt -0.67 -0.96 1 -0.97 0.97 1 Optimized STD 0.51 1.06 1.14 2.59 0.72 1.41 linear rule Ser. Corr. 0.94 0.41 0.92 0.39 0.92 0.93 Corr. with yt -0.85 -0.56 1 0.44 0.95 1 142 4.6. Welfare evaluation: optimized v.s. linear interest rate rules and inflation volatility. Strictly inflation targeting seems far away from the efficient policy rule. Other rules with a lagged interest rate also could reduce the volatility of output and inflation. 4.6.2 Welfare evaluation The comparison of inflation-output volatility pairs is an ad hoc mea- sure of efficiency of a policy rule. Table 4.7 provides the conditional and unconditional expected discounted utility of private agents defined by equa- tion (4.58). Policy rules with high coefficient of lagged interest rate (Rules h, i, k,m) have both higher conditional and unconditional social welfare. In most of cases, including asset price in Taylor rules could enhance social wel- fare. And the traditional Taylor rule and strict inflation targeting rule lie in the range of the second preferable choices. Table 4.7: Welfare results Rule Conditional Welfare Unconditional Welfare Welfare Losta Welfare Lostb IT -61.3968 22.82 -61.4608 28.95 TRI -63.0322 186.36 -63.2317 206.04 TR -61.336 16.74 -61.3977 22.64 TRI + αq > 0 -61.3016 13.3 -61.3358 16.45 TRI + αq < 0 -61.7567 58.81 -61.9513 78 TR+ αq > 0 -61.3369 16.83 -61.4008 22.95 TR+ αq < 0 -61.3456 17.7 -61.4103 23.9 TR + inertia 1 -61.2562 8.76 -61.2846 11.33 TR + inertia 2 -61.2718 10.32 -61.314 14.27 TR + high output response -61.3864 21.78 -61.4723 30.1 TR + low output response -61.2676 9.9 -61.3232 15.19 TR + piw -63.1706 200.2 -63.4479 227.66 Optimized linear rule -61.1686 0 -61.1713 0 Notes: Losta,bis defined as the difference between the optimized linear interest rule and the corresponding simple interest rate rule, with a multiplier of 100. 143 4.6. Welfare evaluation: optimized v.s. linear interest rate rules 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 a b c d e f g h i j k l m R standard deviation of π s ta nd ar d de vi at io n of y Figure 4.2: Policy frontiers: simply linear interest rate rules versus Ramsey policy. The letters on the graph are the corresponding policy rules described in table 4.5. ”R” stands for the Ramsey policy. 144 Chapter 5 Conclusions Home bias in equity portfolios has been a puzzle for decades. The main questions addressed in chapter 2 are: (1) Can standard international mone- tary business cycle models with nominal bonds deliver home bias in equity?; (2) Can optimal home equities be obtained while ensuring that the equilib- rium equities are insensitive to preference parameter values?; (3) Are the relevant covariance ratios in the proposed model consistent with those in the data sample? It has been shown that the answers to these questions depend on the kind of bonds available in the international financial markets and whether monetary shocks have real effects on fundamentals. This chapter quantitatively investigates portfolio choice in a medium- scale, two-country monetary business cycle model with sticky prices and sticky wages. The main mechanism for home bias in equity portfolio lies in two key covariance ratios. One is the covariance ratio between relative labor income that is orthogonal to real exchange rates (or terms of trade), and relative returns to national equities that is orthogonal to real exchange rates. The other is the covariance ratio between relative returns on bonds that is orthogonal to real exchange rates, and relative returns to national equities that is orthogonal to real exchange rates. The model shows that either in a model with complete asset markets or with incomplete asset markets, major shocks considered in the literature, such as, technological shocks, mone- tary shocks, investment efficiency shocks, government expenditure shocks and preference shifts, either separately or jointly, contribute to generating a negative moderate covariance ratio for hedging human capital income risk. This covariance ratio generated by the model is also consistent with the data. Nevertheless, the latter covariance ratio for hedging real exchange rate risk depends on the type of bonds traded internationally. Numerical experiments show that a model with nominal bonds and monetary shocks hardly deliver a robust and close-to-zero covariance ratio for hedging real exchange rate risk , whereas a model with inflation-indexed bonds or with- out monetary shocks can present such a close-to-zero covariance ratio that is consistent with the data. The motivation for adapting inflation-indexed bonds stems from the recent development in bond markets. This is the main 145 Chapter 5. Conclusions contribution of this work. Table 5.1: Optimal bond assets-GDP ratio: inflation-indexed bonds γ = 0.50 γ = 0.85 γ = 1 γ = 1.1 γ = 1.5 γ = 2 γ = 3 σ = 0.5 1.16 0.67 0.50 0.39 0.01 0.41 1.06 σ = 1 0.90 0.28 0.07 0.06 0.49 0.91 1.54 σ = 1.5 0.78 0.08 0.14 0.27 0.71 1.12 1.73 σ = 2 0.70 0.04 0.27 0.40 0.84 1.24 1.83 σ = 5 0.50 0.32 0.55 0.68 1.11 1.49 2.04 σ = 10 0.41 0.43 0.67 0.80 1.21 1.58 2.11 Note: This table reports the ratio of bond assets to GDP in the benchmark model. σ is the coefficient of relative risk aversion and γ is the trade elasticity. Table 5.2: Bond-GDP ratio Country Debt assets - GDP ratio (%) Debt liabilities - GDP ratio (%) External debts - GDP ratio (%) Austria 51.6 68.8 17.1 Canada 20.2 51.2 31.0 Denmark 36.6 75.6 39.0 France 43.9 48.4 4.5 Germany 43.7 44.3 0.6 Italy 29.9 41.7 11.8 Japan 30.2 22.9 -7.4 Spain 23.2 36.3 13.1 UK 126.3 139.6 13.3 US 18.3 26.9 8.5 Average 34.3 41.1 6.8 Data source: Journal of International Economics, Lane and Milesi-Ferretti (2007), adapted by permission. The other contribution of this chapter is exploring the second moments both for portfolios and aggregate variables. Net foreign asset positions ex- hibit counter-cyclical movement with output both in the data and in all models studied here, either with complete markets or with incomplete asset markets. The model also generates sizable international co-movements for aggregate variables. Besides the findings mentioned above, there are other factors worth ad- dressing. First, a model with technological shocks and government expen- 146 Chapter 5. Conclusions diture shocks generates robust super home-bias in equities. A similar super home-bias in equities is obtained when incorporating only technology shocks and preference shocks into a model. It is shown that super-home bias dis- appears when additional shocks, say, monetary shocks, are present in the model. Second, the bond assets-GDP ratio produced by the model varies significantly across different values of preference parameters, although the model could generate robust and reasonable home bias in equities. Ta- ble 5.1 shows this bond assets-GDP ratio in the benchmark model, while the data in table 5.2 displays reasonably-stable bond assets-GDP ratios for many countries. This remains an unsolved puzzle in this work and will likely draw further attention from other researchers. Suggestions for fu- ture work include: (1) taking into account the phenomena of home bias in equities and of reasonable debt-GDP ratios in a unified framework; (2) investigating portfolio choice in a multiple-country environment that incor- porate international propagation and portfolio adjustments responding to shocks in other countries with different country sizes; (3) studying optimal portfolio choice under different stages of financial integration and financial development; (4) exploring cross-sectional implications between the volume of inflation-indexed bonds and equity home bias and validating the mecha- nism developed in this chapter in a further step. How does international risk sharing change with financial integration? Chapter 3 investigates this issue in a center-peripheral framework. I find that financial integration indeed enhances international risk sharing for pe- ripheral countries, but it not necessarily the case for the central country. The main mechanism lies in demand effect induced by country size. When the central country is large, the external demand for assets is low and partial financial integration could provide a better risk sharing for the core country. There are several important aspects untouched in this chapter. First, I focus on consumption volatility. For a complete welfare analysis, I need to resort to higher order approximations to obtain first-order dynamics of assets. This will be left in my ongoing work (Yu 2012). Second, I investigate equity assets only and ignore other type of assets such as debt assets. Future work should include this type of assets and explore asset structures and risk sharing. When both wages and prices are determined by staggered nominal con- tracts, and some credit-constrained agents can only issue non-state con- tingent nominal debt, chapter 4 shows how to conduct optimal monetary policies. In the benchmark model, interest rate should decrease when asset price rises but the response is very small. In terms of welfare maximiza- tion, credit frictions provide policy-makers with another channel to drive 147 Chapter 5. Conclusions down the volatility of consumption and labor supply, which is omitted in a credit-frictionless economy. In the Ramsey policy, output and asset price volatilities are relatively small because external finance premium is positively co-moves with output, which is opposite, however, in real models (flexible price and wage setting and real debt). I also compare several linear interest rate rules according to their inflation- output-gap volatility frontiers and social welfare induced by each rule. Rules with a higher response to lagged interest rate perform better than rules without responding to the lagged interest rate. 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Labor income is the compensation of employees from the OECD total labor cost on OECD.stat database. Notice that the share labor income in total income from non-corporate firms and gross mixed income is relatively small, less than 10% in many countries, even less than 5% in some countries. Because of data unavail- ability of mixed income in quarterly frequency, I take only use of compensation of employees. Bilateral trade volume is constructed as follows. Let exi,j,t be the current value of export (in U.S. dollars) from country i to country j at period t. When country i is the home country, the weight of country j in the foreign country coun- terpart is exi,j,t+exj,i,t∑ j(exi,j,t+exj,i,t) . Data on exi,j,t is from the Direction of Trade of IMF database. Nominal exchange rate is from taken from the International Financial Statis- tics of IMF database. A.2 Annual data on national accounts and net foreign assets Annual data series on real GDP, the GDP deflator, export and import prices, real private consumption, real fixed investment, consumer price index, current ac- count, nominal exchange rate, population, discount rate and treasure-bill rate, which were taken from IMF’s International Financial Statistics (IFS for short) database, for the period 1970 − 2004. Employment data is taken from OECD’s national accounts (OECD.stat database) for the same period. Money supply for U.S., Japan and Canada, using M1 definition of money, were taken from IMF’s IFS database. U.K. doesn’t have M1 definition of money, and I take the closest measure of narrow money, M0, instead, which was drawn from the online database in Bank of England (table code is LPMAVAE ). The rest of countries in the data 160 A.3. Estimating returns on human capital sample are all members of the Euro-Zone, and, instead, I use the growth of narrow money (M1) in EU-16 countries, which is drawn from OECD’s national accounts, as a proxy measure of money growth in these countries. Home and Foreign counterpart are constructed as follows. When U.S. is the Home country, I then constructed trade-weighted aggregates for Japan, U.K., Ger- many, France, Italy, Canada, Spain, Austria and Denmark. The trade weights were constructed by adding up exports from Home country to Foreign country i and exports from Foreign country i to Home country and dividing by the total home trade with the rest of world in the data sample. Of course, the sum of trade weight is equal to one. Export data, in current U.S. dollars, were taken from IMF’s Direction of Trade (1980-2004, CD-ROM) and historical Direction of Trade (1970−1978, Computer file. Sources: STUDYNO = 07628 International Monetary Fund, DIRECTION OF TRADE (Computer file), 2nd release, Washington, DC, International Monetary Fund (producer), 1979, Ann Arbor, MI: Inter-University Consortium for Political and Social Research (distributor), 1979.). The bilateral trade data in year 1979 was filled by using interpolation method. The foreign coun- try is constructed by aggregating the rest of economies in the sample using trade weights described above. Real GDP, real private consumption, real fixed investment and real money balance series are purchasing power parity exchange rates adjusted in 2000 U.S. dollars. Consumption based real exchange rate dada is defined using consumer price index weighted by trade weight. Real imports and exports series are deflated by import and export prices, respectively. Data on foreign equity as- set positions and foreign bond asset positions were taken from Lane and Lane and Milesi-Ferretti (2001) and Lane and Milesi-Ferretti (2007). A.3 Estimating returns on human capital This section displays the method of estimating returns on human capital. From the asset pricing perspective, labor income in an economy can be considered as dividends from human capital. As in the main text, wht denotes the real labor income in the home country. Let vt stands for the real price of human capital in the home country. Following Campbell (1996), I assume that dividend-price ratio is stationary and its unconditional expectation is constant, say, at its long-run deterministic steady state. Let x̃ represent the log of xt. Then the realized log of gross return on human capital has a form of, r̃n,t+1 ≡ log(wht+1 + vt+1)− log(vt) = log ( 1 + vt+1 wht+1 ) + w̃ht+1 − ṽt = κ+ ρṽt+t + (1− ρ)w̃ht+1 − ṽt +O (‖ ‖) (A.1) with κ ≡ log(1 − ρ). ρ characterizes the steady state inverse of gross return to human capital, 1ρ ≡ v+whv . Then up to a first-order approximation, the log of realized gross return to human capital can be expressed as the sum of a constant 161 A.3. Estimating returns on human capital and a weighted average of prices and dividends. Let the log of dividend-price ratio be ζ̃t ≡ w̃ht− ṽt. Then the log of return to human capital (A.1) can be written as, r̃n,t+1 = κ+ ζ̃t − ρζ̃t+1 + ∆w̃ht+1 +O (‖ ‖) (A.2) where ∆w̃ht+1 ≡ w̃ht+1 − w̃ht. Since the dividend-price ratio is stationary, ζ̃t+j doesn’t explode as j increases. Writing equation (A.2) forward recursively yields that the current price-dividend ratio is a weighted average of future log of returns to human capital adjusted by labor income growth, ζ̃t = ∞∑ j=0 ρj ( r̃n,t+1+j −∆w̃ht+1+j ) − κ 1− ρ +O (‖ ‖) (A.3) As in Campbell (1996), I assume further that the conditional expected return to financial wealth equals the conditional expected return to human capital within a country, that is, Et {r̃n,t+j} = Et {r̃a1,t+j} for all j > 0. Taking expectations on both sides of price-dividend ratio (A.3) conditional on the information up to period t and substituting Et {r̃a1,t+j} for Et {r̃n,t+j}, obtains, ζ̃t = Et ∞∑ j=0 ρj ( r̃a1,t+1+j −∆w̃ht+1+j )− κ1− ρ +O (‖ ‖) (A.4) Next, substituting ζ̃t in equation (A.4) into equation (A.2) yields an expression for the log of return to human capital r̃n,t+1. Thereby, the unexpected return to human capital can be written as a linear combination of labor income growth and returns to financial wealth, r̃n,t+1 − Etr̃n,t+1 = −ρ ( ζ̃t+1 − Etζ̃t+1 ) + ∆w̃ht+1 − Et∆w̃ht+1 +O (‖ ‖) = −ρ(Et+1 − Et) ∞∑ j=0 ρj ( r̃a1,t+2+j −∆w̃ht+2+j ) + (Et+1 − Et)∆w̃ht+1 +O (‖ ‖) = (Et+1 − Et) ∞∑ j=0 ρj∆w̃ht+1+j − (Et+1 − Et) ∞∑ j=1 ρj r̃a1,t+1+j +O (‖ ‖) (A.5) The equation above shows that increases in expected future labor income growth (the first term on the right hand side at the bottom line) cause a positive return on human capital, while rises in expected future financial wealth returns (the second term on the right hand side at the bottom line) have a negative effect on returns to human capital since dividend flows from human capital are discounted at a higher 162 A.3. Estimating returns on human capital rate. To estimate the right hand side of the last line of equation (A.5) and to obtain the innovations to human capital returns, I adapt a vector autoregressive approach of Campbell. I put the real stock index return r̃a1,t as the first element of a vector zt and real labor income growth ∆w̃ht as the second element of the vector zt. Other variables observed at the end of period t, denoted by xt, which contain useful information to forecast future stock returns and labor income growths, are stacked following these two variables. Thereby, zt ≡ [r̃a1,t,∆w̃ht, xTt ]T . Suppose that zt follows a first-order VAR process, zt+1 = AZt + εt+1 (A.6) where A is the companion matrix of the VAR and innovations εt+j for j > 1 are orthogonal to zt. As in Campbell (1996), xt contains a relative bill rate (the difference between 3-month Treasury bill rate and its one-year backward moving average) and the yield spread between long (5-year) and short (3-month) term government bonds. These two variables are considered to be informative about future stock returns and labor income growth in the literature. Let e1 be a vector with the same dimension as zt, whose first element is one and whose other elements are all zero, or in another word, r̃a1,t = e T 1 zt. Similarly, let e2 be a vector which picks up the second element in zt, that is, ∆w̃ht = e T 2 zt. The data used here are quarterly data which is described in the data appendix. The foreign country variables are constructed as follows. First, labor income and stock index returns are deflated by CPI and then I remove seasonal components of each series using X-12. Second, express all domestic variables in the home currency using bilateral real exchange rates. Third, using the bilateral trade as weights, take a weighted average of the corresponding variable in other countries to obtain an artificial foreign variable. Each series is demeaned and therefore no constant term in the VAR above. I estimate the VAR processes in the home country and its foreign counterpart separately since stock returns and labor income growth are mainly determined by its own characteristics within a country. Coeurdacier and Gourinchas (2011) estimate a similar process like equation (A.6) using internationally differenced variables and other variables, such as log consumption expenditure, relative bond returns and cyclical external imbalances nxa in Gourinchas and Rey (2007). Before estimating the VAR process in equation (A.6), we need to decide the optimal lags. The optimal lags for most of countries in the sample (i.e., USA, Japan, Germany, France, Italy, Denmark and Australia and their foreign counterparts, UK, Canada, and the foreign counterpart of Spain) is one. I then chose the first-order VAR specification for all countries in the sample. As in the literature, the steady state return parameter is set to ρ = 0.98 in the quarterly data in order to reflect the risk-premium observed in the data on asset prices. After obtaining the coefficient A, then the conditional expectations of vector zt becomes Etzt+1+j = A j+1zt with 163 A.4. Alternative currency pricing: pricing-to-market j ≥ 0. The expected changes in returns on stocks can be written as, (Et+1 − Et) ∞∑ j=1 ρj r̃a1,t+1+j = e T 1 ρA(I − ρA)−1εt+1 (A.7) with an identity matrix I. The expected change in labor income growth then becomes, (Et+1 − Et) ∞∑ j=0 ρj∆w̃ht+1+j = e T 2 (I − ρA)−1εt+1 (A.8) Thereby, the innovations in the return on human capital can be expressed as, r̃n,t+1 − Etr̃n,t+1 = (eT2 − eT1 ρA)(I − ρA)−1εt+1 (A.9) Analogous to Coeurdacier and Gourinchas (2011), the estimation show that innovations to the returns on human capital are quite volatile, with a similar order as returns on stocks. A.4 Alternative currency pricing: pricing-to-market Producer currency pricing assumes a complete exchange rate pass-through. Re- cent evidence from micro data (see, Gopinath and Rigobon 2008; Gopinath, Itskhoki and Rigobon 2010) shows that 90% (97%) of U.S. imports (exports) are priced in dollars.43 Therefore, there is producer currency pricing in exports and local cur- rency pricing in imports for U.S.. In this section, we consider another extreme case in which firms take use of local currency pricing instead of producer currency pricing, and subsequently the exchange rate pass-through is zero. Assume that goods markets are segmented across borders. Firms in both coun- tries can set their prices in terms of local currency, which implies that the law of one price doesn’t hold even for an individual variety. The main difference between these two pricing strategies is the intermediate producers’ maximization problems, market clearing conditions for goods and the evolvement of prices. In the following analysis, except for the currency in pricing, the environment considered in this sec- tion is the exactly same as in section 2.1, and therefore we just list the equations which are different from those in the producer currency pricing. 43Gopinath, Itskhoki and Rigobon (2010) report the fraction of nondollar priced im- ports from most of counties is small. For instance, Germany is 40%, Japan 21%, UK 20% and Canada 4%. 164 A.4. Alternative currency pricing: pricing-to-market A.4.1 Intermediate producers Intermediate producer j obtains nominal profits from both markets in period t, Dp,j,t ≡ (PH,j,t − Pw,t) ( PH,j,t PH,t )−θp YH,t + (EtP ∗H,j,t − Pw,t) ( P ∗H,j,t P ∗H,t )−θp Y ∗H,t (A.10) where the first term on the right hand side denotes the revenues from sales of intermediate goods at Home country minus cost of wholesale goods and the second term shows revenues from abroad net of cost of wholesales, all of which are in terms of Home currency. Producer j chooses P̃H,j,t and P̃ ∗ H,j,t to maximize its expected discounted profits (2.21). The optimal prices are set such that the sum of expected discounted profits from selling intermediates at Home (Foreign) equals the expected discounted cost from purchasing whole goods at Home (Foreign) given the random duration of price readjustment. It follows that the optimal prices are given by, P̃H,j,t = θp θp − 1 Et {∑+∞ τ=0 ξ τ pQt,t+τYH,t+τP 1+θp H,t+τ/Xp,t+τ } Et {∑+∞ τ=0 ξ τ pQt,t+τYH,t+τP θp H,t+τ } P̃ ∗H,j,t = θp θp − 1 Et {∑+∞ τ=0 ξ τ pQt,t+τY ∗ H,t+τP ∗θp H,t+τPH,t+τ/Xp,t+τ } Et {∑+∞ τ=0 ξ τ pQt,t+τEt+τY ∗H,t+τP ∗θpH,t+τ } Foreign intermediate producers have similar rules for optimal prices P̃F,j,t and P̃ ∗F,j,t. It then follows that total real dividends accumulated by Home intermediate firms are given by, dp,t ≡ ∫ 1 0 dp,j,tdj = YH,tpH,t + Y ∗ H,tp ∗ H,tst − ( YH,ts p H,tpH,t Xp,t + Y ∗H,ts p∗ H,tpH,t Xp,t ) Dividends to intermediaries consist of total revenues from sales of intermediates in Home and Foreign goods markets net of total cost of purchasing wholesale goods. spH,t represents the price dispersion in the home market and s p∗ H,t is its foreign counterpart. 165 A.4. Alternative currency pricing: pricing-to-market A.4.2 Market clearings and the competitive equilibrium The market clearing conditions for each good at each country are as follows, CH,t + IH,t = YH,t C ∗ H,t + I ∗ H,t = Y ∗ H,t CF,t + IF,t = YF,t C ∗ F,t + I ∗ F,t = Y ∗ F,t The supply of wholesale goods links to the demand for it via the following equation, Yt = ∫ 1 0 YH,j,tdj + ∫ 1 0 Y ∗H,j,tdj = s p H,tYH,t + s p∗ H,tY ∗ H,t The Home goods price dispersion in Home market is measured by sp∗H,t, which evolves as, sp∗H,t = (1− ξp)p̃−θpH,t + ξppiθpH,tsp∗H,t−1 Home goods in the Foreign market and Foreign goods in the Home and Foreign markets have similar price dispersion evolvements. The aggregate dividends at Home country distributed to equity holders have a form of, dt = αpH,tYH,t Xp,t − pI,tIt + dp,t Labor markets and financial markets are exactly the same as those in the bench- mark model with producer currency pricing. The competitive equilibrium consists of a sequence of prices and of allocation rules for households, wholesalers and inter- mediate producers such that (1) households’ decision rules solve their own utility maximization problems given the prevailing asset prices, goods prices and wages; (2) The decision rules for firms solve their profit maximization problems; (3) Goods markets, labor markets and asset markets clear each period. A.4.3 Steady state portfolios Notice that under pricing to market, exchange rate pass-through is zero and consequently real exchange rate doesn’t perfectly move along with terms of trade even up to a first order approximation. Consequently, even though a bond asset whose interest payment is indexed to locally produced goods does completely hedge terms of trade fluctuations, it doesn’t yet hedge real exchange rate risk completely, and vice versa. We then evaluate the different components of optimal shares of home equities. Analogous to the benchmark model, home equity holdings are ex- tremely sensitive to the preference parameters when nominal consols are traded in the international financial market. The home household holds 412% of foreign eq- uities when trade elasticity is 2 and the coefficient of relative risk aversion is σ = 2. However, the component of optimal equity shares hedging human capital income risk is similar to the benchmark model. 166 A.5. Foreign currency forward contract A.5 Foreign currency forward contract In Engel and Matsumoto (2009a), they show that foreign exchange forward contracts can hedge exchange rate risk. This section aims to examine whether their claims hold in the model studied in this chapter. The asset menu now available to households in both countries becomes equity assets and foreign exchange forward contracts. First, let’s write the household’s budget constraint in local currency as, PtCt +Mt +QtψH,t+1 + EtQ∗tψF,t+1 = ψH,t(Qt +Dt) + ψF,tEt(Q∗t +D∗t ) + (Et −Ft)ψ̃t +WtHdt +Mt−1 + Trt (A.11) where Ft is the delivery price of a forward contract and Et is the spot exchange rate at maturity of the contract. ψ̃t stands for the units of forward contracts purchased by the home household. When ψ̃t ≥ 0, the home household has a long position in a forward contract. The optimal condition for a foreign currency forward contract gives, 0 = Et [ v(t) Uc,t+1 Pt+1 (Et+1 −Ft+1) ] (A.12) where Et −Ft denotes the payoff from a long position in a forward contract on one unit of foreign currency. Table A.1: Optimal home equity shares held by home households: nominal consols and local currency pricing γ = 0.50 γ = 0.85 γ = 1 γ = 1.1 γ = 1.5 γ = 2 γ = 3 σ = 0.50 3.54 1.53 1.00 0.71 -0.16 -0.84 -1.65 σ = 1.00 3.54 1.53 1.00 0.71 -0.16 -0.84 -1.65 σ = 1.50 3.54 1.53 1.00 0.71 -0.16 -0.84 -1.65 σ = 2.00 3.54 1.53 1.00 0.71 -0.16 -0.84 -1.65 σ = 5.00 3.54 1.53 1.00 0.71 -0.16 -0.84 -1.65 σ = 10.00 3.54 1.53 1.00 0.71 -0.16 -0.84 -1.65 Note: The number shown above is the optimal share of home equities held by home residents. Each unit of consol bond pays one unit of local currency each period. Producers use local currency pricing. There’s home bias in con- sumption and no physical capital. Shock process parameters are σ2m = 2σ 2 a, ρa = 0.905 4 and ρm = 0.357. Other parameters are the same as in the bench- mark model. Let the real returns on one unit of foreign currency from a long position in a forward contract be, rc,t ≡ st pi∗t − ft pit (A.13) 167 A.5. Foreign currency forward contract where ft ≡ FtP ∗ t−1 Pt−1 , which has a similar interpretation as real exchange rates. The delivery price of a forward contract in equation (A.12) then is determined by the following condition, Et [Λt,t+1rc,t+1] = 0 (A.14) Rewrite Household’s budget constraint (A.11) in terms of real consumption, Ct + qtψH,t+1 + stq ∗ tψF,t+1 +mt = wtH d t + (qt + dt)ψH,t + st(q ∗ t + d ∗ t )ψF,t + rc,tψt + mt−1 pit + trt (A.15) with ψt ≡ ψ̃tP∗t−1 , the number of forward contracts on foreign currency in terms of real foreign consumption. As usual, net foreign asset position,nfat+1, here is defined as, nfat+1 = qt (ψH,t+1 − 1) + stq∗tψF,t+1 Budget constraint (A.15) can be written as, nfat+1 = wtH d t + dt − Ct + qt−1(ψH,t − 1)r1,t + st−1q∗t−1ψF,tr2,t + rc,tψt = wtH d t + dt − Ct + r2,tnfat + αTt rx,t where αt ≡ [qt−1(ψH,t − 1), ψt]T and excess returns on home equity and forward contract, rx,t ≡ [r1,t − r2,t, rc,t]T . I redo the experiment and obtain the optimal equity shares. The experiment shows that a model with foreign currency forward contracts coincides with a model with one-period nominal bonds, both of which are almost the same as a model with nominal consols. Optimal shares of home equities are quite sensitive to trade elasticity and relative risk aversion. 168 Appendix B Appendix to Chapter 3 B.1 Log-linearization B.1.1 Budget constraints The linearized budget constraint can be written as, 3 4 ÑWB,t+1 + 1 4 ÑWC,t+1 = 3 4β ÑWB,t + 1 4β ÑWC,t + 3 4 α̃′BR̂x,t + 1 4 α̃′CR̂x,t + ΓA1∆ẐB,t + ΓA2∆ẐC,t + ΓA3∆ĈB,t + ΓA4∆ĈC,t (B.1) 1 4 ÑWB,t+1 + 3 4 ÑWC,t+1 = 1 4β ÑWB,t + 3 4β ÑWC,t + 1 4β α̃′BR̂x,t + 3 4β α̃′CR̂x,t + ΓA2∆ẐB,t + ΓA1∆ẐC,t + ΓA4∆ĈB,t + ΓA3∆ĈC,t (B.2) where ΓAs with s = 1, 2, 3, 4 are functions of structural parameters in the model, which are given as, ΓA1 = 1 2 [ −ΓY 1 + ΓY 3 2 + ΓP1(1− α1)− ΓP3 2 (1 + α1 − 2α1α2) ] ΓA2 = 1 2 [ −ΓY 1 − ΓY 3 2 + ΓP1(1− α1) + ΓP3 2 (1 + α1 − 2α1α2) ] ΓA3 = 1 2 [ 1 + ΓY 2 + ΓY 4 2 − ΓP2(1− α1) + ΓP4 2 (1 + α1 − 2α1α2) ] ΓA4 = 1 2 [ ΓY 2 − ΓY 2 2 − ΓP2(1− α1)− ΓP4 2 (1 + α1 − 2α1α2) ] where ΓY s and ΓPs with s = 1, 2, 3, 4 are defined as follows, ΓY 1 = 2α1γ(1 + η) 2α1γ(θ + η) + 1− θ ΓY 2 = (1− θ)(2α1γσ − 2α1 + 1) 2α1γ(θ + η) + 1− θ ΓY 3 = γ(1 + η) [α1(2α2 − 1) + 1] γ(θ + η) [α1(2α2 − 1) + 1] + 1− θ ΓY 4 = (1− θ) [σγ + (σγ − 1)α1(2α2 − 1)] γ(θ + η) [α1(2α2 − 1) + 1] + 1− θ 169 B.1. Log-linearization ΓP1 = 1 + η η + θ 1 Γ1 ΓP2 = [ (1− θ)σ η + θ + 2α1 − 1 ] 1 Γ1 ΓP3 = 1 + η η + θ 1 Γ2 ΓP4 = [ (1− θ)σ η + θ + α1(2α2 − 1) ] 1 Γ2 Γ1 and Γ2 are defined as, Γ1 = 2(1− α1) [ 2α1γ + 1− θ η + θ ] Γ2 = [α1(2α2 − 1)− 1] { γ [α1(2α2 − 1) + 1] + 1− θ η + θ } Notice that Γ1 > 0 and Γ1 < 0 and thereby, ΓP1 > 0, ΓP2 > 0, ΓP3 < 0. ΓY 1 > 0, ΓY 3 > 0. A special case. When trade elasticity is equal to unity γ = 1 and there is no home bias in consuming peripheral goods for peripheral countries α2 = 1 2 , terms of trade can automatically adjust the value of production in the peripheral countries, which implies that when labor supply is inelastic (or the economy is an endowment economy), equities in the peripheral countries are perfect substitutes. B.1.2 First-order approximation to Euler equations Euler equations for portfolio choices can be approximated by the following equations, (σ − ζ)∆ĈB,t + ∆P̂B,t = Et [ σ∆ĈB,t+1 + ∆P̂B,t+1 ] (B.3) (σ − ζ)∆ĈC,t + ∆P̂C,t = Et [ σ∆ĈC,t+1 + ∆P̂C,t+1 ] (B.4) where relative consumer prices ∆P̂B,t ≡ P̂A,t − P̂B,t and ∆P̂C,t ≡ P̂A,t − P̂C,t are defined as, ∆P̂B,t = ΓB1∆ẐB,t + ΓB2∆ẐC,t − ΓB3∆ĈB,t − ΓB4∆ĈC,t ∆P̂C,t = ΓB2∆ẐB,t + ΓB1∆ẐC,t − ΓB4∆ĈB,t − ΓB3∆ĈC,t where ΓBs, s = 1, 2, 3, 4 are defined as, ΓB1 = 1− 2α1 2 ΓP1 − ΓP3 2 (α1 − 2α1α2) ΓB2 = 1− 2α1 2 ΓP1 + ΓP3 2 (α1 − 2α1α2) ΓB3 = 1− 2α1 2 ΓP2 − ΓP4 2 (α1 − 2α1α2) ΓB4 = 1− 2α1 2 ΓP2 + ΓP4 2 (α1 − 2α1α2) 170 B.1. Log-linearization In deriving relative prices ∆P̂B,t and ∆P̂C,t, I have used the fact that prices of products in each country can be expressed as functions of relative consumption and relative technology, P̂BB,t = [ ΓP1 − ΓP3 2 , ΓP1 + ΓP3 2 ] [ ∆ẐB,t ∆ẐC,t ] − [ ΓP2 − ΓP4 2 , ΓP2 + ΓP4 2 ] [ ∆ĈB,t ∆ĈC,t ] (B.5) P̂CC,t = [ ΓP1 + ΓP3 2 , ΓP1 − ΓP3 2 ] [ ∆ẐB,t ∆ẐC,t ] − [ ΓP2 + ΓP4 2 , ΓP2 − ΓP4 2 ][ ∆ĈB,t ∆ĈC,t ] (B.6) Substituting relative prices ∆P̂B,t and ∆P̂C,t into the linearized Euler equations and I obtain, Et [ (σ − ΓB3)∆ĈB,t+1 − ΓB4∆ĈC,t+1 ] = −Et [ ΓB1∆ẐB,t+1 + ΓB2∆ẐC,t+1 ] + (σ − ζ − ΓB3)∆ĈB,t − ΓB4∆ĈC,t + ΓB1∆ẐB,t + ΓB2∆ẐC,t (B.7) Et [ (σ − ΓB3)∆ĈC,t+1 − ΓB4∆ĈB,t+1 ] = −Et [ ΓB1∆ẐC,t+1 + ΓB2∆ẐB,t+1 ] + (σ − ζ − ΓB3)∆ĈC,t − ΓB4∆ĈB,t + ΓB1∆ẐC,t + ΓB2∆ẐB,t (B.8) B.1.3 Solving the dynamic system Let xt ≡ [ÑWB,t; ÑWC,t] and yt ≡ [C̃B,t; C̃C,t]. A four-equation system can be rewritten as follows, A [ xt+1 Et(yt+1) ] = B [ xt yt ] + ft (B.9) with A ≡ 3 4 1 4 0 0 1 4 3 4 0 0 0 0 σ − ΓB3 −ΓB4 0 0 −ΓB4 σ − ΓB3 , B ≡ 3 4β 1 4β ΓA3 ΓA4 1 4β 3 4β ΓA4 ΓA3 0 0 σ − ζ − ΓB3 −ΓB4 0 0 −ΓB4 σ − ζ − ΓB3 ft ≡ ΓA1 ΓA2 ΓA2 ΓA1 ΓB1 ΓB2 ΓB2 ΓB1 [ ∆ẐB,t ∆ẐC,t ] + 0 0 0 0 −ΓB1 −ΓB2 −ΓB2 −ΓB1 Et [∆ẐB,t+1] Et [ ∆ẐC,t+1 ] + 3 4 1 4 1 4 3 4 0 0 0 0 [ ξA,tξB,t ] 171 B.1. Log-linearization Let Ã ≡ A−1B and f̃t ≡ A−1ft. I can rewrite the dynamic system (B.9) as,[ xt+1 Et(yt+1) ] = Ã [ xt yt ] + f̃t (B.10) Following Blanchard and Quah (1980), applying Jordan decomposition to Ã and obtain Ã = VΛV−1 where Λ is a diagonal matrix with eigenvalues of Ã, λi, i = 1, 2, 3, 4, on its diagonal in a descending order. V is a matrix whose columns are the corresponding orthog- onal eigenvector to eigenvalue λi. These four eigenvalues can be written explicitly as, λ1 = 1− ζ σ − [α1(2α2−1)− σ(θ−1) η+θ ](α1−2α1α2) γ−α21γ(2α2−1)2+ (α1(2α2−1)−1)(θ−1)η+θ λ3 = 1 β λ2 = 1− ζ σ + (2α1−1)[σ(θ−1)η+θ −2α1+1] 2(α1−1)[2α1γ− θ−1η+θ ] λ4 = 1 β Observe that λ2 < 1 when 1 2 ≤ α1 ≤ 1 and λ3 > 1, λ4 > 1. λ1 < 1 holds for a large range of parameter values. Let [ x2,t+1 Et(y2,t+1) ] ≡ V−1 [ xt+1 Et(yt+1) ] . The system (B.10) can be written as, [ x2,t+1 Et(y2,t+1) ] = [ Λ11 0 0 Λ22 ] [ x2,t y2,t ] + ˜̃ ft (B.11) with ˜̃ ft ≡ V−1A−1ft. Λ11 is a diagonal matrix with the first two less-than-one eigenvalues on its main diagonal and Λ22 is a diagonal matrix with the last two larger-than-one eigenvalues on the diagonal. The lower partition of system (B.11) then becomes, y2,t = −M∆Ẑt −Nξt (B.12) where M ≡ Λ−122 (I2 − Λ−122 ρ)−1[V−1A−1F ]L and N ≡ Λ−122 [V−1A−1G]L. [·]L de- notes the bottom row of a matrix. F and G are two constant matrices, given by, F ≡ ΓA1 ΓA2 ΓA2 ΓA1 ΓB1(1− ρ) ΓB2(1− ρ) ΓB2(1− ρ) ΓB1(1− ρ) , G ≡ 3 4 1 4 1 4 3 4 0 0 0 0 Substituting y2,t+1 back into xt+1 and yt+1, I obtain the solution for controls yt+1, yt+1 = D1ξt+1 +D2εt+1 +D3 [ xt+1 ∆Ẑt ] (B.13) 172 B.1. Log-linearization Ds, s = 1, 2, 3 are defined as, D1 ≡ − [[ V−1 ] 22 ]−1 N D2 ≡ − [[ V−1 ] 22 ]−1 MΦ D3 ≡ − [[[ V−1 ] 22 ]−1 [ V−1 ] 21 , [[ V−1 ] 22 ]−1 Mρ ] and Φ ≡ [ 1 −1 0 1 0 −1 ] . Unexpected excess returns follow as, R̂B,t+1 − R̂A,t+1 = R11ξt+1 +R21εt+1 (B.14) R̂C,t+1 − R̂A,t+1 = R12ξt+1 +R22εt+1 (B.15) where R11, R21, R12 and R22 are constant and are given by, R11 = −Γ′RB2D1 R21 = Γ′RB1Φ− Γ′RB2D2 R12 = −Γ′RC2D1 R22 = Γ′RC1Φ− Γ′RC2D2 and ΓRB1 = [ ΓP1−ΓP3 2 − ΓY 1+ΓY 32 ΓP1+ΓP3 2 − ΓY 1−ΓY 32 ] ΓRB2 = [ ΓP2−ΓP4 2 − ΓY 2+ΓY 42 ΓP2+ΓP4 2 − ΓY 2−ΓY 42 ] ΓRC1 = [ ΓP1+ΓP3 2 − ΓY 1−ΓY 32 ΓP1−ΓP3 2 − ΓY 1+ΓY 32 ] ΓRC2 = [ ΓP2+ΓP4 2 − ΓY 2−ΓY 42 ΓP2−ΓP4 2 − ΓY 2+ΓY 42 ] Now I write relative marginal utility as, − σ∆ĈB,t+1 −∆P̂B,t+1 = D11ξt+1 +D21εt+1 + otherst+1 (B.16) − σ∆ĈC,t+1 −∆P̂C,t+1 = D12ξt+1 +D22εt+1 + otherst+1 (B.17) where D11 = Γ ′ UB1D1 D21 = Γ ′ UB1D2 − Γ′UB2Φ D12 = Γ ′ UC1D1 D22 = Γ ′ UC1D2 − Γ′UC2Φ and ΓUB1 ≡ [ −σ + ΓB3 ΓB4 ] ΓUB2 ≡ [ ΓB1 ΓB2 ] ΓUC1 ≡ [ ΓB4 −σ + ΓB3 ] ΓUC2 ≡ [ ΓB2 ΓB1 ] The last term otherst+1 represents a collection of terms which are predetermined 173 B.2. Impulse responses at date t+ 1. Substituting excess returns in equation (B.14)-(B.15) and relative marginal util- ities in equation (B.16)-(B.17) into the second-order approximation to Euler equa- tions, yields, Et {[ D11ξt+1 +D21εt+1 0 0 D12ξt+1 +D22εt+1 ] [ R11ξt+1 +R21εt+1 R12ξt+1 +R22εt+1 ]} = 0 (B.18) Now use ξi,t+1 ≡ α̃′iR̂x,t+1, i = B,C to get rid of ξt+1 in the equations above, rearrange related terms and then arrive at a solution for portfolio α̃, which is sum- marized in result 1 and 2. Once obtaining the zeroth-order portfolio α̃, I substitute it into the solution to control variables in equation (B.13) and obtain a solution to the rational expectations equilibrium. Proposition 4 reports this solution in the main text. B.2 Impulse responses Relative labor supply ∆N̂i,t ≡ N̂A,t − N̂i,t can be written as, ∆N̂B,t = 1 θ + η {( Γ′NCB − σ [ 1 0 ]T) (D1H̃ +D2)− ( Γ′NZB − σ [ 1 0 ]T) Φ } εt ∆N̂C,t = 1 θ + η {( Γ′NCC − σ [ 0 1 ]T) (D1H̃ +D2)− ( Γ′NZC − σ [ 0 1 ]T) Φ } εt with ΓNZB ≡ [ (1− α1)ΓP1 − ΓP32 (1 + α1 − 2α1α2) (1− α1)ΓP1 + ΓP32 (1 + α1 − 2α1α2) ] ΓNCB ≡ [ (1− α1)ΓP2 − ΓP42 (1 + α1 − 2α1α2) (1− α1)ΓP2 + ΓP42 (1 + α1 − 2α1α2) ] ΓNZC ≡ [ (1− α1)ΓP1 + ΓP32 (1 + α1 − 2α1α2) (1− α1)ΓP1 − ΓP32 (1 + α1 − 2α1α2) ] ΓNCC ≡ [ (1− α1)ΓP2 + ΓP42 (1 + α1 − 2α1α2) (1− α1)ΓP2 − ΓP42 (1 + α1 − 2α1α2) ] B.3 Financial autarky When financial markets in each country are segmented from each other, trade in goods must balance in each period. The dynamic system then reads ,[ ∆ĈB,t ∆ĈC,t ] = − [ ΓA3 ΓA4 ΓA4 ΓA3 ]−1 [ ΓA1 ΓA2 ΓA2 ΓA1 ] [ ∆ẐB,t ∆ẐC,t ] 174 B.3. Financial autarky Substituting this solution into price equation (B.5) and (B.6), I obtain a solution to prices. I then substitute prices into the log-linearized production functions and labor market clearing conditions to get a solution to consumption and output in each country. 175 Appendix C Appendix to Chapter 4 C.1 A complete set of equilibrium conditions c−σt = βRt+1Et { c−σt+1 pit+1 } (C.1) qtc −σ t = βEt { c−σt+1qt+1(1− δ) + j kst+1 } (C.2) f1t = φ̃− 1 φ̃ w̃tc −σ t (1−τ lt ) ( wt w̃t )φ̃ ldt +α̃βEt (pit+1 piχ̃t )φ̃−1( w̃t+1 w̃t )φ̃−1 f1t+1 (C.3) f2t = l θ t ( wt w̃t )φ̃ ldt + α̃βEt (pit+1 piχ̃t )φ̃( w̃t+1 w̃t )φ̃ f2t+1 (C.4) f1t = f 2 t (C.5) Yt = at(k e t ) α0(ldt ) 1−α0 (C.6) bet+1 = κEt ( qt+1k e t+1pit+1 Rt+1 ) (C.7) cet + qt [ ket+1 − (1− δ)ket ] + τ let + betRt pit = (1− τkt ) ( Yt Xt − wtldt ) + bet+1 (C.8) qt(c e t ) −σ =(cet ) −σϕtκEt ( qt+1pit+1 Rt+1 ) + Et { γβ(cet+1) −σ [qt+1(1− δ) +(1− τkt+1) α0Yt+1 Xt+1ket+1 ]} (C.9) (cet ) −σ = (cet ) −σϕt + Et ( γβ (cet+1) −σRt+1 pit+1 ) (C.10) (1− α0)Yt Xtldt = wt (C.11) it = kt+1 − (1− δ)kt (C.12) 176 C.1. A complete set of equilibrium conditions qt = ψ ( it kt ) + 1 (C.13) yt = ct + c e t + it + c(it) + gt (C.14) x1t = φ φ− 1 1 Xt ytp̃ −φ−1 t + βαEt { c−σt+1 c−σt ( p̃t p̃t+1 )−φ−1( piχt pit+1 )−φ x1t+1 } (C.15) x2t = ytp̃ −φ t + βαEt { c−σt+1 c−σt ( p̃t p̃t+1 )−φ( piχt pit+1 )1−φ x2t+1 } (C.16) x1t = x 2 t (C.17) Ft = yt − yt Xt spt (C.18) Dt = itqt − ( ψ 2 ( it kt )2 kt + it ) (C.19) spt = (1− α)p̃−φt + α ( pit piχt−1 )φ spt−1 (C.20) τt = τ h t l d twt + τ f t (Ft +Dt) + τ k t ( Yt Xt − wtldt ) + τ lht + τ le t (C.21) nt+1 Rt+1 + τt = gt + nt pit (C.22) 1 = α(piχt−1/pit) 1−φ + (1− α)p̃1−φt (C.23) Yt = s p t yt (C.24) w1−φ̃t = α̃w 1−φ̃ t−1 (pi χ̃ t−1/pit) 1−φ̃ + (1− α̃)w̃1−φ̃t (C.25) lt = l d t s w t (C.26) swt = (1− α̃) ( w̃t wt )−φ̃ + α̃ ( wt−1 wt )−φ̃( pit piχ̃t−1 )φ̃ swt−1 (C.27) kt+1 = k e t+1 + k s t+1 (C.28) Fiscal Policy (C.29) Monetary Policy (C.30) ln(at/ā) = ρaln(at−1/ā) + at (C.31) ln(gt/g) = ρgln(gt−1/g) + g t (C.32) 177 C.2. The deterministic steady state C.2 The deterministic steady state The deterministic steady state in this system tries to replicate the long-run average values for variables of interest in the U.S.. The long-run average annualized GDP deflator over 1951− 2009 is 1.0344, then pi = 1.03441/4. Following Mendoza, Razin and Tesar (1994), income tax rates are set as τk = 0.407 and τh = 0.258. I set τf = τk, and lump-sump taxes are not applicable, τ le = τ ls = 0. R = pi β p̃ = ( 1− αpi(1−χ)(φ−1) 1− α ) 1 1−φ X = φ φ− 1 p̃ −1 1− αβpi(1−χ)(φ−1) 1− αβpi(1−χ)φ sp = (1− α)p̃−φ 1− αpi(1−χ)φ w w̃ = ( 1− α̃pi(1−χ̃)(φ̃−1) 1− α̃ ) 1 φ̃−1 sw = (1− α̃)(w/w̃)φ̃ 1− α̃pi(1−χ̃)φ ψ = −1 + ϕκβ + βγ [ 1− δ + (1− τk)α0(1+se)spδsiX ] δ − ϕκδβ − βγ [ (1− δ)δ + (1− τk)α0(1+se)spδ22siX ] q = ψδ + 1 sk = si δ + ψδ2/2 ce y = (1− τk)α0s p X − (δ + κ− κβ)q sk 1 + se c y = 1− si − sg − ce/y j = q(1/β − 1 + δ)sesk 1 + se 1 c/y l = ( φ̃− 1 φ̃ (1− τ l)(1− α̃βpi(1−χ̃)φ̃) 1− α̃βpi(1−χ̃)(φ̃−1) sw X (1− α0)sp w/w̃ 1 c/y ) 1 1+θ ld = l/sw 178 C.2. The deterministic steady state y = ( (ld)1−α0 ( sk 1 + se )α0 1 sp ) 1 1−α0 ce = ce y y c = c y y k = sky ke = sk 1 + se y ks = sesk 1 + se y g = sgy be = κqkeβ Y = ysp F = (1− sp/X)y i = δk D = iq − (i+ ψδ2k/2) w̃ = (1− α0)yspsw Xl(w/w̃) τ = τ lwld + τf (F +D) + τk(Y/X − wld) n = g ( 1− τ ysg ) 1 1/pi − 1/R 179
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Essays in international portfolio choice and monetary policy Yu, Changhua 2012
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Title | Essays in international portfolio choice and monetary policy |
Creator |
Yu, Changhua |
Publisher | University of British Columbia |
Date Issued | 2012 |
Description | Empirical evidence shows that equity home bias is a prevailing fact for most countries, but standard international monetary business cycle models with nominal bonds hardly generate equity home bias for plausible preference parameter values. By incorporating inflation-indexed bonds, I show in chapter 2 that international monetary business cycle models can explain home bias in equities. Inflation-indexed bonds can hedge real exchange rate risk and domestic equities serve as a hedge against domestic labor income risk conditional on real exchange rates. Moreover, this chapter accounts for counter-cyclical movements of net foreign asset positions. These results are robust in environments either with complete markets or with incomplete markets. How does international financial market integration alter international risk sharing? Chapter 3 develops a tractable center-periphery model with portfolio choice to investigate this issue. I compare three stages of financial integration. The first stage is financial autarky. The second stage is the central country becomes financially integrated with peripheral countries, but there is no financial integration between peripheral countries. The third stage is all financial markets are integrated into each other. From financial autarky to partial financial integration, volatility of consumption in all countries drops. From partial financial integration to full financial integration, volatility of consumption in peripheral countries decreases; however, consumption volatility rises in the central country when the central country is relatively large. When peripheral countries are large, the degree of international risk-sharing for all countries increases in the process of financial integration. What's the optimal monetary policy in an economy with financial frictions? Chapter 3 investigates optimal monetary policies in a dynamic stochastic general equilibrium model with sticky prices, sticky wages and credit-market imperfections. Credit frictions distort allocations and prices, and generate large volatility of aggregate variables via the financial accelerator. Policy-makers can take advantage of a debt deflation channel to push down volatility of endogenous variables through the financial decelerator. In the optimized linear interest rate rule, interest rate decreases in asset prices but the response is quantitatively small. This optimized rule exhibits high persistence. Within a class of simple linear interest rate rules, a strict inflation-targeting rule has a larger welfare loss. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-09-25 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-ShareAlike 3.0 Unported |
DOI | 10.14288/1.0073188 |
URI | http://hdl.handle.net/2429/43265 |
Degree |
Doctor of Philosophy - PhD |
Program |
Economics |
Affiliation |
Arts, Faculty of Vancouver School of Economics |
Degree Grantor | University of British Columbia |
GraduationDate | 2012-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-sa/3.0/ |
AggregatedSourceRepository | DSpace |
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