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Essays on exchange rate volatility and optimal monetary policy Xu, Juanyi 2004

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Essays on Exchange Rate Volatility and Optimal Monetary Policy by Juanyi Xu B.A., Economics, Zhejiang University, P. R. China, 1996 M.A., Political Economics, Zhejiang University, P.R.China, 1998 M.A. Econ., University of British Columbia, Vancouver, 1999 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Department of Economics) THE UNIVERSITY OF BRITISH COLUMBIA December ' , 2004 © Juanyi Xu, 2004 11 Abs t r ac t This thesis consists of three essays on exchange rate behavior and optimal monetary policy in open economy. The first essay proposes a framework to explain why the nominal and real exchange rates are highly volatile and seem to be disconnected from macroeconomic fundamentals. Two types of foreign exchange traders, rational traders and noise traders with erroneous stochas-tic beliefs, are introduced into a dynamic general equilibrium model with sticky prices. The presence of noise traders creates deviations from uncovered interest parity. As a result, ex-change rates can diverge significantly from fundamental values. Combined with local currency pricing and consumption smoothing behavior in an infinite horizon model, the presence of noise traders can help to explain the "exchange rate disconnect puzzle". The second essay explores the optimal monetary policy response to domestic and for-eign technology shocks in an open economy with vertical structure of production and trade. Through the vertical linkage in production, any stage-specific productivity shock in one coun-try has a trans-border spillover effect on the other country via vertical trade. So when choosing the optimal monetary rules, each monetary authority should respond to both home and foreign productivity shocks. Another finding is that the flexible exchange rate can not replicate the flexible price equilibrium even under producer currency pricing due to price stickiness in multiple stages. Finally, the exchange rate in such an environment will be more stable than that of an economy without vertical structure of production and trade. The third essay analyzes the determination of monetary policy in a world with a dollar standard, defined here as an environment in which all traded goods prices are set in US dollars. This generates an asymmetry whereby exchange rate pass-through into the US CPI is zero, while pass-through to other countries will be positive. I find that in such an economy, the US is essentially indifferent to exchange rate volatility in setting monetary policy, while the rest of the world places a high weight on exchange rate volatility. More importantly, in a Nash equilibrium of the monetary policy game between the US and the rest of the world, the preferences of the US dominate. Despite this, the US loses from the dollar's role as an Abstract iii international currency due to the absence of exchange rate pass-through even though US preferences dominate world monetary policy. Juanyi Xu. juanyixu@interchange.ubc.ca iv Contents Abstract ii Contents iv List of Tables vii List of Figures viii Acknowledgements ix Summary x 1 Noise Traders and the Exchange Rate Disconnect Puzzle 1 1.1 Introduction 1 1.2 The model 5 1.2.1 Households, Firms and Government 6 1.2.2 Foreign Exchange Market 9 1.2.3 Equilibrium Condition 15 1.3 Model Solution 16 1.3.1 Log-linearization 16 1.3.2 Model 1: Exogenous Entry 19 1.3.3 Model 2: Endogenous Entry 22 1.4 Results 23 1.4.1 Exogenous Case 24 1.4.2 Endogenous Entry 29 1.5 An Extension - Tobin Tax 30 1.6 Conclusions and Subsequent Research 34 Contents v 2 Optimal Monetary Policy with Vertical Production and Trade 36 2.1 Introduction 36 2.2 Basic Model 40 2.2.1 Household 41 2.2.2 Finished goods stage 43 2.2.3 Intermediate goods stage 45 2.2.4 Stochastic shocks 47 2.2.5 Equilibrium 47 2.3 Solution 48 2.4 Optimal money rules 50 2.5 Conclusion 56 3 Global Monetary Policy under a Dollar Standard 58 3.1 Introduction 58 3.2 The two-country model 63 3.3 Solving the Model 66 3.4 Optimal Monetary Policy 70 3.5 Endogenous Currency Pricing 79 3.6 Conclusions 83 Bibliography 84 A Appendices of Chapter 1 89 A.l Optimal Pricing Schedule of Firms 89 A.2 Entry Condition of Noise Traders 89 A.3 A Symmetric Steady State 90 A.4 Model Solution 91 A.4.1 Log-linearized System 91 A.4.2 Derivation of Equations 94 A.5 The simulation of Var(vt) = XVar(st) 95 A.6 Entry Condition of Traders with Tobin Tax 95 A.7 Numerical Undetermined Coefficient Method 96 Contents vi B Appendices of Chapter 2 102 B.l Price index and individual demand 102 B.2 Expected welfare 103 B. 3 Optimal money rules 105 C Appendices of Chapter 3 107 C. l Optimal pricing setting 107 C.2 Model solution 107 C.3 Proof of Proposition 4 109 C.4 Proof of Proposition 5 110 vii Lis t o f Tables 1.1 Parameter Values 24 A . l Exogenous Case (A = 1, a = 2) 98 A.2 Exogenous Case (A = 1.5, a = 2) 99 A.3 Endogenous Case(A = 1.5, a = 2)a 100 A. 4 The impact of Tobin Tax 101 B. l The Optimal Price Policies for Foreign Firms 0 106 C. l The optimal monetary rule in Nash game 112 C.2 The weight on exchange rate volatility in monetary policy decision" 112 viii Lis t of Figures 1.1 Timing of Model 9 B. l The Structure of the Economy 106 C . l The reaction curves for a i , b\ ( p = 4, n = 0.5 and A = 6) 113 C.2 The reaction curves for a,2, 62 ( p = 4, n = 0.5 and A = 6) 113 ix Acknowledgements I am extremely grateful to my supervisor, Professor Michael B. Devereux for advice, guid-ance and encouragement. I am also greatly indebted to other members of my supervisory committee, Professor Paul Beaudry and Professor Francisco Gonzalez for research direction and discussion. I would like to thank Professor Fabrice Collard, Professor Patrick Francois, and Professor Angela Redish for helpful suggestions and discussions at various stages of this dissertation. I am also grateful to my classmates, colleagues and friends, Martin Berka, Lilia Karnizova, Genevieve Verdier, Jacob Wong for helpful suggestions, discussions, and support. I thank the seminar participants at U B C macrolunch, the Bank of Canada, the BIS, McGil l University, S F U , S U N Y Buffalo, Tufts, University of Guelph, the 2003 meetings of the Cana-dian Economics Association and the 2004 meeting of the Western Economic Association for comments and helpful discussions. I am solely responsible for any errors and misinterpreta-tions. X S u m m a r y This thesis consists of three essays on exchange rate behavior and optimal monetary policy in open economy. A central puzzle in international macroeconomics over the last 20 years is that real ex-change rates are volatile and persistent. Furthermore, the exchange rate seems to "have a life of its own", being disconnected from other macroeconomic variables. The answer to this puzzle will help us to understand whether a fixed exchange rate regime will be more desirable in an open economy. The first essay proposes a framework to explain why the nominal and real exchange rates are highly volatile and seem to be disconnected from the macroeconomic fundamentals. Two types of foreign exchange traders, rational traders and noise traders with erroneous stochastic beliefs, are introduced into the dynamic general equilibrium framework of the new open economy macroeconomic literature. The presence of noise traders creates deviations from the uncovered interest parity. As a result, exchange rates can diverge signifi-cantly from the fundamental values. Combined with local currency pricing and consumption smoothing behavior in an infinite horizon model, the presence of noise traders can help to explain the "exchange rate disconnect puzzle". Then it is shown that the excess exchange rate volatility caused by the presence of noise traders can be reduced by the 'Tobin tax' type of exchange rate policies. The second essay explores the optimal monetary policy response to domestic and foreign technology shocks in an open economy with vertical structure of production and trade. That is, countries use imported intermediate goods as an input to produce export goods. In other words, countries are linked sequentially in the production of final goods via trade. Thus, any stage-specific productivity shock in one country has a trans-border spillover effect on the other country via vertical trade. So when choosing the optimal monetary rules, each monetary authority should respond to both home and foreign productivity shocks. Another finding is that the flexible exchange rate can not replicate the flexible price equilibrium even under producer currency pricing due to price stickiness in multiple stages. Finally, the exchange rate in such an environment would be more stable than that of an economy without Summary xi vertical structure of production and trade. These findings suggest that the changes in the trade pattern in the global economy over the last thirty years might affect the international optimal monetary policy rules and values of exchange rate flexibility. So the monetary policy maker should take into account the impact of the changes in the trade pattern when making decisions. The third essay analyzes the determination of monetary policy in a world with a dollar standard, defined here as an environment in which all traded goods prices are set in US dollars. This generates an asymmetry whereby exchange rate pass-through into the US CPI is zero, while pass-through to other countries will be positive. I show that monetary policy in such a setting does seem to accord with popular discussion. In particular, the US is essentially indifferent to exchange rate volatility in setting monetary policy, while the rest of the world places a high weight on exchange rate volatility. More importantly, in a Nash equilibrium of the monetary policy game between the US and the rest of the world, the preferences of the US dominate. That is, the equilibrium is identical to one where the US alone chooses world monetary policy. Despite this, I find surprisingly that the US loses from the dollar's role as an international currency. Even though US preferences dominate world monetary policy, the absence of exchange rate pass-through means that US consumers are worse off than those in the rest of the world, where exchange rate pass-through operates efficiently. Finally, the conditions for a dollar standard to exist is derived. 1 Chap te r 1 Noise Traders and the Exchange R a t e Disconnect Puzz l e 1.1 Introduction A central puzzle in international macroeconomics over the last 20 years is that real exchange rates are volatile and persistent. Furthermore, as Flood and Rose (1995) have elegantly documented, the exchange rate seems to "have a life of its own", being disconnected from other macroeconomic variables. For example, Mussa (1986), Baxter and Stockman (1989) and Flood and Rose (1995) all find that both nominal and real exchange rates are highly volatile, especially when compared to macroeconomic fundamentals, such as relative price level, consumption and outputs. Exchange rate volatility also varies substantially over time. Obstfeld and Rogoff (2000a) state this kind of "exceedingly weak relationship between the exchange rate and virtually any macroeconomic aggregates" as the "exchange rate disconnect puzzle". This irregularity casts some doubts on the traditional monetary macroeconomic model of exchange rates, which assumes that purchasing power parity (PPP) holds. With PPP, the "expenditure-switching" effect of exchange rate changes will lead to substitution between domestically-produced goods and internationally-produced goods. It implies that exchange rate volatility will be transferred to macroeconomic fundamentals. Nevertheless, empirical evidence 1 indicates that nominal exchange rate changes are not fully passed through to goods prices. Motivated by this evidence, Betts and Devereux (1996, 2000) introduce local currency pricing into the baseline Redux model developed by Obstfeld and Rogoff (1995). They assume that firms can charge different prices for the same goods in home and foreign markets and that the prices are sticky in each country in terms of the local currency. This allows the real exchange rate to fluctuate, and delinks the home and foreign price levels. Although the new open economy macroeconomic models with sticky prices, imperfect 'See Engel (1993, 1999) and Parsley and Wei (2001) for details. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 2 competition and local currency pricing can generate volatile exchange rate movements2, they typically predict a strong counterfactual relationship between the real exchange rate and relative consumption 3 . A monetary shock simultaneously raises domestic consumption (by more than it raises foreign consumption) and creates a (temporary) depreciation of home currency. Consequently, these models almost generically predict a strong positive correlation between depreciation and relative consumption, which is not observed empirically. 4 One explanation for this discrepancy might lie in the fact that the nominal exchange rate is also an asset price, and therefore will be inevitably affected by imperfections in the financial markets. These imperfections may include herd behavior, momentum investing and noise traders. Working together with sticky prices, these are all important reasons to explain why the real exchange rate persistently deviates from the level predicted by the fundamentals-based models. A large body of evidence has documented strong heterogeneity in market participants' expectations in the foreign exchange markets 5 . Evans and Lyons (2002) show that most of the short-run exchange rate volatility is related to order flow, which also reflects the heterogeneity in investors' expectations. Although financial economists care about high frequency data, while international macroeconomists focus more on low frequency data, it is still surprising how little the microstructure of real world foreign exchange markets has been considered in the macroeconomic theory of exchange rates. This raises another question: if exchange rate volatility is caused by erroneous beliefs and could be reduced without incurring costs due to other macroeconomic volatilities, then floating exchange rates may be too volatile and costly from a welfare point of view. How-ever, it is impossible to make any policy recommendations in the absence of a welfare-based model which can explain exchange rate volatility and its relationship with macroeconomic 2 A high risk aversion coefficient of household (about 5) is usually required in these models to reproduce the data's volatility of real exchange rate relative to output. See Chari, Kehoe and McGratten (2002). 3See, for example, Chari, Kehoe and McGrattan (2002). 4Benigno and Thoenissen (2003) report the correlation between bilateral exchange rate and bilateral relative consumption for seven countries (Canada, France, West Germany, Italy, Japan, U.K. and U.S.) for the periods starting from 1970 until 2002. The cross-correlation varies between —0.45 and 0.42. 5See Frankel and Froot (1987,1990), Chinn and Frankel(1994), Taylor and Allen (1992) and Cheung and Wong (1998) for details. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 3 fundamentals. Therefore, our paper intends to propose a new approach to study exchange rates, that combines the macroeconomic model of exchange rates and the microstructure approach of foreign exchange markets. This approach is implemented within a specific model, where noise traders are introduced into the new open economy macroeconomic framework. The combination is helpful for understanding the behavior of exchange rates and their relationship with macroeconomic fundamentals. It also gives more rigorous microeconomic foundations to the "noise trader" approach and enriches the new open economy macroeconomic framework with a more realistic setting of the microstructure of foreign exchange market. In addition, it provides a well-defined framework for policy evaluations, especially for those policies that are designed to control non-fundamental volatilities. We adapt the over lapping-generation noise trader model of De Long et al. (1990). Two types of foreign exchange traders are introduced into the general equilibrium framework. One type is the "rational/informed trader", which has rational expectations about future investment returns, while the other type cannot forecast the future returns correctly and is called the "noise trader". The results from the model show that when the number of noise traders increases, so does the exchange rate volatility. Nevertheless, the volatilities of macroeconomic fundamentals (except for the net foreign assets) are completely independent of the noise component on the foreign exchange market. Therefore, our model can generate a relative volatility of real exchange rate to output close to the data, even for a low risk aversion coefficient. Moreover, since in this model nominal and real exchange rate fluctuations can be generated by erroneous belief of noise traders, our model does not predict a strong comovement of exchange rates and fundamentals. Therefore, it is possible to explain the "exchange rate disconnect puzzle" by the approach suggested in this paper. The basic intuition behind our results is as follows. The heterogeneity in beliefs among foreign exchange traders creates the basis for trading volume and deviations from the un-covered interest parity. Arbitrage does not eliminate the effect of noise here because noise Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 4 itself creates risk: short-horizon investors must bear the risk that they may be required to liquidate their positions at a time when asset prices are pushed even further away (by noise traders) from the fundamental values than when investment was made. Therefore, exchange rates can diverge significantly from the fundamental values. The greater the number of noise traders, the more volatile will be the exchange rates. However, why is the exchange rate volatility not transferred to macroeconomic funda-mentals? Normally, there are two channels through which the exchange rate affects macroe-conomic variables: the expenditure-switching effect and the wealth effect (through firms' profits). Under the assumption of local currency pricing, the expenditure-switching effect is eliminated as the relative price of home and foreign goods does not change. Although the wealth effect still exists, it turns out to be quite small quantitatively. This is because the wealth effect of exchange rate change is spread out over current and future periods through intertemporal consumption smoothing, and so tends to be very small. Many economists have suggested that increasing the trading cost on the foreign exchange market might reduce the exchange rate volatility. To understand the effect of this kind of exchange rate policies, the size of the noise component is endogenized by introducing a heterogenous entry cost for noise traders. Only noise traders having entry costs that are sufficiently low will choose to enter the foreign exchange market. We find that given the number of potential noise traders, increasing the entry cost will reduce exchange rate volatility. We also analyze a 'Tobin tax' type of exchange rate policy suggested by Tobin (1978) and Eichengreen, Tobin and Wyplosz (1995) in an extension of the baseline model. We find that a Tobin tax will decrease the exchange rate volatility, however, the size of the impact of a Tobin tax on exchange rate volatility depends crucially on the structure of the foreign exchange market and the interaction of the Tobin tax with other trading costs. The microstructure of the exchange rate market in this paper follows the noise trader literature, especially the work of Jeanne and Rose (2002), which also focuses on the rela-tionship between exchange rate volatility and noise traders. However, the macroeconomic part of their model is a simple monetary model of exchange rates with PPP. Neither nominal Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 5 rigidities nor pricing to market is considered. Moreover, intertemporal optimizing agents and profit maximizing firms are not considered in their model. Another feature of their model is that it is a partial equilibrium model without explicit welfare specifications for households, so rigorous policy analysis is impossible. This paper is also closely related to the new open economy macroeconomic literature. The paper that is closest, in spirit, to our analysis of exchange rate disconnect puzzle is Devereux and Engel (2002). They stated that the key ingredients to explain the exchange rate dis-connect puzzle include: local currency pricing to eliminate the expenditure-switching effect, a special structure of international pricing and product distribution to minimize the wealth effect, incomplete international financial markets, and stochastic deviations from the uncov-ered interest parity. The analysis in this paper differs in the following aspects. First, more microeconomic foundations of noise traders are explored. Both noise traders and rational traders in our model are risk averse and utility maximizing agents, therefore, policy analysis is possible in our model. Second, we show that, the wealth effect of exchange rate changes may be quite small, quantitatively, in an infinite horizon model. Therefore, the exchange rate disconnect puzzle can be explained even without a specific assumption of production and distribution structure to remove wealth effects. This paper is organized as follows. In Section 2, we construct a model that embeds noise traders into a new open economy macroeconomic framework. Both the exogenous entry and endogenous entry specifications are explored. Section 3 features of the solution to the model are discussed. Section 4 gives the results of the model. Section 5 extends the baseline model to analyze the implications of Tobin tax. The paper concludes with a brief summary and suggestions for subsequent research. 1.2 The model The world economy consists of two countries, denominated by home and foreign. Each country specializes in the production of a composite traded good. Variables in the foreign country are denoted by an asterisk. In addition, a subscript h denotes a variable originating Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 6 from the home country; a subscript / denotes a variable used in the foreign country. This model is analogous to most new open economy macroeconomic models except for i the foreign exchange market. Each country is populated by a large number of atomistic households, a continuum of firms that set prices in advance, and a government (a combined fiscal and monetary authority). However, we assume that home and foreign households can only trade nominal bonds denominated in their domestic currency. Although home households cannot access the international bond market, the foreign exchange traders can carry out international bond trading to maximize their utility. Thus, besides the infinitely lived household, a second type of representative agent is introduced into the model, namely, the foreign exchange trader, who lives in an overlapping-generation demographic structure. Hereafter, a superscript H denotes households and a superscript T stands for traders. In the foreign country, for simplicity, it is assumed that only one type of representative agent is present; the foreign household.6 1.2.1 Households, Firms and Government The lifetime expected utility of the home representative household is: ( oo t U=o 4- — —J—L 1-p l - e \ P t ) ! + </>< (1.2.1) Subject to PtCf + Bt+1 + Mt = WtLt + Ut + Mt-i + Tt + Bt(l + rt) (1.2.2) where is the time t composite consumption of home households, composed by a continuum of home goods and foreign goods; both are of measure 1. Let Cj denote the composite consumption of traders, then Cf + = Ct, where Ct is the composite consumption of the home country and is given by: 7 -1 (1.2.3) introducing foreign exchange trader into the foreign country will not change the main results. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 7 where Ch,t = ( j^ C ^ t W ^ ^ ) "~ 1 , C / , t = ( j^ C / , t ( j ) ^ d j j 9 - 1 , and the weight w e (0,1) determines the home representative agent's bias for the domestic composite good. Note that 0 is the elasticity of substitution between individual home(or foreign) goods and 7 is the elasticity of substitution across home and foreign composite goods. Pt is a consumption based price index for period t, which is defined by: Pt uP1^ + (1 - U)P}?] ^ (1.2.4) where Ph,t = ^ ( i ) 1 " 0 * ) ^ and P u = PU{j)l-6dj) ^ . In each period every household is endowed with one unit of time, which is divided between leisure and work. His income is derived from the labor income WtLt, profits from domestic goods producers (which is assumed to be owned by domestic households) n^, interest received on domestic bonds Bt(l + rt) and lump-sum government transfer Tt. Solving the household's problem, the optimality conditions can be written as: ' M < v _ vn> ( L 2 . 5 ) p« / 1 - T+kri The first order conditions of the foreign households are entirely analogous, except that foreign household's consumption is denoted by C 4*, as there is only one type of representative agent in the foreign country. We assume firms have linear technologies, for each home good i, yt(i) = Lt(i). It is also assumed that, due to high costs of arbitrage for consumers, each individual monopolist can price discriminate across countries. Furthermore, as in Betts and Devereux (1996) and Chari, Kehoe and McGrattan (2002), we assume local currency pricing: firms set prices (separately) in the currencies of buyers. Finally, prices are assumed to be set one period in advance and cannot be revised until the following period. That is, the home monopolist sets Ph,t{i) and Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 8 Pht(i) optimally at the end of period t—l, and these prices cannot be changed during time t. Appendix A . l gives the derivation of the optimal pricing schedule of firms. The firms will just set the price so that it equals to a mark-up over the expected marginal cost and a risk premium term arising from the covariance of the firm's profits with the marginal utility of consumption: 6 Et^\DtWtCt) 6 Et-^DtWtCt) 0 - 1 Et-i[DtCt] 1 0-1 Et-i [DtStC;] ( - ' Pf,t = e Et-tiDWCt] D, e Et-^Dtwta Etl^tbt] (1-2.9) where Dt and D* denote the pricing kernels households used to value date t profits. Because all home firms are assumed to be owned by the domestic households, it follows that in equilibrium Dt is the intertemporal marginal rate of substitution in consumption between time t—l and t: Dt is defined analogously. St is the nominal exchange rate at time t. The home government issues the local currency, has no expenditures, and runs a balanced budget every period. The nominal transfer is then given by: Tt = Mt- Mt-i (1.2.11) The stochastic process that describes the evolution of the domestic money supply is: Mf = IHMU (1.2.12) log(/it) - (1-2-13) where e M ; t ~ N(0, o\) is a normally distributed random variable. The stochastic process of money supply in the foreign country is entirely analogous. Also, the home monetary shock and the foreign monetary shock are assumed to be independently distributed, that is, C o u ( £ M , £ * ) = 0. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 9 1.2.2 Foreign Exchange Market Foreign Exchange Traders Following closely the work of De Long et al. (1990) and Jeanne and Rose (2002), the foreign exchange traders are modelled as overlapping generations of investors who decide how many one-period foreign nominal bonds to buy in the first period of their lives. Traders have the same taste, but differ in their abilities to trade in the foreign bond market. Some of them are able to form accurate expectations on risk and returns, while others have noisy expectation about future returns. The former are referred as the "rational trader" and the latter as the "noise traders". Hereafter, the informed trader is denoted by a superscript I and the noise trader is denoted by a superscript A .^ Two specifications of the model are developed. In the first specification, the number of incumbent noise traders is exogenously determined. In the second one, the traders have to pay a fixed entry cost to trade on the foreign exchange market. The introduction of an entry cost helps to endogenize the noise component of the market. This makes the policy analysis possible as policy makers can affect the composition of traders through the entry cost. In the foreign exchange market, at each period, a generation of foreign exchange traders is born. The continuum of the traders is indexed by i € [0,1]. Assuming that in each generation of traders, Nj of them are rational traders, and 1 — Nj are noise traders. The timing of the model is illustrated in Figure 1.1. Figure 1.1: Timing of Model t t+1 Action 1 Action 2 Action 3 Action 1: Time t foreign exchange trader i is born; Time t shocks and nominal interest rates are revealed; The time t born trader i decides if he should enter the foreign bond market. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 10 Action 2: He decides the number of foreign currency bonds B^t+1(i) to purchase based on his expectation about future exchange rate St+i- To finance his purchase, he borrows B^t+1(i)St from the home bond market. Action 3: Time t + 1 exchange rate St+i is revealed, so the return of his investment in terms of home currency is realized, which equals to St+iB^ t + 1 ( i ) ( l + r-j + 1 ). He pays back the principle and interest of his borrowing(.B£ t + 1 ( i ) 5 t ( l + rt+i)), gets the excess return, consumes, and dies. Let <plt denote the dummy variable characterizing the market-entry condition of period t born foreign exchange trader i. If <p\ = 0, trader i will not enter the foreign bond market and if ipl = 1, he will enter. At the beginning of period t, trader i will enter the market as long as the expected utility of entering the market is higher than that of not entering: Et(Ui \<p\ = l)> E\{Ui | ^ = 0) (1.2.14) A foreign exchange trader who has entered the foreign bond market maximizes a mean-variance utility function: max}Ei(C?+1(i)) - f Varj(Cf + 1 (z)) (1.2.15) Subject to P t + 1 C f + 1 = [B*htt+1(i)(l + r*t+1)St+1 - B*hit+1(i)St{l + rt+0] - Pt+1* (1.2.16) where B £ t + 1 ( i ) denotes the amount of one-period foreign currency bonds held by trader i from time t to time t + 1, a is the absolute risk aversion coefficient, the cost c, reflects the costs associated with entering the foreign bond market for trader i. The entry costs may include tax, information costs for investment in the foreign bond market, and other costs when investing abroad. 7 To formalize this heterogeneity, here we follow the specification used by Jeanne and Rose (2002). Rational traders are assumed to have a larger stock of knowledge about the economy and thus, do not need to invest in the 7These costs may be modelled in many ways. In this paper, the entry costs are assumed to be resource-consuming in the sense that it consumes the composite consumption good. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 11 acquis i t ion of informat ion. T h e i r entry costs are therefore zero. For noise traders, they do not have a na tura l ab i l i ty to acquire and process the informat ion about the economy and therefore have to pay an entry cost that is greater than zero. A l t h o u g h the preferences of the noise traders are the same, the noise traders are assumed to be dis t inguished from each other by their entry costs. W i t h o u t loss of generality, the noise traders are indexed by increasing entry costs: Ci for i € [0,1 - Ni] (1.2.17) where a > 0 is the curvature parameter and c is the parameter character iz ing the scale or level of the entry cost of the noise traders. Thus , the noise trader at the left end of the cont inuum (i near 0) tends to have a lower entry cost and the noise trader towards the right end of the con t inuum (i near 1 — A/j) has a higher entry cost. Optimal demand for foreign bond Once the traders have decided to enter the market , the op t ima l demand for foreign bonds pf each type of traders can be derived. Subs t i tu t ing E q u a t i o n 1.2.16 into E q u a t i o n 1.2.15, gives: m a x El BT where pt+i B\ h,t+l (0 t+1 St(i + rt+i)pt+i - (k :Var\ B h,t+l St(l+rt+i)pt+i - a S t +i(l+r*+i) •St(l+r t + i) (1.2.18) is the excess re turn . W e now discuss the informat ion structure of traders. Specifically, we make the following assumptions about the subjective d i s t r ibu t ion over pt+i- T h e ra t iona l traders can predict pt+i correctly; whi le the noise traders cannot predict the future excess re turn correctly. T h a t is, for informed traders: Eilpt+i] = Et[pt+1] Varllpt+i] = Vart[pt+i] (1.2.19) (1.2.20) Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 12 For noise traders, following the work of De Long et al. (1990), we assume: E»[pt+1] = Et\pt+1]+vt (1.2.21) VarFlpt+i] = Vart[pt+1] (1.2.22) Var(vt) = XVar(st) where A 6 (0, +oo) (1.2.23) where vt is assumed to be i.i.d and normally distributed with zero mean. A can be considered as a parameter characterizing the relative magnitude of noise traders' erroneous beliefs to exchange rate volatility. From Equations 1.2.21 and 1.2.19, it can be seen that, compared to the rational trader's expectation, the noise traders' expectation of pt+\ based on time t information is biased from the true conditional expectation by a random error. Nevertheless, noise traders can correctly forecast the conditional variance of the exchange rate. From Equation 1.2.23, another as-sumption is made that the unconditional variance of vt is proportional to the unconditional variance of the exchange rate itself. This assumption helps to tie down the scale of the volatil-ity of noise traders' erroneous beliefs.8 Solving Equation 1.2.18, the optimal bond holding of trader i is given by: BUiii) = „ . " " T ^ , (1-2-24) E\[pt+i\ aTU~Sl +n+i)Vart[pt+i) Therefore, informed traders and noise traders differ in their optimal bond holding. Also, from Equation 1.2.24, the lower the expected excess return, the higher the risk (excess return volatility) and the risk coefficient, the less bond traders (both rational traders and noise traders) will hold. Thus, the traders account for risk when taking positions on assets. At the margin, the return from enlarging one's position in an asset that is mispriced (the expected excess return) is offset by the additional price risk (the volatility of excess return) that must be borne. 8The logic behind this assumption is that the bias in noise traders' expectation must be related to the volatility of the exchange rate itself, otherwise noise traders might expect the future exchange rate to be volatile even under a fixed exchange rate regime. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 13 Equilibrium condition of the foreign exchange market Analysis with no entry costs W e first analyze a simple case where c = 0. T h u s , all the noise traders wil l enter the market and the noise component of the market is exogenously determined by the number of existing noise traders (1 — Nj) on the market. So the aggregate demand for foreign bonds by foreign exchange traders of the home country can be denoted as: B*h,t+1 = NjB^ + il-N^B^ Et St(l+rt+i) 1 a7wr( 1 + rt+i)Vart(pt+i) + (1 - Nrfvt (1.2.25) =>Et St+1(l + r*t+1) _i I St(l+rt+1) + (1 - NT)vt - a - ^ - ( l + rt+1)Vart(Pt+1)Blt+1 = 0 (1.2.26) First Part Second Part Endogenous entry of noise traders W e now endogenize the composit ion of traders who enter the market in each period by introducing positive entry costs for noise traders. T h e entry decision for informed traders is tr iv ial . T h e y bear no entry cost a n d always enter the foreign bonds market in equi l ibrium. A noise trader, however, enters if and only if E q u a t i o n 1.2.14 is satisfied. A s shown in A p p e n d i x B .2 , for trader i, this condit ion takes the form: d < l E l i P t : l ) ] \ = GB? (1.2.27) where GB? is the gross benefit of entry for noise traders. It increases wi th the expected 2aVart(pt+i) t excess return and decreases wi th the condit ional t ime t + 1 exchange rate volatility. Note that in our general equi l ibr ium setting, bo th terms are functions of the number of incumbent noise traders. Let c*t = GB? be the cut-off value of entry cost. F r o m E q u a t i o n 1.2.27, for noise trader i , if a < c\, ip\ — 1; if Cj > c£ , <p\ = 0. T h e number of incumbent noise traders nt is then given by: Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 14 Apparently, the number of active noise traders on the market increases with the square of the expected excess return and the number of existing noise traders, and decreases with the entry cost, the risk aversion coefficient a, and the excess return volatility. The economic intuition behind Equation 1.2.28 is as follows. The presence of more active noise traders creates higher expected excess return and incentives for other noise traders to enter the market, however, the extra volatility brought about by their entry will reduce the gross benefit of entry for noise traders. In equilibrium, the two effects balance and no more noise traders will enter. Substituting Equation 1.2.28 into B^t+1 = NjB^^+ntBJ^'^, we can derive the equilib-rium condition of the foreign bond market when the entry decision of traders is endogenized: Et 5t+i(l + r ? + 1 ) -St(l + r t + i) , St(l+rt+1) j a F ; ; j TVart(pt+i)Bhtt+1 Ni+(Zy(l-Nj) Pt+1 ( 1 - J V » F i r s t P a r t Second Part (1.2.29) Equations 1.2.26 and 1.2.29 represent the interest parity conditions in this economy. Note that the uncovered interest parity does not hold in this model. The last two terms in Equations 1.2.26 and Equation 1.2.29 show the deviation from the uncovered interest parity when noise traders are present in the market. This deviation consists of two parts: the expectation error of the noise traders, and the risk premium term, since the foreign exchange traders are risk averse. In our model, as in De Long et al. (1990), the noise traders can "create their own space": the uncertainty of the noise traders' expectations over the future exchange rate increases the risk borne by informed traders engaged in arbitrage against noise traders. The aversion to this risk will severely limit arbitrage, especially in an overlapping-generation framework. Short-horizon investors must bear the risk that they may be required to liquidate their positions at a time when asset prices are pushed even further away (by noise traders) from the fundamental values than when the investment was made. Therefore, as shown in Section 3, exchange rate can diverge significantly from the fundamental values. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 15 1.2.3 E q u i l i b r i u m Cond i t i on Equilibrium for this economy is a collection of 26 sequences (Pt, P t*, Pn,u Ph,v Pf,t> Ph,v Ct,C?,CtH,C;, Ch,u C*hp Cu, C*KV Su rt, r*t, Dt, D*t, Wt, Wt*, Bt, B*t, B*ht, Lu L*t) satisfying 26 equilibrium conditions. They include the six household optimality conditions (Equations 1.2.5, 1.2.6, 1.2.7 and their foreign counterparts), the definition of the price indexes (Equa-tion 1.2.4 and its foreign analogy), the definition of the pricing kernel (Equation 1.2.10 and its foreign analogy), the interest parity conditions (Equation 1.2.26 or Equation 1.2.29), the four individual demand equations, the four pricing conditions, and the four market clearing conditions for the bonds and goods markets: B t + l = StB*u+l V i (1.2.30) B*t+Blt = 0 (1.2.31) Lt = Ch,t + C*hit (1.2.32) L\ = Cftt + C*u (1.2.33) Finally, the budget constraint of the foreign exchange traders. PtCf = B*Kt(\ + r*t)St - Bfc , t S t - i ( l + n) Exogenous Entry (1.2.34) PtCj = [B*h:t(l + r*t)St - B*hitSt-i(l + rt)] - PtJ^d Endogenous Entry (1.2.35) i=0 And the home country aggregate consumption equation: Ct = C f + Cj Exogenous Entry (1.2.36) nt Ct = C" + Ct +J2°i Endogenous Entry (1.2.37) i=0 Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 16 Then the above two equations, the budget constraints of the home households 1.2.2, Equa-tion 1.2.30 and its one-period lag can be combined to get the national budget constraint of the home country: where I L = UJ PtCt = WtLt + n t + StB*hit(l + r*t) - StB*htt+1 (Ph,t-wt)(^) 7ct + (p*h,st-wt)(^) 1 ct (1.2.38) (1.2.39) 1.3 Model Solution The model can be solved by log-linearization around a non-stochastic, symmetric steady state (as described in Appendix A.3), where net foreign assets are zero 9, all prices are equal, and the exchange rate is unity. Given the log-linearized system, the deviations of the exchange rate and the macroeconomic variables from their t — l expectations are solved in terms of exogenous money supply shocks and the expectation error shocks. 1 0 1.3.1 Log-l inear izat ion The log-linearization of the model is quite standard, except for the interest rate parity condi-tions (Equations 1.2.26 and 1.2.29). Since the non-linearities of the interest parity equations are important for understanding the dynamics of the economy, especially for the exchange rate, the variance term and expectation error term will be kept through second-order approx-imation when the parity equations are log-linearized. 1 1 9The financial market are incomplete in our model since the home and foreign household only have access to non-state contingent domestic currency nominal bonds. If there are no foreign exchange traders, then there is a unit root in the net foreign assets in this kind of model. However, when the foreign exchange traders are present, the net foreign assets are zero at the steady state. Please see Appendix A.4) for detail. 10Hereafter, xi = log(Xt) — log(X), dXt = Xt — X, where X is the non-stochastic steady state value of variable Xt-1 1The detailed model solution, including the log-linearization of the system and the derivation of equations, is given in Appendix A.4). Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 17 Linearizing the interest parity condition for the exogenous entry specification (1.2.26) gives at = Et(st+i) - f3(drt+i - dr*t+1) + (1 - Nfivt - a ( 1 +^)SVart[st'+i}dBlt+1 (1.3.1) Linearizing the interest parity condition for the endogenous entry specification (1.2.29) gives: 1 ( l + f)5 st = Et(st+i) - P(drt+i - dr*t+l) + j^nm - a - Vart[st%i]dB^t+1 (1.3.2) where nt, the number of incumbent noise traders is given by 1 2 : {Et(st-+1) - st - P(dr*t+1 - dn+i) + vt? (1 - Nj) nt = 7TT? 7—r-r (1.3.3) 2aVart{st+i) c Similar to Equations 1.2.26 and 1.2.29, Equations 1.3.1 and 1.3.2 show that the biased expectation of noise traders causes a stochastic deviation from uncovered interest rate parity. This deviation is composed of two parts: the noise traders' expectation errors and a risk pre-mium term. The former, as discussed intensively by Devereux and Engel (2002), is different from the traditional risk premium term that arises from the risk aversion of households. It captures the fluctuations in the exchange rate due to the variation of noise traders' misper-ceptions. As one would expect, the greater the number of noise traders, the greater is the impact of the noise traders' expectation error on the exchange rate. For example, when a positive expectation error shock occurs, the noise trader will have a higher demand for foreign bonds and foreign currency, which leads to a domestic currency depreciation. Therefore, this term tends to increase exchange rate volatility. In contrast to Devereux and Engel (2002), there is also a risk premium term in the parity condition because of the assumption that traders are risk averse. Intuitively, when exchange rate volatility increases, traders would not hold the foreign bonds unless compensated for bearing the extra risk. Consequently, the price of the foreign currency (risky asset) should fall. Thus, the risk aversion of traders (both informed traders and noise traders) will tend to reduce exchange rate volatility. 1 2 Hereafter , the curvature parameter of entry costs a is set to be equal to 1. T h e model can be easily extended to the case where a > 1 or 0 < a < 1, and the main results w i l l not change. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 18 Note that from the log-linearization of the pricing equation of firms, we may get the price index for the home and foreign country: pt = ±(Et-i[wt] + Et-i[w*t] + Et-i[st]) (1.3.4) p* = \iEt-x\wt\ - Et-^st] + Et-X[wt]) (1.3.5) Equations 1.3.4 and 1.3.5 establish that in an expected sense, P P P holds. This is not surprising, as the prices can fully adjust to all shocks after one period. Solve for T — 1 Expectations Taking a linear approximation of the budget constraint of home household 1 3, using the pricing indexes (Equations 1.3.4 and 1.3.5), the relationship between ct and c f , and the fact that (as will hold in equilibrium) in an expected sense, any initial change in net foreign assets is persistent, gives: £ U ( c f - c\) = ^ - i ( - g - ) + (1 - 7 ) ^ - i K - w*t ~ it) + -^{^ - l)dBt (1.3.6) Use f = jj — 1 and the fact that = 0 1 4 , we get: Et-x(<? ~ c?) = (1 - l)Et-i(wt ~ wt - s't) + -~dBt (1.3.7) Equation 1.3.7 shows that the relative home consumption increases in changes of the initial net foreign assets position and decreases in the expected terms of trade, as long as the elasticity of substitution between home and foreign composite goods 7 is greater than 1. From the linear approximation of the goods market clearing conditions (Equations 1.2.32 and 1.2.33), and the labor supply equation 1.2.6 and its foreign equivalent, using price indexes (Equations 1.3.4 and 1.3.5), and taking expectations at t — 1, gives: Et-i{wt - w*t - it) = ^—Et-xic? - c*t) (1.3.8) 1 3 After the home money market equilibrium condition Mt = Mt-i + Tt and the profit condition 1.2.39 are imposed. 1 4 This is because the traders' income is derived from the product of the foreign currency bond holding [Bh t+i) a n d the excess return (pt+i)- At the steady state, both the bond holding and the excess return are equal to zero. If we log-linearize the budget constraint of the traders (Equation 1.2.16) around the steady state, dCr = 0. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 19 Equations 1.3.8 and 1.3.7 give a relationship between the initial net foreign assets and the expected relative consumption: Et-itf - $) = ~ d B t (1.3.9) where o = 1 — ^ . A n increase in the home country's net foreign assets leads to an expected increase in the home relative consumption. Finally, hereafter, we assume that the elasticity of the money demand e = l . 1 5 In equilib-rium, given the random walk assumption of money supply process and the log money utility function, a very convenient property is that the nominal interest rate will be constant. This is because, if the log of the money stock follows a random walk, so does the log of the term Pt(C?)p. Using this fact and pricing indexes (Equations 1.3.4 and 1.3.5), from the linear approximation of the money demand function and its foreign equivalent we could get: Et-i(mt - m*t) = pEt-!(c? - c*) + £ f - i ( s t ) (1.3.10) Therefore, in an expected sense, the exchange rate is consistent with the standard monetary model. 1.3.2 Model 1: Exogenous Entry Hereafter, let x£+j = x{+j — Et~\(xt+j), j > 0 denote the deviation of a variable from its date t — 1 expectation. Then, the log-linearized home household's budget constraint minus its t — 1 expectation gives: c} - c* + j^dBt+1 = st (1.3.11) The right-hand side of Equation 1.3.11 represents the relative wealth effect of an unanticipated shock to the exchange rate through firms' profits. This relative wealth increase will then be spread between an increase in relative home consumption and net foreign assets accumulation. 1 6 A n estimate of the consumption elasticity of the demand for money(equal to i in the model) is very close to unity, as reported by Mankiw and Summers(1986). Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 20 Using Equation 1.3.9 (updated to period t + 1) and Equation 1.3.11, we may establish that: (cf - c*) + ^(4i - = *~t (1-3-12) This equation gives a relationship between current relative consumption, expected period t + 1 relative consumption, and the unanticipated shock to the exchange rate. It represents the constraints on these three variables implied by the intertemporal current account. Then, substituting the log-linearized intertemporal optimality equations into the interest parity condition, we may obtain the consumption-based interest parity condition: pEticC+x - ct) + Et(pt+i - pt) = pEt{cf+1 - cj) + Et{pf+1 - p*t) +Et(st-+x) ~ it + (1 - Nj)vt - ail+p)SVart[St'+x}dB*hj+x (1-3.13) where the left-hand side is the domestic nominal interest rate and the first two terms on the right-hand side represent the foreign nominal interest rate. Using the price indexes (Equations 1.3.4 and 1.3.5) and subtracting Equation 1.3.13 from its t — 1 expectation, we may get the relationship between current relative consumption and anticipated future relative consumption. Et{cf+1 - cf+1) = (cf - c*) - - [it - (1 - Nfivt + a^^-Vart[s{+x]dBlt+l) (1.3.14) In this equation, expected consumption growth in the home country decreases in response to an unanticipated exchange rate depreciation, since this generates an unanticipated real de-preciation, and therefore reduces the home country's real interest rate. From Equation 1.3.13, a positive shock to foreign exchange traders' expectations of the future exchange rate will increase the home real interest rate and lead to an increase in expected consumption growth of the home country. The last term in Equations 1.3.13 and 1.3.14, which denotes a risk pre-mium term due to the risk-aversion of the foreign exchange traders, tends to reduce the real interest rate. Therefore, it has a negative effect on the expected home relative consumption growth. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 21 Finally, the relation between relative money supply and relative consumption can be derived from the money demand equations 1 6 : mt-m*t=p{c} -c*t) (1.3.15) Putting Equations 1.3.15, 1.3.12, 1.3.14 and 1.3:11 together, we can get a system of equilibrium conditions that characterizes {it, c~t — ct, dBt+i}. We may solve for the deviation of the exchange rate from its t - 1 expectation (it) in terms of the exogenous money shock and expectation error of the noise traders. - 1 + 5 + £ V a r t ( * t + i ) g it = (mt-mt) - + + * , . Al-Nj)vt (1.3.16) p+ f + <j>Vart(st+i) p+j+<pVart(st+i) where « ( 1 ± ^ £ ( 1 3 1 7 ) 2 r From Equation 1.3.16, the variance of the future exchange rate deviation, Vart(st~+i) can be solved 1 7 . Let Vart(st~+i) = Vs, Vs is given by the following implicit function: l+f+!y„N 2 p+f+4>vs Vs = — \ -2-^ \Var{mt) + Var (nit*)] (1.3.18) Note that the coefficient cj) is associated with the risk-aversion of traders. The higher the risk aversion coefficient, the lower will be the exchange rate volatility. For both types of traders, their aversion to risk prevents exchange rate volatility from increasing too much. To see this, p+i+^vs ) 0 1 1 right-hand side of Equation 1.3.18 is decreasing in Vs and the denominator 1 — (p+gr+(pV) (1 — Nj)2X is increasing in Vs. Can the exchange rate display 'excess volatility' in this model? When p = 1, the coefficient of (rht — m*) is exactly 1 in Equation 1.3.16. Therefore, with no noise traders, the exchange 16Notice that rht = e^,t, and mj = e^ t. We use rht and m* for notational convenience. 17Since St is linear in rht, m* and vt and the monetary shocks and expectation error shocks are normally distributed with zero mean and constant variance, Vart(sf+i) = Var(st~+i) = constant s V„ Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 22 rate volatility will be equal to that of the fundamentals. If noise traders are present on the market, the exchange rate volatility may be much higher than the fundamental volatility, even when p = 1. What are the responses of the macroeconomic fundamentals such as consumption, labor and wage to the exogenous monetary shocks and expectation error shocks? From the log-linearized goods market clearing condition, labor supply condition and the money demand condition, ib ™t = ^{mt+m*t) + rht (1.3.19) 2.p h = ^-(rnt + m*t) (1.3.20) Zp ct = -riit c*t = -pm*t (1.3.21) Therefore, the volatilities of the macroeconomic fundamentals are only decided by the volatil-ity of the relative monetary shock and the values of the parameters, but not by the volatility of the expectation error and the number of incumbent noise traders in the market. Note that from Equations 1.3.11 and 1.3.15, the net foreign assets are given by: PC 1 dBt+i =—[s~t --(mt - m*t)] (1.3.22) i p Thus, the volatility of the net foreign assets will be affected by the number of incumbent noise traders. 1.3.3 Model 2: Endogenous Entry The endogenous entry case is similar to the exogenous entry case except for the interest parity equation. Substituting the log-linearized intertemporal optimality conditions into the endogenous interest parity condition (Equation 1.3.2), we may get the consumption-based interest parity condition: pEt{ct+i - ct) + Et{p{+i - pt) = pEt{cf+1 - c*) + Et(pf+1 - p*t) +Et(st'+1) - at + l-ntvt - a^0-Vart[st+1}dBnit+i (1-3.23) Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 23 Using the price indexes, we may find a condition analogous to Equation 1.3.14: £ t ( c f + 1 - df+1) = (cf - ct) - l-[st - 1-rnvt + a{-^j^Van[st+1}dBlt+J (1.3.24) where 1 8 2aVart(st+i) c Equations 1.3.15, 1.3.12, 1.3.11 and Equations 1.3.24, 1.3.25 give the solution of the endogenous entry model, and the derivation is entirely analogous to Equation 1.3.16 : - _ - _ -*) 1 + ' + % V a r t ^ f {W+i ) - it + vt}2 (1 - Nj) st (mt r n t ) p + ^ + ^ V a r t { s . + ^ + p + ^ + ^ V a r t i s , + ^ 2 a V a n { s t ~ + l ) N j 5 vt (1.3.26) where + = 2Nj ~f ( L 3 - 2 7 ) Analogous to the exogenous entry case, when p = 1 and no noise traders are present (Ni = 1), it = (rht—rnt), exchange rate volatility will be identical to that of the fundamental. When there are noise traders in the foreign exchange market, the exchange rate may diverge significantly from the fundamental values. The expression for net foreign assets, consumption, labor, and wage are exactly the same as in the exogenous case. 1.4 Results Equations 1.3.16 and 1.3.26 are too complicated to be solved analytically, so the numerical undetermined coefficient method described in Appendix A.7 is used to solve for it, Vs and Ea. Table 1.1 gives the parameter values that are used in the numerical simulation. We choose f3 = 0.94, which produces a steady state real interest rate of six percent, about the average long-run real return on stocks. The parameters n and ip are set so that the elasticity of labor supply is 1 and the time devoted to work is one quarter of the total time in the steady 18Here we use the fact that nominal interest rate are constant and it = st — Et-i(st). Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 24 Table 1.1: Parameter Values Exogenous Case Preferences Final goods technology Money Growth Process Foreign exchange traders Steady State Values Endogenous Casea Foreign exchange traders (3 = 0.94, p = 2, e = 1, T? = 58.2, -0 = 1 9 = 11, 7 = 1.5, u = 0.5 corrie^e*,) = 0, o^ = cr2, = 0.01 c = 0, A 7 / € [0,1], A = 1,1.5 Uss = H*ss = 1, M s s = M*s = 2 c> 0, Ni e [0,1], A = 1.5 "Other parameters in the endogenous case are the same as in the exogenous case. state. The business cycle literature has a wide range of estimates for the curvature parameter p. Chari, Kehoe, and McGrattan(2002) set p = 5 to generate a high volatility of the real exchange rate. In our model, a high exchange rate volatility can be obtained without high risk aversion of households, so it is set to equal 2. For the final goods technology parameters, the elasticity of substitution between domestic produced goods 8 is set to 11 following Betts and Devereux(2000). This gives a wage-price mark-up of about 1.1, which is consistent with the finding of Basu and Fernald(1994). The elasticity of substitution between home goods and foreign goods 7 is set to be 1.5, following Chari, Kehoe, and McGrattan(2002) and Backus, Kehoe, and Kydland(1994). 1 9 1.4.1 Exogenous Case We first solve for the exogenous entry case. Tables A . l and A.2 illustrate the results of the simulations, for different values of A. The first ten rows show the changes in the volatilities of the exchange rate and the net foreign assets when the number of noise traders increases from 0 to 1. The last three rows report the volatilities of the macroeconomic fundamental variables, given the calibrated parameter values. 19Note that, other parameters, such as the money supply process, number of informed traders on the market, and entry costs, are not fully calibrated. However, this will not affect the main conclusions of the paper. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 25 From Tables A . l and A.2, three important findings are: First, the exchange rate volatility increases when the number of noise traders increases, while the volatilities of macroeconomic fundamentals remain constant. Moreover, the exchange rate volatility is much higher than that of the macroeconomic fundamentals. Second, from the functional form of si, listed in the first column of Tables A . l and A.2, the impact of fundamental monetary shocks on the exchange rate (coefficient of rfit or m*) decreases in the number of noise traders. Meanwhile, the effect of expectation error on exchange rate (coefficient of vt) increases when more noise traders are present on the market. Third, exchange rate volatility is higher when the magnification coefficient A increases. Therefore, the critical implication is that 'disconnection' does exist between the exchange rate and the macroeconomic fundamentals in this model. Thus, the presence of noise traders in the foreign exchange market, combined with local currency pricing, generates a degree of exchange rate volatility that may be much higher than that of the underlying fundamental shocks. In other words, the "exchange rate disconnect puzzle" may be explained by the approach suggested in this paper. To understand intuitively why the disconnect puzzle can be solved in such a model, we may examine the case without noise traders. Obviously, the presence of local currency pricing tends to remove the expenditure-switching or substitution effects of exchange rate movements. Nevertheless, with just local currency pricing, the dynamic model will not generate a highly volatile exchange rate and the disconnection, as an exchange rate shock also affects the home real interest rates through the interest rate parity condition. Rewriting Equation 1.3.14 by omitting the expectation error and the risk premium term, gives: pEt(c?+1-cf+1)=p(c?-c*t)-s~t (1.4.1) Together with Equation 1.3.12, Equation 1.4.1 illustrates why exchange rate volatility is limited without noise traders. When a depreciation of home currency occurs, the domestic currency value of foreign sales will increase, giving rise to an increase in home wealth. Equa-tion 1.3.12 indicates that this positive wealth effect increases both current and future relative Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 26 consumption in the home country. Meanwhile, as an arbitrage condition, the interest parity condition (Equation 1.4.1) im-plies that a depreciation of home currency today will reduce the relative real interest rate in the home country and change the path of consumption, so that the current home consumption will increase, relative to the expected future consumption (holding foreign consumption con-stant). If the change in exchange rate is large and a disconnection between consumption and exchange rate is needed (i.e., the change in current consumption has to be small), Equation 1.4.1 suggests that the expected future consumption has to drop a lot. Nevertheless, Equa-tion 1.3.12 implies that the future relative consumption of the home country should increase when a depreciation of home currency occurs. Therefore, with no noise traders, the only possible way to explain the difference between exchange rate volatility and the fundamental volatility is by introducing a high value of p, which is exactly the mechanism emphasized by Chari, Kehoe, and McGrattan(2002). When the noise traders are introduced into the interest parity condition, we can see from Equation 1.3.14 that now a large increase of exchange rate and a small change in current consumption do not necessarily imply a large drop of expected future consumption, because the presence of the expectation errors and the risk premium term of noise traders also drive wedges between the home and foreign real interest rates. This could be called the "level effects" created by the noise traders. The presence of noise traders also creates a "volatility effect", which is due to the as-sumption that the volatility of vt itself is proportional to the exchange rate volatility. This assumption "magnifies" the response of the exchange rate to the expectation error of noise traders. When the nominal exchange rate volatility increases, so does the expectation error volatility, which further increases the exchange rate volatility until the system reaches an equilibrium where Var(vt) = A Var (st). Thus, A is the parameter characterizing this magnifi-cation effect. The higher A, the higher is the exchange rate volatility. Therefore, volatile exchange rates can be obtained in this model. Still, why is the high volatility not transferred to other macroeconomic variables (except for the net foreign assets)? Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 27 Normally, there are two channels through which the exchange rate affects other macroeco-nomic variables: expenditure-switching effects and wealth effects. Since the prices of the import goods are assumed to be sticky in terms of the local currency, the relative price of home-produced goods to foreign-produced goods will remain unchanged when the exchange rate changes. Therefore, the expenditure-switching channel is completely shut down in our model. 2 0 With regard to the wealth effect, from Equation 1.3.12, the increase in wealth that comes from an unexpected depreciation will be spread between an increase in relative home con-sumption and the net foreign assets accumulation. From Equation 1.3.15, however, the increase in relative home consumption is limited by the relative money shocks due to the real balance effect. Therefore, the net foreign assets will absorb most of the wealth increase. This is actually shown in Tables A . l and A.2, when the volatility of the nominal exchange rate increases, so does the volatility of the net foreign assets. However, the magnitude of the volatility of the net foreign asset and the expected future relative consumption are small quantitatively, especially when compared to that of the exchange rate. That implies the wealth effect is also quite small quantitatively. From Equation 1.3.12, £ t ( 4 i - <4+i) = £ft - ( c? - c~t)] ( L 4 - 2 ) It can be seen that the volatility of the change in expected future consumption is quantita-tively small because ~ is small given reasonable parameter values. 2 1 The economic intuition is that the consumption-smoothing behavior of infinitely lived households limits the wealth effect in this model. When a shock leading to an exchange rate depreciation occurs, the households increase their holdings of net foreign assets. This increase will be spread over 2 0 In another paper, the assumption of local currency pricing is relaxed and the baseline model is analyzed under producer currency pricing or a mixture of local currency pricing and producer currency pricing (i.e., the exchange rate pass-through is between 0 and 1). As expected, we find that with positive exchange rate pass-through, the volatilities of macroeconomic fundamentals (consumption, labor and wage) depend on the exchange rate volatility. The higher the exchange rate pass-through, the higher is the correlation between exchange rate volatility and the fundamental volatilities. 2 1For current calibration, J = = 0.0429. Recall that a = 1 — ^ f f i , so as long as the elasticity of substitution between home and foreign goods 7 is greater then 1, a is greater than 1. Thus,^ < 0.06. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 28 many future periods because the households want to smooth their future consumption. The increase in the expected consumption of next period is then quite small. Therefore, the more risk averse are the households (the higher p), the bigger will be a — 1 — ^l+ff (suppose that 7 > 1), and the smaller will be the wealth effect. Moreover, in the monetary model of exchange rate without noise traders, the monetary shocks lead to movements in both macroeconomic fundamentals and exchange rates, as shown by the following equations:2 2 i t = ±±l(rnt-m*t) (1.4.3) P ' r c f -c*t = -p(mt-m*t) (1.4.4) Therefore, it generically predicts a strong comovement and a high and positive correlation between the exchange rate and relative consumption. 2 3 From empirical evidence, however, there is no clear path in the observed cross-correlation. Chari, Kehoe and McGrattan (2002) find that this correlation is negative for U.S. and Europe while it ranges between small and positive to somewhat negative for other country pairs. Nevertheless, in our model, since exchange rate movements can be generated by the ex-pectation error shocks, our model does not predict a strong comovement of the exchange rate and relative consumption. The functional form of sj (listed in the first column of Tables A . l and A.2 shows that the exchange rate can move even when the realization of the fundamen-tals shocks are equal to zero. Furthermore, as shown by the last column of Table A.2, the cross-correlation between the exchange rate and relative consumption decreases when more noise traders are present on the foreign exchange market. Intuitively, this is because the in-troduction of noise traders generate deviations from the uncovered interest parity condition and thus breaks the link between the real exchange rate and relative consumption. Therefore, we may get a small and positive correlation in our model. 2 2 With no traders on the foreign exchange market, Equation 1.3.16 could be rewritten as Equation 1.4.3. 2 3For example, in Chari, Kehoe and McGrattan (2002) the correlation is equal to 1. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 29 1.4.2 Endogenous Entry The exogenous entry specification gives important implications of the model, however, a natural question is what can the monetary authorities do to get rid of the excess volatility in the nominal exchange rate? So in this section, we consider ways to endogenize the entry of noise traders, which will help to evaluate the implications of policies that target the non-fundamental risk. Table A.3 illustrates the simulation result of the endogenous entry specification: First, the exchange rate disconnection still holds in this specification. Second, given the number of noise traders in the market: 1 — Nj, increasing the entry cost c (within a reasonable domain of c 2 4 ) will reduce the exchange rate volatility. The first finding is not surprising. In Equation 1.3.24, as in Equation 1.3.14, the presence of noise traders generates a wedge between home and foreign real interest rates. This wedge, by analogue, is also composed of two parts, the expectation error of incumbent noise traders and the risk premium term. The only difference is that now the number of incumbent noise traders is endogenously decided, and as is the expectation error part. Nevertheless, this does not alter any of the theoretical analysis in Section 1.4.1. This wedge creates the "level effects" and the "volatility effects", which in turn imply a degree of exchange rate volatility that is much higher than the fundamental volatility. Meanwhile, the expenditure-switching effect is eliminated because of the L C P pricing behavior. The wealth effect is quantitatively small because of the households' consumption smoothing behavior in an infinite horizon model. Therefore, exchange rate volatility will not be transferred to the macroeconomic fundamentals except for the net foreign assets. The second finding is quite interesting and has important policy implications. Although the model is complicated and can only be solved numerically, this result is quite intuitive. The higher the entry cost, the fewer noise traders will enter the market and therefore fewer noise components will be present. Thus, it shows that the exchange rate policies that aim at eliminating the non-fundamental risk can be justified theoretically. It also suggests possible 2 4 c e (0,0.25], as the steady state consumption in this model is 0.25. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 30 approaches the monetary authorities may apply to reduce the excess exchange rate volatility, to discourage the entrance of noise traders by increasing the entry cost or to 'educate' the market to reduce the number of noise traders on the foreign exchange market. Furthermore, it suggests monetary authorities could reduce this kind of excess exchange rate volatility by commitments to low exchange rate volatility. In this way, the volatility of expectation error of noise traders will be reduced and so will the exchange rate volatility. A self-contained equilibrium with low exchange rate volatility would be then established. 1.5 An Extension - Tobin Tax Tobin (1978) and Eichengreen, Tobin and Wyplosz (1995) suggest that an international transaction tax on purchases and sales of foreign exchange would be one way to "throw sand in the wheels of super-efficient financial vehicles". They argue that a transaction tax might diminish excess volatility. Even a small transaction tax would deter the fast round trip into a foreign money market. 2 5 A Tobin tax is different from the entry cost we analyzed in the benchmark model. First, it is a common cost for both rational and noise traders. Second, it is not a fixed cost, but increases with the amount of foreign currency bond traded. In this extension, we extend the benchmark model to include a transaction tax to analyze the implication of the Tobin tax on exchange rate volatility in our model. When a transaction tax is imposed, the trader i's problem can be written as: 2 6 max.E\{Cj+ 1{i)) - ar\{Cj+l{i)) (1.5.1) 2 5 A small transaction tax would be a negligible consideration in long-term portfolio or direct investments in other economies. Relative to ordinary commercial and transportation costs, it would be too small to have much effect on commodity trade. 2 aThe transaction cost could not be modelled as linear in B ^ t + l ( i ) , because this would imply that trader i will gain when selling foreign bonds. Thus, we assume a convex transaction cost. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 31 Subject to 2 7 Pt+iC&i = [Blt+im+r;+i)St+i-Blt+l(i)St(l+r^^ (1.5.2) where r is the rate of the transaction tax on foreign bond trading. Solving the traders' problem, we could get: Bh* = E\{Pt+i] ( 1 5 3 ) From Equation 1.5.3, it can be seen that the introduction of a Tobin tax reduces the bond trading of both types of traders. This is quite intuitive, as foreign exchange traders will tend to trade less foreign currency bonds when there is a tax on transactions. When there are only transaction costs, it can be shown that the traders will always choose to enter the market. This is because the transaction cost is convex in the bonds traded, the traders can always choose to hold a small amount of foreign bonds and get a positive expected utility, regardless of how large is r. Therefore, similar to the benchmark model, two cases are analyzed in this extension. In the exogenous entry case, we focus on the transaction cost only. In the endogenous entry case, we assume that noise traders have to pay two costs to trade in the foreign exchange market: the transaction cost and a fixed information cost as in previous sections. However, the informed traders only need to pay a Tobin tax. The analysis of the second case will help to understand the role of the Tobin tax in the economy. Using Equations 1.5.3, we could get the interest parity condition when there exists a transaction tax in the foreign exchange market. For the exogenous entry case: PrdB* ,,, (i + f)S st = Etisf+i) - [3(drt+1 - drt+i) + (1 - Nfivt - g ^ - aK ^ ' Vart[s{+l]dBlt+l (1.5.4) When the noise traders have to pay both the transaction cost and the fixed information cost, the gross benefit of entry for noise traders can be derived: GB t ^ i ) ] 2 p (1.5.5) 2aVart(pt+i) + 2r[^0^ 2 7We assume that the Tobin tax is a real tax and is resource-consuming in the sense that it consumes the composite consumption good. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 32 From Equation 1.5.5, it can be seen that the Tobin tax reduces the gross benefit of entry for noise traders. Therefore, increasing the transaction cost will deter the noise traders from entering the market. Given that, we could get the interest parity condition for the endogenous entry case: 1 PTdB*hi,, (l + r)S st = Et(stll)-0(drt+1-dr;+1) + — n t v t - N i § ^ (1.5.6) where nt, the number of incumbent noise traders is given by: , {Et(st'+i) - st - (3(dr;+1 - drt+1) + vt}2 (1 - Nj) nt = dnt « z (1.5.7) 2aVart(st+1) + ^f+WT C Analogous to Equations 1.3.1 and 1.3.2, Equations 1.5.4 and 1.5.6 give the deviation from the uncovered interest parity. This deviation is composed of three parts. Besides the expec-tation error term and the risk premium term, there is an extra term that comes from the transaction tax. Even in the absence of noise traders, this term still exists. As emphasized by Eichengreen, Tobin and Wyplosz (1995), this term creates room in the interest parity condition and expands the autonomy of monetary policies. To find out if the introduction of the Tobin tax will reduce excess exchange rate volatility, we solve the extended model by the approach described in Section 1.3.28 Then solution of the exogenous entry model is given by: . 1 + ? + £Vart(st~+i) + S r z it = (rnt-m;)p+f + + ^ + p + , + (1-5.8) where q(l + r)SCa P*C a * = 2 f c - 2 ^ T T ^ ) f ( L 5 - 9 ) 2 8The only equation that has been changed besides the interest parity condition is the home country aggre-gate consumption equation, which now becomes: Ct = C" + C? + TB*hit+i Exogenous Entry nt Ct = C" + Ct + TB*htt+i + YlCi Endogenous Entry i=0 Once we log-linearize the above equations around the steady state, the log-linearized equation remain un-changed. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 33 From Equation 1.5.8 we can solve for the exchange rate volatility V s : Vs = — [Var(mt) + Var(rfit*)} (1.5.10) It can easily be shown that if the other variables are kept constant, the numerator on the right-hand side of Equation 1.5.10 decreases in r, the rate of transaction tax, while the denominator increases in r . Since Equation 1.5.10 is an implicit function of Vs, we solve it numerically to get the relationship between Va and r , which is given in Table A.4. It can be seen that the higher the rate of the transaction tax, the lower is the exchange rate volatility. Intuitively, this is because the introduction of the transaction tax reduces the bond trad-ing. In our model, when an exchange rate change occurs, the real balance effect prevents the current consumption from increasing/decreasing more than the changes in the relative real money supply, so the bond holding of households will absorb most of the wealth effect caused by the exchange rate change. If the bond trading is deterred by the transaction tax, in equilibrium, the exchange rate volatility must decrease. For the endogenous entry model, we can derive the solution analogously using Equation 1.5.6: - - . ,1 + f + jVart(sf+l) + ^r | 1 where 2 9 = {Et(sf+1)-st+vt}> ( 1 - i V , ) 2aVart(st~+1) + jsf^r c a(l + f)SCo ,. P2C o * = 2Nj f ^ = 2^,5(1+ f)f ( L 5 ' 1 3 ) Solving Equation 1.5.11 numerically by the approach described in Appendix A.7, we find that exchange rate volatility also decreases in the transaction tax, as in the exogenous case. The results are given in Table A.4. 2 9The derivation of Equation 1.5.12 is analogous to that of Equation 1.3.25. Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 34 In the endogenous entry case, the transaction cost reduces the exchange rate volatility through two channels. First, as in the exogenous case, it reduces the bond trading of both types of traders, which in turn decreases the exchange rate volatility. Second, as shown by Equation 1.5.5, the Tobin tax reduces the gross benefit of entry for noise traders, which consequently reduces the noise component of the foreign exchange market. Therefore, the mechanism through which the transaction cost affects the exchange rate volatility is different when the noise component on the market is endogenously determined. The effect of the Tobin tax will thus be different as well. This can be seen from Table A.4, for the same level of increase in T, the decrease in the exchange rate volatility in the endogenous entry case is larger than that in the exogenous entry case. This finding has important policy implications. It shows that the impact of a Tobin tax on exchange rate volatility depends crucially on the structure of the foreign exchange market and the interaction of the Tobin tax with other trading costs. 1.6 Conclusions and Subsequent Research In this paper we present a model of exchange rate determination which combines the new open economy macroeconomics approach and the noise trader approach for exchange rate behavior. This model emphasizes the interaction of the macroeconomic fundamentals of exchange rate and the microstructure channel through which exchange rates are determined. The latter is often ignored by conventional macroeconomic research on exchange rate and the literature on policy evaluation. Therefore, our work has important implications for understanding exchange rate behavior and exchange rate policies. Two important and promising findings from this model are: 1. Models that take both the macroeconomic and microeconomic factors of exchange rate determination into consideration can explain the "exchange rate disconnect puzzle". 2. The exchange rate volatility caused by irrational market behavior or non-fundamental shocks could be controlled by exchange rate policies. We analyze two kinds of policies. One focuses on the entry cost of noise traders, while the other is a 'Tobin tax' type of exchange rate policy. We find that both policies Chapter 1. Noise Traders and the Exchange Rate Disconnect Puzzle 35 can reduce the exchange rate volatility. However, the effect of a Tobin tax on exchange rate volatility depends crucially on the structure of the foreign exchange market and the interaction of the Tobin tax with other trading costs. Thus, subsequent research should focus on the policy implication of this model. What kind of exchange rate regime is better when non-fundamental shocks to the exchange rates are present - flexible or fixed? If the real exchange rate volatility is primarily affected by non-fundamental factors and most exchange rate volatility is useless, then would the fixed exchange rate regime or a single currency area be better than the flexible exchange regime? This model could also be used to evaluate the welfare implications of exchange rate policies such as the Tobin tax or other policies that discourage the entry of noise traders. These policies are discussed widely, but due to the lack of a welfare-based model which can explain exchange rate volatility and its relationship with macroeconomic fundamentals, they have not been evaluated on a welfare basis. The presentation of explicit utility and the profit maximization problem in our model allows for the possibility of answering these questions based on rigorous analysis. Although this model could help to explain the exchange rate disconnect puzzle, it is still not a fully developed model that could be used to explain all empirical features of exchange rate. To explain the persistence of the real exchange rate, more persistent price setting or 'sticky' information of traders would be needed. For example, the information structure of the noise traders could be changed so that the expectation error is more persistent. 36 Chapter 2 Optimal Monetary Policy with Vertical Production and Trade 2.1 Introduction The debate on optimal monetary policy has been at the heart of international macroeconomics for many years.1 Friedman (1953) and later Mundell (1961) argue that the flexible exchange rate can act as an efficient mechanism for dealing with country-specific shocks when the adjustment of domestic price levels is sluggish. But recent studies of monetary policy in utility-based open economy models have reached varying conclusions about the desirability of flexible exchange rates and the way monetary authorities respond to foreign shocks. Obstfeld and Rogoff (2002) argue that an inward-looking monetary policy, in which the monetary authorities respond solely to their domestic shocks delivers the best possible out-come, and flexible exchange rate can replicate the flexible price equilibrium. However, De-vereux and Engel (2003) show that the optimal monetary policy and exchange rate regime choice critically depend on the currency of export pricing. If prices are set in the currency of producers (PCP), and the pass-through from exchange rate to consumer prices is complete, then the flexible exchange rate is a central part of the optimal monetary policy. But if the prices are set in the currency of buyers (LCP), and do not respond to movements in ex-change rates, then the monetary authorities should keep exchange rate fixed, so they should respond to both home and foreign shocks and the optimal monetary policy cannot replicate the flexible price equilibrium. Corsetti and Pesenti (2001) have a similar conclusion. They analyze how the degree of exchange rate pass-through to prices affect the optimal monetary 'This chapter is based on the joint work with Kang Shi. Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 37 policy and show that the optimal monetary regime is pegged exchange rate in the extreme case where the exchange rate has no impact on consumer price. Tille (2002) emphasizes the importance of the nature and sources of shocks on the optimal monetary policy design. He shows that the value of exchange rate flexibility is much smaller when shocks are sector-specific and argues that the sectoral structure of the economy and the source of shocks significantly affect the international monetary policy and its welfare implication. Therefore, these literatures suggest that the presence of local currency pricing (or incom-plete exchange rate pass-through) and sectoral shocks may significantly change the existing wisdom on international monetary policy based on P C P pricing. However, a notable feature of these literatures is that they focus on an environment where all the trade in goods between countries occurs in one stage. In reality, countries can trade not only finished goods but also intermediate goods, even in more stages. We will follow Hummels et al. (1998) and use the term "vertical trade" to describe this vertical structure of production and trade. More specifically, vertical trade occurs when a country uses imported intermediate goods as an input to produce export goods. This definition captures the idea that countries are linked sequentially in the production of final goods (or finished goods). Hummels et al. (1998) analyze data from 10 O E C D countries and find a strong statistical correlation between the increase in the share of vertical trade in total trade and the rise of the share of trade in GDP. The increase in the vertical trade are found to account for more than 25 percent of the increase in the total trade in most O E C D countries. In some smaller countries such as Canada and Netherlands, the share of vertical trade in total trade approaches 50 percent. Feenstra(1998), Hummels e ta l . (2001), and Y i (2003) also argue that the vertical structure has been a more and more important feature of today's global production and trade. Huang and Liu (1999) show that a vertical chain of production can generate different monetary transmission mechanism in a closed economy. However, the vertical structure of production and trade has yet been remarkably over-looked in the new open economy macroeconomics literature. Obstfeld and Rogoff (2000b) Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 38 and Devereux and Engel (2004) introduce the trade of intermediate goods, but limit the trade of finished goods and focus on the importance of the relative price adjustment of intermediate goods. Nevertheless, the vertical structure of production and trade will influence the inter-national transmission mechanism of productivity shocks and thus affect the way monetary authorities respond to country-specific and stage-specific productivity shocks and desirability of flexible exchange rate in optimal monetary policy. Therefore, in this paper we will explore optimal monetary policy in an open economy with vertical production and trade. To address this question, we introduce two stages of production and trade into a standard two-country general equilibrium model with sticky prices. There are two vertical stages in each country, one is the finished goods stage, the other one is the intermediate goods stage. Each stage in each country has a stage-specific productivity shock. At each stage, there is a continuum of firms, each firm produces differentiated goods. The production of finished goods requires a basket of distinct variety of domestic intermediate goods and a basket of distinct variety of imported intermediate goods. The production of intermediate goods only needs labor. To highlight the impact of this trade pattern on the optimal monetary policy, we maintain the assumption that firms set price in-the currency of producers (PCP) and thus the pass-through of exchange rate changes to import prices is complete. Our model is simple enough to be solved analytically, so the policy evaluation will be based on rigorous welfare comparison. Nevertheless, it incorporates the main feature of the vertical structure of production and trade we want to emphasize in this paper and we find that this trade pattern does have an important impact on the decision of optimal monetary policies in open economies. The main findings of this paper can be described as follows: First, when both finished goods and intermediate goods are tradable, any stage-specific productivity shock in one coun-try has a positive trans-border spillover effect on the other country via vertical production and trade. This effect changes the way monetary authority reacts to other country's productivity shock, and each monetary authority should respond positively and partially to both home and foreign productivity shocks. This is different from from Obstfeld and Rogoff (2002) and Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 39 Devereux and Engel (2003), where the optimal monetary policy based on PCP requires mon-etary authorities to respond only to domestic shocks. Second, a vertical production structure generates multiple price stickiness. Thus, flexible exchange rate cannot adjust the terms of trade to the efficient level in both stages simultaneously. So the flexible price equilibrium cannot be replicated by flexible exchange rate in our model, even in the situation with PCP pricing and complete exchange rate pass-through. Finally, we find that the exchange rate volatility in this economy is lower than that would be obtained in an economy without vertical structure of production and trade. This implies that the multiple stages of production and trade lead to a more integrated global economy so that smaller exchange rate adjustments are needed in response to the country-specific shocks. This paper is closely related to Devereux and Engel (2003). Devereux and Engel (2003) investigates how the price setting affects the optimal monetary policy and exchange rate flexibility, we adopt their approach to derive optimal monetary rules. Our departure is that we allow for vertical production and trade, so that we can explore the difference in the international transmission mechanism of productivity shocks and the limitation of the exchange rate as an adjustment mechanism for nominal rigidity in such an environment. As to the emphasis on multiple stages of production and trade, this paper is also related to Huang and Liu (2003). They try to reconcile the controversy of the welfare consequence between PCP and LCP by modelling multiple stages of production and trade. Our anal-ysis differs because we allow for stage-specific shocks and focus on the reaction of optimal monetary policy to these shocks. Devereux and Engel (2004) also build a new open economy macroeconomic model with intermediate goods trade. They develop a view of exchange rate policy as a trade-off between the desire to smooth fluctuations in real exchange rates (so as to ensure international risk sharing) on one hand, and the need to allow flexibility in the nominal exchange rate (so as to facilitate relative price adjustment) on the other hand, and optimal nominal exchange rate volatility will reflect these competing objectives. In our paper, as emphasized, we introduce vertical structure of production and trade into the new open macroeconomic framework. Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 40 Therefore, we focus on the international transmission mechanism of stage-specific productiv-ity shocks via trade and the impact of multiple-stage price stickiness on the desirability of flexible exchange rate. This paper is organized as follows. Section 2 presents the model. Section 3 gives the solution of the model and also solves the flexible price equilibrium as a benchmark. Section 4 analyzes the optimal monetary policy response to stage-specific productivity shocks and its welfare implication. Section 5 concludes. 2.2 B a s i c M o d e l The world consists of two countries of the same size, denoted as the home country and the foreign country. Each country has one unit of population, they derive utility from aggregated consumption (composed of home finished goods and foreign finished goods), real balance, and leisure. There are two stages of production in each country, one is the finished goods stage, the other is the intermediate goods stage. Note that an important assumption which differ our model from the literature is that we assume both goods are tradable. Each stage has a stage-specific productivity shock. At each stage, there is a continuum of firms indexed on the interval [0,1]. Each firm produces differentiated goods and therefore has some monopolistic power. The production of finished goods requires a basket of distinct variety of domestic intermediate goods and a basket of distinct variety of imported intermediate goods. The production of the intermediate goods only requires labor. Figure B . l gives the structure of the economy. Al l firms set prices before the realization of the shocks, and the prices are in the currency of producer. For simplicity, we abstract from any dynamics by considering a single period model with uncertainty. The structure of events within the period is as follows. First, before the period begins, households can trade in the bond market for a full set of nominal state-contingent bonds. Then the monetary authorities choose optimal monetary rules, given the cross-country risk-sharing rule, taking into account the way in which firms set prices, as well as the dis-Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 41 tribution of stochastic productivity shocks. Following this, firms set prices, given the state-contingent discount factors, the expected demand, and the expected marginal costs. After the realization of stochastic shocks, households work and choose their optimal consumption baskets, production and consumption take place, and the exchange rate is determined. The detailed structure of the economy in the home country is described below. The foreign country is entirely analogous. From now on, foreign variables and foreign currency prices will be indicated by an asterisk. In addition, a subscript h denotes a variable originating from the home country; a subscript / denotes a variable used in the foreign country. 2.2.1 Household The representative households in the home country maximizes the following expected utility 2: Cl~p M U = E ( r r - p + X ^ - - V L ) (2.2.1) where C = 2 C | C | Ch = [J Ch(i)^}>^ (2.2.2) Here C is the aggregate consumption, Ch is the sub-aggregate consumption of a continuum of home finished goods indexed by [0,1], Cf is the sub-aggregate consumption of a continuum of imported foreign finished goods. $p is the real money balances, and L is the home labor supply. A > 1, is the elasticity of substitution between differentiated home(foreign) finished goods, p is the inverse of the intertemporal elasticity of substitution and is assumed to be greater than 1. From Appendix B . l , we can derive the CPI price index and individual demand for finished goods i in the domestic market and in the foreign market, respectively P = p2PJ (2.2.3) ch(i) = \(™T\y)~lc (2.2.4) 2The adoption of this utility function will give us a closed form solution. It is also used in Obstfeld and Rogoff (2002), Devereux and Engel (2003), and Devereux, Shi and Xu (2003). Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 42 CZ(i) = \ { ^ r \ ^ r l C (2.2.5) Home and foreign households can trade a full set of state-contingent nominal bonds, thus, the budget constraint of the home households for a particular state of the world z is written as: P(z)C(z) + M(z) + ]T q(QB(S) = W(z)L{z) + U(z) + B{z) + M0 + T(z) (2.2.6) That is, consumers derive income from the labor income W(z)L(z), the payoff of the state-contingent securities B(z), the profits from their ownership of all home firms I7(z), the initial money balance Mo, and the lump-sum transfer from the government T(z). They choose how many state-contingent bonds to purchase before the period begins, with q(£) and B(£) representing the price and holding, respectively, of a security paying off 1 unit of home currency in state £ G Z, where Z is the set of states. Then the households will choose the money holding, the consumption and the labor supply. It is assumed that the government repays any seignorage revenue through the lump transfer, so that Mo — M(z) + T(z) = 0. The specific money supply process will be discussed in later sections. The trade in state-contingent nominal assets across countries will lead to the following optimal risk-sharing arrangement: C-P n*-p where S is the nominal exchange rate, and P* = P^p*2 is the foreign price level. T is the state-invariant weight and equals l . 3 Equation (2.2.7) implies that one dollar can get the same marginal utility of consumption across countries. Therefore, the real exchange rate is equal to the ratio of marginal utilities of consumption across countries. In addition, household 3 T represents the ratio of the Lagrange multiplier associated with the home household's budget constraint to the Lagrange multiplier associated with the foreign household's budget constraint. It is also a condition capturing the initial distribution of wealth. Devereux and Engel (2003) shows that T equals 1 in a symmetric equilibrium. Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 43 optimization problem gives rise to the money demand equation: M = XPC, (2.2.8) and the implicit labor supply function: W = T)PC. (2.2.9) Equation (2.2.8) and (2.2.9) imply that the nominal wage is proportional to the amount of money in circulation. 2.2.2 Finished goods stage There is a continuum of firms indexed by i £ [0,1] in the finished goods stage in the home country. Each firm i produces home finished goods Y(i) out of home and foreign intermediate goods according to the following production function:4 where OF is the finished goods stage specific shock in the home country and Xh(i) (Xf(i)) is a basket of distinctive variety of intermediate goods produced in the home (foreign) country. Cost minimization Each finished goods producer i takes the prices of intermediate goods as given, so the unit cost to produce finished goods i can be derived as: where Ph is the price index of home intermediate goods denominated by home currency, and Pfh is the price index of foreign intermediate goods which are sold in the home country, denominated by foreign currency. From the cost minimization problem, we can derive finished goods producer i's demand for the basket of distinct variety of home and foreign intermediate goods: 4Here, we assume the production of finished goods requires no labor input. This assumption helps us to solve the model analytically. However, the result will not change even if we allow for labor inputs in this stage. Y{i) = 26FXh(i)12X}{i)12 (2.2.10) A = PhHSPjS* 9F (2.2.11) Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 44 1 SP* Xf{i) = -2{~f^rlY{i) (2.2.13) Finished goods price We assume that in each country the finished goods producer i sets its price in the currency of producer, thus the law of one price holds in each individual final good and Purchasing Power Parity holds in the CPI. From Equation (2.2.4) and (2.2.5), the total demand for home finished goods i is: 5 Y(i) = Ch(i) + C*h{i) = {^)-\^r'C (2.2.14) Given the demand structure and the unit cost function of finished goods, we can derive the optimal pricing policies for finished good i, - E\AC1~P] - E[^-Cl-P}Pt}'Pih* ™ = A § i c ^ r = A E[ci>] (2-2-15) ^ ( 0 = ^ (2-2.16) Where A = is the markup for finished goods pricing. The term E[^jC1~p] represents a risk premium term arising from the covariance of firm i's profit with marginal utility of consumption, where the fluctuation of the exchange rate S directly affects firm's pricing decisions. Meanwhile, intermediate goods stage productivity shocks in both the home country and foreign country will affect the price of home finished goods though F/, and Pjh. For the home finished goods producer, the foreign intermediate goods productivity shock 8*t affects the preset price in the same way as the home intermediate goods productivity shock 0/. This implies that a positive stage specific shock in the foreign intermediate goods stage will generate a substitution from home intermediate goods to foreign intermediate goods. Meanwhile, it also has a positive trans-border spillover effect on the home finished goods. 5For simplicity, we have used the fact that C = C* in the total demand function for final goods. Since the price of final goods are preset in producer currency, purchasing power parity (PPP) holds. Thus, we could derive C = C* from the risk-sharing condition (2.2.7). Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 45 Imposing symmetry, we can drop out the subscript i. The optimal pricing schedules of foreign finished goods firms can be derived analogously and are listed in Table B . l . 2.2.3 Intermediate goods stage It is assumed that there is a continuum of firms indexed by j € [0,1] in the intermediate goods stage in each country. The home intermediate goods firm j uses the following linear technology, subject to the stage-specific shock 9j: Xh(j) = OjLiJ) (2.2.17) Demand structure One basket of distinct variety of the home intermediate goods is given by: Xh = [[ Xh(j)*T]& (2.2.18) Jo where <j> represents the elasticity of substitution between intermediate goods in the home country. The demand for the intermediate good j in this setting can be derived as: XhU) = [^ rV**h (2-2.19) where Ph(j) is the price of the intermediate good j in the home country. Then we may derive the total demand for a basket of distinct variety of home intermediate goods from the home finished goods and the foreign finished goods firms, respectively: xh = i ( ^ ) - 1 1 v m = \{^ri j \ ^ r \ ^ r x c \ d i (2.2.20) x*h = £ Y f { i ) d i = l ^ ' 1 j y ^ ' ^ ' 1 ^ * (2-2-21) Equations (2.2.20) and (2.2.21) show that the demand structure of intermediate goods would be affected by aggregate consumption C , the finished goods stage shock Op (or 6*F ) through A(A*), and the nominal exchange rate S. Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 46 Intermediate goods prices We assume the prices of intermediate goods are preset in PCP, but we allow for pricing to market. That is, the intermediate goods producer sets two prices, one is for the sales in the domestic market, and the other is for the foreign sales6. That is, the law of price may not hold in the intermediate goods stage. The two pricing policies for home intermediate goods producer j can be derived from the following profit maximization problem. # / ^ \ - i . Jo max ETl(j) Ph(j),P'hf(j) \ ( ~ W PhU) i + 2 p p* Ph Pf(i) » ' f i di di}} This yields PkU) = 4> OF E[-^F-PhfU) = <t> w g - g c 1 - ^ E 6% J (2.2.22) (2.2.23) where 0 = PC is the marginal utility of consumption of the home household, which is used as the stochastic discount factor as all the domestic firms are owned by households. 4> = is the markup for intermediate goods price. Note that, for the foreign finished goods producers, the price of home intermediate goods in terms of the foreign currency is Ph!g^• If the intermediate goods firms can set price flexibly, then the price will simply be a fixed markup over the unit labor cost. With the sticky price, there is an additional risk premium term arising from the covariance of marginal cost with the term " which represents the demand risk from its buyer-the finished goods producers. Thus, the productivity shock in the foreign finished goods stage will also have a positive spillover effect on the home intermediate goods as it increases the demand for these goods. Imposing symmetry, we can drop out the 6This PCP pricing with pricing to market is used in Devereux, Shi and Xu (2003). With this assumption, we can solve the model analytically. Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 47 subscript j . The optimal pricing schedules of foreign intermediate goods firms are reported in Table B . l . 2.2.4 Stochastic shocks We assume the final goods stage shock Op and intermediate goods stage shock Oj are: 0F = exp(u), Oi = exp(v) (2.2.24) where u and v is mean zero and normally distributed with a variance-covariance matrix E = ( ° u CT"2V ). A similar assumption is made for the foreign productivity shocks. Our setup allows the productivity shock to be specific to a particular stage in a particular country. This nests a special cases where productivity shock are country-specific (u = v, u* = v*). 2.2.5 Equilibrium The market clearing condition of finished goods market, money market and labor market are trivial. Note that the intermediate goods market clearing condition in the home country is given by: 0IL = Xh + X * h = l - ^ + 1 - ^ ^ - (2.2.25) 2 ph Ph 2 Phj_ Pt v ; s 1 Thus, given the stochastic processes Op, 0*F, Oj and 6}, the 17 variables C, C*, P, P*, Ph, Pf, Ph, Phf, PJ, Pfh, S, A, A*, W, W*, L and L*, are determined by 17 equations: risk-sharing condition (2.2.7), money demand (2.2.8) and its foreign equivalent, labor supply (2.2.9) and its foreign equivalent, home finished goods pricing equations (2.2.15)and its foreign equiv-alent, intermediate goods pricing equation (2.2.22), (2.2.23) and their foreign equivalents, intermediate goods market clearing condition (2.2.25) and its foreign analogy, the unit cost function of finished goods (2.2.11) and its foreign equivalent, and the two CPI price indices. Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 48 2.3 Solution The model is log-linear and the underling productivity shocks are log normal, so we may solve all the endogenous variables in closed form7. We focus on the variables which are determined after the shocks are realized. The nominal exchange rate can be derived from the risk-sharing condition and money clearing condition (2.2.7) and (2.2.8): (2.3.1) From Equation (2.2.7)-(2.2.9), we have: W = SW* (2.3.2) Since the law of one price and PPP holds in CPI level, we have: C = C* = C^J^yP (2.3.3) Ph and Pj? are set before the realization of the shocks, so the consumption in both countries are determined by the home money supply and the foreign money supply. If all the prices can respond to the ex-post value of stochastic shocks, then clearly money will be completely neutral and the economy will be at the efficient flexible price equilibrium8. To measure the desirability of flexible exchange rate, we will use the terms of trade and the expected utility under the flexible price equilibrium as a benchmark. Putting the home and foreign finished goods price indices together, using the risk-sharing condition, the pricing equation of intermediate goods and the labor supply function, we may get the solution for the home and foreign consumption in the flexible price equilibrium. C = C* = ( A ^ r & ^ f l J ^ f l J J ^ (2.3.4) TThis implies that consumption, labor, prices and exchange rate all follow log normal distribution. 8In the economy with two stages of production and trade, there are two sources of distortion associated with monopolistic competition, which lower the efficient expected utility level. However, this inefficiency cannot be eliminated by the monetary policy, so it is not our focus. Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 49 From the intermediate goods market clearing condition (2.2.25), we may solve for the home and foreign labor in the flexible price equilibrium. L — L* = J^C1-p = ( ^ ) _ * ) j _ i ( ^ / « J ) ^ (2.3.5) \<f>T) Note that both the consumption and the employment are functions of geometric weighted average of productivity shocks. The terms of trade under flexible prices in both stages can also be derived: £2_!E, £ & _ f t (2.3.6) To achieve the efficient allocation, the terms of trade in both the finished goods stage and the intermediate goods stage must be adjusted completely according to the corresponding relative productivity shocks9. Following Obstfeld and Rogoff (1998, 2002), it is assumed that the utility derived from real balances is small enough to be neglected. From Equation (2.3.5), the expected utility for the representative consumer in the home country can therefore be measured by the following expression E{^—- - nL) = K £ C 1 - ' ' (2.3.7) where K — (1 " ^ W ) ^ ~ ^ < 0. Since consumption has the log normal property, we may rewrite ECl~p = exp (1 - p)(Ec + ^ c 2 ) , thus, maximizing the expected utility level is equivalent to maximizing UQ = Ec+ ^-j^cr2. Therefore, we can measure the expected utility level in the flexible price equilibrium as Uo(flex) = + l~^rWl + "u* + ^ + + l°uv + 2ou*v*} (2.3.8) In the absence of nominal rigidities, not surprisingly, the equilibrium is independent of the monetary policy. The expected utility level is composed by two parts: a constant function of parameters and the impact of stochastic productivity shocks on the expected utility. When 9In the flexible price equilibrium, the relative price of finished goods to intermediate goods in each country will also respond completely to the corresponding relative productivity shocks. Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 50 p = 1, the expected utility is only affected by the mean of the log consumption. How-ever, when p > 1, the risk-averse household will care about the uncertainty brought by the stochastic productivity shocks, and the expected utility is decreasing in the volatilities of the stochastic productivity shocks. 2.4 Optimal money rules In this section, we study the optimal monetary policy rules in response to stage-specific productivity shocks. The independent monetary authority in each country sets the following monetary rules to maximize the expected utility of the domestic households:1 0 m = mo + diu + a2U* + a$v + a^v* (2.4.1) m* = m o + bin + b2u* + b3v + b4v* (2.4.2) From now on, let x = InX. The policy parameter vectors [ai, a2,03, and [61,62,63,64] will be determined by the international monetary Nash game between two independent monetary authorities. It is assumed that they can commit to their monetary rules and the rules are announced before the firms set their prices. As shown in Devereux and Engel (2003), the expected utility level in a stochastic environ-ment is a function of variances and covariance terms of the log home and foreign consumption and the log exchange rate. Thus, once we solve for the consumption and the exchange rate, we may rewrite the expected utility as functions of the monetary policy parameters. In log terms, we may write Equations (2.3.1) and (2.3.3) as s - E s = m-m* (2.4.3) c-Ec = c* - Ec* = ^ - (m + m*) (2.4.4) 2p 1 0 T h e log-linear money supply rule we choose here is a general form of pol icy rules for the models wi th log normal stochastic shocks. Thus, the log money supply follows random walk. Th i s property wi l l give us a constant nominal interest rate and shut down the effect of the nominal interest rate on the real balance when the real balance takes log form in the ut i l i ty function. Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 51 where Ex denotes the conditional expectation of the variable x before the period begins. The detailed derivation of the variance and covariance terms of c and s in terms of the moneatry policy parameters are given in Appendix B.3. We now turn to the derivation of the objective functions of monetary authorities. Using the intermediate goods market clearing condition (2.2.25), we have: EL = JL-EC1-" (2.4.5) Thus, the objective function of the monetary authority in the home country is exactly the same as Equation (2.3.7). From equation (2.3.7), the expected utility level depends mainly on the mean and the variance of the log consumption, so the key step is to solve for Ec. The detailed derivation is given in Appendix B.2. -ln[X4>n2} (2 - p) _2 1 2 1 2 2 2 2 Ec = — -—cc - —os - — [ou + ou. +ov + crv. + ouv + CT U .W .J + T~ Wsu — o-su-} + — [crsv — asv*] + — [o-cu + Ocu* + Ocu + Ccv*] (2.4.6) 4p 4p Ip Given the log normal property of the consumption and the functional form of the expected utility (2.3.7), the optimization problem of the monetary authority is equivalent to maximiz-ing UQ = Ec+ ^-^•o2, so r/n can be derived as -ln[\jyn2} 1 2 1 2 1 2 , JI , JI , * , „ , „ 1 = Yp 2 c ~ 4p * ~ 4p^ " u " v v * u v a u* v'> + -^[o~su — Osu'\ + T^I 0 "^ _ asv*\ + 2p~lacu + acu' + °~cv + Ocv*] (2.4.7) We may rewrite the expected utility function as a function of policy parameter vectors o and b. For the home households, consumption variance and exchange rate variance reduce utility, while the covariances of consumption and productivity shocks, a positive covariance of the exchange rate and home productivity shocks or a negative covariance with foreign productivity shocks increases utility. The optimal monetary rule will be a trade-off between these costs and benefits and the effects of monetary policies on both the consumption and the exchange rate will be considered. Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 52 A Nash equilibrium of the international monetary game between countries is characterized by the following conditions: max U0(a,bN) (2.4.8) a maxU0r(aN,b) (2.4.9) b The objective function UQ is identical to Uo since home and foreign have identical consumption and employment in equilibrium. Appendix B.3 gives the following solution to this monetary Nash game. 3 ~ 4' o? = l 4' 3 4' oJ = 1 4 bn 1 ~ 4' bn2 = 3 4' •63 = 1 4' bn4 = 3 4 With sticky price, an optimal monetary policy response to a positive productivity shock must be expansionary, so as to shift demand to meet the potential aggregate supply. The magnitude of the response, however, depends on the expenditure switching effect of the exchange rate changes under the P C P pricing regime. As shown in Obstfeld and Rogoff (2000b,2002), Corsetti and Pesenti (2001), Devereux and Engel (2003), with P C P pricing and full exchange rate pass-through, the optimal monetary policy requires the home monetary policy authority to respond only to the home productivity shock. That is, if there are two country-specific shocks 6 and 6*, the policy parameter vector would be [ai = 1, a2 = 0] and [61=0,62 = 1]. Nevertheless, the solution for the Nash game in our model implies that the optimal mon-etary policy requires the home monetary authority to respond positively to both home and foreign productivity shocks, though the response of the home monetary authority to domestic productivity shocks exceeds that of the foreign monetary authority. Home monetary author-ity responds positively to foreign productivity shocks because foreign productivity shocks have a trans-border spillover effect, which is induced by the vertical structure of production and trade. For instance, if a positive stage-specific productivity shock occurs in the foreign Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 53 finished goods stage, it not only increases the potential supply of foreign finished goods, but also increases the demand for home and foreign intermediate goods. In this sense, home intermediate goods producers will benefit from a positive foreign productivity shock. That is, any stage-specific shock in one country would affect both countries like a "world shock", so both countries should respond positively to this shock. Meanwhile, under PCP pricing, the conventional expenditure-switching effect is still play-ing a role in our model. Specifically, a positive stage-specific shock in one country also gen-erates a direct substitution effect from imported goods to domestic produced goods in this particular stage. Intuitively, the substitution effect of a home stage-specific shock on home goods should dominate its indirect positive transborder spillover effect on the foreign country, which requires a nominal exchange rate depreciation in the home country. To achieve this, the response of the home monetary authority to foreign shock should be less than that of foreign monetary authority. From the solution to the international monetary game, we can have the following propo-sitions. Proposition 1 Under the flexible exchange rate regime, the exchange rate is more stable when there is vertical production and trade. From the optimal monetary policy rules, we may solve for the log exchange rate: s = l-u - ]-u* + l-v - l-v* (2.4.10) 2 2 2 2 v ' Given the assumption that the shocks in one country have the variance-covariance matrix E = ( "2 °u2 ) U ! it follows that the variance of optimal exchange rate is CT2 + auv. When there is no correlation between the finished goods shock and the intermediate goods shock in one country (auv = 0), the exchange rate volatility is just CT2. However, when there is only one stage of production and trade and the price is set under PCP pricing 1 2 , the optimal policy parameter vectors are a = [1,0] and b = [0,1]. This implies that the variance of n For simplicity, we assume that a\ = CT2 = CT2. = CTJ. = CT2, and —CT2 < cruv = a u ' v < CT2. 12See Devereux and Engel (2003) for the detail derivation. Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 54 optimal exchange rate is 2o2 if the volatilities of both home and foreign country-specific shocks are equal to o2 as well. Even if ouv > 0, the exchange rate volatility is still lower than that under the economy without vertical structure of production and trade. Therefore, our findings suggest that the floating exchange rate regime is more stable in a world with vertical production and trade. The economic intuition is as follows. With vertical structure of production and trade, the production of the world import and export goods is more diversified, but each country is more integrated with other countries. Therefore, there is no need for large exchange rate changes to adjust the world economy according to the relative shocks. Our model shows that the implication of flexible exchange rate for optimal monetary policy depends on whether shocks are country or stage specific. If the policy makers are not able to observe the nature of shocks, the misconduct of monetary policy might lead to inefficient outcome and higher exchange rate volatility. Proposition 1 also suggests an empirical prediction that the increase in the vertical trade may reduce the exchange rate volatility under a floating exchange rate regime. Proposition 2 The optimal solution cannot replicate the flexible price equilibrium and the flexible exchange rate cannot deliver the economy to the efficient level. When there is only one stage of trade, the conventional wisdom regarding the welfare implication of the flexible exchange rate is that it can bring the economy around the obstacle of nominal rigidities, if the prices are preset in PCP or there is complete exchange rate pass-through. That is, the optimal monetary policy can replicate the flexible price equilibrium. However, this conclusion does not hold when there is more than one stages of production and trade. The intuition is straightforward. Under PCP specification, flexible exchange rate can adjust the terms of trade to the efficient level - the level under the flexible prices equilibrium, so the world resource can be allocated efficiently. However, in a world with vertical structure of production and trade, there are two terms of trades between home and foreign country. Thus, the flexible exchange rate cannot adjust the relative prices of both finished and intermediate goods to the efficient level simultaneously. There still exists Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 55 welfare loss even when monetary authorities optimally respond to both home and foreign productivity shocks. For instance, under the sticky price equilibrium, the term of trade in the finished goods SP* stage is -p*-. Since Ph and Fjf are both predetermined, the term of trade will be proportional to the exchange rate changes which is different from the term of trade under the flexible price equilibrium | f . Similarly, the term of trade in the intermediate goods stage is not equivalent to its flexible price level, either. This intuition is quite similar to the argument of Erceg, Henderson and Levin (2000). They find that when both prices and wages are sticky, the allocation with flexible prices and wages cannot be restored by the optimal monetary policy. That is, if there is multiple stickiness, one policy instrument cannot deal with all nominal rigidities efficiently. In our model, even if the exchange rate policy can adjust the relative prices between home and foreign country, there still exists misallocation between finished goods and intermediate goods. Given the optimal monetary rules, we can compare the maximized expected utility level under the sticky price equilibrium with the expected utility level under the flexible price equilibrium: Uo(flex) - Uo(sticky) = JLfo* + ^ + a 2 + a2 _ 2 ( T ^ _ 2(Tu.v.] > 0 (2.4.11) lop We may use this welfare difference to measure the desirability of flexible exchange rate regime. The higher the welfare difference, the lower the value of the exchange rate flexibility in the open economy with nominal rigidities. Obviously, the vertical structure of production and trade reduces the value of exchange rate flexibility. If there are more stages of vertical production and trade, the desirability of flexible exchange rate for the optimal monetary policy will be even smaller, as more nominal rigidities are present in the economy. Equation 2.4.11 also implies a higher correlation between finished goods shock and in-termediated goods shock in each country increases the value of exchange rate flexibility. When the shocks in two different stages of one country are perfectly correlated, the optimal monetary rules can replicate the flexible price equilibrium 1 3, but the response of monetary 1 3 The welfare difference between the flexible price equilibrium and the sticky price equilibrium is zero. Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 56 authorities to shocks are still different from that in the standard open economy literature. Thus, our findings suggests that the introduction of vertical structure and trade does affect the international transmission mechanism of productivity shocks and the optimal monetary policy design in an open economy. 2.5 Conclusion This paper examines the optimal monetary policy in a world with vertical production and trade by introducing two stages of production and trade into the standard utility-based open economy macroeconomic models. We find that when both finished goods and intermediate goods are tradable, there exists trade-induced trans-border spillover effect of productivity shocks, and this may change the way monetary authorities respond to other countries' productivity shocks. We also find that the flexible exchange rate cannot bring the economy back to the efficient level even under the P C P pricing or the complete exchange rate pass-through case, and the floating exchange rate regime in our model implies more stable exchange rates than in the economy without vertical production and trade. This is quite different from the classical argument for flexible exchange desirability in optimal monetary policy in open economy, suggested by Friedman (1953), Mundell (1961) and recent new open economy macroeconomics literatures. Our findings suggest that the changes in the trade pattern in the global economy over the last thirty years might affect the international optimal monetary policy rules and values of exchange rate flexibility. So the monetary policy makers should take into account the impact of the changes in the trade pattern when making decisions. To verify the theoretical results of the model, some empirical work can be done in the future research to check if the increase of the share of the vertical trade over the last thirty years reduces the exchange rate volatility under a floating exchange rate regime. One advantage of this paper is that all the results can be derived analytically. Never-theless, to make the model tractable, the number of the stages of production and trade we introduced into the economy has to be limited. In the subsequent research, with the help Chapter 2. Optimal Monetary Policy with Vertical Production and Trade 57 of some numerical methods, more stages of production and trade can be considered and the optimal monetary policy in a more integrated global economy can be explored. We conjec-ture that if the number of stages is larger and the world economy is more integrated, the exchange rate flexibility will pay a less important role in the adjustment of relative prices, so the exchange rate volatility will be even smaller. In the extreme case where the number of vertical stages is large enough, a flexible exchange rate regime will be close to a fixed exchange rate regime. Chapter 3 58 Global Monetary Policy under a Dollar Standard 3.1 Introduction // the dollar were ever displaced by the euro, [ the US ] .. would lose the enormous freedom it now enjoys in running macro-economic policy. Ambrose Evans Pritchard, Daily Telegraph, October 10, 2003. The US dollar occupies a unique role in the world economy.1 The dollar resembles an international currency, in the sense that it acts as a means of exchange in international goods and asset trade, a store of value in international portfolios and official foreign exchange reserves, and a unit of account in international commodity pricing. BIS estimates of foreign exchange turnover show that the dollar is used as one side of about 90 percent of daily foreign exchange rate transactions. According to Eichengreen and Mathieson (2000), 60 percent of world foreign exchange reserves are held in US dollars2. Bekx (1998) estimates that over 50 percent of world exports in 1995 were denominated in US dollars, approximately four times the share of the US in total world exports. This predominance of the US dollar has been described by McKinnon (2001, 2002) as a world dollar standard. While the formation of the euro area has generated speculation about the stability of the current dollar role (e.g. Portes and Rey, 1998), at present there seems little evidence of significant change in the use of the dollar in trade and finance. 1This chapter is based on the joint work with Michael Devereux and Kang Shi. 2This share has undoubtedly gone up since the advent of the euro, because all intra-European foreign reserves held in DM's and other European currencies are no longer part of measured reserves. Chapter 3. Global Monetary Policy under a Dollar Standard 59 How does the special role of the US dollar influence monetary policy making in the US and the rest of the world? The quotation above suggests that the US has an advantage in policy making due to the fact that the rest of world holds dollars, and sets prices in dollars. Indeed many commentators argue that there is an enormous welfare gain to the US from having its currency used so widely (e.g. Liu (2002)). The literature on the international monetary system has developed a theory of 'hegemonic stability' based on the idea that the policies of one country play a central role in maintaining the smooth working of the international monetary system (Eichengreen (1995)). According to this theory, US monetary policy may be determined without regard to international constraints, while monetary policy in the rest of the world must take account of US policies. This paper examines the determination of optimal monetary policy in an asymmetric world economy, where the currency of one country (e.g. the US dollar) plays a predominant role in trade 3 . While the evidence cited above illustrates the multi-dimensional role of an international currency, we focus on one particular aspect of this role - the importance of the currency in international export good pricing. We define a reference currency as one in which the prices of all world exports are pre-set. Many authors have noted (e.g. Goldberg and De Campa (2003)) that prices of imported goods sold in the US economy tend to be much less affected by exchange rate fluctuations than do imported good prices in non-US countries. This suggests that prices of a large fraction of exports to the US are pre-set in US dollar terms (which we refer to as local currency pricing, or LCP), and do not react quickly to movements of the exchange rate 4 . However, exports to other countries may have their prices pre-set in the currency of the original producer (producer currency pricing, or PCP), and hence import prices are more 3In the recent international macroeconomics literature, considerable attention has been devoted to the determination of optimal monetary policy under sticky prices. See Benigno and Benigno (2003), Devereux and Engel (2003), Obstfeld and Rogoff (2002), among many other papers. But most of this literature is that it focuses on symmetric environments, where all countries are identical in economic structure. 4Bachetta and Van Wincoop (2002) and Kenen (2003) note that the US dollar is used as an invoice currency for the overwhelming majority of US imports, but for other OECD countries, imports are mainly invoiced in foreign currency. Chapter 3. Global Monetary Policy under a Dollar Standard 60 sensitive to exchange rate movements. In this sense, the monetary policy problem will be asymmetric. The optimal monetary policy in the US will reflect the fact that there is little pass-through from exchange rates to prices of consumer goods in the US economy. On the other hand, for other countries, the optimal monetary policy will take account of high pass-through from exchange rates to prices 5 . How does the asymmetry in international export good pricing affect the optimal monetary policy outcome? We show that, at one level, the monetary policy context and outcomes implied by the model are quite closely in accord with popular wisdom about the position of the US in the world economy. In particular, the monetary authorities of the reference currency place a very low weight on exchange rate volatility in their monetary policy loss function 6 . By contrast, the monetary authorities of rest of the world will be much more concerned with exchange rate volatility. This seems to well approximate the observed indifference of the US to the exchange rate in monetary policy-making. A second feature of the outcome is that the reference currency country follows a more stable monetary policy than that of the rest of the world. More importantly though, we find that the monetary policy game between the reference currency country and the rest of the world has a key sense in which the reference country acts as a 'hegemon'. A Nash equilibrium of this game is identical to one in which world monetary policy (for both countries) is determined by the preferences of the reference currency country alone. That is, the Nash equilibrium of the asymmetric game is the same as that which would obtain were the reference currency monetary authority to choose both its own and the rest of the world's monetary rules to maximize its own welfare. In this sense, the asymmetry in international pricing gives the reference country a dominant role in 5In reality, there is considerable difference between the pass-through of exchange rate changes to import goods prices and final goods prices. In this paper, we abstract from this difference. In fact, the optimal monetary policy is more focused on the pass-through to final consumer goods. It would be possible to allow for a high rate of exchange rate pass-through into import good prices in combination with low pass-through into consumer good prices, without changing the results of the paper at all. This is shown in Devereux, Engel, and Tille (1999). 6In the model, the loss function is endogenously derived from the nature of the environment facing each country. Chapter 3. Global Monetary Policy under a Dollar Standard 61 international monetary policy determination. World monetary policy is designed according to its preferences, and, even if it could play a more explicitly dominant role (by acting as a 'Stackelberg Leader' in monetary policy determination), it would not wish to deviate from the Nash equilibrium. For the rest of the world however, the outcome is quite different. In general their monetary authorities would wish to alter the determination of both their own and the reference country monetary rules, were they capable of playing a more dominant role. Despite this, there are no gains to international monetary policy coordination. Since the reference country enjoys the best possible outcome in a Nash equilibrium, any alternative monetary policy configuration chosen by a 'world monetary authority' is not incentive compatible, except in the trivial case where all the weight is given to reference country welfare. Hence, our model supports the view that, under the dollar standard in international goods pricing, US monetary policy has a predominant role. A natural question to ask then is how much the US gains from this. How much better off are US residents due to the special place of the dollar in export price setting? 7 The surprising answer is that US residents are not better off, but rather are worse off. Expected utility for residents of the reference currency country, where pass-through from the exchange rate to the CPI is zero, is lower than that of the rest of the world, where there is full pass-through. While this may seem inconsistent with the result that the US determines world monetary policy, the explanation is that the asymmetric pricing means that the welfare outcomes are asymmetric. Even if monetary policy were determined by a world social planner with equal weights on both regions, welfare of the reference country would differ from that of the rest of the world. Why does the dollar standard hurt the US? The reason is that when export pricing is done in terms of the US dollar, it prevents an efficient response of relative prices to underlying real 7 Our model excludes some of the factors which would be important in the overall quantitative accounting of the gains from the dollar standard. In particular, there are no offshore holdings of currency in the model, so there is no seigniorage revenue earned on foreign money holdings. Nevertheless, we can do a welfare analysis by comparing expected utility in the reference currency country and in the rest of the world, because other than the special role of the reference currency, the model is otherwise symmetric. / Chapter 3. Global Monetary Policy under a Dollar Standard 62 shocks. A n efficient monetary policy will generally want to employ both expenditure level (affecting total aggregate demand) and expenditure switching (affecting the relative demand for one country's goods) effects. When import prices do not respond to the exchange rate, monetary policy cannot be used to generate expenditure switching effects. This has a welfare cost for the residents of the reference currency economy. Hence, in our model, the dollar standard is costly for the US economy. Where does the special role of the reference currency come from? There is a considerable literature on the determinants of 'international currency'. A n early contribution by Krugman (1984) argues that there may be multiple equilibria due to 'snowballing effects', whereby if one currency becomes the accepted standard, then all participants in international markets have an incentive to use this currency. On the other hand McKinnon (2003) argues that the special role of the U.S. dollar arose partly from the record of low inflation and stable monetary policy that the U.S. economy followed in the Post WWII period. In a later section of the paper, we extend the model to allow exporting firms the choice of currency in which to set prices, and investigate the conditions under which there is an equilibrium where exporters in both countries will use the currency of a single country for price setting8 Our results suggest that both the Krugman multiple equilibria explanation and the M c K -innon policy-determined explanation are important elements in the selection of a reference currency. We show that, in the equilibrium of the monetary policy game, the reference currency country's monetary authority will follow a more stable (lower variance) monetary policy. As a result, this tends to lock in an equilibrium where exporters in both countries use this currency in which to set prices. But the reason that the reference currency mone-tary authorities follow such a rule is precisely because the currency is used as a reference in international trade pricing. This implies however that there are other equilibria where either another currency will play the role of the reference currency, or no country's currency does, so traded goods pricing is symmetric (either L C P or P C P ) across countries. 8Although many other factors are likely to be important in the acceptability of an international currency, the choice of currency for pricing will remain one important channel. Chapter 3. Global Monetary Policy under a Dollar Standard 63 The paper is structured as follows. The following section develops the main model, which is only a slight extension of Devereux and Engel (2003). Section 3 derives the solution of the model for given monetary policy rules. Section 4 derives the optimal rules in Nash equilibrium of a game between monetary authorities. Section 5 extends the model to allow for the endogenous choice of currency in which to set prices. Section 6 concludes. 3.2 The two-country model We construct a simple two-country model of trade and exchange rate determination. Firms set prices in advance, by assumption 9 . There is a continuum of home goods (and home population) and foreign goods (foreign population) of measure n and (1 — n) respectively. Individual home (or foreign) goods are substitutable in preferences with elasticity A, but there is unit elasticity of substitution across the home and foreign categories of goods. The expected utility of home agents is Cl~p M where C = n " n ( l - n ) - * 1 " " ^ ^ - " , Ch = (£ Ch{i)l-Xdi)^, and p > 1. Here C is aggregate consumption, Ch is consumption of the home sub-aggregate, yr are real money balances, with P — PfiPJ~n being the home CPI, and L is the home labor supply. There is only a single period in which events take place 1 0 . The structure of events within the period is as follows. First, before the period begins, households can trade in a full set of nominal state-contingent bonds. This means that house-holds can offset any risk that is associated with monetary policy uncertainty, as well as risk 9Since the model has been well covered in previous papers, here we will only briefly sketch out its main elements. 1 0This may seem to be an extreme assumption, but in fact it is entirely innocuous, given the asset market structure. Extensions to an infinite horizon are quite trivial, and since there exists markets for risk-sharing across countries, this would leave all the results unchanged. Just so as to avoid time subscripts in the notation, we focus on a one-period problem. Chapter 3. Global Monetary Policy under a Dollar Standard 64 due to country-specific productivity shocks (see below). The outcome of this stage is that households will enter the period with their revenue stream governed by an optimal risk shar-ing rule. Then the monetary authorities choose optimal monetary rules, given the optimal risk sharing rule, but taking into account the way in which firms set prices, as well as the distribution of country-specific technology shocks. Following this, firms set prices in advance, contingent on state-contingent discount factors, and the demand and marginal conditions that they anticipate will hold. After the realization of stochastic technology shocks, house-holds choose their optimal consumption baskets, production and consumption takes place, and the exchange rate is determined. Trade in state-contingent nominal assets across countries will lead to the following optimal risk sharing arrangement: TPC = SP*C*P, (3.2.1) where S is the nominal exchange rate, and P* = p* n p*( 1 _ n ) i s the foreign price level n . Optimal financial markets lead to the equalization of the marginal utility of money across countries, up to a state-invariant weighting T. If the countries were entirely ex-ante identical, then obviously T would equal unity. But given the differences in pricing policies, countries are not necessarily the same, ex-ante. In this case, T will be chosen so as to reflect that different positions of the two countries in the initial competitive market in state-contingent assets. Given the structure of preferences, we can show that the that the value of T will be 12 In addition, household optimization gives rise to the money demand rules: M = XPCP, (3.2.3) 1 1 This condition says that optimal risk sharing will equate the marginal utilities of money across countries in each state of the world. The condition is a familiar one - see for instance Chari et al. (2002). For a rigorous proof of this condition, see Devereux and Engel (2003). 1 2For a proof, again see the appendix of Devereux and Engel (2003). Chapter 3. Global Monetary Policy under a Dollar Standard 65 and the implicit labor supply conditions given by W = nPC". (3.2.4) Since monetary policy is determined after financial markets have closed, the monetary authorities take T as given in their evaluation. We delay the discussion of optimal monetary rules until the next section. Firms face demand for their good from consumers in both their domestic country and abroad. Firms have linear technologies, producing output from labor alone, but are subject to unpredictable (at the time of price setting) technology shocks in production. Firms can price-discriminate across national markets, and households have no ability to re-sell goods across countries. In addition, there is an asymmetric pricing structure. Home firms set prices for both the home market and the foreign market in terms of the home currency. But foreign firms set prices for export in terms of the home country currency. Hence, the foreign firms engage in LCP when selling abroad, whereas the home firms follow PCP. In this sense, the home currency is the 'reference currency' in all international trade, because all international traded goods have their prices set within this currency. The Appendix C . l outlines the details of the optimal pricing policies of firms. The following equations give the prices set by the representative home and foreign firm for the goods sold in home and foreign markets, respectively; ptW*SCx-f\ P* = A 6: J. (3.2.8) Chapter 3. Global Monetary Policy under a Dollar Standard 66 In these equations, A represents the markup subscript h, f represents the price of the home good in the foreign market etc, and 6 represents the home country productivity co-efficient. These equations indicate that optimal prices depend on the joint distribution of marginal cost (^-), the exchange rate, and consumption (or aggregate demand). We assume that 6 can be represented as 0 = exp(u), (3.2.9) where u is mean zero and normally distributed. A similar assumption is made with respect to the foreign productivity shock. A n asterisk over the price means that the price is denominated in foreign currency. Hence, all home goods prices are denominated in home currency, while only foreign goods sold in foreign markets are denominated in the foreign currency. Given this convention, then the price indices for each country are as follows: P = P f t P j r (3-2.10) n P}fl~n. (3.2.11) The set of equations given by (3.2.1) and (3.2.2), in combination with (3.2.3) and (3.2.4) (with analogous conditions for the foreign economy), the pricing equations (3.2.5)- (3.2.8), and the price indices (3.2.10) and (3.2.11) give 12 equations that may be solved for the distribution of the variables C, C*, W, W*,P, P*, Phh, Phf, Pfh, P*ff, S, and T. 3.3 Solving the Model Because the model is log-linear and the underlying technology shocks are log-normal, we may solve for the exact distribution of air endogenous variables in closed form (the details are in the Appendix C.2). The solution allows a dichotomy between variables that are determined in advance of the realization of technology shocks, i.e. Phh, Phf, Pfh, P)f, and T, and variables determined after the shocks have occurred; i.e. C,C*,W,W*, and S. Chapter 3. Global Monetary Policy under a Dollar Standard 67 The risk-sharing condition (3.2.1), in combination with the money demand equation (3.2.3) and the analogous condition for the foreign country implies a solution for the exchange rate: s = r ^ . (3.3.1) Substituting this solution back into the money market clearing conditions then implies that .1 M a * rhhrfh C=[-7^rrd> (3-3.2) C* = [- 1V± . (3.3.3) 1,. n n 7~»* 1—n J v ' .1 MnM*^~n\x Lv p n p* 1 X ^hf^ff This implies that home country consumption is independent of the realization of the foreign country money supply. This follows directly from the fact that the home country CPI is predetermined, given that both home goods and imported goods in the home market have prices pre-set in home currency. But with full exchange rate pass-through into foreign imported goods, foreign country consumption is affected by home country monetary shocks. In log terms, we may write these equations as s-E(s) = [m - E(m)] - [m* - E(m*)} (3.3.4) c-E{c) = -p[m-E{m)] (3.3.5) c*-E(c*) = -{n[m-E(m)} + (l-n)[m* - E(m*)}} (3.3.6) P where E denotes the mathematical expectation, and small-case letters denote logarithms. Equations (3.3.4) and (3.3.6) can be solved for the variance of the exchange rate and consumption. But first we need to set out the monetary policy rules. We make the following assumption regarding the determination of monetary policies: m = mo + a\u + C12U* (3.3.7) m* =m*Q + blU + b2u* (3.3.8) Thus, the money supply is a log linear function of the shocks in each country, where the parameters of the rules, a\,a2, and 61,621 have yet to be determined. These rules are Chapter 3. Global Monetary Policy under a Dollar Standard 68 perfectly general, because given that the model is log linear, and the shocks log-normal, the optimal form of monetary rules must be log-linear. Monetary policy will be chosen to maximize expected utility for each country. In order to evaluate expected utility, it is necessary to determine expected consumption and employment. These will be affected by the stochastic structure of the model, given ex-ante optimal price setting. Using (3.2.5)- (3.2.8), along with the labor supply equations, and the risk sharing condition (3.2.1), we may set out the following two conditions which implicitly determine the mean values of C and C*. 1 = ^ r l " " [ g ( f ™ f ) ) 1 1 " n ( 3 - 3 - 9 ) 1 = A 7 ? r - [ E ( ^ ) i ? ~ ^ ) ] (3.3.10) E(C* p) Using the properties of the log-normal distribution, we may re-write Equations (3.3.9) and (3.3.10) in terms of the mean and variances of log consumption and exchange rates. This gives E(c) = - i l m T 1 - ^ ) - 2-^ol - + + + p 2 2p p E(c*) = - i ln(r-"A„) - ^ 4 ~ *^<Z ~ n<jl + (' " n)al' p 2 2p 2p | no-c*u + (1 - n)o-c.u. + n(l - n)(osu - osu.) 3 P P Mean (log) consumption of the home country is determined only by home consumption variance, the variance of technology shocks, and the covariance of consumption with tech-nology shocks. Equations (3.3.5) and (3.3.11) imply that both the mean and variance of home consumption is independent of foreign monetary policy. On the other hand, mean consumption of the foreign country depends both on consumption variance, the covariance of consumption with technology, and on exchange rate variance and covariance with technology shocks. Why is it that exchange rate volatility affects expected foreign consumption, but not Chapter 3. Global Monetary Policy under a Dollar Standard 69 home consumption? This is because exchange rate volatility affects foreign import prices, and through this, the average level of pre-set prices. It will therefore affect mean consumption in the foreign country. Note that from (3.3.6) and (3.3.12), foreign consumption will clearly be influenced by both home and foreign monetary policy rules. Since mean consumption depends on the variance and covariance properties of consump-tion and the exchange rate, we can derive a welfare measure for policy makers solely based on these second moments. Assume that monetary authorities in each country are concerned with the expected utility of consumption and dis-utility of labor supply, but ignore the utility of real money balances 1 3 . Thus, the home country monetary authority chooses its monetary rules to maximize E(jz~p - v L ) - (3-3-13) From the properties of the price setting equations in the home and foreign countries, and the labor market clearing condition, we can establish that E(L) = ^E{Cl-p) + ^^EiC*1-"^ (3.3.14) Ary Xn Combining (3.3.13) and ( 3.3.14), we may write expected home country utility as E(U) = A - ^ A - I K I - P ) ^ - , ) _ (l-n)(X-l)mctl_P) ( 3 3 1 5 ) (1 — pjA A Since the log-normal distribution satisfies ECl~p = exp |(1 - p)[E(c) + ^ u 2 ] } , (3.3.15) ultimately depends only on the second moments of consumption and the exchange rate. These in turn depend on the monetary rules (3.3.7)-(3.3.8). Flexible Price Equil ibrium It is useful to show the allocation that would obtain in an economy with fully flexible prices. If all prices could respond to the ex post value of technology shocks, then money would be neutral. The asymmetry in pricing would be irrelevant, because from (3.2.5)-(3.2.8), with ex-post price setting, the law of one price would hold across markets. Consumption 13Obstfeld and Rogoff (2002) give a justification for this assumption. Chapter 3. Global Monetary Policy under a Dollar Standard 70 and employment would be equalized across countries. The expressions for consumption and employment in the flexible price equilibrium are: c = c* = (\r})--p{ene*1-nyp. (3.3.16) L = L* = {\ny-'{ene*l-n)1-if. (3.3.17) Productivity shocks affect consumption in each country in proportion to country size, and reduce (increase) employment in each country as p > 1 (p < 1). 3.4 O p t i m a l M o n e t a r y P o l i c y We now examine the optimal monetary rules chosen by independent monetary authorities in each country. Monetary policy is chosen with commitment, in the sense that monetary authorities choose the parameters of a monetary rule to maximize expected utility of the domestic agent, taking into account the way in which prices are set. A natural objective of the monetary authorities would be to design optimal monetary policy so that the economy replicates the flexible price allocation 1 4 . But given the way in which prices are set, this is not possible. We show this in the following proposition. Proposition 1 No feasible monetary rule can replicate the flexible price world allocation. Proof: From (3.3.16) and (3.3.17), in order to achieve the flexible price response of con-sumption, the home country must follow a monetary rule in which a\ — n and 0 2 = 1 — n. But if the home country follows this rule, then the foreign country must follow the same rule, 1 4While we might anticipate that there would also be a strategic externality by which each country would attempt to use monetary policy to improve its terms of trade vis a vis the other country, this does not arise here because monetary rules are determined ex-ante, before private sector prices have been set. The strategic externality by which monetary policy may improve one country's welfare by influencing the terms of trade can only be effective for a surprise monetary shock, taking private sector expectations (i.e. prices) as given. For the same reason, there is no incentive to pursue inflation surprises in this model, because the monetary authorities follow rules with commitment. These issues are discussed in Obstfeld and Rogoff (2002). Chapter 3. Global Monetary Policy under a Dollar Standard 71 if it wishes to achieve the flexible price response of foreign consumption (see (3.3.6)). But in this case, neither country achieves the flexible price equilibrium response of employment. To see this, note that employment is determined by P 9P* 6L = n—C + (l-n)—C* (3.4.1) E"hh rhf When the two countries monetary rules follow the conjectured policy, the exchange rate (3.3.4) is constant. Then the right hand side of (3.4.1) is proportional to (en6*1~n)p. But then L cannot satisfy (3.3.17). A n equivalent argument holds for L*. Monetary authorities choose their optimal rules to maximize expected utility, taking into account the determination of prices, consumption, employment, and exchange rates, but taking as given the coefficient of optimal risk-sharing r 1 5 . In order to define an equilibrium of the monetary policy game between countries, it is convenient to reformulate the objective functions (3.3.15) in the following way. Define expected utility in the home country as: EU(a, b) = cpnT^X - ^ ~n^TX* (3.4.2) A Likewise, expected utility for the foreign country monetary authority is EU*(a, b) = fa-nX* - " r - ^ ^ X (3.4.3) A where X and X* are defined as X = eexp[(l-p)(-\ol-fp + ^)} (3.4.4) v* rs \ti v 1 2 n ( l - n ) 2 <?l CTC~.U (o-su - CTSU. ) . . X = e e x p [ ( l - p ) ( - - < T ^ — — — + — + n ( l - n ) ], (3.4.5) 4>n, <Pi-n, a n d 0 are constant functions of parameters 1 6, and CT2 = no\ + (1 — n)o2., o~cu — no-cu + (1 - n)cr c u » , Oc-u = noc*u + (1 - n)oc*u*. 15An alternative possibility is to have monetary policy rules chosen before the ex ante asset market trading. We did solve the model in that case. Although the actual form of the solutions are altered, the qualitative results (in particular the asymmetries) do not differ from those presented below. 1 6In particular, <j>n = ^  - 5 , 0 i - „ = ^  - i f , and 0 = exp ln(Arj)] Chapter 3. Global Monetary Policy under a Dollar Standard 72 Although home consumption is independent of the foreign monetary rule (shown above), its welfare does depend on the foreign monetary rule, because expected home country em-ployment is affected by foreign monetary policy. Thus, the home country is not indifferent to the rule followed by the foreign monetary authority. A special case of (3.4.2) and (3.4.3) arises when p = 1. Then expected employment is constant in both countries (see (3.3.14)). Therefore, the monetary authorities are concerned solely with maximizing expected utility of their own consumption. For the home country, this is equivalent to using monetary rules to maximize: ( - ^ c 2 - y + ^ ) . (3.4.6) By contrast, for the foreign country, in the case p=l, the relevant objective function is (-\ac* ~ U<"l~n^2s - y + <Vu + n(l - n){asu - asu.). (3.4.7) Then home utility depends (negatively) on consumption variance, but is increasing in the covariance of consumption and productivity shocks. An optimal monetary rule will trade off these costs and benefits, making consumption positively co-vary with u and u*. Because there is no pass-through into the home economy, its monetary authority is indifferent to exchange rate variance. For the foreign country, exchange rate variance does have welfare consequences. Exchange rate variance reduces foreign utility 1 7 '. But positive covariance of the exchange rate and home productivity shocks, or a negative covariance with foreign productivity, raises foreign utility 1 8. An optimal monetary rule for the foreign country therefore has to take account of effects on both consumption and the exchange rate. 1 7This occurs because exchange rate variance raises the mean foreign price level, for any expected value of the money stock, and hence reduces expected foreign consumption. 1 8The intuition behind this is that the exchange rate generates an expenditure switching effect in the foreign economy. A positive home (foreign) technology shock requires a depreciation (appreciation) in the exchange rate in order to increase foreign country demand for home (foreign) goods. The equivalent channel does not work in the home country, because there is no expenditure switching at the consumer level. Chapter 3. Global Monetary Policy under a Dollar Standard 73 Proposition 2 When p = 1, the home country is indifferent to exchange rate volatility, while the foreign country places a negative weight on exchange rate volatility. Proof: See above discussion In the more general case with p > 1, the home country is no longer completely indifferent to exchange rate variability. But for all reasonable parameter values and shock distributions, the home country places less weight on exchange rate variability than does the foreign country. Table C.2 illustrates the impact of exchange rate volatility on expected utility, for each country, for various values of p, and a given calibration of other parameters. In all cases, the home country is less affected by movements in exchange rate variance. A Nash equilibrium in the monetary game between countries is defined in the standard way, as the pair a n , 6™ which solves: m&xEU(a,bn) (3.4.8) a maxEU*(an,b) (3.4.9) b The simple form of the model in fact allows us to solve for the exact solutions to the monetary policy rules. By observing (3.4.2) and (3.4.3), we notice an important asymmetry in the monetary policy game. The home country rules a\, a% affect the composite expression X*, defined above, but the foreign country monetary rules 61 ,62; do not affect X. Moreover, note that X* enters linearly, with a negative sign, in both the home and foreign country's objective functions. Hence, in choosing its monetary rules, the foreign monetary authority indirectly chooses to maximize home expected utility. The first order conditions characterizing the Nash equilibrium can be written as (for both ai ,c i2 and 61,62 respectively): T T - 0 < 3 - 4 ' u > Chapter 3. Global Monetary Policy under a Dollar Standard 74 Using the property of optimal risk sharing from equation (3.2.1), we may establish that where the second equality follows from the definition of X and X*. Now substituting into the first order conditions (3.4.10) and (3.4.11), we arrive at the characterization of the optimal monetary reaction functions for each country: Sen = n JA - (1 - p) [n + (1 - n)[n + p(l - n)]]} + n[(l - n)(l - p)}2h (3.4.13) 6a2 = (1 - n)[A - n ( l - p)] + n[(l - n)(l - p)]2(62 - 1) (3.4.14) 6 i = "I'M 1 } > i - 1) (3-4-15) pn + (1 - n) = n ( p - l ) ( o 3 + 1)4-1 pn + ( l - n ) v ; where 5 = A - n ( l - p){l + (1 - n)[p(l - n) + n]} Equations ( 3.4.13) and ( 3.4.14) describe the home country's first order conditions, while (3.4.15) and (3.4.16) describe the foreign country's conditions. The solution to (3.4.13)-(3.4.16) is a Nash equilibrium in the monetary rules. Table C . l describes the solution. From the Table, we may establish that a) n < a\ < 1, 0 < ci2 < (1 — n), b) bi < 0, 62 > 1, and c) ai + 0,2 = 1, 61 + 62 = 1-In the special case with p = 1, we have ai = n, a2 — 1 — n, and 61 = 0,62 = 1- Thus, the home country adjusts monetary policy to both the home and foreign shocks according to their weight in world G D P , and the foreign country focuses only on its own domestic shock. Note that in this case, given our assumption that u and u* are i.i.d., it follows the home country monetary variance is lower than that of the foreign country. In addition we note that the variance of the exchange rate is lower than would occur were there to be no world reference Chapter 3. Global Monetary Policy under a Dollar Standard 75 currency. Given the Nash equilibrium described above, exchange rate variance is 2n2o-\. It is easy to show that, if exchange rate pass-through into both home and foreign countries was complete, then the Nash equilibrium would be a\ = l,a2 = 0, and b\ =0 ,62 = 1. In that case, exchange rate variance would be 2a2. Table C . l shows the solution for ai,a2 and 61,62 in the more general case where p > 1. Figure C . l and Figure C.2 illustrate the reaction curves, for the responses to both shocks. Each is upward sloping. The same general properties of the solution described above still apply. We now focus on the welfare outcomes of the monetary policy game. Using the solu-tions of Table C . l , and the description of the Nash equilibrium, we now state the following proposition: Proposition 3 A Nash equilibrium is identical to an outcome where the home economy determines world monetary policy rules. The proposition says that a Nash equilibrium is asymmetric, in the sense that it gives the same allocation as if the home economy was choosing both its own and the foreign econ-omy's monetary rules. Equivalently, in the Nash equilibrium, the foreign economy indirectly maximizes expected home country utility as well as its own utility. The proof of the proposition is straightforward. From the objective function (3.4.2), note that X is independent of 61 and 62, and home expected utility is linear in X*. Since, in a Nash equilibrium, the foreign monetary authority chooses 61 and 62 to maximize a linear function of X*, it's choice is also the optimal choice of 61 and 62 for the home economy. Note that the proposition specifically does not hold in the reverse direction. The Nash allocations for a\ and a2 do not maximize foreign country welfare. Hence, the foreign country experiences negative welfare externalities in a Nash equilibrium. The key ingredient in this asymmetry is that home consumption is independent of foreign monetary rules. As a result, the foreign monetary policy influences home utility only to the extent that it influences expected employment in the home country. Since the monetary rules 61 and 62 that maximize foreign utility are identical to those which minimize expected home Chapter 3. Global Monetary Policy under a Dollar Standard 76 employment, these rules are then the optimal rules from both the home and foreign country perspective. This also means that the monetary rules governing the world economy are identical to those that would hold were the home country a 'Stackelberg Leader', choosing its monetary policy in advance, taking account of the reaction of the foreign country. Figures C . l and C.2 illustrate the equilibrium in terms of the reaction curves (3.4.13)-(3.4.16) for both a\,bi and 02,62, respectively. For p > 1, the home country's reaction curve slopes upward in both Figures. The foreign country's reaction curve is also upward sloping, and steeper than that of the home country. Point N represents the Nash equilibrium. Since N is a global optimum for the home country, its iso-utility lines can be illustrated as converging to a maximum at N. For the foreign economy, point F represents the global optimum, equivalent to the allocation that would obtain if the foreign monetary authority could choose both home and foreign monetary policy rules. The foreign iso-utility lines converge to a maximum at F . Utility of the foreign country at N is less than at F . In Figure C . l , relative to the Nash equilibrium, the foreign country would like to increase 61 towards zero, and increase a\. Hence, it would like the home country to react more to its own shock, and for itself to react less (in absolute terms) to the home shock. If the foreign economy could act as a Stackelberg leader, it would choose point S. Figure C.2 illustrates the determination of a2,b2. Again, the points N and F represent the global optimal allocations for the home and foreign country respectively, with the first being the Nash equilibrium. If the foreign economy could act as a Stackelberg leader, it would reduce the weight placed on its own productivity shock in its monetary rule, which would have the effect of reducing the home country a2 coefficient. A corollary of the proposition is that there is no gain from international monetary policy coordination, except in the trivial case where the social welfare function used for coordination places all weight on the home economy welfare. The Nash equilibrium is therefore efficient. While the foreign economy could gain from an allocation chosen by an equal weighting world monetary authority, the home economy would lose. In Figures C . l and C.2 we could illustrate Chapter 3. Global Monetary Policy under a Dollar Standard 77 a contract curve, or set of efficient monetary coefficients, indexed by different weights on home and foreign welfare in evaluating the world optimum. The Nash equilibrium is at one end of this curve, where the weight on home utility is one. This equilibrium of the monetary policy game has many features that seem to resemble the description of US monetary policy under the de facto world 'dollar standard'. It is widely acknowledged that the US pays little attention to the exchange rate in its monetary policy. But compared to many other countries, the US follows a more stable path of monetary policy. More importantly, the US does have an advantage over the rest of the world in setting monetary policy, due to the special role of the dollar. In our model, this advantage is quite extreme in the sense that world monetary policy completely reflects US preferences. The situation has some parallels in the historical literature on 'Hegemonic Stability' of the International Monetary System (see Eichengreen (1995) for instance). While this theory was specifically designed to interpret the stability of fixed exchange rate systems, in this model, the US acts as a hegemon, but within a decentralized world Nash equilibrium with flexible exchange rates. Nevertheless, just as in the traditional theory, our model predicts that the hegemon has no interest in international consultation in monetary policy making, and would not support a move towards international monetary cooperation. Much popular discussion of the role of the US dollar in the world economy goes even further than this. It is frequently asserted that US residents gain from the role of the US dollar as the international reference currency. There is a wide range of popular explanations for how these gains might come about - some argue that the role of dollar allows the US to run current account deficits without limit, that it gives it the ability to dictate world monetary policy, or that it allows it to dominate the world oil market. By contrast, most economists (e.g. Krugman (1999)) estimate that the gains that the US gets from the dominance of the dollar are modest, mainly accounted for by seigniorage revenue on offshore dollar holdings, and are a very small percentage of total US fiscal revenue. We now address the question of the welfare gains to a reference currency. In our model, there are no offshore currency holdings, so the primary source of benefit is not present. Chapter 3. Global Monetary Policy under a Dollar Standard 78 Nevertheless, the fact that the outcome of the monetary policy game is asymmetric means that welfare levels are different for the home and foreign countries. Since the rest of the model is perfectly symmetric, the difference in welfare gives an exact measure of the gains from having an international currency. It might be thought that this question has already been answered by Proposition 2. The role of the reference currency leads the home country to be placed on one end of the utility contract curve. It would then seem that the reference currency country is always better off. But this conclusion is incorrect. Since the game itself is asymmetric, welfare levels would differ even if each country's preference were given equal weight in world monetary policy making. In order to assess the gains to having a dominant currency, we must compare levels of expected utility between the home and foreign countries. This comparison gives a surprising result. Proposition 4 In a Nash equilibrium, expected utility for the home country is always lower than that of the foreign country. Proof: See Appendix C.3. Although the home country's preferences dominate world monetary policy making, home country residents are actually worse off than those of the foreign country, in the equilibrium of the monetary policy game. The ownership of an international reference currency bestows costs rather than benefits - residents of the reference country have lower expected utility. Although the result may seem surprising, the explanation is quite intuitive. The absence of exchange rate pass-through into the home economy inhibits the usefulness of monetary policy. As is described above, an ideal monetary policy rule is one which achieves both expenditure level effects and expenditure switching effects. The foreign country can use both channels in designing a monetary rule, because the exchange rate affects relative prices. The home country can't do this - since the exchange rate does not affect the demand for home goods relative to foreign goods, monetary policy can affect only the level of spending. Given the absence of pass-through into the home economy, home output is not adjusted efficiently Chapter 3. Global Monetary Policy under a Dollar Standard 79 to home and foreign technology shocks. As a result, expected utility is lower than that of the foreign country, where output can be affected by the exchange rate. 3.5 E n d o g e n o u s C u r r e n c y P r i c i n g So far it has just been assumed that the home currency is used as a reference for interna-tional pricing. In principle, this decision should be endogenous. The set of forces leading to the adoption of an international 'vehicle' currency have been discussed extensively in the literature on international monetary economics (see Matsuyama et al. (1991) McKinnon (2002), Krugman (1984), Rey (2001)). Many factors, such as economic size, history, capital flows, and economic policy may be part of the explanation. Moreover, the presence of 'net-work externalities' in the choice of standard may give rise to multiple equilibria. Krugman (1984) notes that while economic size is likely to be an important factor, there may also be a 'snowballing' effect, whereby even if countries are of similar size, if one currency becomes ac-ceptable in exchange then all countries will have an incentive to support this outcome 1 9. This suggests that the US dollar standard may be due to historical accident as much as current fundamentals. McKinnon (2002) takes a different view however. He stresses the importance of US monetary policy, arguing that the US dollar's role as a world currency resulted from low and stable US inflation rates in the post-WWII international system. In this section, we present a brief analysis of the determination of the reference currency for international trade. We illustrate the sense in which the asymmetric pricing outcome examined in the previous section can be an equilibrium of the model where the currency of pricing is endogenous20. We assume that firms can choose which currency they would like to set their price in. They do this taking into account that whatever their choice, they will then choose a nominal 19Economic size does not play any significant role in our model, because a) there are no non-traded goods, so all countries are fully open, and b) each country produces a measure of goods equal to its population, so the terms of trade is independent of size. 2 0Our analysis doubtless omits many important factors that determine the role of an international currency, but highlights one potentially important factor, within the context of this model. Chapter 3. Global Monetary Policy under a Dollar Standard 80 price to maximize expected discounted profits. In addition to this however, the firm incurs a cost of adjusting prices, ex-post. We assume that these costs arise only when the price facing consumers is adjusted. We might think of these as menu-changing costs, or customer resistance costs, that require management services on the part of the firm. If the firm sets the price in the local currency of the buyer, it will never face these costs, as the price will be independent of the state of the world. But if the price is set in the exporting firms' own currency, prices facing the foreign consumer will be dependent upon the exchange rate. We handle this in the following simple way. Assume that if the firm sets prices in the consumers currency (LCP), then it faces no additional cost. But if it sets prices in its own currency, then it faces a fixed (nominal) cost given by S21. This is thought of as a cost of ex-post adjustment that comes from the exchange rate pass-through into the importing countries CPI. The presence of this fixed cost per se will therefore encourage the firm to set prices in the currency of the consumer (LCP). On the other hand, the level of expected (discounted) profits, gross of fixed costs, will depend upon whether prices are pre-set in the producers currency or consumers currency. Using the same demand and cost structure from the model set out above, we may define the expected discounted profits on foreign sales for a home firm that sets its export price in terms of its own currency (PCP) as E[dn(i)pcp] = E[d(Phf(i) - ^)X*hPCP(i))} (3.5.1) where Xf^p(i) ^ (^y^C*. If the firm chooses alternatively to set its price in terms of foreign currency (LCP), it faces expected discounted profits given by E[d*(i)LCP] = E[d{SP*hf(i) - ^)X*hLCP(i))} (3.5.2) 2 1 We think of this being part of the technology. That is, there is a technology whereby firms who wish to set export prices in their own domestic currency must incur a fixed management cost <5. Competitive 'managerial' firms provide this management services by combining home and foreign varieties in the same manner that consumers do Chapter 3. Global Monetary Policy under a Dollar Standard 81 where X^{i) = {^)~X^Cr. The Home country firm will set its price in its own currency if the expected profit differ-ential from doing so exceeds the expected menu cost. Thus it follows PCP whenever E[dir(i)PCP] - E\dTt{i)Lvy] > 6 LCP] (3.5.3) The sequence of actions within a period is now described as follows. First, firms choose the currency in which prices are set. Following this, the monetary authorities in each coun-try choose their optimal rules. Then firms choose the actual prices of goods. Finally, the technology shocks are realized, and consumption, output and exchange rates are determined. In Devereux, Engel and Storgaard (2003), it is shown that the left hand side of (3.5.3) may be approximated by the following dffA(A - 1) VarjlnS) W ' Cov(ln—,lnS) (3.5.4) where d and ff denote the discount factor and profits in a deterministic economy, respectively. The intuition behind this condition is straightforward. Since profits are convex (linear) in the exchange rate when the firm following PCP (LCP), a higher exchange rate variance will encourage the firm to follow PCP. But if the covariance of the exchange rate and marginal cost ^ is positive, expected costs will be higher under PCP. If the right hand side of (3.5.4) is positive, the firm would wish to set prices in its own currency (PCP), in the absence of menu costs of price change. Thus, the condition (3.5.3) becomes VarQnS) _ W Cov(\n—,\nS) A ( A - l ) The equivalent condition for the foreign firm is A(A - 1) Var(\nS) _ „ W* . _,' 2 L+Cou(]n— , l n S ) dm > dn' (3.5.5) (3.5.6) where, to maintain symmetry, we assume that the fixed cost facing the foreign firm is identical to that of the home firm. If condition (3.5.5) (condition (3.5.6)) is not satisfied, then the home firm (foreign firm) will instead set prices according to LCP. If we define Z = A(A-l)Jff j ^ a r 2 l n g ) — Cov(ln ^ , In S) Chapter 3. Global Monetary Policy under a Dollar Standard 82 Z_ = max jo, A(A — l)dir VarQns^ + Cov(ln ^ r , In 5)j | , from these conditions, we can estab-lish the following proposition. Proposition 5 There exists a positive menu cost 6 € (Z_, Z) such that all home firms follow PCP, and all foreign firms follow LCP. Proof: See Appendix C.4. The proposition says that there exists a positive menu cost 5 such that the asymmetric pricing structure outlined in the previous section is an equilibrium. In this equilibrium, the home firm will choose PCP, while the foreign firm will choose LCP. Following this, the monetary authorities choose their optimal rules in exactly the same way described in the previous section. The key intuition is that the way in which the monetary rules are set acts so as to lock in the asymmetric pricing policies of home and foreign firms. In an equilibrium of the monetary policy game outlined in Section 3.4, the home country's monetary policy rule tends to target both home and foreign productivity shocks. Since they are independent of each other, the home country's money supply is less volatile than that of the foreign country. As a result, Ccw(ln^, lnS) is less than — Ccw(ln ln5), because in the simple form of the model, the variability of the wage rate is completely determined by the variance of the domestic money stock. Thus, for relatively small menu costs of price change, it is more likely that the home firm will wish to set its price in its own currency, while the foreign firm will wish to set its price in the home currency. This is a variant of the result of Devereux, Engel, and Storgaard (2003), which shows that firms would wish to set their export prices in the currency of the country which had the lowest variance of money growth. While the proposition establishes that the outcome of the previous section is an equilib-rium of a game in which firms choose the currency of pricing, in general there will be other equilibria of this game. For instance, if all firms choose LCP, then from the results of Dev-ereux and Engel (2003), and Devereux, Engel and Storgaard (2003), the optimal monetary rules chosen by home and foreign countries will in fact support global LCP as an outcome. Likewise, because the model is entirely symmetric, if Proposition 5 holds, then there must be Chapter 3. Global Monetary Policy under a Dollar Standard 83 an alternative equilibrium where, if foreign firms follow PCP, and home firms choose LCP, this is supported as an equilibrium in the monetary policy game. These results suggest that both the Krugman (1984) multiple equilibrium hypothesis, and the McKinnon (2002) fundamentals hypothesis, may be part of the explanation for the dollar standard. Given the presence of asymmetric pricing, the endogenous decisions of monetary authorities respond in a certain way so as to confirm the pricing decisions of firms. Nevertheless, there may be other equilibria which would also be self-confirming in the sense that they would induce different monetary policy rules. 3.6 C o n c l u s i o n s An almost universal characteristic of the international monetary system is the role of a dominant currency. In the classical theory of 'hegemonic stability', the economic policies of the dominant currency determine the stability of the international system, typically by adherence to the 'rules of the game' in a fixed exchange rate system. But in the decades since floating exchange rates, the US dollar has remained a pre-eminent currency in international trade and finance - leading to a de facto dollar standard. 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"Can Vertical Specialization Explain the Growth of World Trade?", Jour-nal of Political Economy, February, 52-102. 89 Appendix A Appendices of Chapter 1 A. l Optimal Pricing Schedule of Firms A l l goods are imperfect substitutes in consumption, so each ind iv idual f irm has some market power determined by the parameter 6. T a k i n g prices for all indiv idual goods as given, the opt imal demand function of the consumer for each indiv idual good can be derived, which implies that in each period the consumer allocates a given level of total consumpt ion among the differentiated goods. c « ( , ) ="GSfey(£T c ' c ' < w - ( i - ) G S w ) ' ( £ ) , f t (AL I ) T h e price setting problem of monopolist i is to maximize expected profit condit ional on t — l information, by choosing Ph,t(i) and P^t(i). T h a t is, f irm i solves max Et.! {Dt[Phtt(i)Ch,t(i) + StP^t(i)C^t(i) - WtLt}} (A.1.2) Ph,t(l)iph,t(l> subject to = Ch,t(t) + C*htt(i) (A.1.3) and the downward-sloping demand functions for Ch,t(i) and C^t(i), as in E q u a t i o n A.1 .1 and the foreign analogue. Note that Ph,t(i) and P^ii) are denoted in the home and foreign currency, respectively. Us ing the fact that all prices are preset at t ime t — l a n d apply ing symmetry, we can derive the opt imal pric ing schedule of firms. A.2 Entry Condition of Noise Traders T h i s appendix derives the entry condit ion (Equat ion 1.2.27) for noise traders to enter the foreign bond market. T h e noise trader i will enter the foreign b o n d market if a n d only if E q u a t i o n 1.2.14 holds. If trader i does not enter the market, his expected uti l i ty is given by: EM | <p\ = 0) = 0 (A.2.1) Appendix A. Appendices of Chapter 1 90 While if he enters, it is given by: E\(U\ | ^ = 1) = El max El B*htt+1(i)St(l + rt+1) t+i Pt+i ~ (k 2 1 B*ht+1(i)St(l + rt+1) n+i Substituting the optimal demand for foreign bonds of noise traders, ( + 1 ( i ) = —sr into the above equation, we may establish that Equation A.2.2 is equivalent to: (A.2.2) Pt+i EtN\Pt+i] p^-j- ( l+r t +1) Van [pt+i ] El Er[pt+i]pt+i I aVart[pt+i] Ci Et\pt+i}pt+i aVart[pt+i} > 0 (A.2.3) By the property of noise traders' subjective expectation of pt+i, Equation A.2.3 can be rewritten as: ft (EtN[Pt+i})2 or: aVart[pt+i - Ci {E?[Pt+i]Y 2a(Vart[pt+i})2 [EtN(pt+i)}2 Vart[pt+i] > 0 Ci < 2aVart(pt+i) (A.2.4) (A.2.5) A.3 A Symmetric Steady State In a non-stochastic steady state, all shocks are equal to zero. Hereafter, steady state values are marked by overbars. As the consumption is constant at the steady state, the steady state world interest rate r is tied down by the intertemporal optimality equation (Equation 1.2.7): 1- /3 (A.3.1) From the pricing equation, at the steady state, all the prices are equal and steady state exchange rate 5 = 1 . Then the steady state excess return p = Sg^f^ —1 = 0. From Equation 1.2.24, we will have 1 B*(i) = o V i e [0,1] (A.3.2) The economic intuition behind A.3.2 is that traders are not going to hold foreign bonds because they know that the excess return will be zero, and thus no trade takes place. The only way that no trade will occur in equilibrium is for the uncovered interest parity to hold. 1 N o t e that since vt = 0, only rat ional traders are present on the market. Appendix A. Appendices of Chapter 1 91 Therefore, from the bond market clearing condition, net foreign assets are zero. B = B* = 0 (A.3.3) The steady state values of other variables are straight forward. Since B = 0, a closed-form solution exists for the steady state, in which the countries have identical outputs, consumption and real money holdings: L = L* = C = C* — (Jzj) (A.3.4) p p* \1-PJ AA Model Solution The full model can be described by the 26 equilibrium conditions listed in Section 2.3. To solve the model, we take a log-linearization around the initial non-stochastic steady state described in Appendix C. Given the log-linearized system, the deviations of the exchange rate and the macroeconomic variables from their t — 1 expectations are solved in terms of exogenous money supply shocks and the expectation error shocks. A.4.1 Log-linearized System Money demand function rht - Pt = ~ yedrt+i m*t ~ Pt = ~ yedr*t+i (AA1) Labor supply function wt = i/tlt + pcf + pt w*t = + pc*t + Pt (A.4.2) Euler equations -PdrM = pcf - pEtc{t+1 +pt - Etpt+i - Pdr*t+1 = pc* - pEtcf+1 +p*t - Etpf+1 (A.4.3) Home household budget constraint (w = \) Pt + c f = \{p'htt + ch,t) + \{plt + <t + St) + ppgdBt - -pgdBt+i (AAA) Pricing Equations Ph,t = Et-i[wt] pi t = Et-i[wt] - Et-i[st] p},t = £ ( - i K ] + Et-i[£t] p)<t = Et-iK] (A.4.5) Appendix A. Appendices of Chapter 1 92 Price Indexes (UJ — \) pt = \ph + \vh p* = \PU + \p% (A'4-6) Individual Goods Demand Ch,t = ~l(Ph,t ~Pt) + ct c*ht = - 7 K , t _ P t ) + ct ( A . 4 . 7 ) c},t = -l(p'f,t ~ Pt) + ct c)t = - 7 ( P / , t - Pt) + <?t (AA.?,) Market Clearing Conditions dBt+1 = SdB*htt+1 (A.4.9) k = \ch,t + \4,t ft = \CU + \4,t (AAA°) Home Country Aggregate Consumption (for both exogenous entry and endogenous entry cases) &t = ^ + ^ - (A.4.11) Budget Constraints of Traders Exogenous Entry Case: The budget constraint of foreign exchange traders is given by: Pt+iCT+i = [(1 + r*t+1)St+1 - (1 + rt+1)St] = B*htt+1St(l + rt+1)pt+1 (A.4.12) Linearizing the above equation around B*. = 0 and p = 0 gives dC^+1 = 0. Endogenous Case: The budget constraint of foreign exchange traders is given by nt nt Pt+iCj+l = Blt+1{(l+r;+1)St+1-(l+rt+1)St]-Pt+1^ = B^iStU+rt+i W-Pt+ i X> i=0 i=0 (A.4.13) Linearizing the above equation around Bn = 0, p = 0 and n = 0 gives dCf+l = 0. Interest Parity Conditions Before linearizing the interest parity equation, an approximation 2 is used to rewrite the excess return in log-terms: '5 t + i ( l + r? + 1 )" S t + 1 ( l + r*+1) P t + 1 ~ St(l + rt+1) St(l + rt+1) st+i + ln(l + r*+1) - s t - ln(l + rt+i) (A.4.14) 2If £ is small enough, then ln(l + This approximation of the excess return is widely used in the finance literature. Appendix A. Appendices of Chapter 1 93 Exogenous Case: Using A.4.14, Equation 1.2.26 can be rewritten as: st = Et(st+i) + ln(l + r* + 1 ) - ln(l + r t + 1 ) + (1 - Ni)vt - a-±-(l + rt+1)Vart(st+i)B*ht+1 (A.4.15) Linearizing the above equation around the steady state, but using the second-order approx-imation to approximate the variance term (as the second order terms are important for understanding the dynamics of the model), we have it = Et(st-+1) - 0(drt+1 - dr*t+1) + (1 - Ni)vt - a ( 1 +^)S'Vart[si+l}dBlt+1 (A.4.16) Endogenous Entry Case: Using A.4.14, Equation 1.2.29 can be rewritten as: st = Et(st+i) + In 1 + r t + 1 ) - ln(l + r t + 1 ) + vt - a ' Vart[st+i} A / + nt Ft+\(A/ + nt) (A.4.17) Linearizing above equation around the steady state, but using the second-order approximation to approximate N™+nt Vt and the variance term, we have it = Et{s{+i) - P(drt+i - dr? + 1) + jj-ntvt - a ( 1 ±£S'Vart[st+i]dBlt+1 (A.4.18) where3 {Et\st+i} - s t + ln(l + r? + 1 ) - ln(l + r t + 1 ) + vt}2 1 - Nj 2aVart[st+i} c {Et(st+i) - it ~ P(dr*t+1 - dn+i) + vt}2 (1 - JVj) (A.4.20) 2aVart(st+i) c Money supply processes mi+i =rht + e^t rn*t+l = m*t + e*)t (A.4.21) where e*_t ~ N(0,a2) and e^t ~ N{0,a2£>i). 3Note that at the steady state 5 = 1. Here we also use the fact that approximating ln(l + r( + 1 ) — ln(l+rt+i) around the steady state gives: ln(l + r*+ 1) - ln(l + r t + i ) = -0(dr't+1 - drt+i) (A.4.19) Appendix A. Appendices of Chapter 1 94 A.4.2 Derivation of Equations Derivation of Equations 1.3.14 and 1.3.24 From the log-linearized intertemporal op-timality conditions A.4.3, we have: p(cf - c*) - PEt(cf+1 - cf+1) + Et-iiat) - Etisf+r) = -0{drt+1 - dr*t+1) (A.4.22) Equation A.4.22 minus its datet — 1 expectation gives: p(cf - c*) -PEt{cf+l - c*+1) - Et(sf+1) = -f3(drt~+1 - drf+1) (A.4.23) Taking t — l expectation of Equation A.4.16 and using the fact that at Et-\(vt) — 0, 4, give: Et-i{at) = Et-^st+i) - (3Et.x{drt+1 - dr*t+l) - a ( 1 ^ ^ a r f o ^ l ^ - i ^ t + i ] (A.4.24) Equation A.4.16 minus Equation A.4.24 gives: -f3(drt~+1 - drf+l) = st - Et{st~+1) - (1 - Ni)vt + J 1 V a r [ s f + 1 ] d B l t + 1 (A.4.25) Equation A.4.23 and A.4.25 give Equation 1.3.14. The derivation of Equation 1.3.24 is analogous. The only difference is that when deriving the analogy of A.4.24, we conjecture that Et-i(ritVt) — constant5. Since this term only affects the level of exchange rate and we are interested in the exchange rate volatility, it could be assumed to equal 0. Derivation of Equations 1.3.16 and 1.3.26 To derive Equation 1.3.16, first substituting Equation 1.3.14 into Equation 1.3.12 (c-„ _ - ; ) . [(•+y* - y - ygy -r f -wui ( A 4. 2 6 ) r Substituting A.4.26 into 1.3.15, m - < = ^ [ ( 1 + ^)st - ±(1 - Nt)vt + < L ^ ^ V a r t [ S f + 1 ] d B l t + 1 ) (A.4.27) Note that from Equations 1.3.11 and 1.3.15, dBt+\ = ^p[st — p-(rht — m*)]. Using the fact that dBt+i = SdB^ t + 1 = dB^ t + 1 , we could get: (mt-m*)[l + 2 pJo-rT)Van{St+l)] = + — 2 J+^Vmr*^)^ a (l-Nj)vt (A.4.28) o + r 4 A t this stage, we conjecture that Vart[st+i] = Vart[«t+i] = constant = VA. This conjecture is verified in Section 3.2. 5 From 1.3.25 and the functional form of si (Equation A.7.1), our conjecture can be easily verified. Appendix A. Appendices of Chapter 1 95 R e w r i t i n g E q u a t i o n A.4 .28 gives E q u a t i o n 1.3.16. T h e der ivat ion of E q u a t i o n 1.3.26 is entirely analogous. A.5 The simulation of Var(vt) — XVar(st) Fi r s t , for a given d i s t r ibu t ion of fundamentals, Lo(e^t, £ £ , t ) , the variance of the exchange rate when the expectat ion errors of the noise traders are zero can be calcula ted. It can denoted as o2S0. T h e n , we assume that the stochastic expectat ion error vt is given by: yt = \A<£t (A.5.1) where et is a r andom variable which satisfies the following three condi t ions : 6 Cov(et, e M l t ) = Cov(et, e* i t) = 0 o\ = \ (A.5 .2) E q u a t i o n A . 5 . 1 implies a2 = A<720. G i v e n A . 5 . 1 , and the d i s t r ibu t ion of fundamentals £ o ( £ M * > e M t ) ' t n e v a r i a n c e ° f exchange rate: a2 can be computed . L e t i t be denoted as Compare a2± and a20, i f K e - > 0 (A.5.3) T h e procedure stops at this point , otherwise, we w i l l redefine the stochastic process of vt as: vt yfiolet (A.5.4) Not ice that now a2 = Xcr^. U s i n g A . 5 . 4 and L o ( £ / i , t , ^ , t ) , uncond i t iona l exchange rate vo la t i l i ty could be computed again and wou ld be called a22. If [o^ — c r 2 2 | < e, and e —> 0, the procedure stops here, otherwise, the procedure described above w i l l be repeated to get • • •» "2„. < + 1 U I l t i l < + 1 - <Zn < £-A.6 Entry Condition of Traders with Tobin Tax W h e n the traders only need to pay t ransact ion cost to trade i n the foreign exchange market , trader i w i l l enter the market i f and only if: E\(U\ | 4 = 1) > E\(V\ | <p\ = 0) = 0 (A.6.1) 6 Cov(e/1^t,vt) = Ccn)(ellit,vt) must be equal to zero, as vt is some noise and should not have any funda-mental content. Appendix A. Appendices of Chapter 1 96 Or El max < El :\2 Substituting B^*t+1 B*hM1(i)St(l + rt+i) B*htt+1(i) Pt+l - T 7T t+1 -Vart Blt+1(i)St(l+rt+1) ^ ± ^ + a ^ ( 1 + r t + l ) V a n l p t + l ] shown that Equation A.6.2 is equivalent to: Pt+i (A.6.2) into the above equation, it can be Pt+i c I Pt+l s t ( f + L ) + a ^ ( 1 + r m ) y a r t ( p t + i ) . - r - aVart{pt+i)[ where c = [Ei(pt+i)? > 0 2 [ s f e l +aP7i-A1 + rt+i)Vart(Pt+1)}' It can be shown that the terms in the big bracket of Equation A.6.3 are equal to: St(l + n) Pt+i (A.6.3) (A.6.4) ]2}>o T + St(l + rt) t+i Vart(pt+i) > 0 (A.6.5) Therefore, regardless of how large is the rate of Tobin tax (r), the traders will always enter the foreign bond market. This is because the transaction cost is convex in the bond traded, the trader can always choose to hold a small amount of foreign bonds and get a positive expected utility. When the noise traders has to pay two costs to trade on the foreign exchange market, for noise trader i, he will enter the market if and only if: \2 Ej{ max {E* B ; , t + iW B*hit+1(i)St(l + rt+1) B*t+1(i) Pt+l ~ T Ci 2 ' Btt+1(i)St(l+rt+1) > 0 P t + i r ^ ' 2 j -i. -1_ n+i (A.6.6) Following the steps in Appendix A.2, we could get the following entry condition for noise trader i: \E?(Pt+i)]2 -Pt+i }}>0 a < GB 2aVart(pt+1) + 2T[St(^;t+i)) A.7 Numerical Undetermined Coefficient Method (A.6.7) This section gives details for the undetermined coefficient method used to solve for the func-tional form of it in Equation 1.3.26. First, guess a functional form for if. st =ao + airht + a2m* + a3Vt-ra4rnt2 + a5rn* +a6vf-\-airhtVt + asm*vt + a$rhtm* (A.7.1) Appendix A. Appendices of Chapter 1 97 Given that, we could get Et(sf+i) and Vart(sf+i) easily. Using the facts that rfit = e^t, m* — £* t and Cou(eA1)t,e* t) = 0, and that vt is noise and should not have any fundamental content, gives: Cov(mt,m*) — Cov(rht,vt) = Cov(m*t,vt) = 0 (A.7.2) Therefore, Et(sf+i) = ato*^ + a5a2. + a§<y2v (A.7.3) To get the conditional variance of the exchange rate, the properties of the normally distributed variables 7 and the fact that the three random variables are independent are used. Vart(sf+i) = a\Var(rht) + a2Var(ml) + a\Var(vt) + a\Var(rht2) + a\Var(m*2) a2Var(vt) + a2Var(rhtvt) + ot\Var(m*vt) + ag(rhtm*) + Covariance terms 2 2 , 2 2 , 2 2 , o 2 4 , r i 2 4 = + a2°ei + a i v c r v + 2a4<7^ + 2a£a£. +2a26a$ + a27<T2yv + a\a2e.a2v + ago***. (A.7.5) That is, Vart(sf+i) — Var(si+i) = constant = Vs and Et(sf+i) = E(si~+i) = constant = Es. Using the parameterized Es and Vs from Equations A.7.3 and A.7.5, we might solve for §t from Equation 1.3.26 given any exogenous shocks rht, m\ and vt- To test if our initial guess is a good guess, we can do simulations and regress the it we get from above process on rht, mi a n < ^ vt- If the coefficients (a's) are close enough to the initial guess, the process is stopped. Otherwise, the above procedure will be repeated. This method is actually an undetermined coefficient method, and is also known as the "parameterized estimation approach" in numerical methods. 7 Notice that, if xt is normally distributed with variance cr2., then E[(xt)2k] = (2k - l)(<rl)k E[(xt)2k+1] = 0 where k = 1,2, • • • n (A.7.4) Table A . l : Exogenous Case (A = 1, a = 2) No. of Noise Trader St Var(it) Increase of Var(s~t) Var{dBt+i) 0 st = 0.9562mt - 0.9562mt* 0.0183 0.00% 2.4861E-04 0.1 st = 0.9562mt - 0.9562mJ + 0.0912«;t 0.0184 0.84% 2.4861E-04 0.2 si = 0.9561mt - 0.9561mf* + 0.1824ut 0.0189 3.43% 2.7598E-04 0.3 st = 0.9560mt - 0.9560mt* + 0.2736wt 0.0198 8.06% 3.2484E-04 0.4 s~t = 0.9559mt - 0.9559m(* + 0.3647« t 0.0211 15.27% 4.0097E-04 0.5 s~t = 0.9557mt - 0.9557mt* + 0A557vt 0.0231 26.06% 5.1497E-04 0.6 st = 0.9553mt - 0.9553mt* + 0.5464wt 0.0260 42.31% 6.8646E-04 0.7 si = 0.9548mt - 0.9548mt* + 0.6367wt 0.0307 67.71% 9.5470E-04 0.8 st = 0.9540mt - 0.9540m(* + 0.7263vt 0.0385 110.68% 1.4083E-03 0.9 st = 0.9523mt - 0.9523m(* + 0.8141vt 0.0538 194.14% 2.2895E-03 1.0 st = 0.9482mt —0.9482mJ +0.8964vt 0.0916 400.69% 4.4702E-03 Consumption Var(ct) = Var(c*t) = 0.0025 Home wage Var(wt) = 0.0163 Home Labor Var{lt) = 0.0013 C D OO Table A .2: Exogenous Case (A = 1.5, a = 2) No. of Noise Traders Var(it) Increase of Var(si) Var(dBt+1) Corr(st,c~t - ct) 0 st = 0.9562mt - 0.9562mJ 0.0183 0.00% 2.3979E-04 1.0000 0.1 st = 0.9562mt - 0.9562mt* + 0.0912vt 0.0185 1.26% 2.5308E-04 0.9937 0.2 si = 0.9561mt - 0.9561mt* + 0.1824vt 0.0192 5.23% 2.9502E-04 0.9747 0.3 si = 0.9559mt - 0.9559mt* + 0.2736vt 0.0206 12.58% 3.7265E-04 0.9422 0.4 st = 0.9557mt - 0.9557mt* + 0.3645t;t 0.0228 24.77% 5.0127E-04 0.8948 0.5 si = 0.9553mt - 0.9553m* + 0.4553wt 0.0265 44.85% 7.1327E-04 0.8301 0.6 si =-0.9546mt - 0.9546mJ + 0.5455wt 0.0329 80.00% 1.0845E-03 0.7441 0.7 si = 0.9532mt - 0.9532mt* + 0.6344wt 0.0459 150.75% 1.8314E-03 0.6295 0.8 si = 0.9494mt - 0.9494mt* + 0.7191i>t 0.0803 339.33% 3.8224E-03 0.4737 0.9 st = 0.9351mt - 0.9351mt* + 0.7831wt 0.2183 1094.09% 1.1790E-02 0.2830 1.0 st = 0.9024mt - 0.9024m;T + 0.8047wt 0.5695 3014.60% 3.2060E-02 0.1691 Consumption Var{ct) = Var{c*t) = 0.0025 Home wage Var(wt) = 0.0163 Home Labor Var(lt) = 0.0013 a. 3 fi' s s, CO Table A.3: Endogenous Case(A = 1.5, a = 2)a Ni = 0.1 c = 0.1 a = 0.15 c = 0.25 c = oo Ni = 0.2 c = 0.1 c = 0.15 c = 0.25 c = oo Var(s~t) 0.7381 0.3697 0.1197 0.0168 Var(st) 0.2505 0.0728 0.0299 0.0171 Var{dBt+i) 4.23E-02 2.12E-02 6.80E-03 2.11E-04 Var(dBt+1) 1.43E-02 4.10E-03 1.60E-03 2.199E-04 Mean(n) 0.1741 0.1216 0.0931 0 Mean(n) 0.1743 0.1663 0.1749 0 Ni = 0.4 a = 0.1 a = 0.15 a = 0.25 a = oo Ni = 0.5 c = 0.1 a =0.15 a = 0.25 a = oo Var(st) 0.0313 0.0239 0.0199 0.0173 Var(st) 0.021 0.0201 0.0193 0.0173 Var{dBt+1) 1.60E-03 1.10E-03 8.00E-04 2.246E-04 Var{dBt+1) 8.00E-04 7.00E-04 6.00E-04 2.256E-04 Mean(n) 0.2938 0.2834 0.2629 0 Mean(ri) 0.3067 0.2888 0.2604 0 Ni = 0.8 c = 0.1 c = 0.15 c = 0.25 a = oo Ni = 0.99 a = 0.1 a = 0.15 a = 0.25 a = oo Var(st) 0.0173 0.0171 0.0169 0.0173 Var(s't) 0.0173 0.0173 0.0173 0.0173 Var(dBt+1) 3.00E-04 3.00E-04 2.00E-04 2.00E-04 Var(dBt+1) 2.00E-04 2.00E-04 2.00E-04 2.00E-04 Mean(n) 0.1494 0.14 0.1259 0 Mean(n) 0.0076 0.0072 0.0065 0 Consumption Var(c\) = Var(c*t) = 0.0025 Home wage Var(wt) = 0.0163 Home Labor Var(lt) = 0.0013 a Here we only consider a reasonable range of c: cC (0,0.25], as the steady state consumption in this model is 0.25. Appendix A. Appendices of Chapter 1 101 EH a C M O u p. C S i—1 C O C O CM CN r H Cs co co co CN r H II oo 0 0 0 0 oo oo d t - t - b-II i—i r H r H r H r H r H r H r H r H r H o o o O O II O O o O O d d d d d d d d d d 0 0 0 0 . * CN CM CM r - l —: C O r H O S oo m O S cs O S os <-J b- C D C O C D || H r H r H r H || r H H r H r H r H II o o O o II O O o o O d d d d d d d d d d t -to in m T f C O C O C O C O r H C O o o o o o t~ t- b- C O II CM CM CM CN CM || r H r H H H r H II o p O O o II o O o O O d d d d d d d d d d r H d „—* , " 0 0 i> C D m | | oo 0 0 0 0 CM C N CM CM CN CM CN ICJ 1 - t - t~ 11 CM CN CN CM CN II r H H r H r H r H II H o O o o O O o O o O e d d d d d CM | | O d d d d in e i—) | | —; in C O CM e n m" f * r H T f r H r H CD CO C O C D : S r H o O S O S 0 0 •< | | CM C N CN C N | | CM r H r H O O o O P | | O o o o of a d d d d O •< o d d d d u V cn fl C S oo co T f r H o C O o CM 0 0 C O O CM CN CN <N CN m r H O S 0 0 a II CO co C O co co CA || C O CM CN r H r H <p o o o o o fl II p O O O O itox d d d d d eric d d d d d S- O CO •v CO ( * OS m CN 0 0 CN a C O T f C O C N m in m m T f W t~ o i - CM oo || T f T f T f T f T f II in T f CN C N r H II o o o o o o o o o o Va d d d d d 'to d © d d d f -C3 cs _! C O C O C O CN C O m r H t- o O S —^^  o CJ) 0 0 t~ m o C O T f co C O II 0 0 i^ f - | | m r H 1~ CM CM © o o o o CM CM r H r H O -> d d d d d d d d d d I—1 i-H o co os CO T f r H r H CM T f O S CO oo co OS in 0 0 in C O O T f II i—i i—i o o O S || C O r H o> C N II CM CM CN CM r H II t - C O C D C O d d d d d d d d d d o o in C O oo T f o CO C O m C O co II O S CM iS C O CM T f r~ o m II C O C O T f co in co CM oo in m . m in II CM CM C N CN r H d d o d d & 'ax | SS (H o in 1 0 in O in O in in g II d II CN a II d H r H CM t- II II II II t- II II II II To (- t- To (- t- (-Appendix B Appendices of Chapter 2 102 B.l Price index and individual demand The aggregate consumption in the home country is composed by home finished goods and foreign finished goods: C = ic\c) (B. l . l ) Thus, the consumption-based price index P , which is the minimum nominal expenditure to purchase one unit of aggregate consumption, can be found by solving the following minimiza-tion problem: min Z = PhCh + PfCf (B.1.2) S.t ic\c) = 1 (B.1.3) So the CPI price index is given by: P = P*PJ (B.1.4) Takeing the prices for the home finished goods and foreign finished goods as given, the consumer allocates a given level of aggregate consumption among the home and foreign sub-aggregate finished goods: max C = 2c ]c ) (B.1.5) {Ch,C/} s.t PhCh + PfCf = PC (B.1.6) Thus, we have r - 1 P C r - 1 P C m i 7) C f c - 2 f l T ' C f ~ 2 P j ( B - L 7 ) Similarly, given the definition of the sub-aggregate consumption of the home and foreign finished goods Ch — [J0L C/i(i)~s~di]^ and Cf = [JQ Cf(i)^~di]T=T the price indexes for sub-aggregate home and foreign finished goods are given by : Ph = [[1 P h ^ d t ] ^ , Pf = [fX P / W 1 - ^ ] ^ (B.1.8) Jo Jo Appendix B. Appendices of Chapter 2 103 Then the optimal demand for individual finished goods can be derived as: Ch(i) = [ ^ p ] - * C f c , C,(i) = [ ^ ] - A C , (B.1.9) The price index and individual demand for intermediate goods can be derived analogously. B.2 Expected welfare Solving for Ec First, substituting the optimal pricing schedules Ph and Pf into the defi-nition of the price indices Ph and Pf, we have 1 Ph = (\4>)2— i i ' (B.2.1) [E&->][E{2^)]tlE{Slg=l)]i _ ^Ei^C^mif^^^iE^^^)^ Pi = (A<£) 5 \ ?' e-£-JL (B.2.2) [E&-P)[E{S2^)]i[E{S-^=>)]l Putting the above two equations together, we have fftpy - \<p [ECi-P]2 (B.2.3) Using the labor supply function to eliminate the predetermined prices on both sides, we have 3SL^Bf-i%))k[E{^-"1-[ECl-W 1 = W r ^ i - a i a — ( B - 2 - 4 ) Using the properties of log-normal distribution, taking log of (B.2.4), we have 0 = \0g[X4>T]2} + ^ [EC + i<72 + ^0~l + \^J2V + ^G2C - Osu - Osv - 0 ^ - 0 ^ + 0 ^ + Puv] + \ l E c + \ 0 l + \al> + \°l -Oca- OcA + \{EC + + \°l + \°l ~ °~cu> ~ Vcv] + - [ £ / C + -os + -ou. + -ov* + -oc + o-su* + Osv' - ocu* — ocv* - ocs + ou*v*\ -2[(1-p)Ec+^-^-a2} (B.2.5) 1For simplicity, we have used the fact that C = C* here. Appendix B. Appendices of Chapter 2 104 The next step is to derive the mean of the log consumption in terms of variance and covariance of the log consumption, the log exchange rate, and the productivity shocks. -ln[AeV] (2 - p) _2 1 2 1 r 2 , JI , JI , JI , „ , „ i Ec = — —o-c - — <Ts - — [au + <Ju. +av +av. + auv + au*v*\ + — \o~su - 0~su*] + ~ <J™*1 + 2p^acU a c U * + °~cv + Gov*} (B.2.6) Solving for E L The intermediate goods market clearing condition implies BjL -Xh + Xh- -j-— + -^-pj (B.2.7) Substituting Ph ,Phf, Ph and Pf into the above equation, we have L = 1 PCPhHsPjh)1* EC^=f.) E(&-r) 2\<j> Qi OF E\wshe*-'- -=*--•----i 2A0 Bi eF £ [ - 4 p i B [ ( ^ c , p ] x 1 P*cm*(P;)*S ) £ ( C H + r r ^ - 7 i ^ i . . . „- , . i (B.2.8) Using the labor supply function to eliminate the predetermined terms, and then taking ex-pectation, we have EL = J^-ECl-p (B.2.9) Xcfrq Expected welfare We may rewrite the expected utility of the home country as EU = ^ - ( I - P K A - I W - I ) ^ - , ( R 2 > 1 0 ) (1 - p)X(p Since ECl~p = exp(l — p)(Ec+ ^^-a2), the monetary authority's problem is equivalent to maximize UQ = Ec + K^a2, U q = ln[XW\ _ 1^2 _ ^ -±.[o* + o\. + a2v + erg. + a w + cr^ . ] + —[<T s u - cr s u.] + _ asv] + 2~^l<Jcu ~*~ CTcu* ~*~ a c u + CTc,;^ (B.2.11) Appendix B. Appendices of Chapter 2 105 B.3 Optimal money rules Variance and covariance terms Given the money rules and the solution for the exchange rate and the consumption (2.4.1)-(2.4.4), we have o2c = -~ {(ai + h)2ol + (a2 + 6 2 ) 2 a 2 . + (a 3 + h)2o2v + (a 4 + b4)2<4 2(ai + 6i)(a3 + h)ouv + 2(a2 + 62)(a4 + bi)ou*v*] (B.3.1) o2 = [ ( 0 1 - h)2al + ( 0 2 - 6 2 ) 2 c„. + ( 0 3 - h)2al + (a 4 - 64)2<r2. 2(ai - &i)(a3 - 63)^™ + 2(a2 - 62X^ 4 - 64)cru»l)»] (B.3.2) osu = (ai - &i)<7 2 + (03 - &3 ) t fw ; , cr««* = (a2 ~ foWl* + ( 0 4 - & 4 ) ^ u * « * (B.3.3) CTSU = ( « 3 - M ^ 2 + (ai - 6 i ) ( T u t „ crs„. = (a 4 - 64)cr2. + (a 2 - b2)ou*v* (B.3.4) o-cu = ^-(ai+bi)ol-r-^-(a3-rb3)(7uv, ocu. = ^-(a 2 + 62)o-2. + ^ -(a 4 + 6 4 )a u . „ . (B.3.5) 2/9 Ip Ip Ip ocv = •^(a3 + b3)ol + ^(ai-rbi)ouv, acv* = ^(044-64)^. + -^{a2 + b2)ou*v* (B.3.6) Substituting the above expressions into Equation (B.2.11), we may express the objective functions of monetary authorities as functions of policy parameters. Solution to the Nash game Since the objective functions of the home and the foreign monetary authorities are identical, the reaction functions are given by: dU0(a,bN) ai dU0(aN,b) = 0, V* = 1,2,3,4 (B.3.7) = 0, V* = 1,2,3,4 (B.3.8) bi Thus, from these 8 reaction functions, we may derive the solution to the international mone-tary game. Substituting these optimal policy parameters into Equation (B.2.11), we may get the maximized expected utility level under the sticky price equilibrium. U0{sticky) = ~ l n ^ n 1 + ^f[ol + o\. + a 2 4- a2. 4- 2ouv 4- 2<TU.„.] - T ^ - [ ( 7 « + °l* + °l + ol. - 2auv - 2 < w ] (B.3.9) lop From Equation (B.3.9) and (2.3.8), Equation (2.4.11) is obvious. Appendix B. Appendices of Chapter 2 106 Table B . l : The Optimal Price Policies for Foreign Firms" P* f A-l E[C1-f] Pf P}S 1 , p * * B l — 9 ^ — 1 a The prices with asterisk are in term of foreign currency. Figure B . l : The Structure of the Economy Home Household Assets Labor Foreign Household Home F ina l Goods *\ Home Intermediate Goods Foreign F ina l Goods Labor Foreign Intermediate Goods W Appendix C Appendices of Chapter 3 107 C.l Optimal pricing setting The optimization problem of each firm is to maximize the discounted expected profits, taking the individual demand function as given. Home firms set both the domestic price and export price in the currency of the producer (PCP). The home firm i's problem is then: max E[dir{i)\ = max E[d{{Phh{i) - ^-)Xh(i) + (Phf(i) - ^-)X*h(i))} (C.l . l ) Phh,Phi Phh,Phf v 0 Where ci = PC~P is the stochastic discount factor, Xh{i) = nCh(i) is the total sales of firm i to home residents and X^(i) = (1 — n)C^(i) is the total sales to foreign residents. Foreign firms set both the domestic and the export price in the currency of the consumer (LCP). The foreign firm i's problems is: max E[<T**(i)) = max E[d*((-^- - -^)Xf(i) + (P*ff(i) - —)X}(i))} (C.1.2) Substitute the risk-sharing condition 3.2.1, the labor supply function 3.2.4 and its foreign equivalent into the first order conditions derived from home and foreign firms' optimization problem, we can derive the optimal pricing policies of firms (3.2.5)-(3.2.8). C.2 Model solution Solving for Ec From the price index (3.2.10) and pricing equations (3.2.5) and (3.2.6), we have r WC1~p Mnr VI W'SC1-" M l - n rhhr}h - A E{C1~P) [ } Using the risk-sharing conditions (3.2.1) and (3.2.4) and its foreign equivalent, taking out the predetermined terms, we have Equation (3.3.9) of the paper: 1 ~ XR]T EW^) Now using the fact that the solution for consumption and exchange rate will be log-normal, and taking logs, we may get the expected (log) consumption (3.3.11): Ec = -1 ln[ - 2-^al - n ° l + {l ~ n)°l* + ^ + ( l - n ) a ^ p 2 2p p Appendix C. Appendices of Chapter 3 108 Similarly, we can derive Equations (3.3.10) and (3.3.12). Solving for EL and EU Home goods market clearing condition implies PC . , P * C * lPhh 0 L = n _ ^ + ( l _ n ) - (C.2.4) s Substituting the pricing equations (3.2.5) and (3.2.7) into (C.2.4), we get PC E(C1~P) „ .SP*C* E(C*1-") . r i n c . Using the labor supply equation (3.2.4) and risk sharing condition (3.2.1), and taking expec-tation, we can get Equation (3.3.14) of the paper: EL = —EC1-" + ^^EC*V-rir Xn Xr] Analogously, we can get: EL* = T - 1 ^E{Cl-p) + ^-^E{C*l-p) (C.2.6) A?7 Xn Then we can get the expected home country utility function (3.3.15) and its foreign equivalent: E W = A - ( l - n ) ( A - l ) ( l - p ) t _ ^ l l r - i ^ ( C 2 . 7 ) ( l - p ) A A Calculating the variances and covariances From Equations (3.3.4)-(3.3.6) and mone-tary policy rule (3.3.7) and (3.3.8), we can solve for the variances and covariances terms in Equation (3.3.11) and (3.3.12). °l = (ai " btfo-l + (aa - b2)2o2u. (C.2.8) °l = j2\<A°l + *\°l>\ (C2.9) 4 = l [ ( n a i + (1 - n)bx)2o-2u + (na2 + (1 - n)b2)2o2u.} (C.2.10) P °L = ~ai4. ala' = -wl* (C.2.11) P P 4 „ = i[na, + (1 - n)h}a2u, a2.u. = J [na 2 + (1 - n)^]^. (C.2.12) Appendix C. Appendices of Chapter 3 109 osu = (ai - h)o-1, <W = (o2 - b2)crl. (C.2.13) Using the relationship ECl~p = exp {(1 - p)[E{c) + ^ y ^ ] } (C2.14) we can express the expected home and foreign expected utility as functions of monetary policy parameters (01,02,61,62). C . 3 P r o o f o f P r o p o s i t i o n 4 When p = 1, using Equations (3.4.6) and (3.4.7), given the optimal monetary rules that ai = n, a2 = 1 — n, 61 = 0 and 62 = 1, the expected utility for the home country and foreign country are, respectively EU=-^Y^[al + *l.} (C.3.15) EU* = -^^[al + al.\ (C.3.16) Thus, EU < EU*. When p > 1, using equation (3.4.2) and (3.4.3) and the fact that T = {^Y, we can simplify EU and EU* as : EU = A - ( A - l ) ( l - p ) r x , (i - PM EU* = X~^~_1)ll~P)X' (C.3.18) Since is negative, to prove EU < EU* is equivalent to prove T > 1. We denote X = - \ ° l - \ ° l + °cu (C.3.19) X* = -|<£ - n ( 1 ~ n ) ^ 2 - \al + + n( l - n)(crs„ - (C.3.20) Thus, X = Qexp[(l-p)X] (C.3.21) X * = Qexp[{l-p)X*], (C.3.22) Appendix C. Appendices of Chapter 3 110 Since p > 1, to prove T > 1 is equivalent to prove X < X*. Substituting the optimal monetary rules for the general case listed in Table C . l into Equations (C.3.19) and (C.3.20), we may have X* - X = Ao2u + A*(fa (C.3.23) Where 2pz(nlp- 1) + 1 A*= }\r "){a\*2ha2-(l-n)(n(p-l)2 + 2p-l)-a2n(p-l^^ 2pl(n(p- 1) + 1) (C.3.25) Given the properties of the optimal policy coefficients ( n < a\ < 1, b\ < 0, ai — b\ > 0, 0 < a2 < 1 — n and a2 — b2 < 0), we can show A > 0, A* > 0 (C.3.26) That is, X < X* and X > X*. Therefore, T = > 1 (C.3.27) Thus, EU < EU* when p > 1. Q.E.D. C . 4 P r o o f of P r o p o s i t i o n 5 We can prove Proposition 5 in two steps. First, we prove that the two conditions under which the home firms choose PCP and the foreign firms follow LCP can be satisfied. That is: la29-Cov(ln™,s)>Z (C.4.28) 1 „ W* -a2s+Cov(ln—,s)<Z (C.4.29) where Z = ~= \(x-i)'s = Then we prove there exists a positive menu cost 5 such that both conditions hold simultaneously. Step 1 To prove that both (C.4.28) and (C.4.29) can be satisfied is equivalent to prove the following inequality: W W* Cov(ln —,s) < -Cov{\n—,s) (C.4.30) Appendix C. Appendices of Chapter 3 111 Using the labor supply function W = nPC and money demand function M = \PCP, we can rewrite Equation (C.4.30) as Cov(s,m-u) < -Cov{s,m* -u*) (C.4.31) Given the monetary policy rules (3.3.7) and (3.3.8), (C.4.31) becomes: Cov[(a\ —bi)u+(a,2 — &2)w*, (a\ — l)u+a2U*] < -Cov\(a\ -b\)u+ (a 2 — 02)11*, b\u+ (62 — l)w*] (C.4.32) Using the property that u and u* are i.i.d, (C.4.32) could be rewritten as: (ai - 6i)(ai + 61 - l)o2u + (02 - b2)(a2 + b 2 - l)o2u. < 0 (C.4.33) From the optimal monetary rules listed in Table C . l , we have a\ + a 2 = 1, 61 + 62 = 1, n < a\ < 1, b\ < 0, 0 < 02 < (1 — n) and ^ 2 > 1, this implies (ai - 6i)(oi + 61 - 1) = (02 - b2)(a2 + b2 - 1) < 0 (C.4.34) Thus, we show that the two conditions (C.4.28) and (C.4.29) hold. Step 2 We need to show there exists a positive menu cost 6 ( or Z) such that Equations (C.4.28) and (C.4.29) hold. Defining the left-hand side term in Equations (C.4.28) and (C.4.29) as Z\ and Z2, respectively, and using Equation (C.4.28) and the properties of optimal policy parameters (a's and 6's), we have Zi = ^[(ai-b1)2al + (a2-b2)2ol.}-[(a1-b1)(ai-l)ol + (a2-b2)a2ol.} = (01 - 61) (^~^ + l-ai)o* + (02 - b2) (^=^ - a2)o\. > 0 (C.4.35) + From the proof in Step 1, we have Z2 < Z\. Therefore, there must exist a positive Z 6 (max{0, Z2}, Zi) such that both Equations (C.4.28) and (C.4.29) hold. Q.E.D. Appendix C. Appendices of Chapter 3 112 Table C . l : The optimal monetary rule in Nash game Parameters p > 1 p — 1 ~ \pn+{l-n)\6i-n{p-l)S2 U l [pn+(l-n)]S-n(p-l)S2 \pn+(l-n)}53 tt2 [pn+{l-n)]5-n(p-l)52 , -n(p-l)S3 U 1 [pn+(l-n)]S-n(p-l)52 i [pn+(l-n)\6-n(p-l)S2+n(p—l)S3 ° 2 \pn+(l-n)]6-n(p-l)62 Where c5 = A - n ( l - p ) { l + (1 - n)[p( l - n) + n]} <5i = TI {A - (1 - p)[n + (1 - n)[/j(l - n) + n]]} <52 = n[(l - n)(l - p)}2 53 = (1 - n)[X - n(l - p)] and 61+63 = 6 Table C.2: The weight on exchange rate volatility in monetary policy decision" (p > 1, n = 0.5, al = al. - 0.0004, A = 1.1 ) Weight p= 1.5 p = 2 p = 3 p = 4 p = 6 p = 8 p= 15 Home -0.04 -0.114 -0.341 -0.683 -1.710 -3.2 -12.190 Foreign -0.23 -0.364 -0.716 -1.182 -2.456 -4.184 -13.825 Ratio 0.185 0.317 0.476 0.579 0.696 0.765 0.882 a. The weight on exchange rate volatility in monetary policy decision for home and foreign monetary authorities are measured by 3EU _ (1-n) dX* dEU* _ 1 (1-n) dX* del % dc2 do2 [\-p x J 9<r l where > 0, and Y is endogenously determined by equation 2.2. n 1 - n 0 1 Appendix C. Appendices of Chapter 3 113 Figure C . l : The reaction curves for a\, bi ( p = 4, n = 0.5 and A = 6) Figure C.2: The reaction curves for 02, 62 ( P — 4, n = 0.5 and A = 6) 

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