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Essays on new open economy macroeconomics Shi, Kang 2006

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Essays on New Open Economy Macroeconomics by Kang Shi B.A. , Management Engineering, Zhejiang University, P. R. China, 1994 M . A . Econ., University of British Columbia, Vancouver, 2001 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y The Faculty of Graduate Studies (Economics) T H E U N I V E R S I T Y O F BRITISH C O L U M B I A July 2006 © Kang Shi, 2006 11 Abstract Thi s thesis consists of three essays on the issues related to endogenous currency of pr ic ing i n ut i l i ty-based open economy models. The first essay endogenizes bo th the currency of l iab i l i ty denominat ion and the currency of export pr ic ing to s tudy t w i n dol lar iza t ion in East A s i a n economies, a phenomenon where firms borrow i n U S dollars and also set export prices i n U S dollars. Th i s essay shows that twin dol lar izat ion is an op t ima l strategy for a l l firms when exchange rate flexibility is l imi ted , and it can reduce the welfare loss caused by a fixed exchange rate regime. The second essay examines the role of U S dollar as an o i l currency and its impact on the wor ld dollar s tandard and the global economy in a two-country general equi l ibr ium model w i t h s t icky prices. W h e n the o i l price is denominated i n U S dollars, U S firms bear less exchange rate risk than foreign firms when facing o i l price shocks. A s a result of this asymmetry, a l l the firms have an incentive to set export prices i n dollars, this w i l l generate an endogenously determined dollar s tandard in international goods pr ic ing . In such a case, the households i n the U S are better off than those i n the rest of the wor ld i n t e rm of welfare, though it is costly for the U S to have a dollar standard. The t h i rd essay re-examines the issue of the degree of exchange rate flexibil i ty i n an open economy monetary pol icy game. It focuses on an environment where the currency of pr ic ing is endogenous, and monetary authorities take into account the way i n which firms make this choice. It is shown that there is a unique equi l ibr ium to the monetary policy game, where a l l firms follow P C P , and Friedman's classic defense of flexible exchange rates is upheld. Meanwhile , an alternative method to sustain flexible exchange rates is also provided. K a n g Shi . kangshi@interchange.ubc.ca iii Contents Abstract n Contents iii List of Tables vi List of Figures vii Acknowledgements viii S u m m a r y i x 1 T w i n Dol larizat ion and Exchange Rate Pol icy 1 1.1 Introduction 1 1.2 Stylized Facts 5 1.3 Basic Model 7 1.3.1 Households 9 1.3.2 Non-traded Goods and Imported Goods 11 1.3.3 Export Firms 11 1.3.4 Monetary Policy 17 1.3.5 Equilibrium 18 1.4 Export Firms' Optimal Strategies 19 1.4.1 Model Solution 20 1.4.2 The Optimal Strategies 24 1.4.3 Numerical Results 28 1.4.4 Discussion 30 Contents iv 1.5 Welfare Implication 31 1.6 Conclusion 33 2 Oil Currency and the Dollar Standard 35 2.1 Introduction 35 2.2 Basic Model 38 2.2.1 Households 39 2.2.2 Oil Market 41 2.2.3 Production 42 2.2.4 Monetary Shocks 42 2.2.5 Endogenous Currency of Pricing 43 2.2.6 Equilibrium 45 2.3 Model Solution 45 2.3.1 The Currency of Pricing . . . 45 2.3.2 Optimal Pricing Schedule 46 2.3.3 A Closed-form Solution 47 2.3.4 Welfare Comparison 48 2.4 The Case with Active Monetary Policies 50 2.5 Conclusion 56 3 Flexible Exchange Rates and Endogenous Currency of Pricing 57 3.1 Introduction 57 3.2 Basic Model 59 3.2.1 The Timing of Events 60 3.2.2 Household 60 3.2.3 Production 62 3.2.4 Endogenous Currency of Pricing 63 3.2.5 Monetary Authorities 65 3.3 A Unique Equilibrium with Flexible Exchange Rates 66 3.3.1 Monetary Rules chosen after the Pricing Decision Is Made 66 3.3.2 Monetary Rules Chosen before the Pricing Decision Is Made 71 Contents v 3.4 Sustaining the Flexible Price Equilibrium with Restricted Monetary Policy Rules 73 3.4.1 Case 1: Al l Firms Follow P C P 74 3.4.2 Case 2: Al l Firms Follow L C P 75 3.5 Conclusion 76 Bibl iography 77 A Appendices of Chapter 1 81 A . l Approximation of Expected Profit Functions 81 A.2 The Steady State 82 A.3 Model Solution 83 A.4 The Euler Equation 84 A. 5 The Proof of Proposition 1 85 B Appendices of Chapter 2 89 B. l Proof for Proposition 2 89 B.2 The Labor Market Clearing Condition 90 B.3 Expected Utility for the Special Case p — l 90 B.4 Proof for Proposition 3 91 B. 5 The Case with Productivity Shocks and Active Monetary Authorities 92 C Appendices of Chapter 3 96 C. l Firms 96 C . l . l Preset Prices 96 C.1.2 CI and fi* 96 C.2 The Proof of Binary Game 97 C.3 Equilibrium under Restricted Monetary Policy Rules 98 C.3.1 Case 1 {PCP,PCP} 98 C.3.2 Case 2 {LCP,LCP} 99 C.3.3 Case 3 {PCP, LCP} 99 List of Tables A . l The Currency Denomination of Export Invoicing of Selected Countries (%)a . . 86 A.2 The Currency Denomination of External Debt of Selected Countries (%)b . . . 86 A.3 Standard Deviation of Daily Exchange Rate Fluctuations against the Dollar (%)c 87 A.4 Calibration Parameters 87 A . 5 The Impact of 7 on the Distribution of Firms fl d 87 A.6 Welfare Comparison 87 A.7 Variables Comparison e 88 A. 8 Welfare Comparison with Increasing w ? 88 B. l The Optimal Monetary Parameters in Three Cases 95 C. l Three Optimal Pricing Policies 101 v i i List of Figures 1.1 T i m i n g of Events 3.1 T i m i n g of Events 6 0 3.2 B i n a r y Game 7 2 viii Acknowledgements I am extremely grateful to my supervisor, Professor Michael B. Devereux for advice, guidance and encouragement. I am also greatly indebted to other member of my supervisory committee, Professor Angela Redish and Professor Henry Siu for research direction and discussion. I would like to thank Professor Paul Beaudry, Professor Patrick Francois, Professor Amartya Lahiri, and Professor Juanyi X u for helpful suggestions and discussions at various stages of this dissertation. I am also grateful to Geoffery Dunbar and Min Lu for helpful suggestions, discussions, and support. I thank the seminar participants at U B C macro-lunch, S F U , Hong Kong University of Science and Technology, the 2005 meetings of the Canadian Economics Association and the U B C Target Workshop for comments and helpful discussions. I thank the INE project fellowship for financial assistance. I am solely responsible for any errors and misinterpretations. ix Summary Thi s thesis consists of three essays on new open economy macroeconomics. The first essay develops a smal l open economy general equi l ib r ium model w i t h nominal rigidities to study t w i n dol lar izat ion i n East A s i a n economies, a phenomenon where firms borrow i n U S dollars and also set export prices i n U S dollars. The model endogenizes bo th the currency of l iab i l i ty denominat ion and the currency of export pr ic ing. We show that the key factor that affects the firm's dol lar izat ion decision is exchange rate policy. T w i n dol lar izat ion is an op t imal strategy for a l l firms when exchange rate flexibility is l imi ted , which implies that a fixed exchange rate regime w i l l cause t w i n dol lar izat ion. Furthermore, we find that twin dol lar izat ion can dampen the welfare loss caused by the fixed exchange rate regime, as it helps to cushion the economy against the domestic nominal risk. The second essay examines the role of U S dollar as an o i l currency and its impact on the wor ld dollar s tandard and the global economy i n a two-country general equ i l ib r ium model w i t h s t icky prices. W h e n the o i l price is denominated i n U S dollars, U S firms bear less exchange rate risk than foreign firms when facing o i l price shocks. A s a result of this asymmetry, a l l firms i n the wor ld w i l l set their export prices in U S dollars, which implies a dollar standard in internat ional goods pr ic ing. We also find that, however, i n the presence of active monetary policy, a dollar standard is not necessarily the equi l ib r ium of firms' choices. T h e nature of the equi l ibr ium w i l l depend cr i t ica l ly on the relative sizes of external shocks. F ina l ly , we find that households in the U S are always better off than those i n the rest of the wor ld in the s i tuat ion of having a dollar standard. Th i s is because the welfare gain, which comes from the cost advantage of U S firms, exceeds the welfare cost of having a dollar s tandard, which is due to the lack of exchange rate pass-through. Recent l i terature has emphasized that the degree to which exchange rate adjustment should be part of op t imal monetary pol icy i n an open economy depends cr i t ica l ly on the way i n which Summary x export good prices are set. If export goods are priced in the producer 's currency ( P C P ) , then monetary pol icy should use exchange rate adjustment. B u t i f export goods prices are set i n the consumer's currency ( L C P ) , then op t imal monetary pol icy should keep the exchange rate constant. T h i s paper examines the determination of op t imal monetary pol icy i n an environ-ment where the currency of pr ic ing is endogenous, and monetary authorities take into account the way i n which firms make this choice. It is shown that there is a unique equi l ib r ium to the monetary pol icy game, where a l l firms follow P C P , and Friedman's classic defense of flexible exchange rates is upheld. Furthermore, we show that there is a simple method to sustain flex-ible exchange rates by restrict ing monetary authorities to respond only to domestic economics conditions when monetary authorities cannot take account of the currency of pr ic ing decision. In this case, restr ict ing the form of monetary rule acts as an equ i l ib r ium selection mechanism. Chapter 1 i Twin Dollarization and Exchange Rate Policy 1.1 Introduction This paper is motivated by a frequently observed phenomenon in East Asian economies. On the one hand, the vast majority of lending to these emerging markets is denominated in foreign currency, especially in US dollars. On the other hand, most export goods in these economies are priced in US dollars as well. For instance, in Thailand, approximately 76 billion dollars was recorded as raised by Thai firms on international debt markets between 1992 and mid-1997, while only 3.2 percent was denominated in Thai baht. Meanwhile, about 90 percent of Thai export goods were priced in US dollars.1 This phenomenon represents a coexistence of liability dollarization and export pricing dollarization,2 which we refer to as "twin dollarization". While considerable attention has been paid to the macroeconomic implications of liabil-ity dollarization and of export pricing dollarization, there are few papers which relate them together and investigate a common cause. Intuitively, they should be related to each other. Firms have an incentive to match the currency of their sales revenue with the currency of their debt denomination so that they can use sales revenues to hedge their foreign debts against exchange rate risk. This could explain the coexistence, but it would still not reveal the cause of twin dollarization. In this paper, we try to analyze the "twin dollarization" phenomenon formally and answer the following questions: Why do firms want to borrow in dollars and set export prices in dollars at the same time? Furthermore, if twin dollarization can be rational-ized as an optimal strategy for firms, what is the inducement? Finally, what are the welfare 'See Choi and Cook (2004) and Cook and Devereux (2004). 2Cook and Devereux (2004) use the term "dollar currency pricing" to describe the latter fact. Chapter 1. Twin Dollarization and Exchange Rate Policy 2 implicat ions of twin dollarizat ion? To address these questions, we propose a smal l open economy stochastic general equi l ibr ium model w i t h nomina l rigidities. The economy is subject to two types of external shocks, foreign demand shocks and world interest rate shocks. T h e key feature of the model is that both the currency of l iabi l i ty denominat ion and the currency of export pr ic ing are endogenous. Our analysis concentrates on the behavior of export firms, which depend highly on the wor ld market for bo th the sale of products and the supply of inputs . 3 T h e export firms are monopolist ic competi t ive and supply differentiated goods to the rest of the world . Furthermore, they can set export prices either in domestic currency (peso) or i n foreign currency (dollar). Whatever ^currency they choose, firms must set their export prices before the state of the wor ld is realized, and the prices cannot be adjusted unt i l the next period. F i r m s impor t intermediate goods and finance their purchases by borrowing from international lenders. 4 T h e loan is assumed to be contracted either in domestic currency or i n foreign currency ex ante as wel l . Therefore, four feasible (pure) strategies are available for firms, and each of them represents a combinat ion of the currency of l iab i l i ty denominat ion and the currency of export pr ic ing. Us ing this framework, we find that exchange rate pol icy is a key factor for firms' dol lar iza-t ion decisions. W h e n exchange rate flexibil i ty is low, t w i n dol lar iza t ion is an op t ima l strategy for a l l firms. In other words, a fixed exchange rate w i l l cause t w i n dol lar iza t ion. We also find that, as exchange rate flexibil i ty increases, the degree of dol lar izat ion decreases. T h a t is, fewer firms w i l l choose to borrow in dollars and set export prices i n dollars. A s a result, a floating exchange rate w i l l lead a l l firms to borrow i n the domestic currency and set export prices i n the domestic currency. Under intermediate exchange rate regimes, the currency of export pr ic ing and the currency of l iabi l i ty denomination can be different. In some cases, mult iple equi l ibr ia may exist for firms' op t imal currency strategies due to the presence of a strategic complementari ty among firms. T h e in tu i t ion behind these findings is straightforward. T h e op t ima l strategy for the export f irm is the one that delivers the highest expected profit, which reflects the firm's consideration 3 This is one of main features of export firms in the East Asian economies.' 4Most Asian emerging market economies are export-oriented, and the export sectors are capital intensive. Thus, international lending to these countries is most often used by export firms. Chapter 1. Twin Dollarization and Exchange Rate Policy 3 of bo th the demand for its export goods and marginal cost. In a stochastic environment, exchange rate f lexibil i ty determines how the absorption of external shocks is d iv ided into changes of the domestic interest rate and the exchange rate. W h e n exchange rate f lexibil i ty is high, the exchange rate is volatile and the domestic interest rate is relatively stable. Thus , the export firm w i l l choose to borrow i n pesos to avoid exchange rate risk, as it implies a more stable marginal cost. Meanwhile , by setting prices i n pesos, the f irm can stabil ize the demand for its export goods, as i n this case the relative prices of export goods can be adjusted by exchange rates in the face of demand shocks. Therefore, the op t ima l strategy for a l l firms is to borrow i n pesos and set export prices i n pesos when exchange rates are flexible. In the case of low exchange rate flexibility, exchange rate vola t i l i ty is low and the domestic interest rate vola t i l i ty is high. Therefore, the export firm can stabilize the marginal cost and avoid the volatile domestic interest rate by borrowing i n dollars. Regarding the currency of pr ic ing, due to the lack of exchange rate flexibility, setting prices i n pesos cannot help the f irm to stabilize the demand for its export goods. Meanwhile , when the firm sets export prices i n domestic currency, its demand is direct ly sensitive to exchange rate movement, which results in an increase i n the firm's expected cost, ceteris paribus. So given a low exchange rate flexibility, the firm is more concerned w i t h expected cost than revenue and should choose to set export prices i n dollars. Therefore, tw in dol lar izat ion is an op t ima l strategy for a l l firms under a fixed exchange rate regime. For intermediate levels of exchange rate flexibility, the ranking of firms' currency strategies also depends on the decisions of other firms. Therefore, the interact ion among firms w i l l lead to mixed strategy or mult iple equi l ibr ia . O u r findings suggest that twin dol lar izat ion is op t ima l for firms under a fixed exchange rate regime, but is it a beneficial arrangement for the whole economy i n terms of welfare? Fol lowing Schmit t -Grohe and Ur ibe (2004), we use a per turbat ion method to calculate welfare, which is measured by the representative household's lifetime expected ut i l i ty . In our model , the equi l ibr ium under a flexible exchange rate regime implies a higher welfare than that generated by a fixed exchange rate regime, so there is always a welfare loss associated w i t h a fixed exchange rate regime. Nevertheless, tw in dol lar izat ion can deliver a higher welfare than other currency strategies under a fixed exchange rate regime. In this sense, it dampens the welfare Chapter 1. Twin Dollarization and Exchange Rate Policy 4 loss caused by the fixed exchange rate arrangement. The reason is simple. Under a fixed exchange rate regime, the exchange rate cannot insulate the economy from external shocks, so the economy is subject to both real shocks and nominal uncertainties. But with twin dollarization, some domestic nominal risk can be evaded and thus social welfare is improved. Our paper provides a new angle to study the dollarization phenomenon in East Asian emerging market economies. We relate two aspects of dollarization and study their common cause, while most of the recent literature focuses solely on the macroeconomic implication of liability dollarization. For example, Calvo and Reinhart (2000) show that currency devaluation will lead to the deterioration of the balance sheets of firms with foreign currency debts, which causes real contraction in developing countries.5 In this paper, we emphasize that it is the fixed exchange rate regime that causes twin dollarization, instead of Calvo and Reinhart (2002)'s suggestion that liability dollarization leads to the "fear of floating". In this sense, we show a new linkage between exchange rate policy and dollarization. We also show that there exists a welfare gain from twin dollarization under a fixed exchange rate regime. This is in contrast to the welfare implications of liability dollarization in most of the financial crisis literature, which typically emphasizes the welfare losses caused by foreign currency debt. This paper is also closely related to two other lines of literature. With respect to endogenous currency of pricing, we follow the approach used by Devereux, Engel and Storgaard (2004). They endogenize the currency of export pricing and show that exporters wish to set prices in the currency of the country with a relatively stable monetary policy.6 Our paper differs from theirs in two key dimensions. First, we focus on a small open economy and study how exchange rate flexibility affects firms' currency of export pricing. Second, we illustrate a linkage through which the currency of debt denomination can affect the firm's marginal cost and the currency choice of export pricing. Another line of research that is related to this paper is the literature on endogenous liability 5Also see Aghion, Bachetta, and Bannerjee (2000, 2001), Calvo (2000), Cespedes et al. (2004), Cook (2000), and Krugman (2000) 6Engel (2005) discusses the parallels between the choice of invoicing when prices are sticky and the optimal degree of exchange rate pass-through when prices are flexible. Chapter 1. Twin Dollarization and Exchange Rate Policy 5 denomination.7 Our work differs in several aspects. First, most papers in the literature argue that foreign currency debt exists because of market or institutional failure in emerging market economies, see for example, Jeanne (2000, 2003), and Caballero and Krishnamuthy (2003). Instead, we show that dollarization can be the result of firms' optimization behavior and that exchange rate policy is a key factor affecting firms' decisions. Second, our general equilibrium model setting has a natural advantage over the recent literature based on partial equilibrium or reduced form models, since it allows for welfare analysis.8 Finally, we investigate the common cause of two aspects of dollarization, instead of analyzing liability dollarization in isolation. This paper is organized as follows. Section 2 describes stylized facts of twin dollarization in East Asian economies. Section 3 presents the basic setup of the model. Section 4 solves the model and shows how the firms' dollarization decisions can,be affected by exchange rate policy. Section 5 discusses the welfare implication. Section 6 concludes.. 1.2 Stylized Facts A frequently observed fact in East Asian economies is that firms borrow externally in foreign currency and set their export goods prices in foreign currency as well. Since the US dollar is the dominant foreign currency, we refer to this phenomenon as "twin dollarization". It reflects the fact that, in contrast to the practise of firms in developed countries, firms in emerging market economies seldom use their own currency in goods and financial market trading with the rest of the world. Evidence of "twin dollarization" is given in Tables A . l and A.2. Table A . l reports the 7There is also a presumption in the literature that developing countries cannot borrow in local currency. Some call it the "original sin". This wing of the literature suggests that if somehow we could eliminate this "original sin" then currency and debt crises problems of the developing world would go away. Here we make a different point: there are well defined economic reasons why firms in developing countries may not want to borrow in local currency even if they could. 8Chammon and Hausman (2002) also argue that the central bank's preference may affect a firm's choice of liability denomination, in a reduced form model. Chapter 1. Twin Dollarization and Exchange Rate Policy 6 use of US dollar in the currency of export pricing (invoicing currency) in selected countries.9 For East Asian economies, we report data for Korea, Thailand, Malaysia, and Indonesia. The selected developed countries are the US, Germany, Japan, the U K , France, and Italy. The striking feature of Table A . l is that East Asian economies seldom use their domestic currency as the currency of export pricing. In particular, for Korea and Thailand, almost all of their export goods are priced in foreign currency, mostly in US dollars. By contrast, the share of domestic currency invoicing is much higher in the developed countries. Even in Italy, about 40 percent of exports are priced in Lira. Table A.2 presents the currency composition of external debt for the same set of countries. From Table A.2, we can see that the external debts of these East Asian economies are mainly denominated in foreign currency, while for developed countries, the share of domestic currency denominated external debt is much higher. Although the data in Table A.2 includes both public debt and private debt, statistics show that the share of private debts to total external debts is high in some East Asian economies. For example, data from the Japanese Ministry of Finance show that 81.3% of Thailand's external debts and 98.5% of Korea's external debts were private debts at the end of 1996. Therefore, it is reasonable to infer that most private debts raised by firms in East Asian economies are denominated in foreign currency. This argument is also well supported by other observations in Thailand and Korea: Of the approximately 76 billion dollars recorded as raised by Thai firms on international debt markets between 1992 and mid-1997 (according to the IFR Platinum database), about 3.2% was denominated in Thai baht. During the same period, less than 1% of the approximately 144 billion dollars recorded as raised by Korean firms was denominated in Korean won (Choi and Cook 2004). The detailed currency composition of external debt is not reported in Table A.2, but the dominant use of the US dollar in emerging market debt denomination is clear. From Table A . l and Table A.2, we can see that firms in East Asian emerging market economies tend to follow both export pricing dollarization and liability dollarization, which contrasts with borrowing and export pricing behavior in developed countries. Thus, the special 9 Data for invoicing currency are not available for every country, so we will focus on the available data set in selected countries. Nevertheless, we believe these countries are representative for East Asian economies and developed economies " Chapter 1. Twin Dollarization and Exchange Rate Policy 7 role of the U S dollar i n the global economy could not solely account for the twin dol lar izat ion phenomenon. Some other common characteristics in East A s i a n economies are needed as an explanat ion. A possible explanat ion for this phenomenon is the exchange rate regimes i n these economies. Before the 1997-98 A s i a n crisis, East A s i a n economies like H o n g K o n g , Indonesia, Korea , Malays ia , Phi l ippines , Singapore, Taiwan, and T h a i l a n d a l l pegged their exchange rates to the U S dollar. The exchange rate regimes ranged from a currency board hard peg i n Hong K o n g to a s l id ing or crawling peg i n Indonesia. A l t h o u g h these pegs were often not openly admit ted or were disguised as currency baskets, the common adherence to the dollar is easy to recognize (Table A . 3 ) . O n the other hand, the selected developed countries adopted a flexible exchange rate regime. Table 3 also reports the pre-crisis exchange rate fluctuations of selected East A s i a n currencies and some major currencies against the U S dollar . A s shown i n the first column of Table A . 3 , the volati l i t ies of East A s i a n currencies were usual ly much lower than those of the major currencies. C a n the low exchange rate flexibility in East A s i a n Economies expla in the twin dol lar izat ion phenomenon? How does exchange rate flexibility affect firms' debt-borrowing and price-setting behavior? In the next section, we construct a smal l open economy stochastic general equilib-r i u m model w i t h nomina l rigidities to study the linkage between exchange rate regimes and twin dol lar izat ion. 1.3 Basic Model We consider a smal l open economy populated by two kinds of private agents: households and firms. T h e representative household owns firms and receives the average profit. There is a domestic currency we cal l peso and a foreign currency we ca l l dollar . The household consumes both domestic non-traded goods and foreign goods. T h e domestic non-traded goods are homogenous and produced by a competit ive non-traded goods sec tor . 1 0 T h e export firms 1 0 This assumption implies that export goods are not sold to the domestic market, reflecting the fact that export firms are highly dependent on the world market. We can allow the domestic household to consume the domestically produced traded goods, and it just quantitatively alters the degree of dollarization given an exchange rate regime. Chapter 1. Twin Dollarization and Exchange Rate Policy 8 are monopolis t ic competi t ive, using local labor and impor ted intermediate goods to produce differentiated goods, which are sold to the rest of wor ld . E a c h export f i rm sets its price to exploit its monopoly power, and chooses the currency i n which to set export goods prices. Whatever currency is chosen, the f irm must set its export price before the state of the wor ld is realized. It is assumed that the export f i rm has to borrow external ly to finance the purchases of impor ted intermediate goods. F ina l ly , the currency of l iab i l i ty denominat ion must be chosen by the f irm ex ante as w e l l . 1 1 The economy is subject to two kinds of external shocks: foreign demand shocks and world interest rate shocks. Mone ta ry pol icy (or the exchange rate regime) is represented by a simple domestic interest rate targeting rule, which is used to determine how the absorption of external shocks is d iv ided into changes of the domestic interest rate and the exchange rate. Th i s rule reflects the central bank's preference. Figure 1.1: T i m i n g of Events Export firms Export firms set Period t starts, Export firms Consumption and production choose currencies prices, workers shocks occur borrow debts take place, debts are paid, set wage The t iming of events is shown in Figure 1. Before the start of per iod t, export firms choose the currency in which to set their export good prices and the currency i n which to contract their external debts. After that, they set prices to maximize expected discounted profits, based on the stochastic discount factors, and the anticipated market demand and marginal cost. Meanwhi le , workers preset their wages. 1 2 After per iod t starts, external shocks occur, export firms w i l l borrow to buy intermediate goods for product ion, households w i l l supply labor, and choose their op t imal consumption baskets and domestic bond holding, then consumption and 1 1 The import of intermediate goods for firms' production reflects the dependence on foreign capital of this economy. 1 2There are two types of nominal rigidities in this model, the sticky price and the sticky wage. If prices are flexible, then the firm would be indifferent about the currency of export pricing, so the price stickiness is a key assumption. But the sticky wage assumption is just for the simplification of our analysis, and our results can still hold under a flexible wage. Chapter 1. Twin Dollarization and Exchange Rate Policy 9 production take place, and the exchange rate is determined. Finally, the debts are paid back to lenders at the end of period t. The detailed structure of the economy is described below. Where appropriate, foreign currency (dollar) prices are indicated with an asterisk. 1.3.1 Households The small open economy contains a unit interval [0,1] of households indexed by j. Each house-hold supplies, in a monopolistic fashion, a distinctive variety of labor service. The expected utility of household j is given by EU(j) = Et^P^i^-^ 21 ~ VL.U)) (1-3.1) where C is a consumption index defined across domestic non-traded goods and foreign goods; Et is the expectation operator conditional on information at time t; f3 is the discount factor; p is the inverse of the elasticity of intertemproal substitution; 77 is a scale parameter for the disutlity of labor supply. The consumption index C is defined as follows C t = ^ ( l - o ) ! - ^ ^ ( L 3 - 2 ) where CN is the homogenous domestic non-traded goods produced by a competitive non-traded sector, CF is the consumption of foreign goods, and a is the share of imported foreign goods in the total consumption expenditure of domestic households. The Cobb-Douglas form of equation (1.3.2) implies a unit elasticity of substitution between domestic goods and foreign goods in consumption. 1 3 The consumer price index for domestic households is derived as Pt = PJfTPft (1.3.3) where Px and Pp are the prices of domestic non-traded goods and imported foreign goods respectively. Each household decides his consumption, domestic nominal bond purchase, and labor sup-ply every period. The household derives income from wages, profits from export firms, 1 4 and 1 3 This structure has been adopted by many papers in the open-economy literature. For instance, see Obstfeld and Rogoff (1998, 2002), Devereux and Engel (2003), Corsetti and Pesenti (2001). 1 4 A s the non-traded sector is perfectly competitive, the profit from this sector is zero. Chapter 1. Twin Dollarization and Exchange Rate Policy 10 returns on domestic bond holding. Household j's budget constraint can therefore be written as: PtCt(j) + Bt+1(j) = Wt(j)Lt(j) + (1 + it)Bt(j) + Uxt (1.3.4) where Bt(j) represents the household j's holding of domestic bonds, Wt{j) is the nominal wage, and it is the domestic nominal interest rate between period t — 1 and t. Tlx is the average profit from the export sector. It is assumed that households have no access to the international bond market, but export firms can finance their purchases of imported intermediate goods through the international capital market. This assumption implies that the standard uncovered interest rate parity condition(UIRP) does not hold in our model since there are no arbitrage opportunities for households.15 The aggregate labor service hired by an export or non-traded goods firm can be defined as Lt = [ Lt(j)6-rdj]^ (1.3.5) Jo Therefore, each household j faces a downward-sloping labor demand curve with elasticity 6 > 1, Lt{j) = l^-r6Lt (1.3.6) where W is the production-based wage index and is given by Wt = [JQ Wt(j)i~e]T=°. We assume that the household sets nominal wages one period ahead and, ex post, supplies the amount of labor that firms demand at the predetermined nominal wage. Therefore, the optimal preset nominal wage is [J)~ e-iE. J J ^ L ) { ' Without uncertainty, equation (1.3.7) would simply imply that the marginal utility of real wage equals a fixed markup over the marginal disutility of labor. The household's optimal choice of consumption results in the standard Euler equation: 7 1 ^ = W p * % % J (1-3.8) 1 + it+i Pt+iCl+1(j) 1 5 This assumption captures the fact that households in emerging market economies can seldom smooth con-sumption by holding foreign assets. This imperfect financial market also implies that there might exist an interest rate differential between the domestic interest rate and the world interest rate even when exchange rates are fixed. For example, see Lahiri and Vegh (2003). Chapter 1. Twin Dollarization and Exchange Rate Policy 11 Finally, the demands for non-traded goods and foreign goods are given by: CNt(j) = d - a ) ^ l , . CFt(J) = a ^ 1 (1-3.9) In a symmetric equilibrium, Ct(j) = Ct , Wt(j) = Wt, and Lt(j) = Lt, V? € [0,1], so we can drop the subscript j. 1.3.2 Non-traded Goods and Imported Goods We assume that the non-traded goods sector has a linear production technology: YNt = LNt (1.3.10) Since the sector is perfectly competitive, the price of non-traded goods equals the firm's mar-ginal cost. Pm = Wt (1.3.11) The foreign good has a constant price of one in terms of dollars in the world market. Thus, the price of imported foreign goods in terms of domestic currency (peso) is given by Ppt = St, where 5 is the nominal exchange rate of dollar in terms of peso. Therefore, the domestic CPI price index is Pt = W}~aS? and the de gree of exchange rate pass-through into the domestic CPI equals a. 1.3.3 Export Firms The export sector is monopolistic competitive and contains a unit interval [0,1] of firms indexed by i. Each firm i in this sector sells a differentiated good to the world market, and faces a downward-sloping demand function Y&(PZt(i)) = ( ^ ) - A ^ (1-3.12) *xt *-xt where P £ t ( i ) is the price that the foreign consumer pays for the export good i in period t, P*t is the price index for all export goods sold in the world market. Xt is assumed to be a stochastic foreign demand shift term, following a log-normal distribution with mean x and variance a\ in period t. Without loss of generality, let P*t(i) and P*t be denominated in dollars. 1 6 The 1 6Here, we just use Pxt(i) to denote the price of export good i, whether the export goods prices is preset in pesos or dollars is endogenously determined and will be discussed in later section. Chapter 1. Twin Dollarization and Exchange Rate Policy 12 demand structure implies that foreign demand for the aggregate domestic export good is unit elastic, and each firm's own elasticity of demand is A, where A > 1. In the export sector, the firm produces with both local labor and imported intermediate goods. The production technology for a typical export firm i is given by 1 7 Yxt(i) = ALxt(i)l-"It{i)" (1.3.13) where A = ^ u ^ l ^ - u I S a constant productivity term, and It(i) represents the imported intermediate goods used in the production of good ?. w is the share of imported intermediate goods in the production of export goods. It also represents the capital dependence of this small open economy on the rest of the world. The aggregate output in the export sector is then defined as YXT = [JQ Y x t ( i ) ~ d i ] * ^ . D e b t C o n t r a c t s We assume export firms must finance their purchases of imported intermediate goods through external borrowing. In the beginning of every period, the firm must borrow from the interna-tional capital market, represented here by a continuum of international lenders, and repay the principal and interest when the export goods are sold at the end of each period. Two types of debt contracts are provided by international lenders.1 8 One is contracted in pesos, and the other is contracted in dollars. The export firm can borrow in either pesos or dollars, but the currency of debt contracting must be chosen ex ante. The existence of external shocks implies that the firm has to take some risk in financing, which is contingent on the debt contract. The amount of the loan (in terms of dollars) that firms need to borrow is equal to the import bill q*It, where q* and It are the dollar price and the quantity of imported intermediate goods re-spectively. Here, q* is assumed to be exogenously given in the world market and to be constant over time. 1 7 This production technology for emerging market economies has been used in previous literature. See, for example, Devereux and Poon (2004). 1 8 T h e process of contract negotiation between international lenders and firms is not modelled, since our focus is to study how the firm chooses a debt contract, given the menu of contracts. We assume that the contracts are exogenously posted by the international lenders, and firms' choices in this small open economy have no effect on the design of contracts. Chapter 1. Twin Dollarization and Exchange Rate Policy 13 It is assumed that, if the firm borrows in pesos, the repayment is subject to the ex post gross domestic interest rate 1 + it+i, where it+i is the domestic interest rate between period t and t + l . 1 9 In other worlds, one cannot write a peso contract in terms of a fixed domestic interest rate. This assumption captures the fact that borrowing through short-term local currency debt makes the firm vulnerable to shocks to the domestic interest rate at which the debt is rolled over. 2 0 With a peso contract, to import one unit of intermediate good, the firm will borrow q*St-i in pesos, where St-i is the exchange rate in period t — 1. Therefore, in this case, the firm's total cost in terms of pesos to use one unit of imported intermediate good is given by qF° = q*(l + it+l)St-i (1.3.14) If the firm borrows in dollars, the repayment is at the gross world interest rate 1 + ?"j + 1 , 2 1 where r%+1 is the world interest rate between period t and t + I, and it is assumed to follow a stochastic process given by r*t+1 = f* .+ et (1.3.15) where et is an i.i.d. shock with mean zero and variance <x2 in period t, and r* is the steady state world interest rate. Therefore, with a dollar contract, the firm's total cost in terms of pesos to use one unit of imported intermediate good is given by qf°llar = q*(l+r*t+1)St (1.3.16) That is, when the export firm finances import purchases with dollar debt, it is vulnerable to both the exchange rate risk and the world interest rate shock. 1 9 In general, we can assume that the gross return for a peso contract and a dollar contract are functions h(it+i,rl+1, St, St-i) and /(it+i, rj+i, St, St-i), respectively. As long as h(.) is more sensitive to the domestic interest rate than /(.), the results in our model will still hold. 2 0Mckinnon and Schnabl (2004) explain why firms in developing countries cannot borrow at a fixed interest rate. This is because these firms tend to be small, without well-developed accounting systems, and cannot issue bonds in their own name. Therefore, even firms with longer term projects must roll over short-term bank loans, or, at best, borrow at medium term with the variable interest rate tied to short rates. 2 1 Since international lenders are risk-neutral, the expected return from each kind of contract should be equalized. Thus, Et-i{St-i{l + it+i)-gi} = [1 -HrJ+i] = 1 + r*. This condition can be used to determine the level of exchange rates or the dynamic path of exchange rates, and has no direct impact on the firms' decisions since the firms' dollarization decisions only depend on the second moments of aggregate variables. Chapter 1. Twin Dollarization and Exchange Rate Policy 14 Given the export firm's production technology, marginal cost is given by MCt = Wt ~wq?, where qt can be either gf e s o or qdollaT. As qt varies with the way the firm finances its debts, the marginal cost will also be affected. For simplicity, we normalize q* = 1, so that the marginal cost under a peso contract and a dollar contract, respectively, can be derived as follows: MCPeso = Wi-^Y + it+JSt.!]", MCtMar = W}-^[{1 + rt+1)Stf (1-3.17) Since the nominal wage is predetermined, the change in the marginal cost depends com-pletely on the change in the domestic interest rate, the exchange rate, or the world interest rate. As the firm has to choose the currency of debt denomination ex ante, each firm will choose the debt contract that delivers a higher expected profit. Nevertheless, the expected profit is also affected by the currency of export pricing. Therefore, when the export firm chooses the currency of debt contracting, the currency of export pricing should be considered as well. T h e C u r r e n c y o f E x p o r t P r i c i n g Whatever currency it chooses, the firm must set its export price before the state of the world is realized. Given the debt contract that the firm chooses, that is, given the marginal cost, if the export firm sets its price in pesos, then the expected discounted profit in period t — 1 is M P " ° ( i ) - Et-iWPZrii) - M C ( ) ( ^ ) - ^ ] (1.3.18) where dt is the stochastic discount factor, which is equal to the marginal utility of consumption -p^v, as all export firms are assumed to be owned by domestic households. P^so(i) is the price set by firm i in terms of pesos. P*t is the export sector's price index, which is taken as given by all firms. 2 2 If the export firm sets its price in dollars, then the expected discounted profit in period i — 1 is r>*dollar / - \ v" EndMar{i) = Et-ildtiStP^ii) - MCt){ '* (1-3.19) "xt "xt where P * ^ ° " a r ( i ) is the price set by firm i in terms of dollars. 2 2Since not necessarily all firms will choose the same currency of export pricing, we will just use P£t to denote the aggregate price index. How P£t is determined will be given by equation (1.4.5) in Section 1.4. Chapter 1. Twin Dollarization and Exchange Rate Policy 15 The optimal price policies which maximize the expected profits in equations (1.3.18) and (1.3.19) can be derived as follows, respectively: uPesor\ \ Et-l(MCtS£Zt) - ^dollar,^ _ \Et-\(MCtZt). P * W = A Ek-tftZt) ' P x t { l ) ~ X Et^(StZt) ( L 3 - 2 0 ) where A = is the markup, and Zt = d t P * t A - 1 X t represents a market demand factor. Using the above optimal price policies, we can compute the expected discounted profits as: E]JPeso = xlEt^iSfZt^lEt^iSfZtMCt)}1-* (1.3.21) Endoiiar = A [ j B t _ 1 ( 5 i z t ) ] A [ E t _ 1 ( Z 4 M C t ) ] 1 - A (1.3.22) where A = X Z I ( X 3 T ) ~ A - From now on, we will omit the time subscripts in the approximation of expected profit functions. Without explicit statement, all the expectation and variance operators are conditional on the information set in period t — 1. S t r a t e g i e s a n d P a y o f f Since the firm has to choose both the currency of debt contracting and the currency of export pricing ex ante, the set of feasible pure strategies for each firm is given by 6 = [s i , s 2 ,S3 ,S4] = [(p,p),(p,d),(d,p),(d,d)] (1.3.23) For each strategy Si G 0, i € (1,2,3,4), the first letter represents the currency of export pricing, and the second represents the currency of debt contracting. The expected profit associated with the strategy s; is defined as EU(si). Thus, the optimal strategy Sj for the firm can be determined by the following condition: EU(si) > EIL(s-i), Vsi e 6 (1.3.24) To find the optimal strategy, following Devereux, Engel and Storgaard (2004), we simplify the expected profit as a function of aggregate variables by using a second order approximation approach. From now on, lower-case letters are the natural logs of their upper-case counterparts, that is, x = ln(X). Defining 7T ( S J ) = lni?II(s;)-n* as the difference between the log of expected Chapter 1. Twin Dollarization and Exchange Rate Policy 16 profit and the common component in the log of expected profit for any strategy Si E 0 , 2 3 we may rewrite the expected profit associated w i t h each strategy as : 2 4 A 2 1 - A n(p,p) = y o f + — — o2mcp + A ( l - X)cov{mcp, s) + (1 - \)cov(mcp, z) (1.3.25) A 2 1 - A n(p, d) = y C T s "I 2—°rncd + ^ (1 — \)cov(mcd, s) + (1 — X)cov(mcd, z) (1.3.26) <d, p) = ^cr2 + l-^o2mcp + (1 - \)cov{mcp, z) (1.3.27) 7r (d , d) = ^o-2 + ^-^<r2mcd + (1 - A)co«(mcd, z) (1.3.28) The approximated expected profits are functions of condi t ional variance and covariance terms of the log exchange rate, marginal cost, and market demand. Since a l l terms are af-fected by monetary pol icy rules, we can express the above conditions as functions of the under lying monetary pol icy parameters. Nevertheless, before solving the model , we can derive the following Lemmas from equations (1.3.25)-(1.3.28). L e m m a 1 Given a debt contract, the firm will set export price in pesos if and only if iff 2 — cov(mCk, s) > 0, k=p,d (1.3.29) T h e in tu i t ion behind this op t imal condi t ion for export pr ic ing is as fo l lows . 2 5 W h e n the export firm sets export prices in domestic currency, exchange rate f lexibi l i ty can stabilize the demand by adjusting its relative prices. Therefore, as shown by equation (1.3.18), the firm's expected profit is convex in the exchange rate. W h e n the export f i rm sets prices i n foreign currency, exchange rate changes w i l l only have wealth effects on the firm's revenue, so the expected profit is linear i n the exchange rate, as i l lustrated by equation (1.3.19). Hence, higher exchange rate volat i l i ty w i l l encourage the firm to set prices i n domestic currency. Meanwhile , when the firm sets export prices in domestic currency, its demand is direct ly sensitive to 2 3Here, II* is In^+^al + Xcov(z,s), and ln£^ is the log of steady state profit. 2 4 F o r the detailed derivation, see the Appendix. 2 5 This optimal condition for export pricing was first derived by Devereux, Engle and Storgaard (2004). Chapter 1. Twin Dollarization and Exchange Rate Policy 17 exchange rate movements, which results in an extra covariance term between the marginal cost and the exchange rate. Th i s w i l l increase the firm's expected cost, ceteris paribus. Thus , these two effects have opposite impacts on the firm's choice of the export pr ic ing currency, as shown by equation (1.3.29). Since, i n our model , the covariance term between marginal cost and the exchange rate varies w i t h the debt contract, the export pr ic ing currency decisions also w i l l depend on the choice of debt contract. L e m m a 2 When the firm sets export prices in pesos, it will borrow in pesos if and only if 7;°~mcP + Xcov(mcp, s) + cov(mcp, z) < ^ c r ^ + \cov(mcd, s) + cov(mcd, z) (1.3.30) When the firm sets export prices in dollars, it will borrow in pesos if and only if \<Tmcp + cov(mcp, z) < i c r ^ + cov(mcd, z) (1.3.31) Equat ions (1.3.30) and (1.3.31) show that it is op t imal for the firm to choose the currency of the debt contract, which has a smaller marginal cost effect on the expected profit. W h e n the firm sets export prices i n pesos, the marginal cost effect of the debt contract, i.e., the terms associated w i t h marginal cost in the expected profit function, is composed of three terms: the variance of marginal cost; the covariance of the marginal cost and the exchange rate; and the covariance of the marginal cost and the market demand. The first two terms are positive and reduce the expected profit as they both increase the expected cost, while the effect of the th i rd te rm on the expected profit is ambiguous, as it varies w i t h other firms' strategies. W h e n the firm sets export prices in dollars, as the expected profit is linear i n exchange rate, no covariance term exists between the marginal cost and the exchange rate. Therefore, the marginal cost effect w i l l only involve the first and the th i rd terms. 1.3.4 Monetary Policy Fol lowing recent literature (Woodford 2003 and C l a r i d a et al 2000) i n abstract ing from the de-tails of the monetary mechanism, we s imply assume that the monetary author i ty is commit ted to a domestic interest rate targeting rule l + i m = ( l + f * ) ( f ^ , 7 > 0 (1.3.32) Chapter 1. Twin Dollarization and Exchange Rate Policy 18 where f* is the steady state wor ld interest rate. The parameter 7 represents the degree of exchange rate f lexibil i ty or the coefficient of exchange rate intervention. A s long as 7 > 0, there is a determinate equi l ibr ium value for the nomina l exchange rate. 7 is exogenously given and measures the preference of pol icy makers. T h e higher is 7, the closer the monetary rule approximates a pegged exchange rate regime, where the target for exchange rate peg is SQ. W h e n 7 approaches zero, it represents a flexible exchange rate regime. 1.3.5 E q u i l i b r i u m G i v e n the stochastic processes of shocks (X, e) and the exchange rate regime (7), a symmetr ic equi l ibr ium has the following properties: (a) Households preset wages and choose consumption and domestic bond holdings to maximize expected u t i l i t y subject to their budget constraints; (b) The export f i rm chooses the op t imal strategy Si to maximize the expected profit; (c) G iven the op t ima l currency strategies, firms set prices to maximize expected discounted profit; and (d) Labor , goods and bond markets clear. In equi l ibr ium, we have Ym = Cm = (1 - (1.3.33) "Nt Yxt = { f1 Yxt(i)^di}^ = | i (1.3.34) Jo rxt LNt + / Lxt{i)di = Lt (1.3.35) J O Bt = 0 (1.3.36) The first two equations represent the goods market clearing condit ions for non-traded goods and export goods, respectively. The th i rd equation is the domestic labor market clearing condi t ion, and the last one is the domestic bonds market clearing c o n d i t i o n . 2 6 2 6 In a symmetric equilibrium, we will have Bt(j) = 0, V7. Chapter 1. Twin Dollarization and Exchange Rate Policy 19 1.4 Export Firms' Optimal Strategies In this section, we analyze how monetary policy affects firms' decision. Since the firm's expected profit is a function of the second moments of endogenous aggregate variables (the exchange rate and other macroeconomics variables), to find the optimal strategies of firms, we have to solve the general equilibrium model to get the endogenous aggregate variables. Besides monetary policy, the determination of these variables also depends on the market structure -the distribution of firms who choose different optimal strategies. In other words, every export firms' payoff will be affected by other firms' optimal strategies. In equilibrium, the distribution of firms must be supported by each firm's optimal decision. To solve for the equilibrium, we first assume that there exists a distribution of firms with dif-ferent optimal strategies and derive the aggregate variables given the conjectured distribution. Then every firms' optimal strategy can be determined. Finally, to find out if the conjectured distribution is an equilibrium, we must check if these strategies support that distribution. We denote the distribution of firms as Q = ( / i i , u.2, u2,/14), where / X j € [0,1] is the endogenously determined number of firms who choose the optimal strategy S j € 0 , and Ef=1/ii = 1. For example, in an economy, if all firms set export prices and borrow in dollars, then the economy can be represented by fl = (0,0,0,1). We solve the model by log-linearizing around a non-stochastic, symmetric steady state, which is described in the Appendix. The Appendix also gives the complete solution of the model, so we will just outline the important and intuitive steps of the solutions. Given the log-linearized system, the deviations of the exchange rate and other macroeconomic variables from their t—1 expectations are solved in terms of external shocks (Xt, et). From now on, kt — log(Kt) — log(K), where K is the non-stochastic steady state value of variables Kt, and kt+j = h+j - Et-ih+j, 3 > 0-Chapter 1. Twin Dollarization and Exchange Rate Policy 20 1.4.1 M o d e l Solution In this subsection, we focus on deriving consumption, the exchange rate and other endogenous variables given the dis t r ibut ion of firms il.2T T h e output and net income of export sector G i v e n the d is t r ibut ion of firms fl, there are four types of firms i n the export sector, and the demand for the goods produced by each type of firms is given by ppp v Yg=({$r)-Xj£ (1-4.1) ppd y Yxpf = (i^)-X^ (1.4.2) ^tPxt Pxt *dp y rxt rxt Yxf = ( ^ C X £ - (1.4.4) rxt rxt where Pxt and Yxl are the prices and demand (or output) for the firms who choose the op t imal strategy S j . T h e prices w i t h an asterisk are i n terms of dollars. The export sector's price index P*t i n terms of dollars is then given by ppp pPd , P*xt = M ^ f ) 1 - * + i M ^ f ) X " A + ^{P*ft)l-X + ^(P'it)1-'}— (1.4.5) Log-l inear iz ing the above equation, we have pit = MI(P£? - s t ) + M d t - s t ) + M 3 P ; f + ( 1 . 4 . 6 ) A s a l l prices pxt (or p**') are preset i n per iod i - 1 , pxt - Et-ipxt = 0. Thus , p*t = pxt — Et-ipxt = — {pi + P2)St- Us ing the market clearing condi t ion Yxt = , the deviat ion of (log) output of the export sector from its t — 1 expectation is given by yxt = xt~ V*xt = xt + (MI + M2 )st (1.4-7) 2 7Note that Q should be time variant, but we focus on a stationary equilibrium, so we drop the time subscript for a Chapter 1. Twin Dollarization and Exchange Rate Policy 21 where u\ + fi2 is the number of firms who set their export prices in pesos. Equation (1.4.7) shows that the more firms choose to set prices in pesos, the more stable export output will be. If all firms set export prices in dollars, that is, fi\ + u2 = 0, then the exchange rate cannot adjust the terms of trade to stabilize output, so the export sector output will fully reflect the demand shock Xt-In equilibrium, the household's expenditure is affected by the net income of the export sector, which is equal to wage income plus the profit from the export sector. For simplicity of presentation, we define the net income of export firm i as Gxt(i), and it is given by Gxt(i) = WtLxt(i) + Uxt(i). (1.4.8) For firm i, WtLxt{i) = (1 - tu)MCt(i)Yxt{i) and Uxt(i) = Pxt(i)Yxt(i) - MCt{i)Yxt{i), thus we have Gxt{i) = Pxt(i)Yxt(i) - uMCt(i)Yxt(i) (1.4.9) Given the distribution of firms f2, the total net income of the whole export sector can be rewritten as 4 Gxt = J2^G*t t 1 - 4 - 1 0 ) 8=1 where GsJt is the net income of the firm who chooses the optimal strategy S j . As shown in the Appendix, log-linearizing equation (1.4.10) yields the following relationship. gxt = b§t + it - £et (1.4.11) where b = ea — (a — l)[v + 7(1 — v)]u> + (1 — e) and £ = (a — l)wv. Note that a = A_^AA_1^, the steady state ratio of output to net income in the export sector (^-)- e = /X3 + u.4, is the number of firms who set export prices in dollars, v = ji2 + Hi, is the number of the firms who borrow in dollars. Therefore, (e, v) can be used to measure the degree of dollarization in the economy. For example, (e = l,v = 1) represents an arrangement of twin dollarization. Intuitively, b measures the total effect of exchange rate changes on the net income of the export sector. There are three channels through which the exchange rate can affect Gxt- The first one is the wealth effect ea. For those firms who sell goods in dollars, an exchange rate depreciation will increase the nominal profit in terms of domestic currency. The second channel Chapter 1. Twin Dollarization and Exchange Rate Policy 22 is the cost effect, (a — l)[v + 7(1 — v)]u. This is because an exchange rate depreciation causes an increase in the cost of intermediate goods import, which reduces net income. The third mechanism is the demand effect (1 — e). 2 8 For those firms who set price in peso, an exchange rate depreciation increases the demand for their goods and thus their net income. Therefore, adding these three effects together, we have the total effects of exchange rate changes on net income. Nevertheless, the sign of b is ambiguous and it depends on the exchange rate regime 7 and the distribution of firms Q in equilibrium. From equation (1.4.11), we can also see that net income will be affected by the foreign demand shock X and the world interest rate shock e directly. When more. firms borrow in dollars, the world interest rate shock has a bigger effect on the net income of export sector. In particular, if all debts are borrowed in pesos, then the world interest rate has no impact on the economy. T h e b u d g e t c o n s t r a i n t In equilibrium, the household's budget constraint in period t is given by ptct = wtLt + n x t (1.4.12) The market clearing conditions for the non-traded goods and labor market implies WtLm = (1 — a)PtCt, and Lt = Lm + Lxt, so equation (1.4.12) can be rewritten as aPtCt = WtLxt -f Tlxt = Gxt (1.4.13) It implies that the import expenditure of households is equal to the net income of the export sector. Log-linearizing and using the CPI price index Pt = Wl~aS? and equation (1.4.11), we have ct + (a- b)st = x t - £ e t (1.4.14) Equation (1.4.14) implies that external shocks to the economy will be absorbed by changes of current consumption and the current exchange rate. Since the household has no access to international bond market, the current account dynamics are completely shut down in this model. The term a — b represents the distribution of shock absorption between changes of current consumption and changes of exchange rate. 2 8 W e assume unitary income elasticity of foreign demand in the model, so we get 1 — e, otherwise, it should be multiplied by an elasticity term. Chapter 1. Twin Dollarization and Exchange Rate Policy 23 T h e E u l e r equation Combining the Euler equation (1.3.8) with the domestic interest rate targeting rule (1.3.32), we have 1 0 „ r PtCPt P E t [ - ^ r - \ (1.4.15) ( l + r * ) ( f ^ U f t f i C f + i Again, log-linearizing the above equation, and using the CPI price index and the preset wage equation (1.3.7), we have pct + {a + 7)s t - Et(^ct+l + st+i) = 0 (1.4.16) a As shown by equation (1.4.14), the external shocks are fully absorbed by changes in current consumption and the current exchange rate. Therefore, with the assumption that the shocks Xt and tt are i.i.d, it is easy to show that Et(^ct+i + h+i) = 0 (see the Appendix for details), which yields pct + (a + 7) s t = 0 (1.4.17) This implies that the changes of the current exchange rate have two effects on the household's intertemporal consumption smoothing: depreciation of the exchange rate increases the im-ported goods price and thus the domestic CPI through the term ast, which reduces current consumption; Meanwhile, it raises the domestic interest rate through the parameter 7, which also decreases current consumption. The magnitude of the total effect depends on the degree of openness a and exchange rate flexibility 7. Note that it is independent of the distribution of firms 17. T h e determination of exchange rate and consumption Putting equations (1.4.14) and (1.4.17) together, we may solve for c~t and st, ^ = I ^ 1\ - * £ ' ) > * = -( I W , * . ^ ^ ~ ^ ( ) ( 1 A 1 8 ) (pb + j) - a{p- 1) a{p - 1) - [pb + 7) Using the domestic interest rate targeting rule (1.3.32), we can derive the deviation of the domestic interest rate from its i — 1 expectation 1 + ^ + 1 = ~f /°' ( h J . ^ £ t ~ ^ (1A19) a{p- 1) - (pb + j) The above equations give the determination of consumption, exchange rate and domestic interest rate, which are all still conditional on CI. Note that b and £ are both functions of Q. Chapter 1. Twin Dollarization and Exchange Rate Policy 24 Although the equilibrium values of b and £ cannot be determined yet, from equations (1.4.18) and (1.4.19), we still can roughly find out how the aggregate variables respond to external shocks in two extreme cases. As 7 —> 0, external shocks are mainly absorbed by the flexible exchange rate, thus consumption is less affected by external shocks. Meanwhile, the domestic interest rate is quite stable. As 7 —> 0 0 , the exchange rate is almost fixed and cannot insulate consumption from external shocks. In this case, the domestic interest rate can absorb part of external shocks. But, without current account dynamics, households cannot spread the shocks over to future periods, therefore, current consumption becomes volatile. Equations (1.4.18) and (1.4.19) show that consumption, the exchange rate and the domestic interest rate depend critically on exchange rate flexibility 7, but the effects of Q ( b, £) on these endogenous variables are relatively small, especially in the two extreme cases. 1.4.2 T h e O p t i m a l Strategies Given the above solution, we can calculate all conditional variance and covariance terms of the (log) exchange rate, consumption, marginal cost and other variables in period t — 1 by using the property that covt-i{xt,yt) = covt-i{xt,yt) = covt-i{xuyt)^xuyt- Therefore, we can express firms' payoff functions as a simple function of second moments of external shocks (CT2,<72), and then find the optimal strategies. For simplicity, we drop the time subscript in the following analysis. Using equations (1.3.17) and (1.3.32), we can express all conditional variance (covariance) terms associated with the (log) marginal cost (mcp, mcd) as follows, amcP = ~/2u>2(j2 cov(mcp, s) = 7WCT2 cov(mcp, z) = ju>cov(s, z) (1.4.20) amcd — a ; 2 ( c r s + °~e )> cov(mcd, s) = uj(a2s + crse), cov(mcd, z) = u(cov(s, z) + cov(e, z)) (1.4.21) where exchange rate volatility CT2 and the covariance between the exchange rate and the world interest rare shock oS(L can be derived as, ^ - u . - i i V / ' - 5 * ^ ' - ^ - r V ^ {L422) Chapter 1. Twin Dollarization and Exchange Rate Policy 25 Using the fact that Z = -^P*x~lX, we have z = [7 - (A - 1)(1 - e)]s + x. Thus, the covariance terms cov(s,z) and cov(z,e) can be derived as cov(s,z) = [7 - (A - 1)(1 - e)}*2 + a { p _ 1 } _ { f ) b + j } 4 (1-4.23) caviz, e) = " P f " ( , A " 1 i ( i " e ) 1 ^ 2 (1-4-24) To further simplify the payoff functions, we may rewrite the second moments as amcd = ^bio-2 cov(mcd, s) = u)b2a2, cov(s, z) = <j)\o-2s, cov(s, e) = faa2 (1.4.25) where h = 1 + b2 = 1 + ^ , = ^ a n d «£2 = £ 2 ^ 1 . Substituting these variance and covariance terms into equations (1.3.25)-(1.3.28), we can rewrite the approximated payoff function for each strategy as a linear function of a2. Note that 61, 62, 4>i a n d 4>i a r e a n functions of 7, b and f. 7 r (p ) P ) = i [ A 2 + 7 V ( 1 - A) + 2 7 w ( l - A)A + 2(1 - \)^4>x\a 2s (1.4.26) 7r(p, d) = - [ A 2 + w%(l - A) + 2w62(l - A)A + 2(1 - A)w(0i + <£ 2)]CT 2 (1.4.27) <d,p) = I [A + 7 ^ ( 1 - A) + 2(1 - A ) 7 ^ i ] a 2 (1.4.28) 7r(d,d) = i[A + w 2 6i(l - A) + 2(1 - A)w(0i + 0 2 )K 2 (1.4.29) From Lemmas 1 and 2 (equations 1.3.29-1.3.31), we may derive the following conditions. 7v(p,p) > n(d,p), if 7 < ^ (1.4.30) n(p,d) >7r(d,d), t / 62 < ^ (1.4.31) ir{p,p) > 7r(p,d), t / w(7 2 -fei) + 2 A ( 7 - 6 2 ) + 2 ( 7 < A 1 - ^ 1 - 0 2 ) < O (1.4.32) Chapter 1. Twin Dollarization and Exchange Rate Policy 26 w{d,p)>Tr(d,d), if w ( 7 2 - bi) + 2(70! - 0i - <h) < 0 (1.4.33) Therefore, we may establish the following proposition which describes the optimal strategies when 7 approaches zero and 7 approaches infinity. 2 9 Proposi t ion 1 There always exist two critical values 7^ and 7^ (7^ < "1H)> such that for any 7 € (0,7L), the optimal strategy for each firm is (p,p), that is, all firms set export prices in pesos and borrow in pesos. The distribution of firms is Q, = [1,0,0,0]. For any 7 S ( 7 ^ , 0 0 ) , the optimal strategy for each firm is (d, d), that is, all firms set export prices in dollars and borrow in dollars, and Q, = [0,0,0,1]. The proof is given in the Appendix. Here, we focus on equations (1.3.29)-(1.3.31) to explain the intuition behind the proposition. First, we look at the case where exchange rate flexibility is low. From Lemma 1, we know that there are two effects which have opposite impacts on firms' choices of export pricing currency. When the export firm sets price in domestic currency (foreign currency), revenue is convex (linear) in the exchange rate. Hence, exchange rate volatility encourages the firm to set prices in its own currency (peso). Meanwhile, when the firm sets price in pesos, the positive covariance between marginal cost and the exchange rate will reduce firm's expected profit. As exchange rate flexibility decreases, regardless of which currency the firm chooses to contract its debts, the first effect will be dominated by the second, as 7 gets bigger and CT2 gets smaller. 3 0 As a result, the firm will always choose to set prices in dollars when the exchange rate is fixed. 3 1 Why do firms borrow in dollars when exchange rate flexibility is low? The impact of exchange rate flexibility on the firm's choice of debt contract is mainly through the borrowing 2 9 W e assume w < 0.5 here since it is consistent with the calibration of ui for most emerging market economies. 3 0 This is because Cov(s, mcp) = 701a2 > Cov(s,mcd) = u>(<72 + <rse) > 0, when 7 —* 00 and <r2 —> 0. 3 1 In our model, since we use a general interest rate rule, <r2 is always positive, even in the fixed exchange rate regime where 7 approaches infinity. To make our result hold when CT2 = 0, we can change our model slightly according to Devereux, Shi and Xu (2004). Assume that the firm incurs a cost of adjusting prices ex-post and this cost arises only when the price facing consumers is adjusted. If the firm sets prices in its own currency, it faces a fixed nominal cost. Under such an environment, the optimal pricing condition can be rewritten as icr 2 — cov{md,s) > 5C, where 8C is a positive function of the menu cost. With this slight revision, our result holds even in the case <r2 = 0. Chapter 1. Twin Dollarization and Exchange Rate Policy 27 cost channel. In our model, the borrowing cost of a peso contract is subject to ex post domestic interest rate risk, while the borrowing cost of a dollar contract depends on the world interest rate shock and exchange rate risk. As shown by equations (1.4.18) and (1.4.19), when exchange rate flexibility is low, the domestic interest rate will absorb the foreign demand shock and the world interest rate shock, and thus becomes quite volatile. Hence, borrowing in pesos implies a higher expected cost than borrowing in dollars, which is only subject to the world interest rate shock (exchange rate risk is almost zero). Therefore, firms tend to borrow in dollars as exchange rate flexibility decreases. When exchange rate flexibility is high (7 is low), from equations (1.4.18)-(1.4.19), we know that the exchange rate absorb most foreign demand shocks and world interest rate shocks, so the domestic interest rate is stable and the exchange rate is volatile. Therefore, borrowing in pesos implies a stable borrowing cost for firms. As for the currency of price setting, with high exchange rate flexibility, the first term in equation (1.3.29) is always positive and the second term depends on the cost structure and the way in which firms borrow. If firms borrow in pesos, the covariance between marginal cost and the exchange rate is smaller than that under the dollar debt contract, which is subject to exchange rate risk. Hence, setting export prices in pesos gives firms a higher expected profit. Intuitively, this is because, under peso currency pricing, exchange rate changes can adjust the firm's relative prices, and then the firm can stabilize the demand for its export goods. Therefore, when exchange rate flexibility is high, firms will borrow in pesos and also set export prices in pesos. Proposition 1 shows that exchange rate policy is the key factor affecting firms' dollarization decisions. From equations (1.4.30) - (1.4.33), however, we find that ui, the share of imported intermediates in the production of export goods, also can play a role in firms' dollarization decisions. If u> is small enough, then it requires a lower exchange rate flexibility (a bigger 7) to induce all firms to choose twin dollarization. If ui is big, however, then firms will choose to set prices in dollars and borrow in dollars even in the case where 7 is small. In other words, the magnitude of u will affect the impact of exchange rate flexibility on the degree of dollarization. The intuition is straightforward, a higher u> implies a higher share of borrowing cost in the total cost of production for export firms. As firms have to finance import purchases by external borrowing, when u> is bigger, the benefits of twin dollarization will be bigger. Thus, even a Chapter 1. Twin Dollarization and Exchange Rate Policy 28 small 7 will cause export firms to choose twin dollarization. The dependence of export firms on the world market in the supply of inputs (measured by w in this model) is also an important characteristic of East Asian economies that affects the degree of dollarization. In the above discussion, we focus on the optimal strategies for firms in two extreme cases where 7 < 7L and 7 > 7#. Now, we analyze the optimal strategies for firms in the intermediate case, where 7L < 7 < 1H- AS the optimal strategies for firms in intermediate cases cannot be solved analytically, we resort to a numerical solution. 1.4.3 N u m e r i c a l R e s u l t s C a l i b r a t i o n Our model has only a few parameters that need to be calibrated (Table A.4). The coefficient of risk aversion p, is set to 2 as is commonly assumed in the literature. The discount factor B is calibrated at 0.96, so that the steady state annual real interest rate is 4%. The elasticity of substitution across individual export goods A is chosen to be 7, which implies a steady state markup of 16%. This is slightly higher than the common value of 10% (e.g., Basu and Fernald 1997) used in the literature for industrial economies. As pointed out by Cook and Devereux (2001), however, markups are usually higher in emerging markets. 3 2 Following Devereux and Poon (2004), we set the share of intermediate goods in production w=0.4, which is consistent with the estimates for intermediate imports as a fraction of G D P in Braggion et al (20 03). 3 3 a is set to equal 0.4, which implies that the share of non-traded goods in the consumer price index is set to 0.6. This is close to the evidence cited in Schmitt and Uribe (2000) for Mexico, and by Cook and Devereux (2001) for Malaysia and Thailand. With ct = 0.4 and u — 0.4, the total expenditure on imported goods (including the imported intermediate goods in the export sector) is about half of the GDP. Thus, the steady state debt to (GDP) ratio is about 17%.34 The standard deviation of (log) foreign demand shock ax is set to 4%, so the standard deviation of log G D P is about 2%. Finally, the standard deviation 3 2Rotemberg and Woodford (1998) set A = 7.66, Cook and Devereux (2001) choose A = 6. 3 3 In our model, u> also represents capital intensity in traded goods production. 3 4 T h e value in Cook and Devereux (2004) is about 27.5%, but the debt in their model includes the debt of the non-traded goods sector. Chapter 1. Twin Dollarization and Exchange Rate Policy 29 of the world interest rate shock e is set to 1%.35 Given the calibration of parameters, we can solve for the two critical values JL and 7#-. Table A.5 lists the changes of fl, the distribution of firms when 7 increases. Recallthat m is the number of firms who choose the strategy S j . To highlight the extent of dollarization, we also report the degree of export pricing dollarization e and the degree of liability dollarization v separately in the last two rows of Table A.5. From Table A.5, we can see that, as exchange rate flexibility decreases (7 increases), both e and v increase, implying that the degree of dollarization increases with 7. The intuition is straightforward. As exchange rate flexibility decreases, the benefit of peso currency pricing and peso debt decreases, which induces more firms to set prices in dollars or to borrow in dollars. This finding also suggests that a policy that increases exchange rate flexibility may help to reduce the degree of dollarization in the economy, especially in some intermediate exchange rate regimes, as in these regimes the distribution of firms fl is sensitive to exchange rate flexibility.36 We also find that multiple equilibria exist for some 7 due to the strategic complementarity among firms. For instance, for 7 G (13.368,14.412), two equilibria are present.3 7 The first is fl = [0,1,0,0], where all firms follow the strategy (p,d). The second is fl = [0,0,0,1], where all firms choose (d, d). The existence of multiple equilibria comes from the fact that one firm's dollarization decision will change the profit ranking of different strategies for all firms. Fox example, when 7 G (13.368,14.412), the difference between payoffs, of strategies (p, d) and (d, d) is small, and it decreases with e, the degree of export pricing dollarization. When e increases, the sign of \a2 — cov(mcd,s) will go from positive to negative. Therefore, a slight change of e will affect the relative ranking of Tr(p,d) and 7r(d,d). Intuitively, if one more firm set its price in dollars, it will increase the benefit of all firms who choose the dollar currency pricing, implying that the strategy (d, d) can deliver a higher payoff than strategy (p, d). As a result, all firms will follow the strategy (d, d). Obviously, in these intermediate exchange rate 3 5 W e set 6 = 11 and 77 = 1 in Table A.4, these two parameters have no impact on firms' dollarization decisions, but will be used for the welfare calculation. 3 6 T h e strategy (d,p) is never an optimal strategy in the calibrated model. 3 7Another equilibrium is possible with Q = [0,1 -114 (7) , 0, \n (7)] , where fj,4 (7) is the number of firms choosing the strategy (d, d). In this equilibrium, the strategy (p, d) and (d, d) are indifferent for firms, and /J4 (7) is a function of 7. Nevertheless, it cannot be a stable equilibrium due to the strategic complementarity among firms. Chapter 1. Twin Dollarization and Exchange Rate Policy 30 regimes, strategic complementari ty has a b ig impact on the d is t r ibu t ion of firms. Nevertheless, exchange rate flexibil i ty is s t i l l the key factor for firms' dol lar izat ion decision. 1.4.4 D i s c u s s i o n In the previous subsection, we have shown that, i f exchange rate intervention (7) is sufficiently large, then twin dol lar izat ion w i l l be an opt imal strategy for a l l firms. In other words, a fixed exchange rate w i l l cause t w i n dol lar izat ion. Th i s finding is consistent w i t h the observation i n East A s i a n emerging market economies, but is in contrast to the "fear of floating" literature, that argues that l iabi l i ty dol lar izat ion causes the fear of floating. In our model , i t is the fixed exchange rate itself that leads to both l iab i l i ty and export pr ic ing dol lar izat ion. In this sense, our finding builds a new linkage between fixed exchange rate regimes and dol lar izat ion. Recently, fixed exchange rate regimes have been blamed for the over-borrowing and even financial t u rmoi l i n many East A s i a n economies. Fox example, M c k i n n o n and P i l l (1998) argue that fixed exchange rates encourage excessively risky, unhedged external debts. Cook and Devereux (2001) explore the linkage between exchange rate regimes and capi ta l inflow in a smal l open economy. T h e y show that fixed exchange rates can lead to government subsidies to encourage international borrowing. Our result differ from their i n that we focus on s tudying the impact of fixed exchange rate regimes on firms' currency choices of bo th debt contracting and export pr ic ing. O u r research is also related to the literature on endogenizing l i ab i l i ty denominat ion. T h a t l i terature emphasizes that l iab i l i ty dol lar izat ion is caused by market or ins t i tu t ional failure in emerging market economies. For example, Jeanne (2000) argues that foreign currency debt arises because of commitment or signall ing problems at the firm level. Jeanne (2003) also emphasizes the lack of monetary pol icy credibili ty. Cabal lero and K r i s h n a m u t h y (2003) at-tr ibute the presence of foreign currency debt to the lack of domestic financial development. B y contrast, we show that dol lar izat ion can be an op t ima l arrangement for private agents, and is main ly affected by exchange rate policy. In addi t ion, our u t i l i t y based general equi l ibr ium model sett ing has a natural advantage over the recent l i terature that relies on par t ia l equilib-r i um or reduced form model , since it allows us to study the relevant welfare impl ica t ion . O f course, the key contr ibut ion of our paper to the li terature is that we investigate the common Chapter 1. Twin Dollarization and Exchange Rate Policy 31 cause of both types of dollarization, rather than liability dollarization alone. Our findings imply that twin dollarization is optimal for firms under a fixed exchange rate regime, but is it a beneficial arrangement for the whole economy in terms of welfare? We will answer this question in the next section. 1.5 Welfare Implication In this section, we discuss the welfare implications of firms' dollarization decisions for the economy. The welfare measurement we use here is the conditional expected lifetime utility of the representative household at time zero. Following Schmitt-Grohe and Uribe (2004), the expected lifetime utility is computed conditional on the initial state being the deterministic steady state, which is the same for all policy regimes and for the distribution of firms. 3 8 To measure the magnitude of welfare differential across regimes, we define Ck as the percentage change of deterministic steady state consumption that will give the same conditional expected utility EU under regime k, which is conditional on given 7 and fl. That is, Ck is given implicitly by: 1 _ p l V J =EUk (1.5.1) . 1-/3 where a bar over a variable denotes the deterministic steady state of that variable. If Ck > 0(< 0), the welfare under regime k is implied to be higher (lower) than that of the steady state case. Higher values of Ck correspond to higher welfare. The welfare EUk is computed by taking second order Taylor approximations of the struc-tural equations around the deterministic steady state. The system of equations is solved using a perturbation method described in Schmitt-Grohe and Uribe (2004).39 The values of structural parameters are those used in Section 4. To find out whether or not twin dollarization is beneficial for the economy, we report the welfare effect of firms' endogenous dollarization decisions (Table A.6). We know from Proposi-tion 1 that firms will choose to set prices in pesos and borrow in pesos under a flexible exchange 3 8 This choice of initial state has the advantage of ensuring that the economy starts from the same initial point for all policy regimes considered. 3 9 Matlab codes for welfare calculation are available to download at kang Shi' web site, http://grad.econ.ubc.ca/kangshi. Chapter 1. Twin Dollarization and Exchange Rate Policy 32 rate regime and choose twin dollarization under a fixed exchange rate regime. Therefore, we also focus on these two extreme exchange rate regimes, which are represented by 7 = 0.01 and 7 = 900, respectively. From Table A.6, we can see that given any distribution of firms fl, a flexible exchange rate regime is superior to a fixed exchange rate regime in terms of welfare. Therefore, if monetary policy is chosen ex ante endogenously to maximize the welfare of the economy, twin dollarization can never be an optimal strategy for firms in equilibrium. Under a fixed exchange rate regime, however, twin dollarization is not only optimal for firms, but also a beneficial outcome for the economy. We consider this economy in an initial equilibrium where the exchange rate is floating and all firms choose the strategy (p,p). In the case where the firm's pricing and borrowing behavior is given, if the government switches the exchange rate regime from flexible to fixed, then the welfare loss will be equivalent to about 0.214% steady state consumption. If, however, the firms' endogenous behavior is considered, then firms will choose to set export prices in dollars and borrow in dollars, and the welfare loss is about 0.091% steady state consumption. In other words, twin dollarization increases welfare, given that a fixed exchange rate is in place, and the gain in this case is about 0.123% steady state consumption. This implies that endogenizing the currency choices of firms' export pricing and debt de-nomination will help to improve the welfare for the economy given an exchange rate regime. The welfare gain comes from the fact that firms can adjust their currency choices or dollariza-tion decisions according to the changes in exchange rate regime. Table A.7 compares the mean and variance of consumption, labor and other variables for fl = [1,0,0,0] and fl = [0,0,0,1], under a fixed exchange rate regime. From Table A.7, we can see that twin dollarization delivers much higher expected consumption and lower consumption volatility than does the strategy (p,p) under a fixed exchange rate regime, which thus improves the economy's welfare. Intu-itively, twin dollarization can help the economy to evade the volatile domestic interest rate. As a result, the mean and the volatility of marginal cost tend to be lower, which generates higher and more stable profit for export firms. Since export firms are assumed to be owned Chapter 1. Twin Dollarization and Exchange Rate Policy 33 by households, this implies higher expected consumption and low consumpt ion v o l a t i l i t y . 4 0 Therefore, t w i n dol lar izat ion is not only an op t imal strategy for firms, but also delivers higher welfare for the whole economy. Since ui is an important parameter for firms' dol lar izat ion decisions, we investigate the welfare consequences of changes i n OJ under a fixed exchange rate regime (Table A . 8 ) . We find that, C d d i the welfare when al l firms choose twin dol lar izat ion, does not greatly vary, w i t h ui. Meanwhi le , ( p p , the welfare when al l firms choose the strategy (p,p), sharply decreases w i t h u>. Therefore, the welfare gain of twin dol lar izat ion w i l l rise i f ui increases. Since w measures the share of impor ted intermediate goods i n the product ion of export goods, the higher u, the more debts firms need to borrow from abroad. Thus , t w i n dol la r iza t ion w i l l help to avoid welfare losses for economies that rely heavily on the internat ional capi ta l market. Our results imp ly that t w i n dol lar izat ion can br ing welfare gains to the economy i n some environments. Th i s is i n contrast to the welfare implicat ions of l i ab i l i ty dol lar izat ion i n most of the financial crisis literature, which typica l ly emphasizes the welfare losses caused by foreign currency debt. 1.6 Conclusion Thi s paper studies twin dol lar izat ion - l iab i l i ty dol lar izat ion and export pr ic ing dol lar izat ion in East A s i a n economies. We develop a smal l open economy general equi l ib r ium model w i t h nomina l rigidities, where bo th the currency of export pr ic ing and the currency of l iab i l i ty denominat ion are endogenous. Our ma in findings are that firms' dol lar iza t ion decisions depend cr i t ica l ly on exchange rate flexibility, and twin dol lar izat ion w i l l be an op t ima l strategy for a l l firms i f exchange rate flexibil i ty is l imi ted . Hence, our paper builds a new linkage between fixed exchange rate regimes and dol lar izat ion. Furthermore, we find that t w i n dol lar izat ion 4 0 In the case with twin dollarization, the labor supply is also higher. Thus, higher consumption also can be attributed to the change of wage income, but, the increase of wage income cannot be the main source of consumption increase, since the magnitude of labor supply change is not big enough to account for the increase of consumption. Note that real wage is almost without any change in two cases. On the other hand, the higher labor supply leads to more disutility, which decreases welfare, but this effect is dominated by the increase of consumption. Chapter 1. Twin Dollarization and Exchange Rate Policy 34 can br ing welfare gains to the economy under a fixed exchange rate regime. T h i s contrasts w i t h the welfare implicat ions of l iab i l i ty dol lar izat ion i n most of the financial crisis literature. O u r model can be extended i n a number of ways. For example, i t may be interesting to consider the op t imal debt contract design i n our model , so that we can explore the strate-gic interaction between firms' l i ab i l i ty dol lar izat ion and export pr ic ing dol lar iza t ion. Th i s interaction w i l l be especially interesting i f we allow the export output to be the collateral for external borrowing. A n d , once extended to incorporate the debt contract negotiation process, our model also can analyze the effect of default risk and country spread on firms' dol lar izat ion decisions and the welfare of the economy. 35 Chapter 2 Oil Currency and the Dollar Standard 2.1 Introduction Ever since 1971, when the US broke the link between the dollar and gold that had been agreed to at the Bretton Wood Conference at the end of World War II, the dollar has been a global monetary instrument that the United States, and only the United States, can produce by fiat.1 Although the United States has never imposed explicitly its dollar hegemony on other countries, the dollar has been used in the global economy as a reserve currency and a reference currency for international goods pricing and asset trading. McKinnon (2001, 2002) uses the term "dollar standard" to describe the dominance of the US dollar in the world economy.2 Recently, increased attention has been paid to the dominance of the US dollar and its impact on the global economy. For example, Eichengreen (2004) argues that the current international system is composed of a core and a periphery, where the US is the core and the rest of world is the periphery. As the center country, the US can continue running current account deficits because the periphery, Asian and Latin American emerging markets economies are happy to accumulate dollars and resist the appreciation of their currencies against the dollar. In a sense, world trade is a game in which the US produces dollars and the rest of the world produces goods that the dollar can buy. Different from the "core and periphery" literature, which focuses on the role of US dollar as an international reserve currency, Devereux, Shi, and X u (2004) investigate one particular aspect of the multi-dimensional role of the dollar standard - a reference currency for international good pricing, which represents a situation where all firms set their export prices in US dollars.3 They show that in such a setting, the US monetary 'This chapter is based on the joint work with Michael Devereux and Juanyi Xu. 2Also see McKinnon and Schnabl (2003). 3 That is, for the US, the export prices are set in the producer's currency (PCP), while for the rest of the world, the export prices are set in the buyer's currency (LCP). Chapter 2. Oil Currency and the Dollar Standard 36 authori ty dominates i n the equi l ibr ium of internat ional monetary pol icy game. Nevertheless, U S residents are actual ly worse off than those i n the rest of the wor ld , due to the lack of an efficient expenditure-switching mechanism. In other words, it is costly for the U S to have such a dollar standard. T h e literature analyzes different aspects of the dollar s tandard and their impl ica t ion for the world economy, but it leaves us w i t h two intr iguing questions: W h a t causes a dollar standard? A n d what are the welfare consequences for the U S and the rest of the wor ld if the dollar s tandard is endogenously determined? The emergence of an international currency may be a t t r ibuted to a lot of factors, such as economic size, history, capi ta l flows, and economic policies. K r u g m a n (1984) notes that while economic size is l ikely to be an important factor, there is also a "snowball ing" effect, whereby even i f the countries are of s imilar size, if one currency becomes acceptable i n exchange, then a l l countries w i l l have an incentive to support this outcome. Th i s suggests that the dollar standard may be caused by a historical accident as much as by current fundamentals. M c K i n n o n (2002) takes a different view. He stresses the importance of the U S monetary policy, arguing that the U S dollar 's role as an international currency is a result of low and stable U S inflat ion rates i n the p o s t - W W I I international system. In this paper we want to argue that the dollar hegemony might be caused by a geopoli t ically constructed peculiar i ty that c r i t ica l commodities, most notably o i l , are denominated i n dollars. Since o i l has been the most important energy for product ion and life today, when the o i l price is denominated i n U S dollars, i t creates incentives for exporters i n bo th the U S and the rest of the world to set their goods prices i n dollars, which i n t u rn leads to the dominance of the dollar in the world goods market and financial market. His tor ica l evidence is consistent w i t h this explanat ion. For example, the history of o i l being extensively used i n both the U n i t e d States and the world economy coincides w i t h the growing path of the U S dollar as an international currency. To address our questions and explain our story, we w i l l develop a two-country new open economy macroeconomic model, where the U S dollar is assumed to be the o i l currency. The key features i n our model are: (a) the currency of export pr ic ing is itself endogenous, so the emergence of a dollar s tandard w i l l be the result of firms' op t imal choices; (b) o i l is introduced Chapter 2. Oil Currency and the Dollar Standard 37 as a required productive input and its world price is denominated in US dollars. We analyze two cases. In the benchmark case, both home and foreign countries are subject to monetary supply shocks and world oil price shocks. In such a setting, when the oil price is quoted in US dollars, the oil price shock will have an asymmetric impact on US and foreign firms, as US firms bear less exchange rate risk than foreign firms. This asymmetry will lead all firms in the world to set their export prices in US dollars, which implies a dollar standard in international goods pricing. In the extended case, besides the oil price shock, country-specific productivity shocks and active monetary policies are also introduced. We find that the optimal monetary policies can offset the asymmetry caused by the US dollar's role as an oil currency. Therefore, a dollar standard is not necessarily the equilibrium of firms' choices. Given different relative sizes of external shocks, the equilibrium can be quite different. If productivity shocks are relatively small, then a dollar standard can still be an equilibrium choice for firms. Moreover, contrary to Devereux, Shi, and X u (2004), when there is a dollar standard in international goods pricing, we find that households in the US always are better off than those in the rest of the world. The net welfare difference between the US and the foreign country is composed of two components: the pass-through effect, which represents the cost of having a dollar standard; and the cost-advantage effect, which represents the benefit of having the US dollar as the oil currency. Our result suggests that the special role of the US dollar as an oil currency can help to build a dollar standard in international goods pricing. This paper is closely related to the literature on endogenous currency of pricing. We follow the approach in Devereux, Engel, and Storgaard (2004). They endogenize the firms' currency of export pricing and show that the exporters wish to set their prices in the currency of the country with a relatively more stable monetary policy.4 Our paper differs from theirs, as we focus on an asymmetric environment where one currency has a special role. To introduce the 4 Corsetti and Pesenti (2002) analyze optimal monetary policies when the currency of pricing setting is itself endogenous. They show that there can be multiple equilibria. In one equilibrium, firms set prices using PCP, and an optimal monetary policy maintains a flexible exchange rate, while in another equilibrium, firms set prices using LCP, and the equilibrium is a fixed exchange rate. Chapter 2. Oil Currency and the Dollar Standard 38 effect of an oil price shock on the economy, we use the production approach,5 That is, we assume that firms need oil to make their production plants work. This approach has been used recently by Leduc and Sill (2003). They develop a standard sticky-price, dynamic general equilibrium model, where monopolistically competitive firms use capital, labor, and oil to produce, to study the impact of oil price shocks on US output volatility. This paper is organized as follows. Section 2 presents the benchmark model. Section 3 solves the model and shows that a dollar standard is the equilibrium of firms' choices. Section 4 extends the benchmark model to include the country-specific productivity shocks and active monetary polices. We find that there exist multiple equilibria of the game between firms and monetary authorities. Section 5 concludes. 2.2 Basic Model The world economy consists of two countries, which will be referred to as the home country (the United States) and the foreign country. There is a continuum of home goods (and home population) and foreign goods (foreign population) of measures n and 1 — n respectively. In each country, households maximize expected lifetime utility, taking prices and wages as given. Each firm is monopolistic competitive and use oil and labor to produce differential goods. In this section, we will focus on the baseline model where there only exist two shocks, the money supply shocks and the oil price shocks. Country-specific productivity shocks will be introduced later. For simplicity, we abstract from any dynamics by considering a single period model with uncertainty.6 The structure of events within the period is as follows; Before the period begins, households can trade in a full set of nominal state-contingent bonds. Then the firms choose 5 The oil price shock also has a wealth effect on the economy. For example, Krugman (1985) develops a simple theoretical model to study the effect of an oil price increase on the US dollar. The model shows that the effect depends on a comparison between a direct balance of payment burden caused by oil price increase and a indirect balance of payments benefits of O P E C spending in investment. 6 The assumption of a static model is certainly restrictive, however, it is a common assumption, either explicit (Obstfeld and Rogoff 2000, 2002), or implicit by shutting down the dynamics through the functional forms and asset structure (Corseetti and Pesenti 2001, Devereux and Engel 2003). Chapter 2. Oil Currency and the Dollar Standard 39 the currency in which they set their export goods prices, given the cross country risk-sharing rule, and taking into account the way in which they set prices, as well as the d is t r ibut ion of the stochastic monetary shocks and the o i l price shocks. Fol lowing this, firms set prices i n ad-vance, contingent on the state-contingent discount factors, and the demand and marginal cost conditions that they anticipate to hold. After the realization of stochastic shocks, households work and choose their op t imal consumption baskets, product ion and consumption take place, and the exchange rate is determined. The detailed structure of the home country is described below. T h e foreign country has an identical structure. Where appropriate, foreign variables are indicated w i t h asterisk. 2.2.1 Households T h e representative household i n home country maximizes the following expected ut i l i ty : 7 C 1 _ p M U = E(T—p+X\n--VL) (2.2.1) C n C^~N 1 A —1 A where C = n » » ( 1 _ ^ ) i - n > Ch = [f^n~*Ch(i)~di}*^, and A > 1. Here C is home aggregate consumption, composed of home goods and foreign goods w i t h weights of n and 1 — n respec-tively. Note that the elasticity of subst i tut ion between home and foreign goods is unit . Ch is the home sub-aggregate consumption of a cont inuum of home goods indexed by [0, n], and A is the elasticity of subst i tut ion between home ind iv idua l goods. $p is the real money balance, and L is the costly labor effort. We assume that p, the inverse of inter- temporal elasticity parameter of subst i tut ion is greater than 1. rj and x a r e positive constant scale parameters. E is the expectation operator defined across a l l possible states of nature. T h e specification of (2.2.1) allows us to derive a closed form solution of the model . F r o m the consumption structure, we may derive the consumption-based price index, P = P£PJ-n (2.2.2) 7 To give a closed form solution, our modelling approach is based on special assumptions, about both func-tional form and the stochastic environment, which are also used in Obstfeld and Rogoff (2000), Devereux and Engel (2003), and Devereux, Shi, and Xu (2004). Chapter 2. Oil Currency and the Dollar Standard 40 where Ph and Pj represent the prices for home goods and foreign goods in the home country, respectively. The consumption structure of foreign country is analogous, but w i t h prices and quantities denoted by asterisks. Home and foreign households can trade a full set of nomina l state-contingent bonds, thus the budget constraint of the home household for a part icular state of the wor ld z can be wri t ten as: P{z)C{z) + M(z) + ] T Z{z)B{z) = W(z)L(z) + U(z) + B{z) + M0 + T(z) + n ° " ( z ) (2.2.3) T h a t is, the household obtains the wage income, w i t h W being the nomina l wage, gets the payoff of contingent securities (B) and receives the profit from household's ownership of home goods firms (n) and the revenue from the share of world o i l endowment Uml, the in i t i a l money balance M o , and lump-sum transfers from the government (T ) . The household chooses how many state-contingent securities to purchase before the period begin, w i t h £ z and Bz repre-senting the price and holding, respectively, of a security paying off 1 uni t of home currency i n state z S Z, where Z is the set of states. T h e n the household also chooses the holding of money, consumption and labor supply. We assume that government repays any seignorage revenue through a lump sum transfer, so that Mo — M(z) + T(z) = 0. W e w i l l focus on the stochastic money supply in this section, active money supply rule w i l l be discussed later. The trade i n state-contingent nominal assets across countries w i l l lead to the following risk-sharing condit ion: C~P C*~P ^ = T — , (2.2.4) where S is the nominal exchange rate, and P* = p*npjl~n i s the foreign price level. Equa t ion (2.2.4) implies that one dollar can get the same marginal u t i l i ty of consumption across coun-tries, or the ratio of marginal uti l i t ies of consumption is equal to the real exchange rate. T is the state-invariant weight , 8 from the appendix for Devereux and Enge l (2003), we may show EC^~p) r = (2.2.5) 8 T represents the ratio of the Lagrange multiplier on the home household budget constraint to the Lagrange multiplier on the foreign households budget constraint. Chapter 2. Oil Currency and the Dollar Standard 41 In addition, the home household's optimization gives rise to the money demand function: M = xPC, (2.2.6) and the implicit labor supply schedule: W = rjPC. (2.2.7) Therefore, the nominal wage is proportional to the money in circulation. Combining the money market equilibrium for the home and foreign countries with cross country risk-sharing condition (2.2.4), we can derive the exchange rate as: M S = Y—. (2.2.8) 2.2.2 Oil Market We treat oil as a direct input in the production process suggested by Mork and Hall (1980) and Leduce and Sill (2003). It is assumed that the oil endowment is owned by a third party such as O P E C . Both home and foreign firms have to import oil from the O P E C and it is assumed that the oil price is determined by an exogenous shock and is quoted in the US dollar.9 The supply of oil will be adjusted to equal to the total demand by both home and foreign firms, and the excess stock of oil endowment will be stored for future use. We also assume households in both home and foreign countries get some lump sum transfer of the revenue of oil endowment Hml from O P E C , which represents the net effect of oil wealth emphasized by Krugman (1985). 1 0 The oil price Q is quoted in home currency (the U.S. dollar) and follows a log normal distribution. \nQ = q, q~N(0,aq). (2.2.9) For the foreign firms, they face an oil price ^ , in terms of foreign currency. Therefore, the oil price shock will have an asymmetric effect on home and foreign firms. 9 For simplicity, we do not model the supply side of world oil market and the pricing behavior of O P E C explicitly in this paper. There is strong evidence that the oil price can reasonably be argued as exogenous during the period 1948-1972. See Hamilton 1983,1985. 1 0 Krugman (1985) showed that the distribution of oil wealth and the expenditure of oil revenue may have an impact on the exchange rate and welfare. Nevertheless, in our model, the complete asset market assumption completely eliminates this effect. Chapter 2. Oil Currency and the Dollar Standard 42 2.2.3 Production Each firm i in our economy has the following production technology. Y(i) = AL(i)1-aO{i)a (2.2.10) where O represents the amount of oil used in the production, and A = ~ (i_a)i-a is a con-stant parameter. Since we assume the Dixit-Stiglitz consumption structure, each firm faces a downward-sloping individual demand curve and chooses its optimal price along the demand curve. The cost minimization problems for home and foreign firms give us marginal costs for home and foreign firms in term of their own currency, respectively. MC = W^Q* (2.2.11) MC* = W * 1 _ a ( § ) a (2.2.12) Equations (2.2.11) and (2.2.12) imply that the oil price shock has a direct impact on the firm's production cost. Moreover,foreign firms may have a disadvantage in the cost of production, as they have to bear extra exchange rate risk when using oil, while the US firms do not need to bear this extra exchange rate risk. 1 1 We can show that this asymmetry in cost structure will affect the endogenous currency of pricing decision for both home and foreign firms later. 2.2.4 Monetary Shocks In the baseline model, neither home or foreign monetary authorities are active. The money supply in each country just follows a log normal stochastic process. M = exp(w) (2.2.13) M* = exp(w*) (2.2.14) n W e assume that firms are not able to hedge against exchange rate risk in foreign exchange market.. In our model, the main mechanism that generates a dollar standard comes from the firms' concern about extra exchange rate risk of using oil. In reality, some firms can hedge against the exchange rate risk, but this usually requires an extra management costs. In this sense, our assumption that firms cannot hedge exchange rate risk is not far away from the reality. Chapter 2. Oil Currency and the Dollar Standard 43 where the terms u and u* represent uncontrollable disturbances to money supply. We assume that u ~ iV"(0,<72), and u* ~ N(0,a2,), and o\ — a\». Note that money supply shock and o i l price shock are independent, i . e . , 1 2 Note that cov(u,q) = cov(u*,q) = 0. 2.2.5 E n d o g e n o u s C u r r e n c y of P r i c i n g W h e n a firm sells abroad, it can set its export prices i n its own currency ( P C P ) or in buyer's currency ( L C P ) . Whatever currency i t chooses, it must set the price before the state of the wor ld is k n o w n . 1 3 So the op t imal currency of pr ic ing decision depends on the difference between the expected profit under P C P and the expected profit under L C P . Fol lowing the approach used i n Devereux, Engel , and Storgaard (2004), we can derive home and foreign firms' op t imal conditions for currency of pr ic ing decision. For each firm i i n the home country, who sells a differentiated good to the foreign market, the firm faces a C E S demand curve, rhf rhf where P^jii) is the price the foreign consumer pays for home-produced goods i. P^ is the price index for a l l home goods purchased by the foreign consumer, and P* is the foreign country consumer price index. W i t h o u t loss of generality, let P^j(i), Pj*j- and P* be denominated in foreign currency. If the home firm i sets its price in its own currency (the currency of producer, P C P ) , then the expected discounted profits is PPcp(i) P* EnPc* = E[d(SP^(i) - MC)(^-Al)-x—C*} (2.2.16) 1 2 The mean of log money supply has no impact on the result of our model, for simplicity we assume a simple money rule in which the mean of log money supply equals zero. 1 3 If the firms can freely reset their price when shocks are realized, firms would have no special preference on the currency in which they set their export goods prices, thus the currency of pricing decision for firms would not matter in the economy, and there is no chance for one currency to become a reference currency for international pricing. Also, if prices are flexible, the currency of pricing oil would not be a concern for firms. Chapter 2. Oil Currency and the Dollar Standard 44 where d = -p^ is the marginal utility of home households, which is used as the stochastic discount factor for home firms. MC is the marginal cost of home firms. If the home firm i sets its price in the foreign currency (the currency of buyer, L C P ) , then the expected discounted profit is P*Lcp(i) P* EIlL<* = E[d(P*Lhf{i) - MC){-^)-x—C*] (2.2.17) The home firm will set its price in its own currency if the expected profit differential 0 is positive. Thus, it follows P C P whenever n = EnPcp - EnLcp > o (2.2. is) In Devereux, Engel, and Storgaard (2004), it is shown that the above optimal condition can be be approximated as below: i<72 - ccw(ln(MC), s) > 0 (2.2.19) The equivalent condition for the foreign firm to follow P C P is given by ^cr2 + cov(ln(MC*), s) > 0 (2.2.20) That is, if a firm chooses its export price optimally, then up to a second order approximation, its decision depends only on the variance of the exchange rate and the covariance of exchange rate with marginal cost, and is independent of the variance of market demand, the financial market structure and the prices of all other firms. In our model, due to the complete asset market assumption, exchange rate movement is fully determined by money supply shocks and will not be affected by the degree of exchange rate pass-through. Thus, all firms in one country will choose the same pricing strategy. In Devereux, Engel, and Storgaard (2004), if both countries have identical monetary stability, firms are indifferent between P C P and L C P . But this result will not hold in our model, as the oil price shock will generate an asymmetric effect on firms' marginal cost, which we discuss in detailed in the following section. Chapter 2. Oil Currency and the Dollar Standard 45 2.2.6 E q u i l i b r i u m G i v e n the stochastic processes {u, u*, q}, a symmetr ic equi l ib r ium is a collection of allocations {C, C*, L, L*, M, M*, B(z), B*{z)u. }, and price system {P, P*, W, W*, Ph, Pf, P*, P*}, S, £(z), T} such that • G i v e n the price system, {C, C*, L, L*, M, M*, B(z), B*(z)} solve the consumer's op t imal i ty problem, subject to his budget constraint. • {Ph, Pf, P^ Pf} solve the ind iv idua l firm's op t ima l pr ic ing problem. • The strategy of currency of pr ic ing for each firm is op t imal . • The labor, goods, and money markets clear. 2.3 Model Solution To derive the solution to the baseline model , firstly, we have to find the op t ima l currency of pr ic ing decision for the firms. T h e n use the op t ima l pr ic ing policies of firms to solve the endogenous variables contingent on the realizations of monetary shocks. F ina l ly , we calculate the expected welfare for the home and foreign consumers. 2.3.1 T h e C u r r e n c y of P r i c i n g F r o m A p p e n d i x B , we can derive the expected profit differential for bo th home and foreign firms as follows: 0 = a a 2 > 0, fl* = -cto 2u < 0 (2.3.1) Thi s implies that a l l firms i n the home country w i l l choose P C P and a l l firms i n the foreign country w i l l choose L C P , and the home currency (US dollar) would be the reference currency for internat ional goods pr ic ing. Thus, we can state the following proposi t ion. 14B(z) and B*(z) are the holding of state-contingent nominal bond for home and foreign households in the state of 2, and £(z) is the price of bond in the state z Chapter 2. Oil Currency and the Dollar Standard 46 P r o p o s i t i o n 2 In the world where cr2 = a 2 , , all home and foreign firms will, choose oil cur-rency as the reference currency for international goods pricing. This represents a dollar stan-dard in the world economy. Proposition 2 shows that there is an asymmetric equilibrium in our model. According to Devereux, Engel,and Storgarrd (2004), if a firm chooses its export price optimally, then its decision depends only on the trade-off between the variance of exchange rate and the covariance of the exchange rate with the marginal cost. If countries are identical in all aspects, then firms in home and foreign countries follow the same pricing policy. There exist three symmetric equilibria, and firms are indifferent between P C P and L C P . That is, the difference of expected profit between choosing P C P and choosing L C P is zero. However, in this paper, when the oil price is denominated in US dollars, for home firms, only the labor cost will be affected by changes of exchange rates. For foreign firms, both the labor cost and the oil cost would be affected by exchange rate changes. As (B.1.5) and (B.1.6) show, in absolute value, the covariance between home firms' marginal cost and exchange rates is less than the covariance between foreign firms' marginal cost and exchange rates. Thus, neither home or foreign firms will be indifferent to their currency of export pricing. The home firms will prefer P C P to L C P , while foreign firms would prefer to choosing L C P . This implies that all firms will wish to set their export prices in the US dollars. Since the economy is symmetric in all aspects expect that the oil price is denominated in US dollars, we may ascribe the emergence of US dollar as the reference currency for international goods pricing to the special role of US dollar as an oil currency. For simplicity, we impose the condition that a\ — <72„ in Proposition 2. However, we can show that all firms will want to set their export prices in the US dollar as long as < 1 _ 1 2 a -This implies, a dollar standard can even exist in the case where monetary policy in the foreign country is slightly more stable than that in the US. The more the economies rely on oil, the more chance a dollar standard will exist. 2.3.2 O p t i m a l P r i c i n g Schedule When the dollar is used as the reference currency for international goods pricing, we can derive the optimal pricing policy for home and foreign firms for goods sold in home and foreign Chapter 2. Oil Currency and the Dollar Standard 47 markets, respectively. - E[MCC1-p o-^'W^l <232) - E[MCC*l-p^ E[C* = x ^ » w j ( 2 3 3 ) f f E[C*l~p} Pfh ~ A E [ c l _ p ] (2.3.5) where A represents the markup, subscript hf represents the price of home goods sold in the foreign market. A n asterisk over the price means the price is denominated i n foreign currency. G iven these prices, we can derive the price index for each country as follows: P = PhhP}hn (2-3-6) P* = i ^ r P f f 1 ' " (2.3.7) Th i s imp ly that the home C P I is completely predetermined and independent of external shocks, but there is positive exchange rate pass-through into the foreign C P I . T h i s is an important channel through which a dollar standard may affect the global economy. 2.3.3 A C l o s e d - f o r m S o l u t i o n A s the model is log-linear and the underlying monetary and o i l price shock are log-normal, we may solve for the exact d is t r ibut ion of a l l endogenous variables i n a closed-form. T h e solution allows a dichotomy between variables that are determined in advance of the real izat ion of shocks (i.e. Phh,Phf,Pfh,Pff), a n d T, and variables determined after the shocks have occurred (i.e. C,C*,W,W*, and S). Equa t ion (2.2.8) shows that exchange rate movements are fully determined by the realiza-t ion of money supply shocks. Combin ing it w i t h the market clearing conditions, we can get home and foreign consumption. C = [ - — ^ r - ] ' (2.3.8) * rhhrfh Chapter 2. Oil Currency and the Dollar Standard 48 1 MnM*^~n^ i C* = [- ]P. (2.3.9) L -v- p i p * 1 - n J v ' 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 foreign goods in the home market have prices preset in home currency. But with full exchange rate pass-through into imported goods prices, foreign country consumption is affected by the home country monetary shocks. 2.3.4 Welfare Comparison We now compare the welfare of a representative household in the home country with that in the foreign country and check if the home household would gain when a dollar standard exists. It is assumed that the welfare of the household can be measured as 1 5 • E ( ^ - V L ) (2.3.10) As shown in Devereux and Engel (2003), the expected utility of the household in a stochas-tic environment is a function of variances and covariance terms of log consumption and log exchange rate. Thus, given the solution to the consumption and the exchange rate, we may rewrite the welfare in terms of the variance of monetary and oil shocks. From properties of the pricing setting equations in home and foreign countries, and the labor market clearing condition in the home country, in Appendix B, we can establish that E(L) = (1 - a){^E(C1~p) + ^^E(C*l-p)T} (2.3.11) \r) Xn Combining (2.3.10) and ( 2.3.11), we may rewrite the expected utility of home household as E(U) = A ~ ~ Q ^ A ~ ~ ^EjC1-") - t1 ~"X 1 ~ Q ) ( A ~ 1 ) r £ ( C * 1 ~ p ) (1 - p)\ A (2.3.12) Since the log-normal distribution satisfies ECl~p = exp |(1 - p)[E(c) + ^o- 2 ] } , (2.3.12) ultimately depends only on the second moments of consumption and the exchange rate. To 1 5Obstfeld and Rogoff ( 1998, 2002) argue that the utility of real balance is small enough to be neglected. Chapter 2. Oil Currency and the Dollar Standard 49 derive these second moments, we rewrite the equations which give the closed-form solution to consumption and the exchange rate i n log terms as: s-E(s)=m-m* (2.3.13) c-E{c) = -m (2.3.14) P c* -E(c*) = -\nm+ (1 - n)m*) (2.3.15) P where small-case letters denote logarithms. The expected u t i l i ty of foreign household can be derived i n the same way. To explain the model well , we focus on a special case where p = 1, then give the result for general case where p > 1. W h e n p = 1, the expected labor supplies for bo th countries are a function of constant parameters , 1 6 EL = EL* = ^=^L. Thus , the expected u t i l i t y for the home country only depends on the mean of log consumption, the variance of log consumption has no impact on the expected u t i l i ty of households. EU = Ec-?—^ (2.3.16) A F r o m the money demand function c = In x + m — p and the assumption Em = 0, we may find that, when p = 1, the expected ut i l i ty for home country is fully determined by the expected log price level. EU= -\nX- -Ep. (2.3.17) A The expected u t i l i ty for foreign country can be derived in the same way, EU* = — In x — — Ep*. F r o m A p p e n d i x B , we have Ep = In V - + V^-o* + y o\ (2.3.18) Ep* = InAr? 1-" + 0—^-a2 + %-o\ + (1 - n ) a a 2 (2.3.19) 6In the special case where p = 1, T = 1. Chapter 2. Oil Currency and the Dollar Standard 50 It is straightforward to show that Ep < Ep*, and EU - EC* = (1 - n)a<72 > 0 (2.3.20) Equation (2.3.20) implies that, the home household has higher expected utility than the foreign household in the situation of having a dollar standard. This welfare difference between the US and the foreign country is obviously attributed to the special role of US as an oil currency, as the model is symmetric in all aspects except that the oil price is quoted in US dollars. This welfare gain increases with the share of oil cost in total production cost (a). Because these results can hold in the more general case when p > 1, we conclude in the following proposition. P r o p o s i t i o n 3 A households in the home country has higher welfare than that in the foreign country in the situation of having a dollar standard in international goods pricing, i. e. EU > EU*. Proof: See Appendix B. When p > 1, the expected home utility will be a function of mean of log consumption and variance of log consumption. We can find that the welfare gain for US households decreases with risk-aversion coefficient p. 2.4 The Case with Active Monetary Policies In this section, besides the oil price shock and the money supply shock, we introduce country-specific productivity shocks for each country. We also assume monetary authorities are active and can optimally respond to these shocks. Thus, we can analyze how the optimal monetary policies affect our finding in this section. Now, there exists a sequential game between firms and monetary authorities. The firms choose the currency in which they set their export prices before monetary authorities announce their monetary rules, 1 7 Then, given the currency of pricing, each monetary authority chooses its optimal monetary rules in an international non-coordinated game. Note that both firms and monetary authorities make decision before the state of the world is realized. 1 7 If we change the timing of the action of firms and monetary authorities and let monetary authorities move first, then monetary authorities can use policy to indirectly choose the currency of pricing of firms, see Devereux, Shi, and Xu (2005). Chapter 2. Oil Currency and the Dollar Standard 51 In the economy with country-specific productivity shocks, the production technology for the representative firm in the home country is given by. Y(i) = >4(flL(t)) 1- aO(i)Q (2.4.1) where 9 is a country-specific shock in the home country, following a log normal distribution. 6 = exp(z), z ~ N(0,o2z) (2.4.2) The country-specific shock 6* in the foreign country is analogously distributed, and cov(z, z*) = 0 and a2 = a2.,. We assume that both monetary authorities can respond optimally to the oil price shock q, country-specific productivity shocks z and z*. Also they commit to their money rules when shocks are realized as follows. m = aoq + a\z + a2z* + u (2.4.3) m* = b0q + biz + b2z* + u* (2.4.4) where {a, b} are policy parameters and will be discussed later. The term u and u* are the disturbances to money supply, representing the financial innovation or implementation error for policy makers. Thus, the stability of monetary policy in each country not only depends on the response to real shocks, but is also affected by the shocks from nominal sectors. The equilibrium in this extended case is a bit different from that in the baseline model. We have to take account of the interaction between monetary authorities and firms: (a) Given firms' currency of pricing, monetary authorities in home and foreign countries choose the policy parameters {a, b} by solving the international monetary Nash game. m&xEU{a,bn) (2.4.5) a maxEU*{an,b) (2.4.6) b (b) The optimal policy parameters {a, b} from Equations (2.4.5) and (2.4.6) must support both home and foreign firms's pricing strategies. Chapter 2. Oil Currency and the Dollar Standard 52 To find the equilibrium, we express the expected profit differential of firms' currency of pricing decision as functions of policy parameters {a, b}. From Appendix B, we can establish f2(o, b) = i(oo - b0)[a0 -b0- 2((1 - a)a0 + a)}a2q + \(ai~ 6i)[oi - 61 - 2(1 - a)(oi - l)]a2z + ^{a2-b2)[a2-b2-2{l-a)a2)]a2z.+aa2u (2.4.7) ^*(a>b) = ^(ao - b0)[a0 -bQ + 2(b0 - aa0 + a)]a2 + ^(ai - &i)[ai - 61 + 2(6i - QOI)]CT 2 + i ( a 2 - 6 2 ) ^ 2 - & 2 + 2 ( 6 2 - a a 2 - ( l - a ) ) ] < 7 2 . - a a 2 (2.4.8) Thus, there exist four feasible pricing specification for home and foreign firms. They are {PCP, PCP}, {LCP, LCP}, {PCP.LCP}, and {LCP, PCP}. For simplicity, we focus on the case where p = 1, but the results also hold in the more general case where p > l . 1 8 Table B . l gives the optimal policy parameters for optimal monetary rules in these cases. { P C P , P C P } If all firms in both home and foreign counties follow PCP, the optimal mon-etary policy requires that the home (foreign) authority fully respond to its own productivity shock but ignore the foreign (home) productivity shock. Since the oil price shock is a common shock to both countries, the optimal monetary policy requires both home and foreign monetary authorities to respond in the same way. To fully eliminate the effect of the, oil price shock on firms' marginal cost, for one percent increase in the oil price, the nominal wage should decrease by j~: so the monetary authorities should reduce the money supply by j z ^ percent. Now, we may check whether P C P is indeed an optimal pricing strategy for firms in home and foreign countries, if the monetary policy parameters are chosen as described above. From equations (2.4.7) and (2.4.8), we may establish: fi(o, b) = a2z + aal > 0 (2-4.9) Cl*{a, b) = (1 - 2a)c72 - a a 2 (2.4.10) 1 8 The value of risk aversion parameter p only affects the welfare level and has no impact on the currency of pricing decisions for firms and the equilibrium of the game. Especially, in the symmetric pricing specification, the optimal monetary rules are independent of the parameter p. Chapter 2. Oil Currency and the Dollar Standard 53 It is straightforward to show that a l l home firms w i l l wish to follow P C P , but the op t imal pr ic ing strategy for foreign firms depends on the relative size of p roduc t iv i ty shocks to monetary supply shocks. So the case where a l l firms choose P C P would be an equ i l ib r ium i f the following condi t ion holds > — — (2.4.11) 1 - 2a y J If there is no extra exchange rate risk (a = 0) of using o i l , a l l foreign firms w i l l follow P C P . In the case w i t h extra cost of using o i l , when there is the produc t iv i ty shock is relative large, the net benefit of choosing P C P can outweigh the extra exchange rate risk, P C P w i l l s t i l l be an op t imal strategy for foreign firms. { L C P , L C P } If a l l firms i n bo th home and foreign countries follow L C P , the op t ima l mon-etary pol icy requires the home (foreign) authori ty to adjust their money supplies to home and foreign shocks according to their weights i n world output . T h i s eliminates the effect of pro-duc t iv i ty shocks on the exchange rate. The pol icy response to the o i l price shock is the same as that i n the symmetr ic P C P case. Thus , the variance of op t ima l exchange rate i n this case is completely affected by the disturbance to money supply. A g a i n , from Equat ions (2.4.7) and (2.4.8), we establish: Cl(a,b) = eta 2 > 0 (2.4.12) fl*(a,b) = - C K 7 2 < 0 (2.4.13) Thi s implies L C P is not the op t imal pr ic ing strategy for home firms. So the case where a l l firms choose L C P is not an equi l ibr ium when monetary authorities choose policies after the currency of pr ic ing decision has been determined. { P C P , L C P } If a l l home firms follow P C P and a l l foreign firms follow L C P , i n this asym-metric case, the home authori ty responds to home and foreign p roduc t iv i ty shock according to their weight i n wor ld output, but the foreign authori ty reacts less to the home product iv i ty shock and more to its own product iv i ty shock . 1 9 If the share of o i l cost i n product ion cost a 1 9 T h e optimal monetary parameters listed in Table B . l is for this case where p — 1, the optimal parameters for the more general case where p > 1 depends on p, but they give the same intuitions. Chapter 2. Oil Currency and the Dollar Standard 54 converges to zero, the foreign country would completely responds to its own shock. This is the exact case that Devereux, Shi, and X u (2004) show. Now we check if the pricing strategy for home and foreign firms is optimal. fi(a, b) = n(2 - n)(l - a) 2cr 2 + ao\ > 0 (2.4.14) fi*(a, 6) = n 2 ( l - afa\ - aa2u (2.4.15) We find that all home firms will wish to follow PCP, but the optimal pricing strategy for foreign firms also depends on the relative size of productivity shocks to monetary supply shocks. So the case where all home firms follow P C P and all foreign firms follow L C P would be an equilibrium if the following condition holds 4 < ^ 7 7 ^ 2 ( 2 - 4 - 1 6 ) cr£ nz(l - ay That is, when the relative size of productivity shock is less than some critical value, the extra exchange rate risk would be the main concern for foreign firms when they choose the currency of pricing. Now we conclude the three cases and state the following proposition. 2 0 2 P r o p o s i t i o n 4 > nt(fLa)2 > there is a unique equilibrium for the game between firms and 2 monetary authorities that all the firms follow PCP; If jz^ < ffr < ra2(i_a)2 > there will be two equilibria: one is the case that all firms follow PCP, the other is the case that all home firms follow PCP and all foreign firms follow LCP. The latter case represents a dollar standard. If 2 2f < 1 _ a 2 a , there exists a unique equilibrium that all home firms follow PCP and all foreign firms follow LCP, that also represents a dollar standard. Proposition 4 implies that the equilibrium can be drastically different depending on the relative size of productivity shocks. If productivity shocks are relatively small, then a dollar standard can be an equilibrium. Intuitively, when there are big technology shocks, the variance of the optimal exchange rate increases. So this encourages both US firms and foreign firms to follow PCP, thus giving a flexible exchange rate regime. Given the size of shocks, an increase in a will increase the possibility of a dollar standard in international goods pricing. A n increase in 2 0 For any reasonable values of parameters, we always have l " 2 a < ni^"_a^t • Chapter 2. Oil Currency and the Dollar Standard 55 n is going to reduce the distance between two threshold values of , which in turn lowers the possibility of multiple equilibria. To analyze the welfare gain of the home household in the situation of having a dollar standard, we can derive the expected utility of the home household and the foreign household under a dollar standard, respectively. The detailed derivation is given in Appendix B. EU = -lnA(^)1-" + n( l - n)(l - afa\ + ~~al (2.4.17) EU* = - l n A ( ^ - « + „ ( ! - n ) 2 ( l - afa\ + ^ ^ a 2 u + ( I ^ d ^ ( 2. 4.18) This gives the difference between home and foreign households's expected utility: EU - EU* = (1 - n)aal - n2{\ - n)(l - a)2a2z (2.4.19) The welfare of households in both countries are only a function of the variance of productivity shocks and monetary shocks, as the optimal monetary policy fully eliminates the effect of oil price shock (common shock) on the welfare. The welfare differential between the US and the foreign country consists of two components: the first term is cost-advantage effect, which represents the benefit of having the US dollar as an oil currency; the second one is the pass-through effect, which represents the cost of having a dollar standard, as suggested by Devereux, Shi, and X u (2004). From equation (2.4.19), we can state the following proposition. P r o p o s i t i o n 5 In an environment with active monetary policies, the households in the U.S. are better off than those in the rest of the world when there is a dollar standard. 2 The condition of have a dollar standard as a possible equilibrium is given by < ni^i_a^i, which will ensure that EU > EU*. That is, as long as there is a dollar standard, the cost-advantage effect always dominates the pass-through effect. Thus, there must exist a welfare gain for the US households under the dollar standard. Our result suggests that the US economy does gain from the role of the US dollar as an oil currency.2 1 2 1 In the case where all firms choose PCP, US households are also better off than those in the rest of the world. Chapter 2. Oil Currency and the Dollar Standard 56 2.5 Conclusion The paper analyzes the impact of the U S dollar as an o i l currency and its impl ica t ion for the wor ld dollar s tandard in international goods pr ic ing i n a two-country general equi l ibr ium model w i t h s t icky prices. Our model gives a different insight about the dollar s tandard from the "core and periphery" literature. We relate the U S dollar 's role as an o i l currency to the dollar s tandard i n internat ional goods pr ic ing. We find that, when the currency of export pr ic ing is itself endogenous and o i l is required for product ion, a l l firms have an incentive to set their export prices in U S dollars i n the presence of o i l price shocks, as the o i l price is denominated i n U S dollars. B u t when there are active monetary policies, a dollar s tandard is not necessarily the equi l ibr ium of firms' choices, since the op t imal monetary pol icy can affect firms' currency choices of export pr ic ing. G iven different relative sizes of external shocks, the equi l ibr ium can be quite different. If product iv i ty shocks are relatively smal l , then a dollar s tandard is s t i l l an equi l ibr ium. We also find that the households i n the U S always are better off than those in the rest of the wor ld i n the s i tuat ion of having a dollar s tandard i n internat ional goods pr ic ing. For future research, we w i l l endogenize the currency of o i l pr ic ing. A l s o , the complete asset market assumption can be relaxed' and the effect of o i l price shocks on the wor ld wealth d is t r ibut ion can be explored. 57 Chapter 3 Flexible Exchange Rates and Endogenous Currency of Pricing 3.1 Introduction T h e debate on fixed versus flexible exchange rates has been at the heart of internat ional monetary economics for many years. 1 Fr iedman (1953) and later M u n d e l l (1961) made the case for flexible exchange rates as efficient in responding to country specific shocks when domestic price levels cannot change quickly enough. Since country specific shocks i n general require changes i n relative prices (the terms of trade), it is more efficient to allow these changes to take place quickly v i a nomina l exchange rate adjustment than a slower and potent ial ly more costly process of nomina l price adjustment. Recent studies of monetary pol icy in ut i l i ty-based open economy models have reached varying conclusions about the desirabil i ty of flexible exchange rates. Obstfeld and Rogoff (2000, 2002) develop models where an op t imal monetary pol icy focuses main ly on domestic economic conditions, and an adjustable exchange rate is a key feature of the op t ima l response to shocks. O n the other hand, Corset t i and Pesenti (2001, 2002), and Devereux and Enge l (2003) reach quite different conclusions. In their analysis, monetary pol icy should respond to bo th domestic and foreign shocks, and i n some cases, an op t ima l monetary pol icy should keep the exchange rate fixed, even i n the presence of country specific p roduc t iv i ty shocks. T h e key difference between these two sets of results lies i n the assumptions about price setting and the response of prices to the exchange rate. Obstfeld and Rogoff assume that export goods prices are set i n the currency of the producer ( P C P , or producer 's currency pricing), so that an unanticipated exchange rate depreciation leads to a rise i n the price of impor ted goods 'This chapter is based on the joint work with Michael Devereux and Juanyi Xu. Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 58 facing final consumers, causing a tilting of world demand towards the depreciating country's goods. This 'expenditure switching' is a key part of the optimal response to country specific shocks. Corsetti and Pesenti (2001, 2002) and Devereux and Engel (2003) on the other hand, point out that in many low inflation countries, nominal prices of final goods seem to be quite unresponsive to changes in the exchange rate. In light of this, they assume that export goods prices are set in the currency of the final consumer (LCP, or local currency pricing). When pricing is done in this way, there is no short run impact of a currency depreciation on the price of imported goods facing consumers. The expenditure switching mechanism cannot work. As a result, an optimal monetary policy response, even to country specific shocks, is consistent with a fixed exchange rate. What determines the currency in which exporting firms will set their prices? In Corsetti and Pesenti (2002) and Devereux, Engel, and Stoorgard (2004), this decision is made endogenous. Firms are allowed to choose whether they set prices in domestic currency or foreign currency when selling abroad. Corsetti and Pesenti (2002) show that this can give rise to multiple equilibria - in one equilibrium, firms will set prices in producer's currency, and an optimal monetary policy maintains a flexible exchange rate, while in another equilibrium, firms set prices in local currency, and the equilibrium is a fixed exchange rate. Devereux Engel, and Storgaard (2004) examine the nature of the determination of the currency of pricing, and derive a simple second-order condition which governs this choice. They show that the stability of monetary policy is a critical determinant of the currency of pricing. This paper re-examines the issue of the degree of exchange rate flexibility in an optimal monetary policy game and explores the methods to sustain flexible exchange rates, when the currency of export pricing is endogenous. The key innovation in the first part of this paper is to allow monetary authorities to take into account the way in which firms make their decisions about the currency in which they set prices. In order to sustain this equilibrium, monetary authorities must be able to commit to follow rules which are consistent with the currency of pricing decisions that firms follow. The paper highlights one very sharp result. When monetary authorities take into account the currency of pricing decision in their choice of monetary policy rule, there is a unique equi-Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 59 l ib r ium i n which a l l firms set prices in producer's currency. 2 In welfare terms, the equi l ibr ium w i t h P C P is better. Since it ensures that relative prices change efficiently, it sustains the outcome of the flexible price economy. In this equi l ibr ium the exchange rate is flexible, and i n essence supports the Fr iedman argument for flexible exchange rates. Nevertheless, this may not be a realistic description of monetary policy, since product mar-ket structure is normal ly thought to be outside the purview of monetary policy. A n alternative is to restrict the form of monetary rules itself. In the second part of this paper, we show that, i f monetary authorities are precluded from reacting to foreign p roduc t iv i ty shocks, then their op t imal policies w i l l rule out the equi l ibr ium w i t h L C P , and the unique equi l ibr ium i n this case is the one w i t h P C P and flexible exchange rates. In a sense, restr ict ing the form of the monetary pol icy rule acts as an equi l ibr ium selection mechanism. T h e key insight is that there is a social benefit of exchange rate vola t i l i ty that is not necessarily internalized i n the pol icy makers' decision, when the currency of pr ic ing is taken as given. B y restricting monetary pol icy rules to focus only on domestic economic conditions, the pol icy makers are forced to follow rules that lead to volati le exchange rates. G i v e n this exchange rate volati l i ty, price sett ing rules adjust to ensure efficient exchange rate pass-through. T h i s organizat ion of this paper is as follows. Section 2 lays out the model . Section 3 derives the equi l ib r ium when monetary authorities can take into account the way i n which the firms set export prices. Section 4 discusses the equi l ibr ium when the money pol icy rule is restricted. Section 5 concludes. 3.2 Basic Model T h e wor ld economy consists of two countries, labelled as home and foreign. E a c h country is populated by a large number of atomistic households, a cont inuum of firms that choose the currency of pr ic ing and set prices in advance, and a monetary author i ty which chooses opt i-m a l money rules to maximize the representative household's expected ut i l i ty . B o t h countries 2When monetary policy is chosen conditional on the currency of pricing, then there are two Pareto ranked equilibria. Due to the problem of coordinating expectations of price setters and the actions of monetary au-thorities, the economy can be stuck in an equilibrium with low pass-through, fixed exchange rates, and social welfare lower than the flexible price equilibrium. Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 60 specialize i n the product ion of a composite traded good. 3.2.1 The Timing of Events A l l events take place wi th in a single period. For simplici ty, we abstract from any dynamics . 3 T h e structure of events w i th in the per iod is described i n F igure 3.1. A t the beginning of the period, households can trade i n a full set of nomina l state-contingent bonds. Fol lowing this, the monetary authorities choose op t imal monetary rules, given the cross country risk sharing condi t ion, but tak ing into account the currency of pr ic ing decisions of firms, the way in which firms set op t imal prices, and the dis t r ibut ion of stochastic technology shocks. F i r m s then choose the currency i n which they set their goods prices, and having chosen this, set prices i n advance, based on their stochastic discount factors, and ant icipated demand and marginal cost conditions. After the realization of technology shocks, households work and choose their op t imal consumption baskets, product ion and consumption takes place, and the exchange rate is determined. Figure 3.1: T i m i n g of Events H H s trade M A s choose F i r m s choose state-contingent op t imal currency of bonds monetary rules export pr ic ing F i r m s set Technology Consumpt ion and prices shocks product ion take occur place, exchange rate is determined The detailed structure of home country is described below. T h e foreign country has an identical structure. Where appropriate, foreign real variables and foreign currency prices are indicated w i t h an asterisk. 3.2.2 Household The representative household in the home country maximizes the following expected u t i l i ty : U = E ( ^ + Xln-p--vL) (3.2.1) 1 — p P 3Because we are considering a complete market environment, and there are no predetermined state variables, an infinite horizon model would have the same results as those below. Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 61 where C = n t f ^ i - n C h = [""* / " C f c W ^ d t ] ^ (3.2.2) nn(l - np n J0 Here C is aggregate consumption, C/, is the consumption sub-aggregate over a continuum of home goods indexed by [0, n], jr is real money balances, and L is labor supply. We assume that the inverse of the intertemporal elasticity of substitution in consumption p > 1 and the elasticity of substitution between individual goods A > 1. The foreign consumption sub-aggregate is analogously defined, but over a range of goods indexed by [n, 1]. n and \ a r e positive constant parameters. From the consumption structure, we may derive the CPI price index P = P£P}-n (3.2.3) where Ph and Pf represent the price for home goods and foreign goods in home market, respectively. Home and foreign households trade a full set of nominal state-contingent assets. Thus, the budget constraint of the representative home household for a particular state of the world z is written as: P(z)C{z) + M(z) + £{t)B{t) = W(z)L(z) + n(z) + B{z) + M0 + T(z) (3.2.4) tez That is, home households' income is derived from wage income (WL), payoff of state-contingent securities (B), the profits from home monopolistic firms which are assumed to be owned by home households (n), the initial money balance MQ, and lump-sum transfers from the government (T). Before the state of the world is realized, households purchase state-contingent securities. Note that £ t and Bt represent the price and quantity, respectively, of a state-contingent bond paying off 1 unit of home currency in state t S Z, where Z is the set of all states. After the state is realized, households choose their holding of money, consumption and labor supply. We assume that government repays any seignorage revenue through a lump-sum transfer, so that M 0 - M(z) + T(z) = 0. The specific money supply rules will be defined later. In the appendix of Devereux and Engel (2003), it is shown that the trade in the state-contingent nominal assets across countries leads to the following optimal risk sharing arrange-ment: n-p .. Q*-P Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 62 where 5 is the nomina l exchange rate, and P* = p*npjl~n j s the foreign price level. Equa t ion (3.2.5) implies one dollar can get the same marginal u t i l i ty of consumpt ion across countries. The risk-sharing parameter Y is state-invariant, but its level is determined by the outcome of state-contingent securities trade. A s shown by the appendix of Devereux and Enge l (2003), the risk-sharing parameter satisfies r = ——f,—r. (3.2.6) In addi t ion, the household's opt imiza t ion problem gives rise to the money demand and labor supply conditions: M = XPC, (3.2.7) W = vPC- (3.2.8) Thi s means that the nominal wage is proport ional to the money holding, W = -M (3.2.9) X P u t t i n g Equa t ion (3.2.5), the home household's op t imal i ty condi t ion and their foreign analogy together implies that the nominal exchange rate is determined by M ^ = T — (3.2.10) 3.2.3 P roduc t ion It is assumed that firms are monopolis t ical ly competi t ive and each of them produces a differ-entiated good i by using a linear technology: Y{i) = 6L(i) (3.2.11) where 9 is a home country-specific technology shock, following a log-normal dis t r ibut ion: 9 = exp(u), u ~ N(0, al) (3.2.12) Therefore, the marginal cost for the home firm is then, W MC = — (3.2.13) Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 63 A s the ind iv idua l goods are differentiated, each firm has monopol is t ic power characterized by the elasticity of subst i tut ion between ind iv idua l goods A. Prices of ind iv idua l goods are assumed to be set before the state of the world is realized. It is also assumed that, due to high costs of arbitrage for consumers, each ind iv idua l monopolist can price discr iminate across countries and can set the price in term of the buyer's currency (local currency, L C P ) or its own currency (producer currency, P C P ) . T h e opt imiza t ion problem of each firm is to choose the currency of pr ic ing and then to maximize the discounted expected profits, t ak ing the ind iv idua l demand function as given. The endogenous currency of pr ic ing w i l l be discussed in detai l i n next subsection. The op t imal pr ic ing schedules given the choice of pr ic ing currency are given as follows. P h h=Aib^r (3-2-14) P*fCP = Pkh ( 3 2 1 5 ) r i y c ' - p s - ' i P . L C P = *x"l e J ( 3 . 2 . 1 6 ) .El E[Cfl where A == is the markup, and Phh represent the prices of home goods sold i n home countries; PhfCP and PhfCP represents the prices of home goods sold in foreign country in term of producer currency ( P C P ) and buyer currency ( L C P ) respectively. T h e problem of the firms in the foreign country is entirely analogous and their prices schedule are given in Table C . l . 3.2.4 Endogenous Currency of P r i c ing We now describe how firms choose the currency i n which they w i l l set prices for export ing abroad. W e say the firm follows Producer Currency Pricing, or P C P , i f it sets its export price i n the domestic currency. If the firm chooses to set export price i n the currency of the buyer, it follows Local Currency Pricing, or L C P . Whatever currency it chooses, it must set the price before the state of the wor ld is known. Take a firm i i n the home country selling a differentiated Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 64 good to a foreign market, it faces a downward sloping demand function Phf^ , p* X(Ph*f(i)) = (-^r V C * (3-2.17) Phj{i) is the price the foreign consumer pays for the home goods i. P^j is the price index for all home goods purchased by the foreign consumer, and P* is the foreign country consumer price index. Without loss of generality, let P^j(i), P^j and P* be denominated in foreign currency. If a home firm sets its price in its own currency (producer currency, P C P ) , then the expected discounted profit is PPcp(i) p* EUPCP = E[d(Ppfcp(i) - M C ) ( ^ ) " A - C 1 . (3.2.18) where d is the stochastic discount factor. It equals to the marginal utility of consumption, as all home firms are assumed to be owned by home households. If the home firm set its price in the foreign currency (buyer's currency, L C P ) , then the expected discounted profit is P*L?(i) P* EIILCP = E[d{SP*Lhf{i) - MC){ p{ rX-^C*} (3.2.19) Phf yhf The home country firm will set its price in its own currency if the expected profit differential is positive. That is, it follows P C P if and only if EUPCP - EULCP > 0 (3.2.20) In Devereux, Engel, and Storgaard (2004), it is shown that Equation (3.2.20) can be ap-proximated by the following inequality ^cr2 - cov{\n(MC), s) > 0 (3.2.21) From now on, small-case letters denote logarithms, that is, x = log(X). The equivalent condition for the foreign firm is ^ ( T 2 + cov(ln(MC*), s) > 0 (3.2.22) That is, if a firm chooses its currency of pricing optimally, then up to a second-order ap-proximation, its decision depends only on the variance of exchange rate and the covariance of exchange rate with the marginal cost, and is independent of the variance of market demand and the prices of all the other firms. For simplicity, we define the left-hand side of 3.2.21 and 3.2.22 as fi and fi* respectively. Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 65 3.2.5 M o n e t a r y A u t h o r i t i e s Following a number of recent papers, we assume the monetary authority in each country is concerned with the expected utility of consumption and the disutility of labor effort, but ignores the utility of real money balances.4 Thus, the home country monetary authority chooses monetary policy to maximize E(^—^ - nL) (3.2.23) 1 ~ P Each monetary authority can commit to an optimal rule. The model and the shocks are log-linear, so without any loss of generality, we can write the form of the monetary rules ( for both home and foreign authorities) in the log-linear representation given by m = a\u + a2u* (3.2.24) m* = biu + b2u* (3.2.25) These rules are unrestricted in that they allow the monetary authority to respond freely to both home and foreign shocks. We will also discuss restricted rules in the later section. The policy feedback rule parameters [a,b] are determined by the following international monetary Nash game max EU{a,bn) (3.2.26) a m&xEU*(an,b) (3.2.27) b Given the money rules, the log of exchange rate can be determined by s = lnT + (ai - h)u + (a 2 - b2)u* (3.2.28) Note that, since the monetary authorities act after the trade in state-contingent assets, they take the the risk-sharing parameter T as given. 5 4See for instance, Obstfeld and Rogoff (2002), Corsetti and Pesenti (2001), and Devereux and Engel (2003). 5In Devereux and Engel (2003), it is shown that allowing monetary authorities to choose rules before asset trade makes no difference to the results. Note, because our model is symmetric, F = 1 in equilibrium in any cases. Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 66 3.3 A Unique Equ i l ib r ium w i t h Flexible Exchange Rates T h e key innovation of the model described above is to allow monetary authorities to take account of the way in which firms make their decisions about the currency i n which they set prices. The equi l ibr ium for this model can defined as follows:' G i v e n the stochastic process {6, &*}, a symmetr ic equi l ibr ium is a collection of allocations {C, C*, L, L*, M, M*, B(z)}, price system {P, P*,W,W*, Ph, Pf, P^, PJ, S, T}, and monetary authorities pol icy {a, b}, such that • G iven the price system, {C, C*, L, L*,M, M*, B(z)} solves the household's opt imal i ty problem, subject to his budget constraint. • {Ph, Pf, P^, Pf} solves the ind iv idua l firm's op t imal pr ic ing problem. • The firm's currency of pr ic ing decision is op t imal given op t ima l monetary rules. • {a, b} solves the international monetary Nash game and maximizes the representative household's expected uti l i ty. • T h e labor, goods, and money markets clear. 3.3.1 M o n e t a r y R u l e s chosen after the P r i c i n g D e c i s i o n Is M a d e We first construct an equi l ibr ium condit ional on the op t imal monetary pol icy feedback para-meters. T h e properties of the economy w i l l differ depending on whether the home firm, as well as the foreign firm, follows P C P as opposed to L C P . Because i n equ i l ib r ium a l l firms w i t h i n a country w i l l follow the same pr ic ing strategy, we impose this at the beginning, wi thout any loss of generality. We w i l l construct the equi l ibr ium i n the following manner. F i r s t , we compute the mean and variance of (log) consumption (for both countries), as a function of the policy rule parameters, i n the case where a l l firms i n both countries follow P C P , where a l l firms in both countries follow L C P , or where home firms follow P C P and foreign firms follow L C P (and vice versa). For each case, we may compute expected u t i l i ty of home and foreign agents. We then briefly characterize the outcome to an op t imal monetary pol icy game, where monetary authorities take the currency of pr ic ing decisions as given. F ina l ly , we define the game in Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 67 which governments choose policy rules taking into account that their choices will influence the currency of pricing of firms, and show that a flexible exchange rate is a unique equilibrium of this game. C a s e 1. A l l F i r m s F o l l o w s P C P When all firms follow PCP, we may compute the mean and variance of (log) consumption, as functions of the underlying feedback parameters Ec = - i ln(A„) - 2-^ot - ^f^a2 - (™2+ (* ~ p K " 2 0 2p 3 2p {nociL + (1 — n)acu') , . a s u — osu* /„„„•> + -—2f-L^ J—^-+n{l-n) — — (3.3.1) 2 (nai + (1 - n)bi)2ol + ( « « 2 + (1 - n)fo) ac — 2 (3.3.2) where the other variance and covariance terms in Equation (3.3.1) except o\ and a2,, are also functions of policy parameters, as shown in the appendix Devereux and Engel (2003). Then, as outlined in their appendix, with all firms following PCP, expected utility for the home country may be written as: EU = 6 e x p ( l - p)[-U - ^f^o2 + 2 c 2p 2p {no-cu + (1 - n)acu-) ,crsu--asu*. . . H : \-n{l — n) 1 (3.3.3) P P The expected utility of foreign country is analogous. Then, with all-firms following PCP, a Nash equilibrium to the game in which each monetary authority chooses its feedback parameters to maximize the expected utility (but taking the currency of pricing decision as given) is defined as max EU{a,bn) (3.3.4) a s.t. Home firms choose PCP Foreign firms choose PCP m&xEU*{an,b) (3.3.5) b Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 68 s.t. Home firms choose PCP Foreign firms choose PCP Solving for the above equilibrium, the optimal policy rules for this game are a =[1,0], b =[0,1] (3.3.6) The intuition behind this equilibrium is straightforward (see Devereux and Engel 2003). If each monetary authority focuses its monetary policy only on the domestic productivity shock, then this allows for output, consumption, and the terms of trade to respond to the shocks exactly as they would in a flexible price equilibrium. For terms of trade adjustment, it is necessary that the exchange rate responds to productivity shocks, and in order to facilitate this, monetary policies must respond to shocks in different ways across countries. Finally, we may now check whether P C P is indeed an optimal pricing strategy for firms in the home and foreign countries, if monetary policy is chosen as described in 3.3.6. As shown in Appendix C.1.2, using conditions 3.2.21 and 3.2.22, we may establish that fl = fl* = ^(a2u + a2u,)>0 (3.3.7) where fl and fl* are expected profit differential, as on the left hand side of Equations (3.2.21) and (3.2.22). Hence, all firms will wish to follow PCP, and P C P is an equilibrium when monetary authorities choose policies after the currency of pricing decision has been determined, and therefore choose policies as in Equation (3.3.6). Given the optimal policy parameters, we may compute the expected utility in this equilib-rium, identical across countries, as .2^2 , n _ „\2„2 EU = EU* = e e x P { ( i ^ ) 2 ( n ^ + ( 1 2 - n ) t T " - ) } (3.3.8) where 0 = A ^(i-p)A ^(^V)^ < 0 is a constant function of parameters. C a s e 2. A l l F i r m s F o l l o w L C P If all firms in both countries follow L C P , then the mean and variance of log consumption for the home country can be written as Ec = -I ln(A„) - 2-^a\ - (™2+ ( ! - " ) « £ ) + ( ™ c u + (1 - n ) ^ . ) p 2 2p p Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 69 al = a ' ° l + (3.3.10) where, again, the covariance terms acu and crc1i» are also functions of policy parameters {a, b}, as shown in appendix of Devereux and Engel (2003). In this case, home consumption variance depends only on home monetary policy rules, because there is no pass-through of exchange rate changes into the domestic price level. The appendix then shows that expected utility may be represented as: E U = E C 1 - " [ X - n ^ - p ) ^ - 1 ) ] - V-W-VECr1-' (3.3.11) A(l — p) X E U * = £ C * i - P [ A - ( l - n ) ( l - p ) ( A - l ) _ "(A - 1 ) E c l _ p A(l - p) A (3.3.12) The expected utility is a combination of two separate function of consumption variance and the covariance between consumption and productivity shocks. JSC 1"" = T e x P ( l - p){-\al - (™5 + (* ~ + (™cu +{1-n)acu.)] ( 3 . 3 J 3 ) EC*'-" = Texp( l - p)[-\al. - (™3 + (* " + + (1 - n)ac,u.)} (JJ g u ) where T = (Xrj)^. Thus, the monetary rules that represent an equilibrium of the Nash game in which monetary authorities choose monetary policies taking as given the L C P currency of pricing in both countries, can be derived as a=[n,l—n], b = [n, 1 — n] (3.3.15) In this case, monetary authorities follow identical policies in the two countries. The intu-ition behind this (see Devereux and Engel 2003), is that the exchange rate does not play any allocational role, given zero pass-through in each country. A n optimal monetary rule there-fore does not try to attain the terms of trade adjustment that would occur in a flexible price economy, and so the equilibrium has zero exchange rate variance. Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 70 Again, from Appendix C.1.2, we may use Equations (3.2.21) and (3.2.22) to establish that fl = fl* = 0 (3.3.16) Hence, weakly, L C P is also an equilibrium when monetary authorities choose policies after the currency of pricing decision has been determined. Since both L C P and P C P constitute equi-libria where policy is chosen takingthe pricing decision as given, there are multiple equilibria in this case. 6 Expected utility in this equilibrium can then be computed as EU = EU* = 6exp(l - p){in2-pn}<rl + [(l-f-p(l-n)}al.} 2p Case 3. H o m e F i r m s Follow P C P , Foreign F i r m s Follow L C P Now let us look at the asymmetric case. If the home firm follows P C P , and the foreign firm L C P , there will be zero pass-through of exchange rate changes into the home economy, but full pass-through into the foreign economy. As shown in Devereux, Shi, and X u (2004), we may compute the mean and variance of (log) consumption in the home and foreign economy, respectively, as Ec = -1 I n f a r 1 - ] - ^ a 2 - ™l + ( ; ~ n ) a l ' + ™™ + ( 1 " n ) a c u ' (3.3.18) p 2 2p p E(c*) = - i mor-A,) - i ^ o g . - «±f±<Z - n a l + {\ - n)°l* p 2 Zp Zp | nac*u + (1 - n ) g - c . u . + n ( l - n)(asu - cr S M . ) 3 vZ = ^l*Wu + a22o-2u.} (3.3.20) a2. = ^[(na, + (1 - n)bx)2o2u + (na2 + (1 - n)b2)2a2u.} (3.3.21) J2 where the variance and covariance terms other than o\ and o\» are all functions of policy parameters. 6 This was first pointed out, although in quite a different modelling framework, by Corsetti and Pesenti (2001). Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 71 Therefore, we also can derive expected utility for the home and foreign agent in the following way. E{U) = A - n ( A - l ) ( l - ^ ) g ( c l _ p ) _ ( l - n ) ( A - l ) r j E , ( c t l _ P ) (1 - p)A A E U , = A - ( l - n ) ( A - l ) ( l - p ) g ( c r l _ P ) _ n { X - l l ^ l E { c l _ p ) (1 — p)A A Following Devereux, Shi and X u (2004), we may then obtain the equilibrium of the Nash game in which the home and foreign monetary authorities choose policy rules, taking the {PCP, LCP} pricing configuration as given. The rules for the general case where p > 1 are quite complicated, but the rules for the case where p = 1 is quite straightforward. a=[n,l-n], b = [0,1] (3.3.24) In this case, the home authority responds approximately equally to home and foreign shocks, while the foreign authority puts more weight on its domestic shock. Now we may ask if in fact the {PCP, LCP} configuration is an equilibrium for the economy. The answer is no. From Appendix C.1.2, we may show that fl > 0, Q* > 0 (3.3.25) In each country, firms would wish to set prices in their own currency. Hence in fact, the {PCP, LCP} configuration is not an equilibrium, if monetary policy rules are chosen taking the currency of pricing as given. 3.3.2 Mone ta ry Rules Chosen before the P r i c i n g Decis ion Is M a d e Now we focus on the game where each monetary authority chooses its policy before the currency of pricing decision is made. In general, the maximization problem is quite complex, because the monetary authorities' choice of monetary rule will partly determine the pricing policies of firms in both countries, and hence will lead to a switching across pass-through outcomes that makes the decision discontinuous. But we can circumvent these difficulties by defining a simpler game, in which the choice set of the monetary authorities in each country is simply binary. Although this seems excessively restrictive, in fact it is not really, since we show that Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 72 this game supports the constrained Pareto optimum of the world economy, which is the flexible price allocation. 7 The game that we focus on allows the monetary authority of each country to choose either the Nash equilibrium monetary rules of the ex-post {PCP, PCP} game defined above, or the Nash equilibrium rules of the ex-post {LCP, LCP} game defined above. The binary game is then defined in the matrix in the Figure below. Figure 3.2: Binary Game Foreign Monetary Authority b = [0,1] b = [n, 1 - n] Monetary Authority a = [ n A _ n ] (PCP, PCP) (PCP, PCP) (PCP, PCP) (LCP, L C P ) P r o p o s i t i o n 6 The {PCP, PCP} configuration, and flexible exchange rates represent the unique equilibrium of the monetary policy game when monetary authorities take account of the currency of pricing. The payoffs of this binary game are given in Appendix C.2. Once we know them, then the proof is straightforward. Say that Foreign is following b = [n, 1 — n]. Then if home follows a = [1,0], it generates a switching to all PCP, and then a = [1,0] is clearly better for home. If foreign is following b = [0,1], then the optimal policy for home is to follow a = [1,0]. C o r o l l a r y 1 The equilibrium of the ex-ante monetary policy game supports the full flexible price equilibrium. The logic of the proposition is two-fold. First, it is clear that the outcome under {PCP, PCP} is preferable to both countries, since it supports the full flexible price equilibrium, whereas the outcome under {LCP, LCP} prevents exchange rate pass-through, does not allow optimal TWe have not explored the possibility of equilibria in the game where monetary policy rules are unrestricted, and policy makers take account of the currency of pricing decision, but clearly in welfare terms, these equilibria would be worse than the unique equilibrium of the game defined here. Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 73 terms of trade adjustment, and hence gives lower expected ut i l i ty . B u t the key feature of the result is that by commi t t ing to follow the op t imal monetary rules that are an equi l ib r ium of the ex-post { P C P , P C P } game (case 1 above), the monetary authorit ies w i l l actual ly ensure that { P C P , P C P } constitutes the equi l ibr ium price configuration. Hence, by t ak ing into account the way in which firms choose the currency of pr ic ing, the monetary authorit ies support producer currency pr ic ing and a t ta in the flexible price outcome i n the wor ld economy. A s a by product, it implies that op t imal monetary policy i n an environment where the currency of pr ic ing is endogenous restores the Fr iedman case for flexible exchange rates, and satisfies the central results of the op t imal currency area literature. 3.4 Sustaining the Flexible Price Equilibrium with Restricted Monetary Policy Rules W h e n monetary pol icy is chosen condi t ional on the currency of pr ic ing, and monetary pol icy rules are unrestricted, then there are two Pareto ranked equi l ibr ia . Due to the problem of coordinat ing expectations of price setters and the actions of monetary authorities, the economy can be stuck i n an equi l ibr ium w i t h low pass-through, fixed exchange rates, and social welfare lower than the flexible price equi l ibr ium. One way to el iminate this bad equ i l ib r ium is to require monetary authorities to take account of the currency of pr ic ing decision. We have show this possibil i ty i n our model where the monetary authorities move 'first ' i n the game. If monetary pol icy rules were set before the firms make their currency of pr ic ing decision as the sequence of actions described i n Figure 3.1, then there is a unique equ i l ib r ium to the pol icy game, characterized by P C P , and flexible exchange rates, wh ich coincides w i t h the flexible price wor ld equi l ibr ium. However, this equi l ibr ium requires that monetary pol icy take account of the s t ructural determination of pr ic ing. To the extent that our model captures a p r ic ing decision that may evolve more slowly over t ime than we actual ly allow for i n the simple t i m i n g structure here, it may be unrealistic to assume that monetary pol icy can have the degree of commitment and far-sightedness necessary to sustain such a rule. R e a l wor ld monetary pol icy mak ing does not usually consider product market structure to be w i t h i n its influence. For these reasons, it may Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 74 be that the timing sequence of Figure 3.1 is not the most relevant way to describe monetary policy making. As an alternative, we consider placing restrictions on the monetary policy rule itself in order to select the P C P pricing equilibrium of the monetary policy game. Assume that mone-tary authorities choose monetary policy after firms' currency of pricing decision, but they are restricted to follow rules that focus only on domestic productivity shocks. What will be the outcome in this case? In principle it would seem undesirable to restrict monetary authorities from taking into account all shocks. However, in the presence of coordination externalities in price-setting, this conclusion may no longer hold. Hence we define a restricted monetary policy rule, as follows: m - nu. in* — bn* (3.4.1) Under these rules, monetary policy can respond only to domestic shocks. The policy feedback rule parameter vector {a,b} is determined by the same international monetary Nash game. We present the analysis as follows. First, assume a given currency of pricing. Then derive the outcome of an international monetary Nash game, where monetary authorities take the currency of pricing as given. Finally, use the solution of the monetary game to check whether or not the given currency of pricing configuration is consistent with the optimal strategies of home and foreign firms. 3.4.1 Case 1: A l l F i rms Follow P C P When all firms follow PCP, Appendix C.3 shows that the equilibrium of a Nash game between monetary authorities using restricted monetary rules monetary rules is given as: a = l , 6 = 1 (3.4.2) This is in fact equivalent to (3.3.6). Since, in the case of unrestricted monetary policy, in a Nash equilibrium with PCP, monetary authorities choose to respond to the domestic shock alone, then the restriction on the form of monetary rules is irrelevant. We can use the same logic as above to show that the monetary rules given by (3.4.2) are consistent with P C P pricing behavior on the part of firms. Given (3.4.2), Appendix C.3 shows Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 75 that fl = fl* = j (o2. + o\») > 0, so both home and foreign firms will choose PCP. Hence {PCP, PCP} is an equilibrium of the game with restricted monetary rules. 3.4.2 Case 2: All Firms Follow LCP What happens if all firms in both countries follow L C P ? Appendix C.3 shows that in this case, the restricted optimal monetary rules are a = n, 6=1— n (3.4.3) In this instance, the optimal monetary rules differ from the unrestricted rules under L C P . Monetary authorities offset their own national shock, in proportion to their country weight. But as described in (3.3.15) above, they would strictly prefer to offset the foreign shock as well, since in the absence of exchange rate pass-through, they would wish to use monetary policy to replicate the flexible price response of consumption. What does this monetary rule imply for the currency of pricing decision? Note that unlike the economy with L C P and unrestricted setting of monetary policy, where the exchange rate is fixed in equilibrium, under the restricted money rule, there is inevitably fluctuation in exchange rate, since by design monetary authorities cannot respond to shocks in the same way. But the presence of exchange rate volatility will encourage firms to follow P C P pricing rules. Again, using (3.2.21), we may establish that fl = "(2~n)a\ + a2.. > 0 and fl* = rSa2 + ( l - n K i + n ) ^ > Q T h i s i m p n e s t h a t all firms in both countries will choose PCP. Thus {LCP, LCP} is not an equilibrium when monetary authorities can only use the restricted money rules. ' ;. From the analysis of the above two cases,8 we may state the following proposition. P r o p o s i t i o n 7 The {PCP, PCP} configuration, and flexible exchange rates represent the unique equilibrium of the international monetary Nash game, where the currency of pricing is endogenous, and monetary policy rules are restricted as in (3.4.1). 8 We have not allowed for the possibility of asymmetric pricing equilibria, where one country follows L C P and another country follows PCP. From Appendix C.1.2 however, it is shown that such an asymmetric outcome cannot be an equilibrium. The reason is that if one country follows a P C P rule, then the monetary policy game will imply sufficient exchange rate flexibility that firms in both countries will choose PCP. Hence the asymmetric outcome is ruled out for the same reasons as in Case 2. Chapter 3. Flexible Exchange Rates and Endogenous Currency of Pricing 76 C o r o l l a r y 2 The full flexible price equilibrium can be sustained by the restricted monetary rules.9 The key effect of the restrictions on monetary rules is that they ensure exchange rate ad-justment. W i t h o u t any restrictions, there is an equi l ibr ium where the monetary authorities find it op t ima l to eliminate exchange rate adjustment. G i v e n L C P pr ic ing decisions, exchange rate vola t i l i ty hinders cross country consumption risk-sharing, while at the same t ime, has no benefit i n terms of relative price adjustment (because there is no pass-through to domestic prices). B y ensuring that monetary policy cannot deliver absolute exchange rate stability, the restricted rules cause a l l firms to follow P C P pr ic ing decisions, leading to full exchange rate pass-through. The ensuing op t imal monetary pol icy then brings both efficient international consumption risk-sharing, and efficient relative price adjustment. 3.5 C o n c l u s i o n T h i s paper re-examines the issues of the degree of exchange rate f lexibi l i ty i n an op t imal monetary pol icy game, when the currency of export pr ic ing is endogenous. We show that if monetary pol icy can take account of the currency of pr ic ing decision, then there is a unique equi l ibr ium to the monetary pol icy game, where a l l firms follow P C P , and Fr iedman's classic defense of flexible exchange rates is upheld. Nevertheless, we also realized that this may not be a realistic description of monetary policy, since product market structure is normal ly thought to be outside the purview of monetary policy. A n alternative is to restrict the form of monetary rules itself. 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[54] Schmitt-Grohe, Stephanie and Martin Uribe (2004), "Solving Dynamic General Equilib-rium Models Using a Second-Order Approximation to the Policy Function", Journal of Economic Dynamics and Control, 28, 755-775. [55] Woodford, Michael (2003), Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press, 2003. 81 Appendix A Appendices of Chapter 1 A. l Approximation of Expected Profit Functions As shown in the first paper, the profits under domestic currency pricing are given by EUpeso = X[E(SXZ)\X[E(SXZMC)]1-X (A. l . l ) where A = JZT[{JX^)~x, Z = dP*x~lX, MC depends on the choice of debt contract. This expression may be rewritten as: X\Eexp(ln Z) exp(A In S)}X[Eexp(ln Z) exp(A In S) exp(lnMC)] l~ x (A.1.2) Now using the second order approximation, we have E exp(ln Z) exp(A In S) « exp(£ ' In Z) e x p ( A £ In S) 1 A 2 x{ l + -var(hiZ) + —var (In S) + Xcov(\nZ,In S)} (A.1.3) Using the same approximation for the expression Eexp(\n Z) exp(A InS) exp(ln MC), we get an approximation for EWeso as follows: 1 A 2 ^ [ 1 + -varQn Z) + — var(\n S) + Xcov(\n Z, In S)]x 1 A 2 1 x [1 + -var(\n Z) + —var(\n S) + -var (hi MC) + Accw(ln Z, In S) +cov(\n Z, In MC) + A cot; (In S, In MC)}1~X, (A.1.4) where £ = Aexp[E(\n Z)\ exp[A£(lnS)\ exp[(l - X)E(\nMC)]. Taking logs, we get expected discounted profits equal to 1 1 A 2 1 — A EYpeso K \nJ2+[-var(\nZ) + -jvar(\nS) + —-—var(\nMC)] + {Xcov(\nZ,hS) + A(l - A)cow(lnMC,ln5) + (1 - X)cov(\nZ,In MC)} (A.1.5) where l n j ^ is the steady state profit. Therefore, using the same approximation, we can rewrite Elidoiiar = x[E(SZ)}x[E(ZMC)}1-x as: EUdoiiar _ \nJ2+[^var(\nZ) + ^var(\nS) + ^-^var(\nMC)) + {Ac<w(ln Z, In S) + (1 - A) cow (In Z, In MC)} (A.1.6) 'Here, we use the approximation ln(l + x) RS x. Appendix A. Appendices of Chapter 1 82 A.2 The Steady State The model in the steady state can be described as the following system. There are 13 steady state variables {P, W, S, r,MC, C,L, LX, LN, PX, YX, Z, UX } . PC = WLN + WLX +UX (A.2.1 W = §r)PC» L = LN + LX WLN = (\-ct)PC p = w^iS)0" X MC = WL-W[S(l +r)\ (A.2.2) (A.2.3) (A.2.4) (A.2.5) (A.2.6) (A.2.7) WLX = (l-u)YxMC (A.2.8 Pxt = \MC (A.2.9 Z = p-xC-pP^-xX (A.2.10 Tlx = PXYX - MCYX (A.2.11 l + r = i (A.2.12 5 = 1 (A.2.13 Given the foreign demand X and the structure parameters /?, a, w, A, p and n, the steady state variables can be analytical solved. Note that, we use K to denote the steady state value of variable Kt in the model. Appendix A. Appendices of Chapter 1 83 A.3 Model Solution G i v e n the dis t r ibut ion of firms fl = ( M I , p 2 , M 3 ; M4 ) i w e c a n express the export sector's price index P* i n terms of dollar as below: 2 ppp ppd . . . . ^ 1 _ A = M^f)1-'+M^-)1-'+»sp;dp + ^ P:M (A.3.I) Log-l inear iz ing the export sector's price index, we have PI = Mi (PT -S)+ M2(PT -S)+ p3pxdp + P A P X M (A.3.2) A s a l l prices p § p , P x d p and are preset i n per iod t—1, it implies that pp —Et-ipx = 0, thus p*x = p*x - Et-wl =-(in + p2)s (A.3.3) where p\ + p2 is the number of firms who set their export prices i n pesos. U s i n g the market clearing condi t ion Yx = -pr, the deviat ion of the log export output from its t — 1 expectation is given by yx = x -p* = x + (/xi + P2)s (A.3.4) The log forms of ind iv idua l demand for the firms who choose the strategy Si are given by, respectively, yPP = - \ \ p p p - s - P x } + x - P x (A.3.5) ypd = -X\fxd-s-Px} + x - P x (A.3.6) yiP = -mdv-Vx]+x-p*x (A.3.7) 2 /f = -X\pxdd-px]+x-px (A.3.8) The net income for an export firm who chooses the strategy (p, p) is given by GPP = {PPP_ wMCp)Y™ = [P? - ^ 1 J U ( ( 1 + i t + i J S t - i H i ^ (A.3.9) Note that, i n the steady state MC = (1 + f ) " ^ 1 ^ , fl = ^ ( 1 + r)uWl-uYx, PX = ^ ( 1 + f ) W W ' 1 - W ) and G = ( ^ - w ) ( l + f ) a ' i y 1 _ w F 2 : . Tak ing the log-l inearization, we have 3 P P = ap™ - (a - 1)[(1 - + w 7 s + ust-i] + ypp (A.3.10) where a = > 1. After eliminate the predetermined terms, we have ~gpp = -(a-l)u1s + ypf (A.3.11) 2For simplicity, we drop the subscript t for all variables. Appendix A. Appendices of Chapter 1 84 Following the same technique, we can have the following derivatives of the net income for the export firm who chooses the other strategy, Gpd = (PPD - ojMCD)YPD = [Ppd - UJWx-w{{1 + r*t+1)StY)YPD (A.3.12 gpd = afd - (a - 1)[(1 - w)w + u(s + e)j + yf (A.3.13 ~gpd = _ { a _ l M s + e)+yPd (A.3.14 Gdp = ( S P * * - ujMCP)YDP = [SP;dp - w W 1 - w ( ( l + it+i)St-iT]YdP (A.3.15 gdp = a(s + P*xdp) - (a - 1)[(1 - u)w + wys + usf-i] + €P (A.3.16 gdP = as - (a - l)ur/S + yf (A.3.17 Gf = (SP*dd - ujMCD)YDD = [SP*M - + r*t+l)StT]YDD (A.3.18 gdd = a(S + - (a - 1)[(1 - + w(« + e)]] + (A.3.19 3 d d = as - (a - l)w(s + e) + yf (A.3.20 As the total net export income is given by Gx = p\Gxp + U2Gpd + paGip + P4Gdd, we can express 9x = M i f l P P + M~gpd + M~9dp + t*4~gdd (A.3.21 Using the fact that yx = p\yvp + p,2ypd + M32/xP + ^42/? = x + (pi + p2)s, we can have gx = bs + x - £e (A.3.22 where b = ea - (a — l)[v + 7(1 - v)]u> + (1 - e) and £ = (o - \)OJV. Note that e = p$ + U4 and A.4 The Euler Equation Given ct + (a — b)s~t = xt—£,et, we conjecture that the solution to consumption and the exchange rate is only determined by current shocks and can be given by, ct = a\ft, . h = a2ft {AAA) Appendix A. Appendices of Chapter 1 85 where ft = xt—£,et, a\ and a2 are function of parameters and will be determined later. Since all shocks are i.i.d., and by the definition of xf+j = x{+j — Et-i(xt+j), j > 0, we have EtXt+\ = 0. This implies that Etft+i = 0. Therefore, we have Et{-ct+i + St+i) = ( - a i + a2)Etft+i = 0 (A.4.2) a a From equations 1.4.16, it yields pct + (a + j)st = 0 (A.4.3) Combined with ct + (a — b)§t = Xt — &t, we can solve for a\ and a2 and verify our conjecture. Therefore, we prove that Et(^Ct+i + st+i) = 0. A . 5 The Proof of Proposition 1 As 7 —> 0, we conjecture that ft = [1,0,0,0], and then check if the optimal strategy for each firm support this distribution. Given ft, we have e = 0 and v = 0, thus b\ > 1, b2 = 1, cpl — A and 4>2 = 0. From conditions (1.4.30), (1.4.31) and (1.4.32), we have ir(p,p) > iv(d,p), ir(d,p) > 7r(d, d) and ir(p,p) > ir(p, d). This implies, given ft, (p,p) will be the optimal strategy for all firms and no firm would deviate from the distribution ft = [1,0,0,0]. So ft = [1,0,0,0] is an equilibrium distribution of firms when 7 —> 0. Obviously, this equilibrium may hold for any 7 < ji, where 7L is the maximum value of 7 which can satisfy conditions (1.4.30), (1.4.31) and condition that ir(p,p) > n(d,d). As 7 —> 0 0 , we conjecture that the distribution of firms ft = [0,0,0,1], that is, e = 1 and v = 1. As b2 increases with 7, then from conditions (1.4.30) and (1.4.31), we can show 2 2 ?r(p,p) < n(d,p), n(p,d) < ir(d,d). As 7 is big enough, approximately, for « pal+fco*' 2 £ 2 2 4>i « 7 - p g 2 + f 2 g 2 . 02 » p w h e r e £ = ( a _ 1)W> substituting them in the inequality 1.4.33, we can have It implies that n(d,d) > 7r(d,p). 3 Therefore, (d,d) would be the optimal strategy for all firms, and this is consistent with our conjecture ft = [0,0,0,1]. This result will hold not only in the case where 7 —> 0 0 , but also in any case where 7 > ju, where JH is the minimum value of 7 which violate conditions (1.4.31) and (1.4.33) and satisfy the condition n(p,p) < n(d,d). 3 This inequality will hold for most reasonable values of parameters p and ui and sizes of shocks. Appendix A. Appendices of Chapter 1 86 Table A . l : The Currency Denominat ion of E x p o r t Invoicing of Selected Countries (%) a Countries Observation year US dollar Own currency Korea 1995 88.1 < 2.2* Thailand 1996 91.7 1.3 Malaysia 1996 66 < 20* Indonesia 1995 90 — United States 1996 98.0 98.0 Germany 1994 9.8 76.4 Japan 1995 52.7 35.7 United Kingdom 1992 22.0 62.0 France 1995 18.6 51.7 Italy 1994 23.0 40.0 " Source: The data for developed countries are from Tavlas (1995). The data for Korea and Thailand are from Cook and Devereux (2004) and Mckinnon and Schnabl (2003). Malaysian data are from Coldberg and Till (2005). Indonesia data are from World Bank. The data with asterisk are calculated by the author using data from above resources. Table A . 2 : The Currency Denominat ion of Ex te rna l Debt of Selected Countr ies (%) b Countries Observation year share of foreign currency debt Korea 1995 98.5 Thailand 1995 98.8 Malaysia 1995 93.3 Indonesia 1995 98.7 United States 1993-1998 30 Germany 1993-1998 69 Japan 1993-1998 64 United Kingdom 1993-1998 56 France 1993-1998 59 Italy 1993-1998 86 Source: The data for selected emerging market economies are from Goldstein and Tuner (2004). The data for selected developed countries are from Eichengreen, Hausmann and Panizza (2003). Appendix A. Appendices of Chapter 1 87 Table A .3 : Standard Devia t ion of D a i l y Exchange Ra te F luc tua t ions against the Dol la r (%) c Count r ies P re -c r i s i s P r e - c r i s i s E x reg ime H o n g K o n g D o l l a r 0.02 C B A Indones ian R u p i a h 0.17 C P K o r e a W o n 0.22 C P M a l a y s i a n R i n g g i t 0.25 L i m i t e d f lex ib i l i t y w . r . t . U S do l la r P h i l i p p i n e Peso 0.37 D e facto peg to U S do l la r (or f ixed) S ingapore D o l l a r 0.20 D e facto m o v i n g b a n d to U S do l la r T h a i B a h t 0.21 D e facto peg to U S do l la r (or f ixed) J a p a n Y e n 0.67 I F Deutsche M a r k 0.60 I F Swiss F r a n c 0.69 I F 0 Source: The data of pre-crisis exchange rate flexibility are from Mckinnon and Schnabl (2003); the classification of exchange rate regimes are from Reinhart and Rogoff (2002). C B A = Currency board; CP = Crawling peg; IF = Independently floating. Table A . 4 : Ca l ib ra t ion Parameters P a r a m e t e r s value P a r a m e t e r s value P a r a m e t e r s value P 2 P 0.96 a 0.4 A 7 0.4 4 % 1 % e 11 V 1 Table A.5: The Impact of 7 on the Di s t r ibu t ion of F i r m s ft 7 (0 ,1 .004) (1.004,1.025) (1.025,13.368) (13.368,14.412) (14.412,oo) ( P P W 1 1-M2(7) 0 0(0) 0 (pd)M2 0 ^2(7) 1 1(0) 0 ( d p ) ^ 3 0 0 0 0(0) 0 ( d d ) / i 4 0 0 0 0(1) 1 e = [i3 + u4 0 0 0 0(1) 1 V = H2 + u4 0 ^2(7) 1 1 1 ^ 2 ( 7 ) is the number of firms which choose the strategy S2=(p,d) where 7 e (1.004,1025), and it is also a increasing function of 7, which satisfies with /i|(1.004) = 0 and ^ {l-Q'ib) = 1. Table A . 6 : Welfare Compar i son 7 = 0.01 7 = 900 C (p,p) Q. = ( 1 , 0 , 0 , 0 ) - 0 . 0 0 9 % - 0 . 2 2 3 % (d,d) Q = ( 0 , 0 , 0 , 1 ) - 0 . 0 1 2 % - 0 . 1 0 % Appendix A. Appendices of Chapter 1 88 Table A . 7 : Variables Compar i son e (7 = 900) ft = [1 ,0 ,0 ,0 ] ft = [ 0 , 0 , 0 , 1 ] (P,P) (d,d) c -0 .2715 -0 .0004 °l 0.0047 0.0016 L -0 .0035 -0 .0009 °l 9.16 x 1 0 ~ 4 9.2 x 1 0 - 4 MC 0.0036 0.0018 0.0026 3.4 x 1 0 ~ 5 P 0.0048 0.0018 W 0.0105 0.0039 e X = EXt — X, where X is the steady state variable of X. Table A.8: Welfare Compar i son w i t h Increasing ui f (7 = 900) c UJ = 0.36 w = 0.4 w = 0.45 ui = 0.48 £PP - 0 . 1 6 % - 0 . 2 2 3 % - 0 . 4 6 % - 0 . 8 5 % - 0 . 0 9 9 7 % - 0 . 1 0 % - 0 . 1 0 1 % - 0 . 1 0 2 % 0 . 0 6 1 % 0 . 0 9 1 % 0 . 3 5 9 % 0 . 7 4 8 % ! Note that, C p p a « d Cdd represent the measure of welfare for the two extreme cases, ft = [1,0,0,0] and CI = [0,0,0,1] , respectively. C,dd — C,pp can be defined as the welfare gain of twin dollarization under a fixed exchange rate regime. Appendix B Appendices of Chapter 2 89 B.l Proof for Proposition 2 Firstly, we rewrite the equations (2.2.8), (2.2.11), and (2.2.12) in log term. s = logT + u-u* (B . l . l mc = (1 - a) log - + (1 - a)u + aq (B.l.2 X mc* = (l-a) log - + (1 - a)u* + a(q - s) (B.1.3 X where small-case letters denote logarithms. Using the fact that cov(q,u) = cov(q,u*) = cov(u,u*) = 0. Equations (B.1.1)-(B.1.3) give us: l^ = \(al + al.) (B . l .* cov(mc,s) = (1 — ct)o~u (B.l.5 cov(mc*, s) = — (1 + ot)a\, + a{a\ + a2.,) (B.l.6 So from Equation (2.2.19) and (2.2.20), we have the expected profit differentials fl and fl* fi = \°* ~ cov{mc, s) = ( Q - ^)al + ^a2u, (B.1.7 1 1 1 fl* = - a s + cov(mc*, s) = (- - a)o2u - -a 2u, (B.1.8 If a2. = a2.,, then we have fl = ^as — cov(mc, s) = aa\ > 0 (B.1.9 fl* = i<j s + cov(mc*, s) = aal < 0 (B.l.10 If al 7^ al», the condition to hold the above results need a2 1 ., < — - — (B.l.11 al. 1 - 2a J V QED. Appendix B. Appendices of Chapter 2 90 B .2 The Labor Market Clearing Condition Given the production technology for firms, we can write the labor market clearing condition in home country as L = ( l - a ) ^ Y (B.2.1) and PC p*f* Y = npTh + {l~n)~^r (R2-2) Using the implicit labor supply condition W = rjPC and the pricing equation for home goods in home market and foreign market, _~E[MCC^P] Phh-X E [ c l _ p ] (B.2.3) } E[C*l-p\ we can rewrite the labor market clearing condition as T . MC PCE\Cl-P] „ W 1 MC SP*C*E[C*1~P] L = ( l - a n — — l- L + ( l - a ) ( l - n ) — — — l— (B.2.5) V ; r)PCP \E[MCCl-P] VPCP \E\MCC*1-"} Using the risk-sharing condition that TPCP = SP*C*P and taking the expectation, we may have . ' EL = (1 - a){^E[Cl-p] + ^-^E[C*l-p)T] (B.2.6) Xn Xn B.3 Expected Utility for the Special Case p = 1 From the price index and pricing equation, we have P = P h h n P f h 1 - n = XE[MC]nE[MC*S]1-n (B.3.1) Since MC = W l - a Q a = I "X C 1 _ Q ( - ) l - a M l - a Q a (B.3.2) Thus, MC* = W * x - a ( ^ ) a = ( ^ ) i - Q M * i - Q ( ^ ) Q (B.3.3) p = \(7])l-aE\Ml-aQa]nE[M*1-aQaSl-a}l-n = \CL)l-aE[Ml-aQa] (B.3.4) X X Using the log normal property of these shocks and taking log, we have ^ 2 "2~"9 = p = In A(^) 1 "" + ^ - ^ a 2 + ^ (B.3.5) Appendix B. Appendices of Chapter 2 91 The price index in foreign country is given by Ph. S We denote P* = where P* = fif\n{Plf)l-n (B.3.6) Z = PHf{Pffy-n = XEIMC^EIMC*}1-11 (B.3.7) We can rewrite it as Z = \(^)1-aE[M1-aQa}nE[M*1-aQaS-a)1-n (B.3.8) Thus In Z = In A ( ^ ) 1 - + + y ^ 2 + (1 - n)aol (B.3.9) Since p* = In Z — n In S, we have 2 u 2 " 9 Sp* = InZ = l n A ( ^ ) 1 - " + ^^o-l + + (1 - n ) a ^ (B.3.10) So that, we have Ep < Ep*, EU > EU* (B.3.11) Q E D . B.4 Proof for Proposit ion 3 When p > 1, using T = ^ % - p ) , we can simplify EU and Ef /* as : E U = A - ( A - l ) ( l - p ) E c ( 1 _ p ) 4_ ( l - p ) A U{7* = A ~ ( A ~ 1 ) , ( 1 ~ p ) E C * ( 1 - p ) (B.4.2) ( l - p ) A Since A ~ ^ 1 1 1 j ^ ~ p ) is negative, to prove EU > EU* is equivalent to proving F < 1 or EC^1'^ < EC*^-p). Thus, we only need to show the following condition holds Ec+^o^ES + ^ o t . (B.4.3) From the Equation (2.3.14) and (2.3.15), we have I P °l = -A (B.4.4) 2 1 - , 2 n ( l - n ) _ 2 , . o> = u (B.4.5) Appendix B. Appendices of Chapter 2 92 From home and foreign money demand functions, we have Ec - Ec* = -(Ep* - Ep) (B.4.6) P From the pricing equation and price index, we have P - \,n^i-aE[Ml~aQaCl-p]nE[M*1~aQaSl~aCl~p)l~n D* _ o-n w VM-aE[M1-aQaC*l'p)nE[M*1-aQaS-aC*1-p]1-n Using the log normal distribution property of underline shocks, we can derive Ep = l n A ( ^ ) - + {^L\ol + ^ - {^al (B.4.9) Vsi-a , ( p ~ a ) 2 , . 2 , " 2 2 ( ^ ~ 1 ) 2 _ 2 , r 2 n ( l - n ) ( p - l ) + ( l - n ) Q l _ 2 Ep* = I n A ( j t ) 1 - + <r* + ^ - a» + [ p Thus, (B.4.10) and Ec - Ec* = -^[2n(l - n)(p - 1) + (1 - n)a]cr£ (B.4.11) P - ac2) = ±(p - l )n( l - n)a2u (B.4.12) ^ _ = n(l-n)(P-l).+ ( l - n ) a a l ( B 4 1 3 ) Thus, Equation (B.4.3) holds. Q E D B.5 The Case with Productivity Shocks and Active Monetary Authorities Given the production technology with country-specific shock, the marginal cost for home and foreign firm in log terms are, respectively, mc = (1 — a) \n- + (l-a)(m-z) + aq (B.5.1) X mc* = (1 -a) In 1 + (1 - a)(m* - z*) + a(q - s) (B.5.2) We express the log marginal cost and log exchange rate as a function of policy parameters {a, b}: ' . s = In F + (a 0 - b0)q + (ai - h)z + (a 2 - b2jz* + u - u* (B.5.3) Appendix B. Appendices of Chapter 2 93 mc = (1 - a) In - + (1 - a){a0q + (ai - l)z + a2z* + u) + aq (B.5.4) mc* = (1 - a) In - + (fr0 - cuan + a)q + [b\ - aa\\z + [b2 - aa2 - (1 - a)\z* + u* - au (B.5.5) This gives the following variance terms. \<% = h°o - bvfo-l + (ai - h)2a2 + (a 2 - b2)2a2, +a2u + o2u,\ (B.5.6) cov{mc,s) = (ao-6o)[(l-a)ao+Q]a 2 +(l-a)(ai-6i)(ai-l)a 2 +(l-a)(a 2 -62)a2a 2 , - | -( l-a)a- 2 (B.5.7) cov{mc*, s) = (b0-aao+a){a0-b0)a'2+(ai-bi)(bi-aa1)a'2+(a2-b2){b2-aa2-(l-a))a2z.-(l+a)al (B.5.8) Thus, the expected profit differentials between P C P and L C P for home and foreign firms are given by: Vt(a, b) = |(ao - b0)[a0 -bo- 2((1 - a)a0 + a)]a2q + ^{ax - h)[ai - h - 2(1 - a)(a x - l)]a 2 + \(a2-b2)[a2-b2-2{l-a)a2)}a2z,+aa2u • (B.5.9) fi*(a,b) = \(ao - b0)[a0 - b0 + 2(b0 - aa0 + a)]a2 + ^(ai-~ - &i + 2(&i - aai)]<r2 + ^(a2 - b2)[a2 - b2 + 2(b2 - aa2 - {1 - a))]o2z, - aal (B.5.10) Since the solution to the symmetric case P C P and L C P are the same as Devereux and Engel (2003), we focus on the asymmetric case (PCP,LCP) . When p = 1, EU is equivalent to —Ep and EU* is equivalent to —Ep*. Thus, we can express the expected utilities for home households and foreign households as a function of monetary policy feedback parameters {a, b}. i EU= -{lnX(^-)1'a + ^[((l-a)ao + a)2a2 + (l-a)2(a1-l)2a2z X £ 1 — n. 2 ^ +(1 - a ) 2 a 2 a 2 + (1 - a ) 2 (a 2 - 1)2<T2. + (1 - a) 2a 2]} (B.5.11) +(1 - a)2a\a2z. + (1 - afa2u} + -y-[((l - a)a0 + a)2a2 EU* = -{In A ( ^ ) 1 - " + ^[((1 - a)a0 + a)2a\ + (1 - a)2(ax - 1 ) V X 2 1 — n. 2 [ + ( - a a i + 6i) 2a 2<r 2 + ( - a a 2 + 6 2 - (1 - a))2a2z, +a2u + a2a2u]} (B.5.12) +(1 - a)2o^cr2. + (1 - afal] + —— [(-aa0 + b0 + a)2a\ lNote that, we have assumed a\ = a\ Appendix B. Appendices of Chapter 2 94 The Nash solution to the international monetary game can be derived as a 0 = ——, . a\ = n, ai — 1 — n (B.5.13) 1 — a a bo = —, b\ = na, a2 = 1 — na (B.5.14) 1 — a It is very straightforward that the monetary solutions support ( P C P , L C P ) . We may derive the expected util i t ies for home and foreign households as: E U = _ { l n A ( ^ ) 1 - * + „(l - n)(l - afa\ + ^ y ^ a 2 } (B.5.15) EU* = -{In Xty*-* + n( l - nf(l - afo\ + ^ y ^ . 2 + ^ ~ ^ + ^ ^ (B.5.16) Thi s gives the welfare gain for home households: EU - EU* = - n 2 ( l - n)(l - afo\ + (1 - n)ao-2 (B.5.17) Appendix B. Appendices of Chapter 2 95 Table B . l : T h e O p t i m a l M o n e t a r y P a r a m e t e r s i n T h r e e Cases Parameters ( P C P ; L C P ) ( L C P , L C P ) ( P C P , L C P ) a a a " 0 1 _ Q 1 _ Q i-a a\ 1 n n a 2 0 1-n 1-n bo a cx a 1—a 1—a 1—a b\ 0 n n a 62 1 1-n 1 — na Appendix C Appendices of Chapter 3 96 C.l Firms C . l . l Preset Prices Listed in Table C . l . C.1.2 Q and Q* Here we express the expected profit differential between P C P and L C P for home firms and foreign firms and Q* as functions of the underling monetary policy parameters respectively. If the monetary authorities follow the unrestricted money rules, we have a2 = (a, - hfal + (a2 - b2fa2u. ( C . l . l ) cov(mc, s) = (ai - h)(ai - l)er£ + (a2 - b2)a2o-\. (C.1.2) cov(mc*,s) = (ai - 6 i ) b i ^ + (a 2 - b2)(b2 - l)a2u. (C.1.3) Thus, fi = \{ai - 6i)(2 - a i - h)al + i ( a 2 - b2){-a2 - b2)a2u. (C.1.4) & = \(ai~ & i ) ( « i + h)ol + \{a2- b2)(a2 +b2- 2)a2u. (C.1.5) Therefore, under ( P C P , P C P ) specification, the solution to international monetary game is a = [1,0] and b = [0,1], we have fi[(l, 0), (0,1)] = fT[ ( l , 0), (0,1)] = ~(at + al) > 0 (C.1.6) Under ( L C P , ' L C P ) specification, the solution to international monetary game is a = [n, 1 — n] and b — [n, 1 — n], we have fi[(n, 1 - n), (n, 1 - n)] = n*[(n, 1 - n), (n, 1 - n)] = 0 (C.1.7) Under ( P C P , L C P ) specification, the solution to international monetary game is a little complicated, see Devereux, Shi and X u (2004) for more detail. The optimal monetary policy rules in this case have the following properties: n < a x < l , 0 < a 2 < ( l - r a ) , &i < 0, b2 > 1 (C.1.8) o i + 02 = 1, h + b2 = 1 (C.1.9) Appendix C. Appendices of Chapter 3 97 which give us Q[a,b] = i ( a i - h){2 ~ a x - bM + a2u.) > 0 (C.1.10) Q*[a,b] = i ( a i - 6 i ) ( a i + h)(a2u + u\.) > 0 ( C . l . l l ) This is because a\ — b\ > 0 and 2 — o i — b\ > 0 and o i + b\ > 0. But, if monetary authorities follow restricted money rules, we then have n=l-a{2-a)ol+l-b2cT2u. (C.1.12) W = \a2a2u + \b{2-b)o-2u, (C.1.13) C . 2 T h e P r o o f o f B i n a r y G a m e The binary game where each monetary authority chooses its policy before the currency of pricing decision is made is defined in the matrix of chapter 3. From this binary game, it can be shown that the {PCP, P C P } configuration with the monetary rules (a=[l,0], b=[0,l]) and flexible exchange rates represent the unique equilibrium of the monetary policy game when monetary authorities take account of the currency of pricing. The proof will be straightforward if we can show the following payoff inequalities EU\\C0U0,l) > ^ [ n ^ l - n ] , [ n , l - n ] (C.2.1) • ^ [ 1 , 0 ] , [ n , 1 -n ] > - ^ [ n , 1 - n ] , [ n , 1 - n ] (C.2.2) C.2.1 follows directly from the comparison between 3.3.8 and 3.3.17. To show C.2.2, suppose that the foreign monetary authority follows b = [n, 1 — n], then if the home country chooses a = [1,0] and given the {a=([l,0], b=[n,l-n]} configuration, we have ft > 0, Q* > 0 (C.2.3) So, all the firms will follow P C P . So under this situation, the expected utility for home and foreign country are given by TO.[»,x-»] = e e x p d - p ) A i a l ^ ' ' (C.2.4) where A\ = — n2(l — n)2 — n ( l — rifp — np + n2 + 2np(\ — n)2 and A2 = —(1 — n ) 4 — n ( l — n) 3 p — (1 — n)p + 2(1 - n ) 3 + 2n(l - n)2p. Nevertheless, if the home monetary authority choose a = [n, 1 — n] instead of a = [1,0], then the home and foreign monetary rules are the optimal rules associated with the equilibrium of { L C P , L C P } , so the expected utility for this case is = @ ^ - p ) ( w 2 - ^ + ( ( 1 2 - / - ( 1 - ^ - } (C.2.5) We can show A i - (n2 - np) = n ( l - n)2(p + pn - n) > 0 (C.2.6) Appendix C. Appendices of Chapter 3 98 A2 - [(1 - n)2 - (1 - n)p) = n ( l - n)2{p + pn - n) > 0 (C.2.7) Therefore, C.2.2 holds. Thus, say that foreign is following b = [n, 1 — n]. Then if home follows o = [1,0], it generates a switching to all P C P , and then a = [1,0] is clearly better for home since C.2.2 holds. Similarly, if foreign is following b = [0,1], then the optimal policy for home is to follow a = [1,0]. So the {PCP, P C P } configuration with the monetary rules (a=[l,0], b=[0,l]) and flexible exchange rates represent the unique equilibrium of the ex-ante monetary policy game, which also supports the full flexible price equilibrium. C.3 Equilibrium under Restricted Monetary Policy Rules We now use the same approach to find the equilibrium under restricted monetary policy rules. The restriction of the monetary policy rules will affect the level of the expected utility by changing the variance and covariance terms. C.3.1 Case 1 {PCP,PCP} First, we may derive the exchange rate and consumption in log terms: s-E{s)=m-m* (C.3.1) c_Ec=nm+(l-n)m* P Under the restricted monetary policy rules, from C.3.1 and C.3.2, we can derive the following variance and covariance terms. Jl _ „ 2 „ 2 , L2„2 ji _ n 2a 2o-g + (1 - nfb2o2u. 3 u u*' c ~2 ' (C.3.3) „ _ naal (1 - n)ba2u. acu — , acu. = (U3.4J P P asu = aa2, asu* = —bcr2. (C.3.5) Substituting the above terms into 3.3.3, and solving the international monetary game, we have a = l , 6 = 1 (C.3.6) Then using C.l .12 and C.l .13, we find n = n* = ±(a2u + a2u,)>0 (C.3.7) Thus, all firms choosing P C P is an equilibrium and the optimal monetary policy associated with this equilibrium ensures the flexible exchange rate where s — E(s) = u — u*. The expected utility associated with this equilibrium is exactly the same as in 3.3.8. Appendix C. Appendices of Chapter 3 99 C.3.2 Case 2 {LCP,LCP} Under L C P , we may get the home and foreign countries log consumption: Tfl 771 * c - E c = - c*-Ec* = — (C.3.8) P P Under the restricted monetary policy rule, from C.3.8, we can derive the following variance and covari-ance terms. „ 2 „ 2 1,2 „ 2 (C.3.9) p 2 ' c p 2 Ocu = — , crcu. = 0 (C.3.10) P ba2 c c *u = 0, cr c . u - = — — (C.3.11) P Using a similar approach as in above sections, we can derive a = n, 6 = 1 - n (C.3.12) This gives us ^ n(2 — n) , (1 — n)2 9 „ ^ n2 , 1 — n2 9 „ „ „, ° 2 a* + —Y~^a*->0' n =-3-^  + — a " - > 0 (C-3'13) This implies, all the firms will follow P C P . Thus { L C P , L C P } is not an equilibrium when monetary authorities can only use the restricted money rules. C.3.3 Case 3 {PCP, LCP} Under this asymmetric pricing specification, the exchange rate and consumption in log terms are: s- Es = m-m* (C.3.14) Tfl 1 c-Ec=— c* - Ec" =-[nm + (1 - n)m*] (C.3.15) P P Under restricted monetary policy rules, we may solve for the variances and covariances terms. a2 = a2a\ + b2a2u, (C.3.16) (C.3.17) P2 * - ^ W . + ( . - . ) W . . | ( C . , 1 8 , a2cu = -aal = 0 (C.3.19) P na 2 2 (1 -n)b 2 °u>  a c ' u - = <V (C.d.20) Appendix C. Appendices of Chapter 3 100 aau = aal, °~sw = -ba\. (C .3 .21) Using the same methodology, we can derive the following solution a = nX-(l-p)[n+(l-n)(P-pn + n)] ^ ft = 1 A - n(l - p)[l + (1 - n)(p -pn + n)] It can be shown that n < a < 1. 1 It gives us « = + \*l > 0, ft* = y ^ + \a\. > 0 (C.3.23) In each country, firms would wish to set prices in their own currency. Hence, the { P C P , L C P } configu-ration is not an equilibrium if restricted monetary policy rules are chosen by the monetary authorities, taking the currency of pricing as given. 1In the special case with p = 1, we have a = n. Appendix C. Appendices of Chapter 3 101 Table C . l : T h r e e O p t i m a l P r i c i n g P o l i c i e s P r i c e { P C P , P C P } {LCP, L C P } { P C P , L C P } ~ P U A ^ 1 " ^ ' n " c r V V C ' ^ P S " 1 l . Q - 1 p r V V C l l Z i l l p* £di \ g l » 1 \ s g l fl I 

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