On the Welfare Benefits of an International Currency

On the Welfare Benefits of an International Curreny Prakash Kannan Researh Department International Monetary Fund Otober 2006 Abstrat Is it benefiial for a ountry s urreny to be used internationally? And, if so, an we quantify the benefit? Over the last 50 years or so, the US dollar has held the dominant position as the main medium of exhange in international transations i.e., as the dominant international urreny. Sine the emergene of the euro as a viable alternative international urreny, there has been great interest in the onsequenes of a transfer of the dollar s premier international role to the euro. Building on reent advanements in the literature on searh models of money, this paper presents a novel model-based approah towards assessing the welfare benefits assoiated with the international use of a ountry s urreny. Apart from the familiar benefits assoiated with seignorage, the model highlights a new hannel that operates through the terms of trade. In the alibration exerise arried out in this paper, the benefits originating from this hannel are quantitatively signifiant. The welfare gain for the Euro area in having the euro internationally used ranges from 1.7% to 2.1% of onsumption depending on relative inflation rates. Of this gain, the benefits aruing to seignorage are on the order of 0.5% to 0.7%. The rest of the world is not indifferent as to whih urreny irulates as the dominant international urreny. Conditional on their urreny not being used internationally, their preferene is for the dominant international urreny to be the one with the lowest inflation rate. Keywords: International Curreny, Searh models of money JEL Classifiations: E40, F33 Email: pkannan@imf.org. Corresponding address: 700 19th Street, NW, Washington, D.C. 20431. I would like to thank Mik Devereux, Robert Hall, Emeri Henry, Romans Pans, Esteban Rossi-Hansberg, Mihele Tertilt, Mark Wright, Joanne Yoong and espeially Pete Klenow for valuable omments on this and earlier versions of this paper. The views expressed in this paper are solely mine and do not neessarily reflet those of the International Monetary Fund. 1

1 Introdution National monies, though often a symbol of national pride, are in reality an inonveniene. From the viewpoint of the oasional tourist right up to the multinational orporation, dealing in different urrenies with different ountries has real osts. In ross-border transations, however, while there exist almost as many urrenies as there are ountries, eonomi fores have led to the use of only a handful of urrenies. Over the last 60 years or so, the US dollar has held the dominant position as the main urreny of trade invoiing. The share of world trade denominated in US dollars far exeeds the share of the US in world trade. In Asia, the share of dollar-denominated trade ranges from 52% to 84% aross ountries, while similar estimates for EU ountries, for trade outside the EU area, are in the range of 20% to 71%. 1 Some onsensus has emerged on the fators underlying the hoie of whih urreny is used as the medium of exhange in international transations what we term an international urreny. 2 The size of the issuing ountry, its share of world trade and the stability of the urreny s value are some of the key fators that have emerged from the analysis in this literature. 3 The mapping of these fators into outomes, however, is not one-toone. A reurrent theme in the study of international urrenies is the presene of network externalities. 4 The benefit that agents derive from using a partiular urreny inreases with the number of other agents who use it. These network externalities introdue multiple equilibria and inertia in the hoie of an international urreny. 5 From a normative perspetive, however, the question of the welfare benefit of international urrenies has reeived less attention. Is it benefiial for a ountry s urreny to be used internationally? And, if so, an we quantify the benefit? These questions are not merely of aademi interest. The emergene of the euro as a viable international urreny has sparked great interest as to whether the US dollar s position as the dominant inter- 1 See Goldberg and Tille (2005). 2 Note that this use of the term is different from the notion of a vehile urreny. For ontributions in that literature see Hartmann (1997) and Devereux and Shi (2005). 3 For early ontributions, see Kindleberger (1981), MKinnon (1979), and Krugman (1984). 4 These network externalities have been elaborated upon in more reent papers on international urrenies suh as Dowd and Greenaway (1993), Matsuyama, Kiyotaki and Matsui (1993), Krugman (1980), Rey (2001), and Wright and Trejos (1996,2001). 5 See Chinn and Frankel (2005) and Kannan (2006). 2

national urreny will be transferred to the euro, bringing to mind the mid-20th entury transition from the pound sterling to the dollar as the key international urreny. Portes and Rey (1998) make an attempt to measure the welfare impat of suh a transition. They estimate that the diret and indiret seignorage benefits that the US gains by having its urreny irulate abroad is roughly 0.2% of GDP. Should the euro beome the dominant international urreny as a result of reduing the transation osts in its finanial markets, they estimate an additional benefit of 0.2% of GDP aruing to the Euro area. 6 In this paper, we present a novel model-based approah in assessing the benefits of an international urreny. The modeling approah taken in this paper draws on reent developments in the literature on searh models of money whih originated in the seminal papers of Kiyotaki and Wright (1989,1993). In partiular, the model builds upon the framework presented in Lagos and Wright (2005). While Lagos and Wright analyze monetary equilibria in a losed-eonomy setting, this paper looks at equilibria in an open-eonomy framework. Agents meet and trade with other agents domestially as well as internationally. The interation ours through both entralized markets where homogenous goods are traded as well as deentralized markets where differentiated goods are traded. In the baseline version of the model, exporters optimally hoose whih urrenies to invoie their trade in. A urreny beome international if sellers deide to use it to invoie their international transations. The fators that influene the hoie of the seller are threefold: the behavior of other sellers, the relative size of the trading partner and the inflation rate of the urreny. In ommon with the existing literature on international urrenies as mentioned earlier, the behavior of other sellers in this model introdues an externality into the invoiing deision of a partiular seller. The surplus that arues to invoiing in a given urreny inreases if other sellers invoie in that urreny too. The behavior of other sellers, however, ease to matter in trades with large ountries. When trading with a large enough ountry, it beomes optimal to always invoie in the domesti urreny of the buyer regardless of what other agents are doing. In this model, therefore, the size of a ountry also plays a role in determining whether its urreny is used internationally or not. The resulting model is analytially tratable for the same reasons as Lagos and Wright, and thus 6 Other attempts at measuring the welfare benefit of an international urreny inlude Cohen (1971). Cohen undertakes a detailed exerise in omputing the monetary value of the benefit to the international use of the pound sterling in finanial markets as well as well as trade. 3

failitates the omputation of equilibrium values and the analysis of welfare. 7 The model features two hannels through whih a gain in welfare an our for the residents of the issuing ountry when their urreny is used in international transations. The first hannel is via seignorage revenues. The external demand for real balanes in the domesti urreny leads to a transfer of resoures to the issuing ountry. The benefit obtained through seignorage is inreasing with the rate of inflation. The ountry that has an internationally used urreny, therefore, has the ability to levy an inflation tax on foreign residents. The seond hannel, whih has not reeived any attention in the literature, is one that operates through trade. As more people use the domesti urreny, its value in terms of the quantity of goods that an be purhased for a unit of the urreny inreases. In other words, the issuing ountry experienes a terms-of-trade improvement. The intuition for how the terms-of-trade hannel operates is as follows. As in most monetary models, agents in this model have to deide on how to alloate resoures between present onsumption and real money balanes. If every transation that the agent an expet to ondut in the next period is arried out in the same urreny, then the deision is made based on the rate of inflation and the share of the surplus the buyer expets to get in eah transation. 8 The higher the rate of inflation, the lower the money balanes held by the agent as she would prefer to divert more resoures towards present onsumption. The same effet would our if she got a lower share of the surplus from transations in the deentralized market. If instead, different transations that the agent expets to arry out next period are invoied in different urrenies, the agent has now another dimension to her hoie. She now has to alloate resoures between present onsumption, balanes in the domesti urreny and balanes in a foreign urreny. If the inflation rates and the share of the surplus are the same for both urrenies, the agent will hold balanes in eah urreny as a funtion of the proportion of transations she expets to arry out in that urreny next period. Sine she now has to deide between multiple urrenies, the fration of her resoures that she invests in balanes in any one urreny will be less than the fration of resoures she would have invested had all her next-period trade been onduted in one 7 The ompliations assoiated with searh models that had divisible holdings of money ame from the endogenous distribution of money holdings. The tehnial ontribution of Lagos and Wright was to introdue a framework where the distribution of money holdings is degenerate at the end of every period. 8 The influene of the buyer s share of the surplus in eah transation on the deision regarding how muh money balanes to hold was a novel ontribution of Lagos and Wright (2005). 4

urreny. The surplus in any trade, however, is inreasing in the amount of real balanes brought into the trade. 9 Consequently, ompared to an equilibrium where the agent has to invest in multiple urrenies, the value of the domesti urreny in an equilibrium where the agent s urreny is aepted everywhere is higher. The model developed in this paper allows for a straightforward alibration and thus a quantifiation of the welfare effets desribed above. An important set of parameters in the model are meeting parameters, whih are assoiated with the probabilities that an agent meets another agent either from her own ountry or from another ountry. In the alibration exerise arried out in this paper, we take advantage of a natural interpretation of these parameters by mapping them to bilateral trade shares. We partition the world into three groups: the US, the Euro area and the rest of the world. We ompare onsumption equivalent welfare hanges aross three equilibria. The first is where the US dollar is the sole medium of exhange used in international transations. The seond is one where there are two international urrenies, the dollar and the euro. Transations with the Euro area are arried out in euros while transations with the US are done in dollars. Transations with the rest of the world, however, are still onduted in dollars. In the third equilibrium studied, the international use of the euro inreases as now trade with the rest of the world is also arried out in euros. The welfare benefit for the Euro area in having their urreny internationally used ranges from 1.7% to 2.1% of onsumption depending on the inflation rates in other parts of the world. Of this gain, the benefits aruing to seignorage are on the order of 0.5% to 0.7%. What is a gain for the Euro area is a loss for the US. The net gain, however, is not zero sine inflation rates and ountry sizes differ. An interesting result obtained is that when inflation rates differ aross ountries, the rest of the world is not indifferent as to whih urreny irulates as the dominant international urreny. Conditional on their urreny not being used internationally, the preferene for the rest of the world is for the dominant international urreny to be the one with the lowest inflation rate. In the baseline model, agents were only allowed to transat in the urreny of invoie. We extend the model in the last setion of this paper to allow agents to use any urreny 9 The surplus is inreasing sine for all monetary equilibria in the model, agents will hold less than the effiient quantity of real balanes as long as inflation is higher than that presribed by the Friedman Rule and the buyer s bargaining power is less than one. 5

in their portfolio. We introdue, however, a transation ost for using urrenies that are not the urreny of invoie so that the individual identities of the urrenies still matter. Transation osts have two effets in this model. Apart from the diret effet of reduing welfare anytime a non-invoie urreny is used, they also have an indiret effet in terms of the alloation of resoures towards money balanes. Higher transation osts divert resoures to present onsumption and away from money balanes as it redues the value of the money in the next period. As we would expet, the magnitude of the welfare hanges in this model is inreasing with the size of the transation osts. However, even at a transation ost of 10%, the magnitudes are muh smaller than in the baseline model. The welfare benefit for the Euro area in having the euro internationally used ranges from 0.03% to 0.26% of onsumption depending on the level of the transation osts. The paper is organized as follows: Setion 2 and 3 desribe the general environment and the deision problems faed by the agents in the model. Setion 4 disusses the speifi details of the baseline model. Setion 5 provides a quantitative estimate of the welfare impat of a hange in the international use of a urreny. Setion 6 presents the extension of the baseline model where all urrenies an be used in any given transation. Quantitative estimates of the welfare impat are then alulated. Setion 7 provides a disussion of the results and onludes. An appendix ontains the omplete proofs of all the lemmas and propositions referened in the text and also inludes some robustness heks. 2 Environment There are 3 ountries in the model, A, B and C, eah with their own respetive urrenies, α, β and γ. In eah ountry i, there is a ontinuum of infinitely lived agents of measure n i, with i n i = 1. Time is disrete in this model. 2.1 Goods and Tehnology Eah period is divided into two subperiods. During the first subperiod, agents from all ountries partiipate in a deentralized market where they randomly meet agents from their own ountry or from another ountry. The probability of an agent from ountry i meeting 6

an agent from ountry j is µ ij. 10 Only one bilateral meeting per agent ours during this sub-period. In the seond subperiod, agents trade in a entralized market. There are two types of ommodities that are produed: a single type of homogenous good and a set of differentiated goods. All agents onsume the homogenous good, but eah agent derives utility from only a subset of differentiated goods. Homogenous goods are produed in the entralized market, whereas differentiated goods are produed by individual agents in the deentralized market. Eah agent has a tehnology that allows her to produe only one type of differentiated good, whih is not a member of the set of differentiated goods that she derives utility from. Both homogenous goods and differentiated goods are assumed to be perishable and thus have to be onsumed immediately. 11 The features embedded in the setup desribed above provide the motivation for trade in the deentralized market and generate an environment where money is essential to ondut transations. The assumption that agents annot produe the type of differentiated good that they derive utility from makes trade potentially welfare enhaning. A single oinidene of wants, where either party in a bilateral meeting desires the good produed by the other, ours with probability σ. Meanwhile, the anonymous nature of the meetings between agents rules out any form of redit arrangements, making money essential to failitate transations between agents (see Koherlakota (1998) and Wallae (2001)). We assume that there is no double oinidene of wants in the deentralized eonomy, and as suh, there is no barter trade. 2.2 Preferenes Agents have preferenes for differentiated goods and homogenous goods, u (q) and U (x), that are C 3, stritly inreasing and stritly onave. We also assume that u(0) = 0 and lim q u (q) =. The variables q and x refer to the quantities of the differentiated good and the homogenous good that are onsumed. There is a ost in utility terms assoiated with the prodution of both homogenous goods and differentiated goods, v (y) and (q). 10 µ ij is not independent from µ ji sine the total measure of meetings between agents from both ountries have to math up: n iµ ij = n jµ ji. 11 The perishability assumption is not ruial. Introduing prodution and the ability to store the homogenous good in the form of apital, along the lines of Aruoba and Wright (2003), will not hange any of the results as long as laims on apital are not used for transations. 7

In this paper, we will assume that the funtional form for both the ost funtions are linear, i.e. v (y) = y and (q) = q. The funtional form assumption for the disutility assoiated with produing the differentiated good is purely for the onveniene of obtaining simplified expressions in most ases. The linearity assumption for the ost of produing the homogenous good, however, plays a more signifiant role in the setup whih will beome evident later. Lastly, agents disount between periods with a disount fator of δ. 2.3 Money There is a money-issuing authority in eah ountry that issues the domesti urreny. Money in this environment is perfetly divisible and storable agents an arry any nonnegative quantity of money. We will denote the vetor of urreny holdings of an agent from [ ]. ountry i as Mt i = m i α,t m i β,t m i γ,t Sine agents an arry any non-negative quantity of money, we need to keep trak of the distribution of money holdings in the eonomy. Let ( ) M i t be the CDF of money holdings by agents in ountry i at time t. The total money F i t supply of urreny at time t, m,t, is then: n i i m i,tdf i t ( m i,t ) = m,t (1) Eah period, the money supply grows at a rate g,t, whih is deterministi. As suh, m,t+1 = (1 + g,t ) m,t. The money supply injetions are transferred in a lump sum fashion to the agents in the eonomy during the entralized market subperiod. 2.4 Timing The timing of ativities during a period is as follows: At the beginning of eah period, agents enter the deentralized market where they produe and onsume differentiated goods. At the end of the deentralized market subperiod, the agents interat in a entralized market environment. In this subperiod, the agents deide how muh of the homogenous good to produe and adjust their portfolio of urrenies. 3 Deision problems of agents We now desribe the speifi deision problems of the agents in the model. We begin by desribing the deision problem of the agents in the entralized market, and then, we desribe the deision problem in the deentralized market. 8

3.1 Deision problem of agents in entralized markets In the entralized market, agents deide on the quantity of the homogenous good to produe as well as the quantity of money balanes to bring forward to the next period. Let s denote the nominal prie of a homogenous good produed in ountry i as P i units of urreny i. Aordingly, the prie of this good faed by an agent in ountry j in terms of urreny j is e j/i P i, where e j/i is the exhange rate between the urreny of ountry j and the urreny of ountry i (quantity of urreny j needed to purhase 1 unit of urreny i). Given that homogenous goods produed in all ountries are idential and that there are no transportation osts, arbitrage in the goods market will neessitate that the law of one prie holds so that e j/i P i = P j. Let W i ( M i t ; I i t) denote the value funtion for an agent from ountry i who enters the entralized market with urreny holdings, M i t. I i t refers to information known to the trader at time t that will be relevant for the trader s deision. The information set onsists of the distribution of money balanes of agents in other ountries, their respetive rates of money growth and the invoiing pattern in next period s deentralized market. Based on the setup desribed above, we have the following: subjet to W i ( Mt i ; It) i = max U ( x i) y i + δv i ( Mt+1; i I i ) x i,y i,mt+1 i t+1 x i + φ i t e i/,t m i,t+1 yt i + φ i t e i/,t m i,t + g i,t φ i t m i,t /n i where the ontinuation value, V i ( ), is the value funtion assoiated with the deentralized market in the next period. The other variables used above are desribed as follows: x i and y i are the onsumption and prodution of the homogenous good respetively, φ i t = 1/P i t is the inverse of the nominal prie of a homogenous good produed in ountry i at time t and g i,t is the growth rate of money supply for ountry i. g i,t φ i t m i,t /n i represents the per-apita lump-sum transfer of newly-reated money. (2) 3.2 Deision problem of agents in deentralized market We next desribe the deision problem faed by the agents in the deentralized market. In a bilateral meeting where there is a single oinidene of wants, agents enter into a 9

bargaining game to deide the quantity of goods and money exhanged. Let q i,j ( M i, M j) be the amount of differentiated goods and d i,j ( M i, M j) the amount of money exhanged when a buyer from ountry i meets a seller from ountry j and trade is denominated in urreny. At this point, we allow for these quantities to depend on the money balanes of both the buyer and the seller. 12 The prie of the differentiated good in urreny units is therefore di,j q i,j max q i,j,d i,j [ q i,j. The bargaining problem will take on the Nash form as follows: [ u [ q i,j ( M i, M j)] + W ( M i I D i,j ( M i, M j) ; It i ) ( W M i ; It)] i θ ( M i, M j) + W ( M j + I D i,j ( M i, M j) ; It i ) ( W M j ; It i )] 1 θ (3) The notation D i,j ( M i, M j) represents the vetor of money transfers from buyer i to [ ]. seller j in all the possible urrenies: D i,j = I is a 3x3 matrix with ones on the diagonal elements representing the transation urrenies and zeros elsewhere. The produt of I and D then aptures the amount in the respetive urrenies that needs to be subtrated or added to the money balanes of the buyer and the seller respetively. θ represents the bargaining power of the buyer. For the analysis that follows, we will assume that the buyer and the seller have equal bargaining power, so θ = 0.5. The bargaining problem above is subjet to a onstraint that states that the amount of money transferred annot be more than the amount of that partiular urreny that the buyer possesses. We will refer to this onstraint as the urreny onstraint. In the baseline model, only the urreny of invoie is allowed to be used. So, the urreny onstraint is d i,j m i where is the urreny of invoie and d i,j = 0 for all other urrenies. In the extension presented in Setion 6, we relax this restrition and allow all urrenies to be used. d i,j α d i,j β d i,j γ 4 The baseline model In this setion, we disuss the speifi details of the baseline model. disussing the invoiing deisions made by agents. We begin by 12 q and d an also depend on time. Here, we have suppressed the time subsripts. Later on, we will be restriting ourselves to stationary equilibria where q is onstant. 10

4.1 Strategi deisions by agents During the entralized market subperiod, agents make strategi deisions regarding the invoiing of their trade should they meet a potential buyer in the next period. The agent s strategy will be a funtion that maps elements from the set of nationalities of agents to the set ontaining the three possible invoiing urrenies. 13 We will denote the strategy of an agent from ountry i who deides to invoie her trade with an agent from ountry j in urreny with the notation s ji based on the denomination that gives her the largest surplus. = 1. An agent s deision on the urreny of invoie will be Here, we introdue a restrition in the strategy spae of agents that we feel is empirially relevant. All domesti trade has to be onduted in domesti urreny. While this restrition rules out the study of some interesting phenomena suh as dollarization, this is not the fous of this paper. This restrition basially guarantees a positive demand for balanes in the domesti urreny so that urrenies do not go out of irulation. While we believe that the restrition is empirially plausible, it does have impliations on the welfare omputations whih will be further disussed in Setion 7. We now introdue some terminology that we will be utilizing in later setions of the paper: If a urreny is used as a urreny of invoie in international transations, we will term that urreny an international urreny. If the proportion of international transations that are arried out in a partiular equilibrium is higher than in another equilibrium, we will say that the urreny is more internationally used in the equilibrium where the proportion of international transations invoied in it is higher. Buyers faed with an invoiing deision by a potential seller deide whether to enter into the bargaining game or not. We denote a buyer from ountry i s deision to trade when faed with an agent from ountry j who is denominating her trade in urreny with b ij. This variable is equal to one if the buyer agrees and zero otherwise. 13 More generally, we an define the domain of the agents strategy set to be the money balanes held by the buyer. As it turns out, the distribution of money balanes within eah ountry is degenerate and so we lose nothing by defining the domain just to be the set of nationalities. 11

4.2 Value funtion in the deentralized market Based on the setup above, we an write the value funtion for an agent from ountry i as she enters the deentralized market as V i ( Mt i ; It i ) = µ ij σs ij b ij j { q +µ ij σs ji b ji j,i + 1 j {u ( ) ( q i,j + W M i I D i,j ; It)} i df j t + W ( M i + I D j,i ; It i )} ( ) df j t M j t 2µ ij σ W ( Mt i ; It i ) ( ) M j t (4) where the dependene of q i,j and d i,j on M i and M j is suppressed. In any given meeting where there is a single oinidene of wants, an agent an either be a buyer or a seller in the transation. The first term in the expression above is the expeted value of being a buyer for an agent from ountry i when she meets an agent from ountry j. µ ij is the probability of meeting that agent, σ is the probability of the single oinidene of wants, s ij represents the invoiing strategy of the seller and b ij represents the strategy of the buyer. Sine we allow for the quantities to depend on the money balanes of the seller, we need to find the expeted value of the seller s money balanes. Hene, we integrate over the entire distribution of the seller s money holdings. The seond term aptures the expeted value for the agent from ountry i of being a seller, while the third term aptures the value in the event where there is no single oinidene meeting or no meeting at all. 4.3 Equilibrium We begin by stating the definition of an equilibrium for this eonomy. { } An equilibrium in this eonomy is a set of strategies, s ij, b ij i, j,, pries { φ i } t, e / i,,, quantities in the entralized market { x i t, yt, i Mt+1} i i, t, quantities in the deentralized market q i,j,t, di,j,t i, j,, t, distribution of money holdings Ft i ( ) { } M i t i, t and value funtions { Vt i, Wt i } i, t suh that 1. Strategies form a pure strategy Nash Equilibrium. 2. Value funtions and quantities in the entralized market solve (2) and (4) taking pries and the distribution of money holdings as given. 12

3. Agents solve the symmetri bargaining problem in the deentralized market as stated in (3) taking pries as given. 4. Market for money balanes lears: i n i m i,t dft i ( ) m i,t = m,t, t. 5. Market for homogenous goods lears: i y i = i x i i. For the purposes of this paper, we will be restriting ourselves to stationary equilibria, where the quantity of differentiated goods traded in the deentralized market is onstant over time, q i,j,t = qi,j,t+1 t. To solve for the equilibria of this eonomy, we first derive some useful results in the entralized market problem. Following that, we solve for the quantities of goods and money exhanged in the bilateral meetings in the deentralized market. 4.4 Centralized Market Problem In this subsetion, we derive some useful properties assoiated with the entralized market. From equation (2) in the previous setion, we an substitute for y i t in the objetive funtion from the budget onstraint to get W ( Mt i ; It) i ( = φ i t ei/α,t m i α,t + e i/β,t m i β,t + e i/γ,tm i ) [ ( γ,t + g i,t m i,t /n i + max U x i ) x i t] [ ( + max φ i t ei/α,t m i α,t+1 + e i/β,t m i β,t+1 + e i/γ,tm i γ,t+1)] M i t+1 + max M i t+1 [ δv ( M i t+1 ; I i t+1)] From equation (5), we see that the agent s hoie of next period s money holdings is independent of the respetive values in the urrent period. This feature of the model is a result of the linear funtional form assumption for the disutility assoiated with the prodution of the homogenous good, whih removes any wealth effets from the agent s hoie. The value funtion above also has another useful property in that it is linear with respet to urrent money balanes. We will exploit this feature in the next subsetion. 14 In any equilibrium, the following two first order onditions must hold for all i and : U ( x i t) = 1 (6) φ i ( te i/,t δv m i M i t+1 ; It+1 i ) = if m i > 0 (7) 14 The restrition that 0 < y will not be imposed now, but will be verified during the quantitative analysis. x i t (5) 13

where V m i denotes the partial derivative of V with respet to the balanes held in urreny. Equations (6) and (7) have the standard interpretation of equating marginal benefits to marginal osts. In equation (6), the marginal utility of onsuming an additional unit of the homogenous good is equated to the marginal ost of prodution, whih in this ase is 1. Similarly, the left-hand side of equation (7) represents the marginal ost assoiated with holding an extra unit of urreny in terms of units of the homogenous good. The righthand side, meanwhile, represents the marginal benefit assoiated with holding an extra unit of the urreny. The benefit omes from the ability to purhase extra units of both the differentiated good and the homogenous good in the next period. 4.5 Bargaining Solutions In the last subsetion, we noted that the value funtion in the entralized market has a useful property in that it is linear with respet to urrent money balanes. This property allows for an easier restatement of the bargaining problem in (3). The bargaining problem between buyer i and seller j trading in urreny an be restated as follows: subjet to max q i,j,d i,j [ u ( q i,j ) φ i e i/ d i,j ] [ q i,j d i,j m i + φ j e j/ d i,j ] where the notation indiating the dependene of q and d on the money balanes of the buyer and the seller have been suppressed. The equilibrium quantity of goods and urreny exhanged will depend on whether the urreny onstraint binds or not. If the urreny onstraint does not bind, then it an be shown that the quantity of goods exhanged is given by the following first-order ondition: u ( q i,j ) ( ) φ j e j/ φ i = 1 (8) e i/ ( ) φ Sine the law of one prie holds for homogenous goods, we have that j e j/ = 1. φ i e i/ Therefore, if the urreny onstraint does not bind, we have q i,j = q where q is the effiient quantity of goods exhanged and is impliitly defined by u (q ) = (q ) = 1. The urreny exhanged, m i, is given by φ i e i/ m i = 1 2 q + 1 2 u (q ) (9) 14

If the urreny onstraint does bind, we have that d i,j = m i and the quantity exhanged is given by ( u q φ i e i/ m i i,j = ( u ) q i,j q i,j ( + u ) + 1 q i,j ) z ( q i,j ) In both the ases when the onstraint binds and when the onstraint does not bind, neither the urreny nor the quantity transferred depend on the money balanes of the seller, so we an drop the seller supersripts from d and q. (10) Furthermore, the quantity transferred does not depend on the buyer s holdings of other urrenies apart from the transations urreny. These results follow diretly from the speifiation of the onstraint in the bargaining problem. 4.6 Optimal quantity of money balanes and differentiated goods exhanged In the previous subsetion, we saw that the amount of goods and urreny that are exhanged in the deentralized markets depend on whether the urreny onstraint is binding or not. In this subsetion, we ombine the results from the bargaining problem and the entralized market problem and onsider the problem of the optimal quantity of money balanes to be brought forward to the next period. Following Lemma 1 in the Appendix, we an rewrite the first-order ondition for both the ase where the urreny onstraint binds ( m i,t+1 <,t+1) mi and where it does not bind ( m i,t+1 m i,t+1) as follows: φ i te i/,t δµ i σu ( q i t+1 ) q i,t+1 + ( 1 µ i σ ) φ i t+1e i/,t+1 φ i te i/,t = δφ i t+1e i/,t+1 for m i,t+1 < mi,t+1 (11) for m i,t+1 mi,t+1 (12) There are two hanges in the notation used above. First, the supersript in the quantity of differentiated goods traded, q, now only shows the nationality of the buyer sine we saw that the money balanes of the seller do not affet the quantities traded. Seond, the meeting parameter, µ i, aptures all the bilateral meetings where the agent from ountry i is the buyer and trade is onduted in urreny, i.e. µ i = j µ ijs ij b ij. For example, if agents in ountry A and B are denominating their trade with ountry C in urreny α, then the relevant meeting parameter in the equation above for an agent from ountry C will be µ C α = µ CA + µ CB. 15

Lemma 2 in the Appendix shows that in any monetary equilibrium, the urreny onstraint in the bargaining problem will be binding. As suh, the relevant first-order ondition in any monetary equilibrium is given by equation (11). It should be noted, however, that we are not guaranteed a unique solution (a unique value for q). Uniqueness is guaranteed if V is onave, i.e. if V m i m i 0. In the setion on quantitative analysis, we will verify the onavity of V empirially. At this junture, however, we will make suffiient assumptions on the utility funtion u ( ) suh that V m i m i 0.15 With a unique solution (a unique value for q), we then have a degenerate distribution of money holdings for all periods. All agents in a partiular ountry demand the same quantity of real money balanes in aordane with equation (10) for a given value of q. We now determine the equilibrium quantity of differentiated goods exhanged. From equation (10), we get q i,t+1 = φi t+1 e i/,t+1z (q). Substituting this into equation (11) and dividing both sides by φ i te i/,t, we get 16 [ ( 1 = δ µ i σ u q i ) ] ( ) 1 z (q) i µi σ + 1 1 + g,t The onstant value of q that solves the equation above gives us the stationary equilibrium quantity of differentiated goods traded in a bilateral math where the buyer is from ountry i and the trade is arried out in urreny, q i. Plugging this value of q into equation (10) gives us the real balanes in urreny demanded by an agent from ountry i. 4.7 Equilibrium invoiing strategies In any equilibrium, we will also need to verify that the agent s deision to invoie their trade in a partiular urreny is optimal given the equilibrium money balanes and strategies of buyers from different ountries. An agent from ountry i s surplus from the deision to invoie her trade in urreny with a buyer from ountry j is q j + W ( M i + I D j). The seller will hoose to denominate her trade in urreny if and only if (13) q j + W ( M i + I D j) q j + W ( M i + I D j) (14) 15 A suffiient assumption in this ase would be to assume that u is log onave (see Lagos-Wright (2005)). This assumption is merely suffiient and not neessary. In most pratial appliations, suh as the one arried out in this paper, uniqueness is still obtained even though u is not log-onave. 16 We have utilized the result derived in subsetion 4.8 whih shows that φ t+1 φ t = 1 1+g,t. 16

for all alternative urrenies,, given the strategies of the other agents. We now present some omparative statis results regarding the surplus of the seller whih will be useful when we analyze the welfare impliations later on. Proposition 1 Let the seller s surplus of invoiing in urreny be denoted as S S q j + W ( M i + I D j). We then have (i) SS q q<q > 0 (ii) qi µ i > 0 (iii) qi g < 0 The results in Proposition 1 highlight the network externalities present in the model. This an be seen by onsidering the problem of an agent from A who is deiding how to invoie his trade with agents from B. Assume initially that agents from ountry C are invoiing their trade with agents from B in α and that the best response for agents in ountry A is to invoie in urreny β with agents from ountry B. Now suppose that agents from ountry C deide to invoie in urreny β. From parts (i) and (ii) of Proposition 1, this inreases the surplus to agents from ountry A, i.e. Sβ S, as µb β will now be higher. If the initial best response of ountry A s agents was to invoie in some other urreny, the move by ountry C agents to invoie in β weakly redues their surplus (stritly redues is the best response was originally urreny α). Potential buyers in this model will always find it optimal to aept an offer to trade in a partiular urreny if she has balanes in that urreny. The proof for this is trivial sine buyers are always guaranteed a positive surplus when entering the bargaining game and there is only one meeting in eah day subperiod. If the buyer does not have balanes in the invoiing urreny, she is indifferent as she gets a net surplus of zero in either ase. We resolve this indifferene by assuming that the buyer deides not to aept the offer to trade. 4.8 Pries The inverse of the prie for the homogenous good φ i t = 1/P i t an be found through the market learing ondition for money balanes: 17

φ i te i/,t n j m j i = φ i te i/,t m,t j φ t = ( ) j n jz q j m,t (15) where we use φ t to denote the inverse prie level for the ountry that issues urreny. Note that the model features neutrality in the sense that hanges in the money supply only affet the general prie level sine the quantity of real balanes is stationary. The inflation rate in this eonomy is given by P t+1 Pt = 1 + g,t+1. The model, however, is not superneutral. Changes in the inflation rate will affet the equilibrium quantity of goods exhanged in the deentralized markets, q i, through equation (13). The following proposition relates the general prie level to the international use of the urreny. Proposition 2 As we move from an equilibrium where urreny is more internationally used to one where it is less internationally used, the prie level in the issuing ountry inreases. This result is similar to the result found in Wright and Trejos (2001) where they show that a urreny s purhasing power at home rises when a urreny is used internationally. At a qualitative level, this provides some theoretial explanation as to why the prie level for the US is a onsistent outlier in ross-ountry inome-prie regressions. 5 Welfare benefits of an international urreny Sine the emergene of the euro as a viable alternative international urreny, there has been great interest in the onsequenes of a transfer of the dollar s premier international role to the euro. Papers by Chinn and Frankel (2005) and Eihengreen (2005) disuss possible senarios related to a loss in the key international role of the dollar. The model developed in the previous setions provides a framework in whih we an disuss the welfare impliations of any suh transitions. To analyze the model s welfare impliations assoiated with a wider international use of a urreny, we study differenes in welfare aross three speifi equilibria of interest, whih we shall label as Equilibrium I, II and III. These three equilibria apture several 18

Figure 1: Pattern of trade invoiing in the equilibria analyzed (arrow from i to j indiates i exporting to j ) possibilities when there are two prominent urrenies in the world, suh as is the ase urrently with the euro and the US dollar. The equilibria that we study are defined by the pattern of trade invoiing, whih is shown in Figure 1. Equilibrium I is one where urreny α funtions as the only international urreny. All international trade in this equilibrium is onduted using urreny α. Equilibrium II is one where all invoiing vis-a-vis ountry B is arried out in urreny β. Trade with ountry C, however, is still arried out in urreny α. Equilibrium III is one where trade with ountry C is now also arried out in urreny β. Thus, the international use of urreny α diminishes as we move from Equilibrium I to Equilibrium III. These are, of ourse, not all the possible equilibria in this model. In general, we ould have all permutations of invoiing between ountries. For any given set of meeting parameters, an equilibrium is admissible if the invoiing patterns are optimal for all sellers as determined by equation (14). In the alibration exerise arried out in this paper, Equilibriums I, II and III are admissible and are the most interesting to analyze when there are two prominent urrenies in the world. The steady state value funtions aross the three equilibria differ not only in their equilibrium quantities, but also in their form. We will use the value funtion of agents in Country B as an illustration of the main fores at work as we move aross equilibria. 19

The effets on agents from other ountries an be assessed in an analogous manner. For Equilibrium I, we an write the steady-state value funtion for Country B as 17 (1 δ) V(I) B = U ( x B ) [ ( ) ] x B + µ BB σ u qβ(i) B qβ(i) B [ ( ) ( )] +µ BA σ u qα(i) B qα(i) A φa (I) m B α(i) ma α(i) [ ( ) ( )] +µ BC σ u qα(i) B qα(i) C φa (I) m B α(i) mc α(i) g α φ A (I) mb α(i) (16) The first two terms in the value funtion represent the utility derived from onsumption of the homogenous good. onsumed. x B is the optimal amount of the homogenous good that is The next three terms represent the utility derived from transations in the deentralized market. The last term in equation (16) aptures the seignorage transfer that has to be made to ountry A due to the demand for balanes in urreny α by agents in ountry B. In the ase of equilibrium II, the steady-state value funtion takes on the following form: (1 δ) V(II) B = U ( x B ) [ ( ) ] x B + µ BB σ u qβ(ii) B qβ(ii) B [ ( ) ] +µ BA σ u qβ(ii) B qα(ii) A φb (II) mb β(ii) + φa (II) ma α(ii) [ ( ) ] +µ BC σ u qβ(ii) B qα(ii) C φb (II) mb β(ii) + φa (II) mc α(ii) (17) When we move to Equilibrium II, we note that onsumption of the homogenous good in the entralized market remains the same as in Equilibrium I. The value obtained from domesti transations, however, has hanged. In Equilibrium II, agents from ountry B an now use urreny β for all their transations, not just domesti transations as in Equilibrium I. As suh, µ B β(ii) > µb β(i). From Proposition 1, holding other parameters onstant, we have that qβ(ii) B > qb β(i). Sine in any monetary equilibrium, q < q, we have that u < 1. Therefore, omparing Equilibriums I and II, we see that the international use of urreny β has inreased the net benefit that agents from ountry B obtain from domesti transations. The value obtained from transations with ountry A and C also hange. The hange omes from the fration of transations where agents from ountry B are buyers. In these 17 The roman numeral subsripts refer to the respetive equilibrium. 20

transations, the relevant omparison is between the buyer s net surplus in Equilibrium [ ( ) ] [ ( ) ] I and Equilibrium II, i.e. between u qβ(ii) B φ B (II) mb β(ii) and u qα(i) B φ A (I) mb α(i). We know that µ B β(ii) > µb α(i) and so, holding everything else onstant, we have that qb β(ii) > q B α(i). We an show from equation (13) that, as long as (1 + g ) > δ, 18 the buyer s surplus is inreasing in q. 19 Therefore, if inflation rates are the same aross both urrenies α and β, the surplus that agents from ountry B get from purhases from agents from ountry A and C inreases. As shown above, the international use of urreny β has inreased the value assoiated with both domesti and international transations for agents from ountry B. The intuition for this result is as follows. In the entralized market, agents have to deide on how to alloate resoures between present onsumption and money balanes. If every transation that the agent an expet to ondut in the next period is arried out in the same urreny, then the deision is made based on the rate of inflation and the share of the surplus the buyer expets to get in eah transation. The higher the rate of inflation, the lower the money balanes held by the agent as she would divert more resoures towards present onsumption. The same effet would our if she got a lower share of the surplus from transations in the deentralized market. If instead, different transations that the agent expets to arry out next period are invoied in different urrenies, the agent has now another dimension to her hoie. She now has to alloate resoures between present onsumption, balanes in urreny α, say, and balanes in urreny β. If the inflation rates and the share of the surplus are the same for both urrenies, the agent will hold balanes in eah urreny as a funtion of the proportion of transations she expets to arry out in that urreny next period. Sine she now has to deide between multiple urrenies, the fration of her resoures that she invests in balanes in any one urreny will be less than the fration of resoures she would have invested had all her next-period trade been onduted in one urreny. The surplus in any trade, however, is inreasing in the amount of real balanes brought into the trade by the buyer. 20 Therefore, ompared to an equilibrium where the agent has to invest in multiple urrenies, the total surplus for eah transation in an equilibrium where the agent s urreny is aepted everywhere is higher. In Equilibrium II, urreny β is internationally used so agents from ountry B do not have to hold balanes of 18 That is, as long as we are away from the Friedman Rule. 19 See equation (23) in the proof for Proposition 1 in the Appendix. 20 See proof for Proposition 1. 21

urreny α anymore. As suh, the surplus they obtain from all purhases that they make inrease. It is this effet that we term the terms-of-trade effet. The higher international use of urreny β also yields another benefit for ountry B agents. Sine all the agents now invoie their trade with B in urreny β, agents from ountry B no longer hold any foreign urreny balanes. Therefore, there is no seignorage transfer to ountry A as in Equilibrium I. For equilibrium III, the steady-state value funtion takes on the following form (1 δ) V(III) B = U ( x B ) [ ( ) ] x B + µ BB σ u qβ(iii) B qβ(iii) B [ ( ) ] +µ BA σ u qβ(iii) B qα(iii) A φb (III) mb β(iii) + φa (III) ma α(iii) [ ( ) ] +µ BC σ u qβ(iii) B qβ(iii) C φb (III) mb β(iii) + φb (III) mc β(iii) ] +g β [φ B m β (III) φ B (III) n mb β(iii) B (18) The net benefit obtained from the entralized market one again does not hange relative to Equilibrium I. The net surplus assoiated with domesti transations and transations with agents from ountry A will also be the same as in Equilibrium II. Transations with agents from ountry C, where agents from ountry B are the sellers, are now invoied in urreny β. However, if the inflation rate of urreny β is the same as that of urreny α, the surplus obtained from sales to agents from ountry C will be the same as in the other two equilibria. This is so beause, if g α = g β, we have that q C β(iii) = qc α(i) sine µ C β(iii) = µc α(i). Agents from ountry C, however, now demand balanes in urreny β. Therefore, agents from ountry B now reeive seignorage revenues eah period, as aptured by the last term in equation (18). φ B m β n B represents the per-apita quantity of real money supply, whih will be larger than the per apita real domesti demand, φ B m B β. Note that the benefit from seignorage inreases with the rate of inflation, g β. 5.1 Quantitative estimates for the baseline model We now proeed to present quantitative estimates regarding the welfare impat as we move aross equilibria. To map the model to the data, we first need to do two things. Firstly, we need to determine the ountries or regions of interest that will map into ountries A, B, and C in our model. Seondly, we need to find a ounterpart in the data for the meeting 22

parameters in the model, µ. A suitable ounterpart for ountries A, B and C are the US, the Euro area and a omposite for the rest of the world, ROW. Although these groupings are not ountries per se, we will ontinue to refer to them as suh to ease the presentation. The ROW omposite will be an aggregate of all the remaining ountries in the world (other than the US and the Euro area), with the exeption of the UK and Japan. These two ountries are exluded as we do not want to onsider ases where the urreny of Country C an be used as an international urreny. What we have in mind when we think of ROW is primarily developing ountries who do not have strong or onvertible urrenies of their own. 21 We next turn our attention to quantifying the meeting parameters. The meeting parameters ome in two types: domesti meetings and international meetings. Let s i be ountry i s share of world output and ψ ij be the share of ountry i s trade that is with ountry j. The meeting parameters for ountry i are then alibrated in the following way: µ ii = s i and µ ij = ψ ij s j. As a ountry s relative output inreases, it follows naturally that the measure of domesti transations inrease as well. International meetings, on the other hand, are governed by two variables: the share of trade onduted with that ountry and that ountry s relative size. The measure of meetings between two ountries an therefore inrease if either the share of trade between the two ountries inrease or if the trading partner s relative size inreases. It an be readily verified that, under this parametrization, the sum of all meeting parameters for a given ountry will be less than one. Data for relative output shares were obtained from the World Development Indiators published by the World Bank while bilateral trade data are from the IMF s Diretion of Trade Statistis. The values used are as at the end of 2002. The estimation of the meeting parameters, however, are subjet to adding-up onstraints that state that the total measure of meetings that agents from ountry i have with agents from ountry j has to be the same as the total measure of meetings that agents from ountry j have with ountry i. 22 Due to these onstraints, of the six international meeting parameters, only three are separately identified. We use the parametrization desribed above to ompute the two international meeting parameters for the US and the parameter 21 What this implies in terms of the model is that in heking the optimal invoiing deision based on equation (14), we do not allow for the urreny of ountry C to be used for international transations. 22 See footnote 10. 23