Search-Theoretic Models of International Currency

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Alberto Trejos is an assistant professor at Northwestern University. Randall Wright is professor of economics at the University of Pennsylvania, and consultant to the Federal Reserve Banks of Cleveland and Minneapolis.

backed by central bank reserves in Search- U.S. dollars (as in Argentina)?

Theoretic • Should more Latin American countries simply abandon their local cur- Models of rency and switch to U.S. dollars (as International Panama did)? Though we do not claim to be ready Currency to provide deﬁnitive answers to all of these important questions at this time, we do Alberto Trejos and think that search-theoretic models are in Randall Wright principle well suited to address these types of issues. More traditional theory, including models that include cash-in-advance earch-based theories of the exchange or money-in-the-utility-function type as- process provide economists with a sumptions, are clearly ill suited in this re- Sway of formalizing the microfounda- gard. They are ill suited because too much tions of monetary economics and to dis- is decided by assumption. In particular, cuss a variety of issues in monetary the- the answer to the question, “Which ory and policy in a new light.1 Many monies circulate in which countries?” is questions that cannot even be formulated determined at the outset by the modeler in more traditional monetary models can when he chooses which money to put be proﬁtably analyzed in these new mod- in which cash-in-advance constraint or in els. In this essay, we propose to review re- which utility function. In contrast, cent developments using search-based search models are designed to determine models to study some international mone- endogenously which monies circulate tary issues. where. There seems little doubt that interna- We begin by reviewing a version of tional monetary economics is of central the model in Matsuyama, Kiyotaki, and importance today. Of the types of ques- Matsui (1993). Basically, it can be de- tions one would ultimately like to answer, scribed as follows. There are two countries, consider these: each of which issues its own currency. There is a meeting or matching technology • Should Europe adopt a common that describes the frequency with which a currency? given individual interacts with other individuals, both locals and foreigners (it is as- • If so, should individual countries sumed that one interacts with individuals continue to issue local currencies? from one’s own country more often than a foreigner interacts with these same indi-

• If a state or province separates from viduals). When two agents meet, one 1 a nation (as Quebec periodically with money and the other with real output See, for example, in addition to the work discussed below, discusses in Canada), how should it for sale, they have to decide whether to Kiyotaki and Wright (1989, design its monetary system? trade. Parameter values, as well as expecta- 1991, and 1993), Aiyagari tions regarding other agents’ behavior, and Wallace (1991 and • Should more Latin American coun- jointly determine this decision and thereby 1992), Williamson and Wright tries adopt currency boards, where determine the realm of circulation for each (1994), Trejos (forthcoming) each unit of local currency is currency. Potentially there are three dis- and Ritter (1995).

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tinct types of equilibria—or regimes—in • How does the fact that a currency which we are interested, where none, one, circulates internationally affect its or both of the currencies circulate interna- purchasing power at home? tionally. This model allows one to answer ques- • Where does an international cur- tions like the following: rency purchase more—at home or abroad? • What features of a country make it possible, or likely, for its currency to • What are the effects on seigniorage circulate internationally? and welfare in each country when one money becomes an interna- • How and when can local currencies tional currency? survive in the presence of a univer- sally accepted international cur- • How are policies designed to maxi- rency? mize either seigniorage or welfare affected by concerns of currency • Does an international currency substitution? emerge naturally as economies become integrated? • How are national monetary policies connected, and what is the scope • What are the costs and benefits to a for international cooperation? country of having its currency serve as an international medium of ex- In the next section, we outline the change? basic assumptions on which the model is built. The third section presents the indi- • An extension of this model in Zhou visible output version of the model in (1994) additionally allows one to Matsuyama, Kiyotaki, and Matsui (1993). study the issue of when agents We then present the divisible output would want to exchange currencies model with bargaining in Trejos and in such a world. Wright (1995b), followed by a discussion of some policy implications. The final sec- There are shortcomings with the mod- tion presents some brief conclusions. els mentioned in the previous paragraph. Perhaps the most obvious is that in these models, as in all of the first-generation THE BASIC MODEL search-based models of money, every ex- The economy consists of two coun- change is assumed to be a one-for-one tries, labeled i = 1, 2. One’s country is im- trade. This simplifying assumption makes portant in that it determines the frequency it possible to discuss the process of ex- with which one interacts with other change—and, in particular, which objects agents. Individuals interact, or meet, bilat- circulate as media of exchange among erally according to a random matching

which agents—without tackling the deter- process in continuous time, and ˆ ij de- mination of the relative values of these ob- notes the Poisson arrival rate at which a jects. Unfortunately, however, it obviously citizen of Country i meets citizens of also makes it impossible to talk about Country j. We assume that ˆ ³ ˆ for 2 ii ji Bargaining was ﬁrst introduced prices or exchange rates. Therefore, we j ¹ i. This simply says that, for example, a into (one-country) search models of money in Trejos and also present the extension of the model in Mexican meets Mexicans more frequently Wright (1995a) and Shi Trejos and Wright (1995b) designed to en- than an American meets Mexicans. 2 (1995b). Other recent applica- dogenize prices using bargaining theory. Each country starts with a large num- tions include Aiyagari et al. This extension allows us to raise a ber of citizens, and the fraction of individ- (1996), Trejos (1994), and whole range of new issues, including the uals from Country i is Ni with N1 + N2 = 1. Shi (1995a). following: Thereafter, both populations grow at the

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118 M AY / J U N E 1 9 9 6 same rate ³ 0. The population sizes and class of models under consideration from the meeting technology parameters are not ones with particular cash-in-advance or independent because we have the identity money-in-the-utility-function assumptions 4 N1 ˆ 12 = N2 ˆ 21 (both sides of the equality imposed exogenously. give the total number of international Fiat money is introduced by the gov- meetings per unit time). Here we take ˆ ij ernment of Country j issuing one unit of as primitive and let the populations be free Currency j to some fraction Mj Î (0,1) of to satisfy this identity. A special case is the its newborn citizens, at each point in time, speciﬁcation actually used in Matsuyama, in exchange for some amount of real out- Kiyotaki, and Matsui (1993), which has put (how much real output depends on

ˆ ii = N1 and ˆ ij = *Nj, with * < . Thus, what assumptions we make). Issuing arrival rates are proportional to country money in this way yields a continuous flow size, but they are smaller across countries of seigniorage, although notice that the than they are within countries. As * gets only time private agents interact with the closer to , the two countries become government is when they first enter the more integrated.3 economy. In the special case where = 0, Agents are distinguished not only by the government simply issues an input of their country and date of birth, but also by currency to some fraction of its citizens their tastes and technologies. As in the at the initial date and then shuts down. 3 In this model one does not typical monetary search model, we need to To keep things tractable, we make the physically travel between one adopt some notion of specialization. Here, following assumptions. First, we assume country and another, nor does for simplicity, it is assumed that there are that an agent holding a unit of currency one choose in any other way to K ³ 3 goods and the population of each always spends it all at once, which could interact with foreigners instead country contains equal numbers of K obviously be guaranteed if we simply say of fellow citizens. It is simply types, where each Type k consumes only that the monetary object is indivisible. that you sometimes meet for- Good k and produces only Good k +1 This implies that no one holding currency eigners in your daily routine. (modulo K; that is, Type K produces Good ever holds less than one unit. Second, we Imagine, for example, a town 1). If we assume that meetings are random assume that, except for the newborn, no on the border populated by with respect to consumption-production agent can produce until after he consumes. both Mexicans and Americans. Americans trade with fellow = ˆ type, then ij ij /K is the rate at which This implies that two agents with currency Americans and with Mexicans, a citizen of Country i meets a citizen of do not trade with each other, and no one although perhaps less frequent- Country j who consumes the good he pro- ever holds more than one unit of money. ly with the latter and vice duces; it is also the rate at which the Hence at each point in time there will be versa. What we are interested citizen of Country i meets a citizen of some agents with one unit of money each, in here is the monetary nature Country j who produces the good he called buyers, and a disjoint group with of these interactions; that is, consumes. no money, called sellers. Let the fraction which currencies get traded by There is no centralized market or auc- of buyers from Country i with Currency j whom? tioneer in this model: All trade is bilateral be denoted mij, and the fraction of sellers 4 The simplifying assumptions in and quid pro quo. The large number of from Country i be denoted mi0 = 1 - mi1 - the text allow us to focus exclu- agents rules out private credit because the mi2. Then the vector m = (mij) completely sively on the use of fiat curren- probability of meeting a particular individ- describes the asset distribution across cies as media of exchange, but ual a second time is zero. Our specifica- agents. in principle they can all be tion for tastes and technologies rules out What happens when a buyer and seller relaxed. For examples of mone- direct barter. We also make the assump- meet depends on which version of the tary search models with credit, tion that goods are nonstorable, which model we consider. In Matsuyama, see Hendry (1992) or Shi rules out commodity money. Hence, trade Kiyotaki, and Matsui (1993), output is in- (1995a); for examples with some direct barter, see Kiyotaki requires the use of some form of fiat divisible and the same across the two and Wright (1991 and 1993) money as a medium of exchange. countries. Thus if you find a seller who or Burdett et al. (1995); and However—and this is the crucial point— can produce the right type of output, there for examples with commodity we do not impose that any particular cur- is no question that you always want to money, see Kiyotaki and Wright rency plays this role in any particular trade your money for his one indivisible (1989), Aiyagari and Wallace transaction. This is what distinguishes the good, and, in particular, you do not care (1991), or Li (1995).

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about the seller’s nationality. In Zhou never hold pesos). Second, note that (1994), output is indivisible but differs trades between a buyer and seller of the across the two countries, and consumers same nationality do not alter m because have tastes that ﬂuctuate randomly, which they leave the aggregate distribution un- means they generally do care about the changed. These considerations imply that seller’s nationality. In Trejos and Wright the steady-state equations are (1995b), a unit of output is the same

across countries, but output is perfectly di- (1) ij mi0mji - ij mii mj0 j + (Mi - mii ) = 0, visible and the amount of output that you

get for your money needs to be negotiated. (2) ijmi0mjj i - ijmijmj0 - mij = 0, This means that you might care about the nationality of the seller if the (endoge- for i ¹ j. Given and the identity

nous) prices differ across countries. mi0 = 1 - mii - mij, these can be solved for In any case, the important thing from the steady-state asset distribution m. our vantage is whether a buyer from Consider equation 1 (equation 2 has a Country h with Currency i trades with a similar interpretation). The ﬁrst term says

citizen of Country j because this deter- that mii increases when a seller from Coun- mines the steady-state distribution of as- try i meets a buyer from Country j with sets across the population. In this article, Currency i (given the maintained assump- we consider only equilibria where trade al- tion that when a seller from Country i ways occurs if a buyer with Currency i meets a buyer with Currency i, they always

meets a seller from Country i (pesos deﬁ- trade). The second term says that mii de- nitely circulate in Mexico and dollars creases when a buyer from Country i with deﬁnitely circulate in the United States). Currency i meets a seller from Country j if

Then the key issue is whether trade occurs they trade (that is, if j = 1). The ﬁnal term

when a seller meets a buyer with foreign says that mii increases when the fraction of currency—that is, whether exchange oc- the newborn agents in Country i who re-

curs when a buyer with Currency i meets ceive currency, Mi, exceeds mii. Steady state a seller from Country j ¹ i. requires the net change to be zero. We distinguish the different types of Given the asset distribution, the idea is possible equilibria—or different regimes— to use dynamic programming to solve the

as follows. Let j = 1 if within Country j we individual decision problems concerning see Currency i ¹ j in circulation, in which when to trade. The next section considers 5 In the model in Matsuyama, case we call Currency i an international cur - the model with indivisible output, where Kiyotaki, and Matsui (1993), rency; and let = 0 if in Country j we do every trade is a one-for-one swap. The sec- these are the only possible j regimes, except for possibly not see Currency i in circulation. Then a tion after that considers the case where out- equilibria where at most one of regime is a list of values for = ( 1, 2). put is divisible and agents bargain. the monies has value or equilib- The four possible regimes are = (0,0), ria that are the same except for (1,1), (0,1), and (1,0). In the first case relabeling in the sense that we there is no international currency; in the THE MODEL WITH call American money pesos and second case both currencies are interna- INDIVISIBLE OUTPUT Mexican money dollars. In the tional; and in the final two cases only one Here we analyze the model with indi- model in Trejos and Wright currency is international. Because the last visible output. This is a useful first step, in (1995b), there are other candi- two are mirror images, we focus for now that it allows us to determine the circum- date regimes, where whether only on the latter, = (1,0). Hence there stances under which the different curren- Seller j takes Currency i are three cases to consider, with either 0, 1, cies will be used in transactions between depends on who the buyer is 5 (say, Mexican sellers take dol- or 2 international monies. different agents without simultaneously lars from Americans but not Given a regime, one can determine the solving the bargaining problems facing from other Mexicans); but steady-state values of mij, independent of these agents. For simplicity, we are inter- these candidate regimes can whether output is divisible, as follows. ested here only in steady-state equilibria, never be equilibria under rela- First, note that i = 0 implies mij = 0 (if where the asset distribution and trading tively weak restrictions. Americans never trade for pesos then they strategies are constant with respect to

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120 M AY / J U N E 1 9 9 6 time. We also restrict attention to pure- call that here we look only for equilibria strategy equilibria. where sellers always want to trade for local Let the utility from consuming a unit of currency, although it still has to be one’s consumption good be U and the disu- checked that this is rational. The interest- tility of producing one’s production good be ing issue is whether a seller wants to trade

C, where 0 £ C < U. Let Vij denote the ex- for foreign currency. Whether he wants to pected lifetime utility for a buyer from trade for foreign money depends on the

Country i with Currency j, and Vi0 expected following incentive condition: lifetime utility for a seller from Country i with no money. Let V = (Vij). We call these (6) Vij - C > Vi0 Þ i = 1. the value functions. They satisfy the following ﬂow versions of the Bellman equa- (7) Vij - C < Vi0 Þ i = 0. tions from dynamic programming, That is, a seller accepts foreign money just

(3) rVi0 = ( iimii + ijmji)(Vii - Vi0 - C) in case the value of holding foreign money minus production costs exceeds the value + ( m + m ) (V - V - C) ii ij ij jj i ij i0 of remaining a seller (and waiting for local money).

(4) rVii = ( iimi0 + ijmj0 j)(U + Vi0 - Vii) An equilibrium here can be deﬁned as a list ( ,m,V) satisfying equations 1–7. For

(5) rV ij = ( ii mi0 i + ij mj0)( U + Vi0 - Vij ), each possible regime we will solve equations 1–5 and check equations 6–7; if where r is the rate of time preference. The equations 6–7 are satisfied, this regime value functions are indexed by the agent’s constitutes an equilibrium. In this way we nationality, i, but not by his (consump- can determine the parameter values that tion/production specialization) type, k, be- allow the existence of each type of equilib- cause we will only consider equilibria that rium. For simplicity, and to facilitate com- are symmetric in the sense that all types parison with Matsuyama, Kiyotaki, and use the same strategy and receive the same Matsui (1993), we consider here only the payoff in equilibrium. limiting case where C ® 0. The Bellman equations are interpreted as follows. Equation 3 says that the flow value of being a seller from Country i is the NO INTERNATIONAL rate at which one meets local or foreign buy- MONEY ers with Currency i multiplied by the gain Consider first the regime with no in- from trade in such a meeting, plus the rate at ternational money, and, therefore, no in- which one meets local or foreign buyers ternational trade: = (0,0). In this case with Currency j multiplied by the gain from equations 1–2 imply mii = Mi and mij = 0 trade if such a meeting results in trade (that for j ¹ i. To see when this regime actually is, if i = 1). Equation 4 says that the flow constitutes an equilibrium, insert m and value to holding domestic money is the rate = 0 into equations 3–5 to yield at which one meets local sellers (who always take local currency) or foreign sellers (8) rVi0 = iiMi(Vii - Vi0) who take local currency (which they do if

i = 1) multiplied by the gain from trading. (9) rVii = ii(1 - Mi )(U + Vi0 - Vii ) Equation 5 has a similar interpretation.6 = - + - Whether exchange occurs in a meet- (10) rVij ij(1 Mj )(U Vi0 Vij ). 6 One can prove that there are ing depends here exclusively on the seller no currency exchanges in pure- because the buyer wants to trade in every Note that in this regime, citizens of Coun- strategy equilibrium; that is, meeting where the seller can produce his try i never actually hold Currency j ¹ i, there is no trade when an desired consumption good—he gets an in- but Vij gives the value they would achieve agent with Currency 1 meets divisible unit of output in every trade. Re- if they were to accept Currency j. an agent with Currency 2.

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We need to check two things: the (14) rVi0 = ( iimii + ijmji)(Vii - Vi0) maintained hypothesis that agents always + ( m + m )(V - V ) accept local currency and the incentive ii ij ij jj ij i0 conditions in equations 6–7 that say, in this case, that they do not accept foreign (15) rVii = ( iimi0 + ijmj0)(U + Vi0 - Vii) currency. The former is satisﬁed for all parameter values in this case. Hence it re- (16) rVij = ( iimi0 + ijmj0)(U + Vi0 - Vij). mains only to check Vij £ Vi0. Simple algebra implies that this holds if and only if These equations imply that Vii = Vij. That is, buyers from either country are in- 2 iiMi(1 - Mi) different between holding Currency i and (11) ij(1 - Mj) £ . r + ii Currency j ¹ i. So the currencies are per-

fect substitutes. It also follows that Vii >

One thing we can conclude from this Vi0, and so equations 6–7 are satisﬁed for analysis is that for this regime to constitute all parameter values. Hence, this regime an equilibrium we require ij to be small rel- always constitutes an equilibrium. The in- ative to ii. If one meets foreigners fre- tuition is that if agents believe the two quently enough, then it is not rational to re- currencies will be accepted by all sellers ject foreign money even if you expect local then it is always rational for them to ac- sellers will reject it. Hence a very open econ- cept both currencies themselves. omy is not likely to settle on an equilibrium where foreign money does not circulate. An- other thing we can conclude is that for this ONE INTERNATIONAL regime to constitute an equilibrium, we re- MONEY quire Mj to be big. If there are few foreign We now turn to the regime where citi- buyers and many foreign sellers, then it is zens of Country 1 accept Currency 2, but easier to spend foreign money, and so you notviceversa, =(1,0).Onecansolveequa- should accept it even if you expect local sell- tions 1–2 for m11 = M1 and m21 = 0. The dis- ers will reject it. Other parameters have sim- tribution of Currency 2 holdings is given by: ilarly reasonable effects.

12(1 - M1)M2 (17) m12 = , TWO INTERNATIONAL + 12 + 21(1 - M1) MONIES Now turn to the regime with two in- ( + )M ternational currencies, = (1,1). The ﬁrst (18) m = 12 2 . 22 + + - thing to do is to solve equations 1–2 for 12 21(1 M1) mij. Routine algebra yields the steady state for citizens of Country 1, which can be de- From these one can show that, for given scribed by values of Mi, m10 is lower and m20 higher in this regime than in the other regimes. Also,

( + 21)M1 + 12M2 mi0 is decreasing in both M1 and M2. (12) m10 = 1 - + 12 + 21 The value functions for agents from Country 1 satisfy

(13) m11 =

(19) rV10 = 11m11(V11 - V10) ( + 12+ 21)+ 21[( + 21)(1- M1)+ 12(1- M 2)] M1 ( + 12+ 21)[ + 21(1- M1)+ 12(1- M 2)] . + ( 11m12 + 12m22 )(V12 - V10), The steady state for citizens of Country 2 is described by reversing the subscripts. No- (20) rV11 = 11m10(U + V10 - V11) tice that mi0 is decreasing in both M1 and M2.

The value functions now satisfy: (21) rV 12 = ( 11 m10 + 12 m20 )( U + V10 - V12 ),

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and for agents from Country 2 they satisfy Figure 1a Regions in (M ,M ) Space where (22) rV = ( m + m )(V - V ) 1 2 20 21 12 22 22 22 20 the Different Equilibria Exist Fixed Price Model (23) rV21 = 21m10(U + V20 - V21) For 11 1, 12 0.15, 21 0.15, 22 1, r 0.1, 0.02

(24) rV22 = ( 21m10 + 22m20)

(U + V20 - V22 ).

One thing that follows immediately from these equations is that Vi2 > Vi0, so citizens from both countries are willing to accept Currency 2. Hence, to show when this regime constitutes an equilibrium, we need to check the incentive constraints V11 ³ V10 (to check that it is rational for Country 1 sellers to accept Currency 1) and that V21 £ V20 (to check that it is rational for Country 2 sellers to reject Currency 1). The ﬁrst condition is satisﬁed if and only if

11m10 11m12+ 12m 22 (25) ³ ; 12m20 r+ 11m10+ 12 m20 the second is satisfied if and only if 11 holds, which, in particular, is true m m + m when ij is small relative to ii. (26) 21 10 £ 21 12 22 22 . + + 22 m20 r 21m10 22 m20 2. = (1,1) (dollars and pesos both circulate in both countries) is al- These can be interpreted as saying that it ways an equilibrium, and in this is relatively easy for a buyer from Country regime Vi1 = Vi2 (dollars and pesos 1 to find a Country 1 seller, but relatively are perfect substitutes); hard for a Country 2 buyer to find a Country 1 seller. 3. = (1,0) (dollars circulate in both countries, pesos only in Mexico) is an equilibrium if and only if equa- MODEL SUMMARY tions 25–26 hold. Here we summarize what has been learned from the preceding analysis. Given In Figure 1a we examine the set of our maintained assumptions, there are equilibria for different values of M1 and three qualitatively different types of equi- M2, holding the other parameters fixed. If libria. There is at most one equilibrium of a region in the figure contains the label each type (that is, each implies a unique ( 1, 2 ), then regime = ( 1, 2 ) is an m and V). Properties of the different equilibrium in this region. Notice that regimes include: equilibrium = (1,1) exists for all M1 and

M2. Notice also that equilibrium = (0,0)

1. = (0,0) (dollars circulate only in exists only as long as M1 and M2 are nei- America, pesos only in Mexico) is an ther too big nor too small. When Currency equilibrium if and only if equation i becomes too scarce or too abundant, citi-

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Figure 1b unless these fundamentals take on the right values. Regions in ( , ) Space where 11 12 An interesting extension is the one an- the Different Equilibria Exist alyzed by Zhou (1994), who considers a Fixed Price Model version of the model where agents differ- For M 0.75,M 0.75, 0.1, 2.5,r 0.1, 0.02 1 2 21 22 entiate between home and foreign goods. A given agent gets utility, u, from his preferred good and < u from the other good, where his preferred good ﬂuctuates between domestic and imported, and back again, according to Poisson processes. Agents may consume only preferred goods or some of both goods in equilibrium, depending on parameter values and on the type of equilibrium. In this model there

can also be currency exchanges in pure strategy equilibria. For example, when a Mexican holding a peso but having a taste for American goods meets an American holding a dollar but having a taste for Mexican goods, there is an incentive for them to swap monies in any regime other than = (1,1) (because in that regime the

currencies are perfect substitutes). In any case, one shortcoming of the model as it stands is that every trade is a zens from Country i find that either buy- one-for-one swap. This of course makes it ing or selling locally is too difficult and impossible to discuss the way nominal find it too tempting to take foreign money prices and exchange rates depend on vari- that can be spent easily. The figure also ous components of the model. The exten- tells us about the existence regions for sion in the next section remedies this. equilibria = (1,0) and = (0,1); Cur- rency 1 will be a unique international currency as long as it is not too abundant and THE MODEL WITH Currency 2 is not too scarce. DIVISIBLE OUTPUT In Figure 1b we examine different val- We now turn to the model with divisi-

ues of 11 and 12, holding the other pa- ble output. It is still assumed that when rameters ﬁxed. We see that as Country 1 buyers spend their money, they spend it

becomes more open ( 12 is large relative all, but now the amount of output they get

to 11), it is more difﬁcult to sustain an for their money will be determined

equilibrium where 1 = 0, or where 2 = 1. endogenously. When a seller produces q The opposite implications occur as local units of output for a buyer (of the right trade in that country becomes easier (that type), the latter enjoys utility, u(q), while

is, as 11 becomes larger). the former suffers disutility, c(q). With For many parameter values there exist no loss in generality, we can normalize multiple equilibria. Hence, which regime c(q) = q, as long as we also renormalize we end up in depends to a large degree on u(q). This simply means that agents bar- expectations. At the same time, fundamen- gain in terms of utils and not physical tals—that is, preferences, technology, and units of output. We assume that u(0) = 0,

the values of M1 and M2—exert an impor- u¢(0) > 1, u¢(q) > 0 for all q > 0, u²(q) < 0 for tant inﬂuence. At least in some cases, a all q > 0, and there exists qˆ > 0 such that particular regime cannot be an equilibrium u( qˆ ) = qˆ .

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We need to appeal to some form of bi- By construction, the seller always lateral bargaining theory to determine q. wants to trade at the take-it-or-leave-it For example, one could use the general- offer. It is possible, however, that a buyer ized Nash bargaining solution. This says may prefer to not trade at these terms. For that when a buyer from Country i with example, although an American seller will Currency j meets a seller from Country h, always take pesos at some price, a buyer if they trade at all, the quantity q solves with pesos may prefer to wait until he the following problem: meets a Mexican seller if the seller is expected to be willing to agree to a suffi- (27) ciently better price. A buyer with Cur- 1- max[u(q) + Vi0 - Vij] [- q + Vhj - Vho] rency j is willing to trade with a seller from Country i if and only if the utility he subject to u(q) + Vi0 - Vij ³ 0, derives from consuming qij exceeds the and - q + Vhj - Vh0 ³ 0. value to keeping his money and spending it elsewhere. Hence the following incentive In this problem, q maximizes the so-called condition must be satisfied if regime is Nash product, which is the payoff to the to constitute an equilibrium: buyer, u(q) +Vi0, minus his threat point, Vij, all to the power , times the payoff to the (29) u(qij ) > Vjj Þ i = 1 seller, - q + Vhj, minus his threat point, Vh0, all to the power 1 - . The payoffs are what (30) u(qij ) < Vjj Þ i = 0. the agents get if they trade, the threat points ar e what they get if they do not trade, and The value functions for buyers satisfy represents the relative bargaining power in the hands of the buyer. The constraints say (31) rVii = iimi0[u(qii) - Vii] that agents must get a greater payoff from + ijmj0 j[u(qji) - Vii] trading than from not trading.7 For simplicity, we assume here that the buyer has all the bargaining power: (32) rVij = iimi0- i[u(qij) - Vij] = 1. This effectively means that he can + m [u(q ) - v ], make a take it or leave it offer to the seller. ij j0 jj ij Two things follow immediately from this. First, assuming that the buyer wants to for j ¹ i. We write equations 31 and 32 in trade, he offers to exchange his money for this way to facilitate comparison with the the quantity that makes the seller indiffer- model in the previous section; in particu- ent between accepting and rejecting (that lar, see equations 4 and 5. However, is, the second constraint will bind). Sec- strictly speaking, equations 31 and 32 are ond, this implies that the seller never gets not exactly correct without some modifica- any of the gains from trade, and therefore tion in the framework, for the following

Vi0 = 0. Hence when a buyer from Country reason. Consider, for example, the case i with Currency j meets a seller from j = 0, which says that Currency i does Country h, if they trade at all, then the not circulate in Country j. Then equation quantity is given by 31 says that a buyer from Country i with Currency i who meets a seller from Coun-

(28) qhj = Vhj. try j cannot trade. But, given the bargaining rules, he always could offer to trade

Note that this quantity depends on the and get Vji > 0 (the amount a seller from nationality of the seller and the currency Country j is willing to give for Currency i), 7 See Osborne and Rubinstein being used, but not on the nationality of and in principle we should allow him that (1990) for an extended discus- the buyer. Also note that the nominal option. In fact, one can show that even if sion on Nash bargaining and its price of output in Country h in terms of agents believe that j = 0, a buyer from relation to explicit strategic bar-

Currency j is phj = 1/qhj. Country i with Currency i who meets a gaining games.

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seller from Country j will offer to trade.8 for j ¹ i. In this regime, no transactions ac-

This means that j = 0 can never be an tually occur at qij for j ¹ i, but qij but tells equilibrium in the model as it stands. us how much one could get with Currency However, this result is really just an j from Country i sellers. artifact of our simplifying assumptions It is not hard to show that, as long as this

that the buyer gets to make a take it or equilibrium exists, qii and qji ar e both de-

leave it offer and there is no possibility of cr easing in Mi, and are independent of Mj direct barter. Relaxing either of these al- for j ¹ i. Also, q11 > q22 if and only if 11

lows j = 0 to be an equilibrium for some (1 - M1) > 22 (1 - M2). Because one unit

parameter values, although it also compli- of Currency i buys qii units of real output, cates the model signiﬁcantly. We can we can say that one unit of Currency 1 is

maintain the simplicity of the present set wo r th e = q11 /q22 = p22/p11 units of Curren c y

up and still generate equilibria with j = 0 2. This is the exchange rate that would be in another (albeit perhaps less satisfactory) implied by purchasing power parity.9 Th e

way, as follows. First, expand the bargain- exchange rate e falls with M1 and rises

ing game so that a seller has to decide with M2. Also, e rises with 11 and falls

whether to enter negotiations with a buyer with 22 . Of course, international diffe r - before the buyer gets to make his take it or ences in utility and production functions leave it offer. Then, given certain addi- would also inﬂuence e. We have kept tional technical assumptions, it will be an these the same across countries purely for equilibrium for the seller to refuse to trade notational simplicity.

given that he believes j = 0. In this case, To see when this regime actually con- equations 31 and 32 do describe the value stitutes an equilibrium, we must check the functions as functions of . See Trejos and maintained hypothesis that Currency i cir- Wright (1995b), for details. culates in Country i and the incentive con-

Deﬁne the vectors q = (qij), V = (Vij ), ditions in equations 29–30. The former

and m = (mij ). Then an equilibrium is now can easily be seen to hold for all parameter a list ( , m, V, q) satisfying equations 1–2 values. The latter, which says that individ- and equations 28–32. Henceforth we ig- uals with Currency j do not want to buy nore the value functions because they are from sellers from Country i, holds if and

redundant by virtue of the equilibrium only if u(qij) £ q jj, for i ¹ j. Using equations

condition Vij = qij. Therefore, a steady-state 33–34, we can rewrite this inequality as equilibrium is completely characterized by the regime , the asset distribution m, and the values of the currencies q. (35) 8 We thank Rao Aiyagari for pointing this out to us. NO INTERNATIONAL

9 MONEY Suppose one peso buys q11 which is satisﬁed if and only if ij is small units of output in Mexico and Consider ﬁrst = (0,0). First, note compared with jj. The intuition is similar

one dollar buys q22 units of out- that m does not depend on whether out- to the model with indivisible output. put in America, and suppose put is divisible, so we already know that there is a market in which the asset distribution from the previous currencies can be traded at the section (this is, of course, true in any TWO INTERNATIONAL nominal exchange rate e (that regime). Now equations 31–32 can be re- MONIES is, one peso buys e dollars). arranged to show that prices satisfy Now turn to the regime with two in- Then one peso could be used to buy eq units of output in ternational currencies, = (1,1). One can 22 = = Q America using dollars. m show that in this regime qii qij i; (33) q = ii i0 u(q ) That is, the two currencies are perfect Purchasing power parity holds if ii r + m ii a peso buys the same amount ii i0 substitutes in the sense that they purchase

directly at home and using dol- ijmj0 the same amount from a given seller. If (34) qij = u(qjj), lars abroad: q11eq22. r + ijmj0 there were a market in which agents

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m could trade the two currencies, the mar- (36) q = 11 10 u(q ) 11 + 11 ket clearing price would have to be 1. But r 11m10 the purchasing power parity exchange 21m10 rate, as deﬁned above, is given by e = (37) q21 = u(q11), r+ 21m10 Q1/Q2. In general e differs from unity in this regime: Even though quantities do which are qualitatively the same as in the not depend on which currency the buyer = (0,0) regime. Because m10 is lower in is using, they do depend on the national- this regime than in = (0,0), q11 and q21 ity of the seller. are lower here. Intuitively, the inﬂux of

One can also show that Q1 and Q2 are foreign money inﬂates prices denominated both decreasing in M1 and M2. Thus an in- in the domestic currency. Also, q11 and q21 crease in either money supply increases are decreasing in both M1 and M2 because the price level in both countries. Also, as m10 is decreasing in both M1 and M2. long as the two countries are not too dif- The purchasing power of the interna- ferent in terms of ij and Mj, the effect of a tional currency depends on the seller, as change in M1 is stronger in Country 1 than given by in Country 2 and vice versa. This means 11m10u(q12) + 12 m20u(q22) that the purchasing power parity exchange (38) q12 = r + 11m10 + 12m20 rate e = Q1/Q2 is decreasing in M1 and increasing in M2. To see when this regime constitutes m u(q ) + m u(q ) an equilibrium, we need to check the in- (39) q = 21 10 12 22 20 22 . 22 r + m + m centive conditions in equations 29–30. 21 10 22 20 These hold in this regime if and only if u(Qi) ³ Qj, j ¹ i. If the countries are sym- There is a unique nonzero solution for metric, in the sense that M1 = M2, ii = jj (q12, q22) and both are decreasing in M1 and and ij = ji, these conditions always hold M2. Since m10 is lower and m20 higher in and the equilibrium with two international this regime, q12 is lower here than in the currencies always exists. But if they are suf- = (1,1) regime. Moreover, q12 > q11, with

ﬁciently dissimilar, then the incentive con- the difference increasing in M1 - M2. ditions will be violated and = (1,1) is not Hence, citizens of Country 1 value the in- an equilibrium. The model is very different ternational currency more than their own in this regard from the one with indivisible domestic currency. As long as the coun- output. In that model, equilibrium = tries are not too dissimilar, one can also

(1,1) exists for all parameter values. Here, show that q22 > q12. even if the two countries’ currencies are We still need to check when this perfect substitutes, the two nationalities of equilibrium exists. The maintained sellers are not. For example, if jj is low hypothesis that Currency i always circu- relative to ii then Country j sellers do not lates in Country i holds if and only if 10 give much for (either) currency compared u(q22) ³ q12. Then equations 29–30 re- with Country i sellers, and Country i buy- quire u(q12) ³ q22, so Country 2 buyers ers may prefer to hold on to their cash with Currency 2 buy from Country 1 10 This condition, which says that rather than spend it in Country j. sellers, and u(q21) £ q11, so that Country 1 buyers with Currency 1 do not buy agents from Country 1 with from Country 2 sellers. Each of these Currency 2 buy from Country 2 ONE INTERNATIONAL conditions looks qualitatively like one sellers, can bind for some para- MONEY meter values in this regime. If that has been encountered earlier and this condition were violated, We now turn to the regime where holds under similar circumstances; how- then Mexicans, for example, citizens of Country 1 accept Currency 2, ever, because qij differs across regimes, would sell goods for dollars and but not vice versa, = (1,0). The pur- the parameter values for which the equi- then spend these dollars on chasing power of the national currency is libria exist are quantitatively different. Mexican sellers, but not on given by American sellers.

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Figure 2a in Mi and Mj, and e = Q1/Q2 and e = Q /Q is decreasing in M and Regions in (M ,M ) Space where 1 2 1 1 2 increasing in M . the Different Equilibria Exist 2 For 1, 0.15, 0.15, 1,r 0.1, 0.02 11 12 21 22 3. = (1,0) (dollars circulate in both countries, pesos only in Mexico) exists if a combination of the above

conditions hold. In this regime, qij

is decreasing in Mi and Mj for all i,

j, q12 ³ q11 (Mexicans value dollars more than Mexicans value pesos),

and q22 > q12 (Americans value dollars more than Mexicans value dollars) at least if the two countries are not too dissimilar.

In Figure 2a we examine different val-

ues of M1 and M2, holding the other parameters ﬁxed. Notice that the regime = (1,1) exists for all M and M in this example, al- 1 2 though this is not true in general. Also, be-

cause ij is relatively small, equilibrium = (0,0) exists for all but very high values of

M1 and M2. Equilibria where only one currency circulates internationally exist only if DIVISIBLE OUTPUT the other currency is not too abundant. For example, = (1,0) exists only if M1 is not too SUMMARY big. In Figure 2b we examine variation in

As in the indivisible output model, 11 and 12. Now equilibrium = (1,1) ex- there are three qualitatively different ists in only a small region of the ﬁgure. Equi-

regimes and at most one equilibrium of each librium = (0,0) exists if and only if 12 is type (that is, each implies a unique m and below some cutoff. Also notice that as either

q). Other things being equal, the value of a 11 or 12 increases, it becomes more likely currency is higher if it circulates interna- that equilibrium = (1,0) exists. This is be- tionally and lower if the other money circu- cause as Country 1 increases either its inter- lates internationally. Other properties of the nal economic activity or its openness, sellers different regimes include the following: in Country 1 value money more highly and this makes Country 2 buyers more willing 1. = (0,0) (dollars circulate only in to deal with them. America, pesos only in Mexico) is an equilibrium if and only if is ij POLICY small relative to ii; in this regime,

qii is decreasing in Mi and indepen- In this section we analyze the effects

dent of Mj. of changes in (M1, M2) in the model with

divisible output and endogenize (M1, M2) 2. = (1,1) (dollars and pesos both by modeling the objective functions of the circulate in both countries) is an two governments and the rules for their equilibrium if and only if the two strategic interaction. Because for some countries are not too dissimilar; in questions analytical results are difﬁcult to

this regime, q11 = q12 = Q1 and q21 = derive, we analyze numerically an example

q22 = Q2 (dollars and pesos are per- with u(q)= q. The features that we high-

fect substitutes), Qi is decreasing light are ones that we can either prove an-

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128 MAY / JU N E 1 9 9 6

alytically to hold in general or those which Figure 2b seem to be robust to alternative parameter- Regions in ( , ) Space where izations in examples. 11 12 the Different Equilibria Exist Because Government i is assumed to purchase goods from a fraction Mi of its newborn citizens, per capita seigniorage revenue in real terms is given by Si = Miqii. One can show that S1 ﬁrst increases and then decreases with M1. Also, given that multiple equilibria exist, S1 is greater when Currency 1 is international than when it is not and lower when Currency 2 is international than when it is not. Finally, given

M2, S1 is maximized at different levels of M1 in the different regimes, depending on the extent to which Country 1 is able to export inﬂation abroad. We deﬁne welfare as the average (steady state) utility of private citizens:

(40) Wi = mi0Vi0 + mi1Vi1 + mi2Vi2.

Which regime yields the highest welfare depends on M1 and M2. For instance, if M1 is very low, then W1 is highest in regime = (1,0), where Currency 2 is accepted in Country 1 but not vice versa, because this policies jointly. One way to do this is to as- makes trade easier within the host country. sume that seigniorage is freely transferable Also, = (1,1) may or may not dominate across countries, in which case they maxi-

= (0,0). For instance, if M2 is very low or mize S1 + S2. We denote this outcome by T T very high, then W1 is highest in regime (M1 , M2 ). Or we can assume seigniorage = (0,0). is nontransferable, in which case we use N The next step is to endogenize (M1, M2). the bargaining solution that chooses (M1 , N S S We assume that governments take the re- M2 ) to solve max [S1 - S1 ][S2 - S2 ], S gime as given and restrict their choices to where Sj is seigniorage in Country j when S S 11 policies that allow for the existence of that policy is given by (M1 , M2 ). There may actually be very regime as an equilibrium (that is, we ignore We can summarize our ﬁndings as fol- interesting policies, but we policies aimed at changing a currency’s lows. First, if foreign money circulates in leave such a discussion to realm of circulation).11 We also restrict at- Country i, independently of whether Cur- future research. tention to policies that do not change over rency i circulates abroad, then the welfare 12 This can be explained as fol- time and to steady-state comparisons. maximizing level of Mi is lower than the lows. Welfare in Country 1, for We look for Nash equilibria when each level that maximizes seigniorage. However, example, is W1 = m11 V11 + government seeks to maximize the welfare when foreign money does not circulate in m12 V12, whereas seigniorage is W W S1 = M1V11. When no foreign of its own citizens, denoted (M1 , M2 ). In Country i, welfare and seigniorage are W 12 money is held at home (m = other words, W is maximized at M when maximized at the same value of M . Sec- 12 1 1 i 0), m is proportional to M M W is taken as given and vice versa. We ond, starting from the Nash equilibrium 11 1 2 and maximizing seigniorage is also look for Nash equilibria when each where governments maximize seigniorage, the same as maximizing wel- government seeks to maximize seigniorage, reducing the amount of money in both fare. This result is not particular- S S denoted (M1 , M2 ). Given that they seek to countries increases welfare in both coun- ly robust and does not hold in maximize seigniorage, we also consider the tries. Third, and more interestingly, reduc- generalized versions of the possibility of international policy coordina- ing the amount of money in both coun- model that include barter or tion by letting the governments choose tries increases seigniorage for both other bargaining solutions.

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Figure 3a,b Figure 4a,b Feasible and Equilibrium Feasible and Equilibrium Seigniorage Welfare

For a11 1,a12 0.15,a21 0.15,a22 1, For a11 1,a12 0.15,a21 0.15,a22 1, r 0.1,g 0.02 r 0.1,g 0.02 (a) l (1,0)

(b) l (1,1) (b) l (1,1)

governments. As neither government takes the noncooperative equilibrium between into account the effect it has on the other, welfare maximizers, the noncooperative the noncooperative equilibrium is charac- equilibrium between seigniorage maximiz- terized by too much money. The coopera- ers, the cooperative solution when revenue N N tive equilibrium (M1 , M2 ) involves less is transferable, and the cooperative solution money and more seigniorage. when revenue is nontransferable. In the These results hold in general. We now transferable revenue case, the point T de- describe in more detail the ﬁndings from the picts the seigniorage raised in each country numerical examples. We depict the frontier and not the ﬁnal division of the revenue be- and the outcomes in (S1, S2) space, assum- tween the governments. Figures 4a and 4b ing the regime is = (1,0) in Figure 3a and show the same points in (W1, W2) space. Be- = (1,1) in Figure 3b (the regime with no cause the graphs are drawn using the same international currency is uninteresting for scales, one thing they illustrate is how the this exercise). The points labeled W, S, T, possible values of seigniorage and welfare and N are the payoffs in the four scenarios: vary across regimes.

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The results we wish to highlight are as one currency, one central bank or both. follows. First, though the cooperative solu- While this may help to facilitate trade, it tions are on the frontier in (S1, S2) space, makes it more difficult to have indepen- the noncooperative solutions are inside the dent monetary policies tailored to condi- frontier. This is especially so in regime tions in the individual economies. We = (1,1), where it is easiest to export infla- leave this to future research. tion. Also, Figure 3a shows that in regime = (1,0), when transfers are allowed, the co- REFERENCES operative solution is to concentrate seign- Aiyagari, S. Rao, and Neil Wallace. “Existence of Steady States with iorage collection in Country 2, where it can Positive Consumption in the Kiyotaki-Wright Model,” Review of be done more efficiently. That is, the govern- Economic Studies (October 1991), pp. 901–16. ments print more of the international currency and less of the national currency in _____ and _____. “Fiat Money in the Kiyotaki-Wright Model,” addition to printing less money in total. Economic Theory (Vol. 2, 1992), pp. 447–64. Figure 4 shows how the possible val- _____, _____, and Randall Wright. “Coexistence of Money and ues of (W1, W2) are much higher in regime Interest-Bearing Securities,” Journal of Monetary Economics = (1,1). However, each of the outcomes (June 1996), pp. 397-419. is inside the frontier. Also, when both gov- Burdett, Kenneth; Melvyn Coles; Nobuhiro Kiyotaki; et al. “Buyers and ernments try to maximize seigniorage, one Sellers: Should I Stay or Should I Go?” The American Economic of them may actually end up with less Review, Papers and Proceedings (May 1995), pp. 281–86. seigniorage than when both are trying to Hendry, Scott. “Credit in a Search Model with Money as a Medium of maximize welfare. Symmetrically, citizens Exchange.” Manuscript, University of Western Ontario, (1992). in one country may be worse off in terms of welfare when both governments are try- Kiyotaki, Nobuhiro, and Randall Wright. “A Contribution to the Pure ing to maximize welfare than when both Theory of Money,” Journal of Economic Theory (April 1991), governments are concerned with seignior- pp. 215–35. age, as can be seen in the figure for Coun- _____ and _____. “On Money as a Medium of Exchange,” Journal try 1 in regime = (1,0). Finally, in regime of Political Economy (August 1989), pp. 927–54. = (1,0), the country that issues interna- _____ and _____. “A Search-Theoretic Approach to Monetary Eco- tional money is better off. nomics,” The American Economic Review (March 1993), pp. 63–77. Li, Yiting. “Commodity Money Under Private Information,” Journal of FINAL REMARKS Monetary Economics (December 1995), pp. 573-92. This article has summarized some re- Matsuyama, Kiminori, Nobuhiro Kiyotaki, and Akihiko Matsui. “Toward a cent results in the literature on search- Theory of International Currency,” Review of Economic Studies (April theoretic models of international currency. 1993), pp. 283–307. We think that models in this class will Osborne, Martin, and Ariel Rubinstein. Bargaining and Markets. Academic eventually prove useful for studying im- Press (1990). portant issues in international monetary Ritter, Joseph A. “The Transition from Barter to Fiat Money,” economics. The analysis illustrates the po- The American Economic Review (March 1995), pp. 134–49. tential welfare gains from having one currency (or two currencies that serve as per- Shi, Shouyong. “Credit and Money in a Search Model with Divisible fect substitutes). However, it also indicates Commodities.” Manuscript, University of Queens, (1995a). that there can be a welfare loss in one _____. “Money and Prices: A Model of Search and Bargaining,” country from having a unified currency if Journal of Economic Theory (December 1995b), pp. 467-96. another country is pursuing a particularly Trejos, Alberto. “Money, Prices and Private Information.” Manuscript, bad policy from the former’s point of view. Northwestern University, (1994). It may be interesting to consider versions of the model where the two countries have _____. “Qualitative Uncertainty and the Liquidity of Money,” Economic different preferences or production tech- Theory ( December 1995b), pp. 467-96. nologies or are subject to different shocks _____ and Randall Wright. “Search, Bargaining, Money and Prices,” to analyze the tradeoffs between having Journal of Political Economy (February 1995a), pp. 118–41.

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_____ and _____. “Toward a Theory of International Currency: A Step Further.” Manuscript, Northwestern University, (1995b). Williamson, Steve, and Randall Wright. “Barter and Monetary Exchange Under Private Information,” The American Economic Review (March 1994), pp. 104–123. Zhou, Ruilin. “Currency Exchange in a Random Search Model.” Manuscript, University of Pennsylvania, (1994).

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