1 See, for example, in additionto the work discussed below,Kiyotaki and Wright (1989,1991, and 1993), Aiyagariand Wallace (1991 and1992), Williamson and Wright(1994), Trejos (forthcoming)and Ritter (1995).

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Search-TheoreticModels ofInternationalCurrencyAlberto Trejos and Randall Wright

Search-based theories of the exchangeprocess provide economists with away of formalizing the microfounda-

tions of monetary economics and to dis-cuss a variety of issues in monetary the-ory and policy in a new light.1 Manyquestions that cannot even be formulatedin more traditional monetary models canbe profitably analyzed in these new mod-els. In this essay, we propose to review re-cent developments using search-basedmodels to study some international mone-tary issues.

There seems little doubt that interna-tional monetary economics is of centralimportance today. Of the types of ques-tions one would ultimately like to answer,consider these:

• Should Europe adopt a commoncurrency?

• If so, should individual countriescontinue to issue local currencies?

• If a state or province separates froma nation (as Quebec periodicallydiscusses in Canada), how should itdesign its monetary system?

• Should more Latin American coun-tries adopt currency boards, whereeach unit of local currency is

backed by central bank reserves inU.S. dollars (as in Argentina)?

• Should more Latin American coun-tries simply abandon their local cur-rency and switch to U.S. dollars (asPanama did)?

Though we do not claim to be readyto provide definitive answers to all of theseimportant questions at this time, we dothink that search-theoretic models are inprinciple well suited to address these typesof issues. More traditional theory, includ-ing models that include cash-in-advanceor money-in-the-utility-function type as-sumptions, are clearly ill suited in this re-gard. They are ill suited because too muchis decided by assumption. In particular,the answer to the question, “Whichmonies circulate in which countries?” isdetermined at the outset by the modelerwhen he chooses which money to put in which cash-in-advance constraint or inwhich utility function. In contrast, search models are designed to determineendogenously which monies circulatewhere.

We begin by reviewing a version of the model in Matsuyama, Kiyotaki, and Matsui (1993). Basically, it can be de-scribed as follows. There are two countries,each of which issues its own currency.There is a meeting or matching technologythat describes the frequency with which agiven individual interacts with other indi-viduals, both locals and foreigners (it is as-sumed that one interacts with individualsfrom one’s own country more often than a foreigner interacts with these same indi-viduals). When two agents meet, one with money and the other with real outputfor sale, they have to decide whether totrade. Parameter values, as well as expecta-tions regarding other agents’ behavior,jointly determine this decision and therebydetermine the realm of circulation for eachcurrency. Potentially there are three dis-

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.

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tinct types of equilibria—or regimes—inwhich we are interested, where none, one,or both of the currencies circulate interna-tionally.

This model allows one to answer ques-tions like the following:

• What features of a country make itpossible, or likely, for its currency tocirculate internationally?

• How and when can local currenciessurvive in the presence of a univer-sally accepted international cur-rency?

• Does an international currencyemerge naturally as economies be-come integrated?

• What are the costs and benefits to acountry of having its currency serveas an international medium of ex-change?

• An extension of this model in Zhou(1994) additionally allows one tostudy the issue of when agentswould want to exchange currenciesin such a world.

There are shortcomings with the mod-els mentioned in the previous paragraph.Perhaps the most obvious is that in thesemodels, as in all of the first-generationsearch-based models of money, every ex-change is assumed to be a one-for-onetrade. This simplifying assumption makesit possible to discuss the process of ex-change—and, in particular, which objectscirculate as media of exchange amongwhich agents—without tackling the deter-mination of the relative values of these ob-jects. Unfortunately, however, it obviouslyalso makes it impossible to talk aboutprices or exchange rates. Therefore, wealso present the extension of the model inTrejos and Wright (1995b) designed to en-dogenize prices using bargaining theory.2

This extension allows us to raise awhole range of new issues, including thefollowing:

• How does the fact that a currencycirculates internationally affect itspurchasing power at home?

• Where does an international cur-rency purchase more—at home orabroad?

• What are the effects on seigniorageand welfare in each country whenone money becomes an interna-tional currency?

• How are policies designed to maxi-mize either seigniorage or welfareaffected by concerns of currencysubstitution?

• How are national monetary policiesconnected, and what is the scopefor international cooperation?

In the next section, we outline thebasic assumptions on which the model isbuilt. The third section presents the indi-visible output version of the model inMatsuyama, Kiyotaki, and Matsui (1993).We then present the divisible outputmodel with bargaining in Trejos andWright (1995b), followed by a discussionof some policy implications. The final sec-tion presents some brief conclusions.

THE BASIC MODELThe economy consists of two coun-

tries, labeled i = 1, 2. One’s country is im-portant in that it determines the frequencywith which one interacts with otheragents. Individuals interact, or meet, bilat-erally according to a random matchingprocess in continuous time, and ˆ

ij de-notes the Poisson arrival rate at which acitizen of Country i meets citizens ofCountry j. We assume that ˆ

ii ≥ ˆji for

j ≠ i. This simply says that, for example, aMexican meets Mexicans more frequentlythan an American meets Mexicans.

Each country starts with a large num-ber of citizens, and the fraction of individ-uals from Country i is Ni with N1 + N2 = 1.Thereafter, both populations grow at the

2 Bargaining was first introducedinto (one-country) search mod-els of money in Trejos andWright (1995a) and Shi(1995b). Other recent applica-tions include Aiyagari et al.(1996), Trejos (1994), andShi (1995a).

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same rate ≥ 0. The population sizes andthe meeting technology parameters are notindependent because we have the identityN1

ˆ12 = N2

ˆ21 (both sides of the equality

give the total number of internationalmeetings per unit time). Here we take ˆ

ij

as primitive and let the populations be freeto satisfy this identity. A special case is thespecification actually used in Matsuyama,Kiyotaki, and Matsui (1993), which hasˆ

ii = N1 and ˆij = *Nj, with * < . Thus,

arrival rates are proportional to countrysize, but they are smaller across countriesthan they are within countries. As * getscloser to , the two countries becomemore integrated.3

Agents are distinguished not only bytheir country and date of birth, but also bytheir tastes and technologies. As in thetypical monetary search model, we need toadopt some notion of specialization. Here,for simplicity, it is assumed that there areK ≥ 3 goods and the population of eachcountry contains equal numbers of Ktypes, where each Type k consumes onlyGood k and produces only Good k +1(modulo K; that is, Type K produces Good1). If we assume that meetings are randomwith respect to consumption-productiontype, then ij = ˆ

ij / K is the rate at whicha citizen of Country i meets a citizen ofCountry j who consumes the good he pro-duces; it is also the rate at which thecitizen of Country i meets a citizen ofCountry j who produces the good heconsumes.

There is no centralized market or auc-tioneer in this model: All trade is bilateraland quid pro quo. The large number ofagents rules out private credit because theprobability of meeting a particular individ-ual a second time is zero. Our specifica-tion for tastes and technologies rules outdirect barter. We also make the assump-tion that goods are nonstorable, whichrules out commodity money. Hence, traderequires the use of some form of fiatmoney as a medium of exchange. However—and this is the crucial point—we do not impose that any particular cur-rency plays this role in any particulartransaction. This is what distinguishes the

class of models under consideration fromones with particular cash-in-advance ormoney-in-the-utility-function assumptionsimposed exogenously.4

Fiat money is introduced by the gov-ernment of Country j issuing one unit ofCurrency j to some fraction Mj ∈ (0,1) ofits newborn citizens, at each point in time,in exchange for some amount of real out-put (how much real output depends onwhat assumptions we make). Issuingmoney in this way yields a continuous flowof seigniorage, although notice that theonly time private agents interact with thegovernment is when they first enter theeconomy. In the special case where = 0,the government simply issues an input ofcurrency to some fraction of its citizensat the initial date and then shuts down.

To keep things tractable, we make thefollowing assumptions. First, we assumethat an agent holding a unit of currencyalways spends it all at once, which couldobviously be guaranteed if we simply saythat the monetary object is indivisible.This implies that no one holding currencyever holds less than one unit. Second, weassume that, except for the newborn, noagent can produce until after he consumes.This implies that two agents with currencydo not trade with each other, and no oneever holds more than one unit of money.Hence at each point in time there will besome agents with one unit of money each,called buyers, and a disjoint group with no money, called sellers. Let the fraction of buyers from Country i with Currency jbe denoted mij, and the fraction of sellersfrom Country i be denoted mi0 = 1 − mi1 −mi2. Then the vector m = (mij) completelydescribes the asset distribution acrossagents.

What happens when a buyer and sellermeet depends on which version of themodel we consider. In Matsuyama,Kiyotaki, and Matsui (1993), output is in-divisible and the same across the twocountries. Thus if you find a seller whocan produce the right type of output, thereis no question that you always want totrade your money for his one indivisiblegood, and, in particular, you do not care

3 In this model one does notphysically travel between onecountry and another, nor doesone choose in any other way tointeract with foreigners insteadof fellow citizens. It is simplythat you sometimes meet for-eigners in your daily routine.Imagine, for example, a townon the border populated byboth Mexicans and Americans.Americans trade with fellowAmericans and with Mexicans,although perhaps less frequent-ly with the latter and viceversa. What we are interestedin here is the monetary natureof these interactions; that is,which currencies get traded bywhom?

4 The simplifying assumptions inthe text allow us to focus exclu-sively on the use of fiat curren-cies as media of exchange, butin principle they can all berelaxed. For examples of mone-tary search models with credit,see Hendry (1992) or Shi(1995a); for examples withsome direct barter, see Kiyotakiand Wright (1991 and 1993)or Burdett et al. (1995); andfor examples with commoditymoney, see Kiyotaki and Wright(1989), Aiyagari and Wallace(1991), or Li (1995).

about the seller’s nationality. In Zhou(1994), output is indivisible but differsacross the two countries, and consumershave tastes that fluctuate randomly, whichmeans they generally do care about theseller’s nationality. In Trejos and Wright(1995b), a unit of output is the sameacross countries, but output is perfectly di-visible and the amount of output that youget for your money needs to be negotiated.This means that you might care about thenationality of the seller if the (endoge-nous) prices differ across countries.

In any case, the important thing fromour vantage is whether a buyer fromCountry h with Currency i trades with acitizen of Country j because this deter-mines the steady-state distribution of as-sets across the population. In this article,we consider only equilibria where trade al-ways occurs if a buyer with Currency imeets a seller from Country i (pesos defi-nitely circulate in Mexico and dollarsdefinitely circulate in the United States).Then the key issue is whether trade occurswhen a seller meets a buyer with foreigncurrency—that is, whether exchange oc-curs when a buyer with Currency i meetsa seller from Country j ≠ i.

We distinguish the different types ofpossible equilibria—or different regimes—as follows. Let j = 1 if within Country j wesee Currency i ≠ j in circulation, in whichcase we call Currency i an international cur -rency; and let j = 0 if in Country j we donot see Currency i in circulation. Then aregime is a list of values for = ( 1, 2).The four possible regimes are = (0,0),(1,1), (0,1), and (1,0). In the first casethere is no international currency; in thesecond case both currencies are interna-tional; and in the final two cases only onecurrency is international. Because the lasttwo are mirror images, we focus for nowonly on the latter, = (1,0). Hence thereare three cases to consider, with either 0, 1,or 2 international monies.5

Given a regime, one can determine thesteady-state values of mij, independent ofwhether output is divisible, as follows.First, note that i = 0 implies mij = 0 (ifAmericans never trade for pesos then they

never hold pesos). Second, note thattrades between a buyer and seller of thesame nationality do not alter m becausethey leave the aggregate distribution un-changed. These considerations imply thatthe steady-state equations are

(1) i jmi0mj i − i jmi imj0 j + (Mi − mi i) = 0 ,

(2) ijmi0mjj i − ijm ijmj0 − mij = 0,

for i ≠ j. Given and the identitymi0 = 1 − mii − mij, these can be solved forthe steady-state asset distribution m.

Consider equation 1 (equation 2 has asimilar interpretation). The first term saysthat mii increases when a seller from Coun-try i meets a buyer from Country j withCurrency i (given the maintained assump-tion that when a seller from Country imeets a buyer with Currency i, they alwaystrade). The second term says that m ii de-creases when a buyer from Country i withCurrency i meets a seller from Country j ifthey trade (that is, if j = 1). The final termsays that mii increases when the fraction ofthe newborn agents in Country i who re-ceive currency, Mi, exceeds mii. Steady staterequires the net change to be zero.

Given the asset distribution, the idea isto use dynamic programming to solve theindividual decision problems concerningwhen to trade. The next section considersthe model with indivisible output, whereevery trade is a one-for-one swap. The sec-tion after that considers the case where out-put is divisible and agents bargain.

THE MODEL WITH INDIVISIBLE OUTPUT

Here we analyze the model with indi-visible output. This is a useful first step, inthat it allows us to determine the circum-stances under which the different curren-cies will be used in transactions betweendifferent agents without simultaneouslysolving the bargaining problems facingthese agents. For simplicity, we are inter-ested here only in steady-state equilibria,where the asset distribution and tradingstrategies are constant with respect to

5 In the model in Matsuyama,Kiyotaki, and Matsui (1993),these are the only possibleregimes, except for possiblyequilibria where at most one ofthe monies has value or equilib-ria that are the same except forrelabeling in the sense that wecall American money pesos andMexican money dollars. In themodel in Trejos and Wright(1995b), there are other candi-date regimes, where whetherSeller j takes Currency idepends on who the buyer is(say, Mexican sellers take dol-lars from Americans but notfrom other Mexicans); butthese candidate regimes cannever be equilibria under rela-tively weak restrictions.

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time. We also restrict attention to pure-strategy equilibria.

Let the utility from consuming a unit ofone’s consumption good be U and the disu-tility of producing one’s production good beC, where 0 ≤ C < U. Let Vij denote the ex-pected lifetime utility for a buyer fromCountry i with Currency j, and Vi0 expectedlifetime utility for a seller from Country iwith no money. Let V = (Vij). We call these

the value functions. They satisfy the fol-lowing flow versions of the Bellman equa-tions from dynamic programming,

(3) rVi0 = ( iimii + ijmji)(Vii − Vi0 − C)

+ ( iimij + ijmjj) i(Vij − Vi0 − C)

(4) rVii = ( iimi0 + ijmj0 j)(U + Vi0 − Vii)

(5) r Vi j = ( i imi0 i + i jmj0) (U + Vi0 − Vi j) ,

where r is the rate of time preference. Thevalue functions are indexed by the agent’snationality, i, but not by his (consump-tion/production specialization) type, k, be-cause we will only consider equilibria thatare symmetric in the sense that all typesuse the same strategy and receive the samepayoff in equilibrium.

The Bellman equations are interpretedas follows. Equation 3 says that the flowvalue of being a seller from Country i is therate at which one meets local or foreign buy-ers with Currency i multiplied by the gainfrom trade in such a meeting, plus the rate atwhich one meets local or foreign buyerswith Currency j multiplied by the gain fromtrade if such a meeting results in trade (thatis, if i = 1). Equation 4 says that the flowvalue to holding domestic money is the rateat which one meets local sellers (who al-ways take local currency) or foreign sellerswho take local currency (which they do if

i = 1) multiplied by the gain from trading.Equation 5 has a similar interpretation.6

Whether exchange occurs in a meet-ing depends here exclusively on the sellerbecause the buyer wants to trade in everymeeting where the seller can produce hisdesired consumption good—he gets an in-divisible unit of output in every trade. Re-

call that here we look only for equilibriawhere sellers always want to trade for localcurrency, although it still has to bechecked that this is rational. The interest-ing issue is whether a seller wants to tradefor foreign currency. Whether he wants totrade for foreign money depends on thefollowing incentive condition:

(6) Vij − C > Vi0 ⇒ i = 1.

(7) Vij − C < Vi0 ⇒ i = 0.

That is, a seller accepts foreign money justin case the value of holding foreign moneyminus production costs exceeds the valueof remaining a seller (and waiting for localmoney).

An equilibrium here can be defined asa list ( ,m,V) satisfying equations 1–7. Foreach possible regime we will solve equa-tions 1–5 and check equations 6–7; ifequations 6–7 are satisfied, this regimeconstitutes an equilibrium. In this way wecan determine the parameter values thatallow the existence of each type of equilib-rium. For simplicity, and to facilitate com-parison with Matsuyama, Kiyotaki, andMatsui (1993), we consider here only thelimiting case where C → 0.

NO INTERNATIONALMONEY

Consider first the regime with no in-ternational money, and, therefore, no in-ternational trade: = (0,0). In this caseequations 1–2 imply mii = Mi and mij = 0for j ≠ i. To see when this regime actuallyconstitutes an equilibrium, insert m and

= 0 into equations 3–5 to yield

(8) rVi0 = iiMi(Vii − Vi0)

(9) rVii = ii(1 − Mi)(U + Vi0 − Vii)

(10) rVij = ij(1 − Mj )(U + Vi0 − Vij ).

Note that in this regime, citizens of Coun-try i never actually hold Currency j ≠ i,but Vij gives the value they would achieveif they were to accept Currency j.

6 One can prove that there areno currency exchanges in pure-strategy equilibrium; that is,there is no trade when anagent with Currency 1 meetsan agent with Currency 2.

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We need to check two things: themaintained hypothesis that agents alwaysaccept local currency and the incentiveconditions in equations 6–7 that say, inthis case, that they do not accept foreigncurrency. The former is satisfied for allparameter values in this case. Hence it re-mains only to check Vij ≤ Vi0. Simple alge-bra implies that this holds if and only if

(11) ij(1 − Mj) ≤ .

One thing we can conclude from thisanalysis is that for this regime to constitutean equilibrium we require ij to be small rel-ative to ii. If one meets foreigners fre-quently enough, then it is not rational to re-ject foreign money even if you expect localsellers will reject it. Hence a very open econ-omy is not likely to settle on an equilibriumwhere foreign money does not circulate. An-other thing we can conclude is that for thisregime to constitute an equilibrium, we re-quire Mj to be big. If there are few foreignbuyers and many foreign sellers, then it iseasier to spend foreign money, and so youshould accept it even if you expect local sell-ers will reject it. Other parameters have sim-ilarly reasonable effects.

TWO INTERNATIONALMONIES

Now turn to the regime with two in-ternational currencies, = (1,1). The firstthing to do is to solve equations 1–2 formij. Routine algebra yields the steady statefor citizens of Country 1, which can be de-scribed by

(12) m10 = 1 −

(13) m11 =

M1 .

The steady state for citizens of Country 2 isdescribed by reversing the subscripts. No-tice that mi0 is decreasing in both M1 and M2.

The value functions now satisfy:

(14) rVi0 = ( iimii + ijmji)(Vii − Vi0)

+ ( iimij + ijmjj)(Vij − Vi0)

(15) rVii = ( iimi0 + ijmj0)(U + Vi0 − Vii)

(16) rVij = ( iimi0 + ijmj0)(U + Vi0 − Vij).

These equations imply that Vii = Vij.That is, buyers from either country are in-different between holding Currency i andCurrency j ≠ i. So the currencies are per-fect substitutes. It also follows that Vii >Vi0, and so equations 6–7 are satisfied forall parameter values. Hence, this regimealways constitutes an equilibrium. The in-tuition is that if agents believe the twocurrencies will be accepted by all sellersthen it is always rational for them to ac-cept both currencies themselves.

ONE INTERNATIONALMONEY

We now turn to the regime where citi-zens of Country 1 accept Currency 2, butnotviceversa, = (1,0).Onecansolveequa-tions 1–2 for m11 = M1 and m21 = 0. The dis-tribution of Currency 2 holdings is given by:

(17) m12 = ,

(18) m22 = .

From these one can show that, for givenvalues of Mi, m10 is lower and m20 higher inthis regime than in the other regimes. Also,mi0 is decreasing in both M1 and M2.

The value functions for agents fromCountry 1 satisfy

(19) rV10 = 11m11(V11 − V10)

+ ( 11m12 + 12m22 )(V12 − V10),

(20) rV11 = 11m10(U + V10 − V11)

(21) r V1 2 = ( 1 1m1 0 + 1 2m2 0) (U + V1 0 − V12) ,

( + 12)M2

+ 12 + 21(1 − M1)

12(1 − M1)M2

+ 12 + 21(1 − M1)

( + 12+ 21)+ 21[( + 21)(1−M1)+ 12(1−M2)]( + 12+ 21)[ + 21(1−M1)+ 12(1−M2)]

( + 21)M1 + 12M2

+ 12 + 21

2iiMi(1 − Mi)

r + ii

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and for agents from Country 2 they satisfy

(22) rV20 = ( 21m12 + 22m22)(V22 − V20)

(23) rV21 = 21m10(U + V20 − V21)

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

(U + V20 − V22 ).

One thing that follows immediately fromthese equations is that Vi2 > Vi0, so citizensfrom both countries are willing to acceptCurrency 2.

Hence, to show when this regime con-stitutes an equilibrium, we need to checkthe incentive constraints V11 ≥ V10 (tocheck that it is rational for Country 1 sell-ers to accept Currency 1) and that V21 ≤ V20

(to check that it is rational for Country 2sellers to reject Currency 1). The first con-dition is satisfied if and only if

(25) 1

1

1

2

m

m1

2

0

0

≥ ;

the second is satisfied if and only if

(26) ≤ .

These can be interpreted as saying that itis relatively easy for a buyer from Country1 to find a Country 1 seller, but relativelyhard for a Country 2 buyer to find aCountry 1 seller.

MODEL SUMMARYHere we summarize what has been

learned from the preceding analysis. Givenour maintained assumptions, there arethree qualitatively different types of equi-libria. There is at most one equilibrium ofeach type (that is, each implies a uniquem and V). Properties of the differentregimes include:

1. = (0,0) (dollars circulate only inAmerica, pesos only in Mexico) is anequilibrium if and only if equation

11 holds, which, in particular, is truewhen ij is small relative to ii.

2. = (1,1) (dollars and pesos bothcirculate in both countries) is al-ways an equilibrium, and in thisregime Vi1 = Vi2 (dollars and pesosare perfect substitutes);

3. = (1,0) (dollars circulate in bothcountries, pesos only in Mexico) isan equilibrium if and only if equa-tions 25–26 hold.

In Figure 1a we examine the set ofequilibria for different values of M1 andM2, holding the other parameters fixed. Ifa region in the figure contains the label ( 1, 2), then regime = ( 1, 2) is anequilibrium in this region. Notice thatequilibrium = (1,1) exists for all M1 andM2. Notice also that equilibrium = (0,0)exists only as long as M1 and M2 are nei-ther too big nor too small. When Currencyi becomes too scarce or too abundant, citi-

21m12+ 22m 22

r+ 21m10+ 22m20

21m10

22 m20

11m12+ 12m 22

r+ 11m10+ 12m20

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Figure 1a

Regions in (M1,M2) Space where the Different Equilibria ExistFixed Price ModelFor 11 1, 12 0.15, 21 0.15, 22 1, r 0.1, 0.02

zens from Country i find that either buy-ing or selling locally is too difficult andfind it too tempting to take foreign moneythat can be spent easily. The figure alsotells us about the existence regions forequilibria = (1,0) and = (0,1); Cur-rency 1 will be a unique international cur-rency as long as it is not too abundant andCurrency 2 is not too scarce.

In Figure 1b we examine different val-ues of 11 and 12, holding the other pa-rameters fixed. We see that as Country 1becomes more open ( 12 is large relativeto 11), it is more difficult to sustain anequilibrium where 1 = 0, or where 2 = 1.The opposite implications occur as localtrade in that country becomes easier (thatis, as 11 becomes larger).

For many parameter values there existmultiple equilibria. Hence, which regimewe end up in depends to a large degree onexpectations. At the same time, fundamen-tals—that is, preferences, technology, andthe values of M1 and M2—exert an impor-tant influence. At least in some cases, aparticular regime cannot be an equilibrium

unless these fundamentals take on the rightvalues.

An interesting extension is the one an-alyzed by Zhou (1994), who considers aversion of the model where agents differ-entiate between home and foreign goods.A given agent gets utility, u, from his pre-ferred good and < u from the othergood, where his preferred good fluctuatesbetween domestic and imported, and backagain, according to Poisson processes.Agents may consume only preferred goodsor some of both goods in equilibrium, de-pending on parameter values and on thetype of equilibrium. In this model therecan also be currency exchanges in purestrategy equilibria. For example, when aMexican holding a peso but having a tastefor American goods meets an Americanholding a dollar but having a taste forMexican goods, there is an incentive forthem to swap monies in any regime otherthan = (1,1) (because in that regime thecurrencies are perfect substitutes).

In any case, one shortcoming of themodel as it stands is that every trade is aone-for-one swap. This of course makes itimpossible to discuss the way nominalprices and exchange rates depend on vari-ous components of the model. The exten-sion in the next section remedies this.

THE MODEL WITH DIVISIBLE OUTPUT

We now turn to the model with divisi-ble output. It is still assumed that whenbuyers spend their money, they spend itall, but now the amount of output they getfor their money will be determinedendogenously. When a seller produces qunits of output for a buyer (of the righttype), the latter enjoys utility, u(q), whilethe former suffers disutility, c(q). Withno loss in generality, we can normalizec(q) = q, as long as we also renormalizeu(q). This simply means that agents bar-gain in terms of utils and not physicalunits of output. We assume that u(0) = 0,u′(0) > 1, u′(q) > 0 for all q > 0, u″(q) < 0 forall q > 0, and there exists q̂ > 0 such thatu( q̂) = q̂.

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Figure 1b

Regions in ( 11, 12) Space where the Different Equilibria ExistFixed Price ModelFor M1 0.75,M2 0.75, 21 0.1, 22 2.5,r 0.1, 0.02

We need to appeal to some form of bi-lateral bargaining theory to determine q.For example, one could use the general-ized Nash bargaining solution. This saysthat when a buyer from Country i withCurrency j meets a seller from Country h,if they trade at all, the quantity q solvesthe following problem:

(27)max[u(q) + Vi0 − Vij] [−q + Vhj − Vho]1−

subject to u(q) + Vi0 − Vij ≥ 0,

and − q + Vhj − Vh0 ≥ 0.

In this problem, q maximizes the so-calledNash product, which is the payoff to thebuyer, u(q) + Vi0, minus his threat point, Vij,all to the power , times the payoff to theseller, −q + Vhj, minus his threat point, Vh0,all to the power 1 − . The payoffs are whatthe agents get if they trade, the threat pointsa re what they get if they do not trade, and represents the relative bargaining power inthe hands of the buyer. The constraints saythat agents must get a greater payoff fromtrading than from not trading.7

For simplicity, we assume here thatthe buyer has all the bargaining power:

= 1. This effectively means that he canmake a take it or leave it offer to the seller.Two things follow immediately from this.First, assuming that the buyer wants totrade, he offers to exchange his money forthe quantity that makes the seller indiffer-ent between accepting and rejecting (thatis, the second constraint will bind). Sec-ond, this implies that the seller never getsany of the gains from trade, and thereforeVi0 = 0. Hence when a buyer from Countryi with Currency j meets a seller fromCountry h, if they trade at all, then thequantity is given by

(28) qhj = Vhj.

Note that this quantity depends on thenationality of the seller and the currencybeing used, but not on the nationality ofthe buyer. Also note that the nominalprice of output in Country h in terms ofCurrency j is phj = 1/qhj.

By construction, the seller alwayswants to trade at the take-it-or-leave-itoffer. It is possible, however, that a buyermay prefer to not trade at these terms. Forexample, although an American seller willalways take pesos at some price, a buyerwith pesos may prefer to wait until hemeets a Mexican seller if the seller is ex-pected to be willing to agree to a suffi-ciently better price. A buyer with Cur-rency j is willing to trade with a sellerfrom Country i if and only if the utility hederives from consuming qij exceeds thevalue to keeping his money and spendingit elsewhere. Hence the following incentivecondition must be satisfied if regime isto constitute an equilibrium:

(29) u(qij) > Vjj ⇒ i = 1

(30) u(qij) < Vjj ⇒ i = 0.

The value functions for buyers satisfy

(31) rVii = iimi0[u(qii) − Vii]

+ ijmj0 j[u(qji) − Vii]

(32) rVij = iimi0- i[u(qij) − Vij]

+ ijmj0[u(qjj) − vij],

for j ≠ i. We write equations 31 and 32 inthis way to facilitate comparison with themodel in the previous section; in particu-lar, see equations 4 and 5. However,strictly speaking, equations 31 and 32 arenot exactly correct without some modifica-tion in the framework, for the followingreason. Consider, for example, the case

j = 0, which says that Currency i doesnot circulate in Country j. Then equation31 says that a buyer from Country i withCurrency i who meets a seller from Coun-try j cannot trade. But, given the bargain-ing rules, he always could offer to tradeand get Vji > 0 (the amount a seller fromCountry j is willing to give for Currency i),and in principle we should allow him thatoption. In fact, one can show that even ifagents believe that j = 0, a buyer fromCountry i with Currency i who meets a

7 See Osborne and Rubinstein(1990) for an extended discus-sion on Nash bargaining and itsrelation to explicit strategic bar-gaining games.

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seller from Country j will offer to trade.8

This means that j = 0 can never be anequilibrium in the model as it stands.

However, this result is really just anartifact of our simplifying assumptionsthat the buyer gets to make a take it orleave it offer and there is no possibility ofdirect barter. Relaxing either of these al-lows j = 0 to be an equilibrium for someparameter values, although it also compli-cates the model significantly. We canmaintain the simplicity of the present setup and still generate equilibria with j = 0in another (albeit perhaps less satisfactory)way, as follows. First, expand the bargain-ing game so that a seller has to decidewhether to enter negotiations with a buyerbefore the buyer gets to make his take it orleave it offer. Then, given certain addi-tional technical assumptions, it will be anequilibrium for the seller to refuse to tradegiven that he believes j = 0. In this case,equations 31 and 32 do describe the valuefunctions as functions of . See Trejos andWright (1995b), for details.

Define the vectors q = (qij), V = (Vij),and m = (mij). Then an equilibrium is nowa list ( , m, V, q) satisfying equations 1–2and equations 28–32. Henceforth we ig-nore the value functions because they areredundant by virtue of the equilibriumcondition Vij = qij. Therefore, a steady-stateequilibrium is completely characterized bythe regime , the asset distribution m, andthe values of the currencies q.

NO INTERNATIONALMONEY

Consider first = (0,0). First, notethat m does not depend on whether out-put is divisible, so we already knowthe asset distribution from the previoussection (this is, of course, true in anyregime). Now equations 31–32 can be re-arranged to show that prices satisfy

(33) qii =r +

iim

ii

i

m0

i0

u(qii)

(34) qij =r +

ijmij

j

m0

j0

u(qjj),

for j ≠ i. In this regime, no transactions ac-tually occur at qij for j ≠ i, but qij but tellsus how much one could get with Currencyj from Country i sellers.

It is not hard to show that, as long as thisequilibrium exists, qi i and qj i a re both de-c reasing in Mi, and are independent of Mj

for j ≠ i. Also, q1 1 > q2 2 if and only if 1 1

(1 − M1) > 2 2(1 − M2). Because one unitof Currency i buys qi i units of real output,we can say that one unit of Currency 1 isw o rth e = q1 1/q2 2 = p22/p1 1units of Curre n c y2. This is the exchange rate that would beimplied by purchasing power parity.9 T h eexchange rate e falls with M1 and riseswith M2. Also, e rises with 1 1 and fallswith 2 2. Of course, international diff e r-ences in utility and production functionswould also influence e. We have keptthese the same across countries purely fornotational simplicity.

To see when this regime actually con-stitutes an equilibrium, we must check themaintained hypothesis that Currency i cir-culates in Country i and the incentive con-ditions in equations 29–30. The formercan easily be seen to hold for all parametervalues. The latter, which says that individ-uals with Currency j do not want to buyfrom sellers from Country i, holds if andonly if u(qij) ≤ q jj, for i ≠ j. Using equations33–34, we can rewrite this inequality as

(35)

which is satisfied if and only if ij is smallcompared with jj. The intuition is similarto the model with indivisible output.

TWO INTERNATIONALMONIES

Now turn to the regime with two in-ternational currencies, = (1,1). One canshow that in this regime qii = qij = Qi;That is, the two currencies are perfectsubstitutes in the sense that they purchasethe same amount from a given seller. Ifthere were a market in which agents

8 We thank Rao Aiyagari forpointing this out to us.

9 Suppose one peso buys q11

units of output in Mexico andone dollar buys q22 units of out-put in America, and supposethat there is a market in whichcurrencies can be traded at thenominal exchange rate e (thatis, one peso buys e dollars).Then one peso could be used tobuy eq22 units of output inAmerica using dollars.Purchasing power parity holds ifa peso buys the same amountdirectly at home and using dol-lars abroad: q11eq22.

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could trade the two currencies, the mar-ket clearing price would have to be 1. Butthe purchasing power parity exchangerate, as defined above, is given by e =Q1/Q2. In general e differs from unity inthis regime: Even though quantities donot depend on which currency the buyeris using, they do depend on the national-ity of the seller.

One can also show that Q1 and Q2 areboth decreasing in M1 and M2. Thus an in-crease in either money supply increasesthe price level in both countries. Also, aslong as the two countries are not too dif-ferent in terms of ij and Mj, the effect of achange in M1 is stronger in Country 1 thanin Country 2 and vice versa. This meansthat the purchasing power parity exchangerate e = Q1/Q2 is decreasing in M1 and in-creasing in M2.

To see when this regime constitutesan equilibrium, we need to check the in-centive conditions in equations 29–30.These hold in this regime if and only ifu(Qi) ≥ Qj, j ≠ i. If the countries are sym-metric, in the sense that M1 = M2, ii = jj

and ij = ji, these conditions always holdand the equilibrium with two internationalcurrencies always exists. But if they are suf-ficiently dissimilar, then the incentive con-ditions will be violated and = (1,1) is notan equilibrium. The model is very differentin this regard from the one with indivisibleoutput. In that model, equilibrium =(1,1) exists for all parameter values. Here,even if the two countries’ currencies areperfect substitutes, the two nationalities ofsellers are not. For example, if jj is lowrelative to ii then Country j sellers do notgive much for (either) currency comparedwith Country i sellers, and Country i buy-ers may prefer to hold on to their cashrather than spend it in Country j.

ONE INTERNATIONALMONEY

We now turn to the regime wherecitizens of Country 1 accept Currency 2,but not vice versa, = (1,0). The pur-chasing power of the national currency isgiven by

(36) q11 =r+

11

1

m

1m10

10

u(q11)

(37) q21 =r+

21

2

m

1m10

10

u(q11),

which are qualitatively the same as in the= (0,0) regime. Because m10 is lower in

this regime than in = (0,0), q11 and q21

are lower here. Intuitively, the influx offoreign money inflates prices denominatedin the domestic currency. Also, q11 and q21

are decreasing in both M1 and M2 becausem10 is decreasing in both M1 and M2.

The purchasing power of the interna-tional currency depends on the seller, asgiven by

(38) q12 =

(39) q22 = .

There is a unique nonzero solution for(q12, q22) and both are decreasing in M1 andM2. Since m10 is lower and m20 higher inthis regime, q12 is lower here than in the

= (1,1) regime. Moreover, q12 > q11, withthe difference increasing in M1 − M2.Hence, citizens of Country 1 value the in-ternational currency more than their owndomestic currency. As long as the coun-tries are not too dissimilar, one can alsoshow that q22 > q12.

We still need to check when thisequilibrium exists. The maintainedhypothesis that Currency i always circu-lates in Country i holds if and only ifu(q22) ≥ q12.10 Then equations 29–30 re-quire u(q12) ≥ q22, so Country 2 buyerswith Currency 2 buy from Country 1sellers, and u(q21) ≤ q11, so that Country1 buyers with Currency 1 do not buyfrom Country 2 sellers. Each of theseconditions looks qualitatively like onethat has been encountered earlier andholds under similar circumstances; how-ever, because qij differs across regimes,the parameter values for which the equi-libria exist are quantitatively different.

21m10u(q12) + 22m20 u(q22)r + 21m10 + 22m20

11m10u(q12) + 12m20u(q22)r + 11m10 + 12m20

10 This condition, which says thatagents from Country 1 withCurrency 2 buy from Country 2sellers, can bind for some para-meter values in this regime. Ifthis condition were violated,then Mexicans, for example,would sell goods for dollars andthen spend these dollars onMexican sellers, but not onAmerican sellers.

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DIVISIBLE OUTPUT SUMMARY

As in the indivisible output model,there are three qualitatively differentregimes and at most one equilibrium of eachtype (that is, each implies a unique m andq). Other things being equal, the value of acurrency is higher if it circulates interna-tionally and lower if the other money circu-lates internationally. Other properties of thedifferent regimes include the following:

1. = (0,0) (dollars circulate only inAmerica, pesos only in Mexico) isan equilibrium if and only if ij issmall relative to ii; in this regime,qii is decreasing in Mi and indepen-dent of Mj.

2. = (1,1) (dollars and pesos bothcirculate in both countries) is anequilibrium if and only if the twocountries are not too dissimilar; inthis regime, q11 = q12 = Q1 and q21 =q22 = Q2 (dollars and pesos are per-fect substitutes), Qi is decreasing

in Mi and Mj, and e = Q1/Q2 and e = Q1/Q2 is decreasing in M1 andincreasing in M2.

3. = (1,0) (dollars circulate in bothcountries, pesos only in Mexico)exists if a combination of the aboveconditions hold. In this regime, qij

is decreasing in Mi and Mj for all i,j, q12 ≥ q11 (Mexicans value dollarsmore than Mexicans value pesos),and q22 > q12 (Americans value dol-lars more than Mexicans value dollars) at least if the two countriesare not too dissimilar.

In Figure 2a we examine different val-ues of M1 and M2, holding the other parame-ters fixed. Notice that the regime = (1,1)exists for all M1 and M2 in this example, al-though this is not true in general. Also, be-cause ij is relatively small, equilibrium

= (0,0) exists for all but very high values ofM1 and M2. Equilibria where only one cur-rency circulates internationally exist only ifthe other currency is not too abundant. Forexample, = (1,0) exists only if M1 is not toobig. In Figure 2b we examine variation in

11 and 12. Now equilibrium = (1,1) ex-ists in only a small region of the figure. Equi-librium = (0,0) exists if and only if 12 isbelow some cutoff. Also notice that as either

11 or 12 increases, it becomes more likelythat equilibrium = (1,0) exists. This is be-cause as Country 1 increases either its inter-nal economic activity or its openness, sellersin Country 1 value money more highly andthis makes Country 2 buyers more willingto deal with them.

POLICYIn this section we analyze the effects

of changes in (M1, M2) in the model withdivisible output and endogenize (M1, M2)by modeling the objective functions of thetwo governments and the rules for theirstrategic interaction. Because for somequestions analytical results are difficult toderive, we analyze numerically an examplewith u(q)= q. The features that we high-light are ones that we can either prove an-

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Figure 2a

Regions in (M1,M2) Space where the Different Equilibria ExistFor 11 1, 12 0.15, 21 0.15, 22 1,r 0.1, 0.02

alytically to hold in general or those whichseem to be robust to alternative parameter-izations in examples.

Because Government i is assumed topurchase goods from a fraction Mi of itsnewborn citizens, per capita seignioragerevenue in real terms is given by Si = Miqii.One can show that S1 first increases andthen decreases with M1. Also, given thatmultiple equilibria exist, S1 is greater whenCurrency 1 is international than when it isnot and lower when Currency 2 is interna-tional than when it is not. Finally, givenM2, S1 is maximized at different levels of M1

in the different regimes, depending on theextent to which Country 1 is able to exportinflation abroad.

We define welfare as the average(steady state) utility of private citizens:

(40) Wi = mi0Vi0 + mi1Vi1 + mi2Vi2.

Which regime yields the highest welfaredepends 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 thismakes trade easier within the host country.Also, = (1,1) may or may not dominate

= (0,0). For instance, if M2 is very low orvery high, then W1 is highest in regime

= (0,0).The next step is to endogenize (M1, M2).

We assume that governments take the re-gime as given and restrict their choices topolicies that allow for the existence of thatregime as an equilibrium (that is, we ignorepolicies aimed at changing a currency’srealm of circulation).11 We also restrict at-tention to policies that do not change overtime and to steady-state comparisons.

We look for Nash equilibria when eachgovernment seeks to maximize the welfareof its own citizens, denoted (M1

W, M2W). In

other words, W1 is maximized at M1W when

M2W is taken as given and vice versa. We

also look for Nash equilibria when eachgovernment seeks to maximize seigniorage,denoted (M1

S, M2S ). Given that they seek to

maximize seigniorage, we also consider thepossibility of international policy coordina-tion by letting the governments choose

policies jointly. One way to do this is to as-sume that seigniorage is freely transferableacross countries, in which case they maxi-mize S1 + S2. We denote this outcome by(M1

T, M2T). Or we can assume seigniorage

is nontransferable, in which case we usethe bargaining solution that chooses (M1

N,M2

N) to solve max [S1 − S1S][S2 − S2

S],where Sj

S is seigniorage in Country j whenpolicy is given by (M1

S, M2S).

We can summarize our findings as fol-lows. First, if foreign money circulates inCountry i, independently of whether Cur-rency i circulates abroad, then the welfaremaximizing level of Mi is lower than thelevel that maximizes seigniorage. However,when foreign money does not circulate inCountry i, welfare and seigniorage aremaximized at the same value of Mi.12 Sec-ond, starting from the Nash equilibriumwhere governments maximize seigniorage,reducing the amount of money in bothcountries increases welfare in both coun-tries. Third, and more interestingly, reduc-ing the amount of money in both coun-tries increases seigniorage for both

11 There may actually be veryinteresting policies, but weleave such a discussion tofuture research.

12 This can be explained as fol-lows. Welfare in Country 1, forexample, is W1 = m11 V11 +m12 V12, whereas seigniorage isS1 = M1V11. When no foreignmoney is held at home (m12 =0), m11 is proportional to M1

and maximizing seigniorage isthe same as maximizing wel-fare. This result is not particular-ly robust and does not hold ingeneralized versions of themodel that include barter orother bargaining solutions.

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Figure 2b

Regions in ( 11, 12) Space where the Different Equilibria Exist

governments. As neither government takesinto account the effect it has on the other,the noncooperative equilibrium is charac-terized by too much money. The coopera-tive equilibrium (M1

N, M2N) involves less

money and more seigniorage.These results hold in general. We now

describe in more detail the findings from thenumerical examples. We depict the frontierand the outcomes in (S1, S2) space, assum-ing the regime is = (1,0) in Figure 3a and

= (1,1) in Figure 3b (the regime with nointernational currency is uninteresting forthis exercise). The points labeled W, S, T,and N are the payoffs in the four scenarios:

the noncooperative equilibrium betweenwelfare maximizers, the noncooperativeequilibrium between seigniorage maximiz-ers, the cooperative solution when revenueis transferable, and the cooperative solutionwhen revenue is nontransferable. In thetransferable revenue case, the point T de-picts the seigniorage raised in each countryand not the final division of the revenue be-tween the governments. Figures 4a and 4bshow the same points in (W1, W2) space. Be-cause the graphs are drawn using the samescales, one thing they illustrate is how thepossible values of seigniorage and welfarevary across regimes.

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Figure 3a,b

Feasible and EquilibriumSeigniorageFor a11 1,a12 0.15,a21 0.15,a22 1,r 0.1,g 0.02(a) λ (1,0)

(b) λ (1,1)

Figure 4a,b

Feasible and Equilibrium WelfareFor a11 1,a12 0.15,a21 0.15,a22 1,r 0.1,g 0.02

(b) λ (1,1)

The results we wish to highlight are asfollows. First, though the cooperative solu-tions are on the frontier in (S1, S2) space, the noncooperative solutions are inside thefrontier. This is especially so in regime

= (1,1), where it is easiest to export infla-tion. Also, Figure 3a shows that in regime = (1,0), when transfers are allowed, the co-operative solution is to concentrate seign-iorage collection in Country 2, where it canbe done more efficiently. That is, the govern-ments print more of the international cur-rency and less of the national currency inaddition to printing less money in total.

Figure 4 shows how the possible val-ues of (W1, W2) are much higher in regime

= (1,1). However, each of the outcomesis inside the frontier. Also, when both gov-ernments try to maximize seigniorage, oneof them may actually end up with lessseigniorage than when both are trying tomaximize welfare. Symmetrically, citizensin one country may be worse off in termsof welfare when both governments are try-ing to maximize welfare than when bothgovernments are concerned with seignior-age, as can be seen in the figure for Coun-try 1 in regime = (1,0). Finally, in regime

= (1,0), the country that issues interna-tional money is better off.

FINAL REMARKSThis article has summarized some re-

cent results in the literature on search-theoretic models of international currency.We think that models in this class willeventually prove useful for studying im-portant issues in international monetaryeconomics. The analysis illustrates the po-tential welfare gains from having one cur-rency (or two currencies that serve as per-fect substitutes). However, it also indicatesthat there can be a welfare loss in onecountry from having a unified currency ifanother country is pursuing a particularlybad policy from the former’s point of view.It may be interesting to consider versionsof the model where the two countries havedifferent preferences or production tech-nologies or are subject to different shocksto analyze the tradeoffs between having

one currency, one central bank or both.While this may help to facilitate trade, itmakes it more difficult to have indepen-dent monetary policies tailored to condi-tions in the individual economies. Weleave this to future research.

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