American Economic Association
A Search-Theoretic Approach to Monetary Economics Author(s): Nobuhiro Kiyotaki and Randall Wright Source: The American Economic Review, Vol. 83, No. 1 (Mar., 1993), pp. 63-77 Published by: American Economic Association Stable URL: http://www.jstor.org/stable/2117496 . Accessed: 14/09/2011 06:08
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http://www.jstor.org A Search-TheoreticApproach to MonetaryEconomics
By NOBUHIRO KIYOTAKI AND RANDALL WRIGHT *
The essentialfunction of money is its role as a medium of exchange. We formalizethis idea using a search-theoreticequilibrium model of the exchange process that capturesthe "doublecoincidence of wants problem"with pure barter. One advantage of the frameworkdescribed here is that it is very tractable.We also show that the modelcan be used to addresssome substantive issuesin monetaryeconomics, including the potentialwelfare-enhancing role of money,the interactionbetween specialization and monetaryexchange, and the possibilityof equilibriawith multiplefiat currencies.(JEL EOO,D83)
Since the earliest writings of the classical theoretic equilibrium model of the exchange economists it has been understood that the process that seems to capture the "double essential function of money is its role as a coincidence of wants problem" with pure medium of exchange. The use of monetary barter in a simple and natural way. We exchange helps to overcome the difficulty show that this gives rise to a medium-of- associated with pure barter in economies exchange role for fiat currency. We also where trade is not centralized through some show that the model can be used to address perfect and frictionless market. Many at- some substantive issues in monetary eco- tempts have been made in the literature to nomics. formalize this, with varying degrees of suc- Previously, in Kiyotaki and Wright (1989), cess.' In this paper, we present a search- we used a search-theoretic model to deter- mine endogenously which commodities would become media of exchange, or com- modity money. We were also able to con- struct an equilibrium with valued fiat cur- rency; but since that model was designed * Kiyotaki: Federal Reserve Bank of Minneapolis and Department of Economics, University of Min- primarily to study commodity money, it is nesota, Minneapolis, MN 55455; Wright: Federal not the most tractable framework within Reserve Bank of Minneapolis and Department of Eco- which to discuss fiat money. For instance, nomics, University of Pennsylvania, Philadelphia, PA we only showed that fiat money could be 19104. This is a much revised version of our earlier working paper, "Search for a Theory of Money." Part valued if it has intrinsic properties at least of the work on that project was accomplished while as good as the best available commodity both authors were at the London School of Economics in the spring of 1990, and we are grateful for that institution's hospitality. The National Science Founda- tion, the University of Pennsylvania Research Founda- tion, and the Hoover Institution all provided financial support. We also thank many friends, colleagues, stu- to cite all of the relevant work, we refer the reader to dents, and seminar participants for their input. Peter the survey by Joseph M. Ostroy and Ross M. Starr Diamond, Robert Lucas, Dale Mortensen, David (1990). We would, however, like to mention the contri- Romer, Neil Wallace, and two anonymous referees bution of Robert A. Jones (1976) and the extensions by made some especially useful suggestions, although they Seongwhan Oh (1989) and Katsuhito Iwai (1988). Al- should not be held accountable for the final product. though there are many technical differences, that model The views expressed here are those of the authors and is definitely related in spirit to the search-theoretic not necessarily those of the Federal Reserve Bank of approach we describe here. In particular, there are Minneapolis or the Federal Reserve System. heterogeneous agents and commodities, and in equilib- 1There is a voluminous literature on the founda- rium certain commodities are chosen as media of ex- tions of monetary thecry, and rather than attempting change in order to reduce search costs.
63 64 THE AMERICAN ECONOMIC REVIEW MARCH 1993
money. In Kiyotaki and Wright (1991), we we make some simplifying assumptions in used an alternative search-based model to specifying the basic model that one arguably illustrate the robustness of monetary equi- may want to relax, and we show how to do libria; that is, fiat money can be valued as a so in the Appendix. medium of exchange even if it has intrinsic properties, like its rate of return, that are I. The Basic Model inferior to other available assets. We also constructed an example in that model to The economy is populated by a large show how the use of fiat money can affect number of infinite-lived agents, with total welfare. population normalized to unity. There is However, due to the generality of the also a large number of consumption goods. specification in that paper, we were not able These consumption goods are indivisible and to say much about the features of monetary come in units of size one. We refer to them equilibria, other than that they exist and are as real commodities, to distinguish them robust, and our characterization of welfare from fiat money, which is an object that no did not proceed much beyond a numerical one ever consumes and can be thought of as example. The model to be presented in this a collection of pieces of paper or certain paper can be thought of as a simplified types of seashells, for example, with no in- version of Kiyotaki and Wright (1991). Our trinsic value. A crucial feature of the model first objective is to demonstrate that this is that there is an exogenous parameter x, class of models is actually very tractable. with 0 < x < 1, that captures the extent to Our second objective is to convince the which real commodities and tastes are dif- reader that search-based models can be used ferentiated. In particular, x equals the pro- not just to determine which objects serve as portion of commodities that can be con- media of exchange or to prove the existence sumed by any given agent, and x also equals of valued fiat money, but to address some the proportion of agents that can consume more applied issues in monetary economics any given commodity.2 If a commodity is as well. In particular, we use the model to one of those that can be consumed by an discuss the potential welfare-enhancing role agent, then we say that it is one of his of money, the interaction between special- consumption goods. Consuming one of his ization and monetary exchange, and the consumption goods yields utility U > 0, while possibility of equilibria with multiple cur- consuming other commodities (or money) rencies. yields zero utility. The rest of the paper is organized as Initially, a fraction M of the agents are follows. In Section I we describe the basic each endowed with money while 1- M are model. In Section II we characterize the each endowed with one real commodity, welfare effects of money. Among other where 0 < M < 1. Money may or may not things, the model implies that equilibria have value. If it does, then it is convenient where fiat money is universally acceptable to assume that agents who are initially en- are generally superior to nonmonetary equi- dowed with money are endowed with ex- libria and to equilibria where it is only actly one unit of real balances, so that in partially acceptable. In Section III we intro- order to buy a real commodity they must duce specialization by producers, by assum- spend all of their cash. There are two ways ing that they face a trade-off between to guarantee that this is the case. First, and productivity and the marketability of their output. The model implies that use of money, by making exchange easier, leads to more specialized and, therefore, more effi- 2For example, suppose there are K distinct goods cient production. In Section IV we discuss a and each agent consumes k of them; then x = k 7K. version of the model that allows for multi- Alternatively, suppose there is a continuum of goods indexed by points around a circle of circumference 1 ple fiat currencies. In Section V we con- and each agent consumes goods corresponding to points clude. In order to improve the presentation in a fixed arc; then, x is the length of that arc. VOL. 83 NO. 1 KIYOTAKIAND WRIGHT:SEARCH-THEORETICAPPROACH 65
most straightforwardly, we can simply as- traders meet, exchange takes place if and sume that the monetary object is indivisible, only if it is mutually agreeable, that is, if like the real commodities in the model. Then and only if both agents are at least as well if money trades at all it must trade one-for- off after the trade. Because there is a large one against a real commodity, and each number of anonymous agents, all trade is agent with one indivisible unit of money will quid pro quo (there can be no IOU's or have one unit of real balances. Alterna- other forms of private credit). We also as- tively, we can assume that money is divisi- sume that there is a transaction cost E in ble, determine the price level endogenously terms of disutility, where 0
are referred to as commodity traders, while cepted. This means that x is the probability agents with fiat money are referred to as that a commodity trader located at random money traders. Let ,u denote the fraction of is willing to accept any given commodity, traders who are money traders, so that a and therefore x2 is the probability that two trader located at random has money with commodity traders consummate a barter probability ,u and a real commodity with transaction. This is precisely William Stan- probability 1- ,u. ley Jevons's (1875) "double coincidence of Individuals choose strategies for deciding wants problem" with direct barter: not only when to accept various commodities and fiat do you have to meet someone with some- money in order to maximize their expected thing that you want, this someone also has discounted utility from consumption net of to want what you have.5 transaction costs, taking as given the strate- The next thing to determine is whether gies of others. We look for Nash equilibria. individuals accept money. Let H denote the We restrict attention for the most part to probability that a random commodity trader symmetric equilibria, where all agents and accepts money and let wr be the best re- all real commodities are treated the same, sponse of a representative individual. We and to steady-state equilibria, where strate- will solve the best-response problem using gies and all aggregate variables are constant dynamic programming. Let Vj denote the over time. To construct the set of such payoff or value function for the individual in equilibria, we describe some basic proper- state j, where j = 0, 1, or m indicates that ties that they must satisfy, use these proper- he is a producer, a commodity trader, or a ties to describe an individual trader's best- money trader, respectively. Then, if r > 0 is response correspondence, and determine its the rate of time preference, Bellman's equa- fixed points. tions are given by The first thing to note is that an agent always accepts a real commodity if it is one (1) rVo = a( V1-VO) of his consumption goods, whereupon he immediately consumes it and enters the (2) rV1=J (1-p)x2(U-8+V0-V1) production process. Also, we claim that a commodity trader will never accept a com- + 3ptx maxr( Vm- V1) modity that is not one of his consumption ir goods. This is due to the fact that in a symmetric equilibrium no real commodities (3) rVm=j8(1-pI)Hx(U-E+VO-Vm). are treated as special, and therefore, the probability of a trade offer being accepted by the next agent one meets is independent 5The result that traders never accept commodities of the type of commodity one has. Hence, that are not their consumption goods means that there there is no advantage to trading one real is no commodity money in a symmetric equilibrium. This is not to say that the model cannot have nonsym- commodity for another, and since there is a metric equilibria, in which some real commodities do transaction cost E, unless a commodity is become media of exchange, but only that we restrict going to be consumed it will never be ac- attention to symmetric outcomes here. Commodity money is analyzed in a related model in Kiyotaki and Wright (1989). Although a small transaction cost e guarantees that there will be a double-coincidence problem in a symmetric equilibrium, the double-coinci- dence problem arises without transaction costs in the asymmetric equilibria studied in Kiyotaki and Wright input into production (an assumption we adopted from (1989) and Aiyagari and Wallace (1991). Another way S. Rao Aiyagari and Neil Wallace [1991, 1992]) and the to guarantee that there is a double-coincidence prob- way we distribute the initial endowments. In principle, lem is to assume that a real commodity can only be we could initially endow agents with more than one stored by its producer, as in Kiminori Matsuyama et al. unit of commodities or real balances, but this would (1993), which seems natural if we interpret these com- require solving for the steady-state inventory distribu- modities as services rather than goods. Under this tion and would lead to potentially complicated bargain- assumption, we can dispense with the transaction cost ing problems in bilateral exchange. entirely. VOL. 83 NO. 1 KIYOTAKIAND WRIGHT:SEARCH-THEORETIC APPROACH 67
Equations like these are standard in search theory (formal derivations for a No closely related model can be found in Producers Kiyotaki and Wright [1991], for example). They have the followinginterpretation. Ac- cording to (1), the flow return to a pro- ducer, rVo,equals the rate at which output is produced, a, times the gain from switch- ing from production to exchange, V1- VO. Accordingto (2), the flow return to a com- X , #~~~A-x m modity trader equals the sum of two terms. Commodty 41 Money The first term is the rate at which he meets \Traders J Traders other commodity traders, ,3(1- ,u), times the probabilitythat both want to trade, x2, FIGURE 1. DYNAMIC STRUCrURE OF THE MODEL times the gain from trading,consuming, and switching back to production, U - E + VO- vJ. The second term is the rate at which he the fact that Nm = M (the number of money meets money traders, 13k, times the proba- traders equals the number of agents en- bility that a money traderwants to trade, x, dowed with money),(4) can be reduced to times the gain from accepting money with probability7r, where v is chosen optimally. (5) M= alL/(a + ) Accordingto (3), the flow returnto a money trader equals the rate at which he meets where p = (,u, HI) is defined by7 commodity traders, ,3(1 - ,u), times the _ probabilitythat both want to trade, Hx, (6) =3(l - A) LXII + ( 1 I)X2. times the gain from trading,consuming, and switching to production, U-E + V0-Vm.6 Equation (5) is a quadratic in 1L,and for The above dynamicprogram depends not any M E [0, 1] and l e [0, 1] there will only on the strategies of others, as repre- exist a unique value of ,u = ,u(M, H) in [0,1] sented by HI,but also on A, the proportion satisfying this equation. Furthermore,one of traders holding money. However, A can can show that ,u(O,H) = 0, ,u(1, Hl) = 1, be determined as a function of H and the d,u /dM > O, and d&/dH >O. Given A = initial endowment of money, M. Begin by yt(M,H), the unique steady state is fully letting No, N1, and Nm denote the propor- describedby tions of the populationwho are producers, commodity traders, and money traders. Then the model has a dynamic structure (7) No=p/(a+q.) with transitions as illustrated in Figure 1. To determineits steady state, we equate the N1 = (1l-)a/(a + p) flow out of and into production: However, for the purpose of analyzingthe (4) aNO=3l(1-,u)x2N1+ f3(1-,a) IxNm. above dynamicprogram, ,u summarizesall the agent needs to know about the steady If we use the fact that the Nj's sum to 1 and state. If we insert ,u = ,u(M,Hl) into (1)-(3),
6We have implicitlyassumed that a money trader 7Note that 'pcan be interpretedas consumptionper never accepts a commoditythat is not one of his traderper unit time: it is the rate at which a represen- consumptiongoods, but one can show that this is tative tradermeets commoditytraders, 3{(-1), times alwaystrue in equilibrium.That is, one can show that the probabilitythat a deal is consummated,which is the only time an agent would want to exchangemoney the probabilityour representativetrader has money for a commoditythat is not one of his consumption and they trade, ,uxI, plus the probabilitythat our goods is when money is valueless, in which case he representativetrader has a real commodityand they cannot. trade,(1-,)x'. 68 THE AMERICAN ECONOMIC REVIEW MARCH 1993
agents expect that money will be valueless, so they never accept it, and this expectation 450 is self-fulfilling.The equilibriumwith H = 1 will be called the pure-monetary equilibrium. -IT In this case, agents expect that money will be universallyacceptable, and so they al- ways take it, and this expectation is self- fulfilling.Finally, the equilibriumwith HI= x will be called the mixed-monetary equilib- rium. In this case, agents are indifferent between accepting and rejecting money as o n~~~~~~~~~~~~~~~long as other agents take it with probability H = x, and so partial acceptabilitycan also be self-fulfilling.Alternatively, a symmetric mixed-strategyequilibrium where all agents accept money with probabilityx could be reinterpreted as a nonsymmetric pure- strategyequilibrium, where a fraction x of FIcGURE2. THE BEST-RESPONSECORRESPONDENCE agentsaccept money with probability1 while the rest accept it with probability0.8 then, given M, this dynamic program de- II. Welfare fines a correspondence from fl to best re- sponses, 7r. The set of equilibria is the set of The first thing we want to do in this fixed points of this correspondence. section is to compareutility across the vari- To characterize this set, first note that if ous equilibria,for a given value of M. For fl < x then (1-(3) imply that Vm< Vl, which the purpose of this comparison,we keep implies that the best response is vr = O. In- things tractable by restrictingattention to tuitively, if money is being accepted with a the limiting case where a -> oo. In this case lower probability than a barter offer, then it production is instantaneous, and so all is harder to trade using money than barter, agents are either money tradersor commod- and so the best response is never to ex- ity traders: Nm= M, N1 =1-M, and Au= change a real commodity for money. Sec- M. This makes it relatively easy to solve ond, if II > x, then (l)-(3) imply that Vm> (1)-(3) for the reduced-formpayoffs: Vl, which implies 7r = 1. If money is being accepted with a greater probability than a (8) rJ1 = tfrx + f3xH[MH + (1- M)x]} barter offer, then it is easier to trade using money, and so the best response is to ex- (9) rVm change a real commodity for money when- =qf{rHI`+13x1[MHI+(1-M)x]} ever possible. Finally, if fl = x, then (l)-(3) imply that Vm= V, which implies that 7r where if = (U - E),f(1 - M)x/(r + f3xH). can be anything in [0, 1]. If monetary ex- We can now substitute Hl= O, H = x, and change and barter are equally easy then H = 1 into (8) and (9) and compare utility traders are indifferent between having across equilibriafor commoditytraders and money and real commodities, and they could money traders. accept money with any probability. Based on these results, the best-response corre- 8There can exist non-steady-stateequilibria in this spondence is as shown in Figure 2, and model where the probabilitythat money is accepted there are exactly three equilibria: II = O, varies over time. An example of a "sunspotequilib- fl = and fl = x. rium,"in which the probabilitythat moneyis accepted 1, fluctuatesrandomly over time even thoughthe funda- The equilibrium with fl = O will be called mentals are nonstochasticand time-invariant,is con- the nonmonetary equilibrium. In this case, structedin Kiyotakiand Wright(1990). VOL. 83 NO. 1 KIYOTAKIAND WRIGHT:SEARCH-THEORETICAPPROACH 69
If we let the superscriptsN, M, and P To pursue this, define the welfare crite- refer to the nonmonetary,mixed-monetary, rion and pure-monetaryequilibria, respectively, then we have the following results. First, (10) W = NOVO+ N1V1+ NmVm. commodity traders are equally well off in the nonmonetaryand mixed-monetaryequi- This can be interpreted as the ex ante ex- libria and strictly better off in the pure- pected utility of all agents before the initial monetary equilibrium: VN= Vm < VP. Sec- endowment of money and output is ran- ond, money traders are strictlybetter off in domly distributedamong them. After some the pure-monetaryequilibrium than in the algebra,(10) can be simplifiedto yield mixed-monetary equilibrium and strictly better off in the mixed-monetaryequilib- (11) rW= (U-e)pa/(a + sp) rium than in the nonmonetaryequilibrium: VZ < JQ < V, . Thus, given the initial en- where p = (,u, [I) was defined in (6) dowment of money and real commodities, above.10Now consider maximizingW with all agents are at least weakly better off if respect to M. Since W is increasingin p, we money is acceptablethan if it is not, and all proceed by finding the value ,tu that maxi- agents are strictly better off if money is mizes S with respect to ,u, and then deter- universallyacceptable than if it is only par- mine the optimal value M' from the steady-state condition (5), M = /(a + q). tially acceptable.9 a/.t ' The next thing we want to do is to exam- The result is as follows: if x ? then ine how utility varies with M, and for the = 0, which implies MO = 0; if x <2 purpose of this comparison we return to then AO=(1-2x)/(2-2x), which implies the general case where ac need not be oo. M' > 0. Intuitively, when x ? -, each agent The experimentwe consider is to increase is willing to consume(and therefore accept) the numberof agents initiallyendowed with at least half of the commoditiesproduced in money and to reduce the number initially the economy, and pure barter is not very endowed with real output, so that we can difficult.In this case the role for a medium maintain the tractabilityof the unit-inven- of exchangeis not very important,and it is tory assumption.In either the nonmonetary optimal to endow everyonewith real output equilibriumor the mixed-monetaryequilib- and no one with money.When x < , on the rium, all agents are better off the lower is other hand, pure barter is sufficientlydif- M. The reason is that, in these equilibria, ficult that the introduction of some fiat money does nothing to ameliorate the money improveswelfare, in spite of the fact double-coincidenceproblem, and so it is that, in the experimentunder consideration, better to endow everyone with real con- endowingsome agents with money requires sumption goods rather than intrinsically endowing fewer agents with real output at worthless paper or seashells. The more in- the initial date. We also note that M' is 1 terestingcase is the pure-monetaryequilib- decreasing in x, and that M' -2 as x -O 0. rium, where fiat currencydoes have a gen- Thus, as x shrinks and the double-coinci- uine role to play in facilitatingexchange. dence problem becomes more difficultit is optimal to endow more agents with money. We now turn to a version of the model in 9These results differ from those in Kiyotaki and which money can be interpreted as being Wright (1990), where we assumed that agents initially divisiblerather than indivisibleand investi- endowed with fiat currency would freely dispose of it and produce a new commodity in the nonmonetary gate the welfare implicationsof a particular equilibrium. This made the initial stock of real com- mechanismfor determiningthe price level, modities differ across monetary and nonmonetary equi- libria and therefore made welfare comparisons am- biguous. Following Aiyagari and Wallace (1992), we 10Noticethat rW equals the differenceU - e times assume here that agents initially endowed with fiat aggregate consumption,since 'p is consumptionper money cannot produce until they consume, which keeps trader per unit time and a /(a + 'p) = N1 + Nm is the the initial stock of output constant across equilibria. numberof traders. 70 THE AMERICAN ECONOMIC REVIEW MARCH 1993
P. We look for a pure-monetary equilibrium in which each money trader carries P units of cash and all P units are required to purchase one real commodity. Then real balances are given by M = C/P for any exogenous stock of nominal currency C. Of course, to make P endogenous we need to impose an additional equilibrium condition. Consider the method used by Diamond (1984) in his cash-in-advance search model, which is to impose as an equilibrium condi- tion that the gains from trade for a com- modity trader and a money trader are equalized whenever an exchange is made between them:
O M M? 1 M (12) = U ? + vm . Vm-V, Vo FIGURE 3. VALUE FUNCTIONS AND WELFARE
The left-hand side is the gain from trade for a commodity trader who accepts money, while the right-hand side is the gain for a entails M* = 0 and P = oo, and hence there money trader who acquires one of his con- can be no monetary exchange. sumption goods. Recall that the value of ,u that maximizes If a pure-monetary equilibrium satisfies W is A' = (1-2 x)/(2-2 x), and comparing condition (12), we call it a split-the-surplus this with (13) we find ,u?> ,u*. This means equilibrium. Notice that both sides of (12) the split-the-surplus equilibrium yields a depend on ,ut. If ,u is large, there are many lower value of ,u, and hence a lower value money traders and few commodity traders, of M and a higher value of P for any given so having money is not very desirable; hence, C, than that which maximizes ex ante util- a commodity trader who acquires money ity. However, the split-the-surplus equilib- gets a smaller gain than a money trader who rium is still ex post Pareto optimal. To see acquires one of his consumption goods. why, consider the value functions Vmand V1 Thus, for large ,u the right-hand side of (12) (Vo is proportional to V1 and need not be exceeds the left-hand side, and we need to considered independently). One can show reduce ,u until either (12) holds or we hit that, as functions of M, both are concave, ,u = 0. Inserting the reduced-form payoff and V1 is increasing but Vm is decreasing at functions and simplifying, the unique value M = M* (see Fig. 3). Hence, any movement of ,u that satisfies (12) is given by away from M* that makes commodity traders better off makes money traders worse off, and vice versa. The split-the- (13) p* =(1-2x)/(2-2x) surplus equilibrium is therefore ex post efficient even though it fails to maximize - r/2,3x(1- x). ex ante welfare."
If r <,(3x(1 - 2x) then ,u* > 0, which im- plies a unique M* > 0 satisfying (5), and a finite equilibrium price level P* = C/M*. 11Figure 3 indicates that both commodity and money traders prefer a lower value of M than that which If r ? - be satis- ,3x(1 2x), then (12) cannot maximizes W. This ostensibly paradoxical result can be fied for any value of p. > 0. In this case, we understood by noting that the number of agents of say that the split-the-surplus equilibrium each type varies with M. VOL. 83 NO. 1 KIYOTAKIAND WRIGHT:SEARCH-THEORETICAPPROACH 71
III. Specialization more generally acceptable medium of ex- change, which they will then use to buy An insight dating back at least to Adam whatever consumption goods they desire. Smith (1776) is that specialization is limited Hence, specializationleads to a greaterrole by the extent of the market and that the use for money, while at the same time the use of money encourages specialization by en- of money affords a greater opportunityfor larging the extent of the market. As Smith specializationby facilitatingthe process of puts it: exchange. In order to formalize this, we introduce a trade-off between productivity When the division of labour has been and marketabilityby assumingthat the ar- once thoroughly established, it is but a rival rate in the production process is a very small part of a man's wants which function of the numberof agents willing to the produce of his own labour can consumethe output:a = a(x), where a' < 0. supply. He supplies the far greater The idea is that, by becomingmore special- part of them by exchanging that sur- ized, a producer can increase output per plus part of the produce of his own unit time, a, but only at the cost of reducing labour, which is over and above his the fraction of consumerswho will accept of own consumption, for such parts x.12 the produce of other men's labour as his output in exchange, he has occasion for. Every man thus Before entering the production process, lives by exchanging, or becomes in agents choose x, taking as given the behav- some measure a merchant, and the ior of others. If money is accepted with society itself grows to be what is prop- probability LI and other producers' deci- erly a commercial society. sions imply that a given individualcan con- But when the division of labour first sume a fraction X of their output, his pay- began to take place, this power of off if he chooses x is describedby exchanging must frequently have been very much clogged and embarrassed in its operations. One man, we shall sup- (14) rVO=a(x)[V1(x)-VO] pose, has more of a certain commodity than he himself has occasion for, while (15) rV1(x) another has less. ... But if this latter should chance to have nothing that = (1- )XX[U - e + Vo - Vl(x)] the former stands in need of, no ex- change can be made between them. + f3/xL[ Vm - V1(x)] ... In order to avoid the inconve- niency of such situations, every pru- dent man in every period of society, (16) rVm= (1-,u)Xf1(U--F+ Vo-VM). after the first establishment of the di- vision of labour, must naturally have The choice of x will be made by a producer endeavored to manage his affairs in to maximizethe right-handside of (14), and such a manner, as to have at all times this x is then carriedover to the exchange by him, besides the peculiar produce sector as a state variable. As was the case of his own industry, a certain quantity earlier, this best-responseproblem also de- or such of some one commodity other, pends on ,ut,but ,u will be determined below as he imagined few people would be likely to refuse in exchange for the produce of their industry. 120ne interpretation is that each consumer derives [1937 pp. 22-3] utility from a fixed set of characteristics embodied in some commodities, and a larger value of x implies that the producer's output contains a greater number of Smith is suggesting that specialization, characteristics and hence has a larger potential market. while it may have desirable consequences in Eduardo Siandra (1990) has independently developed a very similar model. Robert King and Charles Plosser terms of productivity, makes barter difficult. (1986) and Harold L. Cole and Alan C. Stockman Whenever they can, specialized producers (1992) provide other analyses of the interaction be- will therefore tend to sell their output for a tween money and specialization. 72 THE AMERICAN ECONOMIC REVIEW MARCH 1993 using the steady-stateconditions as a func- ,A, and rI, we write it as M = M(X,p, F). tion of strategiesand M. Given M, an equilibriumis a solution to Equations(14)-(16) can be solved for VO, X = x(X, A, ) and M = M(X, A, H) with V1, and Vm. In particular, after simplifica- either [I = 0, Fl = X, or I = 1 (since, for tion, we find that any given X, the model has a nonmonetary equilibrium,a mixed-monetaryequilibrium, and a pure-monetaryequilibrium, exactly as (17) a(x)[V1(x)-VO] in the model without endogenous special- ization). In Figure 4 we draw the locus of = xa(x)/l(x) Z(x) points in (,u, X)-space satisfying each of these conditions. Notice that the M = where M(X,,u,H) curve is upward-sloping, shifts to the rightas LIincreases, and goes through ;(x) = [r + /3(1- A)XlH][r + ,3(1 - ,)xX + ,83Axll] (M,O). Also, the X= x(X, tt,ll) curve is upward-sloping,horizontal, or downward- + a(x)[r + /(1 - A)XH + ,3,xfl] sloping dependingon whether [I = 0, X, or 1, and goes through the same intercept and 4 does not depend on x. The individ- (0, XO)in any case. The intersectionof these ual choice of x can be found by maximizing two curves determinesthe equilibriumval- Z(x). If we assume an interiorsolution, the ues of ,t and X in the nonmonetaryequilib- first-order condition Z'(x) = 0 can be rear- rium, the mixed-monetaryequilibrium, or ranged to yield:13 the pure-monetaryequilibrium, depending on whether LI= 0, I = X, or LI= 1.14 xa'(x) Again let the superscripts N, M, and (18) - ( P representthe nonmonetary,mixed-mone- tary, and pure-monetaryequilibria. As can be seen from the diagram,specialization is r + a(x) greatest in the pure-monetaryequilibrium, r + (3l(1-,)Xx + 83/xH lower in the mixed-monetaryequilibrium, and lowest in the nonmonetaryequilibrium: It can be shown that the second-ordercon- XP < XM < XN. The intuition behind this dition Z" < 0 holds if we assumethat a" < 0. result is that when money circulatesthere is Then (18) completelycharacterizes the indi- less of an advantageto having a high value vidual'schoice of x, given X, II, and ,u. We of x, since it does not necessarilyrequire a write x = x(, 4,u [I). double coincidence of wants in order to For a symmetricequilibrium we must have exchange. We can also ask how specializa- X = x, or X = x(X,,u, H). Another equilib- tion depends on M. An increasein M shifts rium condition comes from the steady-state the M = M(X, lu, LI) curves to the right but equation M = a1t/(a + ). Since the right- does not affect the X = x(X, ,L, LI) curves. hand side of this equation depends on X, As can be seen from the diagram,when M increases, the result is an increase in X in the nonmonetaryequilibrium, a decrease in 13One can also maximize the right-hand side of (14) directly by setting
a'(x)[VI(x)- VO] + a(x)Vl(x) = 0 14Notice that the pure- and mixed-monetary equilib- where V1(x) is, from (15), ria must be unique, but since fl = 0 implies that both the X= x(X,,u,fl) curve and the M =M(X, fl) curve are upward-sloping, they could intersect more Vl(x) = rV7(x)/{rx + ,1x2[(1 - p)X + /uLH]}. than once, and there could be more than one nonmon- etary equilibrium. Although examples with multiple Manipulating these equations yields the same first- nonmonetary equilibria can be constructed, we rule order condition as in the text, equation (18). this out in the following discussion. VOL.83 NO. 1 KIYOTAKIAND WRIGHT:SEARCH-THEORETIC APPROACH 73
x
1~~~~~~~~~-
XN
XM r=
M 1 A
FIGURE 4. M AND X CURVES, WHOSE INTERSECTIONS DETERMINE EQUILIBRIUM VALUES OF ,U AND X
X in the pure-monetary equilibrium, and no extent of the market"). In any pure-mone- change in the mixed-monetary equilibrium. tary equilibrium, an increase in ,3 shifts the Roughly speaking, an increase in M in the X= x(X,/,u1) curve down and shifts the pure-monetary equilibrium encourages spe- M = M(X, , 1) curve to the right, resulting cialization because producers can more eas- in a decline in X and therefore an increase ily market their specialized output when in specialization and productivity. As ,( -3 oo, there is more money in circulation.'5 x -* 0, and specialization becomes com- Consider now the effect of increasing the plete. As this happens, barter becomes ex- arrival rate in the exchange sector, ,3, which tremely difficult, and the ratio of the volume can be thought of as reducing the frictions of barter to monetary exchange vanishes.16 associated with trade (or "increasing the In the limit, agents almost always sell their production goods for money and use money to buy their consumption goods; as Robert W. Clower (1965) puts it, "money buys goods 15The discussion of the effect of an increase in M and goods buy money; but goods do not buy takes the real money supply to be exogenous, say, because the monetary object is indivisible. Alterna- goods." In this model, however, there is no tively, we can assume that money is divisible and deter- constraint that agents must use cash. To mine the level of real balances endogenously in pure- monetary equilibrium, given nominal balances, using the split-the-surplus condition discussed above. In terms of Figure 4, we need to shift the M = M(X, A, 1) 16The rate of barter exchange is /3(1 - A)2x2, while curve until the gains from trade for commodity traders the rate of monetary exchange is ,3,(l - A)x. The ratio and money traders are equalized. One can show that of these two is (1- ,u)x /I, which vanishes as x -O 0 there exists a unique split-the-surplus equilibrium, and (note that ,u is bounded below by M). For any finite ,3, it implies a finite price level under appropriate param- however, there will always be some direct barter in eter restrictions, as in Section II. equilibrium. 74 THE AMERICAN ECONOMIC REVIEW MARCH 1993 the contrary,it is because the economy has cepting red money and blue money be rIR settled on the use of a generallyacceptable and '1B. Then Bellman'sequations are de- currencythat specializationbecomes prof- scribedby itable, and it is specializationthat inhibits barter. (19) rVo = a(V1 - VO)
IV. Dual Currency Regimes (20) rV=1,38Cx2(U-e+Vo-V,) In this section we take up the possibility of multiple fiat monies. It is motivated by + 13RXmaXTrR(VR - V) the observation that, in some economies, 7TR there seems to be more than one type of currency in simultaneous circulation. For + f/BXmaxrB(VB- V7) instance,it is possible in certainlocations to 7B have both a domesticcurrency and a foreign currency used in exchange, although per- (21) rVR = YR + JCXrHR(U ? + V0 VR) hapsthe formeris generallyacceptable while the latter is only partiallyacceptable. One (22) rVB = YB + A,CXHB(U? + example is that Canadiandollars are often V0 VB) accepted just across the U.S. border, and vice versa, although the foreign currencies These depend on AR and AB, but as above, are not always accepted by domestic resi- the steady-stateconditions can be solved for dents. Furthermore,this situation can per- unique values of AR and ALB,given strate- sist even if the two currenciesdiffer in terms gies and exogenousvalues of MR and MB.17 of rates of return or other intrinsicproper- Our goal is to constructan equilibriumin ties. which both monies circulate, but with dif- In order to study the phenomenon of ferent acceptabilities:1 = iR > fiB > 0. This dual-currencyequilibria, we assume that requires VR > V1 = VB. Now, VR > V1 fol- there are now two colors of fiat money:red lows immediately from 11R = 1. Further- and blue. To simplify the presentation as more, for the case in which YR = YB= 0, much as possible, we only considerthe case (20)-(22) imply that VB= V, if and only if in which specializationis exogenous,and we assume that both monies are indivisible.If we endow all agents with either one unit of (23) iB=AX red money, one unit of blue money, or one real commodityat the initial date, then all where A = (r+flX,AC+63,R)/(r+I,.xlC + agents will alwayshold one and only one of ,BX,UR). Notice that A >1. If IR =1 and these objects at all future dates as well. We rIB= Ax, we have an equilibriumin which give the monies potentiallydifferent intrin- red money is universally accepted while blue sic propertiesby letting YR and YB denote money is only partiallyaccepted. By conti- flow yields or dividends;that is, each money nuity, we can perturb YR and YB without yields yj "utils" to its bearer per unit time destroyingthe equilibrium,as long as IYRI (if yj < 0 then it can be thought of as a and IYBIare not too great. In particular,we storage cost). Also, let the supplies of the can construct equilibria with 1 = fiR > IIB two monies be MR and MB, with MR+ even though YR< YB. In such an equilib- MB < 1, let AR and AB be the proportions rium, both monies circulate, but the high- of traderswith red money and blue money, return asset is less acceptable or less liquid and let ,uc = 1 - AR-R-UB be the proportion of traderswith real commodities. To formulate 17Theway we write Bellman'sequations implicitly the representativeindivid- assumesthat agents never trade one currencyfor an- ual's best-responseproblem, let the proba- other, which is true in equilibriumbecause such a bilities of random commodity traders ac- trade could not possiblymake both agentsbetter off. VOL. 83 NO. 1 KIYOTAKIAND WRIGHT SEARCH-THEORETICAPPROACH 75 than the low-returnasset. That is, the rea- APPENDIX A son why red money is universallyaccept- able, even though it is dominatedin rate of Here we sketch a version of the model return, is that it has liquidityvalue. If the that makes it precise why agents need to spread YB - YR becomes too big, however, trade, without assuming that they cannot this equilibriumcan no longer exist.'8 consume their own output. This is perhaps more satisfying, but it does entail an in- VI. Conclusion crease in notation. The implicationsof this versionof the model are essentiallythe same We have presented a model of exchange as those described in the text. For simplic- in which the difficultyof pure barter leads ity, we consider only the case where a = oo, to a transactionsrole for fiat currency,and but it should be clear how to handle the we have used the model to address several more general model. issues in monetary economics.19Other ap- Supposethere are K types of agents,with plicationscan also be studied in this frame- equal numbers of each type, and K com- work. In Matsuyamaet al. (1993), an ex- modities, where K 2 3. Agents are special- plicit two-countryversion of a model similar ists in production but generalists in con- to the one presentedhere is consideredand sumption,in the followingsense. Each agent used to investigate some issues relating to can produce only some of the commodities international monetary theory. In Steve -to ease the presentation, suppose each Williamsonand Wright(1991), a "lemons" type can produce exactlyone commodity- problem is introduced into the moslel and but has a need to consume differentthings used to illustratethe role of fiat currencyin at different points in time. In particular, helping to overcomethe frictionsassociated after consumingone commoditya consumer with private information. Siandra (1990) realizes a taste or need for a new commod- considers further the relationshipbetween ity drawnat random.That is, the probability specialization and monetary exchange. that the new commoditywill be j is 1/K for Victor B. Li (1991) pursues some issues any j = 1, 2,..., K. A consumerwith a need relatingto externalities,welfare, and policy. for commodityj gets utility U from consum- Ramon Marimonet al. (1990) use a related ing it and no utility from anythingelse, at model to analyze learning. Aiyagari and least until j is consumed and a new taste Wallace(1992) considerseveral other appli- shock is realized. cations. Although there are many unan- Consider a representative agent. After swered questions and much work remains consumption,with probability1/K he needs to be done, we think that these search- the commodityhe can produce, consumes theoretic models have definitely enhanced immediately,and draws a new taste shock, our understandingof the exchangeprocess, while with probability (K - 1)/K he needs in general, and of money, in particular. somethingelse and must attemptto acquire it through trade. Therefore, the expected value of drawinga new taste shock, say VJ, 18Similar argumentscan be used to show that there = + V1(K- are equilibriain which both monies are universally will satisfy Vn (U + V/K 1)/K. acceptableeven thoughone has a higherrate of return, Eventually,our representativeagent needs and equilibriain which one money circulatesbut the something that he cannot produce. When other does not even though one or the other has a he meets a potential trading partner, the higher rate of return. Exampleswith two circulating probabilitythat this partnerneeds the good fiat currencieshave also been constructedby Aiyagari and Wallace(1992). our representativeagent produces is 1/K, 19Inorder to focus on more substantiveissues, we while the probabilitythat this partner also have neglected many of the technical aspects of has the good our representativeagent needs search-basedexchange models, like the possibilityof is 1/(K - 1), since he must have one of the multiple dynamic equilibria, including sunspot and cyclical equilibria (see Kiyotaki and Wright, 1990; commodities other than the one that he Michele Boldrinet al., 1991;Timothy J. Kehoe et al. himself needs. Hence, the probabilityof a 1991). double coincidence is 1/K(K - 1). This im- 76 THE AMERICAN ECONOMIC REVIEW MARCH 1993 plies that the value functionsfor commodity the above equations implies that Vm- V1 - and money traderssatisfy 7 = 0 if and only if
) + r-1 r ,B-'(1-M) (-- n-V M P(1-M X2(U-? ) K(K-1) f3(1-M)X(U-c-0 )
fiM Notice that HM > 0 and IM < 1 if and only + K 7r(Vm-Vi) K if q < 71. Hence, there exists the same set of three l8(1- M) equilibria for any -q in (0,7t). In terms of rV -= (l M)(U-? +vn vm). Figure 2, an increase in qj shifts the best- K response correspondenceto the right. For -j
and Money Growth," International Eco- Pure Theory of Money,"Journal of Eco- nomic Review, May 1992, 33, 283-98. nomic Theory, April 1991, 53, 215-35. Diamond, Peter A., "Aggregate Demand Li, Victor E., "Money in Decentralized Ex- Management in Search Equilibrium," change and the Optimal Rate of Infla- Journal of Political Economy, October tion," mimeo, Northwestern University, 1982, 90, 881-94. November 1991.
9 "Money in Search Equilibrium," Marimon, Ramon, McGrattan, Ellen and Econometrica, January 1984, 52, 1-20. Sargent,Thomas J., "Money as a Medium Iwai, Katsuhito,"The Evolution of Money: A of Exchange in an Economy with Artifi- Search Theoretic Foundation of Mone- cially IntelligentAgents," Journal of Eco- tary Economics," University of Pennsylva- nomic Dynamics and Control, May 1990, nia CARESS Working Paper No. 88-03, 14, 329-74. 1988. Matsuyama,Kiminori, Kiyotaki, Nobuhiro and Jevons,William Stanley, Money and the Mech- Matsui, Akihiko,"Toward a Theory of In- anism of Exchange, London: Appleton, ternational Currency," Review of Eco- 1875. nomic Studies, 1993 (forthcoming). Jones, RobertA., "The Origin and Develop- Oh, Seongwhan,"A Theory of a Generally ment of Media of Exchange," Journal of Acceptable Medium of Exchange and Political Economy, August 1976, 84, Barter," Journal of Monetary Economics, 757-75. January1989, 23, 101-19. Kehoe, Timothy J., Kiyotaki, Nobuhiro and Ostroy, Joseph M. and Starr, Ross M., "The Wright, Randall, "More on Money as a Transactions Role of Money," in Ben- Medium of Exchange," Federal Reserve jamin M. Friedmanand Frank K. Hahn, Bank of Minneapolis Staff Report 140, eds., Handbook of Monetary Economics, 1991; Economic Theory (forthcoming). Amsterdam: North-Holland, 1990, pp. King, Robert G. and Plosser, Charles I., "Mo- 3-62. ney as the Mechanism of Exchange," Siandra, Eduardo, "Money and Specializa- Journal of Monetary Economics, January tion in Production,"Department of Eco- 1986, 17, 93-115. nomics WorkingPaper No. 610, Univer- Kiyotaki,Nobuhiro and Wright, Randall, "On sity of California,Los Angeles, 1990. Money as a Medium of Exchange," Jour- Smith, Adam, An Inquiry into the Nature and nal of Political Economy, August 1989, Causes of the Wealth of Nations, London: 97, 927-54. W. Strahan and T. Cadell, 1776; reprinted, and , "Search for a Theory New York: Modern Library, 1937. of Money," National Bureau of Economic Williamson,Steve and Wright,Randall, "Barter Research (Cambridge, MA) Working Pa- and Monetary Exchange Under Private per No. 3482, 1990. Information," Federal Reserve Bank of and , "A Contribution to the MinneapolisStaff Report 141, 1991.