The Search-Theoretic Approach to Monetary Economics: APrimer

Transcription

1 The Searh-Theoreti Approah to Monetary Eonomis: APrimer by Peter Rupert, Martin Shindler, Andrei Shevhenko, and Randall Wright Peter Rupert is a senior eonomi advisor at the Federal Reserve Bank of Cleveland; Martin Shindler is a dotoral andidate in eonomis at the University of Pennsylvania; Andrei Shevhenko is an assistant professor of eonomis at Mihigan State University; Randall Wright is a professor of eonomis at the University of Pennsylvania. This paper was written while Shindler, Shevhenko, and Wright were visiting the Federal Reserve Bank of Cleveland. Introdution This paper presents some results in monetary theory derived using very simple game-theoreti models of the exhange proess. The underlying model is a variant of searh theory, a framework that has been used extensively in a wide variety of appliations. This approah is well suited to disussing the proess of exhange and money s role in the proess. The approah utilized here is expliitly strategi, in the following natural sense: When I deide whether to aept in trade a ertain objet other than one I desire for my own onsumption for example, money I must onjeture as to the probability that other agents will aept it from me in the future. This evidently ought to be modeled as a game. In searh theory, the type of game to be onsidered is expliitly dynami, and exhange takes plae in real time. Also, the models allow us to fous preisely on various fritions in the exhange proess that might give money a role in an equilibrium, or effiient, arrangement. Among the fritions are these: Agents are not always in the same plae at the same time; long-run ommitments annot be enfored; and agents are anonymous in the sense that their histories are not publi information. Suh fritions are ruial for a logially oherent theory of money; the approah desribed here helps to larify eah one s role. This approah ontrasts with attempts to model the role of money in a ompetitive equilibrium (Walrasian) model a diffiult task that has met with mixed suess at best. In a ompetitive equilibrium model, the exhange proess is not expliitly modeled. That is, agents start with an initial alloation A and hoose a final alloation B so as to maximize utility, subjet to the latter not osting more than the former, but how they get from point A to point B is not disussed. Does some unmodeled agent (maybe the autioneer) make the neessary trades with a pik-up and delivery servie? Or do the agents trade diretly with eah other? Do they trade bilaterally or multilaterally? In real time, or before prodution and onsumption ativity starts? Do they barter diretly or trade indiretly using media of exhange? The standard ompetitive equilibrium paradigm does not address suh questions. Searh models, in ontrast, are designed with exatly these issues in mind and therefore are logial tools for studying monetary eonomis, as we shall illustrate.

2 I. The Basi Model To model anonymous trade, it is natural to start with a large number of agents formally, we assume a ontinuum. For simpliity, we assume these agents live forever and disount the future at rate r. Thereis a ontinuum of indivisible onsumption goods. To generate gains from trade, we need to assume that agents are speialized. There are many ways to do this, but an easy one is to assume that eah agent i is able to produe just one type of good. The unit prodution ost for any agent is. For onveniene, we assume that these goods annot be stored and so must be onsumed immediately after they are produed. Obviously, this means that onsumption goods annot serve as media of exhange, allowing us to highlight the role of money. To make trade interesting in the model, we need to assume that tastes are heterogeneous. Again, there are many ways to do this, but for simpliity we assume the following: First, given any two agents i and j, write iwj to mean i wants to onsume the good that j produes in the sense that i derives utility u from onsuming what j produes if iwj, and he derives utility from onsuming what j produes otherwise. Then, for any randomly seleted agents, we assume prob iwi, prob jwi x,andprob jwi iw j y. Thefirst assumption, prob iwi, means that no agent ever wants to onsume his own output (whih is why they trade). The seond assumption parameterizes the extent of the basi searh frition: The smaller x is, the lower the probability that a random trader has what you want. However, the third assumption is the important one, sine it parameterizes Jevons (875) famous double oinidene of wants problem: The smaller y is, the lower the probability that a trader who has what you want also wants what you have. Besides the onsumption goods already mentioned, there is another objet alled money, whih onsists of an exogenously fixed quantity of M indivisible units of a storable objet (of ourse, money must be storable to be useful). Holding money yields utility γ: If γ, money pays a dividend, like many real assets, and if γ, then money has a storage ost; γ de- sribes the ase of pure fiat money. Although this last ase may be the most interesting, for generality we allow γ. Initially, one unit of money is randomly alloated to eah of M agents. Although we will relax this later, for now we assume that agents holding money annot produe (one way to motivate this is to assume that after produing, you need to onsume before you an produe again). Thus, no one an ever aquire more than one unit of money, and so an agent always holds either or unit of money. To simplify the presentation, we do not allow agents to freely dispose of money, but this is never binding exept in one ase mentioned below. We now desribe the trading proess. Rather than assuming a entralized (Walrasian) market, here the agents must trade bilaterally. The simplest way to model this is to assume that they meet aording to a pairwise random mathing proess. Upon meeting, a pair deide whether to trade, then part ompany and reenter the mathing proess. Let α denote the (Poisson) arrival rate in the mathing proess that is, the probability of meeting someone in a given unit of time. 2 For reasons disussed later, we assume the history of any agent s past meetings and trades is not known to anyone else. We want to analyze agents individual trading strategies. An agent obviously should never aept a good in trade if he does not want to onsume it, sine goods are not storable. Whenever possible, agents should barter for a good they do want to onsume (the ase of a double oinidene). What needs to be determined is whether an agent should trade goods for money and money for goods. Let denote the probability that the representative agent trades goods for money, and let denote the probability that he trades money for goods. These must satisfy the equilibrium onditions given below. We will say that money is used as a medium of exhange, or irulates, if and only if. Let and be the value funtions (lifetime, disounted, expeted utility) of agents with or units of money. Sine we onsider only stationary and symmetri equilibria, j does not depend on time or on the agent s name, only his money inventories. If we think of time as proeeding in disrete periods of length τ, we an alulate the payoff of holding money as follows: The probability of meeting anyone during this period is approximately ατ by the Pois- Several notions of speialization in the literature are speial ases of this model. For example, in Kiyotaki and Wright (99) or Aiyagari and Wallae (99), there are N goods and N types of agents, where type n produes good n and wants good n mod N. Then x N, and y if N 2, while y if N 2. Alternatively, in Kiyotaki and Wright (99, 993) and muh of the related literature, the events iw j and jwi are independent, and so y x. 2 It is the bilateral rather than the random mathing assumption that is important. Corbae, Temzelides, and Wright (2) show how to redo the model, allowing agents to hoose endogenously whom they meet, rather than meeting at random. Their model shares the basi insights disussed here, although it is ompliated by the need to determine equilibrium meeting patterns as well as equilibrium trades.

3 2 son assumption. 3 If the person you meet an produe (meaning, in the version of the model that we are onsidering here, he does not have money), whih ours with probability M, and you want what he an produe, whih ours with probability x, and you both want to trade, whih ours with probability in equilibrium, then you trade, onsume, and ontinue without money, for a total payoff of u. In all other events (you meet no one, you meet someone with a good you do not want, et.), you simply ontinue with your money, for a payoff of. In all events, you also get γτ from storing the money. Hene, ατ rτ M x u ατx M γτ o τ where o τ is the approximation error assoiated with the Poisson proess and hene satisfies o τ τ as τ. Rearranging, we have rτ ατ M x u γτ o τ Dividing by τ and taking the limit as τ, we arrive at the ontinuous-time Bellman s equation, () r αx M u An analogous argument implies that the value funtion for an agent without money satisfies (2) r αxy M u γ αxm The first term in this expression represents the gain from a diret barter trade, while the seond represents the gain from trading goods for money with probability. Notie that you an only barter when there is a double oinidene of wants and the other person has no money. II. Equilibrium Define the net gain from trading goods for money by, and the net gain from trading money for goods by u. If we normalize αx to redue notation (whih we an always do with no loss of generality by redefining units of time appropriately), we have: (3) (4) M M y u ru γ r M y u r γ r The equilibrium onditions for and are (5) j as j Notie that j depends on, so to see if some andidate and onstitute an equilibrium, one simply inserts the j and heks equation (5). Consider first the ase γ (fiat money). This implies for all parameter values, so we always have. Also, is equal in sign to ˆ,where r ˆ M y u (6) M u Notie that is always an equilibrium: Sine ˆ, implies ˆ, whih implies. Thus, satisfies the equilibrium ondition. Naturally, there is an equilibrium in whih no one aepts fiat money. However, if M y u r M y then ˆ, whih means is an equilibrium as well. So there is also an equilibrium where fiat money irulates as long as is not too big. 4 Finally, if ˆ, there is also a mixed-strategy equilibrium in whih ˆ. In this ase, if other agents aept money with exatly the right probability ˆ, you are indifferent as to aepting or rejeting it, so randomizing is an equilibrium. The above model is essentially that of Kiyotaki and Wright (993), exept that they assume inaddi- tion to γ. This implies that three equilibria neessarily exist,,, and y. In the mixedstrategy equilibrium, money is aepted with the same probability as a good (sine y is the probability of a double oinidene), whih makes you indifferent. When, money must have a stritly greater probability of being aepted than a barter trade for you to be indifferent about aepting money, beause you must inur the prodution ost to get the money. More generally, when γ, we have to determine endogenously; for example, if γ is large, then agents may prefer to hoard rather than spend their money. The results for the general ase are summarized as follows: 3 That is, the probability of meeting one person in a period of length τ is ατ o τ, and the probability of meeting more than one is o τ, where o τ satisfies o τ τ as τ. 4 Alternatively, for a given, we an say that r and M must be relatively small for a monetary equilibrium to exist (agents must be patient and money not too plentiful).

4 M M M M y 3 Proposition. There are five types of equilibria, and they exist in the following regions of parameter spae:. and is an equilibrium iff r r 2 2. and is an equilibrium iff r r 3 3. and is an equilibrium iff r r r 2 4. and is an equilibrium iff r 3 r r 4 5. and is an equilibrium iff r r r 4 where the ritial values of r are given by γ r M y u u γ y u r 2 u γ y u r 3 γ u r 4 These are the only (steady-state) equilibria. Proof: See the appendix. We haraterized the regions where the different equilibria exist in terms of r, but we ould have used some other parameter, suh as. Routine algebra implies r r 2 r 3 r 4. Also, note that our assumption of no free disposal is never binding, exept possibly when, and even then only if γ. 5 More importantly, there are equilibria where andγ ; that is, agents value money and use it as a medium of exhange despite its storage ost. We will have more to say about the eonomis underlying the above results in the next setion, after we introdue a slight variation on the model, sine it will be interesting to ompare the two versions. III. Alternative Speifiation A key assumption in the above model is that agents holding money annot produe; this is what prevents them from aquiring more than a single unit of money. The fat that agents hold either or unit of money is what makes the model so tratable (see below). Although the assumption that agents holding money annot produe is ommon in the literature, it has some undesirable impliations. For example, if two agents with money meet and there is a double oinidene of wants, they annot trade. A related impliation is that as M inreases, the produtive apaity of the eonomy neessarilydereases, whihmakesit diffiult to interpret the effets of hanges in the money. So here we present an alternative model, first disussedby Siandra (993, 996), where agents with money an produe, and we simply impose the ondition that agents an store, at most, one unit of money. The first issue to be resolved is, what happens in a double oinidene when you have money and the other person does not do you barter or pay with ash? 6 We resolve this by allowing agents to play the following simple game: First, with probability β the agent with money is hosen and with probability β the agent without money is hosen to propose either a barter or a ash transation (in priniple they ould also propose not to trade at all, but we ignore this option, whih will always be dominated by proposing barter). Seond, the other agent responds either by aepting, whih exeutes the proposal, or rejeting, whih implies they part ompany (figure ). A strategy for the agent with j units of money, j or, is denoted as ψ j, whih equals the probability that he proposes barter (and so ψ j is the probability he proposes ash). Proposition 2. In a double-oinidene meeting between an agent with and an agent without money, generially the unique subgame-perfet equilibrium in pure strategies is ψ ψ. Proof: See the appendix. Having resolved the ambiguity that arises when both barter and ash are available, we an now derive the Bellman equations. Again, let time proeed in disrete 5 To be preise, we should say what agents do after disposing of their money. We assume here that they annot trade, as they annot produe. Hene, agents will dispose of money and drop out of the trading proess iff. Sine it is easy to see that in any equilibrium and in any equilibrium with, the only ase where disposal ould potentially our is, whih implies γ r. Hene, agents dispose of money if and only if and γ. 6 This is the only ambiguous ase; every other meeting has only one feasible transation (that is, if you enounter a double oinidene and have no money, barter is the only option). The issue did not ome up in the previous setion, beause agents with money annot barter.

5 ατxy ατxy M y ατ ατm y M M y y y M y 4 FIGURE Game Tree Aept (u, u ) Barter ψ Rejet (,) Cash ψ Aept (, ) Buyer proposes β Rejet (,) β Aept (u, u ) Seller proposes Barter Cash ψ ψ Rejet Aept (,) (, ) Rejet (,) periods of length τ. Then (7) rτ ατxy u ατ x u x γτ o τ (8) rτ ατxy u ατm x x o τ where we temporarily reintrodue αx to failitate omparison to equations () and (2) in the previous model. Observe, for example, that now a money holder barters every time he enounters a double oinidene, whih ours with probability ατxy. Indeed, he uses money only when he enounters a single oinidene, whih ours with probability ατ y x, and the other agent does not have money and they both agree to trade, whih ours with probability. Rearranging, we let τ and normalize αx as before, to write the ontinuous-time Bellman equations for the alternative model (9) () r y u r y u u M Although the value funtions are different aross the two models, we ompute j and define equilibrium exatly as in the last setion. The results are as follows. Proposition 3. In the alternative model, where agents with money an produe, there are five potential types of equilibria and they exist in the following regions of parameter spae:. and is an equilibrium iff r ˆr 2 2. and is an equilibrium iff r ˆr 3 3. and is an equilibrium iff ˆr r ˆr 2 4. and is an equilibrium iff ˆr 3 r ˆr 4 5. and is an equilibrium iff ˆr r ˆr 4 γ

6 M M y y 5 where the ritial values of r are given by γ ˆr ˆr 2 γ u ˆr 3 γ ˆr 4 γ u u u These are the only (steady-state) equilibria. Proof: The proof is analogous to that of Proposition and so is omitted for brevity. FIGURE 2 Equilibria in γ r)-spae When Money Holders Cannot Produe o = = r r 4 r 3 o =, o = or o (,) = o = = = o = = o = (,) = ( M)( y )(u ) ( M) y(u ) [ M+( M) y ](u ) The regions of γ r spae where the different equilibria exist in the two models (those where money holders annot produe and those where they an produe) are shown in figures 2 and 3. The same five types of equilibria exist in both models; the regions where they exist are similar but quantitatively different unless y, of ourse, sine the models are idential when there is no barter. In either ase, when γ is very low the only equilibrium is one in whih no one aepts money; if γ is very high, the only equilibrium is one in whih no one spends it. Hene, money irulates if and only if its intrinsi properties are not too bad or too good. Also, both models inlude a region or r 2 r γ FIGURE 3 Equilibria in γ r -Spae When Money Holders Can Produe o = = r ( M)( y )(u ) M( y )(u ) r 4 o =, o = or o (,) o = = = r 2 o = = = or (,) o = = where the unique equilibrium is, as well as other regions where there are multiple equilibria: In one region, we must have, but an be,, or between and ; in another region, we must have, while an be,, or between and. 7 The models differ in that it is atually more diffiult to get money to irulate when money holders an produe; the potential region where is larger when they annot. Intuitively, agents with money are more willing to spend it when they annot produe, sine doing so allows them to barter. If γ, the differenes between the two models are espeially stark: When money holders annot produe, there are always three equilibria,, y,and ; in the other model there are two, and. The intuition is as follows: When money holders an produe, there is no ost assoiated with aquiring money when we set γ, so if there is a stritly positive probability of money being aepted, you should always aept it. This is not true when money holders an- 7 It is well known that there is a strategi omplementarity in the deision to aept money,, but it is less well understood that the same is true of. For some parameters, an agent is more willing to spend money if he believes that others will do the same. This only ours when γ. r 3 r γ

7 M M M M y 6 not produe, beause even when γ, aepting money involves the opportunity ost of giving up your barter option. The version of the model in whih agents with money annot produe is easy to motivate by saying that one must onsume before produing, sine this leads naturally to the result that agents in equilibrium always hold either or unit of money. However, the alternative version where money holders an produe also seems more natural in some respets. For example, when two agents with money meet and there is a double oinidene, they trade. Of ourse, this version does require assuming diretly that agents an only store or units of money. There is not neessarily a right or wrong model, and the hoie should be ditated by how well it addresses a ertain question and how tratable it is in any given appliation. I. Welfare the exhange proess even though money rowds out barter. FIGURE 4 Welfare as a Funtion of M rw rw S rw K rw K rw S rw S In this setion, we disuss welfare, defined by W M (average utility). For the purpose of disussion, we also set γ. Using straightforward algebra, in the two models we have rw K rw S y M y M u u rw K O M K M S M M where the supersript K indiates the ase where money holders annot produe and the supersript S indiates the ase where they an. Notie first that when we ompare two equilibria in either model, other things being equal, W is greater in the equilibrium with the higher. This simply says that the more aeptable money is, the more useful it is. The next thing to notie is that aross the two models, given any we have W S W K with strit inequality as long as y and M ; not surprisingly, agents are better off if we allow money holders to produe. We now want to fous on equilibrium with and onsider maximizing W with respet to M. There- sult for the model in whih money holders an produe is M 2. The intuition is simple. Money is useful beause it failitates trade every time an agent i with money and an agent j without money meet and iwj holds. To maximize the frequeny of suh meetings, we should have half the agents holding money and half of them not: M 2. For the other model, the welfaremaximizing poliy is M 2y 2 2y if y 2,andM if y 2. Hene, the optimal M is lower in this model, simply beause when money holders annot produe, inreasing M rowds out barter. Still, for small y the welfare-maximizing M is positive beause it failitates This disussion ignores the fat that is not an equilibrium for all parameter values. If γ, it is easy to see that in either version of the model, isan r equilibrium if and only if M M y u.the true optimal poliy, then, is the minimum of M and the values given above. In figure 4, we depit welfare in eah type of equilibrium by the urves W j, W j,orw j, where money is aepted with probability,, or. The supersript j K or S refers to the two different models. The urves are drawn only for values of M suh that the equilibria exist. Figure 5 shows welfare as a funtion of y, given that we set M to its welfaremaximizing level (whih depends on y). These figures illustrate various properties, inluding: W is always higher when money holders an produe; and W inreases with aross equilibria in either model.. Essentiality of Money At this stage, it is instrutive to highlight the role of various fritions in the model, to understand what makes money essential. Following Hahn (965), we say money is essential if it allows agents to ahieve alloations they annot ahieve with other mehanisms that also respet the enforement and information on-

8 y 7 FIGURE 5 Welfare as a Funtion of y (optimal M) rw, M rw K M S M M K rw S M S straints in the environment. 8 First, we need some sort of double oinidene of wants problem. For this it is important that not everybody an trade multilaterally. For example, onsider an eonomy with three agents, where eah agent of type n 2 3 produes good n and wants good n mod 3. If all agents meet at the same plae and are able to make multilateral trades, it is feasible for eah agent to produe and onsume in the period: Agent 2 produes for agent, agent produes for agent 3, and agent 3 produes for agent 2. Our restrition to bilateral meetings, given speialized tastes and tehnology, is merely a onvenient way to generate a double-oinidene problem. In the three-agent example, in every bilateral meeting one agent wants what the other produes, while the other does not. Seond, even given a double-oinidene problem, for money to be essential it is also important that agents annot ommit to long-term agreements. Consider the following redit arrangement: produe for anyone you meet who wants your good. This arrangement resembles redit in the sense that agents reeive onsumption today in exhange for nothing but a promise to repay someone in kind at a future date. It is also obviously an effiient arrangement; that is, it generates the maximum possible expeted utility, say W u r, given the normalization αx, where the subsript on W stands for redit. If agents ould y ommit to this arrangement ex ante, they would all agree to do so, and there would be no need for money. Clearly, an imperfet ability to ommit to future ations is important if money is to have an essential role. However, even in the absene of expliit ommitments, ooperative agreements like the redit arrangement an sometimes be enfored by reputational onsiderations if individual ations are publi information. Thus, onsider the arrangement: produe for anyone who wants your prodution good as long as everyone else has done so in the past; as soon as someone deviates from this, trigger to plan X, where plan X is to be determined. Of ourse, plan X must be self-enforing (that is, it must be an equilibrium), and we want the outome of plan X to be suffiiently unpleasant to keep agents from deviating from the effiient arrangement. We will assume here that plan X is to trade if and only if there is a double oinidene of wants, whih generates expeted utility W b y u r, where the subsript b stands for barter. 9 It is in individuals self-interest not to deviate from the redit arrangement in whih they are supposed to produe if and only if W W b, whih simplifies to r r u. As always, if agents are suffiiently patient, the threat of triggering to pure barter supports the effiient outome, and again money is not essential. Of ourse, this assumes that agents trading histories an be observed publily; otherwise, it is not possible for agents to use trigger strategies. When trading histories are private information, the only sustainable outome without money is pure barter. But if there is money, we an do better than pure barter, even when trading histories are pri- 8 For a reent treatment of this problem, see Koherlakota (998). 9 We do not trigger to autarky beause we assume that if two agents want to barter without it being observed, they an; so the worst possible equilibrium is the one where none but double-oinidene trades our. Nothing muh hinges on this; a similar message holds if we an trigger to autarky, and it is, in fat, easier to support redit-like arrangements by triggering to autarky. If the number of agents was small, then even if agents did not observe all other agents histories but only their own, we ould potentially support the effiient arrangement by the following strategy: If ever you diretly observe someone deviate (by not produing for you when you would like him to), stop produing for anyone else. This would set off a hain of agents who observe deviations and would eventually lead the eonomy into autarky. With a large number of agents, however, if I fail to produe for you, there is zero probability that in the future I will meet you or I will meet someone who has met you, and so the hain will never get bak to me. So with a large number of agents, to support redit it does not suffie to have agents observe (only) their own histories. See Araujo (2) for more disussion.

9 M T T 8 vate. Notie that monetary exhange generates lower welfare than the redit arrangement, although higher than pure barter. Money does not do as well as redit beause of the random-meeting tehnology and beause money holdings are bounded, leading to some meetings where I want your good and you do not want mine, but either I have no money or you already have money. In these meetings, monetary exhange will not work, while redit ould work as long as trading histories are publily observable. Even if we relax the upper bound on money holdings, the fat that money holdings are bounded below by zero means that money annot do as well as redit in a random-mathing environment. We onlude that money has an essential role in the model for three reasons: the double-oinidene problem, the lak of ommitment, and private information on trading histories. For an extended disussion of these issues, see Koherlakota (998, 2) and Koherlakota and Wallae (998). I. Pries This setion provides an extension in whih the assumption of indivisible goods is relaxed, although money is still indivisible and so agents will always have either or units of money. Following Shi (995) and Trejos and Wright (995), we will use bargaining theory to determine pries endogenously. For simpliity, we set y, so there is no diret barter. Given that goods are perfetly divisible, let u q be the utility of onsuming q units of one s onsumption good and q the disutility of produing q units of one s prodution good. We assume u, u, u q, q, u q, and q, for q, with at least one of the weak inequalities strit. For future referene, we define q by u q q. Also, there is a ˆq suh that u ˆq ˆq. When a buyer meets a seller who an produe the right good, they bargain over how muh q will be exhanged for the buyer s unit of money, implying a nominal prie p q. Otherwise, the model is exatly idential to that in the previous setion. Letting and denote the value funtions and taking q Q as given, the generalizations of the dynami programming equations an be expressed as () (2) r u Q r M Q These an be easily solved for Q and Q.Taking Q and Q as given, q will solve a bargaining problem. In equilibrium, of ourse, q Q. The bargaining model an be formulated in several different ways. A typial approah is the generalized Nash bargaining solution, (3) q argmax u q Q Q q θ θ where θ is the bargaining power of the buyer and T j is the threat point of the agent with j units of money. Also, the maximization is subjet to u q and q. Here we will set θ 2,andT T 2. 2 FIGURE 6 Monetary Equilibrium in the Divisible-Goods Model q q q e q * q e(q) q f(q) g(q) The bargaining solution in equation (3) defines a mapping q q Q from ˆq into itself. That is, if other agents are giving Q units of output for one This assumption makes it irrelevant whether we use the version in whih money holders an produe or the one in whih they annot, sine they are idential when y. Shi (995) and Trejos and Wright (995) analyze the ase with y, but only for a speial bargaining solution and only for the model where money holders annot produe. Rupert, Shindler, and Wright (forthoming) onsider the general ase. 2 It is also ommon to set the threat points equal to ontinuation values: T j j. Both an be derived from an underlying strategi model; see Osborne and Rubinstein (99) for the bargaining theory. Q

10 M M 2M M 9 unit of money, then a partiular pair bargaining bilaterally will agree to q q Q. An equilibrium is a fixed point, q q Q. In general, we must be areful with the onstraints on the bargaining problem: When y, the onstraints are never binding in equilibrium. However, if y the onstraints may bind; therefore, it is instrutive to proeed allowing for the possibility of binding onstraints. The onstraints an be rewritten q D Q and u q D Q,where D Q Q Q. The former onstraint is satisfied if and only if q f Q, and the latter is satisfied if and only if q g Q, for inreasing funtions f andg. As figure 6 shows, both f and g go through the origin in the Q q plane, and g lies below f and below the 45 line for all Q ˆq.Also,f rosses the 45 line at a unique q ˆq. Hene, our searh for equilibria an be onstrained to the interval q. The first-order ondition for an interior solution to equation (3), taking Q and Q as given, is Q q u q u q Q q This defines a funtion q e Q, also shown in the figure. It too goes through the origin and intersets the 45 line at a unique point q e. Hene, q q Q an be written as q Q min e Q f Q for all q q e, and q Q max e Q g Q for all q q e q.for q q, it does not really matter how we define q Q, whih neessarily is below the 45 line, and we set it equal to q Q. This makes it lear that for all parameter values, q q Q has exatly two fixed points: a nonmonetary equilibrium q, and a unique monetary equilibrium q q e. 3 One important property of monetary equilibrium (that ontinues to hold even if y ) is the following. Reall that q is defined by u q q.thenitis easy to show that e q q and, therefore, q e q, as seen in figure 6. This is signifiant beause q is the effiient outome. More preisely, if we define welfare as before, W M,aftersimplifiation we have (4) rw M u q q Hene, W is maximized with respet to q at q. The result q e q says that in equilibrium q e is too low or, equivalently, the prie level is too high. 4 The eonomi intuition for this result is straightforward. If a seller ould turn the proeeds from his prodution into immediate onsumption, as in a stati or fritionless model, then he would produe until u q q. But in a monetary exhange, the proeeds from prodution onsist of ash that an only be spent in the future when an opportunity to buy omes along. Sine he disounts the future, a seller is only willing to produe less than the amount that satisfies u q q. Indeed, to verify that fritions are driving the result, observe that when r orα, we have q e q. Another question to ask is how q depends on M. One might expet q e M, but it is atually possible to have q e M forsmallm (at least if r is also small). The explanation is that when M is lose to zero, very little trade ours. In this ase, inreasing M inreases the frequeny of produtive meetings between buyers and sellers, whih in turn inreases both and. The net effet on the bargaining solution an be a higher q. However, there is some threshold ˆM an be sure that the value of money eventually begins to fall as M inreases. We an also ask how M affets welfare. It is lear that if a planner an hoose both M and q to maximize 2 suh that qe M forallm ˆM, sowe W, he will hoose M 2 and q q. This is beause M 2 maximizes the number of trades (just as in the previous setion when y ), and q q maximizes the surplus that results from eah trade. However, if the planner an hoose only M and q is determined in equilibrium, then the value of M that maximizes W satisfies the first-order ondition W M s u q M q u q q 2 q e M As the seond term is negative at M, the solution is M o 2. This illustrates the trade-off between providing liquidity (making trade easier), and reduing the value of money (lowering the surplus from eah trade). Reduing the value of money redues welfare beause, as we have already established, q is too low in equilibrium. 3 If y, one an show that for large r there are no monetary equilibria, while for small r there are multiple monetary equilibria. 4 This is true even though bargaining is bilaterally effiient in the sense that the agreement is on the Pareto frontier in eah exhange, taking as given the value of Q that prevails in other exhanges. The point is that all agents would be better off (in an ex ante sense) if they ould get everyone to ommit to inreasing q. A stronger result is atually true: Not only is q e too low aording to the ex ante riterion W, it is also too low aording to the ex post riteria and. That is, buyers and even sellers would be better off if q were bigger. The result that q e q also an be shown for models where agents an hold any amount of money; see Trejos and Wright (995, pp. 33 4).

11 ατx M M M M M M M M 2 II. Dynamis We previously foused on steady states; in this setion, we onsider dynami equilibria in the model with divisible goods. Although one ould do things more generally, 5 we restrit attention to the ase where θ in the bargaining solution (equation [3]). This is equivalent to assuming that agents with money get to make take-it-or-leave-it offers. Hene, they will demand the quantity that satisfies (5) q sine this is the most a produer would give to aquire urreny. In the previous setion s model, this immediately implies by virtue of equation (2), so we have q. Inserting this into equation () we have (6) r q u q q An equilibrium is a q that solves equation (6). Although the model with take-it-or-leave-it offers is interesting in its own right, mainly beause of its simpliity, we use it here to study dynamis. We need to rederive the Bellman equations without limiting our attention to steady state. First, write the disrete time value of holding money at t as t ατ rτ x u t τ t τ γτ o τ where we have reintrodued the utility of holding money γ. Rearranging, we have r t α x u t τ t τ t τ t γ o τ τ τ Taking the limit as τ (7) r t αx, we arrive at γ t u t t where indiates the time derivative. A similar derivation yields (8) r t M t t q t Note that beause of equation (5), the first term on the right side of equation (8) is. Heneforth, we will omit the time argument t when there is no risk of onfusion. Then an equilibrium is a bounded time path for eah of the variables q satisfying (7), (8), and (5) at every point in time. As in steady-state analysis, we want to eliminate the value funtions from (5). To this end, subtrat (7) and (8) to obtain r u q Equation (5) implies q q in the previous equation yields (9) q F q r q q q γ. Inserting this u q γ Any bounded, non-negative path for q solving the above differential equation onstitutes an equilibrium. FIGURE 7 Dynami Equilibria γ q q= F( q) q L qh q To keep the number of ases manageable, assume γ. Figure 7 shows equation (9) for this ase. When γ and large, there are no monetary equilibria; when γ but not too big, there are two monetary steady states, q L and q H. Clearly, q L is stable while q H is unstable. Hene, in addition to the steady states, the set of equilibria is as follows: For all q q L, there is an equilibrium that onverges monotonially up to q L ;forallq q L q H, there is an equilibrium that onverges monotonially down to q L. The former 5 See Coles and Wright (998) and Ennis (999).

12 2 (latter) equilibria are haraterized by deflation (inflation), due simply to beliefs. If agents expet pries to hange, this an be a self-fulfilling prophey. When γ, q L oaleses with the nonmonetary equilibrium q. Hene, in addition to the steady states, the set of equilibria inludes paths starting at any q q H andonvergingdowntoq. III. Extensions and Related Literature In this setion, we provide a short overview of some extensions to and appliations of the above models in the literature. We will disuss some of the papers briefly; others we will merely mention. Our intent is to provide a bibliography rather than a review, so that the interested reader at least knows where to look. 6 The basi searh-theoreti monetary model an be generalized along several dimensions. Speialization is endogenized in more detail in Kiyotaki and Wright (993), Burdett, Coles, Kiyotaki, and Wright (995), Shi (997b), and Reed (999), for example. More general prodution strutures are inorporated in Kiyotaki and Wright (99) and Johri (999). Long-term partnerships, in addition to one-time exhanges, are onsidered in Siandra (996) and Corbae and Ritter (997). arious extensions of bargaining are onsidered by Engineer and Shi (998, 999), Berentsen, Molio, and Wright (forthoming), and Jafarey and Masters (999). We already mentioned redit in setion II, and there are several papers that attempt to have money and redit in the model at the same time: Koherlakota and Wallae (998) assume histories are imperfetly observed over time; Cavalanti and Wallae(999a,b) and Cavalanti, Erosa, and Temzelides (999) assume that the histories of only some agents are observed; and Jin and Temzelides (999) assume only histories of loal neighbors are observed. Following Diamond (99), some papers have bilateral redit and money, with repayment (expliitly or impliitly) enfored by ollateral. These inlude Hendry (992), Shi (996), Shindler (998), and Yiting Li (forthoming). Many papers deal with ommodity money, as opposed to fiat money. The basi idea is to determine endogenously whih of many possible goods beome media of exhange. Kiyotaki and Wright (989) onsider a version of the model where type i onsumes good i and produes i, with N 3 types. The goods are all storable, although at different osts. It is shown that goods with low storage osts may or may not ome to serve as money, depending on parameter values as well as whih equilibrium the eonomy is in; that is, there an be equilibria in whih high-storage-ost goods are used as money. Aiyagari and Wallae (99, 992) generalize this to N types and onsider several appliations. Wright (995) extends the model to allow agents to hoose their type. Renero (994, 998b, 999) onsiders several extensions of the framework. Among other things, he shows that equilibria in whih goods with high storage osts serve as money an have good welfare properties, perhaps surprisingly (the intuition is that there is more trade in suh equilibria). Other related papers inlude Kehoe, Kiyotaki, and Wright (993), Cuadras-Morato and Wright (997), and Renero (998a). There is also a literature on searh models with private information. Williamson and Wright (994) assume there is unertainty onerning the quality of goods. In suh an environment, a generally reognizable money has the potential role of mitigating the informational fritions and induing agents to adopt strategies that inrease the probability of aquiring high-quality output. So money may be valued even if the double-oinidene problem vanishes (that is, even if y ). Trejos (997) presents a simplified version of the model (essentially by setting y ), whih allows him to obtain analytial solutions to the model. Kim (996) endogenizes the extent of the private information problem. Cuadras-Morato (994) and Yiting Li (995b) use a version of this model to study ommodity money. All the above papers assume indivisible goods. Trejos (999) ombines private information with divisible goods and bargaining. elde, Weber, and Wright (999) and Burdett, Trejos, and Wright (forthoming) use ommodity money models with private information to study some issues in monetary history, inluding Gresham s Law. Other related papers inlude Wallae (997b), Williamson (998) and Katzman, Kennan, and Wallae (999). Several papers attempt to model poliy as follows: There is a subset of agents who are subjet to the same searh and information fritions as everyone else 6 A large body of work in searh theory is tangentially related to the approah to monetary eonomis presented here. This brief review annot disuss all suh work, but we do want to mention Diamond (982); although there is no money in that model, it is in some respets quite similar to that in setion II. The version in Diamond (984) does have money, but it is imposed through a ash-in-advane onstraint; so although it in some ways resembles the framework presented here, its spirit is quite different. See also Diamond and Yellin (985). We also mention Jones (976) as well as the extensions by Oh (989) and Iwai (996), whih attempt to build a model along lines similar to those presented here; see Ostroy and Starr (99) for a review of this and related work. Other general disussions that onentrate more on models like the ones in this paper inlude Wallae (996, 997a).

13 22 but at olletively. Call these agents government agents. The idea is to see how government agents exogenously speified trading rules affet the endogenously determined equilibrium behavior of other (private) agents. Papers in this group inlude itor Li (994, 995a), Ritter (995), Aiyagari, Wallae, and Wright (996), Aiyagari and Wallae (997), Li and Wright (998), Green and Weber (996), Wallae and Zhou (997), and Berentsen (2). For example, in itor Li (994, 995a, 997), government agents an tax money holdings when they meet private agents. A key result is that taxing money holdings may be effiient. The reason is that in his model (whih also endogenizes searh intensity) there is too little searh by agents holding money, for standard reasons. Taxing them inreases their searh effort, and this an improve welfare. The mathing model seems a natural one for studying issues related to international monetary eonomis. For example, one an think about parameterizing differenes in the effiieny of eonomi ativity as well as degrees of openness aross ountries in terms of arrival rates. Among the first authors to analyze this in a model with multiple urrenies and multiple ountries are Matsuyama, Kiyotaki, and Matsui (993). They find that several types of equilibria an arise, inluding those in whih one urreny irulates only loally while another emerges as an international urreny; they find other equilibria in whih all urrenies are universally aepted. They ompare these equilibria in terms of welfare. Zhou (997) extends their model to study urreny exhange. These models assume indivisible goods. Trejos and Wright (999) endogenize pries using divisible goods and bargaining. Other examples of models with multiple urrenies inlude Kultti (996), Green and Weber (996), Craig and Waller (999), Peterson (2), and Curtis and Waller (2). Some papers onsider intermediation (in the form of middlemen, for example) as an alternative (or sometimes in addition) to money. An early paper to expliitly onsider intermediation in a searh model without money is Rubinstein and Wolinsky (987). They generate a role for middlemen by speifying exogenously a set of agents who may have a more effiient tehnology for finding buyers than sellers have for finding buyers. Yiting Li (998) is a very different model, in whih private information about the quality of onsumption goods ombined with the existene of a ostly quality verifiation tehnology give rise to a role for intermediation. In Shevhenko (2), intermediation arises from inventory-theoreti onsiderations: Middlemen keep a stok of several goods on hand to inrease the probability that a random buyer will find something he likes. The Shevhenko and Li papers also endogenize the number of intermediaries in the eonomy by means of a free entry ondition. See also Camera (2), Camera and Winkler (2), and Hellwig (2). The framework s most important reent extension is perhaps the relaxation of the strong assumptions on how muh money agents an hold typially, zero or one unit of money, as we assume above. Models that onsider suh an extension an be an order of magnitude more ompliated, but they are obviously more realisti and generate many interesting new results. They are also apable of addressing more traditional poliy questions, suh as the optimal rate of inflation, whih are diffiult to study in models where agents hold only zero or one unit of money. Suh a model is ontained in Molio (999), who allows agents to hold any nonnegative amount of money and to bargain over the quantity of goods as well as the amount of money that is traded in eah bilateral meeting. Beause of the model s omplexity, however, it an only be solved numerially. The numerial analysis generates interesting results on poliy, welfare, the equilibrium distribution of pries, and other issues. Green and Zhou (998b) and Zhou (999) also present a model with divisible money, where several results an be derived analytially. Unlike Molio, they assume that sellers set pries and annot observe buyers money holdings. Although suh an environment ould still have equilibria with a distribution of pries, they only look for equilibria where all sellers set the same prie. Several interesting results emerge, inluding the existene of multiple (indeed, a ontinuum of) steady states, indexed by the nominal prie level. Also, there an be an endogenous upper bound for money holdings: Agents with suffiient ash will not aept more. (Molio s model an also generate this.) Related referenes are Green and Zhou (998a), Zhou (998), Camera and Corbae (999), Taber and Wallae (999), Berentsen (999a,b), and Roheteau (999). Shi (997a) presents an analytially solvable model with perfetly divisible money, but his model is quite different in some dimensions from the rest of the literature. 7 There are other appliations and extensions that annot all be onsidered in this brief review. However, 7 In Shi s model, the deision-making unit onsists of a family with a large number of members (formally, a ontinuum), rather than a single individual. In this framework, family members share money holdings between periods, so every family starts the next period with the same amount of money by the law of large numbers. Appliations and extensions of this model are ontained in Shi (998, 999).

14 23 we want to mention some examples of papers that study evolution or learning in this framework, inluding Marimon, MGrattan, and Sargent (99), Sethi (996), Staudinger (998) and Basçi (999). They are interested in determining whih of the equilibria are more robust; for example, an agents learn to use money? Brown (996), Duffy and Ohs (998, 999), and Duffy (2) ask the same kind of questions, but use laboratory methods with paid human subjets to test them experimentally. Although the results are by no means definitive, they are interesting in that they point to ertain areas where laboratory subjets do not behave as theory predits. However, the most reent experiments (Duffy, 2) produe results that are enouraging from the perspetive of the theory. IX. Conlusion In this paper, we have presented simple versions of the basi searh-theoreti models of monetary exhange. Even these simple models allow a variety of questions to be addressed, and there are a wide range of extensions and appliations. We hope this illustrates the usefulness of the framework for monetary eonomis and will enourage the reader to pursue these issues further.

15 ), M,, M γ M M βψ y βψ y M β β y β M β M β ψ β ψ 24 Appendix Proof of Proposition : For pure strategy equilibria, insert and into equations (3) and (4) and determine the region of parameter spae in whih the inequalities in (5) hold. Consider. For this to be an equilibrium, we require and. Inserting into (3) and (4), one finds and if and only if r r r 4, as stated in the proposition. The other pure strategy ases are similar. For mixed strategies, solve j for j and then determine the region of parameter spae in whih j. Consider and. For this to be an equilibrium, we require and. Now implies y r γ u It is easy to see that iffr r 3 and iff r r 4, and the ondition is redundant. Hene, this equilibrium exists iff r r 3 r 4, as stated. The other mixed-strategy ases are similar. In this way, we obtain the omplete set of equilibria. Proof of Proposition 2: First note that rejeting a barter offer is always stritly dominated by aepting, given u. Now suppose that the seller gets to propose and that he proposes a ash transation. There are three possibilities. First, if the proposal will be rejeted, so the seller would have been stritly better off proposing barter. Seond, if then the proposal will be aepted, but in this ase u (beause u so again the seller would have been stritly better off proposing barter. So a seller would never propose a ash trade over barter exept possibly if. A symmetri argument implies that a buyer would never propose a ash trade exept possibly if. This gives us two ases to onsider: (i), whih implies u whih further implies ψ (sine implies the seller stritly prefers barter), whih is the only ase in whih we an have ψ ; and (ii), whih implies u and ψ, whih is the only ase in whih we an have ψ. Consider ase (ii), where, u ψ and ψ in equilibrium. Note u implies. Suppose ; then the agent without money gets u from proposing a ash transation, whih is stritly less than what he gets proposing barter. So ψ requires. Now the value funtions an be written r ym u y βψ y r y u ym βψ ym ψ u ψ M ψ u ψ Sine we are in ase (ii), we have, u and ψ. Hene, subtrating and and simplifying, we have (2) ru ψ M y u y u This equality is violated for generi parameter values when ψ. Hene there is no equilibrium where sellers propose ash with probability. A symmetri argument for ase (i) implies there is no equilibrium where buyers propose ash with probability. This means that the unique pure strategy equilibrium is for agents to propose barter with probability : ψ ψ. γ