Economic Review 4Thquarter 2000

E C O N O M I C R E V I E W

2000 Quarter 4 Vol. 36, No. 4

Productivity and the Term Structure 2 Economic Review is published quarterly by the Research by Joseph G. Haubrich Department of the Federal Reserve The recent record-setting economic expansion and the accompanying Bank of Cleveland. To receive copies record-setting bull market in stocks are often attributed to Federal Reserve or to be placed on the mailing list,

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¥§¦©¨ ¥ ¥ £ ¨ £ © ! interest rate policy and increased productivity. But if interest rates behave ¢¡¤£ differently when productivity changes, interest rate policy may need to change as well. This Review examines how productivity changes affect the or fax it to 216-579-3050. entire term structure of interest rates, from short-term rates to long-term rates. The article works out a model based on a representative agent Economic Review is also available framework, which considers one person as a price taker who makes electronically through the Cleveland

investment choices depending on prices, and these choices are used to Fed’s site on the World Wide Web:

£ %£ & ¨ £ ' ( ©)© (¥*,+ ( - determine the prices that will clear the markets. The person decides how "#"$" . much to consume and how much to invest in each of two assets—a short-term, risk-free, real bond, and a risky productive investment. A key Research Director: Mark Sniderman feature of the model is that risk and return change over time in a way that is Editor: Paul Gomme directly related to productivity. The results show that productivity plays a Managing Editor: Monica crucial role in how various interest rates interact, but its effect is not simple. Crabtree-Reusser Productivity is not a single factor that affects interest rates uniformly, and Article Editors: Michele Lachman and parameters of the productivity process, such as the duration of a productivity Deborah Zorska increase or the speed of adjustment, affect the term structure differently. These parameters, which might otherwise be overlooked, must be specified Designer: Michael Galka before the appropriate interest rate policy can be identified. Opinions stated in Economic Review The Search-Theoretic Approach to Monetary are those of the authors and not Economics: A Primer 10 necessarily those of the Federal Reserve Bank of Cleveland or of the by Peter Rupert, Martin Schindler, Andrei Shevchenko, and Board of Governors of the Federal Randall Wright Reserve System. The authors present simple versions of the models used in the Material may be reprinted provided search-theoretic approach to monetary economics. They discuss results on that the source is credited. Please the existence of monetary equilibria, the potential for multiple equilibria, and send copies of reprinted material to welfare. They consider models where prices are fixed as well as models the managing editor. where prices are determined endogenously by bilateral bargaining. After discussing the frictions necessary to construct a model with an essential role for money, they conclude the paper by reviewing many extensions and We welcome your comments,

applications in the related literature. questions, and suggestions. E-mail us

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ISSN 0013-0281 The Search-Theoretic Approach to Monetary Economics: A Primer by Peter Rupert, Martin Schindler, Andrei Shevchenko, and Randall Wright Peter Rupert is a senior economic advisor at the Federal Reserve Bank of Cleveland; Martin Schindler is a doctoral candidate in economics at the University of Pennsylvania; Andrei Shevchenko is an assistant professor of economics at Michigan State University; Randall Wright is a professor of economics at the University of Pennsylvania. This paper was written while Schindler, Shevchenko, and Wright were visiting the Federal Reserve Bank of Cleveland.

Introduction are not public information. Such frictions are crucial for a logically coherent theory of money; the approach This paper presents some results in monetary theory described here helps to clarify each one’s role. derived using very simple game-theoretic models of This approach contrasts with attempts to model the the exchange process. The underlying model is a vari- role of money in a competitive equilibrium (Walrasian) ant of search theory, a framework that has been used model—a difficult task that has met with mixed suc- extensively in a wide variety of applications. This cess at best. In a competitive equilibrium model, the approach is well suited to discussing the process of exchange process is not explicitly modeled. That is, exchange and money’s role in the process. The ap- agents start with an initial allocation A and choose a proach utilized here is explicitly strategic, in the fol- final allocation B so as to maximize utility, subject to lowing natural sense: When I decide whether to ac- the latter not costing more than the former, but how cept in trade a certain object other than one I desire for they get from point A to point B is not discussed. my own consumption—for example, money—I must Does some unmodeled agent (maybe the auctioneer) conjecture as to the probability that other agents will make the necessary trades with a “pick-up and deliv- accept it from me in the future. This evidently ought ery service”? Or do the agents trade directly with each to be modeled as a game. other? Do they trade bilaterally or multilaterally? In In search theory, the type of game to be consid- real time, or before production and consumption activ- ered is explicitly dynamic, and exchange takes place ity starts? Do they barter directly or trade indirectly in real time. Also, the models allow us to focus pre- using media of exchange? The standard competitive cisely on various frictions in the exchange process that equilibrium paradigm does not address such questions. might give money a role in an equilibrium, or effi- Search models, in contrast, are designed with exactly cient, arrangement. Among the frictions are these: these issues in mind and therefore are logical tools for Agents are not always in the same place at the same studying monetary economics, as we shall illustrate. time; long-run commitments cannot be enforced; and agents are anonymous in the sense that their histories 11

I. The Basic Model always holds either 0 or 1 unit of money. To simplify the presentation, we do not allow agents to freely dis-

To model anonymous trade, it is natural to start with a pose of money, but this is never binding except in one

¤ large number of agents—formally, we assume a 0 1 case mentioned below. continuum. For simplicity, we assume these agents We now describe the trading process. Rather than

live forever and discount the future at rate r. There is assuming a centralized (Walrasian) market, here the

¤ a 0 1 continuum of indivisible consumption goods. agents must trade bilaterally. The simplest way to To generate gains from trade, we need to assume that model this is to assume that they meet according to agents are specialized. There are many ways to do this, a pairwise random matching process. Upon meeting, a but an easy one is to assume that each agent i is able pair decide whether to trade, then part company and re- to produce just one type of good. The unit production enter the matching process. Let α denote the (Poisson) cost for any agent is c 0. For convenience, we as- arrival rate in the matching process—that is, the prob- sume that these goods cannot be stored and so must be ability of meeting someone in a given unit of time.2 consumed immediately after they are produced. Obvi- For reasons discussed later, we assume the history of ously, this means that consumption goods cannot serve any agent’s past meetings and trades is not known to as media of exchange, allowing us to highlight the role anyone else. of money. We want to analyze agents’ individual trading strate- To make trade interesting in the model, we need to gies. An agent obviously should never accept a good assume that tastes are heterogeneous. Again, there are in trade if he does not want to consume it, since goods many ways to do this, but for simplicity we assume are not storable. Whenever possible, agents should the following: First, given any two agents i and j, barter for a good they do want to consume (the case of write iWj to mean “i wants to consume the good that a double coincidence). What needs to be determined j produces”—in the sense that i derives utility u c

¤ is whether an agent should trade goods for money and π from consuming what j produces if iWj, and he derives money for goods. Let 0 denote the probability that the π utility 0 from consuming what j produces otherwise. representative agent trades goods for money, and let 1

Then, for any randomly selected agents, we assume denote the probability that he trades money for goods.

¡ ¡

prob iWi ¡ 0, prob jWi x, and prob jWi iW j These must satisfy the equilibrium conditions given

y. The ﬁrst assumption, prob iWi ¡ 0, means that no below. We will say that money is used as a medium of agent ever wants to consume his own output (which is π π π

exchange, or circulates, if and only if 0. 0 1 ¤ why they trade). The second assumption parameter- Let V0 and V1 be the value functions (lifetime, dis- izes the extent of the basic search friction: The smaller counted, expected utility) of agents with 0 or 1 units x is, the lower the probability that a random trader has of money. Since we consider only stationary and sym- what you want. However, the third assumption is the metric equilibria, Vj does not depend on time or on the important one, since it parameterizes Jevons’ (1875) agent’s name, only his money inventories. famous double coincidence of wants problem: The If we think of time as proceeding in discrete peri- smaller y is, the lower the probability that a trader who ods of length τ, we can calculate the payoff of holding has what you want also wants what you have.1 money as follows: The probability of meeting anyone Besides the consumption goods already mentioned, during this period is approximately ατ by the Pois-

there is another object called money, which consists of

¤ an exogenously ﬁxed quantity of M 0 1 indivisible 1 Several notions of specialization in the literature are units of a storable object (of course, money must be special cases of this model. For example, in Kiyotaki and storable to be useful). Holding money yields utility γ: Wright (1991) or Aiyagari and Wallace (1991), there are N

If γ 0, money pays a dividend, like many real assets, goods and N types of agents, where type n produces good

¤ ¥ ¦ § ¦

n and wants good n 1 mod N . Then x 1 N, and y 1 if

γ γ

and if 0, then money has a storage cost; 0 de-

¦ ¨ scribes the case of pure ﬁat money. Although this last N ¦ 2, while y 0 if N 2. Alternatively, in Kiyotaki and Wright

(1991, 1993) and much of the related literature, the events ©

case may be the most interesting, for generality we al- ©

iW j and jWi are independent, and so y x.

¥ γ £ low 0. Initially, one unit of money is randomly 2 It is the bilateral rather than the random matching as- allocated to each of M agents. Although we will re- sumption that is important. Corbae, Temzelides, and Wright lax this later, for now we assume that agents holding (2000) show how to redo the model, allowing agents to choose money cannot produce (one way to motivate this is to endogenously whom they meet, rather than meeting at random. Their model shares the basic insights discussed here, assume that after producing, you need to consume be- although it is complicated by the need to determine equilib- fore you can produce again). Thus, no one can ever rium meeting patterns as well as equilibrium trades. acquire more than one unit of money, and so an agent 12

3 π π

son assumption. If the person you meet can produce The equilibrium conditions for 0 and 1 are ¡

(meaning, in the version of the model that we are con- ¡

¢ £ ¢ £

1 0

sidering here, he does not have money), which occurs ¤

π ¢ ∆

with probability 1 M, and you want what he can pro- (5) j 0 1 as j 0

¢ duce, which occurs with probability x, and you both 0 0 want to trade, which occurs with probability π in equi- ∆ π Notice that j depends on , so to see if some candi- librium, then you trade, consume, and continue with- date π and π constitute an equilibrium, one simply ¡ 0 1 out money, for a total payoff of u V . In all other π 0 inserts the j and checks equation (5).

events (you meet no one, you meet someone with a γ Consider ﬁrst the case 0 (ﬁat money). This im- good you do not want, etc.), you simply continue with plies ∆ 0 for all parameter values, so we always 1 ¤

your money, for a payoff of V1. In all events, you also π ∆ π ¡ πˆ have 1. Also, is equal in sign to , where

get γτ from storing the money. Hence, 1 0 0

¡ ¡

¡ rc 1 M ¡ y u c

¡ πˆ

(6) ¢

¡ ¡

1 ¡

ατ π 1 M ¡ u c

¡ ¡ V 1 M x u V 1 1 rτ 0

π πˆ

¡ ¡ ¡ Notice that 0 is always an equilibrium: Since

0 ¤ ¡

¡ ατ π γτ τ ¢

¡ ¡ ¤

1 x 1 M V o π ∆ 1 π π ˆ 0, 0 0 implies 0 , which implies 0 0. Thus,

π 0 satisﬁes the equilibrium condition. Naturally,

τ 0 where o ¡ is the approximation error associated with there is an equilibrium in which no one accepts ﬁat

τ τ

the Poisson process and hence satisﬁes o ¡ 0 as money. However, if τ

0. Rearranging, we have

¡ ¡

1 M ¡ 1 y u

¡ ¡ ¡

c ¤

¡ ¡

τ ατ π γτ τ

¡ ¡ ¡ ¢

r V 1 M x u V V o ¡ 1 0 1 r 1 M ¡ 1 y

πˆ π τ τ then 1, which means 1 is an equilibrium as Dividing by and taking the limit as 0, we ar- 0 rive at the continuous-time Bellman’s equation, well. So there is also an equilibrium where ﬁat money 4

circulates as long as c is not too big. Finally, if πˆ 1,

¡ ¡

α π γ there is also a mixed-strategy equilibrium in which

¡ ¡ ¢ (1) rV1 x 1 M u V0 V1 π πˆ 0 . In this case, if other agents accept money with An analogous argument implies that the value function exactly the right probability πˆ, you are indifferent as for an agent without money satisﬁes to accepting or rejecting it, so randomizing is an equi-

librium.

¡ ¡ ¡ ¡

α α π

¡ ¡ ¡ ¢

(2) rV0 xy 1 M u c xM V1 V0 c The above model is essentially that of Kiyotaki and Wright (1993), except that they assume c 0 in addi-

The ﬁrst term in this expression represents the gain γ tion to 0. This implies that three equilibria neces-

from a direct barter trade, while the second represents π π π

sarily exist, 0, 1, and y. In the mixed- the gain from trading goods for money with probabil- π strategy equilibrium, money is accepted with the same ity . Notice that you can only barter when there is a probability as a good (since y is the probability of double coincidence of wants and the other person has a double coincidence), which makes you indifferent.

no money. When c 0, money must have a strictly greater prob- ¤ ability of being accepted than a barter trade for you II. Equilibrium to be indifferent about accepting money, because you must incur the production cost to get the money. More

Deﬁne the net gain from trading goods for money by γ £ π

generally, when 0, we have to determine 1 en- ¡ ∆ ¡ γ

0 V1 V0 c, and the net gain from trading money dogenously; for example, if is large, then agents may ¡

∆ ¡ α for goods by 1 u V0 V1. If we normalize x 1 prefer to hoard rather than spend their money. to reduce notation (which we can always do with no The results for the general case are summarized as loss of generality by redeﬁning units of time appropri- follows:

ately), we have:

¡ ¡

π ¡ γ 3 That is, the probability of meeting one person in a pe-

¡ ¡

M 1 M y u c ru

¤ ¥

∆ riod of length τ is ατ o τ , and the probability of meeting

¡ (3)

1 π τ τ τ τ ¥ ¤ ¥ ¤ ¥ § ¤

r more than one is o ¤ , where o satisﬁes o 0 as

¡ ¡ ¡ ¡

τ ¤

π γ 0.

¡ ¡

1 M ¡ y u c rc

∆ ¥

¢ (4) 0 r π 4 Alternatively, for a given c ¨ 0, we can say that r and M must be relatively small for a monetary equilibrium to exist (agents must be patient and money not too plentiful). 13

Proposition 1. There are ﬁve types of equilibria, and present an alternative model, ﬁrst discussed by Siandra they exist in the following regions of parameter space: (1993, 1996), where agents with money can produce,

π π and we simply impose the condition that agents can

1. 1 and 0 is an equilibrium iff r r

0 1 £ 2 store, at most, one unit of money.

π π

2. 0 0 and 1 1 is an equilibrium iff r r3 The ﬁrst issue to be resolved is, what happens in

π π ¢ a double coincidence when you have money and the

¤ ¡ 3. 0 1 and 1 0 1 is an equilibrium iff r1

other person does not—do you barter or pay with r r2

cash?6 We resolve this by allowing agents to play the

π ¢ π

¡ 4. 0 ¤ 1 and 1 is an equilibrium iff r 0 1 3 following simple game: First, with probability β the

r r 4 agent with money is chosen and with probability 1 ¡ β

π π

5. 1 and 1 is an equilibrium iff r r r £ 0 1 1 £ 4 the agent without money is chosen to propose either where the critical values of r are given by a barter or a cash transaction (in principle they could

also propose not to trade at all, but we ignore this op-

¡ ¡ ¡

¡ ¡ M 1 M y u c tion, which will always be dominated by proposing

1 u barter). Second, the other agent responds either by

¡ ¡

γ ¡

accepting, which executes the proposal, or rejecting,

1 M ¡ y u c r

2 u which implies they part company (ﬁgure 1). A strat-

¡ ¡ ¡

egy for the agent with j units of money, j 1 or 0,

¡ 1 M ¡ y u c

r ψ

3 c is denoted as j, which equals the probability that he ¡

¡ ψ

¡ ¡

¡ proposes barter (and so 1 is the probability he

γ j

¡ ¡

1 M ¡ 1 y u c

¢ r 4 c proposes cash). These are the only (steady-state) equilibria. Proposition 2. In a double-coincidence meeting between an agent with and an agent without money, Proof: See the appendix. generically the unique subgame-perfect equilibrium in

ψ ψ

We characterized the regions where the different pure strategies is 0 1 1. equilibria exist in terms of r, but we could have used

some other parameter, such as c. Routine algebra im- Proof: See the appendix.

plies r1 r2 r3 r4. Also, note that our assumption Having resolved the ambiguity that arises when both of no free disposal is never binding, except possibly barter and cash are available, we can now derive the

π γ 5 when 0 0, and even then only if 0. More im- Bellman equations. Again, let time proceed in discrete

portantly, there are equilibria where π 0 and γ 0; ¤ that is, agents value money and use it as a medium of exchange despite its storage cost. We will have more ¥ 5 To be precise, we should say what agents do after dis- to say about the economics underlying the above re- posing of their money. We assume here that they cannot sults in the next section, after we introduce a slight trade, as they cannot produce. Hence, agents will dispose of money and drop out of the trading process iff V 0. Since it

1 ¡ variation on the model, since it will be interesting to ¡

is easy to see that V0 0 in any equilibrium and V1 V0 in any π ¨

compare the two versions. equilibrium with 0 0, the only case where disposal could po-

¦ § π ¦ γ tentially occur is 0 0, which implies V1 r. Hence, agents

dispose of money if and only if π ¦ 0 and γ 0. 0 III. Alternative Speciﬁcation ¥ 6 This is the only ambiguous case; every other meeting has only one feasible transaction (that is, if you encounter a A key assumption in the above model is that agents double coincidence and have no money, barter is the only op- holding money cannot produce; this is what prevents tion). The issue did not come up in the previous section, be- them from acquiring more than a single unit of money. cause agents with money cannot barter. The fact that agents hold either 0 or 1 unit of money is what makes the model so tractable (see below). Al- though the assumption that agents holding money cannot produce is common in the literature, it has some undesirable implications. For example, if two agents with money meet and there is a double coincidence of wants, they cannot trade. A related implication is that as M increases, the productive capacity of the economy necessarily decreases, which makes it difﬁcult to inter- pret the effects of changes in the money. So here we 14

FIGURE 1 Game Tree Accept (u–c, u–c)

Barter Reject (0,0) ψ1

1–ψ1 Cash Accept (∆1,∆0) Buyer proposes β Reject (0,0)

1– β Accept (u–c, u–c)

Reject Seller proposes Barter (0,0) ψ0

1–ψ0 Cash Accept (∆1,∆0)

Reject (0,0)

periods of length τ. Then tions for the alternative model

¡ ¡

(9) rV1 y u c

ατ

¡ ¡

(7) V1 xy u c V1

¡ ¡

τ ¡

¡ ¡

1 r ¡

π γ

¡ ¡

1 y ¡ 1 M u V V ¡

¡ 0 1

¡ ¡

ατ π

¡ ¡

1 M ¡ 1 y x u V

¡ ¡ ¡

0 ¡

¡ ¡ ¡ ¢ (10) rV y u c 1 y M V V c

¡ 0 1 0

¡ ¡ ¡

¡ ατ ατ π

¡ ¡

1 xy 1 M 1 y x V1

¡ γτ τ Although the value functions are different across the

o ¡ ∆

two models, we compute j and deﬁne equilibrium ex- ¥

1 ¡

ατ ¡

¡ actly as in the last section. The results are as follows.

¡ (8) V xy u c V

0 1 rτ 0

¡ ¡

ατ π ¡

M 1 y ¡ x V1 c Proposition 3. In the alternative model, where agents

¡ ¡

¡ with money can produce, there are ﬁve potential types

ατ ατ π ¡

1 xy M 1 y x V0

¡ τ of equilibria and they exist in the following regions of

¤ o ¡ parameter space:

where we temporarily reintroduce αx to facilitate com- π π

1. 1 and 0 is an equilibrium iff r rˆ parison to equations (1) and (2) in the previous model. 0 1 £ 2

π π

Observe, for example, that now a money holder barters 2. 0 0 and 1 1 is an equilibrium iff r rˆ3 every time he encounters a double coincidence, which

π π

¤ ¡ occurs with probability ατxy. Indeed, he uses money 3. 0 1 and 1 0 1 is an equilibrium iff rˆ1

only when he encounters a single coincidence, which r rˆ2

ατ

occurs with probability 1 y ¡ x, and the other agent

π ¢ π

4. 0 ¤ 1 and 1 is an equilibrium iff rˆ does not have money and they both agree to trade, 0 1 3

r rˆ4

π which occurs with probability 1 M ¡ . α

τ π π

Rearranging, we let 0 and normalize x 1

5. 1 and 1 is an equilibrium iff rˆ r rˆ £ 0 1 1 £ 4 as before, to write the continuous-time Bellman equa- 15

where the critical values of r are given by FIGURE 3

¡ ¡ ¡

γ γ

M 1 y ¡ u c Equilibria in -Space When Money Holders

¤ r rˆ 1 u Can Produce

γ rˆ2 u

r ٙ ٙ

γ rˆ c

3 r4 r3

¡ ¡ ¡

¡ ¡

1 M ¡ 1 y u c

¢ rˆ 4 c These are the only (steady-state) equilibria. πo = 0, πo = 0 = 1 Proof: The proof is analogous to that of Proposition 1 πo

or π = 1 ∋ and so is omitted for brevity. 1 1 πo (0,1) πo = π = 1 π = 1 ٙ 1 1 r FIGURE 2 2 γ

Equilibria in )-Space When Money Holders ¤ r Cannot Produce ٙ r1 π o = 1 r π = 0 rළ4 rළ3 1 π 1 = 1

or ∋ = 1 π 1 (0,1) πo π = 0 1 γ –(1–M)(1–y)(u–c) 0 M(1–y)(u–c) π o = 0, = 1 πo = 0 π o

or π o = 1 ∋ 1 π1 = 1 π 1 =

π o (0,1) π where the unique equilibrium is 1, as well as other = 1 rළ π1 2 regions where there are multiple equilibria: In one re- π π gion, we must have 1 1, but 0 can be 0, 1, or be- π tween 0 and 1; in another region, we must have 0 1, π 7 while 1 can be 0, 1, or between 0 and 1. rළ1 π o = 1 The models differ in that it is actually more difﬁcult π 1 = 0 to get money to circulate when money holders can pro- π

1 = 1 duce; the potential region where π 0 is larger when ∋ or ¤ = 1 π1 (0,1) π o they cannot. Intuitively, agents with money are more π 1 = 0 willing to spend it when they cannot produce, since

γ γ

–(1–M)(1–y)(u–c) 0 (1–M)y(u–c) [M+(1–M)y](u–c) doing so allows them to barter. If c 0, the dif- ferences between the two models are especially stark: When money holders cannot produce, there are always

π π π ¢

¤ ¡

three equilibria, 0, 0 y , and 1; in the

γ ¡

The regions of ¤ r space where the different equi- π π

other model there are two, 0 and 1. The intu- libria exist in the two models (those where money ition is as follows: When money holders can produce, holders cannot produce and those where they can pro- there is no cost associated with acquiring money when

duce) are shown in ﬁgures 2 and 3. The same ﬁve types γ

we set c 0, so if there is a strictly positive prob- of equilibria exist in both models; the regions where ability of money being accepted, you should always they exist are similar but quantitatively different— accept it. This is not true when money holders can-

unless y 0, of course, since the models are identical when there is no barter. In either case, when γ is very low the only equilibrium is one in which no one ac- ¥ 7 It is well known that there is a strategic complementarity π γ in the decision to accept money, 0, but it is less well under- cepts money; if is very high, the only equilibrium is π stood that the same is true of 1. For some parameters, an one in which no one spends it. Hence, money circu- agent is more willing to spend money if he believes that oth-

lates if and only if its intrinsic properties are not too ers will do the same. This only occurs when γ ¨ 0. bad or too good. Also, both models include a region 16

not produce, because even when c 0, accepting the exchange process even though money “crowds out” money involves the opportunity cost of giving up your barter. barter option. The version of the model in which agents with FIGURE 4 money cannot produce is easy to motivate by saying that one must consume before producing, since this Welfare as a Function of M leads naturally to the result that agents in equilibrium rW always hold either 0 or 1 unit of money. However, S the alternative version where money holders can pro- rW1 duce also seems more natural in some respects. For rW K example, when two agents with money meet and there 1 is a double coincidence, they trade. Of course, this version does require assuming directly that agents can rW S only store 0 or 1 units of money. There is not neces- π sarily a right or wrong model, and the choice should K be dictated by how well it addresses a certain question rWπ and how tractable it is in any given application.

rW S IV. Welfare 0

In this section, we discuss welfare, deﬁned by W K ¡

¡ rW0

MV1 1 M ¡ V0 (average utility). For the purpose of

γ discussion, we also set 0. Using straightforward algebra, in the two models we have M

O MK MS Mළ

¡ ¡

K ¡

¡¤ ¡ ¡

rW 1 M 1 M y M u c

¡ ¡

S ¡

¡ ¡ ¡ ¤ rW y M 1 M 1 y u c π where the superscript K indicates the case where This discussion ignores the fact that 1 is not an γ money holders cannot produce and the superscript S equilibrium for all parameter values. If 0, it is easy π indicates the case where they can. Notice ﬁrst that to see that in either version of the model, 1 is an

¡ rc

equilibrium if and only if M M 1 . The

when we compare two equilibria in either model, other £

¡ ¡ 1 y u c things being equal, W is greater in the equilibrium with true optimal policy, then, is the minimum of M and the the higher π. This simply says that the more accept- values given above. In ﬁgure 4, we depict welfare in able money is, the more useful it is. The next thing to each type of equilibrium by the curves W j, W j, or W j, 0 π 1 notice is that across the two models, given any π we where money is accepted with probability 0, π, or 1. S K

have W W with strict inequality as long as y 0 ¤

The superscript j K or S refers to the two different and M 0; not surprisingly, agents are better off if we

¤ models. The curves are drawn only for values of M allow money holders to produce. such that the equilibria exist. Figure 5 shows welfare π

We now want to focus on equilibrium with 1 as a function of y, given that we set M to its welfare- and consider maximizing W with respect to M. The re- maximizing level (which depends on y). These ﬁgures sult for the model in which money holders can produce illustrate various properties, including: W is always 1 is M 2 . The intuition is simple. Money is useful higher when money holders can produce; and W in- because it facilitates trade every time an agent i with creases with π across equilibria in either model. money and an agent j without money meet and iWj holds. To maximize the frequency of such meetings, we should have half the agents holding money and half V. Essentiality of Money

1 of them not: M . For the other model, the welfare- 2 At this stage, it is instructive to highlight the role

1 2y 1

maximizing policy is M if y , and M 0 2 2y 2 of various frictions in the model, to understand what 1 if y 2 . Hence, the optimal M is lower in this model, makes money essential. Following Hahn (1965), we simply because when money holders cannot produce, say money is essential if it allows agents to achieve al- increasing M “crowds out” barter. Still, for small y the locations they cannot achieve with other mechanisms welfare-maximizing M is positive because it facilitates that also respect the enforcement and information con- 17

FIGURE 5 commit to this arrangement ex ante, they would all agree to do so, and there would be no need for money. Welfare as a Function of y (optimal M) Clearly, an imperfect ability to commit to future actions is important if money is to have an essential role. rW, M However, even in the absence of explicit commitments, cooperative agreements like the credit arrangement can sometimes be enforced by reputational con- siderations if individual actions are public informa- rWS tion. Thus, consider the arrangement: “produce for Mළ anyone who wants your production 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 M S X is to be determined. Of course, plan X must be self-enforcing (that is, it must be an equilibrium), and we want the outcome of plan X to be sufﬁciently un- M pleasant to keep agents from deviating from the ef- S ﬁcient arrangement. We will assume here that plan X is to trade if and only if there is a double coinci-

rWK dence of wants, which generates expected utility Wb

¡ 9 M K y u c ¡ r, where the subscript b stands for barter. It is in individuals’ self-interest not to deviate from

y the credit arrangement in which they are supposed to ¡

0 1 ¡

produce if and only if c Wc Wb, which sim-

¡ ¡

plifies to r r˜ 1 y u c c. As always, if £ agents are sufficiently patient, the threat of triggering to pure barter supports the efficient outcome, and straints in the environment.8 again money is not essential. Of course, this assumes First, we need some sort of double coincidence of that agents’ trading histories can be observed publicly; wants problem. For this it is important that not ev- otherwise, it is not possible for agents to use trigger erybody can trade multilaterally. For example, con- strategies.10 When trading histories are private infor-

sider an economy with three agents, where each agent mation, the only sustainable outcome without money

¢ ¤

of type n 1 ¤ 2 3 produces good n and wants good is pure barter. But if there is money, we can do better

n 1 mod 3 ¡ . If all agents meet at the same place than pure barter, even when trading histories are pri- and are able to make multilateral trades, it is feasible

for each agent to produce and consume in the period: ¥ Agent 2 produces for agent 1, agent 1 produces for 8 For a recent treatment of this problem, see Kocher- lakota (1998). agent 3, and agent 3 produces for agent 2. Our restric- ¥ 9 We do not trigger to autarky because we assume that tion to bilateral meetings, given specialized tastes and if two agents want to barter without it being observed, they technology, is merely a convenient way to generate a can; so the worst possible equilibrium is the one where none double-coincidence problem. In the three-agent exam- but double-coincidence trades occur. Nothing much hinges ple, in every bilateral meeting one agent wants what on this; a similar message holds if we can trigger to autarky, the other produces, while the other does not. and it is, in fact, easier to support credit-like arrangements by triggering to autarky. Second, even given a double-coincidence problem, ¥ 10 If the number of agents was small, then even if agents for money to be essential it is also important that did not observe all other agents’ histories but only their own, agents cannot commit to long-term agreements. Con- we could potentially support the efficient arrangement by the sider the following credit arrangement: “produce for following strategy: “If ever you directly observe someone de- anyone you meet who wants your good.” This ar- viate (by not producing for you when you would like him to), stop producing for anyone else.” This would set off a chain of rangement resembles credit in the sense that agents re- agents who observe deviations and would eventually lead the ceive consumption today in exchange for nothing but economy into autarky. With a large number of agents, how- a “promise” to repay someone in kind at a future date. ever, if I fail to produce for you, there is zero probability that It is also obviously an efficient arrangement; that is, it in the future I will meet you or I will meet someone who has generates the maximum possible expected utility, say met you, and so the chain will never get back to me. So with a large number of agents, to support credit it does not suffice

¡ α

¡ Wc u c r, given the normalization x 1, where to have agents observe (only) their own histories. See Araujo the subscript c on Wc stands for credit. If agents could (2000) for more discussion. 18 vate. Notice that monetary exchange generates lower different ways. A typical approach is the generalized welfare than the credit arrangement, although higher Nash bargaining solution, than pure barter.

¡ θ

¡ ¡ Money does not do as well as credit because of the (13) q argmax u q V0 Q T1

random-meeting technology and because money hold- ¡ 1

¡ ¡ ¤ V1 Q c q T0 ings are bounded, leading to some meetings where I θ want your good and you do not want mine, but either I where is the bargaining power of the buyer and Tj

have no money or you already have money. In these is the threat point of the agent with j units of money.

¡ meetings, monetary exchange will not work, while Also, the maximization is subject to u q ¡ V0 V1 and

¡ θ 1

credit could work as long as trading histories are pub-

V1 c q ¡ V0. Here we will set 2 , and T1 T2 licly observable. Even if we relax the upper bound 0.12 on money holdings, the fact that money holdings are bounded below by zero means that money cannot do as FIGURE 6 well as credit in a random-matching environment. We conclude that money has an essential role in the model Monetary Equilibrium in the Divisible-Goods for three reasons: the double-coincidence problem, the Model lack of commitment, and private information on trading histories. For an extended discussion of these is- q sues, see Kocherlakota (1998, 2000) and Kocherlakota and Wallace (1998). qٙ

VI. Prices f(Q) This section provides an extension in which the assumption of indivisible goods is relaxed, although e(Q) money is still indivisible and so agents will always g(Q) have either 0 or 1 units of money. Following Shi (1995) and Trejos and Wright (1995), we will use bargaining theory to determine prices endogenously. For

simplicity, we set y 0, so there is no direct barter.

Given that goods are perfectly divisible, let u q ¡ be

the utility of consuming q units of one’s consump-

tion good and c q ¡ the disutility of producing q units

¡ of one’s production good. We assume u 0 ¡ c 0 ,

£ £ £ £ £ £

¡ ¡ ¡ ¡

u 0 ¡ c 0 0, u q 0, c q 0, u q 0, and Q

¤ ¤ £ e ٙ ¤

q q* q q

£ 1

c q ¡ 0, for q 0, with at least one of the weak ¤

inequalities strict. For future reference, we deﬁne q

£ £

by u q ¡ c q . Also, there is a qˆ 0 such that

¡ u qˆ ¡ c qˆ . When a buyer meets a seller who can

produce the right good, they bargain over how much q The bargaining solution in equation (13) deﬁnes a

will be exchanged for the buyer’s unit of money, im-

¡ ¤

mapping q q Q from 0 qˆ into itself. That is,

plying a nominal price p 1 q. Otherwise, the model if other agents are giving Q units of output for one is exactly identical to that in the previous section.

Letting V and V denote the value functions and tak- ¥

1 0 11 This assumption makes it irrelevant whether we use ing q Q as given, the generalizations of the dynamic the version in which money holders can produce or the one programming equations can be expressed as in which they cannot, since they are identical when y ¦ 0.

Shi (1995) and Trejos and Wright (1995) analyze the case

¡ ¡

¡¤ ¡

(11) rV1 1 M u Q V0 V1 with y 0, but only for a special bargaining solution and only ¡

¡ for the model where money holders cannot produce. Ru-

¡ ¢ (12) rV M V V c Q 0 1 0 pert, Schindler, and Wright (forthcoming) consider the general

case.

¡ These can be easily solved for V V Q and V

1 1 0 12 It is also common to set the threat points equal to con-

¡ ¡ V Q ¡ . Taking V Q and V Q as given, q will solve

0 1 0 ¦ tinuation values: Tj Vj. Both can be derived from an under-

a bargaining problem. In equilibrium, of course, q lying strategic model; see Osborne and Rubinstein (1990) for Q. The bargaining model can be formulated in several the bargaining theory. 19

unit of money, then a particular pair bargaining bi- along. Since he discounts the future, a seller is only

laterally will agree to q q Q . An equilibrium is a willing to produce less than the amount that satisﬁes

£ £

¡ ¡ ¡ ﬁxed point, q q Q . In general, we must be care- u q c q . Indeed, to verify that frictions are driv- ful with the constraints on the bargaining problem: ing the result, observe that when r 0 or α ∞, we

When y 0, the constraints are never binding in equi- have q q . librium. However, if y 0 the constraints may bind; Another question to ask is how q depends on M.

¤ e therefore, it is instructive to proceed allowing for the One might expect ∂q ∂M 0, but it is actually pos- e

possibility of binding constraints. The constraints can sible to have ∂q ∂M 0 for small M (at least if r is

¡ ¡ ¡

be rewritten c q ¡ D Q and u q D Q , where also small). The explanation is that when M is close to

¡ ¡

D Q ¡ V1 Q V0 Q . The former constraint is satis- zero, very little trade occurs. In this case, increasing

ﬁed if and only if q f Q ¡ , and the latter is satisﬁed if M increases the frequency of productive meetings be-

and only if q g Q ¡ , for increasing functions f and g. tween buyers and sellers, which in turn increases both

As ﬁgure 6 shows, both f and g go through the origin V1 and V0. The net effect on the bargaining solution

¡ in the Q ¤ q plane, and g lies below f and below the can be a higher q. However, there is some threshold

1 e

¢ ˆ ∂ ∂ ˆ

¤ 45 line for all Q 0 qˆ . Also, f crosses the 45 line M such that q M 0 for all M M, so we

2 ¤

at a unique q1 0 ¤ qˆ . Hence, our search for equilibria can be sure that the value of money eventually begins

¤ can be constrained to the interval 0 q1 . to fall as M increases.

The ﬁrst-order condition for an interior solution to We can also ask how M affects welfare. It is clear

¡ ¡ equation (13), taking V1 V0 Q and V0 V0 Q as that if a planner can choose both M and q to maximize

given, is W, he will choose M 2 and q q . This is because

1 M maximizes the number of trades (just as in the

¡ 2

¡ ¡

£ £

¡ ¡ ¡ ¡ ¡ ¡ ¢ V Q c q u q u q V Q c q 0

1 0

previous section when y 0), and q q maximizes

the surplus that results from each trade. However, if

¡ This deﬁnes a function q e Q , also shown in the the planner can choose only M and q is determined ﬁgure. It too goes through the origin and intersects the

e in equilibrium, then the value of M that maximizes W

45 line at a unique point q . Hence, q q Q can be

¥ e satisﬁes the ﬁrst-order condition

¡ ¤ ¡ ¤ written as q Q ¡ min e Q f Q for all q 0 q ,

¥ e

¡ ¤ ¡ ¤ and q Q ¡ max e Q g Q for all q q q1 . For

∂ q q , it does not really matter how we deﬁne q Q ¡ ,

¤ 1 W s

¡ ¡

¡¤ ¡ ¡ 1 2M u q c q which necessarily is below the 45 line, and we set it ∂M

e equal to q Q ¡ 0. This makes it clear that for all pa- ∂

¡ q

¡ ¡

£ £

¡ ¡ ¢ M 1 M ¡¤ u q c q 0

¡ rameter values, q q Q has exactly two ﬁxed points:

∂M a nonmonetary equilibrium q 0, and a unique mone- e 13 1

tary equilibrium q q 0. ¤ As the second term is negative at M 2 , the solution is One important property of monetary equilibrium o 1 M . This illustrates the trade-off between provid- (that continues to hold even if y 0) is the following. 2

¤ ing liquidity (making trade easier), and reducing the

£ £

¡ Recall that q is deﬁned by u q ¡ c q . Then it is value of money (lowering the surplus from each trade).

easy to show that e q ¡ q and, therefore, q q , Reducing the value of money reduces welfare because, as seen in ﬁgure 6. This is signiﬁcant because q is the as we have already established, q is too low in equilib-

efﬁcient outcome. More precisely, if we deﬁne welfare rium.

¡ as before, W MV1 1 M V0, after simpliﬁcation

we have ¥

13 If y ¨ 0, one can show that for large r there are no mon- ¡

¡ etary equilibria, while for small r there are multiple monetary

¡¤ ¡ ¡ ¢ (14) rW M 1 M u q c q equilibria. ¥ 14 This is true even though bargaining is bilaterally efﬁcient

Hence, W is maximized with respect to q at q . The

e e in the sense that the agreement is on the Pareto frontier in result q q says that in equilibrium q is too low— each exchange, taking as given the value of Q that prevails in 14 or, equivalently, the price level is too high. other exchanges. The point is that all agents would be better The economic intuition for this result is straight- off (in an ex ante sense) if they could get everyone to commit forward. If a seller could turn the proceeds from his to increasing q. A stronger result is actually true: Not only is qe production into immediate consumption, as in a static too low according to the ex ante criterion W, it is also too low according to the ex post criteria V0 and V1. That is, buyers and

or frictionless model, then he would produce until even sellers would be better off if q were bigger. The result

£ £

u q ¡ c q . But in a monetary exchange, the pro- that q q also can be shown for models where agents can ceeds from production consist of cash that can only be hold any amount of money; see Trejos and Wright (1995, pp. spent in the future when an opportunity to buy comes 133–4). 20

VII. Dynamics As in steady-state analysis, we want to eliminate the value functions from (15). To this end, subtract (17) We previously focused on steady states; in this section, and (18) to obtain we consider dynamic equilibria in the model with di-

visible goods. Although one could do things more gen-

¡ ¡

¡ ¡ ¡

15 ¡

θ γ ˙ ˙

¡¤ ¡ ¡ ¢ r V V ¡ 1 M u q c q V V erally, we restrict attention to the case where 1 in 1 0 1 0

the bargaining solution (equation [13]). This is equiv- ¡

˙ ˙

£ alent to assuming that agents with money get to make Equation (15) implies c q ¡ q˙ V1 V0. Inserting this take-it-or-leave-it offers. Hence, they will demand the in the previous equation yields

quantity that satisﬁes

(19) q˙ F q

¡ ¡ ¡ ¡

¡ ¤

(15) V V c q

¡ ¡ ¡

1 0 r 1 M ¡ c q 1 M u q

c q ¡ since this is the most a producer would give to acquire currency. In the previous section’s model, this imme- Any bounded, non-negative path for q solving the

diately implies V0 0 by virtue of equation (12), so above differential equation constitutes an equilibrium.

¡ we have V1 c q . Inserting this into equation (11)

we have FIGURE 7

¡ ¡

¡¤ ¡ ¡ ¢ (16) rc q ¡ 1 M u q c q Dynamic Equilibria • An equilibrium is a q that solves equation (16). Al- q though the model with take-it-or-leave-it offers is interesting in its own right, mainly because of its sim- • plicity, we use it here to study dynamics. q = F (q) We need to rederive the Bellman equations without limiting our attention to steady state. First, write the discrete time value of holding money at t as

–γ

¥ ¡ 1 ¡

ατ ¡ π τ

¡ ¡ V t ¡ 1 M x u V t

1 1 rτ 0

¡ ¡ ¡ ¡ ¡

¡ ατ π τ γτ τ

¡ ¡ ¤ 1 x 1 M ¡ V1 t o where we have reintroduced the utility of holding q γ money . Rearranging, we have q

L qH

¡ ¡ ¡

¡ ¡

α π τ τ

¡ ¡ ¡

rV1 t ¡ 1 M x u V0 t V1 t

τ τ

¡ ¡ ¡

¡ V t ¡ V t 1 1 γ o ¡ τ τ ¢ τ

Taking the limit as 0, we arrive at

¡ ¡

α π

¡ ¡ ¡

(17) rV1 t ¡ x 1 M u V0 t V1 t ¡ ¡ To keep the number of cases manageable, assume γ

˙ ¤

V1 t ¡ γ 0. Figure 7 shows equation (19) for this case.

£ γ ˙ When 0 and large, there are no monetary equilib- where V1 indicates the time derivative. A similar γ ria; when 0 but not too big, there are two monetary derivation yields

steady states, qL and qH . Clearly, qL is stable while qH

¡ ¡

¡ is unstable. Hence, in addition to the steady states,

¡ ¡ ¡ ¡ ¢

(18) rV0 t ¡ M V1 t V0 t c q V0 t

¢ ¡ the set of equilibria is as follows: For all q0 0 ¤ qL ,

Note that because of equation (15), the ﬁrst term on there is an equilibrium that converges monotonically

¢ ¡ the right side of equation (18) is 0. up to qL; for all q0 qL ¤ qH , there is an equilibrium Henceforth, we will omit the time argument t when that converges monotonically down to qL. The former

there is no risk of confusion. Then an equilibrium is a

¤ ¡ bounded time path for each of the variables V ¤ V q 0 1 15 See Coles and Wright (1998) and Ennis (1999). satisfying (17), (18), and (15) at every point in time. 21

(latter) equilibria are characterized by deflation (infla- high-storage-cost goods are used as money. Aiyagari tion), due simply to beliefs. If agents expect prices to and Wallace (1991, 1992) generalize this to N types change, this can be a self-fulfilling prophecy. When and consider several applications. Wright (1995) ex- γ

0, qL coalesces with the nonmonetary equilibrium tends the model to allow agents to choose their type.

q 0. Hence, in addition to the steady states, the set Renero (1994, 1998b, 1999) considers several exten-

¢ ¡

of equilibria includes paths starting at any q0 0 ¤ qH sions of the framework. Among other things, he shows and converging down to q 0. that equilibria in which goods with high storage costs serve as money can have good welfare properties, perhaps surprisingly (the intuition is that there is more VIII. Extensions and Related Literature trade in such equilibria). Other related papers include In this section, we provide a short overview of some Kehoe, Kiyotaki, and Wright (1993), Cuadras-Morato extensions to and applications of the above models and Wright (1997), and Renero (1998a). in the literature. We will discuss some of the papers There is also a literature on search models with pri- briefly; others we will merely mention. Our intent is vate information. Williamson and Wright (1994) as- to provide a bibliography rather than a review, so that sume there is uncertainty concerning the quality of the interested reader at least knows where to look.16 goods. In such an environment, a generally recog- The basic search-theoretic monetary model can be nizable money has the potential role of mitigating the generalized along several dimensions. Specialization informational frictions and inducing agents to adopt is endogenized in more detail in Kiyotaki and Wright strategies that increase the probability of acquiring (1993), Burdett, Coles, Kiyotaki, and Wright (1995), high-quality output. So money may be valued even if Shi (1997b), and Reed (1999), for example. More the double-coincidence problem vanishes (that is, even general production structures are incorporated in Kiy- if y 1). otaki and Wright (1991) and Johri (1999). Long-term Trejos (1997) presents a simplified version of the partnerships, in addition to one-time exchanges, are model (essentially by setting y 0), which allows him considered in Siandra (1996) and Corbae and Ritter to obtain analytical solutions to the model. Kim (1996) (1997). Various extensions of bargaining are consid- endogenizes the extent of the private information prob- ered by Engineer and Shi (1998, 1999), Berentsen, lem. Cuadras-Morato (1994) and Yiting Li (1995b) Molico, and Wright (forthcoming), and Jafarey and use a version of this model to study commodity money. Masters (1999). We already mentioned credit in sec- All the above papers assume indivisible goods. Tre- tion II, and there are several papers that attempt to jos (1999) combines private information with divisi- have money and credit in the model at the same time: ble goods and bargaining. Velde, Weber, and Wright Kocherlakota and Wallace (1998) assume histories are (1999) and Burdett, Trejos, and Wright (forthcoming) imperfectly observed over time; Cavalcanti and Wal- use commodity money models with private informa- lace(1999a,b) and Cavalcanti, Erosa, and Temzelides tion to study some issues in monetary history, includ- (1999) assume that the histories of only some agents ing Gresham’s Law. Other related papers include Wal- are observed; and Jin and Temzelides (1999) assume lace (1997b), Williamson (1998) and Katzman, Ken- only histories of local neighbors are observed. Follow- nan, and Wallace (1999). ing Diamond (1990), some papers have bilateral credit Several papers attempt to model policy as follows: and money, with repayment (explicitly or implicitly) There is a subset of agents who are subject to the enforced by collateral. These include Hendry (1992), same search and information frictions as everyone else Shi (1996), Schindler (1998), and Yiting Li (forthcoming). ¥ 16 A large body of work in search theory is tangentially re- Many papers deal with commodity money, as op- lated to the approach to monetary economics presented here. posed to fiat money. The basic idea is to determine This brief review cannot discuss all such work, but we do want endogenously which of many possible goods become to mention Diamond (1982); although there is no money in that model, it is in some respects quite similar to that in section II. media of exchange. Kiyotaki and Wright (1989) con- The version in Diamond (1984) does have money, but it is im- sider a version of the model where type i consumes posed through a cash-in-advance constraint; so although it in

¡ some ways resembles the framework presented here, its spirit good i and produces i 1, with N 3 types. The goods are all storable, although at different costs. It is quite different. See also Diamond and Yellin (1985). We is shown that goods with low storage costs may or also mention Jones (1976) as well as the extensions by Oh (1989) and Iwai (1996), which attempt to build a model along may not come to serve as money, depending on pa- lines similar to those presented here; see Ostroy and Starr rameter values as well as which equilibrium the econ- (1990) for a review of this and related work. Other general omy is in; that is, there can be equilibria in which discussions that concentrate more on models like the ones in this paper include Wallace (1996, 1997a). 22 but act collectively. Call these agents government will find something he likes. The Shevchenko and Li agents. The idea is to see how government agents’ papers also endogenize the number of intermediaries exogenously specified trading rules affect the endoge- in the economy by means of a free entry condition. nously determined equilibrium behavior of other (pri- See also Camera (2000), Camera and Winkler (2000), vate) agents. Papers in this group include Victor Li and Hellwig (2000). (1994, 1995a), Ritter (1995), Aiyagari, Wallace, and The framework’s most important recent extension Wright (1996), Aiyagari and Wallace (1997), Li and is perhaps the relaxation of the strong assumptions on Wright (1998), Green and Weber (1996), Wallace and how much money agents can hold—typically, zero or Zhou (1997), and Berentsen (2000). For example, in one unit of money, as we assume above. Models that Victor Li (1994, 1995a, 1997), government agents can consider such an extension can be an order of magni- tax money holdings when they meet private agents. A tude more complicated, but they are obviously more key result is that taxing money holdings may be ef- realistic and generate many interesting new results. ficient. The reason is that in his model (which also They are also capable of addressing more traditional endogenizes search intensity) there is too little search policy questions, such as the optimal rate of inflation, by agents holding money, for standard reasons. Taxing which are difficult to study in models where agents them increases their search effort, and this can improve hold only zero or one unit of money. Such a model welfare. is contained in Molico (1999), who allows agents to The matching model seems a natural one for study- hold any nonnegative amount of money and to bargain ing issues related to international monetary economics. over the quantity of goods as well as the amount of For example, one can think about parameterizing dif- money that is traded in each bilateral meeting. Be- ferences in the efficiency of economic activity as well cause of the model’s complexity, however, it can only as degrees of openness across countries in terms of ar- be solved numerically. The numerical analysis gener- rival rates. Among the first authors to analyze this in a ates interesting results on policy, welfare, the equilib- model with multiple currencies and multiple countries rium distribution of prices, and other issues. are Matsuyama, Kiyotaki, and Matsui (1993). They Green and Zhou (1998b) and Zhou (1999) also find that several types of equilibria can arise, includ- present a model with divisible money, where several ing those in which one currency circulates only lo- results can be derived analytically. Unlike Molico, cally while another emerges as an international cur- they assume that sellers set prices and cannot observe rency; they find other equilibria in which all currencies buyers’ money holdings. Although such an environ- are universally accepted. They compare these equi- ment could still have equilibria with a distribution of libria in terms of welfare. Zhou (1997) extends their prices, they only look for equilibria where all sellers model to study currency exchange. These models as- set the same price. Several interesting results emerge, sume indivisible goods. Trejos and Wright (1999) en- including the existence of multiple (indeed, a contin- dogenize prices using divisible goods and bargaining. uum of) steady states, indexed by the nominal price Other examples of models with multiple currencies in- level. Also, there can be an endogenous upper bound clude Kultti (1996), Green and Weber (1996), Craig for money holdings: Agents with sufficient cash will and Waller (1999), Peterson (2000), and Curtis and not accept more. (Molico’s model can also generate Waller (2000). this.) Related references are Green and Zhou (1998a), Some papers consider intermediation (in the form of Zhou (1998), Camera and Corbae (1999), Taber and middlemen, for example) as an alternative (or some- Wallace (1999), Berentsen (1999a,b), and Rocheteau times in addition) to money. An early paper to explic- (1999). Shi (1997a) presents an analytically solvable itly consider intermediation in a search model with- model with perfectly divisible money, but his model is out money is Rubinstein and Wolinsky (1987). They quite different in some dimensions from the rest of the generate a role for middlemen by specifying exoge- literature.17 nously a set of agents who may have a more effi- There are other applications and extensions that can- cient technology for finding buyers than sellers have not all be considered in this brief review. However, for finding buyers. Yiting Li (1998) is a very different

model, in which private information about the qual- ¥ 17 In Shi’s model, the decision-making unit consists of a ity of consumption goods combined with the existence family with a large number of members (formally, a contin- of a costly quality verification technology give rise to uum), rather than a single individual. In this framework, family a role for intermediation. In Shevchenko (2000), in- members share money holdings between periods, so every termediation arises from inventory-theoretic consider- family starts the next period with the same amount of money ations: Middlemen keep a stock of several goods on by the law of large numbers. Applications and extensions of this model are contained in Shi (1998, 1999). hand to increase the probability that a random buyer 23 we want to mention some examples of papers that study evolution or learning in this framework, including Marimon, McGrattan, and Sargent (1990), Sethi (1996), Staudinger (1998) and Basçi (1999). They are interested in determining which of the equilibria are more robust; for example, can agents learn to use money? Brown (1996), Duffy and Ochs (1998, 1999), and Duffy (2001) ask the same kind of questions, but use laboratory methods with paid human subjects to test them experimentally. Although the results are by no means definitive, they are interesting in that they point to certain areas where laboratory subjects do not behave as theory predicts. However, the most recent experiments (Duffy, 2001) produce results that are en- couraging from the perspective of the theory.

IX. Conclusion

In this paper, we have presented simple versions of the basic search-theoretic models of monetary exchange. Even these simple models allow a variety of questions to be addressed, and there are a wide range of extensions and applications. We hope this illustrates the use- fulness of the framework for monetary economics and will encourage the reader to pursue these issues further. 24

Appendix functions can be written

Proof of Proposition 1: For pure strategy equilibria,

¡ ¡ ¡

∆

¡ ¡ ¡ π π rV yM u c 1 y 1 M

insert 0 and 1 into equations (3) and (4) and deter- 1 1

¡ ¡

¡ ¡ ¡

βψ β ψ

mine the region of parameter space in which the in-

¡ ¡

y 1 M ¡¤ 1 1 0 u c ¡

π π ¡

equalities in (5) hold. Consider 1. For this

¡ ¡ ¡ ¡

0 1 ¡ βψ β ψ ψ ∆ γ

¡ ¡

∆ ∆ y 1 M ¡¤ 1 1 1 0 1 0 1 to be an equilibrium, we require 0 and 0. In-

0 1 ¡

¡ ¡ ¡

∆

¡ ¡ ¡

π π ∆ rV y 1 M u c 1 y M

serting 0 1 1 into (3) and (4), one ﬁnds 0 0 0 0

¡ ¡ ¡

∆ ¡

βψ β ψ

¤ and 0 if and only if r r r , as stated in the

¡ ¡ 1 1 4 yM 1 1 0 u c

proposition. The other pure strategy cases are sim- ¡

¡ ¡ ¡

¡ βψ β ψ ψ ∆

¡ ¡ ¢ ∆ π yM 1 1 1 0 1 0 0 ilar. For mixed strategies, solve j 0 for j and

then determine the region of parameter space in which ∆ ∆ ¡

Since we are in case (ii), we have 0, u c

π π π 1 0

¢ ¢

¡ ¤ ¡ j 0 ¤ 1 . Consider 0 0 1 and 1 1. For this ψ

and 1 1. Hence, subtracting V1 and V0 and simpli- ∆ ∆ to be an equilibrium, we require 0 0 and 1 0. fying, we have ∆ Now 0 0 implies

(20)

¡ ¡ γ ¡

¡ rc

¡ ¡ ¡ ¡

¡ γ β

¡ ¡ ¡ ¡

ru 1 y M y 1 M 1 u c

y ¡

0 ¡ ψ

¡ 1 M ¡ u c

¡ ¡

0 ¡

¡ ¡

y 1 M ¡ 1 u c π π

It is easy to see that 0 iff r r and 1 iff ¤ 0 ¤ 3 0

∆ This equality is violated for generic parameter values r r4, and the condition 1 0 is redundant. Hence, ψ

when 0 0. Hence there is no equilibrium where

¡ this equilibrium exists iff r r ¤ r , as stated. The 3 4 sellers propose cash with probability 1. A symmetric other mixed-strategy cases are similar. In this way, we

argument for case (i) implies there is no equilibrium obtain the complete set of equilibria. where buyers propose cash with probability 1. This Proof of Proposition 2: First note that rejecting a means that the unique pure strategy equilibrium is for

ψ ψ barter offer is always strictly dominated by accepting, agents to propose barter with probability 1: 0 1

given u c. Now suppose that the seller gets to pro- 1. ¤ pose and that he proposes a cash transaction. There are ∆ three possibilities. First, if 1 0 the proposal will be rejected, so the seller would have been strictly better off proposing barter. Second, if ∆ 0 then the pro-

1 ¤

¡ ∆

posal will be accepted, but in this case 0 u c (be- ¡

∆ ∆ ¡ cause 0 1 u c), so again the seller would have been strictly better off proposing barter. So a seller would never propose a cash trade over barter except ∆ possibly if 1 0. A symmetric argument implies that a buyer would never propose a cash trade except pos- ∆ sibly if 0 0. This gives us two cases to consider: (i)

∆ ∆ ¡

0 0, which implies 1 u c, which further im- ψ ∆

plies 1 (since 0 implies the seller strictly 0 1 ¤

prefers barter), which is the only case in which we can

ψ ∆ ∆ have 1 1; and (ii) 1 0, which implies 0 u c ψ

and 1 1, which is the only case in which we can ψ

have 0 1.

∆ ∆ ψ

Consider case (ii), where 1 0, 0 u c, 0 1

ψ ∆ ¡ and 1 1 in equilibrium. Note 0 u c implies

π π

0 1. Suppose 1 1; then the agent without money

∆ ¡ π π ¡ gets 1 0 1 u c from proposing a cash transac-

tion, which is strictly less than what he gets proposing

ψ π π barter. So 0 1 requires 0 1 1. Now the value 25

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