COORDINATION IN THE USE OF MONEY
LUIS
ARAUJO
BERNARDO GUIMARAES
Agosto
de 2013
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TEXTO PARA DISCUSSÃO 325•AGOSTO DE 2013• 1
Os artigos dos Textos para Discussão da Escola de Economia de São Paulo da Fundação Getulio Vargas são de inteira responsabilidade dos autores e não refletem necessariamente a opinião da
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Coordination in the use of money
Luis Araujoy Bernardo Guimaraesz
April 2013
Abstract
Fundamental models of money, while explicit about the frictions that render money essential, are silent on how agents actually coordinate in its use. This paper studies this coordination problem, providing an endogenous map between the primitives of the environment and the beliefs on the acceptability of money. We show that an increase in the frequency of trade meetings, besides its direct impact on payo¤s, facilitates coordination. In particular, for a large enough frequency of trade meetings, agents always coordinate in the use of money.
Key Words: Money, Beliefs, Coordination. JEL Codes: E40, D83
We thank the associate editor Ricardo Lagos and an anonymous referee for their comments and suggestions. We also thank Federico Bonetto, Gabriele Camera, Raoul Minetti, Stephen Morris, Daniela Puzzello, Randy Wright, Tao Zhu and seminar participants at Dauphine, ESSIM 2011, Michigan State, Rochester, SAET Meeting 2011, São Paulo School of Economics-FGV and the 2011 Summer Workshop on Money, Banking, Payments and Finance at the Federal Reserve Bank of Chicago.
yCorresponding Author. Michigan State University, Department of Economics, and São Paulo School of
Economics-FGV. E-mail address: araujolu@msu.edu.
1
Introduction
The principle that the use of money should be explained by its essentiality is well established among monetary theorists.1 Indeed, a precise description of the frictions (e.g., limited commitment, limited record-keeping) which render money essential is a central element in fundamental models of money, such as random matching, overlapping generations, and turnpike models. However, these models typically exhibit multiple equilibria, including one in which money is not valued, and have no say on which equilibrium will be played. In this paper we step towards …lling this gap by exploring how the primitives of an economy impact agents’ ability to coordinate in the use of money.
We cast our analysis in a search model of money along the lines of Kiyotaki and Wright (1993). The key departure from their environment is the assumption that money is not …at. We let the economy experience di¤erent states over time, and assume that while money is intrinsically useless in a large region, some states have an impact on the characteristics of money. There are states where money is either intrinsically useful or convertible into something useful, and states where either the use of money may involve some intrinsic disutility or money becomes less valuable as a medium of exchange (which is the case, for instance, of a hyperin‡ation). In particular, there are faraway states where either accepting or not accepting money is a strictly dominant action.
Our main result is that there exists a unique rationalizable equilibrium. If the gains from trade are relatively large and agents are relatively patient, agents coordinate in the use of money in all states in which money has no intrinsic utility (and also in some states in which money has negative intrinsic utility). If instead the gains from trade are small and agents are relatively impatient, agents never coordinate in the use of money in states in which money has no intrinsic utility (and also in some states in which money brings positive intrinsic utility).
It turns out that patience, which can also be interpreted as the frequency of trade meetings, critically helps agents coordinate in the use of money – more so than an increase in the gains from trade in a given meeting. Intuitively, an agent that expects to face many trading opportunities is willing to accept money even if he holds relatively pessimistic beliefs as to the acceptance of money in the near future. Since all agents make the same reasoning, pessimistic beliefs on the acceptability of money cannot be part of an equilibrium. As the frequency of trade meetings go to in…nity, agents coordinate in the use of money irrespective of the di¤erence between the utility from consumption and the production cost.
Our paper relates to the literature on equilibrium selection in dynamic games with complete information, in particular, Frankel and Pauzner (2000) (FP), and Burdzy, Frankel, and Pauzner (2001) (BFP).2 We share with these papers both the assumption that there exist faraway states where it is strictly dominant to choose a particular action and the result that the ensuing equilibrium
1Money is essential if it achieves socially desirable allocations which could not be achieved otherwise. Kocherlakota
(1998) and Wallace (2001) are key references.
2It is also related to the literature on equilibrium selection in games with incomplete information, surveyed by
is unique.3 An important di¤erence is that in FP and BFP, an increase in the time discount factor does not help selecting the e¢cient outcome. In their model, if the time discount factor is large enough, the risk-dominant equilibrium is selected regardless of whether it is e¢cient. This suggests that the coordination problem involved in the use of money is markedly di¤erent from that present in other (non-monetary) settings in which coordination matters.
Finally, there is a strand of models within monetary economics that studies how the addition of an intrinsic utility to money may help to reduce the set of equilibria. In overlapping generations models, the focus is on the elimination of monetary equilibria that exhibit in‡ationary paths (Brock and Scheinkman (1980), Scheinkman (1980)). In search models of money, the objective is to characterize the set of …at money equilibria that are limits of commodity-money equilibria when the intrinsic utility of money converges to zero (Zhou (2003), Wallace and Zhu (2004), Zhu (2003, 2005)). A result that comes out of this work is that if goods are perfectly divisible and the marginal utility is large at zero consumption, autarky is not the limit of any commodity money equilibria. This result critically depends on the assumption of a su¢ciently high probability that the economy reaches a state where money acquires an intrinsic utility. In contrast, our results on coordination in the use of money hold even if the probability that money ever acquires intrinsic utility is arbitrarily small and the probability it ever acquires intrinsic disutility is equal to one. Thus, coordination in the use of money is not simply the result of working backwards from a distant future where money is sure to acquire a positive intrinsic utility. What matters to determine whether agents will coordinate or not are the primitives of the environment, i.e., the gains from trade and the rate at which agents meet.
The paper is organized as follows. In section 2, we present the model, deliver our main result, and discuss our assumptions. We also examine how the primitives of the environment impact agents’ ability to coordinate in the use of money. In section 3, we conclude. The appendix contains proofs omitted from the main text.
2
Model
2.1 Environment
Our environment is a version of Kiyotaki and Wright (1993). Time is discrete and indexed by t. There arekindivisible and perishable goods and the economy is populated by a unit continuum of agents uniformly distributed acrossktypes. A typeiagent derives utilityuper unit of consumption of good i and is able to produce one unit of good i+ 1 (modulo k) per period, at a cost c < u.
3One relevant di¤erence is that in a monetary economy, the bene…t of exerting e¤ort in exchange for money
Agents maximize expected discounted utility with a discount factor 2(0;1). There arekdistinct sectors, each sector specialized in the exchange of one good. In every period, agents choose which sector they want to visit but inside a sector they are randomly and pairwise matched. Each agent faces one meeting per period, and meetings are independent across agents and over time. There exists a storable and indivisible object, which we call money. An agent can hold at most one unit of money at a time, and money is initially distributed to a measure m of agents.
In any given period, the economy is in some state z 2 R. The economy starts at z = 0 and the state changes according to a random process zt+1 =zt+ zt, where zt follows a continuous
probability distribution that is independent of z and t, with expected value E( z) and variance Var( z)>0. The state of the economy does not a¤ect the preferences for goods and the production technology. Thus in the absence of money, states could be interpreted as realizations of a sunspot variable. However, the state of the economy a¤ects the characteristics of money if z is either su¢ciently large or su¢ciently small. If an agent holds one unit of money at the beginning of a period in a state z > zb, he can choose between bringing this unit to a trading sector, obtaining an intrinsic ‡ow utility zero; and keeping this unit throughout the period, obtaining an intrinsic ‡ow utility (1 ) (z) > 0. In turn, if an agent holds one unit of money at the beginning of a period in a statez < bz, he can choose between bringing this unit to a trading sector, obtaining an intrinsic ‡ow utility (z)<0; and keeping this unit throughout the period, obtaining an intrinsic ‡ow utility zero. Money provides no intrinsic utility, i.e., (z) = 0, in statesz2[ z;b bz]. We assume that (z) is weakly increasing everywhere and strictly increasing whenever (z) 6= 0. Moreover, limz!1 (z) = , where > c; andlimz! 1 (z) = , where > u. Throughout, we think of b
z as a very large number.
We assume that(1 m)u+mc > . This implies that the net bene…t of bringing money to the trading sector which o¤ers the good one likes, under the belief that money is going to be accepted as a medium of exchange, is larger than the intrinsic utility of money. Therefore, as it will be shown, even though the possibility that money may acquire some intrinsic utility a¤ects society’s ability to coordinate in its use, from an individual perspective, the bene…t of money comes solely from its use as a medium of exchange.
which relaxes the intrinsic uselessness of …at money and another which relaxes the inconvertibility of …at money.
The assumption that money may be intrinsically useful in some states of the world captures the idea that the physical properties of the object used as money may not be invariant over time. For example, the statezmay re‡ect technological developments which allow to convert money into a shiner object, say a jewel. In this case, if an agent holds one unit of money at the beginning of a period in a state z >zb, he can use the technology to convert money into a jewel, in which case he obtains a ‡ow utility (1 ) z zb
1+z bz , where > c. As a result, if z is su¢ciently large, it is
strictly dominant to produce in exchange for money, irrespective of the behavior of other agents. In another direction, the statezmay re‡ect changes in the environment which increase the transaction costs of using money as a medium of exchange, or which make it more costly to carry money into a sector which produces the good one likes. In this case, if an agent chooses to bring money into a trading sector in a state z < bz, he obtains a ‡ow utility (z) = 1+jzjzzbjbzj , where > u. Hence if z is su¢ciently small, an agent is not willing to produce in exchange for money. Under this interpretation of (z), we have (z) =p(z) forz >zb, (z) = 0 forz 2[ z;b bz], and (z) =p(z) forz < zb, wherep(z) = 1+jzjzzbjbzj.
The assumption that money is inconvertible, although common in models of money with explicit microfoundations, is quite restrictive. Wallace (1980), for instance, points out that …at money systems are rare throughout history. In modern economies, the issuer of paper money does not stand ready to convert money into something desirable, and it is arguably unlikely to do so in the foreseeable future, but this does not mean that it will never do so.4 Thus the state z can be interpreted as capturing the commitment of the money issuer to the value of money. Under this interpretations, shocks to z re‡ect changes in the environment a¤ecting the likelihood that the money issuer will be willing to convert money into something desirable, i.e., something which provides an intrinsic ‡ow utility. In states z 2 [ bz;bz], the money issuer does not interfere in the economy. However, in states z >bz, there is a probabilityp(z) = 1+zzzbbz that an agent with money is approached by the money issuer at the beginning of the period and o¤ered to exchange his unit of money into one unit of a commodity which provides a ‡ow utility (1 ) , where > c. As a result, if zis su¢ciently large, an agent is willing to produce in exchange for money, irrespective of the behavior of the other agents. Thus very large values of z capture instances where the money issuer is fully committed with keeping the value of money (e.g., the gold standard system). In states z < zb, if an agent chooses to bring money into a trading sector, there is a probability p(z) = 1+jzjzzbjzbj that his money will be taxed away by the money issuer. The taxation of money is a
4To give a sense as to how convertibility is not an outright impossibility, arguments in favor of the
natural way to capture the e¤ects of in‡ation in an environment with indivisible money (Li (1995), Li and Wright (1998)). In particular, very small values of zcapture a scenario of hyperin‡ation, in which money is essentially worthless as a medium of exchange. Under this interpretation of (z), we have (z) =p(z) forz >zband (z) = 0 forz < zb.5
Summarizing, the existence of di¤erent states of the world are meant to capture technological changes which relax the intrinsic uselessness of money, or institutional changes which relax the inconvertibility of money. We also introduced the possibility of states in which it is strictly dominant not to accept money to make it transparent that our result on the society’s ability to coordinate in the use of money does not hinge on the assumption that, in the long run, the economy will most likely be in states in which money is intrinsically useful. In fact, in what follows we show that our results remain essentially unchanged if we choose E( z) and bz so as to make the probability that the economy will ever reach a state where money generates some intrinsic utility (or can be converted into something useful) arbitrarily small.
2.2 Benchmark Case: Fiat Money
We initially consider the scenario wherezb=1and money is …at. In this case, the random variable z is a sunspot, with no impact on payo¤s. We are interested in the behavior of an agent without money, who is asked to produce in exchange for money. The decision on whether to o¤er money in exchange for a desirable good is trivial in our model, since the most an agent can obtain with one unit of money is one unit of good, and all desirable goods provide the same utility u.
In principle, the behavior of an agent may depend on the history of states and, consequently, on the agent’s belief about the behavior of all other agents in all future periods, and after every possible history of states. A natural approach though if money is …at is to look at equilibria where agents’ behavior do not depend on the history of states. One such equilibrium is autarky, where each agent believes that all other agents will never accept money. Under some conditions on parameters, another equilibrium is money, in which each agent believes that all other agents will always accept money. Let V1;z be the value function of an agent with money in state z if he
believes that all other agents will always accept money, and letV0;z be de…ned in a similar way for
an agent without money. We have
V1;z =m EzV1+ (1 m) (u+ EzV0),
and
V0;z =m[ ( c+ EzV1) + (1 ) V0] + (1 m) EzV0,
5Note that there is no intrinsic disutility associated with bringing money to the market in statesz < zb. Indeed,
where EzV1 is the expected payo¤ of carrying one unit of money into the next period, EzV0
is the expected payo¤ of carrying zero units of money into the next period, and 2 [0;1] is the probability that the agent accepts money. Consider the expression for V1;z. An agent with money
goes to the sector that trades the good he likes. In this sector, there is a probability m that he meets another agent with money and no trade happens. There is also a probability (1 m) that he meets an agent without money, in which case they trade, the agent obtains utility u, and moves to the next period without money. A similar reasoning holds for an agent without money.
Assume that = 1. Then, for all z,
V1;z V0;z = (1 m)u+mc.
It is indeed optimal to always accept money as long as c+ V1;z V0;z, i.e.,
[(1 m)u+mc] c, (1)
which we henceforth assume. Thus, if money is …at and (1) holds, there always exist an autarkic equilibrium and a monetary equilibrium. Kiyotaki and Wright (1993) show that there also exists an equilibrium where agents are indi¤erent between accepting and not accepting money. A similar equilibrium exists here.6
2.3 General Case
We now consider the case where bz is …nite, so there are states where money carries some intrinsic utility. We are particularly interested in the behavior of agents who are asked to produce in exchange for money in states where money yields no intrinsic utility.
Let '(t) denote the probability that any state z0 > z is reached at time s+t and not before, conditional on the agent being in state z in period s. Since zt follows a continuous probability
distribution that is independent of the current state z and the period s, '(t) does not depend on z ors. Proposition 1 summarizes our main result.
Proposition 1 Let bz 2 R. Generically, there exists a unique rationalizable equilibrium in the model. Agents play according to a threshold z such that an agent produces in exchange for money if and only if z z . If
X1
t=1
t'(t) [(1 m)u+mc]> c, (2)
then z < bz. If the inequality is reversed, z >bz.
6These are not the only equilibria in our environment if money is …at. For instance, there exists an equilibrium
Proposition 1 shows that in the region[ z;b bz]agents will always coordinate in the use of money if (2) holds, and will never coordinate in the use of money otherwise. The proof of Proposition 1 is organized in steps. In Lemma 1, we show that agents’ decisions on whether to produce in exchange for money are strategic complements.
Lemma 1 (Strategic complementarity) Consider two scenarios: in case A, agents produce in ex-change for money in states z 2 ZA; in case B, agents produce in exchange for money in states
z2ZB. IfZB ZA, then the net bene…t of producing in exchange for money in case A is at least as large as in case B.
Proof. See Appendix.
The decision about accepting money is actually a decision between an action and an option. The agent can accept money now, which entails a costc but puts her in the position of consuming the desirable good at the …rst time the economy reaches a state where money is accepted by others. Alternatively, the agent can reject money now and wait for another opportunity. In doing so, she saves in e¤ort cost, because the present value of future e¤ort cost is lower than c, but she might miss on an opportunity to spend money quickly. Both the value of accepting money now and the value of the option of accepting money later are increasing in the likelihood that money will be accepted by the other agents. Lemma 1 shows that an increase in the likelihood that money will be accepted by the other agents has a stronger e¤ect on the value of accepting money now than in the value of the option of accepting money later.
Lemma 1 implies that if it is optimal to accept money in statez under the belief that all other agents accept money if and only if z0 z, then it is optimal to accept money in statez under the belief that money is accepted in states z0 z, regardless of the belief about how agents behave in states z0 < z. This result will be instrumental for our proof. It will ensure that when considering
an agent’s belief as to how others behave, there is no loss in generality in restricting attention to beliefs that others follow symmetric cut-o¤ strategies, i.e., strategies in which an agent accepts money if and only if the current state is above than or equal to some threshold state.
Lemma 2 shows that if an agent believes that all the other agents are following a cut-o¤ strategy at state z, then the best response is also a cut-o¤ strategy. This best response is unique and we denote the corresponding cut-o¤ state by Z(z).
Lemma 2 (Optimality of cut-o¤ strategy) For every z 2 R, there exists a unique Z(z) such that if an agent believes that all the other agents follow a cut-o¤ strategy at z, then she produces in exchange for money if and only if z0 Z(z).
Proof. See Appendix.
economy reaches a state where money is accepted, and postponing the e¤ort cost. If an agent believes that all other agents are following a cut-o¤ strategy at some state z, andz0 is much larger than z, it makes sense to produce in exchange for money now, as the probability that money will be spent quickly is relatively high. In contrast, if z0 is much smaller thanz, it makes more sense to wait for a future opportunity of producing in exchange for money, since there will probably be other opportunities of doing so before the agent can spend it. Lemma 2 shows that this intuition applies more generally: if others are following a cut-o¤ strategy at some statez, the payo¤ of producing in exchange for money for an agent is increasing in z0. Hence, the best response to cut-o¤ strategies
is also a cut-o¤ strategy.
Lemma 3 uses Lemmas 1 and 2 to show that if an agent believes that all the other agents are following a cut-o¤ strategy at statez, then her expected payo¤ employing the same cut-o¤ strategy is su¢cient to determine the optimal behavior in state z, even though in general that is not her optimal strategy.
LetV~1(z)be the value of holding one unit of money at the end of periods, under the assumption
that all agents, including the agent herself, follows a cut-o¤ strategy at state z. V~0(z) is similarly
de…ned for an agent holding zero units of money. We have the following result.
Lemma 3 (Optimal behavior) V~1(z) V~0(z) is strictly increasing in z =2 [ z;b bz] and constant in
z2[ z;b bz]. Moreover, in any statez2[ z;b bz], if V~1(z) V~0(z)> c, it is strictly optimal to produce
in exchange for money. Otherwise, it is optimal not to produce in exchange for money.
Proof. See Appendix
The fact thatV~1(z) V~0(z)is independent of z ifz2[ z;b bz]is simple to demonstrate. Indeed,
for all z2[ bz;zb], ~
V1(z) V~0(z) =
X1
t=1
t'(t) R1
z V1;z0 V0;z0 dF(z
0jt) . (3)
In words, conditional on the belief that money is only accepted in states z0 > z, the net expected payo¤ of holding money is given by the discounted probability that the …rst time the economy reaches some statez0 > zafter periodshappens in periods+t(which is given by t'(t)) multiplied by the net expected value of having money in state z0 at the beginning of the period, that is, the expected value of V1;z0 V0;z0. The cdf ofz0 is given byF(z0jt), i.e., the probability that the state
of the economy is below or equal to z0 conditional on the event that the …rst time the economy reaches some state z0 > z after periodshappens in period s+t. Now, if an agent follows a cut-o¤
strategy at zand if she believes that all other agents follow the same cut-o¤ strategy, the value of having money in some state z0 > z is given by
V1;z0 =m Ez0V1+ (1 m) (u+ Ez0V0), (4)
while the value of not having money in some statez0> z is
The expected valuesEz0V0 andEz0V1 are complicated objects, but our analysis is vastly simpli…ed
because for any z0 > z,
V1;z0 V0;z0 = (1 m)u+mc. (6)
We can then substitute (6) in (3) and obtain
~
V1(z) V~0(z) =
X1
t=1
t'(t) [(1 m)u+mc]. (7)
In (4), we used the assumption (1 m)u+mc > . Combined with z 2 [ bz;zb], it implies that an agent always prefers to bring money to the market in states z0 > z instead of keeping money
throughout the period in order to enjoy its intrinsic positive ‡ow utility. In turn, in (5), we used the fact that in states below z, an agent prefers to keep money throughout the period, instead of attempting to use it in some trading post. As a result, the right-hand side of (7) does not depend on z.
Consider now the result thatV~1(z) V~0(z)is strictly increasing inzforz62[ bz;bz]. The intuition
for the proof runs as follows. Whenz < z^, agents can spend money in the interval[z; z^), but it is costly to do so. A lower value of z implies both a larger region where spending money is costly and a higher cost of spending money close to z, hence it reduces the incentives for accepting it. In turn, when z >z^, agents cannot spend money in the interval (^z; z], but they obtain a positive intrinsic ‡ow utility from having money in that region. A larger z implies both a larger region where money brings some intrinsic utility (in case it cannot be spent) and a higher intrinsic ‡ow utility close to z.
The key result of Lemma 3 is that, if an agent believes that all other agents are following a cut-o¤ strategy at statez2[ z;b bz], then in order to know whether she …nds it optimal to accept money in statez, all we need to know is what the agent would choose if she was following a cut-o¤ strategy at statez. Indeed, Lemma 3 shows that if an agent strictly prefers to accept money in statez, then she must be following a threshold strategy at some state Z(z)< z. The proof is by contradiction. If Z(z) z, then V~1(z) V~0(z) > c would imply V~1[Z(z)] V~0[Z(z)] > c since V~1(z) V~0(z) is
weakly increasing in z everywhere. However, using Lemma 1, V~1[Z(z)] V~0[Z(z)]> c implies the
agent would strictly prefer to accept money in state Z(z) if others were following a cuto¤ strategy aroundz Z(z), This contradicts the de…nition ofZ(z)as the cut-o¤ state. Intuitively, compared to accepting money in state z, incentives for accepting money at stateZ(z) z can only increase if all other agents are following a cut-o¤ strategy at z.
We are now ready to prove Proposition 1. The proof uses an induction argument, where at each step strictly dominated strategies are eliminated. We start with the problem of an agent on whether to produce in exchange for money if the current state is large. Owing to the increasing positive intrinsic utility of money in states z > bz, and the fact that limz!1 (z) = > c= , to
action wheneverz > ZH. Consider now the problem of an agent on whether to produce in exchange
for money in the current state isz=ZH. This agent knows that all the other agents always accept
money if z > ZH. Lemma 1 then implies that the worst possible scenario for an agent that accepts
money occurs when all others are playing a cut-o¤ strategy at stateZH. In this case, a lower bound
on V~1(z) V~0(z) is given by (7) in all states z > zb, as it does not include the intrinsic utility of
holding money when money is not accepted as a medium of exchange. We can then use Lemma 3 to conclude that, if (2) holds, an agent …nds it strictly optimal to accept money in state z =ZH.
Now, by continuity on z, there exists some such that the agent strictly prefers to accept money in state ZH . In consequence, once not-accepting money has been ruled out for all z ZH,
accepting money becomes a strictly dominant action for an agent in any state z > ZH . We can
then apply Lemma 3 once more to conclude that, if (2) holds, an agent …nds it strictly optimal to accept money in statez=ZH . Proceeding this way, and using the factV~1(z) V~0(z) is strictly
increasing in zin statesz < bz, we eventually reach a statez < zbsuch thatV~1(z ) V~0(z ) =c.
An analogous reasoning applies if we consider the problem of an agent on whether to produce in exchange for money if the state z is small. Since bringing money to a trading post implies an intrinsic disutility (z), and since limz! 1 (z) = < u, as z ! 1, accepting money in
exchange for production becomes a strictly dominated action. Thus there exists someZLsuch that
not accepting money is a strictly dominant action whenever z < ZL. Consider now the problem of
an agent in state z=ZL. Note that (7) is an upper bound ofV~1(z ) V~0(z ) in all states z < zb
as it does not include the cost associated with bringing money to a trading post. We can then use Lemma 3 to conclude that, if the reverse of (2) holds, an agent …nds it strictly optimal not to accept money in state z = ZL. Now, by continuity on z, there exists some such that the agent
strictly prefers not to accept money in state ZL+ . In consequence, once accepting money has
been ruled out for allz ZL, to not accept money becomes a strictly dominant action for an agent
in any statez < ZL+ . Proceeding this way, and using the factV~1(z) V~0(z) is strictly increasing
inz in states z >zb, we eventually reach a state z >bz such thatV~1(z ) V~0(z ) =c.
Finally, it is immediate that, except in a measure zero set where the expression in (7) happens to be equal to c, the iterative process of eliminating strictly dominated strategies will end up at a unique point z < z^or atz >z^.
2.4 Discussion
money is the belief on its acceptability as a medium of exchange, and not that it may acquire an intrinsic utility.7
Perturbations to the process for z do not signi…cantly a¤ect the equilibrium conditions, as long as the set of probabilities'(t)of reaching nearby states where money is accepted as a medium of exchange does not change much. In particular, the set of probabilities '(t) are very similar if E( z) = 0or if E( z) = , where is a very small (negative or positive) number. Thus, by setting very small and negative andbzsu¢ciently large, we can make the probability that money will have a negative intrinsic utility in the long run arbitrarily close to one, without signi…cantly a¤ecting the condition for money being accepted in all states where it has no intrinsic utility. Thus coordination in the use of money does not come from simply working backwards from some distant future where money acquires positive intrinsic utility with a su¢ciently high probability.8 As we show in the next section, what matters to determine whether agents will coordinate or not in the use of money is the ratio between the utility of consumption and the cost of production (u
c), the quantity of
money in circulation (m) and, most importantly, the rate at which agents meet, captured by the agents’ discount factor ( ).
The assumption that z follows a random walk makes the problem identical at every state z. That simpli…es the analysis, but implies that in the long run, irrespective of the value of bz, the economy will usually be at states outside the [ bz;zb] interval. However, a small modi…cation of the random walk process could rule out this outcome without signi…cantly a¤ecting our results. Consider a process such that E( z) = for any z > 0 and E( z) = for any z < 0. For su¢ciently small, the set of probabilities of reaching a nearby state in the following periods would not be substantially a¤ected, and thus the condition for a unique monetary equilibrium would be very similar to (2). We can then make sure that the economy will rarely be outside of the [ bz;bz] interval by choosing a large enough bz.9
We have considered the case of a continuous state space but our results can be extended to a discrete state space, i.e., the case where the set of states is given by Z. We present this case in the Appendix. The key di¤erence is that, in the discrete case, if the economy is in some state z in the current period, there is a positive probability that it will be in the same state after a couple of periods. This gives rise to an intermediate region of parameters were multiple equilibria exist. In this intermediate region, the possibility of the economy being exactly at the same state in the future might allow agents to coordinate on arbitrary beliefs. However, except for this multiplicity region,
7If we increased the intrinsic utility of money by assuming that >(1 m)u+mc, agents would not spend money
in some faraway states. Lacking the option of accepting money in some future states, agents could choose to accept (but not spend) money now. That would make the problem more complicated, since the decision of the agent with money would not be trivial anymore, without necessarily adding much insight to the problem.
8In turn, by setting very small and positive andbzsu¢ciently large, we can make the probability that money will
have a positive intrinsic utility in the long-run arbitrarily close to one, without signi…cantly a¤ecting the condition for money not to be accepted in all states where it has no intrinsic utility. The sheer fact that money will acquire a positive intrinsic utility does not ensure that agents will coordinate in its use.
results are analogous to the continuous case. Moreover, as the frequency of trade meetings, captured here by the agents’ discount factor, goes to in…nity, the multiple-equilibrium region disappears.
2.5 Coordination and the primitives of the environment
In what follows, we look at (2) in more detail, to examine how changes in the primitives of the environment a¤ect society’s ability to coordinate in the use of money. An increase in the time discount factor can be interpreted as a reduction in the time interval between two consecutive periods. In our environment, since there is one meeting per period, this amounts to an increase in the arrival rate of trading opportunities.10 Thus, in the discussion below, we think of an increase in the time discount factor as capturing an increase in the frequency of trade meetings.
To better understand the role of the primitives, it is useful to rewrite (1) and (2) as
h
(1 m)u
c +m i
1, (8)
and h
(1 m)u
c +m i
>1, (9)
where
=
1 X
t=1
t 1'(t). (10)
The key di¤erence between (8) and (9) is that (8) takes as given that all other agents will coordinate in the use of money, and examines how changes in primitives a¤ect the incentives of an
individual agent to produce in exchange for money; while (9) examines how changes in primitives a¤ects society’s ability to actually coordinate in the use of money. The contribution of our paper lies in uncovering the latter e¤ect. First, a simple inspection of (8) and (9) makes it transparent that these e¤ects are somewhat linked, and the same forces which create incentives to produce in exchange for money under the belief that all other agents accept money; also help to coordinate in its use. However, a more careful analysis unveils a distinctive role to the frequency of trade meetings. In order to make this point clear, let
L u
c; m = (1 m) u c +m, and compare
@ @L u
c; m L(u c;m)=1
=
L u c; m
(11)
1 0Alternatively, we could consider a setting in which is …xed, but agents are randomly and anonymously matched
in pairs n 1 times in each period. An increase in n would then amount to an increase in the arrival rate of trading opportunities. We obtain the same results with this alternative speci…cation, but since the agent’s decision on whether to accept money depends on the state of the economy, when there is more than one meeting in a period, we need to assume that the state of the economy changes across meetings and not across periods. This way, if is the period discount factor and there arenmeetings in a period, then n1 is the discount factor in between meetings.
The analysis then is exactly the same as in section 2.3, with n1 replacing . Asngoes to 1, this discount factor
with
@ @L uc; m
L(u c;m)=1
=
L uc; m 1 X
t=1
t 1'(t)
1 X
t=1
t t 1'(t)
. (12)
The derivative in (11) shows by how much has to increase when there is a unit decrease in
(1 m)u
c +m, in order for an agent to be willing to produce in exchange for money, under the
belief that all other agents accept money. In turn, the derivative in (12) shows by how much has to increase when there is a unit decrease in (1 m)uc +m, in order for all agents to coordinate in the use of money. The key implication is that, if (1 m)u
c +m decreases by one unit, the
required increase in for coordination reasons is smaller than the one required to sustain an agent’s individual incentive to produce in exchange for money. In other words, an increase in the frequency of trade meetings, more so than an increase in uc or a decrease in m, critically helps agents coordinate in the use of money. The intuition runs as follows. If an agent expects to meet many partners in a short period of time, he is willing to accept money even if he believes that a large number of his future partners will not accept money. That a¤ects beliefs of all other agents: they know others will be willing to produce in exchange for money, even if they hold relatively pessimistic beliefs. Consequently, pessimistic beliefs cannot be an equilibrium.
It is particularly interesting to look at the extreme case where converges to one. To do so, consider …rst the scenario where the stochastic process that governs z is symmetric, that is, E( z) = 0. In this case, since the probability that the economy will eventually reach a state to the right of the current state is equal to one (that is, P1t=1'(t) = 1) it must be that converges to one as converges to one. This means that if one restricts attention to the region of parameters where money is an equilibrium in the benchmark model of section 2.2, then as ! 1, money is always the unique equilibrium in the model of section 2.3. Finally, it is possible to construct an example where the probability of ever getting to the region where holding money is a strictly dominant action is arbitrarily small and still, as goes to 1, converges to one. Let E( z) = for some <0. Since zt follows a continuous probability distribution, as approaches zero and goes to
3
Final remarks
The notion of essentiality, by emphasizing that the societal bene…ts of money come from the fact that it achieves desirable allocations, helps identify models in which the use of money is justi…ed on grounds of e¢ciency (Wallace (2001)). However, it does not say much about how agents actually end up coordinating in the use of money, and how do the primitives of the environment impact such coordination. We think of our paper as a step in this direction.
We have chosen to present our analysis in a search model of money along the lines of Kiyotaki and Wright (1993), a choice that is mainly driven by tractability reasons. This choice though comes at a cost as their environment is special in some dimensions, particularly the indivisibility of money and the indivisibility of goods – assumptions that have been relaxed in subsequent papers such as Trejos and Wright (1995), Shi (1995, 1997) and Lagos and Wright (2005). We leave the extension to these settings for future work. Since the environment in Kiyotaki and Wright (1993) combines key elements that matter for the problem of coordinating in the use of money in a tractable manner, we see it as a natural starting point.
References
[1] Brock, W. and Scheinkman, J., 1980, Some Remarks on Monetary Policy in an Overlapping Generations Model, in Models of Monetary Economies, edited by John Kareken and Neil Wallace,Federal Reserve Bank of Minneapolis, Minneapolis.
[2] Burdzy, K., Frankel, D., and Pauzner A., 2001, Fast Equilibrium Selection by Rational Players Living in a Changing World, Econometrica 68, 163-190.
[3] Frankel, D. and Pauzner, A., 2000, Resolving indeterminacy in dynamic settings: the role of shocks,Quarterly Journal of Economics 115, 283-304.
[4] Kiyotaki, N. and R. Wright, 1989, On Money as a Medium of Exchange,Journal of Political Economy 97, 927-954.
[5] Kiyotaki, N. and R. Wright, 1993, A Search-Theoretic Approach to Monetary Economics,
American Economic Review 83, 63-77.
[6] Kocherlakota, N., 1998, Money is Memory,Journal of Economic Theory, 81, 232-51.
[7] Lagos, R. and Wright, R., 2005, A Uni…ed Framework for Monetary Theory and Policy Analy-sis,Journal of Political Economy 113, 463-484.
[9] Scheinkman, J., 1980, Discussion, in Models of Monetary Economies, edited by John Kareken and Neil Wallace,Federal Reserve Bank of Minneapolis, Minneapolis.
[10] Shi, S., 1995, Money and Prices: A Model of Search and Bargaining, Journal of Economic Theory, 67, 467-496.
[11] Shi, S., 1997, A Divisible Search Model of Fiat Money,Econometrica 65, 75-102.
[12] Trejos, A. Wright, R., 1995, Search, Bargaining, Money, and Prices, Journal of Political Econ-omy 103, 118-141.
[13] Wallace, N., 2001, Whither Monetary Economics?,International Economic Review, 42:4, 847-869.
[14] Wallace, N. and T. Zhu, 2004, A Commodity-Money Re…nement in Matching Models,Journal of Economic Theory 117, 246-258
[15] Zhou, R., 2003, Does Commodity Money Eliminate the Indeterminacy of Equilibria?, Journal of Economic Theory 110, 176–190.
[16] Zhu, T., 2003, Existence of a Monetary Steady State in a Matching Model: Indivisible Money,
Journal of Economic Theory 112, 307-324.
[17] Zhu, T., 2005, Existence of a Monetary Steady State in a Matching Model: Divisible Money,
Journal of Economic Theory 123, 135-160.
A
Proofs
A.1 The interpretation of (z) as in‡ation
Forz < bz, the government takes money from a fractionpof the agents with money and randomly redistribute to everyone in the economy without money (including the agents that lost their money). So every period pmunits of money are redistributed among 1 m+pmagents.
Suppose an agent …nishes the current period with money. The probability that she will have this unit of money at the beginning of the next period is
!= 1 p+p pm
1 m+pm=
(1 m)(1 p) +pm
1 m+pm .
For p = 0, we get ! = 1, and for p = 1, ! = m (everyone has the same chance of starting next period with money).
An agent that exerted no e¤ort will have money next period with probability
$= pm
LetV1f andV0f be the value functions for an agent with one and zero units of money, respectively, at the end of the current period, and letV1andV0be the value functions with one and zero units of
money, respectively, at the start of next period. If an agent believes that money is always accepted,
V1f = [!V1+ (1 !)V0],
V0f = [$V1+ (1 $)V0],
and
V1 = m V1f + (1 m)(u+ V
f
0 ),
V0 = m( c+ V1f) + (1 m) V
f
0.
The agent is willing to produce in exchange for money i¤
V1f V0f > c,
which can be rewritten as
(1 m)(1 p)
1 m+pm [(1 m)u+mc]> c. This condition is not satis…ed with p is su¢ciently high.
A.2 Proof of Lemma 1
The economy is in state z in period t. In case A, agents accept money in states z 2 ZA; in case
B, agents accept money in states z 2 ZB, and ZB ZA. De…ne the set Z~ = ZA ZB, i.e., Z~
comprises all states z such that agents accept money in caseA but not in case B. Consider four agents in period t in state z, two in case A (one accepts money, one doesn’t) and two in case B (one accepts money, one doesn’t). We need to show that vA(z) =vA
e (z) vAn(z) is larger than
vB(z) = vB
e (z) vnB(z), where vei(z) is the value of producing in exchange for money in case
i2 fA; Bg, and vi
n(z) is the value of not producing in exchange for money in casei2 fA; Bg.
Consider a particular realization of the process for f zg and for the random variable that de…nes whether agents meet a buyer or a seller. Let t+T be the …rst period after t that the economy is in a state z0 such that either: (i) z0 2 Z~ and the agent in case A …nds it optimal to accept money; (ii) z0 2Z~, the agent in case A …nds it optimal not to accept money and meets a seller; (iii) z0 2 ZB and the agent in case A …nds it optimal to accept money; (iv) z0 2 ZB, the
agent in case A …nds it optimal not to accept money and meets a seller; (v) z0 62Z
A, the agent in
case A …nds it optimal to accept money and meets a buyer.
value of not accepting moneyvBn (z)in using this strategy will be (weakly) lower than its true value.
Hence we will obtain an upper bound for vB(z). Since we want to show that vA(z)> vB(z),
it is enough to prove that vA(z) is larger than this upper bound for vB(z).
Now, note that after t and before t+T, there is no transaction. If the economy is not in one of the above …ve events, then either z 2 Z~ but they meet a buyer and do not accept money, or z62Z~. Then ifz2ZB, the agent in caseA…nds it optimal not to accept money (so no one accepts
money) and they meet a buyer, so there is no transaction. Otherwise z62ZAand either the agent
in case A …nds it optimal to accept money but they meet a seller, or the agent in case A …nds it optimal not to accept money (so no one accepts money). Therefore we can write:
ve(z) = c+ET;z0 Tv1(z0) + and vn(z) =ET;z0 Tv0(z0)
where v1(z0) is the value of having money in state z0 at the beginning of period t+T, v0(z0) is
the value of not having money in state z0 at the beginning of period t+T, and E
T;z0 denotes
expectations of the joint distribution of T and z0, and is whatever utility the agent gets from holding money. Hence
v(z) = c+ET;z0 Tv1(z0) Tv0(z0) +
so we need to show that
ET;z0 T vA1(z0) vA0(z0) v1B(z0) v0B(z0) >0
Note that cand do not appear in the above expression because they are the same in casesAand B.
We will show that wheneverz02Z~, v1A(z0) v0A(z0) v1B(z0) v0B(z0) >0, and in all other cases, vA
1(z0) vA0(z0) v1B(z0) vB0(z0) = 0. Since z0 2Z~ with a positive probability, we get
the claim.
Event (i): z0 2Z~ and they accept money. In caseA the agent is able to spend money, but not in case B.
In caseA:
vA1(z0) = (1 m) u+ Ez0 v0A(z00) +m Ez0 v1A(z00) ,
vA0(z0) = (1 m) Ez0 v0A(z00) +m c+ Ez0 v1A(z00) ,
so
v1A(z0) v0A(z0) = (1 m)u+mc. In caseB:
vB1(z0) = Ez0 vB1(z00) ,
Therefore
v1B(z0) vB0(z0) = (1 m) E v1B(z00) vB0(z00) +mc, and
v1A(z0) v0A(z0) vB1(z0) v0B(z0) = (1 m) u E v1B(z00) vB0(z00) .
We know that v1(z00) v0(z00) u because if the agent with no money does nothing until the
agent with money can spend it, then the di¤erence v1(z00) v0(z00) will be equal to u, where
2 f0;1;2; :::g is the number of periods the agent with money has to wait to spend it. Since that strategy might not be optimal for the agent with no money, we get that u is an upper bound for v1(z00) v0(z00). Since <1, v1A(z0) v0A(z0) v1B(z0) vB0(z0) >0.
The argument has assumed that the agent in caseApays no cost to take money to the trading post, but that is not true if z0 < bz. In that case, either the agent does not take money to the
trading post (so it is as if z0 62 ZA since agents will not be able to use money, and this case is
covered below), or the agent receives
v1A(z0) = (1 m) u+ Ez0( vA0(z00) +m Ez0 vA1(z00) + (z0), (13)
where (z0)<0, hence
v1A(z0) v0A(z0) vB1(z0) vB0(z0) = (1 m) u E v1B(z00) vB0(z00) + (z0),
but if the agent …nds it optimal to pay the cost (z0) and take money to the trading post, then
(1 m) u+ Ez0 v0A(z00) +m Ez0 v1A(z00) + (z0)> Ez0 vA1(z00) ,
which can be written as
(1 m) u Ez0 v1A(z00) + Ez0 vA0(z00) + (z0)>0, (14)
which implies vA
1(z0) vA0(z0) v1B(z0) vB0(z0) >0.
Event (ii): z0 2Z~, they accept money and meet a seller. In caseA:
v1A(z0) = u+ Ez0 v0A(z00) ,
v0A(z0) = Ez0 vA0(z00) ,
so
vA1(z0) vA0(z0) =u In caseB:
v1B(z0) = Ez0 v1B(z00) ,
Therefore
v1B(z0) vB0(z0) = E v1B(z00) v0B(z00) .
Again using (v1(z00) v0(z00)) u, we get that v1(z0) v0(z0) is strictly larger in case A.
The argument has assumed the agent in caseApays no cost to take money to the trading post, but an argument similar to the one for case (i) applies here: either the agent prefers not to take money to the trading post, which is equivalent to z0 62Z
A, or it pays o¤ to incur cost (z0) for the
possibility of using money.
Event (iii): z0 2ZB and they accept money. Then in cases Aand B,
v1(z0) = (1 m)( u+ Ez0( v0(z00) +m Ez0 v1(z00) ,
v0(z0) = (1 m) Ez0 v0(z00) +m c+ Ez0 v1(z00) ,
so
v1(z0) v0(z0) = (1 m)u+mc,
which does not depend on whether we are in caseAorB. If there is a cost (z0), that will be added to the equation forv1(z0) in casesA and B, so it makes no di¤erence.
Event (iv): z0 2ZB, they do not to accept money and meet a seller. Then
v1(z0) = u+ Ez0 v0(z00) ,
v0(z0) = Ez0 v0(z00) ,
hence
v1(z0) v0(z0) =u
in both casesAandB. If there is a cost (z0), that will be added to the equation forv1(z0)in cases
A and B, sov1(z0) v0(z0) is still the same in both cases.
Event (v): z0 62ZA, the agent …nds it optimal to accept money and meets a buyer. Then
v1(z0) = Ez0 v1(z00) ,
v0(z0) = c+ Ez0 v1(z00) ,
so
v1(z0) v0(z0) =c.
Hence v1(z0) v0(z0) is the same in cases A andB.
A.3 Proof of Lemma 2
The economy is in statezin periodt. Agents accept money ifz z . Consider four agents in period t, two in statez (one accepts money, one doesn’t) and two in statez ! (one accepts money, one doesn’t), with ! >0. Denote the value functions of those who start in statezby v0
respectively, and the value functions of those who start in statez ! byve!(z !)and v!n(z !),
respectively. We will show that v0(z) =ve0(z) v0n(z)> v!(z !) =v!
e (z !) v!n(z !).
We will consider a suboptimal strategy for the agent starting in statez ! who does not accept money in periodt: until periodt+T (de…ned below), this agent will accept money in a given state z00 if and only if it is optimal to accept money in state z00+!. This implies that, for any given
state, she will accept money whenever it is optimal for the agent who started in state z to accept money. Consequently, the value of not accepting money v!
n(z) using this strategy will be lower
than its true value. Hence we will obtain an upper bound for v!(z). Since we want to show that
v0(z)> v!(z), it is enough to prove that v0(z) is larger than this upper bound for v!(z).
Consider a particular realization of the process for f zg and for the random variable that de…nes whether agents meet a buyer or a seller. Lett+T be the …rst period aftertwhen the agents who started in state z are in a state z0 such that either: (i) z0 z and z0 2 Z0; (ii) z0 z and z062Z0 and they meet a seller; (iii)z0< z ,z0 2Z0 and they meet a buyer.
We start analyzing the case z 2 [ bz;zb], under which agents will always take money to the trading post when z > bz and will never take money to the trading post when z < zb, so the intrinsic utility of money has no e¤ect on payo¤s. We will deal with the casesz < zband z >zb later.
Denote by Z0 the set of states where it is optimal for the agent to accept money. Note that
after t and before t+T, there is no transaction. If the economy is not in one of the above cases then either z z but they meet a buyer and z0 62Z0 (so they do not accept money); or z < z and they meet a seller (who does not accept money); orz < z but they meet a buyer andz0 62Z0 (so they do not accept money). Therefore we can write:
ve0(z) = c+ET;z0 Tv10(z0) and v0n(z) =ET;z0 Tv00(z0)
where v1(z0) is the value of having money in state z0 at the beginning of period t+T, v0(z0) is
the value of not having money in state z0 at the beginning of period t+T, and E
T;z0 denotes
expectations of the joint distribution of T and z0. Hence
v0(z) = c+ET;z0 Tv10(z0) Tv00(z0) .
Analogously
v!(z !) = c+ET;z0 Tv!1(z0 !) Tv0!(z0 !) ,
so we need to show that
ET;z0 T v01(z0) v00(z0) v1!(z0 !) v0!(z0 !) >0.
A particular realization of the process forf zg leads to a particularz0. Now we consider each
of the three cases mentioned above.
First, consider the case z0 ! z . Then the situation is the same for the agents that started in stateszand z !: all of them are willing to accept money and can spend money (depending on who they meet). For the agent that started in state z
v10(z0) = (1 m) u+ Ez0 v00(z00) +m Ez0 v01(z00) ,
v00(z0) = (1 m) Ez0 v00(z00) +m c+ Ez0 v10(z00) ,
so
v10(z0) v00(z0) = (1 m)u+mc. Analogously
v1!(z0 !) v0!(z0 !) = (1 m)u+mc, and we have
v10(z0) v00(z0) v1!(z0 !) v!0(z0 !) = 0.
Now consider the case z0 ! < z . Then both agents without money are willing to accept money, but only the agent who started in state z is able to spend money. Thus for the agent that started in state z, as before
v10(z0) v00(z0) = (1 m)u+mc, but for the agent that started in state z !:
v1!(z0 !) = Ez0 v1!(z00 !) ,
v0!(z0 !) = (1 m) Ez0 v!0(z00 !) +m c+ Ez0 v!1(z00 !) .
Therefore
v1!(z0 !) v0!(z0 !) = (1 m) E v1!(z00 !) v0!(z00 !) +mc.
As argued above, v1(z00) v0(z00) u because if the agent with no money does nothing until the
agent with money can spend it, then the di¤erence (v1(z00) v0(z00) will be equal to u, where
2 f0;1;2; :::g is the number of periods the agent with money has to wait to spend it. Since that strategy might not be optimal for the agent without money,uis an upper bound forv1(z00) v0(z00).
Since <1,v10(z0) v00(z0)is strictly larger than v!
1(z0 !) v0!(z0 !).
Case (ii): z0 z ,z0 62Z0 and they meet a seller. Those without money cannot sell their good. Again, this case can be split in two.
First, consider the casez0 ! z . Then the situation is the same for the agents who started in states z and z !: all of them can spend money. For the agent who started in statez,
v10(z0) = u+ Ez0 v00(z00) ,
thus
v01(z0) v00(z0) =u. Analogously
v1!(z0 !) v0!(z0 !) =u, so
v10(z0) v00(z0) v1!(z0 !) v!0(z0 !) = 0.
Now consider the case z0 ! < z . Then only the agent who started in state z is able to spend
money. For the agent who started in state z, nothing changes and
v01(z0) v00(z0) =u,
but for the agent who started in state z !;
v1!(z0 !) = Ez0 v1!(z00 !) ,
v0!(z0 !) = Ez0 v0!(z00 !) ,
therefore
v1!(z0 !) v!0(z0 !) = Ez0 v1!(z00 !) v0!(z00 !) .
Again, usingv1(z00) v0(z00) u, we get that
v10(z0) v00(z0)> v1!(z0 !) v!0(z0 !).
Case (iii): z0 < z ,z0 2Z0 and they meet a buyer.
Now, those without money exert e¤ort (facing cost c) and sell their good. Hence for the agent who started in z,
v10(z0) = Ez0 v10(z00) ,
v00(z0) = c+ Ez0 v10(z00) ,
so
v01(z0) v00(z0) =c. Analogously,
v1!(z0 !) v!0(z0 !) =c. Hence v1(z0) v0(z0) is the same in both cases.
In sum, at timet+T, eitherz0 ! < z z0, in which casev01(z0) v00(z0)> v1!(z0 !) v!0(z0 !) orv10(z0) v00(z0) =v!
1(z0 !) v!0(z0 !). Since there is a positive probability thatz0 ! < z z0,
we get that v0(z)> v!(z !). This proves v(z) is increasing inz.
point in the future), so v(z) <0; for very high values ofz, an agent who accepts money is sure that money will be accepted in the next periods, so v(z) >0 as long as ((1 m)u+mc)> c (which is true by assumption). Continuity of payo¤s inzimplies that there exists some Z(z )such that v(Z(z )) = 0. Since v(z) is increasing in z, the optimal response to agents following a cut-o¤ strategy at z is to produce in exchange for money if and only if z Z(z ).
We now turn to the casez >bz. The agent will only take money to the trading post if
(1 m) u+ Ez0( v00(z00) +m Ez0 v10(z00) > Ez0 v01(z00) + (z0),
which implies
(1 m) u Ez0 v10(z00) v00(z00) > (z0).
An argument similar to those leading from equation (13) to equation (14) shows that, in the case z0 ! < z z0, either we get v0
1(z0) v00(z0) [v1!(z0 !) v!0(z0 !)]>0 or the situation is
equivalent to z0 < z . Moreover, if z >zb, agents who accepted money at z and z ! might get some utility from money betweentandt+T. Since ()is increasing inz, this e¤ect only reinforces the result that v(z) is increasing inz.
Lastly, the case z > bz. Again, the argument here is very similar to the ones leading from equation (13) to (14): if it makes sense to take money to the trading post, then in cases z0 ! < z z0, we get that v0(z) > v!(z !). Moreover, since () is increasing in z, in cases
z0 ! > z and both agents are willing to spend money, the agent who started in statez ! faces
a larger cost (z0 !), which reinforces the result that v(z) is increasing inz.
A.4 Proof of Lemma 3
The proof that V~1(z) V~0(z) is independent of z, when z2 [ bz;zb], is in the main text. Consider
now z > bz. In this case, agents accept and spend money if z0 z, they get some positive utility in the interval (z; zb ) and there are no transactions whenz0 <zb. Hence (3) does not hold anymore
and we have
~
V1(z) V~0(z) =
1 X
t=1
'(t)
(
tR1
z V1;z0 V0;z0 dF(z
0jt) +
t 1
X
s=1
tE[ (z
s)]dG(zsjt)
)
.
The integral term is the same as in the case z 2 [ z;b zb]. Since all agents are following a cut-o¤ strategy aroundz, there is no transaction until the …rst period the economy reaches a statez0 z, which happens in period twith probability '(t). The term inside the integral is the net expected value of having one unit of money once the economy reaches a state z0 > z. The second sum is the
expected intrinsic utility the agent will receive before period t. Repeating the steps that lead from (3) to (6), we get
~
V1(z) V~0(z) =
1 X
t=1
'(t)
(
t[(1 m)u+mc] + t 1
X
s=1
tE[ (z
s)]dG(zsjt)
)
Clearly, for a given realization of f zg,zs in increasing inz. Since (zs)is strictly increasing inzs
in the region[bz;1), we get thatV~1(z) V~0(z)is strictly increasing in zin this region. A corollary
to this result is that in casez2 >zband z1 2[ bz;zb],
~
V1(z2) V~0(z2)>V~1(z1) V~0(z1).
Finally, considerz < zb. We have
~
V1(z) V~0(z) =
X1
t=1
t'(t) R1
z V1;z0 V0;z0 dF(z
0jt)
as before, but now,
V1;z0 =m Ez0V1+ (1 m) (u+ Ez0V0) + (z0).
with (z0)<0 whenever z0 2(z; bz). Thus, we have
~
V1(z) V~0(z) =
X1
t=1
t'(t) (1 m)u+mc+E (z0) .
For a given realization of f zg, z0 is increasing in z. Since (z0) is strictly increasing in z0 in the
region z0 < zb,V~1(z) V~0(z) is strictly increasing in z in this region. A corollary to this result is
that in case z2 < bz andz1 2[ z;b zb],
~
V1(z2) V~0(z2)<V~1(z) V~0(z).
We now show that the optimal behavior of an agent in a statez2[ z;b bz]obtains by looking at the di¤erence between V~1(z) V~0(z) and c. Consider …rst the case
~
V1(z) V~0(z)> c. (15)
Let V1(z1; z2; z3) be the value of having money at the end of the period if the current state is
z1, all the other agents are following a cut-o¤ strategy at z2, and the agent is following a cut-o¤
strategy at z3 and letV0(z1; z2; z3) be the payo¤ of having no money in the same situation. There
exists a uniqueZ(z)that is a best response to all agents following a cut-o¤ strategy at z (Lemma 2). HenceZ(z) satis…es
V1[Z(z); z; Z(z)] V0[Z(z); z; Z(z)] =c. (16)
We need to show thatZ(z)< z. Suppose it is not. A corollary of Lemma 1 is thatV1[Z(z); z; Z(z)]
V0[Z(z); z; Z(z)] is decreasing in z (a larger z means money is accepted in less states). Hence
Z(z) zcombined with (16) and Lemma 1 imply
V1[Z(z); Z(z); Z(z)] V0[Z(z); Z(z); Z(z)] c
But Z(z) z combined with the fact thatV~1(z) V~0(z) is weakly increasing inz implies
V1[Z(z); Z(z); Z(z)] V0[Z(z); Z(z); Z(z)] V~1(z) V~0(z)> c
B
The case of a discrete state space
In what follows we consider the case in which the state space is discrete, given by Z, and the random process zt follows a discrete probability distribution that is independent ofzand t, with
expected value E( z) 0 and variance Var( z)>0. In order to consider in a tractable manner both the region of parameters where agents coordinate in the use of money and the region where agents do not coordinate in the use of money, we simplify the the description of the environment by assuming that, in states z >zb, it is strictly optimal to accept money because a special agent in the economy forces other agents to do so.
First, it is straightforward to adapt the proofs of Lemmas 1 and 2 to the discrete case. Thus, as in Lemma 2, if all the other agents are following a cut-o¤ strategy at some state z2Z\[ z;b zb], in which money is accepted in state z0 2Z\[ z;b bz]if and only ifz0 z, then there exists a unique best reply, given by the cut-o¤ strategy Z(z)2Z\[ z;b bz]. Combining Lemmas 1 and 2, it is also straightforward to show that Lemma 3 still holds, that is, in order to assess the agent’s optimal behavior in some statez2Z\[ bz;zb], it is su¢cient to consider the payo¤s induced by a symmetric strategy pro…le where all agents choose the same cut-o¤ strategy z.
The only di¤erence in the case of a discrete state space rests in the implications of Lemma 3. The reasoning runs as follows. As in the case of a continuous state space, since the random process zt follows a probability distribution that is independent of the current state z, it must be that
for all z and z0 in[ z;b bz],
V1(z; z; z) V0(z; z; z) =V1(z0; z0; z0) V0(z0; z0; z0).
First, assume that [V1(z; z; z) V0(z; z; z)] > c. There exists a unique Z(z) that is a best
response to all the other agents following a cut-o¤ strategy at z (adaptation of Lemma 2). By de…nition, Z(z) is the smallestz2Z\[ bz;zb]such that
fV1[Z(z); z; Z(z)] V0[Z(z); z; Z(z)]g c.
Since V1[Z(z); z; Z(z)] V0[Z(z); z; Z(z)] is strictly decreasing in z, there are two possibilities:
eitherZ(z) =zorZ(z)< z. In the latter case, the same reasoning applied to the continuous state space implies that the unique symmetric equilibrium in cut-o¤ strategies is the one in which all agents follow a cut-o¤ strategy at z = zb. If, instead, Z(z) = z, then there exists a symmetric equilibrium in which all agents follow a cut-o¤ strategy at z. Now, since z is generic, then there exists one such equilibrium for each z2Z\[ bz;zb].
In order to pin down the whole region of parameters in which the unique equilibrium has money always accepted in the regionz2Z\[ z;b bz], we only need to consider the problem of an agent in statez 12Z\[ bz;zb], who follows a cut-o¤ strategy at this state and believes that all the other agents are following a cut-o¤ strategy at statez2Z\[ z;b zb]. This agent has an strict incentive to produce in exchange for money if and only if
Denote by (t) the probability of reaching any state strictly larger thanz 1at time s+tand not before. Then, we can rewrite the above condition as
1 X
t=1
t (t)
!
[(1 m)u+mc]> c. (17)
Now, assume that [V1(z; z; z) V0(z; z; z)] < c. Again, there exists a unique Z(z) that is a
best response to all the other agents following a cut-o¤ strategy at z. By de…nition, Z(z) is the largest z2Z\[ bz;bz]such that
fV1[Z(z) 1; z; Z(z) 1] V0[Z(z) 1; z; Z(z) 1]g c.
Since V1[Z(z); z; Z(z)] V0[Z(z); z; Z(z)] is strictly decreasing in z, there are two possibilities:
either Z(z) = z or Z(z) > z. In the latter case, there exists a unique symmetric equilibrium in cut-o¤ strategies, in which all agents follow a cut-o¤ strategy at z =zb+ 1. If, instead, Z(z) = z, then there exists a symmetric equilibrium in which all agents follow a cut-o¤ strategy at z+ 1. Again, since z is generic, then there exists one such equilibrium for eachz2Z\[ bz;zb].
In order to pin down the whole region of parameters in which the unique equilibrium has money never accepted in the region z2 Z\[ z;b zb], we only need to consider the problem of an agent in state z 1 who follows a cut-o¤ strategy at state z 2 Z\[ z;b bz] and believes that all the other agents are following a cut-o¤ strategy at statez 12Z\[ bz;zb]. This agent has an strict incentive not to produce in exchange for money if and only if
[V1(z 1; z 1; z) V0(z 1; z 1; z)]< c.
Denote by +(t) the probability of reaching any state larger or equal thanz 1 at times+tand not before. Then, we can rewrite the above condition as
1 X
t=1
t
+(t)
!
[(1 m)u+mc]< c. (18)
Equations (17) or (18) are versions of (2) with (t)or +(t)instead of '(t). In the continuous case we consider the probabilities of reachingzwhen the economy starts in a state that is arbitrarily close to z. Here, we have to start from the closest state where money is not accepted or the closest state where money is accepted, depending on which strategies we want to eliminate. As there is some distance between them, the probabilities (t) and +(t) will not be the same. Hence there will be a region with multiple equilibria. But as the support of z increases, the discrete distribution gets closer to a continuous distribution, and (t) and +(t) get closer and closer to each other.
As in the continuous case, we can rewrite condition (17) as
