Monetary Trading: An Optimal Trading Mechanism under Limited Record-Keeping∗ Guilherme Carmona† University of Surrey August 5, 2016

Abstract We consider a random-matching, absence-of-double-coincidence-of-wants environment with a limited record-keeping technology: The only feasible trading mechanisms are portable-object mechanisms. The portable object is indivisible and subject to a unit upper bound. We define the monetary trading mechanism as one of such portable-object mechanisms. Both individual preference types and money holdings may be either observable or unobservable. We show that the monetary trading mechanism is an optimal portable-object trading mechanism if and only if money holdings are unobservable and types are observable. Furthermore, we show that the monetary trading mechanism is still an optimal portable-object trading mechanism when both money holdings and types are unobservable but random matching is replaced with direct search. In the cases where monetary trading is optimal, we show that any equilibrium yielding an ∗

I wish to thank Luis Araujo, Aleksander Berentsen, Alex Gershkov, Nobu Kiyotaki, Cyril Mon-

net, Mauricio Prado, Neil Wallace and seminar participants at the Copenhagen Business School and the 3rd African Search and Matching Workshop (Marrakech, Morocco) for very helpful comments. Any remaining errors are, of course, mine. † Address: University of Surrey, School of Economics, Guildford, GU2 7XH, UK; email: [email protected].

1

(average expected discounted) utility strictly higher than that of autarky is observationally equivalent to monetary trading.

Journal of Economic Literature Classification Numbers: E40; C73; D82. Keywords: Monetary trading, portable-object trading mechanisms, incentive compatibility, optimality.

1

Introduction

After Kocherlakota’s (1998a) influential paper, money is regarded to be a recordkeeping device.1 Kocherlakota’s (1998a) result is quite general and, in particular, does not depend on the particular way the allocation of resources in an economy depends on money. In other words, the reason why an intrinsic useless object such as money has value is independent of the way it is used to determine the allocation of resources. This is in contrast to the way this question has been addressed in monetary economics. In fact, in monetary equilibria in standard monetary models (e.g. the random-matching model of Kiyotaki and Wright (1989)), there are specific rules on when and how goods and money are transferred between individuals. Thus, to differentiate, we refer to the latter as monetary trading; more concretely, we define monetary trade to be a specific portable-object trading mechanism where, in particular, each person has a balance, which rises when he gives up goods, and falls when he acquires goods.2 Kiyotaki and Wright (1989) (as well as many other) have introduced explicit trading frictions under which monetary trading is an equilibrium. But are there alternative non-monetary trading mechanisms (i.e. (portable-object) trading mechanisms defined by rules different from those of monetary trading) that overcome those trading frictions in a more efficient way? If yes, then it is not clear why trade is monetary and not determined by the rules of a more efficient non-monetary trading mechanism. If, instead, monetary trading is an efficient way of overcoming the trading frictions 1 2

See also Kocherlakota (1998b). See Section 2.4 for the formal definition of monetary trading we use. The term “portable-object

trading mechanism” is based on Townsend (1987).

2

in those models, the case for monetary trading is then very compelling. Kocherlakota and Wallace (1998) (KW, henceforth) have shown that, under random matching, bilateral trade and a limited record-keeping technology (i.e. each individual does not observe the past behavior of his trading partner), the most efficient trading mechanism is monetary. In this paper we revisit KW’s optimality result for monetary trading to better understand under what conditions does this result hold. We shall argue that monetary trading is optimal if and only if money holdings are unobservable and individual preference types are observable. We depart from KW’s formalization in that their no-commitment condition on trading mechanisms is not explicitly imposed.3 More precisely, we assume, in contrast to KW, that transfers are enforceable and, in particular, we do not require that an individual, after choosing not to produce, be allowed to keep the amount of money that he had before.4 A motivation for our formalization is that it puts consumption and production on par. Indeed, in KW, an individual does not have the choice of keeping the same amount of money if he chooses to consume. Instead, as it is the case in a monetary trading mechanism, the money balance of an individual is reduced after he consumes. Thus, some level of enforcement needs to be assumed to imply that each consumer faces two options in a monetary trading mechanism: either (a) consume and reduce the amount of money owned or (b) do not consume and keep the same amount of money. In fact, such enforcement assumption is crucial to monetary trading: As Camera and Gioffr´e (2014) have shown, monetary trading cannot be sustained in large economies once consumers have the option of consuming without paying. But then such enforcement ability can also be used to give each producer two choices in an alternative, non-monetary trading mechanism: (c) produce and keep the same amount of money or (d) do not produce and reduce the amount of money owned, i.e. an individual, after choosing not to produce, may not be allowed to keep 3

However, KW’s no-commitment condition will hold when an individual’s money holdings are

not observable by his trading partner. 4 We allow, as in KW, an individual to choose not to produce so that trade in any meeting is still decentralized.

3

the same amount of money that he or she had before.5 Under enforceable transfers, the information that each individual has about his trading partner’s preference type and money holdings becomes critical. If money holdings are unobservable and types are observable, then monetary trading is optimal. This is so because private money holdings act as a substitute for no-commitment as producers can claim to have zero money, which in our setting implies that they can leave the match with the same amount of money that they have entered. This allows us to recover KW’s optimality result when types are observable and money holdings are not. More surprising is that this is the only possible case: If either types are unobservable or both types and money holdings are observable, then monetary trading is not an optimal trading mechanism. In this case, there is a non-monetary trading mechanism that is more efficient that a monetary one. Thus, monetary trading is optimal if and only if money holdings are unobservable and types are observable. The assumption that money holdings are their owner’s private information is a reasonable one in a setting like our where individuals are anonymous and, consequently, each individual’s trading history is also his own private information. The assumption of anonymous individuals also justifies the assumption that each individual’s type is his own private information. Monetary trading is then not an optimal trading mechanism. However, we show that this follows from the extreme nature of random matching. Specifically, we consider a direct search setting along the lines of Corbae, Temzelides, and Wright (2003) (CTW, henceforth) where individuals are still randomly matched but only amongst those who choose to locate themselves in a specific location. In this setting, we show that monetary trading is an optimal trading mechanism. While money holdings and types are still private information, they are now effectively known (with probability one); however, money holdings are 5

In a monetary trading mechanism, the reduction of the consumer’s money holdings reflects the

price of the consumption good. Analogously, we can regard the reduction of the balance of an individual after he chooses not to produce as reflecting the price of the good, i.e. the amount of money that the individual must give to obtain the increase in utility resulting from not producing.

4

still private information and, in particular, each producer can still claim to have zero units when he has one unit of money, thus obtaining a form of no-commitment. Our results on the optimality of monetary trading are strong in two senses. First, they hold on general conditions on the elements defining the environment. In particular, unlike the optimality results of KW and CTW (Proposition 3 and Corollary 1 of those papers, respectively), they do not require monetary trading to be such that the first-best quantity level is produced in every match. Second, in the cases where monetary trading is optimal, each equilibrium yielding an average expected discounted utility strictly higher than that of autarky, coincides with a monetary trading mechanism on the equilibrium path. Thus, any equilibrium that dominates autarky is observationally equivalent to monetary trading. In summary, we have shown that: (a) The optimality results of KW and CTW hold in general, i.e. they do not depend on particular choices of parameters, (b) any equilibrium yielding a strictly positive average expected utility is observationally equivalent to a monetary trading mechanism, and (c) these conclusions depend critically on the information that each individual has about his trading partner’s preference type and money holdings. The paper is organized as follows. The model is presented in Section 2. Section 3 contains our result for the random-matching model. The setting and results with direct search are presented in Section 4. Some concluding remarks are in Section 5. The proof of our results are in the Appendix.

2

The model

In this section, we describe the environment, define our equilibrium concept, formalize the optimum problem and define the monetary trading mechanism.

2.1

The environment

The environment is analogous to that of Kocherlakota and Wallace (1998) without memory (thus, also analogous to the environment introduced in Aiyagari, Wallace, 5

and Wright (1996)). Time is discrete and the horizon is infinite. There are N ≥ 3 distinct perishable goods at each date and there is a [0, 1] continuum of each of N types of people. Each type is specialized in consumption and production: A type n person consumes good n and produces good n+1 (modulo N ). Each person maximizes expected discounted utility with discount factor β ∈ (0, 1). In each period, each person can produce a quantity on a set A ⊆ R+ . We assume that A is compact, that 0 ∈ A and that A has at least two elements. Two particular cases are of interest: The indivisible goods case where A = {0, 1} and the divisible goods case where A = [0, a ¯] for some a ¯ > 0. In each period, the utility of producing a ∈ A equals −a and the utility for a type n person of consuming a units of good n equals u(a), where u : R+ → R+ is increasing and twice differentiable, with u(0) = 0, u′′ < 0 and u′ (0) = ∞. We further assume that if a∗ ∈ A solves maxa∈A [u(a) − a], then a∗ > 0.6 In each period, each person can produce a quantity on a set A ⊆ R+ . We assume that A is compact, that 0 ∈ A and that A has at least two elements. Two particular cases are of interest: A = {0, 1} and A = [0, a ¯] for some a ¯ > 0. In each period, the utility of producing a ∈ A equals −a and the utility for a type n person of consuming a units of good n equals u(a), where u : R+ → R+ is increasing and twice differentiable, with u(0) = 0, u′′ < 0 and u′ (0) = ∞. Let a∗ ∈ A solve maxa∈A [u(a) − a] and we assume that a∗ > 0. There is an additional good called money which is neither produced nor consumed. Money is indivisible and perfectly durable and each person can store at most one unit of it from one date to the next. In each period, people are randomly matched in pairs (except in Section 4 where we consider direct search). This matching is such that, for each person, the distribution of partners’ type and money holdings from which an agent’s meeting is drawn matches the demographic distribution of types and money holdings in the entire population of the economy. 6

Due to u(0) = 0, this assumption is equivalent to u(a∗ ) > a∗ for each a∗ ∈ A that solves

maxa∈A [u(a) − a].

6

Each person in a meeting may or may not know his trading partner’s type and may or may not know his trading partner’s money holdings. We consider the resulting four possible cases.

2.2

Direct trading mechanisms

Trade between agents in the economy is described by a trading mechanism. We focus on trading mechanisms that are symmetric across agents (i.e. that treat equally all agents of the same type), stationary (i.e. that do not depend on the time period) and direct (i.e. agents are asked to report their types and money holdings). Such mechanisms are described as follows. In each meeting, each agent observes his trading partner’s variables (type and money holdings) that are observable and then reports those that are his own private information. When types are private information, each agent can claim to be of any possible type; in contrast, when money holdings are private information, each agent reports his money holdings by showing (part of) it, e.g. by placing some of his money on a table. As a function of the observed or reported values of types and money holdings for each agent in the meeting, the producer receives an action recommendation of how much to produce. The action taken by the producer, as well as the (observed or reported) vector of money holdings, determine each agent’s transfer of money. Formally, a symmetric and stationary direct trading mechanism π (a trading mechanism, for short) is defined by a decision function B : {0, 1}2 → A and a transfer function t : {0, 1}2 × A → {−1, 0, 1}2 . The interpretation of the function B is that B(m) describes the recommendation that the producer in an (observed or reported) single-coincidence meeting receives regarding how much to produce when both he and the consumer have observed or reported money holdings m = (mp , mc ) ∈ {0, 1}2 , where mp refers to the producer and mc to the consumer. Throughout the paper, for each vector concerning money holdings, the first coordinate refers to the producer while the second to the consumer. Thus, we often write t = (tp , tc ). For each (m, a) ∈ {0, 1}2 × A and i = p, c, ti (m, a) is the transfer that agent i receives (or gives, if negative) when the (observed or reported) money holdings equal m and the 7

producer’s action equals a. For example, here is how the mechanism works in the case where types are observable and money holdings are not: 1. Two agents meet. 2. Each one observes the type of the other. 3. If the match is a non-coincidence one, then nothing happens. If it is a singlecoincidence one, then: 4. Agents simultaneously report money holdings m ˜ with m ˜ i ≤ mi for each i = p, c (here mi denotes i’s true money holdings). 5. Producer receives a recommended action B(m). ˜ 6. Producer chooses an action a (not necessarily equal to B(m)). ˜ 7. Transfers t(m, ˜ a) are made. The functions t represents transfers of money between the agents, hence it must satisfy tp (m, a) + tc (m, a) = 0 for each (m, a) ∈ {0, 1}2 × A.

(1)

Furthermore, no agent can transfer more money than he has declared to have. Thus, it must be that mi + ti (m, a) ≥ 0 for each i = p, c and (m, a) ∈ {0, 1}2 × A.

(2)

We say that a trading mechanism π is feasible if (1) and (2) hold. Note that the unit upper bound is not imposed on the transfer function but rather on each agent’s next period money holdings defined as follows. For each i = p, c, next period’s money holdings of agent i in a single-coincidence meeting depends on the agent’s current money holdings mi , the reported money holdings m ˜ and the action a chosen by the producer. It is described by the function Ti : {0, 1}×{0, 1}2 ×A → {0, 1} defined by Ti (mi , m, ˜ a) = min{1, mi + ti (m, ˜ a)} for each (mi , m, ˜ a) ∈ {0, 1} × {0, 1}2 × A. (3) 8

Actual trade in the economy depends on the trading mechanism π being used and on the distribution of money holdings in the economy. The latter is assumed to be symmetric across types and is, thus, described by an element q of ∆ = {(q0′ , q1′ ) ∈ R2+ : q0′ + q1′ = 1}, where qm is the fraction of people of each type having m units of money for each m ∈ {0, 1}. We focus on stationary distributions of the Markov chain on the set {0, 1} that π (together with specification of the economy) induces when agents are truthful and obedient.7 The following notion is convenient to describe it: For each m ∈ {0, 1}2 , let T (m) = (Tp (mp , m, B(m)), Tc (mc , m, B(m))). A stationary distribution of the Markov chain on {0, 1} induced by π is q such that q1 = 1 − q0 and





   ∑ ∑ qm qm N −2    q0 = q0  + +   N N N }  | {z m:Tc (m,0)=0 m:Tp (0,m)=0  {z } no-coincidence | {z } | 

producer

consumer



(4)

   ∑ ∑ qm  qm   + + q1  .  N N m:T (m,1)=0 m:T (1,m)=0   p c {z } | {z } | producer

consumer

The terms in parenthesis on the right-hand side of the above equation describe the probability of an individual having zero units of money tomorrow when he has zero or one units, respectively, today. Thus, the right-hand side of the above equation is the fraction of people having zero units of money tomorrow; the requirement that it equals the fraction of people having zero units of money today is the standard stationarity requirement. 7

It is easy to see that any q ∈ ∆ is a stationary distribution of the Markov chain on {0, 1}

induced by π if mi + ti (m, B(m)) ≤ 1 for each m ∈ {0, 1}2 and i = p, c. Moreover, if q ∈ ∆ is a stationary distribution of the Markov chain on {0, 1} induced by π and mi + ti (m, B(m)) > 1 for some m ∈ {0, 1}2 and i = p, c, then either q0 = 0 or q1 = 0. This result will, however, never be used in this paper.

9

Following Okuno-Fujiwara and Postlewaite (1995), we refer to a pair µ = (π, q) as a trading norm when π is a trading mechanism and q ∈ ∆. We say that a trading norm µ = (π, q) is stationary if q is a stationary distributions of the Markov chain on {0, 1} induced by π. Given a stationary trading norm µ = (π, q), the utility each agent receives by being truthful and obedient is described by a function V : {0, 1} → R. Specifically, for each m ∈ {0, 1}, V (m) (which we write as Vm ) gives the expected discounted utility of an agent having m units of money. The function V satisfies [ ] [ ] q0 −(1 − β)B(0, 0) + βVTp (0,0) + q1 −(1 − β)B(0, 1) + βVTp (0,1) (N − 2)βV0 V0 = + N N | {z } | {z } producer no-coincidence (5) [ ] [ ] q0 (1 − β)u(B(0, 0)) + βVTc (0,0) + q1 (1 − β)u(B(1, 0)) + βVTc (1,0) + N {z } | consumer

and

[ ] [ ] q0 −(1 − β)B(1, 0) + βVTp (1,0) + q1 −(1 − β)B(1, 1) + βVTp (1,1) (N − 2)βV1 + V1 = N N {z } | {z } | producer no-coincidence (6) [ ] [ ] q0 (1 − β)u(B(0, 1)) + βVTc (0,1) + q1 (1 − β)u(B(1, 1)) + βVTc (1,1) + . N | {z } consumer

The function V above allows us to verify whether or not an agent in a singlecoincidence meeting has an incentive to be truthful and obedient. In each meeting, each agent has a belief p = (pn′ ,m′ )n′ ∈N,m′ ∈{0,1} on the type and money holdings of her trading partner which depends on what she has observed. Let P (n, m) be the set of beliefs that an individual of type n having m units of money may have depending on what she observes. Specifically, when types and money holdings are observable, then pn′ ,m′ = 1 where n′ is her trading partner’s type and m′ her trading partner’s money holdings; hence P (n, m) consists of the 2N degenerate probability distributions on N × {0, 1}. If only the type is observable, then the agent is certain about the type but (in general) not on the money holdings of her trading partner. Given that q is the distribution of money holdings, which is symmetric across types, then pn′ ,0 = q0 and 10

pn′ ,1 = q1 where n′ is her trading partner’s type. Thus P (n, m) consists of the N such probability distributions on N × {0, 1}. When only the money holdings are observable, then pn′ ,m′ = 1/N for all n′ ∈ N where m′ is her trading partner’s money holdings; therefore P (n, m) is the set of the 2 such probability distributions on N × {0, 1}. Finally, when neither types nor money holdings are observable, then pn′ ,m′ = qm /N for all (n′ , m′ ) ∈ N × {0, 1} and P (n, m) is the singleton set containing this probability distribution on N × {0, 1}. In summary: Types

Money

Beliefs

Observable

observable

pn′ ,m′ = 1 for some (n′ , m′ ) ∈ N × {0, 1}.

Observable

unobservable pn′ ,0 = q0 and pn′ ,1 = q1 for some n′ ∈ N .

Unobservable observable

pn′ ,m′ = 1/N for some m′ ∈ {0, 1} and all n′ ∈ N .

Unobservable unobservable pn′ ,m′ = qm /N for all (n′ , m′ ) ∈ N × {0, 1}. Possible lies also depend on these cases too. Let D(n) be the set of possible types an agent can report when her true type is n and D(m) be the set of possible money holdings an agent can report when her true money holdings are m. Agents cannot lie about a variable when its value is observable, hence D(n) = {n} when types are observable and D(m) = {m} when money holdings are observable. When types are unobservable, each agent can declare to be of any possible type, thus D(n) = {1, . . . , N }. In contrast, when money holdings are not observable, each agent’s report consists of placing some of his money on the table and, thus, an agent cannot claim to have more money than he has. Thus, D(m) = {m ˜ ∈ {0, 1} : m ˜ ≤ m}, i.e. D(0) = {0} and D(1) = {0, 1}. 11

Consider an agent of type n ∈ N and with money holdings equal to m ∈ {0, 1}, who can claim (˜ n, m) ˜ such that n ˜ ∈ D(n) and m ˜ ∈ D(m). For each n, n′ ∈ N and y ∈ A, let un,n′ (y) be a type n agent’s utility from consuming y units produced by a agent of type n′ . Thus,   u(y) un,n′ (y) =

 0

if n′ = n − 1(modN ),

(7)

otherwise.

We have that being truthful and obedient is optimal if, for each n, n ˜ ∈ N , m, m ˜ ∈ {0, 1} with n ˜ ∈ D(n) and m ˜ ∈ D(m), a0 , a1 ∈ A and p ∈ P (n, m), 1 ∑

p(n+1)m′ [−(1 − β)B(m, m′ ) + βVTp (m,m′ ) ]

m′ =0 1 ∑

+

p(n−1)m′ [(1 − β)u(B(m′ , m)) + βVTc (m′ ,m) ] ≥

m′ =0 1 ∑

(8)

p(˜n+1)m′ [−(1 − β)am′ + βVTp (m,(m,m ˜ ′ ),am′ ) ]

m′ =0

+

1 ∑

′ ,m)) p(˜n−1)m′ [(1 − β)un,˜n−1 (B(m′ , m)) ˜ + βVTc (m,(m′ ,m),B(m ˜ ˜ ].

m′ =0

We say that a stationary trading norm µ = (π, q) is incentive compatible if (8) holds.8 We say that a trading norm µ = (π, q) is an equilibrium if (a) π is feasible, (b) µ is stationary and (c) µ is incentive compatible. The set of equilibria is denoted by E.

2.3

The optimum problem

The optimum problem is analogous to the one in Kocherlakota and Wallace (1998) and with a similar interpretation. Its goal is to choose an equilibrium that maximizes the average expected discounted utility of those who have money and those who do 8

Note that in (8) the no-coincidence case has been omitted since considering it explicitly would (∑ ) (∑ ) add the term βVm = βVm for all n ˜ ∈ (n′ ,m′ ):n′ ̸∈{n−1,n+1} pn′ m′ (n′ ,m′ ):n′ ̸∈{˜ n−1,˜ n+1} pn′ m′ ∑ ∑ D(n) to both of its sides. The equality (n′ ,m′ ):n′ ̸∈{n−1,n+1} pn′ m′ = (n′ ,m′ ):n′ ̸∈{˜n−1,˜n+1} pn′ m′ is ∑ clear when types are observable (n = n ˜ in that case) and (n′ ,m′ ):n′ ̸∈{n−1,n+1} pn′ m′ = (N − 2)/N = ∑ ′ ′ (n′ ,m′ ):n′ ̸∈{˜ n−1,˜ n+1} pn m when types are unobservable.

12

not, with weights given by the proportion of people holding and not holding money, respectively. Thus, in this way, every agent is treated in the same way. Given a trading norm µ, let its expected discounted utility function V be written more explicitly as V (µ). The average expected discounted utility is then 1 ∑ 1 ∑ qm qm′ W (µ) = q0 V0 (µ) + q1 V1 (µ) = [u(B(m, m′ )) − B(m, m′ )]. N m=0 m′ =0

The optimum problem is max W (µ). µ∈E

Any of its solutions is called an optimal equilibrium.

2.4

Monetary trading norms

The notion of a monetary trading mechanism that we introduce in this section is effectively as in Kocherlakota and Wallace (1998) and is such that, in certain singlecoincidence meetings, the produces produces and receives money, while the consumer gives money and receives the consumption good. We say that µ = (π, q) is a monetary trading norm if 0 < q0 < 1 and, for some y > 0 such that u(y) > y,   y

if m = (0, 1),

B(m) =

 0 otherwise, and   (−1, 1) if m = (0, 1) and a = y, t(m, a) =  (0, 0) otherwise.

The interpretation is clear: In a monetary trading norm, the producer produces (a strictly positive quantity) when he has (announced) zero units of money and the consumer has (announced) one unit of money. In this case, the producer receives, and the consumer gives, one unit of money. In the remaining cases, the producer does not produce and there is no transfers of money. These cases can be interpreted as follows: When the money holdings are m = (mp , mc ) = (0, 0) or m = (1, 0), the consumer has no money to pay for the good; and when m = (1, 1), then the 13

producer cannot receive more money due to unit upper bound on money holdings. Furthermore, 0 < q0 < 1 means that trade actually takes place under such monetary trading norm as there is a strictly positive probability (equal to q0 (1 − q0 )/N ) that a producer with zero units of money meets a consumer with one unit of money. In the above definition, the actual monetary trading norm depends on q0 and y and, thus, we denote it by µM q0 ,y . We let M = {µM q0 ,y : 0 < q0 < 1, 0 < y < u(y)}

(9)

denote the set of all monetary trading norms.

3

Results

Our main results for the random-matching model are presented in this section: The non-optimality of the monetary trading mechanism when either types are unobservable or both types and money holdings are observable (Section 3.1) and its optimality with observable types and non-observable money holdings (Section 3.2).

3.1

Unobservable types or observable types and money holdings

In the case of either unobservable types or observable types and money holdings, a grim-trigger trading norm yields an higher average expected discounted than any monetary trading norm. We say that µ = (π, q) is a grim-trigger trading norm if q0 = 0 and, for some y > 0 such that u(y) > y,   y if m = (1, 1), B(m) =  0 otherwise, and   (−1, 1) if m = (1, 1) and a ̸= y, t(m, a) =  (0, 0) otherwise. The interpretation is as follows. As long as both the producer and the consumer in a single-coincidence meeting have one unit of money, then the producer produces a 14

strictly positive quantity. Furthermore, no money is transferred so that each individual’s next period money holdings is still equal to one. In contrast, if the producer fails to produce this implies that he loses his money (while it is transferred to the consumer, he cannot keep it due to the unit upper bound on money holdings). Thus, effectively, a producer who fails to produce once will be in an autarkic situation for the remaining time periods. Each individual’s money holdings is, in a grim-trigger trading mechanism, simply a signal of whether or not he has produced in all preceding periods in which he was a producer in a single-coincidence meeting. The distribution of money holdings is such that all individuals start with one unit of money, which is a stationary distribution of the Markov chain induced by any grim-trigger trading mechanism. The actual grim-trigger trading norm depends on y and, thus, we denote it by µG y. We let G = {µG y : 0 < y < u(y)}

(10)

denote the set of all grim-trigger trading norms. Proposition 1 Suppose that either types are unobservable or both types and money holdings are observable. For each µM ∈ M ∩ E, there exists µG ∈ G ∩ E such that W (µG ) > W (µM ). Proposition 1 implies that no monetary trading norm is an optimal equilibrium. This is so because it is possible to support the same production level y in singlecoincidence meetings with a grim-trigger trading norm. Under the latter trading norm, trade, which is efficient due to u(y) > y, occurs at every single-coincidence meeting, hence with probability 1/N . In contrast, in the former monetary trading norm, trade occurs only on those single-coincidence meetings where the producer has zero units of money and the consumer has one unit of money, thus with probability at most 1/(4N ). The following corollary to Proposition 1 summarizes this result. Corollary 1 If µ ∈ M and either types are unobservable or both types and money holdings are observable, then µ is not an optimal equilibrium. 15

3.2

Observable types and unobservable money holdings

We now assume that each agent’s type is observable and that his money holdings are not observable by his trading parter. Under these assumptions, we obtain that all equilibria coincide with some monetary trading norm on the equilibrium path. In particular, this means that, when types are observable and money holdings are nonobservable, grim-trigger trading norms are no longer equilibria. In fact, the truthtelling incentive compatibility constraint fails as any producer in a single-coincidence meeting with one unit of money strictly prefers to report that he has zero units of money. In this way, he does not produce in the current period and, as he neither gains nor loses money, starts next period still with one unit of money. In the case of a monetary trading norm, and unlike in the grim-trigger trading norm, a producer is required to produce only when he has zero units of money. Thus, as he cannot claim to have one unit of money, he cannot take advantage of the possibility of misreporting his money holdings to avoid having to produce. Proposition 2 below shows that this is the only way of avoiding the temptation of individuals misreporting their money holdings and having a strictly positive probability of trade. In fact, it shows that any equilibria that yield a strictly positive average expected discounted utility must coincide with a monetary trading norm on the equilibrium path. Proposition 2 Suppose that types are observable and money holdings are not observable. If µ = (π, q) ∈ E is such that W (µ) > 0, then there exists π ∗ = (B ∗ , t∗ ) such that (π ∗ , q) ∈ M and, for all m ∈ {0, 1}2 , B(m) = B ∗ (m) and T (m) = T ∗ (m). Proposition 2 imposes no conditions on transfers off the equilibrium path (i.e. when a ̸= B(m)). However, whenever a = B(m), the transfer on an equilibrium trading norm must be equal to those of a monetary trading mechanism. Furthermore, the production decisions must coincide with those of a monetary trading mechanism. Thus, in particular, we have that B(m) > 0 if and only if m = (0, 1), i.e. a strictly positive quantity is produced and traded if and only if a producer with zero units of money meets a consumer with one unit of money. 16

It also follows from Proposition 2 that, in the stationary distribution of money holdings, a strictly positive fraction of agents have zero units of money and a strictly positive fraction of agents have one unit of money. This means that there is a strictly positive probability of trade, which is required for the average expected discounted utility to be strictly positive. As a corollary we obtain that there is an monetary trading norm that is an optimal equilibrium. Corollary 2 Suppose that types are observable and that money holdings are not observable. If supµ∈E W (µ) > 0, then there exists an optimal equilibrium µ∗ ∈ M . The corollary says that for any specification of the environment, i.e. a specification of N , β, A and u, there is an equilibrium monetary trading norm that solves the optimum problem. Since a monetary trading norm is specified by q0 ∈ (0, 1) and y ∈ A, this means that there is an optimal fraction of individuals holding zero units of money and an optimal quantity traded in each match. A more precise description of the optimal q0 and y can be given in some special cases (see Section A.8 for details). One is the case where A = {0, a∗ } with u(a∗ ) > a∗ so that the consumption goods are indivisible. In this case, y = a∗ and q0 = 1/2 9 M when the monetary trading norm µM 1/2,a∗ is an equilibrium. When µ1/2,a∗ is not an

equilibrium, then q0 ∈ (1/2, 1) as the incentive compatibility constraints are easier to satisfy the higher q0 is. The optimal q0 is the smallest such value that makes the corresponding monetary trading norm be an equilibrium. Another case is when A = [0, a ¯] with a∗ < a ¯ (recall that a∗ solves maxa∈A [u(a)−a]). As before, if µM a∗ ,1/2 is an equilibrium, then this is the optimal monetary trading norm. ∗ Moreover, when µM a∗ ,1/2 is not an equilibrium, then the optimum has 0 < y < a and

q0 > 1/2. 9

This is analogous to Kiyotaki and Wright (1989). As they have pointed out, a high fraction of

agents with zero units of money implies that trade is infrequent due to the lack of many consumers with positive money holdings. Furthermore, a small fraction of agents with zero units of money also implies that trade is infrequent due to the lack of many producers with zero money holdings. Thus, trade frequency and average expected discounted utility are maximized when q0 = 1/2.

17

Corollary 2 is a version of KW’s optimality result for monetary trading norms in our setting. Indeed, feasibility (namely, condition (2)) requires that someone who reports zero money must receive a non-negative transfer. Thus, KW’s non-commitment condition is satisfied in our setting since each individual can choose not to produce and claim to have zero units of money, thus (at least) keeping his money holdings.

4

Direct search

The assumptions under which monetary trading is optimal, observable types and unobservable money holdings, hold in settings where individuals are not anonymous but cannot observe the trading histories of their trading partners. As we show in this section, the optimality of monetary trading also holds when individuals are anonymous and, therefore, both types and money holdings are unobservable provided that individuals are matched in a more direct way that effectively reveals to each individual the type of his trading partner. To this end we now consider a framework where individuals match in a more direct way in the spirit of Corbae, Temzelides, and Wright (2003). In this setting, while neither types nor money holdings are observable, each individual effectively knows (in the sense of probability one belief) the type and money holdings of his trading partner. However, it is still possible for each individual to lie, namely on their money holdings. As a result, we shall show that monetary trading is an optimal trading norm. Bilateral matching is described by an assignment rule mapping characteristics into characteristics (see Corbae, Temzelides, and Wright (2003, Section 4)). Our interpretation is that a pair of characteristics assigned to each other in this way means that all the individuals having one of the two characteristics go to the same location and the members of one group are randomly matched with elements of the other. The characteristics that (potentially) determine the assignment rule include individuals’ type and money holdings. These two characteristics are enough to describe 18

monetary trading as defined below but additional characteristics are needed to describe other trading mechanisms such as a grim-trigger trading mechanism. The latter is defined so that half of the individuals of each type produce in odd periods and consume in even periods while the other half does the opposite. To allow for the above form of asymmetry, we first partition, for each n ∈ N , the set of individual of type n into K subsets Cn,1 , . . . , Cn,K . The measure Cn,k is αn,k > 0 for each (n, k) ∈ N × {1, . . . , K} and we assume that the fraction of individuals with m ∈ {0, 1} units of money in Cn,k equals qm independently of n and k. An assignment rule is a function ψ : N × {1, . . . , K} × {0, 1} → N × {1, . . . , K} × {0, 1} such that ψ ◦ ψ equals the identity. We do not impose that ψ is measure-preserving, which means that there is a chance that some individuals are actually unmatched under ψ. Thus, ψ(n, k, m) describes the set of individuals that a person with characteristics (n, k, m) may potentially be matched with. As a result, we interpret (n′ , k ′ , m′ ) = ψ(n, k, m) (which implies that ψ(n′ , k ′ , m′ ) = ψ ◦ ψ(n, k, m) = (n, k, m)) as saying that individuals with characteristics (n, k, m) and (n′ , k ′ , m′ ) go to the same location and are then randomly matched. Second, we allow the matching process to depend on a finite-valued state variable ω ∈ {1, . . . , Ω}. The state variable evolves deterministically, with state ω + 1(modΩ) following state ω for each ω ∈ {1, . . . , Ω}. Consequently, we consider a family of assignment rules Ψ = (ψ1 , . . . , ψΩ ). A trading mechanism is now a family of assignment rules Ψ together with a decision function B : {0, 1}2 → A and a transfer function t : {0, 1}2 ×A → {−1, 0, 1}2 ; we write π = (Ψ, B, t). We say that a trading mechanism π = (Ψ, B, t) is feasible if (B, t) satisfy (1) and (2). As before, we refer to a pair µ = (π, q) as a trading norm when π is a trading mechanism and q ∈ ∆. From the perspective of each individual, the trading norm works as follows, assuming that all the other individuals are truthful and obedient and given ω ∈ {1, . . . , Ω}:10 ˆ m) 1. Announce (ˆ n, k, ˆ and potentially be matched with someone with characteris10

In what follows and to simplify the notation, type N + 1 is set to be equal to type 1, type 0 is

type N and state Ω + 1 is state 1.

19

ˆ m). tics (n′ , k ′ , m′ ) = ψω (ˆ n, k, ˆ 11 2. Report (˜ n, m) ˜ to his trading partner. 3. If n′ = n ˜ + 1, then receive recommendation B(m, ˜ m′ ), choose a and resulting transfers t(m, ˜ m′ , a) are made. 4. If n′ = n ˜ − 1, then B(m′ , m) ˜ is chosen and transfers t(m′ , m, ˜ B(m′ , m)) ˜ are made. 5. If n′ ̸∈ {˜ n − 1, n ˜ + 1}, then no production and no transfers take place. The following two examples illustrate. We say that µ is a grim-trigger norm under G direct search if (a) µ = (Ψ, µG y ) with µy ∈ G, (recall (10)) (b) Ω = K = 2, and (c)

for each (n, k, m) ∈ N × {1, . . . , K} × {0, 1}, αn,k = 1/2, ψ1 (n, 1, m) = (n + 1, 2, m), ψ1 (n, 2, m) = (n − 1, 1, m), ψ2 (n, 1, m) = (n − 1, 2, m) and ψ2 (n, 2, m) = (n + 1, 1, m). Thus, individuals’ behavior and the transition of money holdings is as before (part (a)), there are two states of the world and two groups of individuals per type (part (b)), and half of the individuals of each type produce in odd periods and consume in even periods (group 1) while the other half does the opposite (group 2). The definition of a monetary trading norm in this setting is as follows. We say that µ is a monetary trading norm under direct search if (a) µ = (Ψ, µM q0 ,y ) for some µM q0 ,y ∈ M (recall (9)), and (b) for some K, Ω ∈ N and all (ω, n, k) ∈ {1, . . . , Ω} × N × {1, . . . , K}, ψω (n, k, 0) = (n + 1, k ′ , 1) for some k ′ ∈ {1, . . . , K} and ψω (n, k, 1) = ˆ 0) for some kˆ ∈ {1, . . . , K}. Thus, individuals’ behavior and the transition of (n−1, k, money holdings is as before (part (a)), and all individuals with money are consumers and all individuals without money are producers (part (b)). We let the set of all monetary trading norms under direct search be denoted by M ∗ . Consider an individual of type n, belonging to Cn,k and having m units of money. Let (n′ , k ′ , m′ ) = ψω (n, k, m) for some ω and, for convenience, let θ = (n, k, m, n′ , k ′ , m′ ) 11

ˆ m) More precisely, the individual goes to the location where those with characteristics ψω (ˆ n, k, ˆ

are.

20

denote the characteristics of the individuals in the match. The probability that such θ individual meets someone is denoted by αn,k,m and is given by

{ θ αn,k,m

= min

Likewise, we write

} αn′ ,k′ qm′ ,1 . αn,k qm

{ αnθ ′ ,k′ ,m′

= min

} αn,k qm ,1 . αn′ ,k′ qm′

We again focus on stationary distributions of the Markov chain on the set {0, 1} that π (together with specification of the economy) induces when individuals are truthful and obedient. A stationary distribution of the Markov chain on {0, 1} induced by π is q such that q1 = 1 − q0 and, for each (ω, n, k) ∈ {1, . . . , Ω} × N × {1, . . . , K}, q0 =

1 ∑

( θm ) θm qm αn,k,m 1Pm (n′m , m′m ) + αn,k,m 1Cm (n′m , m′m )

m=0

(11)

θ0 θ0 + q0 αn,k,0 1N \{n−1,n+1} (n′m ) + q0 (1 − αn,k,0 ) ′ where, for each m ∈ {0, 1}, (n′m , km , m′m ) = ψω (n, k, m) are the characteristics of

those that are potentially matched with individuals with characteristics (n, k, m), ′ θm = (n, k, m, n′m , km , m′m ), Pm = {(ˆ n, m) ˆ :n ˆ = n + 1, Tp (m, m) ˆ = 0} is the set of

types and money holdings such that if an individual with type n and money holdings m is matched with someone with those types and money holdings, then he is the producer and leaves the match with zero units of money and Cm = {(ˆ n, m) ˆ : n ˆ = n−1, Tc (m, ˆ m) = 0} is analogous. We say that a trading norm µ = (π, q) is stationary if q is a stationary distributions of the Markov chain on {0, 1} induced by π. To define the utility and incentive compatibility of trading norms, we first define the current utility and next period’s money holdings that can arise. Consider a match between two people with characteristics (n, k, m) and (n′ , k ′ , m′ ), which we abbreviate by writing θ = (n, k, m, n′ , k ′ , m′ ). Moreover, suppose that the person with characteristics (n, k, m) claims to be of type n ˜ and to have m ˜ units of money, and that, in the case where this person is the producer in a single-coincidence meeting (i.e. n′ = n ˜ + 1), he chooses action a. The current utility and next period’s money ˜ a) and Tn,m (θ, ˜ a, m) respectively, holdings of the person of type n are denoted by vn (θ, 21

where θ˜ = (˜ n, k, m, ˜ n′ , k ′ , m′ ) is the characteristics that those individuals claim to have (note that these quantities are independent of k). They are given by (recall the definition of un,˜n−1 in (7))     un,˜n−1 (B(m′ , m)) ˜ if n′ = n ˜ − 1,    ˜ a) = −a vn (θ, if n′ = n ˜ + 1,      0 otherwise,     min{1, m + tc ((m′ , m), ˜ B(m′ , m))} ˜ if n′ = n ˜ − 1,    ˜ a) = min{1, m + t ((m, ′ Tn,m (θ, if n′ = n ˜ + 1, p ˜ m ), a)}      m otherwise. Given a stationary trading norm µ = (π, q), the utility each agent receives by being truthful and obedient is described by a function V : Ω × N × {1, . . . , K} × {0, 1} → R such that, for each (ω, n, k, m) ∈ {1, . . . , Ω} × N × {1, . . . , K} × {0, 1}, [ ] θ θ Vω,n,k,m = (1−αn,k,m )βVω+1,n,k,m +αn,k,m (1 − β)vn (θ, B(m, m′ )) + βVω+1,n,k,Tn,m (θ,B(m,m′ )) , (12) where (n′ , k ′ , m′ ) = ψω (n, k, m) and θ = (n, k, m, n′ , k ′ , m′ ). Incentive compatibility requires each individual to find it optimal to be truthful, both when he is matched with someone and also prior to be matched, and to be obedient when he is the producer in a match. Let (ω, n, k, m) ∈ {1, . . . , Ω} × N × {1, . . . , K} × {0, 1} be given and note that being truthful yields Vω,n,k,m for an individual with characteristics (n, k, m) at state ω. He can, however, claim to ˆ m) have characteristics (ˆ n, k, ˆ ∈ N × {1, . . . , K} × {0, 1}, in which case he will potenˆ m). tially be matched with someone with characteristics ψω (ˆ n, k, ˆ This may change ˆ ˆ m, , where θˆ = (ˆ n, k, ˆ n′ , k ′ , m′ ) the probability of being matched, which is now αnθˆ ,k, ˆm ˆ

ˆ m). and (n′ , k ′ , m′ ) = ψω (ˆ n, k, ˆ He can further lie and claim to be of type n ˜ and to have m ˜ units of money. In this case, if matched, the match is between individuals who claim to have characteristics θ˜ = (˜ n, k, m, ˜ n′ , k ′ , m′ ).12 We say that a 12

Recall that the outcome of the match is independent of k; thus we could have written θ˜ =

˜ m, (˜ n, k, ˜ n′ , k ′ , m′ ) with k˜ ∈ {1, . . . , K}.

22

stationary trading norm µ = (π, q) is incentive compatible if, for each (ω, n, k, m) ∈ {1, . . . , Ω} × N × {1, . . . , K} × {0, 1}, ] [ ˆ θˆ ˜ )βV + α (1 − β)v ( θ, a) + βV (13) Vω,n,k,m ≥ (1 − αnθˆ ,k, ˜ ω+1,n,k,m n ˆm ˆm ω+1,n,k,Tn,m (θ,a) ˆ n ˆ ,k, ˆ ˆ m) for all (ˆ n, k, ˆ ∈ N × {1, . . . , K} × {0, 1}, (˜ n, m) ˜ ∈ N × {0, 1} and a ∈ A such that m, ˆ m ˜ ∈ D(m). We say that a trading norm µ = (π, q) is an equilibrium if (a) π is feasible, (b) µ is stationary and (c) µ is incentive compatible. The set of equilibria is denoted by E ∗ . The average expected discounted utility is, for each ω ∈ {1, . . . , Ω}, Wω (µ) =

∑ αn,k qm Vω,n,k,m (µ). N n,k,m

Proposition 3 shows that any equilibrium under direct search yielding a strictly positive average expected discounted utility is such that, in every period, those with zero units of money are producers and those with one unit of money are consumers. Furthermore, in each match, the producer produces a strictly positive quantity which the consumer pays for by transferring his unit of money to the producer. Thus, on the equilibrium path, any equilibrium is monetary. Proposition 3 Under direct search, if µ = (Ψ, π, q) ∈ E ∗ is such that Wω∗ (µ) > 0 for some ω ∗ ∈ {1, . . . , Ω}, then: 1. For each (ω, n, k) ∈ {1, . . . , Ω} × N × {1, . . . , K}, there exists k ′ , k¯ ∈ {1, . . . , K} ¯ 0). such that ψω (n, k, 0) = (n + 1, k ′ , 1) and ψω (n, k, 1) = (n − 1, k, θ 2. For each (ω, n, k) ∈ {1, . . . , Ω} × N × {1, . . . , K}, αn,k,m = 1 where θ =

(n, k, m, ψω (n, k, m)). 3. There exists π ∗ = (B ∗ , t∗ ) such that (Ψ, π ∗ , q) ∈ M ∗ , B(0, 1) = B ∗ (0, 1), T (0, 1) = T ∗ (0, 1) and Wω (µ) = Wω (Ψ, π ∗ , q) for each ω ∈ {1, . . . , Ω}. As in Section 3, Proposition 3 imposes conditions only on the equilibrium path. In particular, it imposes no condition on transfers off the equilibrium path. But as all 23

matches in equilibrium are single-coincidence matches, Proposition 3 does not impose any condition on the production decisions other than on B(0, 1). It is also the case that the stationary distribution of money holdings in any equilibrium with strictly positive average expected utility is such that q0 = q1 = 1/2, which means that the frequency of trade is maximized. It turns out that any equilibrium µ ∈ E ∗ is such that Wω (µ) = WΩ (µ) for all ω ∈ {1, . . . , Ω}, i.e. the average expected discounted utility is independent of ω (see Lemma 13). Given this, we let the average expected discounted utility of an equilibrium trading norm µ be denoted by W (µ). The optimum problem is max∗ W (µ). µ∈E

As before, any of its solutions is called an optimal equilibrium. As a corollary of Proposition 3 we obtain that there is an monetary trading norm that is an optimal equilibrium. Corollary 3 Under direct search, if supµ∈E ∗ W (µ) > 0, then there exists an optimal equilibrium µ∗ ∈ M ∗ . As before, this corollary says that for any specification of the environment (i.e. for each N , β, A and u), there is an equilibrium monetary trading norm that solves the optimum problem. Here a monetary trading norm is specified by q0 ∈ (0, 1), y ∈ A and a family of assignment rules Ψ. We can let the latter be such that Ω = K = 1, which means that assignments are independent of time (Ω = 1) and that all individuals of the same type go to the same location (K = 1). We have that q0 = 1/2 as this is a necessary condition for equilibrium. The optimal quantity is the one that maximizes the surplus u(y) − y created in each match amongst those consistent with equilibrium. In the case where A = {0, a∗ }, then y = a∗ and we have supµ∈E ∗ W (µ) > 0 if and only if βu(a∗ ) ≥ a∗ as in Corbae, Temzelides, and Wright (2003, Corollary 1). In general, we have that y solves max y∈A:βu(y)≥y

[u(y) − y].

24

5

Conclusion

We have considered an economy were trading is difficult due to the absence of double coincidence of wants. Trading is further impaired because the record-keeping technology is limited: Each person’s trading history is private information to the person and information about it can only be conveyed through a portable object (money). In such setting with random matching, monetary trading is an optimal trading mechanism precisely when individual preference types are observable and money holdings are not: Of all the stationary trading mechanisms that satisfy feasibility and incentive compatibility, the monetary trading mechanism is the one that yield the highest average expected discounted utility to the agents of the economy. Moreover, all equilibrium trading mechanisms are monetary on the equilibrium path, i.e. observationally equivalent to a monetary trading mechanism. None of these results hold when preference types are unobservable or when both types and money holdings are observable. However, in settings with anonymous individuals such as the one considered in this paper, it is natural to assume that money holdings are unobservable. If types are also unobservable (again due to anonymity), then it is still true that all equilibrium trading mechanisms are observationally equivalent to a monetary trading mechanism when random-matching is replaced with direct search. Thus, monetary trading is an optimal trading mechanism also in this case. While our emphasis has been on the information that each individual has on others’ types and money holdings, there are other elements that are important for the optimality of monetary trading. One aspect that is important is the relationship between the population size and the individuals’ discount factor. In fact, Araujo (2004) and Carmona (2002b), building on Ellison (1994), have shown that with a finite population and a sufficiently large discount factor then, even when types are observable and money holdings are not, there is a non-monetary trading norm which is more efficient than any monetary trading norm.13 13

Such non-monetary trading mechanism can be written as a portable-object trading mechanism

with the property that each individual’s actions depend only on her own money holdings.

25

Another aspect is the boundedness of the trading mechanisms which, in our case, is obtained by assuming that money is indivisible and subject to a unit upper bound. Without these assumptions, Kocherlakota (2002) has shown that there is a nonmonetary portable-object trading norm which is more efficient than any monetary trading norms. We have emphasized the very basic features of monetary trading and for that it was enough to consider relatively simple environments. Specifically, in our setting, different monetary trading mechanisms differ only on the quantity produced by producers with zero units of money to consumers with one unit of money and on the fraction of individuals having no money in the population; consequently, the optimality of a monetary trading norm concerns the choice of these two variables. By allowing for a richer environment, Deviatov and Wallace (2009) and Deviatov and Wallace (2014) have considered more general monetary trading norms and uncovered additional features that optimal monetary trading norms have. In light of the above, it would be interesting to extend optimality results for monetary trading to a richer class of trading norms and to richer environments. We conjecture that this may be possible, at least in the simple class of environments considered in this paper, by imposing some robustness requirements to equilibrium trading norms.14

A

Appendix

A.1

Properties of grim-trigger trading norms

Let µG y = (π, q) ∈ G be a grim-trigger trading norm. Thus, u(y) > y > 0. We first claim that q = (0, 1) is a stationary distribution of the Markov chain induced by π. In fact, q is a solution of (

1 1 N −2 q0 = q0 + + N N N q1 = 1 − q0 . 14

)

( + q1

0 0 + N N

) ,

See Carmona (2002a) for a (not entirely satisfactory) result in this direction.

26

The expected discounted utility function V G satisfies: V0G = βV0G , −(1 − β)y + βV1G N − 2 G (1 − β)u(y) + βV1G G V1 = + βV1 + N N N Thus, we obtain that V0G = 0, u(y) − y V1G = , and N u(y) − y . W (µG y) = N

A.2

(14) (15) (16)

Properties of the monetary trading norms

Let µM q0 ,y = (π, q) ∈ M be a monetary trading norm. Thus, q0 ∈ (0, 1) and u(y) > y > 0. We first claim that q ∈ ∆ is a stationary distribution of the Markov chain induced by π. In fact, any q ∈ ∆ is a solution of ) ( q0 1 N −2 q0 q0 = q0 + + + q1 , N N N N q1 = 1 − q0 . Fix q0 ∈ [0, 1] and let q1 = 1 − q0 . The expected discounted utility function V M satisfies: −q1 (1 − β)y + β(q0 V0M + q1 V1M ) (N − 2)βV0M βV0M + + , N N N βV1M (N − 2)βV1M q0 (1 − β)u(y) + β(q0 V0M + q1 V1M ) = + + N N N

V0M = V1M

Thus, we obtain that

A.3

u(y) − y , and N q0 u(y) + (1 − q0 )y − V0M ) = β . (1 − β)N + β

W (µM q0 ,y ) = q0 (1 − q0 )

(17)

β (V1M 1−β

(18)

Proof of Proposition 1

Let µM ∈ M ∩ E and let 0 < q0 < 1 and 0 < y < u(y) be such that µM = µM q0 ,y . Define βM =

Ny . q0 u(y) + (1 − q0 )y + (N − 1)y 27

By (8) with n = n ˜, m = m ˜ = 0, a0 = a1 = 0 and any p ∈ P (n, m), it follows that β q0 u(y) + (1 − q0 )y (V1M − V0M ) ≥ y ⇔ β ≥y 1−β (1 − β)N + β and, hence, that β ≥ βM . G G Let µG = µG y . Then µ ∈ G. We next show that µ ∈ E. Let

βG =

Ny u(y) − y + N y

and note that βG ≤ βM since q0 u(y) + (1 − q0 )y + (N − 1)y = q0 (u(y) − y) + N y and q0 (u(y) − y) + N y ≤ u(y) − y + N y. Consider (8). In the case where both types and money holdings are observable, we have that D(n) = {n}, D(m) = {m} and pn′ m′ = 1 for some (n′ , m′ ) ∈ N × {0, 1}. Thus, (8) is equivalent to −(1 − β)y + βV1G ≥ βV0G and, hence, it holds because β ≥ βM ≥ βG . Consider next the case where types are unobservable and money holdings are observable. Let n, n ˜ ∈ N , m, m ˜ ∈ {0, 1}, a0 , a1 ∈ A and p ∈ P (n, m) be given and such that m ˜ = m and pn′ m′ = 1/N for all n′ ∈ N and some m′ ∈ {0, 1}. If m = 0 or if m = 1 and m′ = 0, (8) holds since

G 2βVm N

≥−

(1−β)am′ N

G

m + 2βV . When m = m′ = 1, first N

note that the right-hand side of (8) is, for each a0 and a1 , maximized when n = n ˜ since (1 − β)u(y) ≥ 0 implies that −(1 − β)a1 + βVTGp (1,(1,1),a1 ) N

−(1 − β)a1 + βVTGp (1,(1,1),a1 ) βV1G (1 − β)u(y) + βV1G + ≥ + . N N N

Then (8) holds since, whenever a1 ̸= y, −(1 − β)y + βV1G (1 − β)u(y) + βV1G −(1 − β)a1 + βV0G (1 − β)u(y) + βV1G + ≥ + N N N N which holds since β ≥ βM ≥ βG and, therefore,

β (V1G 1−β

− V0G ) ≥ y. Thus (8) holds

in the case where types are unobservable and money holdings are observable. Finally, consider the case both types and money holdings are unobservable. Let n, n ˜ ∈ N , m, m ˜ ∈ {0, 1}, a0 , a1 ∈ A and p ∈ P (n, m) be given and such that m ˜ =m and pn′ m′ = qm′ /N for all n′ ∈ N and m′ ∈ {0, 1}. If m = 0, (8) holds since 2βV0G N

1 ≥ − (1−β)a + N

2βV0G . N

When m = 1, first note that the left-hand side of (8) is

−(1 − β)y + βV1G (1 − β)u(y) + βV1G (1 − β)(u(y) − y) + 2βV1G + = . N N N 28

Second, note that the right-hand side of (8) is, for each a0 and a1 , maximized when n=n ˜ since Tc (1, (1, m), ˜ B(1, m)) ˜ = 1 for each m ˜ ∈ {0, 1} implies that it equals −(1 − β)a1 + βVTGp (1,(m,1),a ˜ 1) N when n ˜ = n and

+

(1 − β)u(y) + βV1G N

−(1 − β)a1 + βVTGp (1,(m,1),a ˜ 1) N

+

βV1G N

when n ˜ ̸= n; the conclusion then follows by (1 − β)u(y) ≥ 0. Thus, assume n ˜ = n and consider (a1 , m) ˜ ̸= (y, 1). If m ˜ = 0, then the right-hand side of (8) equals

−(1 − β)a1 + βV1G + βV1G 2βV1G ≤ ; N N

hence, (8) holds since (1 − β)(u(y) − y) ≥ 0. If m ˜ = 1, then a1 ̸= y and the right-hand side of (8) equals −(1 − β)a1 + βV0G (1 − β)u(y) + βV1G (1 − β)u(y) + β(V0G + V1G ) + ≤ . N N N Hence, (8) holds since β ≥ βM ≥ βG and, therefore,

β (V1G 1−β

− V0G ) ≥ y. Thus (8)

holds in the case where both types and money holdings are unobservable. It follows from above that µG ∈ E. To complete the proof, it follows by (16) and (17) that W (µG ) =

u(y) − y u(y) − y > q0 (1 − q0 ) = W (µM ), N N

as required.

A.4

Proof of Proposition 2

Let µ ∈ E be such that W (µ) > 0. We start by establishing some lemmas. The following lemma shows that there cannot be monetary transfers when both agents in a single-coincidence meeting claim to have zero units of money. Furthermore, in this case, the producer will not produce if this case happens with a strictly positive probability. Lemma 1 For each i = p, c and a ∈ A, ti ((0, 0), a) = Ti (0, (0, 0), a) = 0. Furthermore, either q0 = 0 or B(0, 0) = 0. 29

Proof. Fix a ∈ A and consider m ˜ = (0, 0). Since π is feasible, (1) and (2) hold. Then ti ((0, 0), a) = m ˜ i + ti ((0, 0), a) ≥ 0 for each i = p, c. This, together with ∑ i=p,c ti ((0, 0), a) = 0, implies that ti ((0, 0), a) = 0 and, hence, Ti (0, (0, 0), a) = 0 for each i = p, c. Suppose that q0 > 0. Then it follows by (8) with p ∈ P (n, m) with

∑ m′

p(n+1)m′ =

1, m ˜ = m = 0, a0 = 0 and a1 = B(0, 1) that −(1 − β)B(0, 0) + βV0 ≥ βV0 , which implies that B(0, 0) = 0. The next lemma shows that a producer in a single-coincidence meeting having one unit of money will start next period with the same one unit of money even if he claims to have zero units. Lemma 2 For each a ∈ A, Tp (1, (0, 1), a) = 1. Proof. Let a ∈ A. We have that tp ((0, 1), a) = 0 + tp ((0, 1), a) ≥ 0 by (2). Hence, 1 + tp ((0, 1), a) ≥ 1 and, thus, Tp (1, (0, 1), a) = min{1, 1 + tp ((0, 1), a)} = 1. Lemma 3 shows that the expected discounted utility of starting with one unit of money is at least as high as that of starting with zero units. Lemma 3 V1 ≥ V0 . Proof. It follows from (8) when p ∈ P (n, m) is such that

∑ m′

p(n+1)m′ = 1,

n=n ˜ , m = 1, m ˜ = 0, a0 = B(0, 0) and a1 = B(0, 1) that 1 ∑

′

qm′ [−(1−β)B(1, m )+βVTp (1,m′ ) ] ≥

m′ =0

Similarly, when 1 ∑

1 ∑

qm′ [−(1−β)B(0, m′ )+βVTp (1,(0,m′ ),B(0,m′ )) ].

m′ =0

∑ m′

p(n−1)m′ = 1, (8) implies that ′

qm′ [(1−β)u(B(1, m ))+βVTc (1,m′ ) ] ≥

m′ =0

1 ∑

qm′ [(1−β)u(B(0, m′ ))+βVTc (1,(m′ ,0),B(m′ ,0)) ].

m′ =0

These inequalities, together with (5) and (6), imply that V1 − V0 ≥

(N −2)β N

q0 β N q1 β + N q0 β + N q1 β + N

(V1 − V0 ) +

30

( ( ( (

VTp (1,(0,0),B(0,0)) − VTp (0,0) VTp (1,(0,1),B(0,1)) − VTp (0,1) VTc (1,(0,0),B(0,0)) − VTc (0,0)

) )

)

) VTc (1,(1,0),B(1,0)) − VTc (1,0) .

For each m ∈ {0, 1} and i ∈ {p, c}, we have that Ti (0, m) = Ti (0, (0, m), B(0, m)) = min{1, 0 + ti ((0, m), B(0, m))} ≤ min{1, 1 + ti ((0, m), B(0, m))} = Ti (1, (0, m), B(0, m)). Thus, letting Mi = {m ∈ {0, 1} : Ti (1, (0, m), B(0, m)) ̸= Ti (0, m)}, we have that ∑ Ti (1, (0, m), B(0, m)) = 1 and Ti (0, m) = 0 for each m ∈ Mi . Let γ = m∈Mp qm + ∑ m∈Mc qm and note that 0 ≤ γ ≤ 2. Furthermore, by what has been shown above, γβ (N − 2)β (V1 − V0 ) + (V1 − V0 ) . N N ( ) ≥ 0 which implies that V1 ≥ V0 . Thus, (V1 − V0 ) 1 − β + 2−γ N V1 − V0 ≥

The following lemma shows that the fraction of agents having zero units of money is strictly positive and strictly below one. Lemma 4 q0 ∈ (0, 1). Proof. Suppose that q0 = 1. Since, by Lemma 1, B(0, 0) = 0 and Tp (0, 0) = Tc (0, 0) = 0, it follows from (5) that V0 = 0. Hence, W (µ) = V0 = 0, contradiction our assumption that W (µ) > 0. Suppose that q1 = 1. Then (4) implies that Tp (1, 1) = Tc (1, 1) = 1 and, by Lemma ∑ 2, Tp (1, (0, 1), 0) = 1. Then (8) with p ∈ P (n, m) such that m′ p(n+1)m′ = 1, n = n ˜, m = 1, m ˜ = 0, a0 = a1 = 0 implies that −(1 − β)B(1, 1) + βV1 ≥ βV1 . Thus, B(1, 1) = 0. Then (6) implies that V1 = 0. Therefore, W (µ) = 0, contradicting our assumption that W (µ) > 0. Thus, it follows from what has been shown above that q0 ∈ (0, 1). Lemma 5 pins down the decision function B. Lemma 5 B(0, 0) = B(1, 0) = B(1, 1) = 0 and B(0, 1) > 0. Proof. It follows from Lemmas 1 and 4 that B(0, 0) = 0. ∑ Condition (8) with p ∈ P (n, m) such that m′ p(n+1)m′ = 1, n = n ˜ , m = 1, m ˜ = 0, a0 = a1 = 0 implies that q0 [−(1−β)B(1, 0)+βVTp (1,0) ]+q1 [−(1−β)B(1, 1)+βVTp (1,1) ] ≥ q0 βVTp (1,(0,0),0) +q1 βVTp (1,(0,1),0) . 31

Since Tp (1, (0, 0), 0) = 1 + tp ((0, 0), 0) = 1 by Lemma 1 and Tp (1, (0, 1), 0) = 1 by Lemma 2, it follows that the right-hand side of the above inequality equals βV1 . Since V1 ≥ V0 by Lemma 3, its left-hand side is at most −(1−β)(q0 B(1, 0)+q1 B(1, 1))+βV1 . Since q0 > 0 and q1 > 0 by Lemma 4, it follows that B(1, 0) = B(1, 1) = 0. Suppose that B(0, 1) = 0. Then, by the above and by Lemma 1, B(m) = 0 for each m ∈ {0, 1}2 . This implies that W (µ) = 0, which contradicts our assumption that W (µ) > 0. Thus, it follows that B(0, 1) > 0. The following lemma, together with Lemma 1, determines the function T on the equilibrium path. Lemma 6 Tp (0, 1) = Tp (1, 0) = Tp (1, 1) = 1, Tc (0, 1) = Tc (1, 0) = 0 and Tc (1, 1) = 1. Furthermore, −(1 − β)B(0, 1) + βV1 ≥ βV0 . Proof. Consider (8) with p ∈ P (n, m) such that

∑ m′

p(n+1)m′ = 1, m = m ˜ =0

and a0 = a1 = 0. By Lemma 1 and Lemma 5, we have that T (0, 0) = 0, B(0, 0) = 0 and B(0, 1) > 0. Thus, we obtain that [ ] q0 βV0 + q1 −(1 − β)B(0, 1) + βVTp (0,1) ≥ q0 βV0 + q1 βVTp (0,(0,1),0) and, hence, by Lemmas 3 and 4, −(1 − β)B(0, 1) + βVTp (0,1) ≥ βVTp (0,(0,1),0) ≥ βV0 . Thus, Tp (0, 1) = 1 and −(1 − β)B(0, 1) + βV1 ≥ βV0 . In particular, it follows that V1 > V0 . Next, consider (8) with p ∈ P (n, m) such that

∑ m′

p(n+1)m′ = 1, m = 1, m ˜ =0

and a0 = a1 = 0. By Lemmas 1 and 2, the right-hand side of (8) equals βV1 . Since B(1, 0) = B(1, 1) = 0 by Lemma 5, the left-hand side of (8) equals β(q0 VTp (1,0) + q1 VTp (1,1) ). Since V1 > V0 by the above and q0 > 0 and q1 > 0 by Lemma 4, then it follows that Tp (1, 0) = Tp (1, 1) = 1. We have that Tp (0, 1) = 1 implies that tp ((0, 1), B(0, 1)) = 1. Hence, by (1), tc ((0, 1), B(0, 1)) = −1 and Tc (0, 1) = 0. Similarly, Tp (1, 0) = 1, together with (1) and (2), implies that tp ((1, 0), B(1, 0)) = 0 and tc ((1, 0), B(1, 0)) = 0. Thus, Tc (1, 0) = 0. Finally, suppose that Tc (1, 1) = 0. By the above, Lemma 1 and (4), we obtain that N q0 = q0 (q0 + 1 + N − 2) + (1 − q0 )(0 + 1) ⇔ (q0 − 1)2 = 0 ⇔ q0 = 1. 32

But q0 = 1 contradicts Lemma 4. Thus Tc (1, 1) = 1. M We now conclude the proof of Proposition 2. Write µM q0 ,B(0,1) = (πB(0,1) , q) and let M π ∗ = πB(0,1) . It follows by Lemmas 1, 5 and 6 that B(m) = B ∗ (m) and T (m) = T ∗ (m)

for all m ∈ {0, 1}2 . Furthermore, q0 ∈ (0, 1) by Lemma 4 and B(0, 1) > 0 by Lemma 5. Finally, B(m) = B ∗ (m) and T (m) = T ∗ (m) for all m ∈ {0, 1}2 together with (17) imply that W (µ) = q0 (1 − q0 ) u(B(0,1))−B(0,1) . Since W (µ) > 0, then u(B(0, 1)) > N B(0, 1) and it follows that (π ∗ , q) ∈ M . This completes the proof of Proposition 2.

A.5

Proof of Corollary 2

Consider the following maximization problem: q0 (1 − q0 )(u(y) − y) (q0 ,y)∈[0,1]×A N q0 u(y) + (1 − q0 )y subject to β ≥ y. (1 − β)N + β max

Since the objective function is continuous and the constraint set is compact and nonempty (for instance, y = 0 satisfies the constraint), then a solution (q0∗ , y ∗ ) exists. We have that supµ∈E W (µ) > 0 and, hence, let µ ∈ E be such that W (µ) > 0. Then, by Proposition 2, W (µ) =

q0 (1−q0 )(u(y)−y) N

0 )y and β q0 u(y)+(1−q ≥ y. Therefore, it (1−β)N +β

follows that q0∗ ∈ (0, 1) and 0 < y ∗ < u(y ∗ ). Thus, to complete the proof, it suffices to show that µ∗ = µq0∗ ,y∗ ∈ M is an equilibrium. Consider (8) with n = n ˜ , m ∈ {0, 1}, m ˜ ∈ D(m), a0 , a1 ∈ A and p ∈ P (n, m). ∑ Suppose first that m′ p(n+1)m′ = 1, i.e. the incentive compatibility constraint of a producer. If m = 1, then (8) holds since βV1M ≥ −(1 − β)(q0 a0 + q1 a1 ) + βV1M . If m = 0, then (8) holds because, whenever a1 ̸= y, q0 βV0M +q1 [−(1−β)y+βV1M ] ≥ q0 βV0M +q1 βV0M ≥ q0 [−(1−β)a0 +βV0M ]+q1 [−(1−β)a1 +βV0M ] 0 )y − V0M ) = β q0 u(y)+(1−q ≥ y (the case where a1 = y is simpler). (1−β)N +β ∑ Suppose next that m′ p(n−1)m′ = 1, i.e. the incentive compatibility constraint of

since

β (V1M 1−β

a consumer. If m = 0, then (8) holds since both of its sides equal βV0M . If m = 1, then (8) holds because q0 (1 − β)u(y) + β(q0 V0M + q1 V1M ) ≥ βV1M . 33

Indeed,

β (V1M 1−β

0 )y − V0M ) = β q0 u(y)+(1−q ≤ (1−β)N +β

u(y) (1−β)N +β

≤ u(y). This shows that µ∗ is

an equilibrium and completes the proof.

A.6

Proof of Proposition 3

Let µ ∈ E ∗ be such that Wω∗ (µ) > 0 for some ω ∗ ∈ {1, . . . , Ω}. The following lemma shows that having one unit of money is not worse than having zero units of money. Lemma 7 For each ω ∈ {1, . . . , Ω}, n ∈ N and k ∈ {1, . . . , K}, Vω,n,k,1 ≥ Vω,n,k,0 . Proof. Let ω ∈ {1, . . . , Ω}, n ∈ N and k ∈ {1, . . . , K}. Moreover, let (n′ , k ′ , m′ ) = ψω (n, k, 0) and θω = (n, k, 0, n′ , k ′ , m′ ). Finally, to simplify notation, we write Vω,m instead of Vω,n,k,m for each m ∈ {0, 1}. It follows from (13) with n ˜=n ˆ = n, kˆ = k, m = 1, m ˜ =m ˆ = 0 and a = B(0, m′ ) that [ ] θω θω Vω,1 ≥ (1 − αn,k,0 )βVω+1,1 + αn,k,0 (1 − β)vn (θω , B(0, m′ )) + βVω+1,Tn,1 (θω ,B(0,m′ ))} . Since [ ] θω θω Vω,0 = (1 − αn,k,0 )βVω+1,0 + αn,k,0 (1 − β)vn (θω , B(0, m′ )) + βVω+1,Tn,0 (θω ,B(0,m′ )) , we obtain that Vω,1 − Vω,0 = cω β(Vω+1,1 − Vω+1,0 ), where cω =

  1

if Tn,1 (θω , B(0, m′ )) = 1 and Tn,0 (θω , B(0, m′ )) = 0,

 1 − αθω n,k,0

otherwise.

(19)

Since (19) holds for each ω ′ ∈ {1, . . . , Ω}, then ( Ω ) ∏ Vω,1 − Vω,0 ≥ cω′ β Ω (Vω,1 − Vω,0 ). ω ′ =1

As 0 ≤ cω′ ≤ 1 for each ω ′ ∈ {1, . . . , Ω} and β ∈ (0, 1), it follows that Vω,1 ≥ Vω,0 . Lemma 8 shows that there is no production on single-coincidence matches where the consumer has no money and also in single-coincidence matches where the producer has one unit of money. 34

Lemma 8

1. If ψω (n, k, 0) = (n + 1, k ′ , 0) for some ω ∈ {1, . . . , Ω}, n ∈ N and

k, k ′ ∈ {1, . . . , K}, then B(0, 0) = 0 and T (0, 0) = (0, 0). 2. If ψω (n, k, 1) = (n + 1, k ′ , m′ ) for some ω ∈ {1, . . . , Ω}, n ∈ N , k, k ′ ∈ {1, . . . , K} and m′ ∈ {0, 1}, then B(1, m′ ) = 0. Proof. Fix some ω ∈ {1, . . . , Ω}, n ∈ N and k ∈ {1, . . . , K}. To simplify the notation, we write Vm instead of Vω+1,n,k,m for each m ∈ {0, 1}. For part 1, consider (13) with n ˜=n ˆ = n, kˆ = k, m ˜ =m ˆ = 0 and a = 0. Since Tp (0, (0, 0), a) = 0 for each a ∈ A by Lemma 1, then (13) is −(1 − β)B(0, 0) + βV0 ≥ βV0 . Thus B(0, 0) = 0. Furthermore, T (0, 0) = (0, 0) by Lemma 1. For part 2, consider (13) with n ˜ = n ˆ = n, kˆ = k, m ˆ = m = 1, m ˜ = 0 and a = 0. Since Tp (1, (0, m′ ), 0) = 1 + tp ((0, m′ ), 0) = 1 by Lemmas 1 and 2, then (13) is −(1 − β)B(1, m′ ) + βVTp (1,m′ ) ≥ βV1 . Thus B(1, m′ ) = 0. The following lemma shows that, on the equilibrium path, there are, with a strictly positive probability, single-coincidence matches between a producer with zero units of money and a consumer with one unit of money. In these matches, the producer produces a strictly positive quantity and money changes hands. Lemma 9

1. There exist ω ∈ {1, . . . , Ω}, n ∈ N and k, k ′ ∈ {1, . . . , K} such that

ψω (n, k, 0) = (n + 1, k ′ , 1). 2. B(0, 1) > 0. 3. T (0, 1) = (1, 0). 4. q0 ∈ (0, 1). ¯ k ′ ) such Proof. It follows by Wω∗ (µ) > 0 and Lemma 8 that there exists (¯ n, k, ¯ 0) = (¯ that ψω¯ (¯ n, k, n + 1, k ′ , 1), B(0, 1) > 0, u(B(0, 1)) − B(0, 1) > 0 and q0 ∈ (0, 1). ¯ m Condition (13) with n ˜ = n ˆ = n ¯ , kˆ = k, ˜ = m ˆ = 0 and a = 0 implies that −(1−β)B(0, 1)+βVω¯ ,¯n,k,T ¯ p (0,1) ≥ βVω ¯ p (0,(0,1),0) . Since Vω ¯ p (0,(0,1),0) ≥ Vω ¯ by ¯ ,¯ n,k,T ¯ ,¯ n,k,T ¯ ,¯ n,k,0 Lemma 7, then Tp (0, 1) = 1. Then (1) implies that Tc (0, 1) = 0, i.e. T (0, 1) = (1, 0).

35

The following three lemma are used to strengthen part 1 of Lemma 9, namely that all single-coincidence matches are between a producer with zero units of money and a consumer with one unit of money. Lemma 10 For all n ∈ N and k ∈ {1, . . . , K}, there exists ω ∈ {1, . . . , Ω} and k ′ ∈ {1, . . . , K} such that ψω (n, k, 1) = (n − 1, k ′ , 0). Proof. We start by showing that Vω,n,k,m ≥ 0 for each (ω, n, k, m). Let V = ˆ m) minω′ ,n′ ,k′ ,m′ Vω′ ,n′ ,k′ ,m′ and consider (13) with (ˆ n, k, ˆ = (n, k, m) and n ˜ ̸∈ {n′ − θ θ )βVω+1,n,k,m + αn,k,m βVω+1,n,k,m ≥ βV for each 1, n′ + 1}. Then Vω,n,k,m ≥ (1 − αn,k,m

(ω, n, k, m). Thus, V ≥ βV and, hence, V ≥ 0. Fix n ∈ N and k ∈ {1, . . . , K} and suppose that ψω (n, k, 1) ̸= (n − 1, k ′ , 0) for each ω ∈ {1, . . . , Ω} and k ′ ∈ {1, . . . , K}. Then, for each ω, Vω,n,k,1 ≤ (1 − θ θ αn,k,1 )βVω+1,n,k,1 + αn,k,1 βVω+1,n,k,1 = βVω+1,n,k,1 . Continuing by induction, we obtain

that Vω,n,k,1 ≤ β Ω Vω,n,k,1 . This, together with Vω,n,k,1 ≥ Vω,n,k,0 ≥ 0, implies that Vω,n,k,1 = Vω,n,k,0 = 0. ˆ m) Consider (13) with the given (n, k), m = 1, ω ∈ {1, . . . , Ω}, (ˆ n, k, ˆ such that ˆ m) (n − 1, 1, 0) = ψω (ˆ n, k, ˆ and (˜ n, m) ˜ = (n, 1). Since Vω,n,k,1 = 0 = Vω+1,n,k,0 , then (13) is 0 ≥ (1 − α)0 + α[(1 − β)u(B(0, 1)) + β0], ˆ m,n−1,1,0) (ˆ n,k, ˆ

which fails due to u(B(0, 1)) > 0 and α = αnˆ ,k, ˆm ˆ

> 0 (the latter, due to

ˆ This contradiction shows that ψω (n, k, 1) = q0 ∈ (0, 1) and αnˆ ,kˆ > 0 for each n ˆ , k). (n − 1, k ′ , 0) for some ω ∈ {1, . . . , Ω} and k ′ ∈ {1, . . . , K}. Lemma 11 For each ω ∈ {1, . . . , Ω}, n ∈ N and k ∈ {1, . . . , K}, if ψω (n, k, 1) = ¯ 1) for some k¯ ∈ (n − 1, k ′ , 0) for some k ′ ∈ {1, . . . , K} then ψω (n, k, 0) = (n + 1, k, {1, . . . , K}. Proof. Let (ω, n, k, k ′ ) be such that ψω (n, k, 1) = (n − 1, k ′ , 0). ¯ 0) for some n Suppose first that ψω (n, k, 0) = (¯ n, k, ¯ ∈ N and k¯ ∈ {1, . . . , K}. θ1 θ0 θ0 Then (11) is q0 = q1 αn,k,1 + q0 (1 − αn,k,0 ) + q0 αn,k,0 . Since q1 > 0 by Lemma 9 θ1 ˆ then (11) and αn,k,1 > 0 (the latter due to q0 ∈ (0, 1) and αnˆ ,kˆ > 0 for each n ˆ , k),

36

¯ 1) for some n fails, a contradiction. This implies that ψω (n, k, 0) = (¯ n, k, ¯ ∈ N and k¯ ∈ {1, . . . , K}. θ1 θ0 θ0 Next suppose that n ¯ ̸= n + 1. Then (11) is q0 = q1 αn,k,1 + q0 (1 − αn,k,0 ) + q0 αn,k,0 ,

¯ 1) for some k¯ ∈ which, by the argument above, fails. Thus ψω (n, k, 0) = (n + 1, k, {1, . . . , K}. Lemma 12 For each ω ∈ {1, . . . , Ω}, n ∈ N , k ∈ {1, . . . , K}, there exist k ′ , k¯ ∈ ¯ 0). {1, . . . , K} such that ψω (n, k, 0) = (n + 1, k ′ , 1) and ψω (n, k, 1) = (n − 1, k, Proof. Fix (n, k) ∈ N × {1, . . . , K} and, by Lemma 10, let (ω, k ′ ) ∈ {1, . . . , Ω} × ¯ 1) {1, . . . , K} be such that ψω (n, k, 1) = (n − 1, k ′ , 0). Then ψω (n, k, 0) = (n + 1, k, for some k¯ by Lemma 11. We write y instead of B(0, 1). It follows from (13) with ˆ m) (ˆ n, k, ˆ = (n, k, 1), n ˜ ̸= {n − 1, n + 1}, m ˜ ∈ {0, 1} and a = 0 that β

Vω+1,n,k,1 − Vω+1,n,k,0 ≤ u(y). 1−β

(20)

ˆ m) Moreover, it follows from (13) with (ˆ n, k, ˆ = (n, k, 0), (˜ n, m) ˜ = (n, 0) and a = 0 that β

Vω+1,n,k,1 − Vω+1,n,k,0 ≥ y. 1−β

(21)

θm For each m ∈ {0, 1}, let αm = αn,k,m where θm = (n, k, m, ψω (n, k, m)). Moreover,

let δ ∈ R be such that α1 = α0 + δ. We then obtain from (12) that Vω,n,k,1 − Vω,n,k,0 = (1 − α1 )β(Vω+1,n,k,1 − Vω+1,n,k,0 ) − δβVω+1,n,k,0 +α1 (1 − β)(u(y) + y) − δ(1 − β)y −α1 β(Vω+1,n,k,1 − Vω+1,n,k,0 ) + δβVω+1,n,k,1 = (1 − α1 )β(Vω+1,n,k,1 − Vω+1,n,k,0 ) − α0 β(Vω+1,n,k,1 − Vω+1,n,k,0 ) +α1 (1 − β)(u(y) + y) − δ(1 − β)y. It follows by (20) and (21), respectively, that (1 − α1 )β(Vω+1,n,k,1 − Vω+1,n,k,0 ) ≤

37

(1 − α1 )(1 − β)u(y) and −α0 β(Vω+1,n,k,1 − Vω+1,n,k,0 ) ≤ −α0 (1 − β)y. Hence, Vω,n,k,1 − Vω,n,k,0 ≤ (1 − α1 )(1 − β)u(y) − α0 (1 − β)y +α1 (1 − β)u(y) + α1 (1 − β)y − δ(1 − β)y = (1 − β)u(y) + α1 (1 − β)y − (α0 + δ)(1 − β)y = (1 − β)u(y). Therefore,

Vω,n,k,1 − Vω,n,k,0 ≤ βu(y) < u(y). 1−β ˆ 0). It then follows Suppose that, for each kˆ ∈ {1, . . . , K}, ψω−1 (n, k, 1) ̸= (n−1, k, β

by Lemma 7 and Lemma 8 that, for some α ˜ ∈ [0, 1] and m ˜ ∈ {0, 1}, Vω−1,n,k,1 = (1 − α ˜ )βVω,n,k,1 + α ˜ βVω,n,k,m˜ ] ≥ βVω,n,k,1 . ˆ m) Consider (13) with the given (ω − 1, n, k), m = 1, (ˆ n, k, ˆ such that (n − 1, 1, 0) = ˆ m). ψω (ˆ n, k, ˆ Since Vω−1,n,k,1 ≤ βVω,n,k,1 , a necessary condition for (13) is βVω,n,k,1 ≥ (1 − α)βVω,n,k,1 + α[(1 − β)u(y) + βVω,n,k,0 ], which fails due to u(y) >

β(Vω,n,k,1 −Vω,n,k,0 ) 1−β

ˆ m,n−1,1,0) (ˆ n,k, ˆ

and α = αnˆ ,k, ˆm ˆ

> 0 (the latter, due

ˆ Hence, ψω−1 (n, k, 1) = (n − 1, k, ˆ 0) for some to q0 ∈ (0, 1) and αnˆ ,kˆ > 0 for each n ˆ , k). kˆ ∈ {1, . . . , K}. ˜ 0) for some k˜ ∈ It then follows from Lemma 11 that ψω−1 (n, k, 0) = (n + 1, k, {1, . . . , K}. An inductive argument then completes the proof of the lemma. ω θ For each ω ∈ {1, . . . , Ω}, n ∈ N , k ∈ {1, . . . , K} and m ∈ {0, 1}, αn,k,m = αn,k,m

where ω = (n, k, m, ψω (n, k, m)). Furthermore, let y = B(0, 1). Lemma 13 establishes the remaining properties of equilibrium trading norms. Lemma 13

ω 1. αn,k,m = 1 for each ω ∈ {1, . . . , Ω}, n ∈ N , k ∈ {1, . . . , K} and

m ∈ {0, 1}. 2. q0 = q1 = 1/2. 3. Vω,n,k,1 =

u(y)−βy 1+β

and Vω,n,k,0 =

−y+βu(y) 1+β

k ∈ {1, . . . , K} and m ∈ {0, 1}. 38

for each ω ∈ {1, . . . , Ω}, n ∈ N ,

4. βu(y) ≥ y. 5. Wω (µ) =

u(y)−y 2

for each ω ∈ {1, . . . , Ω}.

Proof. Let ω ∈ {1, . . . , Ω}, n ∈ N and k ∈ {1, . . . , K} be given. It follows ω ω from Lemma 12 that (11) is q0 = q1 αn,k,1 . Similarly, we obtain q1 = q0 αn,k,0 . Hence,

q0 =

1 1+αω n,k,0

and q1 =

1 . 1+αω n,k,1

ω ω Since q0 + q1 = 1, it follows that αn,k,0 = αn,k,1 =1

and that q0 = q1 = 1/2. This establishes parts 1 and 2 of the lemma. We have that (12) implies that Vω,n,k,1 = (1 − β)u(y) + βVω,n,k,0 , and Vω,n,k,0 = −(1 − β)y + βVω,n,k,1 . Thus, Vω,n,k,1 =

u(y)−βy 1+β

and Vω,n,k,0 =

−y+βu(y) . 1+β

This proves part 3 of the lemma.

We turn to the proof of part 4 and we start by showing that (13) implies that β

Vω+1,n,k,1 −Vω+1,n,k,0 1−β

ˆ m) ≥ y. Specifically, it follows from (13) with (ˆ n, k, ˆ = (n, k, 0),

(˜ n, m) ˜ = (n, 0) and a = 0 that β

Vω+1,n,k,1 −Vω+1,n,k,0 1−β

≥ y. By part 3 of the lemma, it

follows that βu(y) ≥ y. Finally, we establish part 5. We have that Wω (µ) =

∑ αn,k qm ∑ αn,k u(y) − βy − y + βu(y) u(y) − y Vω,n,k,m (µ) = = . N 2N 1 + β 2 n,k,m n,k

This concludes the proof of the lemma. M Write π ∗ = πB(0,1) . It follows by Lemma 9 that B(0, 1) = B ∗ (0, 1) and T (0, 1) =

T ∗ (0, 1). It then follows from Lemma 12 that (Ψ, π ∗ , q) ∈ M ∗ .

A.7

Proof of Corollary 3

We have that supµ∈E ∗ W (µ) > 0 and, hence, let µ ∈ E ∗ be such that W (µ) > 0. Then, by Proposition 3, µ ∈ M ∗ . Let y = B(0, 1). Since µ ∈ E ∗ , βu(y) ≥ y by Lemma 13. In particular, the above shows that there exists y ∈ A such that βu(y) ≥ y. Let

39

y¯ ∈ A be a solution of max[u(y) − y] y∈A

subject to βu(y) ≥ y. Consider monetary µ∗ with B ∗ (0, 1) = y¯ and Ω = K = 1. We next claim that µ∗ is an equilibrium. We have that V1 := Vn,1 =

u(¯ y )−β y¯ 1+β

and V0 := Vn,1 =

−¯ y +βu(¯ y) . 1+β

Note that V1 >

V0 ≥ 0 since βu(¯ y ) ≥ y¯. Consider (13) with m = 1 and some given n ∈ N . Its right-hand side is (1 − ˜ a) + βV β)vn (θ, and we may assume that it is not the case that n ˆ =n ˜ =n ˜ Tn,m (θ,a) and m ˆ =m ˜ = 1; indeed, (13) clearly holds in the case where the individual has been ˜ a) ≤ 0 for any a ∈ A. Since V truthful. Then vn (θ, ≤ V1 in any case, then (13) ˜ Tn,m (θ,a) holds as its left-hand side is V1 and its right-hand side is at most βV1 < V1 . Consider (13) with m = 0 and some given n ∈ N . Then m ˆ =m ˜ = 0 and we may ˜ a) + assume that n ˆ ̸= n or n ˜ ̸= n or a ̸= y¯. If n ˜ =n ˆ and a = y¯, then (1 − β)vn (θ, ˜ a) + βVTn,m (θ,a) = −(1 − β)¯ y + βV1 = V0 and (13) holds. Otherwise, (1 − β)vn (θ, ˜ βVTn,m (θ,a) ≤ βV0 ≤ V0 and (13) also holds. Thus, it follows that µ∗ is an equilibrium. ˜ Finally, we have that W (µ∗ ) =

u(¯ y ) − y¯ u(y) − y ≥ = W (µ). 2 2

This completes the proof.

A.8

Optimal monetary trading norms

This sections provides more detail on the optimal monetary trading norms in the case where the consumption goods are indivisible and in the case where they are divisible. Suppose first that A = {0, a∗ } with u(a∗ ) > a∗ . For each α ∈ [0, 1], let βα∗ ∈ [0, 1) be the smallest β ′ ∈ [0, 1) such that αu(a∗ ) + (1 − α)a∗ ≥ a∗ for all β ≥ β ′ . (1 − β)N + β

40

Thus,

 ∗ ∗  0 if αu(a )+(1−α)a ≥ a∗ , N ∗ βα =  N a∗ −(αu(a∗ )+(1−α)a∗ ) otherwise. (N −1)a∗

(22)

Consider the monetary trding norm µM q0 ,a∗ where 0 < q0 < 1, here denoted by µq0 . It follows by (18) that β q0 u(a∗ ) + (1 − q0 )a∗ (V1 − V0 ) = 1−β (1 − β)N + β and, by Lemma 6, we must have q0 u(a∗ ) + (1 − q0 )a∗ ≥ a∗ (1 − β)N + β and, in fact,

since

q0 u(a∗ )+(1−q0 )a∗ ˆ +βˆ (1−β)N

q0 u(a∗ ) + (1 − q0 )a∗ ≥ a∗ for all βˆ ≥ β ˆ + βˆ (1 − β)N is increasing in βˆ (due to N > 1). Thus, the definition of βq∗0

implies that β ≥ βq∗0 . Conversely, it is easy to see that µq0 is an equilibrium if β ≥ βq0 . Thus, the optimum problem is max

q0 ∈[0,1]:β≥βq0

q0 (1 − q0 )

u(a∗ ) − a∗ . N

(23)

We have that β ≥ βq0 for some q0 ∈ (0, 1) since supµ∈E W (µ) > 0. Since q0 7→ βq0 ∗ ∗ is decreasing, then either β ≥ β1/2 or β ∈ (β1∗ , β1/2 ). In the former case, it follows

from (23) that µ1/2 is an optimal equilibrium. In the latter case, letting α ∈ (1/2, 1) is such that βα∗ = β, we have that µα is an optimal equilibrium. Next suppose that A = [0, a ¯] with a ¯ > a∗ . In this case, let (ˆ y , qˆ0 ) be a solution of q0 (1 − q0 )(u(y) − y) (y,q0 )∈A×[0,1] N 1−β y subject to q0 ≥ N . β u(y) − y max

Note that (ˆ y , qˆ0 ) exist because the objective function is continuous and the constraint set is compact and nonempty. Moreover, note that the constraint is equivalent to 1 −V0 β V1−β ≥ y.

Thus, if

1−β y 1 ≥N , ∗ 2 β u(a ) − a∗ 41

then yˆ = a∗ and q0 = 1/2. If, instead, 1 1−β y 1/2. To see the latter, note first that for each q0 ∈ (0, 1], there exists y > 0 such that the constraint is satisfied. Since yˆ = 0 implies that

q0 (1−q0 )(u(ˆ y )−ˆ y) N

= 0, this implies

that yˆ > 0. Also, it implies that u(ˆ y ) − yˆ > 0 and 0 < qˆ0 < 1. ∗

yˆ a Suppose next that yˆ ≥ a∗ . Then qˆ0 ≥ N 1−β ≥ N 1−β > 1/2, where β u(ˆ y )−ˆ y β u(a∗ )−a∗

the second inequality follows from the concavity of u. We then obtain that qˆ0 = yˆ since, otherwise, it would be possible to decrease qˆ0 , keep yˆ and still N 1−β β u(ˆ y )−ˆ y

satisfy the constraint; this would increase the objective function. yˆ It follows from qˆ0 = N 1−β that yˆ solves β u(ˆ y )−ˆ y

max

( ) y 1−β y 1 − N (u(y) − y) N 1−β β u(y)−y β u(y)−y N

y∈A

and, thus, also solves y max y∈A:u(y)−y>0 u(y) − y

( ) y y2 1−β 1−β 1−N (u(y) − y) = y − N . β u(y) − y β u(y) − y (24)

The derivative of the objective function of (24) at yˆ equals ( )2 [ ] yˆ u(ˆ y ) − yˆ 1−β ′ 1−N 2 − (u (ˆ y ) − 1) = β u(ˆ y ) − yˆ yˆ [ ] yˆ(u′ (ˆ y ) − 1) yˆ(u′ (ˆ y ) − 1) 1 − qˆ0 2 − = 1 − 2ˆ q0 + qˆ0 ≤ 1 − 2ˆ q0 < 0, u(ˆ y ) − yˆ u(ˆ y ) − yˆ where the last inequality follows from qˆ0 > 1/2 and the previous inequality since u(ˆ y ) − yˆ > 0 and u′ (ˆ y ) − 1 ≤ 0 (the latter because yˆ ≥ a∗ and u is concave). But the derivative of the objective function of (24) at yˆ being non-zero implies that yˆ is not a solution to (24), a contradiction. This contradiction shows that yˆ < a∗ . The argument to show that qˆ0 > 1/2 is analogous to the one above. Suppose that qˆ0 ≤ 1/2. As above, yˆ solves (24). The derivation of its objective function at yˆ is 1 − 2ˆ q0 + qˆ0

yˆ(u′ (ˆ y ) − 1) yˆ(u′ (ˆ y ) − 1) ≥ qˆ0 > 0, u(ˆ y ) − yˆ u(ˆ y ) − yˆ 42

where the first inequality follows from qˆ0 ≤ 1/2 and the second because u′ (ˆ y) − 1 > 0 (the latter because yˆ < a∗ and u is concave). But the derivative of the objective function of (24) at yˆ being non-zero implies that yˆ is not a solution to (24), a contradiction. This contradiction shows that qˆ0 > 1/2.

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Kocherlakota, N. (1998a): “Money is Memory,” Journal of Economic Theory, 81, 232–251. (1998b): “The Technological Role of Fiat Money,” Federal Reserve Bank of Minneapolis Quarterly Review, 22(3), 2–10. (2002): “The Two-Money Theorem,” International Economic Review, 43, 333–347. Kocherlakota, N., and N. Wallace (1998): “Incomplete Record-Keeping and Optimal Payment Arrangements,” Journal of Economic Theory, 81, 272–289. Okuno-Fujiwara, M., and A. Postlewaite (1995): “Social Norms and Random Matching Games,” Games and Economic Behavior, 9, 79–109. Townsend, R. (1987): “Economic Organization with Limited Communication,” American Economic Review, 77, 954–971.

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