One or Two Monies? Mei Dongy

Janet Hua Jiangzx

March 30, 2010

Abstract The set of incentive-feasible allocations is examined in a dynamic quasi-linear environment where agents lack commitment and have private information over their idiosyncratic characteristics. When record-keeping is available, the …rst-best allocation is implementable in a set of su¢ ciently patient economies. When record-keeping is limited to one money, this set is strictly smaller – except when private information is absent. When record-keeping is expanded, but limited to two monies, the set of economies for which the …rst-best is implementable corresponds to that of record-keeping, even when private information is present. We further demonstrate that two monies are a perfect substitute for record-keeping.

JEL Categories: E40, F30, D82 Keywords: Record-keeping; Money; Private Information; Limited Commitment; Mechanism Design

The authors would like to thank David Andolfatto, Robert Jones, Alexander Karaivanov, Fernando Martin, Ed Nosal, Chris Waller, Randy Wright, Robert King (the Editor), an anonymous referee and participants at the brown bag seminar at Simon Fraser University, the 2007 Cleveland Fed summer workshop on Money, Banking, Payments, and Finance, the 2008 Midwest Macro Meetings, the 2008 Canadian Economics Association Meetings and the 2008 Econometric Society North America Summer Meetings for helpful suggestions and comments. The views expressed in this paper are those of the authors. No responsibility for them should be attributed to the Bank of Canada. y Currency Department, the Bank of Canada. z Department of Economics, University of Manitoba. x Corresponding author, Email: [email protected].

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1

Introduction

Some form of record-keeping (memory) is generally needed to implement desirable allocations in dynamic environments that are subject to frictions such as limited commitment and private information. Since Ostroy (1973), it has been generally understood that …at money is a record-keeping technology. When other forms of record-keeping are absent, or otherwise too costly to operate, the introduction of a …at-money instrument can be socially bene…cial; see Kocherlakota (1998 a; b). There is by now an extensive literature that examines the role of money and monetary policy in "micro-founded" models of money; see Williamson and Wright (2010), and the references cited therein. Relatively little attention, however, has been paid in this modern literature to study whether one money is su¢ cient to replace the record-keeping technology.1 While multiple monies may coexist in the equilibria of these monetary models, more than one money is usually redundant in performing the record-keeping role. Kocherlakota (2002) is a notable exception. The paper demonstrates that two monies are necessary and su¢ cient to replace the record-keeping technology in the presence of limited commitment when money is divisible, concealable, and in …xed supply. Our paper re-examines Kocherlakota’s results in a quasi-linear environment with heterogeneous agents using the mechanism design approach. We depart from Kocherlakota (2002) along two important dimensions. First, we allow money supply to vary, which we show invalidates Kocherlakota’s result that a second money is essential to overcome limited commitment. Second, we add a new friction to limited commitment: agents hold private information about their types. This new friction makes one money insu¢ cient to replace recordkeeping and restores the essentiality of a second money. The basic model involves two ex ante types of agents and two aggregate states of the economy occurring with equal probabilities. In state 1, one type have higher marginal utility than the other and the reverse is true in state 2. Given that all agents have the same endowment, the optimal allocation requires that in each state, low-valuation agents transfer goods to high-valuation agents. The optimal allocation can be achieved only if it satis…es the participation constraints (hereafter PC) imposed by limited commitment and incentive constraints (hereafter IC) imposed by private information about types. We examine how the planner implements the optimal allocation with each of three memory technologies – record-keeping, one money, and two monies. The memory technology a¤ects how information is transmitted, which in turn determines the speci…c forms of the constraints imposed by existing frictions. Our main results are as follows. First, in contrast to Kocherlakota (2002), one money acts as a perfect substitute for the record-keeping technology when there is only limited commitment. The form of the PCs is determined by the planner’s ability to catch and penalize non-participants. With a record-keeping technology, once an agent refuses to participate, the planner can record the information permanently and impose perpetual autarky as punishment. A properly designed one-money mechanism can also record the same information permanently. Speci…cally, the planner can issue new money to reward participants, and then require a higher monetary entry fee for future participation in the mechanism. If an agent refuses to participate at any point, he cannot pay the future entry fee and will be permanently excluded from the mechanism. Kocherlakota (2002) …nds a di¤erent result owing to his restriction to monetary mechanisms with a …xed money supply, which weakens the planner’s ability to catch and penalize non-participants. Second, one money is not a perfect substitute for the record-keeping technology when there is private information about types. To deal with private information in our model, there are two ways to induce truthful type reporting: ex ante sorting and ex post sorting. Ex ante sorting asks agents to report their types before the state of the economy is realized; after the state is realized, the planner proposes allocations based on earlier reports. Ex post sorting asks agents to report their types after the state of the economy is realized and uses the information to propose allocations. The advantage of ex ante sorting is that it imposes less stringent ICs. However, it relies on the e¤ective transmission of information (about type reports) from before to after the realization of the state. Ex post sorting does not require the transmission of such information, 1 Since Mundell (1961), there has been a large literature on optimal currency area. The literature abstracts from the microfounded role of …at money and focuses on the relative advantages and disadvantages of a single currency versus multiple currencies in facilitating international trade and implementing stabilizing monetary polices. Another strand of literature makes explicit the micro-founded role of …at money and investigates whether multiple currencies can coexist or cocirculate using search theoretic monetary models. Examples are Matsuyama et al. (1993), Trejos and Wright (2001), Camera and Winkler (2003), Camera et al. (2004), and Craig and Waller (2004). In contrast, the goal of our paper is to examine whether multiple currencies (in comparison to a single currency) improve welfare by providing superior record-keeping.

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but it imposes more stringent ICs. A record-keeping technology can generate permanent records of type reports for e¤ective ex ante sorting, while one money cannot. With one money, to use ex ante sorting, the planner can encode type reports along only one dimension: the amount of money balance. After the state is revealed, the planner must rely on checking agents’money balances to infer their earlier reports. However, this will not be e¤ective when money is concealable. In particular, those who were given more money can claim to have reported to be either type. As a result, an allocation can be achieved only if it entails transfers from the type with a lower money balance to the type with a higher money balance. Otherwise, all agents can show the lower money balance and ask for transfers. It then follows that when the planner asks agents to report their types before the state is realized, every agent will report to be the type that is assigned a higher money balance, which invalidates ex ante sorting. With one money, the only way to align incentives is through ex post sorting, which imposes more stringent ICs. Therefore, one money is not su¢ cient to replace record-keeping. Third, we …nd that adding a second money restores ex ante sorting so that two monies become a perfect substitute for the record-keeping technology. Two monies allow the planner to record information (about type reports) along two dimensions: the total money balance and the composition of the two monies. This enables the planner to accurately retrieve original type reports by examining agents’ monetary portfolios. In particular, the planner can encode di¤erent type reports into di¤erent portfolios with a …xed total money balance but di¤erent compositions. An agent cannot lie about earlier type reporting because a change in composition must be accompanied by a change in total money balance. We further demonstrate that two monies acting as a perfect substitute for record-keeping continues to hold in more general environments. Since money is divisible, two monies can generate an in…nite number of di¤erent portfolios, all featuring the same total balance but di¤erent compositions of the two monies. As long as the pieces of information to be recorded are countable, two monies are su¢ cient to replace the record-keeping technology. Besides Kocherlakota (2002), our paper is also related to Kocherlakota and Krueger (1999) in that it shares the feature that a second money improves welfare by serving as a signalling device to deal with private information. However, Kocherlakota and Krueger (1999) build on Trejos and Wright (1995) with indivisible money. The result that there is no need for a third money does not extend to models with multiple types of agents. Owing to the inventory constraint, two monies can provide, at most, two di¤erent monetary portfolios and record two pieces of information. In addition, unlike our paper, Kocherlakota and Krueger (1999) do not focus on comparing money and record-keeping. Instead of fully characterizing the optimal allocation and the required memory technology to implement the allocation, their paper …nds a non-empty set in the parameter space such that two currencies improve welfare over a single currency. The rest of the paper proceeds as follows. Section 2 lays out the physical environment and characterizes the …rst-best allocation. Section 3 introduces limited commitment and shows that one money acts as a perfect substitute for record-keeping. Section 4 adds private information about types and shows that one money is insu¢ cient to replace record-keeping. In this case, adding a second money restores the equivalence between money and record-keeping. Section 5 extends the model by considering N > 2 types of agents and argues in general that two monies are a perfect substitute for the record-keeping technology. Conclusions are provided in Section 6.

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The Physical Environment

The framework is the quasi-linear environment introduced by Lagos and Wright (2005) without the search friction (see Figure 1). Time is discrete and runs from 0 to 1: Each period consists of two stages: day and night. There are two goods, one in each stage. Both goods are perishable. There are two types of agents, labeled by a and b; each is of measure 1. [Place Figure 1 about here] During the day, all agents can produce and consume the day good, and share the same linear preferences over the good. Let z be the amount of production (consumption if z is negative). The disutility of production (utility of consumption if z is negative) is z: At night, each agent is endowed with y units of the night good. Agents’ preferences depend on the realization of a state variable s, which is equal to 1 or 2 with equal probabilities. The state variable is realized at the beginning of the night and is publicly observable. When s = 1, type a’s preferences are given by u(c) and type b’s preferences are given by u(c), where > 1,

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u(0) = 0; u00 < 0 < u0 and u0 (0) = +1: When s = 2, type a have utility u(c) and type b have utility u(c). Note that type a have high marginal utility in state 1 and low marginal utility in state 2 (vice versa for type b). The shock to the state variable is iid across time. We will focus on symmetric stationary allocations, where all agents of the same type are treated in the same way and the two types are treated symmetrically. This implies that agents who have the same valuation of night goods consume the same amount at night and produce the same amount during the following day stage. Let ch and zh represent the night-stage consumption and the next-day-stage production for high-valuation agents (type a if s = 1 and type b if s = 2). Similarly, let c` and z` be the nightstage consumption and the next-day-stage production for low-valuation agents (type a if s = 2 and type b if s = 1).2 The ex ante lifetime utility of agents at a stationary allocation (ch ; c` ; zh ; z` ) is given by W = 12 1 1 [ u(ch ) zh + u(c` ) z` ]. The …rst-best allocation (ch ; c` ; zh ; z` ) maximizes W subject to the resource constraints ch + c` 2y and zh + z` 0, or maximizes3 W (ch ; c` ) =

1 1 21

[ u(ch ) + u(c` )] ;

(1)

subject to ch + c` = 2y;

(2)

zh + z` = 0:

(3)

The …rst-best allocation is characterized by u0 (ch ) = u0 (c` ); ch + c` = 2y and zh + z` = 0: Since > 1; the night-stage allocation features ch > y > c` : The planner can instruct each night-stage low-valuation agent to transfer y c` units of his endowment to high-valuation agents. Since the day-stage production/consumption enters linearly in preferences, any (zh ; z` ) that satis…es (3) would entail no ex ante welfare loss. Welfare at the …rst-best allocation is W = 21 1 1 [ u(ch ) + u(c` )]. The …rst-best allocation can be achieved if agents’types are public information, and agents are able to commit to participating in the mechanism. If agents lack commitment and have private information about their types, the …rst-best allocation can be achieved only if there is some form of memory technology that allows information about agents’types and actions to be passed across time (see Kocherlakota, 1998a; b). In addition to the resource constraints, an implementable allocation must also satisfy the PCs (so that agents have the incentive to stick with the mechanism) and the ICs (so that agents have the incentive to truthfully reveal their private information). The available memory technology determines how e¤ectively information can be transmitted across time and determines the speci…c forms of the PCs and the ICs. The following sections will study and compare the conditions under which the …rst-best allocation can be achieved using three memory technologies: record-keeping, one money and two monies. A record-keeping technology allows the planner to record any information and retain it for all future references. Money is de…ned as a set of durable, perfectly divisible and concealable tokens that can be issued only by the planner. We consider the class of no-commitment direct mechanisms without restricting to a speci…c mechanism (or game form). Detailed examples of the mechanisms will be provided to illustrate how to implement the …rst-best allocation with the available memory technology.

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Limited Commitment

As in Kocherlakota (2002), we …rst assume that agents lack commitment and that agents’ types are public information. With limited commitment, the allocation (ch ; c` ; zh ; z` ) must respect ex post rationality. The available memory technology may a¤ect the forms of the PCs by determining the penalty for nonparticipation. 2 Assume

z = 0 for each agent at the day stage of period 0. u0 (c) > 0 and agents su¤er disutility from day production, it is obvious that it is not optimal to have slack resource constraints. In the following, attention will be restricted to allocations with binding resource constraints. The binding day-stage constraint implies that the lifetime utility W can be simpli…ed as in (1). 3 Since

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3.1

Mechanisms with Record-Keeping

With a record-keeping technology, if an agent refuses to participate at any point in time, the planner can record the non-participation and exclude the non-participant from the mechanism forever. The mechanism can thus impose perpetual autarky as the penalty for non-participation. At the night stage, there are two PCs: u(ch ) + ( zh + W ) u(c` ) + ( z` + W )

u(y) + W0 ,

for high-valuation agents,

u(y) + W0 ,

for low-valuation agents,

where W is de…ned as in (1) and W0 = 12 1 1 [ u(y) + u(y)] is the welfare associated with perpetual autarky. At the day stage, there are also two PCs: zh + W

W0 ,

for agents with high valuation in the previous night stage,

z` + W

W0 ,

for agents with low valuation in the previous night stage.

Note that if ch > c` ; for night-stage high-valuation agents, the day-stage PC implies the night-stage PC; the reverse is true for night-stage low-valuation agents. As a result, it is su¢ cient to use the day-stage PC for high-valuation agents and the night-stage PC for low-valuation agents, which can be rewritten as zh z`

W

W

W0 ;

(4)

u(y)

W0

u(c` )

:

(5)

With limited commitment, an allocation can be achieved if and only if (4), (5), and the resource constraints (2), (3) are satis…ed. The …rst-best allocation can be achieved if and only if W

W0 + W

W0

u(y)

u(c` )

0;

or4 u(y) u(c` ) : [u(ch ) u(y)]

0

Proposition 1 When agents lack commitment, mechanisms with a record-keeping technology can achieve the …rst-best allocation if and only if 0.

3.2

One-money Mechanisms

Suppose that a record-keeping technology is unavailable, making it impossible to directly record and pass along information across time. In this case, the planner uses tokens – which we call money (denoted as $) – as a substitute for the record-keeping technology.5 Money is divisible, concealable as in Kocherlakota 4 If

>

satis…es zh

0,

the combinations of (zh ; z` ) that are consistent with the …rst-best allocation are not unique. Any (zh ; z` ) that

W

W0 and z`

W

W0

u(y) u(c` )

and zh + z` = 0 can achieve the …rst-best allocation. If

=

0,

the

u(y) u(c` ) .

combination of (zh ; z` ) is unique with zh = W W 0 , z` = W W0 If < 0 , the …rst-best allocation cannot be achieved. The second-best (symmetric stationary) allocation (c` ; ch ; zh ; z` ) conditional on is characterized by u(y) [u(ch )

u(c` ) u(y)]

= ;

c` + ch = 2y; zh = W (ch ; c` ) z` = W (ch ; c` )

W0

W0; u(y)

u(c` )

:

The second-best allocation has the feature that ch > ch > y > c` > c` . 5 The planner, however, has access to a contemporaneous memory technology that can remember agents’ actions within a stage.

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(2002), but money supply is allowed to vary. We show that mechanisms using one money can deal with limited commitment as e¤ectively as a record-keeping technology can. As pointed out in Kocherlakota (2002), when money is concealable, it is necessary to establish a monotonically increasing relationship between "proper" behavior and money balances. Here, the proper behavior is to follow the planner’s instructions based on agents’types (which suggest that type b transfer goods to type a if s = 1 and vice versa if s = 2). The planner can reward participants with newly issued money and require an increasing monetary entry or participation fee for future participation in the mechanism. The ever-increasing entry fee allows one-money mechanisms to e¤ectively catch non-participants and cast them into perpetual autarky (see Figure 2 for an example of a one-money mechanism; a detailed description of the mechanism is provided in Appendix A.1, available as supplementary materials). Therefore, one-money mechanisms involve the same PCs as those with a record-keeping technology. [Place Figure 2 about here] Proposition 2 When agents lack commitment, one money acts as a perfect substitute for the record-keeping technology and can achieve the …rst-best allocation if and only if 0. Proposition 2 is di¤erent from Kocherlakota (2002), which shows that one money is not a perfect substitute for record-keeping in the presence of limited commitment. This is because Kocherlakota (2002) restricts to mechanisms with a …xed money supply. Suppose that each agent is endowed with one unit of money in period 0 and no new money is injected thereafter. Without loss of generality, assume that s0 = 1; so that the planner asks type b to transfer goods to type a at night. Suppose that one type b agent does not participate (because he does not want to make the transfer). We check whether the planner can create a permanent record for the non-participant and exclude him from all future participation in the mechanism. For those who participate, the planner has to ensure that each of them enters period 1 night stage with the same money balance so that they are treated equally (whether or not they made transfers in period 0 night stage) as required by the …rst-best allocation. Without injections of new money, every participant will enter the new night stage with 1 unit of money. It then follows that the planner cannot force the non-participant into autarky since he also has $1 and can come back to the mechanism to ask for night-stage transfer if he becomes a high-valuation agent. Hence, one-money mechanisms with a …xed money supply cannot impose autarky as the punishment for non-participation.6 With a …xed money supply, the planner can only exclude non-participants temporarily. For example, the planner can ask high-valuation agents to use money in exchange for goods from low-valuation agents at night, and reverse the exchange pattern in the following day stage. Now, each low (high) valuation agent leaves the night stage with more (less) than $1. By requiring an entry fee higher than 1$ to receive day-stage transfer, the planner can exclude an agent who skipped the previous night stage as a low-valuation agent from the following day stage.7 The PCs become z`

[u(y)

zh

W

u(c` )]= , W0 ,

for night-stage low-valuation agents, for night-stage high-valuation agents.

The …rst-best allocation can be achieved if and only if f

u(y) u(c` )

+W

2[u(y) u(c` )] > u(ch ) u` + (1 )u(y)

W0

0, or

0:

where the superscript "f" denotes a …xed money supply. Since non-participants cannot be forced into perpetual autarky, one money cannot fully replace record-keeping.8 6 With

a …xed money supply, Kocherlakota (2002) concludes that a second money is necessary to restore autarkic punishment for nonparticipation. In addition, the assumption of a …xed money supply also determines that Kocherlakota (2002) must record agents’participation history through decimal expansions of monetary holdings. After participating in a period, an agent’s money holdings develop a new decimal digit. If the agent skips a period, the decimal digits of his money holdings will fall short. Here, the planner can record particpation history by issuing new money to participants. In Figure 2; we use integer increments as an example, but the proposed mechanism carries through with noninteger increments as well. 7 Previous high-valuation agents enter the day stage with less than $1. By imposing $1 entry fee for all subsequent night stages, the planner can exclude from the mechanism forever those who consume ch at night but refuse to participate in the following day stage. The PC for high-valuation agents thus remains the same as before. 8 Note that in Kocherlakota (2002), one-money mechanisms with a …xed money supply can never achieve the …rst-best

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4

Private Information about Types

The above analysis shows that if money supply is allowed to vary, one money acts as a perfect substitute for the record-keeping technology in the presence of limited commitment. In this section, we examine whether that conclusion is robust to the existence of private information about types. To make allocation instructions, the planner needs to know agents’ valuations at the night stage. Because of the structure of the physical environment, knowledge of agents’types is equivalent to knowledge of their valuations. If types are private information, the planner’s allocation instructions will be based on agents’ reported (rather than directly observed) types. With private information, allocations must satisfy the ICs, i.e., agents must have the incentive to truthfully reveal their private information. Now the available memory technology may a¤ect the forms of both the PCs and the ICs.

4.1

Ex Ante and Ex Post Sorting

There are two ways to deal with the incentive problem caused by private information about types. One way is to induce agents to truthfully report their types before the state of the economy is realized, which is labeled as ex ante sorting. The other way is to induce agents to truthfully reveal their types after the state of the economy is realized, which is labeled as ex post sorting. 4.1.1

Ex Ante Sorting

To use ex ante sorting, the planner asks agents to report types before the state of the economy is realized; after the realization of the state, the planner proposes allocation based on earlier type reports. When s = 1, the planner asks those who reported to be type b agents to transfer goods to those who reported to be type a agents. When s = 2, the planner makes the reverse instruction. The planner has to ensure that information about type reports can be e¤ectively transmitted from before to after the state is realized. If such memory technology is available, the IC imposed by private information about types takes the following form 1 1

u(ch ) 2

zh

+

u(c` )

z`

1

2

1

u(c` ) 2

z`

+

u(ch )

zh 2

:

(6)

Since types are permanent, if recorded information can be passed on into the in…nite future, the planner needs only to ask agents to report their types once at the day stage of periods 0 and then uses the information for all future allocations. This is why the lifetime utilities are used in (6).9 One can see that (6) holds as long as ch > c` , which is satis…ed at the …rst-best allocation.10 Using ex ante sorting does not impose extra constraints on zh and z` ; other than the resource constraint (3). Lemma 1 When agents lack commitment and hold private information about their types, ex ante sorting mechanisms can achieve the …rst-best allocation if and only if 0: 4.1.2

Ex Post Sorting

To use ex post sorting, the planner asks agents to report types after the state of the economy is realized, which is equivalent to asking agents to report their valuations of the night goods. In this case, the ICs take the form allocation. Here, the mechanism with a …xed money supply can achieve the …rst-best allocation if agents are su¢ ciently patient. This is due to quasi-linear preferences (or the existence of the day stage) in our model. With a …xed money supply, the only way to induce low-valuation agents to transfer goods to high-valuation agents is to give the former some money in return. In the absence of quasi-linear preferences (or equivalently, the day stage), the mechanism implies that agents start the night stage with heterogeneous money holdings, which is inconsistent with the …rst-best allocation. With qausi-linear preferences, the heterogeneity in money holdings following a night stage can be eliminated during the following day stage. In addition to tractability, quasi-linear preferences provide a new channel for alignment of incentives. 9 Here we focus on stationary allocation, and it su¢ ces if information about type reports can be transmitted only from day to the following night. In that case, the planner needs to ask agents to report their types at each day stage. The incentive constraint becomes u(ch ) zh u(c` ) z` u(c` ) z` u(ch ) zh + + ; 2 2 2 2 which is essentially the same as (6). 1 0 The stationary symmetric second-best allocation also satis…es this condition.

7

u(ch ) + ( zh + W )

u(c` ) + ( z` + W );

u(c` ) + ( z` + W )

u(ch ) + ( zh + W );

which can be rearranged as zh zh

z` z`

[u(ch )

u(c` )]

u(ch )

u(c` )

;

(7)

:

(8)

The …rst constraint ensures that high-valuation agents (type a at s = 1 or type b at s = 2) do not want to imitate low-valuation agents (type a at s = 2 or type b at s = 1). The second constraint ensures that low-valuation agents do not want to imitate high-valuation agents. In comparison to ex ante sorting, ex post sorting imposes extra restrictions on zh and z` . If ex post sorting is used, the …rst-best allocation can be achieved if and only if (ch ; c` ; zh ; z` ) satis…es (4), (5), (7), and (8), as well as the resource constraints (2) and (3). The restriction on is stated in Lemma 2 (refer to Appendix B.1, available as supplementary materials, for the proof). Lemma 2 When agents lack commitment and hold private information about their types, ex post sorting u(ch ) u(c` ) mechanisms can achieve the …rst-best allocation if and only if 1 ( +1)[u(c ) u(y)] > 0 . h

Ex post sorting requires a higher to achieve the …rst-best allocation than ex ante sorting does. However, it does not rely on information transmission from before to after the realization of the state of the economy (or from day to night).11

4.2

Mechanisms with Record-Keeping

With a record-keeping technology, the planner can ask agents to report their types at the beginning of period 0, record the information, and use it to infer an agent’s marginal utilities in all future night stages. For example, if an agent reports to be type a, the planner knows that he has high valuation in state 1 and low valuation in state 2. Hence, ex ante sorting is e¤ective. Proposition 3 When agents lack commitment and hold private information about their types, mechanisms with a record-keeping technology can achieve the …rst-best allocation if and only if 0.

4.3

One-money Mechanisms

Suppose that a record-keeping technology is not available. This subsection investigates whether one money is a perfect substitute for the record-keeping technology. As discussed in Section 3, one-money mechanisms can deal with limited commitment as e¤ectively as a record-keeping technology, and thus share the same PCs as mechanisms with a record-keeping technology. How one-money mechanisms solve private information remains to be examined. In particular, we will check whether one-money mechanisms can e¤ectively use ex ante sorting. With one money, one can encode di¤erent type reports along only one dimension: the amount of money balance. To use ex ante sorting, the planner asks agents to report their types at the beginning of period 0 and gives di¤erent money balances to di¤erent reported types. Without loss of generality, we assume that the planner gives more money to those who report to be type a. For ex ante sorting to be e¤ective, the planner must be able to retrieve agents’initial type reports in the future by checking their money balances and then recommend allocations according to observed money balances. However, since money balances are concealable, the planner is not always able to retrieve agents’ initial type reports. Agents who can show high money balances must be those who reported to be type a; yet those who can show low money balances could either be reported type b or reported type a because the latter can hide part of their money balances. 1 1 Ex

post sorting does require information to be transmitted from night to the following day stage.

8

Therefore, an allocation can be implemented only if it suggests goods transfers from reported type b to reported type a. It then follows that all agents will report to be type a to acquire the high money balance at the beginning of period 0, which invalidates ex ante sorting. The planner needs to use ex post sorting to induce agents to truthfully reveal their types (or, equivalently, their marginal utilities) at night by resorting to variations in production/consumption in the following day stage. The type with high marginal utility can choose to consume more at the night stage, leave the stage with less money, and produce in the following day stage in return for more money. Refer to Figure 3 for an example of a one-money mechanism (and see Appendix A.2, available as supplementary materials, for a detailed description of the mechanism). Ex post sorting requires information to be transmitted from night to day, and money balances are able to achieve that purpose. For example, if st 1 = 1, at the day stage of period t, the planner can infer that those with a low money balance are reported type a; while those with a high money balance are reported type b. [Place Figure 3 about here] Proposition 4 When agents lack commitment and hold private information about their types, one money is insu¢ cient as a perfect substitute for the record-keeping technology. One-money mechanisms can achieve the …rst-best allocation if and only if 1 > 0:

4.4

Two-money Mechanisms

Given that one money is insu¢ cient as a perfect substitute for the record-keeping technology, this subsection introduces a second money and show that two monies constitute a perfect substitute for the record-keeping technology. Label the two monies as "red" (denoted as R$) and "green" (denoted as G$). Two-money mechanisms can deal with limited commitment in much the same way as one-money mechanisms and, hence, the PCs remain the same as those with a record-keeping technology. Unlike one-money mechanisms, two-money mechanisms can restore ex ante sorting. With two monies, the planner can encode and record information along two dimensions: the total money balance and the composition of the two monies. Compared with one-money mechanisms, the information embedded in di¤erent monetary portfolios featuring the same total balance can be preserved and cannot be tampered with by hiding money balances. For example, at the beginning of period 0, the planner can give those reported type a one unit of red money and those reported type b one unit of green money. At the following night stage, the planner can retrieve agents’initial type reports by checking their monetary portfolios. The planner can then recommend transfers of goods from green money holders to red money holders if s = 1 and vice versa if s = 2. Di¤erent from one-money mechanisms, instructions in both states can be implemented because those who are required to make the transfers cannot show the required monetary portfolio to receive the transfer. See Figure 4 for an example of a two-money mechanism (A detailed description of the mechanism is provided in Appendix A.3, available as supplementary materials). [Place Figure 4 about here] Proposition 5 When agents lack commitment and hold private information about their types, two monies act as a perfect substitute for the record-keeping technology. Two-money mechanisms can achieve the …rst-best allocation if and only if 0. A second money is essential and improves welfare over one money when < 1 . When 0 < 1, two monies can achieve the …rst-best allocation, while one money cannot. When < 0 ; the …rst-best allocation cannot be achieved even with the record-keeping technology. The constrained optimal allocation (ch ; c` ) features ch > ch > y > c` > c` . Two monies can achieve the constrained optimal allocation, but one money cannot. Record-keeping and two monies can use ex ante sorting, and the associated incentive constraint is automatically satis…ed as long as ch > c` . The planner can use (zh ; z` ) to deal only with limited commitment. With one money, ex post sorting has to be used, and the ICs lead to extra restrictions on (zh ; z` ), which makes one-money mechanisms less e¤ective.

9

5

Extensions

This section extends the previous results to an environment with N > 2 types of agents and argues in general that two monies are a perfect substitute for the record-keeping technology.

A Model with N > 2 Types of Agents

5.1

Suppose that there are N > 2 types of agents (each type is of measure 1) and N states of the economy indexed by f1; 2; :::; N g. The state of the economy is realized at the night stage and each state occurs with probability 1=N . The night-stage preferences of type i agents in state j are described by ij u(cij ), where ij takes N possible values 1 > 2 > ::: > N > 0 with equal probabilities. For the value of ij , it helps to visualize that the N states lie along a circle in a clockwise sequence of state 1, 2, ..., and N . Type i agents have the highest valuation 1 at s = i, and as they travel clockwise along the circle of the states, ij decreases and reaches ( j i + 1; if j i the lowest value N at s = i 1. Mathematically, ij = v(i;j) where v(i; j) = is N + j i + 1; if j < i the valuation-indicator function and v 2 f1; 2; :::; N g. Attention is again restricted to symmetric stationary allocations where agents with the same v consume the same amount, cv ; at night and produce the same amount, zv ; at the following day stage.12 The …rst-best allocation is characterized by vu

0

(cv ) =

v0

0

0

u (cv0 ) for all v; v 2 f1; 2; :::; N g and

N X

cv = N y:

v=1

Suppose that c1 > c2 > ::: > cB > y > cB+1 > ::: > cN so that B types are night-stage borrowers who consume more than their endowments, and N B types are lenders who consume PNless than their endowments. Any day-stage allocation (z1 ; z2 ; :::; zN ) that satis…es the resource constraint v=1 zv = 0 ish consistent with i PN u(c ) : the …rst-best allocation. The ex ante …rst-best lifetime utility for any type is W N = N1 1 1 v v v=1 With limited commitment and private information about types, an allocation can be implemented only if it satis…es the PCs and the ICs. First consider the PCs. With N types, there are 2N PCs (one night constraint and one day constraint for each v 2 f1; 2; :::; N g), v u(cv )

+ ( zv + W N )

v u(y)

+ W0N ;

zv + W N

W0N ; hP i PN N 1 1 N where W N = N1 1 1 u(y) v=1 v : v=1 v u(cv ) and W0 = N 1 With regard to the ICs, ex ante sorting and ex post sorting are still the two ways to deal with private information. To use ex ante sorting, the planner has to be able to keep and retrieve information about type reports from before to after the realization of the state of the economy. If such information can be transmitted, the following N 2 N constraints ensure that each type i has the incentive to truthfully report his type at the day stage of period 0, 2 3 2 3 N N X 1 1 4X 1 1 5 4 5 for all i 6= i0 2 f1; 2; :::; N g. ij u(cij ) ij u(ci0 j ) N1 N 1 j=1 j=1 Lemmas 3 and 4 state two results when ex ante sorting can be used (see Appendix B.2 and B.3, available as supplementary materials, for the proof). Lemma 3 With N > 2 types of agents, the incentive constraints imposed by ex ante sorting are satis…ed at the …rst-best allocation.13 1 2 As

in the case with two types, it is assumed that each agent has z = 0 at the day stage of period 0. ex ante sorting incentive constraints are also satis…ed at the second-best symmetric stationary allocation.

1 3 The

10

Lemma 4 With N > 2 types of agents, when agents lack commitment and hold private information about their types, ex ante sorting mechanisms can achieve the …rst-best allocation if and only if PN

v=B+1 v

N 0

PB

v=1 v

[u(y)

[u(cv )

u(cv )] u(y)]

:

To use ex post sorting, the following N 2 N constraints must be satis…ed to ensure that agents with do not want to report to have v0 with v 0 = 6 v v u(cv )

+ ( zv + W N )

v u(cv 0 )

v

+ ( zv0 + W N ) for all v 6= v 0 2 f1; 2; :::; N g

Compared with ex ante sorting, ex post sorting relies on the variation of day-stage production/consumption to align incentives. It does not, however, require information about type reports to be transmitted from before to after the state is realized. Lemma 5 states the condition under which the …rst-best allocation can be achieved through ex post sorting (see Appendix B.4, available as supplementary materials, for the proof). Lemma 5 With N > 2 types of agents, when agents lack commitment and hold private information about their types, ex post sorting mechanisms can achieve the …rst-best allocation if and only if N 1

PN

PN

v=2 (N + 1

v=2 (N

v)

v

+1 u(cv

v) 1)

u(cv ) PN u(cv ) + v=1 v [u(cv )

v

u(cv

1)

u(y)]

>

N 0 :

With a record-keeping technology, the planner can use ex ante sorting to deal with private information. To deal with limited commitment, the planner can directly record non-participation and cast non-participants N into perpetual autarky. The …rst-best allocation can be achieved if and only if 0 . In the absence of a record-keeping technology, the planner can use money to record and transmit information across time. One-money mechanisms deal with limited commitment in the same way as mechanisms with a recordkeeping technology do. However, one-money mechanisms cannot use ex ante sorting to deal with private information. The planner has to use ex post sorting and the …rst-best allocation can be achieved if and N N only if < N 1 > 0 . When 1 , a second money is essential and improves welfare over one money. Furthermore, two monies restore ex ante sorting and act as a perfect substitute for the record-keeping technology. Proposition 6 With N > 2 types of agents, when agents lack of commitment and hold private information about their types, two monies act as a perfect substitute for the record-keeping technology. Two-money N mechanisms can achieve the …rst-best allocation if and only if 0 . Essentially, two-money mechanisms deal with limited commitment and private information exactly the same way as two-money mechanisms with two types of agents do (An example of a two-money mechanism is provided in Appendix A.4, available as supplementary materials). Each type hold a monetary portfolio that features the same total money balance but di¤erent compositions of the two monies. Since an in…nite number of such portfolios are available with two monies, the planner can distinguish each type as long as the number of the types is countable. For any N , two monies can serve as a perfect substitute for the record-keeping technology. Note that if money is indivisible, as in Kocherlakota and Krueger (1999), N monies are required to replace the record-keeping technology when there are N types of agents. With indivisible money (and an inventory restriction that one can hold at most one unit of money, a standard assumption in models with indivisible money for tractability), two monies can only provide two di¤erent monetary portfolios. This implies that two monies are not enough to signal N types of agents. Divisibility of money makes two monies more robust as a perfect substitute for the record-keeping technology.

5.2

Two Monies as a Perfect Substitute for Record-keeping

We have shown that two monies are su¢ cient to replace record-keeping in an environment with more than two types of agents.14 Here we develop an intuitive argument to show that the result holds in very general 14 A

related result in Townsend (1987) shows that two types of tokens are enough to distinguish past histories in an environment featuring spatial separation and private information.

11

environments. If money balances are not concealable, there is a one-to-one mapping between records and money balances. Money balances will thus carry the relevant information across time. When money balances are concealable, the one-to-one mapping will be destroyed, since agents can hide money balances. The introduction of a second money solves the problem by encoding di¤erent information into monetary portfolios with the same total balances but di¤erent compositions of the two monies.15 The information content of the monetary "records" remains intact across time. When money is divisible, it is possible to encode any countable pieces of information into di¤erent monetary portfolios so that a third money will not be needed.

6

Conclusion

Recent advances in micro-founded monetary theories seem to have reached a consensus that the role of money is to make up for missing record-keeping technologies. The paper examines whether money serves as a perfect substitute for the record-keeping technology in an environment where ex ante heterogeneous agents lack commitment and have private information about their types. The available memory technology –a record-keeping, one money, or two monies –determines the e¤ectiveness of information transmission and a¤ects the forms of the PCs and the ICs. The …nding is that with a variable money supply, one money is a perfect substitute for record-keeping when there is only limited commitment (in contrast to Kocherlakota, 2002), but ceases to be so when there is private information about types. In the latter case, adding a second money restores money as a perfect substitute for record-keeping. The welfare improving role of a second money lies in the superior ability of two monies to deal with private information. The result that two monies being su¢ cient to replace record-keeping is shown to be robust in more general environments. The results of the paper are developed in a mechanism design context where allocations are based on agents’participation choices and reports on private information. An interesting question is whether the same allocations can be achieved with some form of market mechanisms, for example, competitive equilibrium. A preliminary investigation of this topic can be found in Appendix C (available as supplementary materials), which shows that in the context of competitive markets, two monies provide a signalling device and improve welfare over one money when there is private information. We look forward to future research that will provide a fuller treatment of this important topic.

1 5 Imagine that di¤erent pieces of information are recorded as di¤erent points along a straight line r + g = m in (r; g) space where r and g are the amounts of red money and green money respectively, and m is the total money balance.

12

Figure 1: Environment

1/2 Type a and b: -z 1/2

Day

s=1 Type a: u(c) Type b: u(c) >1 s=2 Type ba: u(c) Type Type u(c 1 ) b: u(c) >1 Night Type b u(c1)

Notes: 1. c is consumption of night good; 2. z is production (consumption if negative) of day good; 3. s is the state variable which is equal 1 or 2 with probability ½; shocks to s are i.i.d. across time.

13

Figure 2: One-money Mechanism with Limited Commitment If type a, receive * goods and $ (t+ ) Pay $ t entry fee

If type a, submit z goods and receive $ (t+1 ) Pay $( t+ ) entry fee

If type b, submit * goods and receive $ (t+ ) money

If type b, receive z goods and $ (t+1)

with 0< <1 Period t day

Period t-1 night

Notes: The figure shows the mechanism when st-1=1; exchange a and b when st-1 =2.

14

Figure 3: One-money Mechanism with Limited Commitment and Private Information

If report to be type a, receive * goods and $ (t+ h ) Pay $ t entry fee

Pay $ ( t+ entry fee

h)

Submit z goods and receive $ (t+1)

If report to be type b, submit * goods and receive $ (t+ l ) with 0< h<

l

Pay $ ( t+ l ) entry fee

<1

Period t-1 night

Receive z goods and $ (t+1)

Period t day

Notes: The figure shows the mechanism when st-1=1; exchange a and b when st-1=2.

15

16

... Pay G$ t entry fee

Period t-1 night (st-1=1)

Submit * goods and receive G$ (t+ )

Receive * goods and R$ (t+ ) (0< <1)

Notes: The figure shows the mechanism when st-1=1; exchange a and b when st-1=2.

Period 0 day

If report to be type b, Receive G$ 1

If report to be type a, Receive R$ 1

Pay R$ t entry fee

Pay G$ ( t+ ) entry fee

Pay R$ ( t+ ) entry fee

Figure 4: Two-money Mechanism with Limited Commitment and Private Information

Period t day

Receive z goods and G$ (t+1)

submit zz goods and Submit R$ (t+1) (t+1) receive R$

References [1] Camera, G., Craig, B., Waller, C. J., 2004. Currency competition in a fundamental model of money. Journal of International Economics 64 (2), 521-544. [2] Camera, G., Winkler, J., 2003. International monetary trade and the law of one price. Journal of Monetary Economics 50 (7), 1531-1553. [3] Craig, B., Waller, C. J., 2004. Dollarization and currency exchange. Journal of Monetary Economics 51 (4), 671-689. [4] Kocherlakota, N., 1998a. The technological role of …at money. Federal Reserve Bank of Minneapolis Quarterly Review 22 (3), 2–10. [5] Kocherlakota, N., 1998b. Money is memory. Journal of Economic Theory 81 (2), 232–251. [6] Kocherlakota, N., Krueger, T., 1999. A signaling model of multiple currencies. Review of Economic Dynamics 2 (1), 231-244. [7] Kocherlakota, N., 2002. The two-money theorem. International Economic Review 43 (2), 333-346. [8] Lagos, R., Wright, R., 2005. A uni…ed framework for monetary theory and policy analysis. Journal of Political Economy 113 (3), 463-484. [9] Matsuyama, K., Kiyotaki, N., Matsui, A., 1993. Toward a theory of international currency. Review of Economic Studies 60 (2), 283-307. [10] Mundell, R. A., 1961. A theory of optimum currency areas. American Economic Review 51 (4), 657-665. [11] Ostroy, J. M., 1973. The informational e¢ ciency of monetary exchange. American Economic Review 63 (4), 597-610. [12] Townsend, R. M., 1987. Economic organization with limited communication. American Economic Review 77 (5), 954–971. [13] Trejos, A., Wright, R., 1995. Search, bargaining, money, and prices. Journal of Political Economy 103 (1), 118-141. [14] Williamson, S., Wright, R., 2010. New monetarist economics: models. In: Friedman, B., Woodford, M. (Eds.), Handbook of Monetary Economics, second edition, forthcoming. [15] Wright, R., Trejos, A., 2001. International currency. Advances in Macroeconomics 1 (1), Article 3.

A A.1

Detailed Descriptions of Mechanisms One-money Mechanism with Limited Commitment

At either stage in each period, an agent chooses action from f0; 1g, where 0 means that the agent does not participate and 1 means that the agent participates. The mechanism speci…es the outcome function at each night stage that maps an agent’s action to an allocation and a monetary receipt. At the day stage of period 0, the mechanism endows each agent with 1 unit of money and assigns 0 production/consumption to each agent. At the night stage of period t 1 for t 1 and 0 < < 1, if a type a agent chooses 1 (by paying t $), the agent is entitled to a transfer of goods consumption ch ) and a monetary receipt t + $;

(which implies

if a type b agent chooses 1 (by paying t $), the agent is entitled to a transfer of goods implies consumption c` ) and a monetary receipt t + $;

17

(which

if an agent chooses 0 (because he cannot or chooses not to pay the entry fee), the agent gets 0 transfer and 0 monetary receipt. At the day stage of period t for t

1,

if a type a agent chooses 1 (by paying t + $), the agent is entitled to a transfer of goods implies production of z) and a monetary receipt t + 1 $;

z (which

if a type b agent chooses 1 (by paying t + $), the agent is entitled to a transfer of goods z (which implies consumption of z) and a monetary receipt t + 1 $; if an agent chooses 0 (because he cannot or chooses not to pay the entry fee), the agent gets 0 transfer and 0 monetary receipt. The mechanism is a no-commitment mechanism. The equilibrium concept that is adopted is Nash Equilibrium. It is straightforward to show that everybody choosing 1 in all stages consists of a Nash Equilibrium. The …rst-best allocation can be achieved if and only if 0.

A.2

One-money Mechanism with Limited Commitment and Private Information

At either stage in each period, an agent chooses action from f0; 1g, where 0 means that the agent does not participate and 1 means that the agent participates. The mechanism speci…es the outcome function at each night stage that maps an agent’s action and type reports to an allocation and a monetary receipt. At each day stage, the outcome function maps an agent’s action and monetary entry fee to an allocation and a monetary receipt. At the day stage of period 0, the mechanism endows each agent with 1 unit of money and assigns 0 production/consumption to each agent. For example, at the night stage of period t 1 for t 1; 0 < h < ` < 1: if an agent chooses 1 (by paying t $) and reports to be type a, the agent is entitled to a transfer of goods (which implies consumption ch ) and a monetary receipt t + h $; if an agent chooses 1 (by paying t $) and reports to be type b, the agent is entitled to a transfer of goods (which implies consumption c` ) and a monetary receipt t + ` $; if an agent chooses 0 (because he cannot or chooses not to pay the entry fee) and reports to be either type, the agent gets 0 transfer and 0 monetary receipt. At the day stage of period t for t

1:

if an agent chooses 1 and pays t + h $, the agent is entitled to a transfer of goods production of z) and a monetary receipt t + 1 $;

z (which implies

if an agent chooses 1 and pays t + ` $, the agent is entitled to a transfer of goods z (which implies consumption of z) and a monetary receipt t + 1 $. if an agent chooses 0 (because he cannot or chooses not to pay either entry fee), the agent gets 0 transfer and 0 monetary receipt. The mechanism is a direct mechanism that recommends allocations based on agents’reports about their types at the night stage (there is no need for type reporting during the day because monetary balances can carry the necessary information from night to day). The equilibrium concept that is adopted is again Bayesian Nash Equilibrium. It is straightforward to show that it is a Bayesian Nash Equilibrium for type a agent to choose 1 and report to be type a at each night stage and type b agents to choose 1 and report to be type b at each night stage. The …rst-best allocation can be achieved if and only if 1.

18

A.3

Two-money Mechanism with Limited Commitment and Private Information

At either stage in each period, an agent chooses action from f0; 1g, where 0 means that the agent does not participate and 1 means that the agent participates. The mechanism speci…es the outcome function that maps an agent’s action and monetary entry fee to an allocation and a monetary receipt except that, at the day stage of time 0, the outcome function maps an agent’s action and reported types to an allocation and a monetary receipt. For example, at the night stage of period t 1 for t 1, 0 < < 1: if an agent chooses 1 and pays t R$, the agent is entitled to a transfer of goods consumption ch ) and a monetary receipt t + R$;

(which implies

if an agent chooses 1 and pays t G$, the agent is entitled to a transfer of goods consumption c` ) and a monetary receipt t + G$;

(which implies

if an agent chooses 0 (because he cannot or chooses not to pay either entry fee), the agent gets 0 transfer and 0 monetary receipt. At the day stage of period t for t

1:

if an agent chooses 1 and pays t + R$, the agent is entitled to a transfer of goods production of z) and a monetary receipt t + 1 R$;

z (which implies

if an agent chooses 1 and pays t + G$, the agent is entitled to a transfer of goods z (which implies consumption of z) and a monetary receipt t + 1 G$. if an agent chooses 0 (because he cannot or chooses not to pay either entry fee), the agent gets 0 transfer and 0 monetary receipt. At the day stage of period 0: if an agent chooses 1 and reports to be type a, the agent is entitled to 0 transfer and a monetary receipt 1 R$; if an agent chooses 1 and reports to be type b, the agent is entitled to 0 transfer and a monetary receipt 1 G$; if an agent chooses 0, the agent gets 0 transfer and 0 monetary receipt. The mechanism is a direct mechanism and makes allocation recommendations based on agents’reports on their types at the beginning of period 0. In all future periods, this information is carried through each agent’s monetary portfolio. As before, the Bayesian Nash Equilibrium concept is adopted. It is straightforward to show that a Bayesian Nash Equilibrium is where type a agents choose to hold 1 R$ at the beginning of period 0, choose 1 and pay the entry fee with R$ in all future stages; and type b agents choose to hold 1 G$ at the day stage of period 0, choose 1 and paying the entry fee with G$ at all future stages. The …rst-best allocation can be implemented if and only if 0.

A.4

Two-money Mechanism with N Types of Agents

At the day stage of date 0, the planner asks agents to report their types, and gives those who report to be type i a monetary portfolio ri R$ + (1 with 0 < ri < 1 for each i 2 f1; 2; :::; N g and 1 two-money mechanism is illustrated in Figure 5.

ri )G$

r1 > r2 > ::: > rN

19

0. The implementation of a

20

...

Pay R$ tri and G$ t(1-ri ) entry fee

Period t-1 night (st-1 =j)

Receive [cv(i,j) –y] goods, Pay R$ (t+ ) ri and R$ (t ) ri and G$ (t+ ) (1-ri) G$ (t+ ) (1-ri ) entry fee (0< <1)

Notes: 1. [cv(i,j) –y] may be negative; in that case, the agent submits [y-cv(i,j)] units of night goods. 2. zv(i,j) may be negative; in that case, the agent receives - zv(i,j) units of day goods.

Period 0 day

If report to be type i, Receive R$ ri and G$(1-ri )

Figure 5: Two-money Mechanism with Limited Commitment and Private information (multiple types)

Period t day

Submit zv(i,j) goods; receive R$ (1+t)ri and G$ (1+t)t(1-ri)

B B.1

Proofs Proof of Lemma 2

Proof. When ex post sorting is used, an allocation (ch ; c` ; zh ; z` ) can be achieved if and only if (4), (5), (7), (8) and the resource constraints (2) and (3) are satis…ed. Note that the PC for high-valuation agents (4) and the IC for low-valuation agents (8) imply the PC for low-valuation agents (5). Note also that only (8) and (4) are in tension with the day stage resource constraint. As a result, we need only to consider (4), (8) and the resource constraints. Together, they imply that the …rst-best allocation can be achieved if and only if W

W0 + W

[u(ch )

W0

u(c` )]

0;

or u(ch ) u(c` ) : ( + 1)[u(ch ) u(y)]

1

The result that

B.2

1

>

0

follows from u(ch ) + u(c` ) > (1 + )u(y):

Proof of Lemma 3

Proof. Given the symmetric structure of the economy (and the focus on stationary symmetric allocations), it su¢ ces to prove that type 1 agents do not want to mis-report to be type 2; 3; :::; N . Other types face the same incentive constraints. The …rst step is to show that type 1 do not mis-report to be type 2; or N X

v u(cv )

2 u(c1 )

+

3 u(c2 )

+

+

N u(cN 1 )

+

1 u(cN ):

(9)

v=1

LHS ( 1 [( 1 = 0. =

RHS )u(c 2 1) + ( 2) + ( 2

3 )u(c2 )

2 3)

+

+(

+

+(

N 1

N )u(cN 1 )

N 1

N )]u(cN )

(

N

+( N 1 )u(cN )

1 )u(cN )

The last inequality holds if 1 > 2 > > N and c1 c2 cN . It follows that (9) is satis…ed at the …rst-best allocation. The second step is to prove that type 1 do not mis-report to be type 3, or N X

v u(cv )

3 u(c1 )

+

4 u(c2 )

+

+

N u(cN 2 )

+

1 u(cN 1 )

+

2 u(cN 2 ):

v=1

LHS RHS = ( 1 3 )u(c1 ) + ( 2 4 )u(c2 ) + ( 3 5 )u(c3 ) + ( 4 +( N 4 N 2 )u(cN 4 ) + ( N 3 N 1 )u(cN 3 ) + ( ( N 1 )u(c ) ( )u(c 1 N 1 N 2 N 2) = 1 u(c1 ) + 2 u(c2 ) N 1 u(cN 3 ) N u(cN 2 ) ( 1 ( 2 N 1 )u(cN 1 ) N )u(cN 2 ) ( 1 N 1 )u(cN 3 ) + ( 2 N )u(cN 2 ) ( 1 )u(c ) ( N 1 N 1 2 N )u(cN 2 ) 0,

21

6 ) u(c4 ) N 2

+ ::: N )u(cN

2)

where the inequalities follow from 1 > 2 > > N and c1 c2 imitate type 3 at the …rst-best allocation. Following similar arguments, it can be shown that all of the rest N > N and c1 c2 cN . 1 > 2 >

B.3

cN . Type 1 thus do not want to 3 constraints are also satis…ed if

Proof of Lemma 4

Proof. Based on Lemma 3, we need only to consider the participation constraints that for v 2 f1; 2; :::; N g; v u(cv )

+ ( zv + W N ) zv + W N

+ W0N ;

v u(y) W0N :

(10) (11)

It is straightforward that for any night-stage borrower (with cv > y), the day-stage PC implies the nightstage PC. For any night-stage lender (with cv y), the night-stage PC implies the day-stage PC. The PCs can be rewritten as v u(cv )

+ ( zv + W N ) zv + W N

v u(y) + W0N for v

W0N for v 2 f1; 2; :::; Bg; 2 fB + 1; B + 2; :::; N g

or v [u(cv )

zv

WN

zv

u(y)]

+ WN

v [u(cv )

(12)

W0N for v 2 fB + 1; B + 2; :::; N g:

The PCs and the day-stage resource constraint allocation can be achieved if and only if N X

W0N for v 2 f1; 2; :::; Bg;

u(y)]

PN

v=1 zv

+ WN

W0N

(13)

= 0 together determine that the …rst-best

+

B X

(W N

W0N )

0;

v=1

v=B+1

which can be rearranged as N 0

PN

v=B+1 v

PB

[u(cv )

v=1 v

B.4

Proof of Lemma 5

[u(y)

u(cv )] u(y)]

:

Proof. The proof is conducted in four steps. Step 1. The …rst step proves that for the night-stage ICs, it is su¢ cient to show that an agent with does not have the incentive to mimic an agent with v 1 or v+1 , or v [u(cv ) v [u(cv )

u(cv+1 )] u(cv 1 )]

(zv (zv

zv+1 ) for v = f1; 2; :::; N 1g; zv 1 ) for v = f2; 3; :::; N g:

v

(14) (15)

Using (14) for v + 1 gives v+1 [u(cv+1 )

Since

v

v+1 ,

u(cv+2 )]

(zv+1

zv+2 ):

(16)

it can be derived that v [u(cv+1 )

u(cv+2 )]

22

(zv+1

zv+2 ):

(17)

Adding (14) for v and (17) gives v [u(cv )

u(cv+2 )]

(zv

zv+2 );

which shows that agents with v do not want to mis-report to have agents with v do not want to claim to have v+3 ; v+4 , etc. Now using (15) for v 1 gives v 1 [u(cv 2 )

Since

v 1

v,

u(cv

1 )]

(zv

v+2 .

zv

2

Similarly, it can be shown that

1 ):

the following result can be derived v [u(cv 2 )

u(cv

1 )]

(zv

2

zv

1 ):

u(cv )]

(zv

2

zv );

(18)

Adding (15) for v and (18) gives v [u(cv 2 )

which shows that agents with v will not claim to have v 1 . Similarly, it can be shown that agents with v do not want to mimic those with v 3 , v 4 , etc. The argument above implies that it is su¢ cient to consider only 2(N 1) ICs: (14) and (15). Step 2: Notice that for v 2 f2; 3; :::; N g, (15) for agents with v and (10) for agents with v 1 imply (10) for agents with v , so that it su¢ ces to consider only one night-stage PC 1

z1 The ICs imply that if c1 required,

c2

u(y)] + W N

[u(c1 )

cN , then z1

z2

which also implies no night-stage PC is required. To sum up, it is su¢ cient to consider the 2(N PN PN constraints v=1 cv = N y and v=1 zv = 0. Step 3: Rewrite the 2(N 1) ICs as: v 1

[u(cv

1)

u(cv )]

zv

zN . If follows that only one day-stage PC is

WN

z1

zv

W0N :

W0N ;

(19)

1) ICs (14) and (15), one PC (19) and the resource

v 1

[u(cv

1)

u(cv )] for v 2 f2; 3; :::N g

where the …rst inequality represents (14) and the second inequality represents (15). Note that only (15) is in tension with the day-stage resource constraint, so that we need only to consider three sets of conditions: (19), (15), and the resource constraints. It then follows that the …rst-best can be achieved if and only if N 1 1 X (N + 1 N v=1

v) v [u(cv

1)

WN

u(cv )]

W0N ;

which can be used to derive

PN

u(cv ) : PN v) v u(cv 1 ) u(cv ) + v=1 v [u(cv ) u(y)] v=2 (N + 1 PN v) v u(cv 1 ) u(cv ) , C Step 4. The last step proves that N > N 0 : De…ne A v=2 (N + 1 PB PN 1 N N A u(y)], and D u(cv )]. Rewrite 1 and N 0 as 1 = A+C D and v=1 v [u(cv ) v=B+1 v [u(y) N D 0 = C. N 1

PN

v=2 (N

N 1

N 0

+1

=

v)

v

u(cv

1)

(A D)(C D) (A + C D)C

23

Since C

D > 0, A > 0, C > 0, it follows that

A

D

N X

=

(N + 1

v)

v

N 1

u(cv

>

1)

N 0

if and only if A

v=2

=

N X

u(cv ) +

v

D > 0:

[u(cv )

u(y)]

v=B+1

(N

1) 2 u(c1 ) +

B X

[ (N

v + 1)

v

+ (N

v)

v+1 ]u(cv )

v=2

N X

+

[ (N

v)

v + (N

v)

N X

v+1 ]u(cv )

v=B+1

(N

v u(y)

v=B+1

1) 2 u(c1 ) +

B X

[ (N

v + 1)

v

+ (N

v)

v+1 ]u(c2 )

v=2

+

N X

[ (N

v)

v + (N

v)

v+1 ]u(c2 )

v=B+1

=

(N

1) 2 [u(c1 )

N X

v=B+1

u(c2 )] + [u(c2 )

u(y)]

N X

v=B+1

> 0:

24

v

v u(y)

C

Decentralization with Competitive Markets

This Appendix illustrates how two monies improve allocations over one money in a competitive equilibrium when there is private information. The planner in the main text is replaced with a monetary authority that issues money and conducts monetary policy. The monetary authority cannot impose lump-sum taxes or entry fees. Neither can it impose non-linear prices, or exclude people from participation in the competitive markets. One implication of these assumptions is that money supply cannot shrink. The monetary authority conducts monetary policy by paying interest on money holdings. Assume that interest is paid at the beginning of each night stage. Interest payments are …nanced by issuing new money. Let i be the nominal interest rate. By …rst solving the competitive monetary equilibrium with one money and then with two monies, we show how two monies allow for more ‡exible monetary policy and generate better allocations.

C.1

Monetary Equilibrium with One Money

Consider …rst the day-stage problem. Let m denote the amount of money that an agent holds at the beginning of the day stage and let W (m) be the associated value function.16 During the day, agents choose production z and money holding to carry to the night stage m. ^ The problem can be speci…ed as W (m) = max z;m ^

1 z + [Vh (m) ^ + V` (m)] ^ 2

s.t. z = m ^

m;

or W (m) = max m ^

m

1 m ^ + [Vh (m) ^ + V` (m)] ^ ; 2

where is the value of money during the day, and Vh (m) ^ and V` (m) ^ are the night-stage value functions for a high-valuation agent and a low-valuation agent, respectively. The …rst-order condition (FOC) is 1 0 [V (m) ^ + V`0 (m)] ^ ; = if m ^ > 0: (20) 2 h Note that since the FOC does not involve m, the choice of m ^ is independent of m; i.e., all agents enter the 0 night stage with the same amount of money. The envelope condition gives W (m) = : At night, an agent becomes a high-valuation agent or a low-valuation agent with equal probabilities. If the agent becomes a high-valuation agent (type a agent at s = 1 or type b agent at s = 2), he is a buyer at the night stage. Let q be the amount of the night good that he purchases and d be the amount of monetary payment. A buyer solves the following problem: Vh (m) ^ = max f u(y + q) + W+ (m ^ + im ^ + q;m

s.t.

d = q and d

d)g

m ^ + im; ^

where is the value of money at night and im ^ is the amount of interest paid by the monetary authority. The subscript "+" is used to indicate variables in the following period. The associated Lagrangian is L = max f u (y + d) + W+ (m ^ + im ^ d

where

h

d)g +

^ h (m

+ im ^

d);

is the Lagrangian multiplier associated with the cash constraint. Let ch = y + q. The FOC is: u0 (ch ) = W+0 (m ^ + im ^

d) +

h

=

+

+

h:

The envelope condition implies that Vh0 (m) ^ = (1 + i)W+0 (m ^ + im ^

d) +

h (1

+ i) = (1 + i) u0 (ch ) :

(21)

1 6 In the main text, W is used to represent the social welfare. By slightly altering the notation, we use W (m) in this section of the appendix to denote the agent’s value function at the day stage.

25

If the agent becomes a low-valuation agent (type a at s = 2 or type b at s = 1), he is a seller at the night stage. The value function is V` (m) ^ = maxu(y

q) + W+ (m ^ + im ^ + d) s.t.

q;d

d = q;

or V` (m) ^ = maxu(y

d) + W+ (m ^ + im ^ + d):

d

The FOC is u0 (c` ) = W+0 (m ^ + im ^ + d) =

+:

(22)

The envelope condition implies that 0

V` (m) ^ = (1 + i)W+0 (m ^ + im ^ + d) = (1 + i)u0 (c` ) :

(23)

To derive the monetary equilibrium, (20), (21), and (23) can be combined to give

Using (22) to substitute for

1 [(1 + i) u0 (ch ) + (1 + i)u0 (c` ) ] = : 2 gives 1 u0 (ch ) (1 + i)[ 0 + 1] = 2 u (c` )

In a stationary equilibrium, =

+

: +

= 1 + i. It follows that u0 (ch ) 2 = 0 u (c` )

1:

The above equation together with the night-stage resource constraint ch + c` = 2y; characterizes the competitive monetary allocation (~ ch ; c~` ) when there is only one money. It is straightforward that c~h < ch and c~` > c` . Compared with the …rst-best allocation, high-valuation agents consume too little and low-valuation agents consume too much. With regard to the e¤ect of monetary policy, money is super-neutral: the nominal interest rate i (and, equivalently, the money growth or in‡ation rate) does not appear in the equations characterizing equilibrium allocations. The e¤ect of in‡ation is completely neutralized by interest payments on money holdings.

C.2

Monetary Equilibrium with Two Monies

This subsection shows how two monies can improve welfare over one money in competitive markets. Imagine that the monetary authority issues two types of monies, e.g. red money and green money. Agents are allowed to hold either type of money. With two monies, the monetary authority can make interest payments contingent on agents’monetary portfolios. In particular, the monetary authority pays interests only on red money when s = 1. Suppose that an agent holds m units of red money. The interest payment consists of i i 2 m units of red money and 2 m units of green money. There is no interest payment on green money. When s = 2, the monetary authority only pays interests on green money. For m units of green money, the interest payment consists of 2i m units of red money and 2i m units of green money. This interest payment scheme ensures that both monies grow at the same rate 1 + 2i . A type j 2 fa; bg agent with money holdings (mjr ; mjg ) at the beginning of the day stage solves the following problem

W (mjr ; mjg ) = max s.t. z +

j r mr

z+ +

1 Vh (m ^ jr ; m ^ jg ) + V` (m ^ jr ; m ^ jg ) 2 j g mg

26

=

^ jr rm

+

^ jg : gm

Use ( r ; g ) to denote the value of red money and green money at the day stage. Let (m ^ jr ; m ^ jg ) denote the agent’s choice of money holdings to carry to the night stage. From the unconstrained problem, the FOCs are ^ jr ; m ^ jg ) ^ jr ; m ^ jg ) 1 @V` (m 1 @Vh (m + 2 2 @m ^ jr @m ^ jr

r;

= if m ^ jr > 0;

(24)

^ jr ; m ^ jr ; m ^ jg ) 1 @V` (m ^ jg ) 1 @Vh (m + 2 2 @m ^ jg @m ^ jg

g;

= if m ^ jg > 0.

(25)

The envelope conditions imply that @W (mjr ; mjg ) mjr @W (mjr ; mjg ) mjg

=

r;

=

g:

Consider a type a agent where s = 1 is realized at the night stage. The agent becomes a high-valuation agent and thus a buyer. Let q denote the amount of the night good that the agent purchases and (dr ; dg ) be the monetary payment made by the agent. The values of the monies at night are represented by ( 1r ; 1g ). Here, the superscript 1 stands for the aggregate state s and the subscript indicates the type of money. The value function is i a i a u(y + q) + W+ (m ^ ar + m ^ ^ dr ; m ^ ag + m 2 r 2 r i a i a 1 1 ^ and dg m ^ ag + m ^ : m ^ ar + m r dr + g dg = q; dr 2 r 2 r

^ ag ) = max Vh (m ^ ar ; m

q;dr ;dg

s.t.

dg )

^ ar units of red money and 2i m ^ ar Note that interest is only paid on red money, and the interest consists of 2i m units of green money. Substitute q from the constraint and de…ne the Lagrangian as

L =

max

dr ;dg

+

u(y +

a ^r r (m

The Lagrangian multipliers ( respectively. The FOCs are

1 r dr

i + m ^r 2

a r;

a g)

i + W+ (m ^r + m ^r 2 i dr ) + ag (m ^g + m ^ r dg ): 2 1 g dg )

+

0

u (ch ) 1 r dr

+

1 g dg .

dg )

are associated with the cash constraints of red money and green money,

u0 (ch )

where ch = y +

i dr ; m ^g + m ^r 2

1 r 1 g

r+ g+

a r a g

= 0;

(26)

= 0;

(27)

Using (26) and (27) gives the envelope conditions

@Vh (m ^ ar ; m ^ ag ) @m ^ ar

= =

i + )+ 2 i u0 (ch )[(1 + ) 2 r+ (1

27

i i + ar (1 + ) + 2 2 i 1 1 ]; r + 2 g g+

a g

i 2 (28)

@Vh (m ^ ar ; m ^ ag ) = @m ^ ag

+

g+

= u0 (ch )

g

1 g:

(29)

When s = 2 is realized at the night stage, the type a agent becomes a low-valuation agent and thus a seller. The notations are de…ned similarly as before. The value function is V` (m ^ ar ; m ^ ag ) = max u(y dr ;dg

2 r dr

2 g dg )

i a i a + W+ (m ^ ar + m ^ g + dr ; m ^ ag + m ^ + dg ) : 2 2 g

The FOCs are u0 (c` ) 0

u (c` ) where c` = y

2 r dr

2 g dg .

2 r 2 g

r+

=

0;

(30)

g+

=

0;

(31)

The envelope conditions are @V` (m ^ ar ; m ^ ag ) @m ^ ar @V` (m ^ ar ; m ^ ag ) @m ^ ag

=

r+

=

r+

= u0 (c` ) i + 2 i 2

= u0 (c` )

2 r;

g+ (1

(32) i + ) 2

i + (1 + ) 2

2 r

2 g

:

(33)

By symmetry, the FOCs and the envelope conditions for a type b agent when s = 1 is realized at night are as follows u0 (c` ) u0 (c` )

1 r 1 g

r+

= 0;

(34)

g+

=

(35)

0;

@V` (m ^ br ; m ^ bg ) @m ^ br

i = u0 (c` ) (1 + ) 2

@V` (m ^ br ; m ^ bg ) @m ^ bg

=

g+

= u0 (c` )

1 r

+

i 2

1 g

;

1 g:

(36) (37)

The FOCs and the envelope conditions for a type b agent when s = 2 is realized are: u0 (ch ) 0

u (ch )

@Vh (m ^ br ; m ^ bg ) @m ^ br @Vh (m ^ ar ; m ^ ag ) @m ^ ag

2 r 2 g

b r b g

g+

=

g+

=

u0 (ch )[

r+

Combining (24) with (28) and (32) gives

28

b g

+ i 2

= 0;

(38)

=

(39)

0;

= u0 (ch ) 2 r

2 g;

i + (1 + ) 2

2 g ]:

(40) (41)

1 0 i u (ch )[(1 + ) 2 2 1 r;

Using (34), (35), and (30) to substitute for

i 2

1 r

+

1 g

and

1 u0 (ch ) i i [(1 + ) + 0 2 u (c` ) 2 2 In this economy,

g+ r+

1 g]

1 + u0 (c` ) 2

2 r,

we have

g+

]+

r+

2 r

1 2

r:

r

:

(42)

r+

represents the exchange rate of the two monies at the day stage. The FOCs also give g+

1 g 1 r

=

r+

2 g 2; r

=

which means that the exchange rate at the night stage in either state should be equal to the exchange rate at the day stage. The exact exchange rate is indeterminate. Without loss of generality, we set the exchange rate at 1 so that the two monies are perfect substitutes. Note that because of the way the monetary authority pays interest, both monies grow at the same rate, and we have r = g = 1+ 2i in a stationary equilibrium. r+ g+ After some algebra, (42) becomes u0 (ch ) u0 (c` )

(2 + i)= 1+i

1

Ar .

(43)

Follow the same steps to combine (25) with (29) and (33) to derive u0 (ch ) u0 (c` )

2+i

(1 + i)

Ag :

(44)

It can be shown that Ar < Ag (recall that < 1). A direct implication is that only one of (43) and (44) can hold at equality. If (43) holds at equality, it implies that (44) is slack, and type a agents do not want to hold any green money when they make portfolio choices at the day stage. If (44) holds at equality, it implies that type a agents would like to hold an in…nite amount of red money when they make portfolio choices at the day stage, which cannot be an equilibrium. Therefore, the only equilibrium is where type a agents hold only red money. The allocation (ch ; c` ) is solved from u0 (ch ) u0 (c` ) ch + c`

(2 + i)= 1+i 2y:

= =

1

;

(45) (46)

By symmetry, type b agents take only green money into the night stage and have the same allocation (ch ; c` ). Now compare the equilibrium allocations with one money and two monies. First, note that when i = 0, both one money and two monies generate the same allocation. However, when i > 0, it can be shown that 1<

(2 + i)= 1+i

1

<

2

;

which implies that c` < c` < c~` < y < c~h < ch < ch . Given that high-valuation agents consume too little and low-valuation agents consume too much at night with one money, two monies can improve welfare for all i > 0. Making interest payments conditional on the type of money held by agents introduces a channel for agents to signal their types (and, equivalently, valuation at the night stage). As type a agents have high valuation at s = 1, when they choose money holdings to take into the night stage, they have the incentive to choose the type of money that can generate a higher return at state 1, which is the red money. Green money generates a higher return at state 2 and, hence, is held by type b agents at the beginning of the night stage. By choosing di¤erent monetary portfolios, agents signal their types (and thus the valuation of night goods); this allows the monetary authority to engage in policies to redistribute resources from sellers to buyers, which improves welfare. Such policies cannot be implemented when there is only one money because

29

all agents hold the same amount of money and thus the same portfolio. In competitive markets, two monies improve welfare over one money by allowing agents to signal their types (or initiating the ex ante sorting), which is consistent with the result from a mechanism design framework.

30