One or Two Monies?
Mei Dong
yJanet Hua Jiang
zxMarch 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.
yCurrency Department, the Bank of Canada. zDepartment of Economics, University of Manitoba.
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 (1998a; 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 record-keeping 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 pun-ishment. 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,
1Since Mundell (1961), there has been a large literature on optimal currency area. The literature abstracts from the
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. Section2 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. Section5extends the model by consideringN >2types of agents and argues in general that two monies are a perfect substitute for the record-keeping technology. Conclusions are provided in Section6.
2
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 0to 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 byaandb; each is of measure1.
[Place Figure 1 about here]
u(0) = 0; u00<0< u0 andu0(0) = +1:Whens= 2, typeahave utilityu(c)and typebhave utility u(c).
Note that typeahave high marginal utility in state1and low marginal utility in state2(vice versa for type
b). The shock to the state variable is iidacross 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. Letchandzhrepresent 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
night-stage consumption and the next-day-night-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 (c
h; c`; zh; z`) is given by
W = 1211 [ u(ch) zh+u(c`) z`]. The …rst-best allocation(ch; c`; zh; z`)maximizesW subject to the
resource constraintsch+c` 2yandzh+z` 0, or maximizes3
W(ch; c`) =
1 2
1
1 [ 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 andzh+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 isW =1211 [ 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.
3
Limited Commitment
As in Kocherlakota (2002), we …rst assume that agents lack commitment and that agents’ types are pub-lic 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 non-participation.
2Assumez= 0for each agent at the day stage of period0.
3Sinceu0(c)>0and agents su¤er disutility from day production, it is obvious that it is not optimal to have slack resource
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(y) + W0, for high-valuation agents,
u(c`) + ( z`+W) u(y) + W0, for low-valuation agents,
whereW is de…ned as in (1) andW0= 1211 [ 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 ifch> 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 W W0; (4)
z` W W0
u(y) 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
0
u(y) u(c`)
[u(ch) u(y)]:
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
4If >
0, the combinations of(zh; z`)that are consistent with the …rst-best allocation are not unique. Any(zh; z`)that
satis…eszh W W0andz` W W0 u(y) u(c`) andzh+z`= 0can achieve the …rst-best allocation. If = 0, the
combination of(zh; z`)is unique withzh=W W0,z`=W W0 u(y) u(c`). 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(c` ) [u(ch) u(y)] = ;
c` +ch = 2y;
zh =W(ch; c` ) W0;
z` =W(ch; c` ) W0
u(y) u(c` )
:
The second-best allocation has the feature thatch> ch > y > c` > c`.
5The planner, however, has access to a contemporaneous memory technology that can remember agents’ actions within a
(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 monoton-ically 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 typebtransfer 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 2is di¤erent from Kocherlakota (2002), which shows that one money is not a perfect substi-tute 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 typeb to transfer goods to typeaat 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 period1 night stage with the same money balance so that they are treated equally (whether or not they made transfers in period0night stage) as required by the …rst-best allocation. Without injections of new money, every participant will enter the new night stage with1unit 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 $1. By requiring an entry fee higher than1$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) u(c`)]= , for night-stage low-valuation agents,
zh W W0, for night-stage high-valuation agents. The …rst-best allocation can be achieved if and only if u(y) u(c`)
+W W0 0, or
f 2[u(y) u(c`)]
u(ch) u`+ (1 )u(y)
> 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
6With 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 Figure2;we use integer increments as an example, but the proposed mechanism carries through with noninteger increments as well.
7Previous high-valuation agents enter the day stage with less than $1. By imposing$1entry fee for all subsequent night
stages, the planner can exclude from the mechanism forever those who consumechat night but refuse to participate in the
following day stage. The PC for high-valuation agents thus remains the same as before.
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. Whens= 1, the planner asks those who reported to be typebagents to transfer goods to those who reported to be typea
agents. Whens= 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) zh
2 +
u(c`) z`
2
1 1
u(c`) z`
2 +
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 periods0and 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 asch> c`, which is satis…ed at the …rst-best allocation.10 Using ex ante sorting does not impose extra
constraints onzh andz`;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.
9Here 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
2 +
u(c`) z`
2
u(c`) z`
2 +
u(ch) zh
2 ; which is essentially the same as (6).
u(ch) + ( zh+W) u(c`) + ( z`+W);
u(c`) + ( z`+W) u(ch) + ( zh+W);
which can be rearranged as
zh z`
[u(ch) u(c`)]
; (7)
zh z`
u(ch) u(c`)
: (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 typeb 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 onzh andz`.
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 mechanisms can achieve the …rst-best allocation if and only if 1 u(ch) u(c`)
( +1)[u(ch) u(y)] > 0.
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 period0, 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 state1 and low valuation in state2. 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 Section3, 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 typea; yet those who can show low money balances could either be reported typebor reported typeabecause the latter can hide part of their money balances.
Therefore, an allocation can be implemented only if it suggests goods transfers from reported type b to reported typea. It then follows that all agents will report to be type ato acquire the high money balance at the beginning of period0, 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 periodt, the planner can infer that those with a low money balance are reported typea;while those with a high money balance are reported typeb.
[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 typeaone unit of red money and those reported typebone 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 ifs= 1and vice versa ifs= 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 Figure4for 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` )featuresch> 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 asch> c`. The planner can use(zh; z`)to deal only with limited
5
Extensions
This section extends the previous results to an environment withN >2types of agents and argues in general that two monies are a perfect substitute for the record-keeping technology.
5.1
A Model with
N >
2
Types of Agents
Suppose that there areN >2types of agents (each type is of measure1) andN states of the economy indexed
byf1;2; :::; Ng. 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 iagents in state j are described by iju(cij), where ij takes N
possible values 1> 2> ::: > N >0with equal probabilities. For the value of ij, it helps to visualize that
theN states lie along a circle in a clockwise sequence of state1,2, ..., andN. Typeiagents have the highest valuation 1 at s=i, and as they travel clockwise along the circle of the states, ij decreases and reaches
the lowest value N ats=i 1. Mathematically, ij= v(i;j)wherev(i; j) =
(
j i+ 1;
N+j i+ 1;
ifj i
ifj < i is
the valuation-indicator function andv2 f1;2; :::; Ng. 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
vu0(cv) = v0u0(cv0)for allv; v02 f1;2; :::; Ng and
N
X
v=1
cv=N y:
Suppose that c1 > c2 > ::: > cB > y > cB+1 > ::: > cN so that B types are night-stage borrowers who
consume more than their endowments, andN Btypes are lenders who consume less than their endowments. Any day-stage allocation(z1; z2; :::; zN)that satis…es the resource constraintPNv=1zv = 0is consistent with
the …rst-best allocation. The ex ante …rst-best lifetime utility for any type isWN = N1 11 hPNv=1 vu(cv)
i
:
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 eachv2 f1;2; :::; Ng),
vu(cv) + ( zv+WN) vu(y) + W0N;
zv+WN W0N;
whereWN = 1
N
1 1
hPN
v=1 vu(cv)
i
andWN
0 =N1
1 1 u(y)
PN v=1 v:
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 followingN2 N constraints ensure that each typeihas the incentive to truthfully report his type at the day stage of period0,
1
N
1 1
2 4
N
X
j=1
iju(cij)
3 5 1
N
1 1
2 4
N
X
j=1
iju(ci0j)
3
5 for alli6=i02 f1;2; :::; Ng.
Lemmas3and4state 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 WithN >2 types of agents, the incentive constraints imposed by ex ante sorting are satis…ed at the …rst-best allocation.13
1 2As in the case with two types, it is assumed that each agent hasz= 0at the day stage of period0.
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
N
0
PN
v=B+1 v[u(y) u(cv)]
PB
v=1 v[u(cv) u(y)]
:
To use ex post sorting, the followingN2 N constraints must be satis…ed to ensure that agents with
v
do not want to report to have v0 withv0 6=v
vu(cv) + ( zv+WN) vu(cv0) + ( zv0+WN)for allv=6 v02 f1;2; :::; Ng
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. Lemma5 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
v=2(N+ 1 v) v u(cv 1) u(cv)
PN
v=2(N+ 1 v) v u(cv 1) u(cv) +
PN
v=1 v[u(cv) u(y)]
> N0:
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 into perpetual autarky. The …rst-best allocation can be achieved if and only if N0. 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 record-keeping 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 only if N1 > N0 . When < N1, 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 WithN >2types 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 mechanisms can achieve the …rst-best allocation if and only if N0 .
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 areN 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 signalN 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
1 4A related result in Townsend (1987) shows that two types of tokens are enough to distinguish past histories in an environment
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 5Imagine that di¤erent pieces of information are recorded as di¤erent points along a straight liner+g=min(r; g)space
s=1
Type
a: u(c)
Type
b: u(c)
>1
Type
b
u(c
1)
1/2
Night
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.
Type
a
and
b: -z
Day
Figure
1
: Environment
1/2
s=2
Type
a: u(c)
Type
b: u(c)
>1
Notes:
The figure shows the mechanism when
s
t-1=1
; exchange
a
and
b
when
s
t-1=2
.
Pay
$ t
entry fee
If type
a
,
receive * goods and
$ (t+ )
If type
b
,
submit * goods and
receive
$ (t+ )
money
with
0< <1
Pay
$( t+ )
entry fee
If type
a
,
submit
z
goods and
receive
$ (t+1 )
If type
b
,
receive
z
goods and
$ (t+1)
Figure
2
: One-money Mechanism with Limited Commitment
Pay
$ t
entry fee
If report to be type
a
,
receive * goods
and
$ (t+
h)
If report to be type
b
,
submit * goods
and receive
$ (t+
l)
with
0<
h<
l<1
Pay
$ ( t+
h)
entry fee
Receive
z
goods
and
$ (t+1)
Figure
3
: One-money Mechanism with Limited Commitment
and Private Information
Submit
z
goods
and receive
$ (t+1)
Period
t-1
night
Period
t
day
Pay
$ ( t+
l)
entry fee
Notes:
P
ay
R$
t
entr
y
fe
e
R
ec
eiv
e
*
go
ods
a
nd
R$
(t
+
)
(0<
<1
)
P
ay
R$
(
t+
)
en
tr
y
fe
e
R
ece
ive
z
go
od
s
and
G$
(t+
1)
Fi
gu
re
4
:
T
wo
-m
o
ney
M
ec
ha
ni
sm
w
ith L
im
it
ed
C
o
mmi
tmen
t
an
d
P
riv
at
e
In
fo
rm
at
ion
Su
bm
it
*
goo
ds a
nd
re
ce
ive
G$
(t
+
)
P
ay
G
$
t
en
try fe
e
N
ote
s:
Th
e
fig
ure
sh
ow
s
th
e
m
ec
ha
ni
sm
w
h
en
s
t-1=1
;
ex
ch
an
ge
a
a
nd
b
wh
en
s
t-1References
[1] Camera, G., Craig, B., Waller, C. J., 2004. Currency competition in a fundamental model of money. Journal of International Economics64(2),521-544.
[2] Camera, G., Winkler, J., 2003. International monetary trade and the law of one price. Journal of Monetary Economics50(7),1531-1553.
[3] Craig, B., Waller, C. J.,2004. Dollarization and currency exchange. Journal of Monetary Economics51 (4),671-689.
[4] Kocherlakota, N., 1998a. The technological role of …at money. Federal Reserve Bank of Minneapolis Quarterly Review22(3),2–10.
[5] Kocherlakota, N., 1998b. Money is memory. Journal of Economic Theory81(2),232–251.
[6] Kocherlakota, N., Krueger, T., 1999. A signaling model of multiple currencies. Review of Economic Dynamics2(1), 231-244.
[7] Kocherlakota, N., 2002. The two-money theorem. International Economic Review43(2),333-346.
[8] Lagos, R., Wright, R., 2005. A uni…ed framework for monetary theory and policy analysis. Journal of Political Economy113 (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 Re-view77(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), Article3.
A
Detailed Descriptions of Mechanisms
A.1
One-money Mechanism with Limited Commitment
At either stage in each period, an agent chooses action fromf0;1g, where0 means that the agent does not participate and1means 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 assigns0 production/consumption to each agent. At the night stage of periodt 1 fort 1 and0< <1,
if a typeaagent chooses1(by payingt$), the agent is entitled to a transfer of goods (which implies consumptionch) and a monetary receiptt+ $;
if an agent chooses0(because he cannot or chooses not to pay the entry fee), the agent gets0transfer and0 monetary receipt.
At the day stage of period tfort 1,
if a type aagent chooses1(by payingt+ $), the agent is entitled to a transfer of goods z(which implies production ofz) and a monetary receiptt+ 1$;
if a type b agent chooses 1 (by paying t+ $), the agent is entitled to a transfer of goods z (which implies consumption ofz) and a monetary receiptt+ 1$;
if an agent chooses0(because he cannot or chooses not to pay the entry fee), the agent gets0transfer and0 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
Informa-tion
At either stage in each period, an agent chooses action from f0;1g, where 0 means that the agent does not participate and1 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 period0, the mechanism endows each agent with1unit of money and assigns0 production/consumption to each agent. For example, at the night stage of period t 1fort 1;
0< h< `<1:
if an agent chooses 1 (by payingt $) and reports to be type a, the agent is entitled to a transfer of goods (which implies consumptionch) and a monetary receipt t+ h $;
if an agent chooses 1 (by paying t $) and reports to be type b, the agent is entitled to a transfer of goods (which implies consumption c`) and a monetary receiptt+ `$;
if an agent chooses0 (because he cannot or chooses not to pay the entry fee) and reports to be either type, the agent gets0transfer and 0monetary receipt.
At the day stage of period tfort 1:
if an agent chooses 1and payst+ h$, the agent is entitled to a transfer of goods z (which implies
production ofz) and a monetary receiptt+ 1$;
if an agent chooses 1 and pays t+ ` $, the agent is entitled to a transfer of goods z (which implies consumption ofz) and a monetary receiptt+ 1$.
if an agent chooses 0 (because he cannot or chooses not to pay either entry fee), the agent gets 0 transfer and 0monetary 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 typea
A.3
Two-money Mechanism with Limited Commitment and Private
Informa-tion
At either stage in each period, an agent chooses action fromf0;1g, where0 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 time0, 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 periodt 1 fort 1,0< <1:
if an agent chooses 1 and pays t R$, the agent is entitled to a transfer of goods (which implies consumptionch) and a monetary receiptt+ R$;
if an agent chooses 1 and payst G$, the agent is entitled to a transfer of goods (which implies consumptionc`) and a monetary receiptt+ G$;
if an agent chooses 0 (because he cannot or chooses not to pay either entry fee), the agent gets 0 transfer and 0monetary receipt.
At the day stage of period tfort 1:
if an agent chooses1and payst+ R$, the agent is entitled to a transfer of goods z (which implies production ofz) and a monetary receiptt+ 1R$;
if an agent chooses 1 and payst+ G$, the agent is entitled to a transfer of goodsz (which implies consumption ofz) and a monetary receiptt+ 1G$.
if an agent chooses 0 (because he cannot or chooses not to pay either entry fee), the agent gets 0 transfer and 0monetary receipt.
At the day stage of period 0:
if an agent chooses1and reports to be typea, the agent is entitled to0transfer and a monetary receipt 1 R$;
if an agent chooses1and reports to be typeb, the agent is entitled to0transfer and a monetary receipt 1 G$;
if an agent chooses0, the agent gets0transfer and 0monetary receipt.
The mechanism is a direct mechanism and makes allocation recommendations based on agents’reports on their types at the beginning of period0. 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 period0, choose1and pay the entry fee with R$ in all future stages; and typebagents choose to hold 1G$ at the day stage of period 0, choose1 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 date0, the planner asks agents to report their types, and gives those who report to be typeia monetary portfolio
riR$ + (1 ri)G$
with 0 < ri < 1 for each i 2 f1;2; :::; Ng and 1 r1 > r2 > ::: > rN 0. The implementation of a
F
ig
ure
5
: Two
-mon
ey
M
e
ch
a
ni
sm
wi
th Li
m
it
e
d
C
om
mi
tme
nt
an
d
P
ri
v
at
e
in
fo
rm
at
ion
(m
ul
tip
le
typ
es
)
P
er
iod
0
day
. .
.
R
ec
ei
ve
[c
v( i,j)–
y]
g
o
od
s,
R$
(t
) r
ia
nd
G$
(t
+
) (1
-r
i)
(0<
<
1)
P
ay
R$
tr
iand
G$
t(1
-r
i)
en
try fee
S
u
b
m
it
z
v(i,j)
goo
ds
;
rec
ei
ve
R$
(1+
t)r
ia
nd
G$
(1
+t
)t
(1
-r
i)
P
ay
R$
(t
+
)
r
ia
nd
G$
(t
+
) (1
-r
i)
en
try f
ee
Pe
ri
od
t-1
ni
gh
t
(s
t-1=j
)
Pe
ri
od
t
d
ay
If r
ep
ort t
o
b
e
ty
pe
i
,
R
ec
ei
ve
R$
r
ia
nd
G
$
(1
-r
i)
N
ote
s:
1.
[c
v( i,j)–
y]
m
ay
be
ne
gat
ive
; i
n
tha
t ca
se
,
the
age
nt s
ub
m
it
s
[y
-c
v( i,j)]
uni
ts
o
f
ni
gh
t
go
od
s.
2.
z
v(iB
Proofs
B.1
Proof of Lemma
2
Proof. Whenex postsorting 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 W0
[u(ch) u(c`)] 0;
or
1
u(ch) u(c`)
( + 1)[u(ch) u(y)]: The result that 1> 0 follows from u(ch) +u(c`)>(1 + )u(y):
B.2
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 type1 agents do not want to mis-report to be type2;3; :::; N. Other types face the same incentive constraints.
The …rst step is to show that type 1do not mis-report to be type 2;or
N
X
v=1
vu(cv) 2u(c1) + 3u(c2) + + Nu(cN 1) + 1u(cN): (9)
LHS RHS
= ( 1 2)u(c1) + ( 2 3)u(c2) + + ( N 1 N)u(cN 1) + ( N 1)u(cN)
[( 1 2) + ( 2 3) + + ( N 1 N)]u(cN) ( N 1)u(cN)
= 0.
The last inequality holds if 1> 2> > N andc1 c2 cN. It follows that (9) is satis…ed at the
…rst-best allocation.
The second step is to prove that type 1do not mis-report to be type3, or
N
X
v=1
vu(cv) 3u(c1) + 4u(c2) + + Nu(cN 2) + 1u(cN 1) + 2u(cN 2):
LHS RHS
= ( 1 3)u(c1) + ( 2 4)u(c2) + ( 3 5)u(c3) + ( 4 6)u(c4) +::: +( N 4 N 2)u(cN 4) + ( N 3 N 1)u(cN 3) + ( N 2 N)u(cN 2)
( N 1 1)u(cN 1) ( N 2)u(cN 2) = 1u(c1) + 2u(c2) N 1u(cN 3) Nu(cN 2)
( 1 N 1)u(cN 1) ( 2 N)u(cN 2) ( 1 N 1)u(cN 3) + ( 2 N)u(cN 2)
where the inequalities follow from 1> 2> > N andc1 c2 cN. Type1 thus do not want to
imitate type3 at the …rst-best allocation.
Following similar arguments, it can be shown that all of the rest N 3 constraints are also satis…ed if 1> 2> > N andc1 c2 cN.
B.3
Proof of Lemma
4
Proof. Based on Lemma3, we need only to consider the participation constraints that forv2 f1;2; :::; Ng;
vu(cv) + ( zv+WN) vu(y) + W0N; (10)
zv+WN W0N: (11)
It is straightforward that for any night-stage borrower (withcv> y), the day-stage PC implies the
night-stage PC. For any night-night-stage lender (with cv y), the night-stage PC implies the day-stage PC. The PCs
can be rewritten as
vu(cv) + ( zv+WN) vu(y) + W0N forv2 f1;2; :::; Bg;
zv+WN W0N forv2 fB+ 1; B+ 2; :::; Ng
or
zv v
[u(cv) u(y)]
+WN W0N forv2 f1;2; :::; Bg; (12)
zv WN W0N forv2 fB+ 1; B+ 2; :::; Ng: (13)
The PCs and the day-stage resource constraint PNv=1zv = 0 together determine that the …rst-best
allocation can be achieved if and only if
N
X
v=B+1
v[u(cv) u(y)]+WN WN
0 +
B
X
v=1
(WN W0N) 0;
which can be rearranged as
N
0
PN
v=B+1 v[u(y) u(cv)]
PB
v=1 v[u(cv) u(y)]
:
B.4
Proof of Lemma
5
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 v
does not have the incentive to mimic an agent with v 1 or v+1, or
v[u(cv) u(cv+1)] (zv zv+1)forv=f1;2; :::; N 1g; (14)
v[u(cv) u(cv 1)] (zv zv 1)forv=f2;3; :::; Ng: (15)
Using (14) for v+ 1gives
v+1[u(cv+1) u(cv+2)] (zv+1 zv+2): (16) Since v v+1, it can be derived that
Adding (14) forv 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 v+2. Similarly, it can be shown that agents with v do not want to claim to have v+3; v+4, etc.
Now using (15) forv 1gives
v 1[u(cv 2) u(cv 1)] (zv 2 zv 1): Since v 1 v, the following result can be derived
v[u(cv 2) u(cv 1)] (zv 2 zv 1): (18) Adding (15) forv and (18) gives
v[u(cv 2) u(cv)] (zv 2 zv);
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 only2(N 1)ICs: (14) and (15).
Step2:Notice that forv2 f2;3; :::; Ng, (15) for agents with vand (10) for agents with v 1imply (10) for agents with v, so that it su¢ ces to consider only one night-stage PC
z1 1[u(c1) u(y)] +WN W0N:
The ICs imply that ifc1 c2 cN, thenz1 z2 zN. If follows that only one day-stage PC is
required,
z1 WN W0N; (19)
which also implies no night-stage PC is required.
To sum up, it is su¢ cient to consider the 2(N 1) ICs (14) and (15), one PC (19) and the resource constraintsPNv=1cv=N y andPNv=1zv= 0.
Step 3:Rewrite the2(N 1)ICs as:
v 1
[u(cv 1) u(cv)] zv zv 1 v[u(cv 1) u(cv)]forv2 f2;3; :::Ng
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
1
N
NX1
v=1
(N+ 1 v) v[u(cv 1) u(cv)] WN W0N;
which can be used to derive
N
1
PN
v=2(N+ 1 v) v u(cv 1) u(cv)
PN
v=2(N+ 1 v) v u(cv 1) u(cv) +
PN
v=1 v[u(cv) u(y)]
:
Step 4. The last step proves that N1 > N0 : De…ne A PNv=2(N + 1 v) v u(cv 1) u(cv) , C
PB
v=1 v[u(cv) u(y)], and D
PN
v=B+1 v[u(y) u(cv)]. Rewrite N1 and N0 as N1 = A+CA D and N
0 =DC.
N
1
N
0 =
SinceC D >0,A >0,C >0, it follows that 1N > N0 if and only ifA D >0:
A D =
N
X
v=2
(N+ 1 v) v u(cv 1) u(cv) + N
X
v=B+1
v[u(cv) u(y)]
= (N 1) 2u(c1) +
B
X
v=2
[ (N v+ 1) v+ (N v) v+1]u(cv)
+
N
X
v=B+1
[ (N v) v+ (N v) v+1]u(cv) N
X
v=B+1
vu(y)
(N 1) 2u(c1) +
B
X
v=2
[ (N v+ 1) v+ (N v) v+1]u(c2)
+
N
X
v=B+1
[ (N v) v+ (N v) v+1]u(c2)
N
X
v=B+1
vu(y)
= (N 1) 2[u(c1) u(c2)] + [u(c2) u(y)]
N
X
v=B+1
v
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. Letibe 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. Letmdenote the amount of money that an agent holds at the beginning of the day stage and letW(m)be the associated value function.16 During the day, agents choose production z and money holding to carry to the night stagem^. The problem can be speci…ed as
W(m) = max
z;m^ z+ 1
2[Vh( ^m) +V`( ^m)] s.t. z= m^ m; or
W(m) = max ^
m m m^ +
1
2[Vh( ^m) +V`( ^m)] ;
where is the value of money during the day, andVh( ^m)andV`( ^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 2[V
0
h( ^m) +V`0( ^m)] ; = ifm >^ 0: (20)
Note that since the FOC does not involvem, the choice ofm^ is independent ofm; i.e., all agents enter the night stage with the same amount of money. The envelope condition givesW0(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 (typeaagent ats= 1 or typebagent ats= 2), he is a buyer at the night stage. Letqbe the amount of the night good that he purchases anddbe the amount of monetary payment. A buyer solves the following problem:
Vh( ^m) = max
q;m+f u(y+q) + W+( ^m+im^ d)g
s.t. d=qandd 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
d f u(y+ d) + W+( ^m+im^ d)g+ h( ^m+im^ d);
where h is the Lagrangian multiplier associated with the cash constraint. Letch=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 6In the main text,W is used to represent the social welfare. By slightly altering the notation, we useW(m)in this section
