A Model in Which Outside and Inside Money are Essential

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. A Model in which Outside and Inside Money are Essential David C. Mills, Jr. 2006-38 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

A MODEL IN WHICH OUTSIDE AND INSIDE MONEY ARE ESSENTIAL David C. Mills, Jr. Federal Reserve Board of Governors The author thanks Ricardo Cavalcanti, James Jordan, Robert King, Tomas Sjostrom, and seminar participants at Michigan State University, the Federal Reserve Board, the Federal Reserve Banks of Cleveland and Philadelphia, the U.S. Naval Academy, Penn State University, and the 2001 Midwest Macroeconomics Conference in Atlanta, GA for helpful comments. I especially thank Neil Wallace for guidance and support. Some of this paper was completed as part of my dissertation at the Pennsylvania State University. All errors are my own. The views in this paper are solely the responsibility of the author and should not be interpreted as re(cid:13)ecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. Address correspondence to: David C. Mills, Jr., Federal Reserve Board, Mail Stop 188, 20th & C Streets, NW, Washington, DC 20551; e-mail: david.c.mills@frb.gov; Fax: 202-872-7533. 1

Abstract Ipresentanenvironmentforwhichbothoutsideandinsidemoneyare essential as means of payment. The key model feature is that there is imperfect monitoring of issuers of inside money. I use a random matching model of money where some agents have private trading histories and others have trading histories that can be publicly observed only after a lag. I show via an example that for lags that are neither too long nor too short, there exist allocations that use both types of money that cannot be duplicated when only one type is used. Inside money provides liquidity that increases the frequency of trades, but incentive constraints restrict the amount of output that can be traded. Outside money is immune to such constraints and can trade for higher levels of output. Keywords: Inside and outside money 2

1. INTRODUCTION Thefactthatcreditrelationshipsamongindividualsareanalternative to money as a means of exchange provides a challenge to any model of money. Ifcreditrelationshipsarerichenough, moneyisnotnecessary to achieve good allocations. Thus, a model of money must contain su(cid:14)cient frictions to establishing rich enough credit relationships for moneytobenecessaryasameansofexchange. Necessaryfrictions,as demonstrated by Ostroy (1973), Townsend (1989), and Kocherlakota (1998), include a lack of commitment to future actions by individuals and some limitations on the ability to monitor individuals’ trading histories. Kocherlakota and Wallace (1998), hereafter KW, formalize these ideas in a random-matching model of money. They model limitations on the ability to monitor trading histories by introducing a random lag into the updating of those histories. They then study how the length of the lag a(cid:11)ects optimal allocations and demonstrate that, in general, optimal allocations involve the use of (outside) money and some form of credit when the expected lag is su(cid:14)ciently long enough. Inside or private money is a particularly interesting type of credit instrument to study because both economists and policy makers have for a long time been interested in whether and how inside money shouldberegulated. Thisgoesbackatleasttothenineteenth-century debatesbetweentheso-calledcurrencyschool,whichadvocatedapublicmonopolyonmoneyissuewithstrictcontrols, andthefree-banking 3

school, which promoted a relatively laissez-faire approach to private note issue. The fundamental concern about inside money is the incentive to overissue. This incentive arises from the fact that inside money is a credit instrument that the issuer promises to redeem in the future in exchange for something that is costly to give up. The incentive to overissue is one reason that Friedman (1959) favored 100% reserve requirements, essentially advocating the elimination of inside money. The recent removal of legal impediments to inside money issue and the advent of technologies that make electronic money feasible suggest that the question of whether and how to regulate inside money remains an important issue today. Thus, to get at the questions concerning the regulation of inside money, one needs a model of money that provides a framework to compare alternative monetary regimes, and one that can explicitly model incentives to overissue. One such model is contained in Cavalcanti and Wallace (1999a,1999b), hereafter CW(a,b). To generate a role for both money and credit, they model limitations on the ability to monitor trading histories in the following way: some people (bankers) can be perfectly monitored via a recordkeeping technology whereas others (nonbankers) cannot be monitored at all because their trading histories are private information. CW(a) show that, because their trading histories are perfectly known, it is incentive feasible for bankers to issue and redeem their own notes as inside money (i.e., bankers can engage in credit relationships), which can then circulate 4

among the nonbankers (who have no access to credit because of their anonymity) as a means of exchange. CW(b) compares the use of outside money with that of inside money and (cid:12)nd that inside money is superiorbecause,inadditiontoduplicatingallocationsthatareimplementable with outside money, it implements allocations that increase thefrequencyoftrades. Thisisbecauseinsidemoneyprovidesinstant liquidity to its issuers. Good allocations in CW(a,b) require issuers of inside money to redeem (each other’s) inside money. If the KW lag is introduced for them, then the longer that lag, the greater the temptation they face to defect from such redemption. If the lag is long enough then inside money cannot work. The lag, of course, does not limit the functioning of outside money. (The lag is in(cid:12)nite in models that have only outside money.) The new idea is to get a role for both inside and outside money. Inside money retains its role as providing liquidity to the issuer, but the lag limits how valuable it can be and, therefore, gives a roleforoutsidemoney. Thelimitedvalueofinsidemoneyissomething like an endogenous credit limit (see Kehoe and Levine 1993). Both KW and CW(a,b) assume that money is indivisible and that holdings are in the set 0;1 . I maintain that assumption, but, in f g order to allow a potential role for both inside and outside money, consider worlds with two distinct monies: either two inside monies, two outside monies, or one of each. I show numerically that there exist background models with a lag that is neither too long nor too 5

short and allocations that can only be implemented using both types of money. That seems to suggest that for some background models, good allocations require both inside and outside money, but so far I have not demonstrated this. The rest of the paper proceeds as follows. In Section 2., I describe the background matching model, which is essentially borrowed from earlier work. Then, as preliminary motivation, I introduce the updating lag for money issuers when there is one inside money in Section 3.. I describe a simple class of inside-money allocations and how the lag a(cid:11)ectswhatisimplementable. TheninSection4.,Iintroducethreealternative monetary arrangements, each with two distinct monies and each corresponding to one kind of monetary system: inside money only, outside money only, and both. I then present the numerical example that demonstrates that the use of both inside and outside money can be required for implementation. Section 5. concludes and the appendix contains a formal characterization of the alternative monetary arrangements. 2. THE MODEL 2.1. Background Environment and Sequence of Events The background environment is the familiar random-matching environment from Shi (1995) and Trejos and Wright (1995). Time is dis- 6

crete and the horizon is in(cid:12)nite. There are S distinct, divisible, and perishable types of goods at each date and there is a [0;1] continuum of each of S specialization types of agents, where S > 2 . An agent whosespecializationtypeissconsumesonlygoodsandproducesonly good s+1 (modulo S), for s = 1;2;:::;S. Each agent maximizes expected discounted utility with a discount factor (cid:12) (0;1). Utility in 2 a period is given by u(c) y where c is the amount consumed and y is (cid:0) the amount produced. The function u is de(cid:12)ned on [0; ), increasing, 1 and twice di(cid:11)erentiable and satis(cid:12)es u(0) = 0, u < 0, and u(0) = . 00 0 1 Moreover, there exists ymax > 0 such that u(ymax) = ymax: In each period, agents are randomly matched in pairs. As is familiar, the random matching along with agent specialization in both consumption and production means that there are no meetings in which there is a double coincidence of wants.1 Rather, meetings are either single-coincidence meetings or no-coincidence meetings. A single-coincidence meeting is a meeting that contains a type s agent (the producer) and a type s+1 agent (the consumer) for some s. A no-coincidence meeting is a meeting in which neither agent produces what the other consumes. 1A lack of double coincidence of wants is another necessary feature of an economy for money to be essential as a means of exchange. 7

2.2. Lack of Commitment and Imperfect Monitoring As mentioned in the Introduction, assumptions regarding a lack of commitment to future actions by agents and some limitations on the abilitytomonitoragents’tradinghistoriesmustbepresentinamodel of money if money is to be essential as a means of exchange. In this model, I assume that agents cannot commit to future actions and I build on the speci(cid:12)cation of imperfect monitoring in CW(b)’s model of inside money. As in CW(b), the society is able to keep a public record of the actions of a fraction B of each specialization type of agent, where B [0;1]: Agents whose histories are a part of the public record are 2 called bankers. Society has no public record for the remaining fraction 1 B of each specialization type, the nonbankers. (cid:0) In CW(b), society’s ability to monitor bankers’ trading histories is perfect. In this model, I allow for imperfect monitoring of bankers’ trading histories. I assume that, similar to KW, the public record is not updated immediately after every action.2 Speci(cid:12)cally, there is a deterministic lag of T periods, where T 0. At the beginning of each (cid:21) date t > T, the bankers’ trading histories are known up through what 2In their model, everyone’s history (B = 1) is updated with probability (cid:26) at each date, whichproducesanaverageupdatinglagof 1. There,thatledtoaslightlysimplerformula- (cid:26) tionofthepayo(cid:11)fromdefectingthanadeterministiclag. Here,themorestraightforward deterministic lag is simpler. 8

they did until the beginning of date t T. All more recent actions are (cid:0) private information to the banker. When two agents meet, the following is common knowledge: each tradingpartner’sspecializationtype,assetholdings(describedbelow), information type (banker or nonbanker), and the past actions of the bankers in the meeting that occurred up to t T periods ago. (cid:0) 2.3. Assets Therearetwodistincttypesofindivisibleandperfectlydurableassets: outside monies and inside monies. Anoutsidemoney isneitherproducednorconsumed. Eachbanker has a technology that enables her to create as many as two distinct objects called notes.3 Because these notes are a type of credit instrument that may circulate as a means of payment among nonbankers, they may serve as inside money. The notes issued by a single agent are distinguishable from those issued by another agent, so that counterfeiting is not a problem. 3Because nonbankers lack commitment and are anonymous, it is not incentive feasible for them to issue inside money, so assuming that they do not have access to a printing technology is innocuous. 9

2.4. Weakly Implementable Allocations and Welfare An allocation describes what happens in all pairwise meetings, conditional on the states of the agents in a meeting. The states of bankers represent some combination of asset holdings and trading histories. The states of nonbankers, whose histories are private, can only represent asset holdings. Initial conditions on the distribution of assets and histories make such allocations su(cid:14)cient to describe the evolution of the economy. The following two-stage game is played in each pairwise meeting. In the (cid:12)rst stage, agents simultaneously announce states. Given the information types of the agents and their announced states from the (cid:12)rst stage and conditional on the absence of a discovered defection, the allocation suggests actions in the meeting. If there has been a discovered defection by either agent in the meeting, then no trade is always suggested. In the second stage, agents simultaneously decide whethertoagreeordisagreetothesuggestedallocation. Ifbothagree, then the suggested allocation is carried out. If at least one agent disagrees, then both leave the meeting without trading. A banker is a defector if she either misrepresents her state in the (cid:12)rst stage, or chooses not to participate in the suggested allocation in the second stage. Suppose an initial defection occurs at date t. For the T 1 periods that follow it, a banker is an undiscovered defector. (cid:0) From period t + T on, a defecting banker is a discovered defector. 10

Defecting bankers are permanently punished with autarky. DEFINITION 1. An allocation is weakly implementable if there is a subgame perfect Nash equilibrium in which agents truthfully announce their states in the (cid:12)rst stage and agree to the suggested actions for their meetings. Weakly implementable allocations can be characterized as those that satisfy a set of feasibility and incentive constraints. The feasibility constraints pertain to agents’ abilities to transfer assets to one another (i.e., whether money can be issued and redeemed or not). The incentive constraints contain both truth-telling constraints about agents’ states in the (cid:12)rst stage of a meeting, and participation constraints at the second stage. The truth-telling constraints pertain to bankers’ revealing their true state (which can represent both asset holdings and histories). Because asset holdings are common knowledgeinameetingandtheirtradinghistoriesareunknown,nonbankers cannot misrepresent their states at the (cid:12)rst stage. Participation constraints imply that agents must prefer to accept the suggested allocation over not accepting it in the second stage of a meeting. I want to focus, whenever possible, on good allocations. I de(cid:12)ne a simple ex ante representative agent welfare criterion|one that treats agents as identical before they are assigned their information types and states|to be the expected discounted utility of the gains from trade over all single-coincidence meetings. The gains from trade are denoted z(y) u(y) y. Welfare is maximized by the production (cid:17) (cid:0) 11

and consumption of y argmax[u(y) y] in every single-coincidence (cid:3) (cid:17) y (cid:0) meeting. The limited ability of the agents to make use of credit arrangements makes this welfare level impossible to obtain. Nonetheless, such a welfare level serves as a benchmark for comparison of alternative monetary arrangements. 3. IMPERFECT MONITORING AND THE VALUE OF INSIDE MONEY Tomotivatehowbothinsideandoutsidemoneymaybeessentialwhen there is imperfect monitoring of trading histories, I (cid:12)rst introduce the updating lag for bankers and show how ex ante welfare declines with the lag when there is only one inside money and no outside money. Theexamplehasassetholdingslimitedtotheset 0;1 andallocations f g restricted to be both stationary and symmetric. Consider an allocation in which the same output level, y (0;y ], (cid:3) 2 is produced in all single-coincidence meetings except when (1) a nonbanker producer has a unit of inside money and when (2) a nonbanker consumer does not have a unit of inside money. The (cid:12)rst exception is implied by the restriction on asset holdings and the nonbanker producer’s participation constraint. The second is a feature of the allocation that implies that nonbanker consumers do not receive gifts.4 4Because banker histories are monitored, it may be implementable for banker producers to produce for (i.e., give gifts to) nonbanker consumers who do not have a unit of inside 12

In addition, suppose that nonbanker consumers surrender a unit of money when they consume y and that nonbanker producers receive a unit of money when they produce y. In single-coincidence meetings between nonbanker consumers and banker producers, the banker producer redeems the unit of inside money and destroys it. In single-coincidence meetings between banker consumers and nonbanker producers, the banker issues a unit of inside money. Because of the symmetry imposed on the allocation, meetings between bankers do not involve the transfer of inside money. Banker histories are used only for the purpose of punishing defecting bankers and do not in(cid:13)uence allocations when there are no discovered defections. This makes the (cid:12)rst stage of the two-stage game innocuous. Finally, I assume that half of the nonbankers within each specialization type start with a unit of inside money, whereas the other half do not. Under the above scheme, it is easy to verify that the distribution of money is a steady state distribution. I can now express the value functions for both nonbankers and bankers that are implied by the above allocation. Let vn denote i the no-defection expected discounted utility of a nonbanker with asset holdings i 0;1 at the start of a period. The value functions for 2 f g nonbankers are 1 B S(1 (cid:12))vn = [B+ (cid:0) ][ y+(cid:12)(vn vn)] (1) (cid:0) 0 2 (cid:0) 1 (cid:0) 0 money. Such allocations may improve welfare. 13

1 B S(1 (cid:12))vn = [B+ (cid:0) ][u(y) (cid:12)(vn vn)]: (2) (cid:0) 1 2 (cid:0) 1 (cid:0) 0 As is well known and easy to verify, these equations have a unique solution with S(1 (cid:12))+[B+1](cid:12) [vn vn] = [B+1 B][u(y)+y] > 0: f (cid:0) g 1 (cid:0) 0 (cid:0)2 Now let vb denote the no-defection expected discounted utility of a banker at the start of a period. The value function for a banker is 1 B S(1 (cid:12))vb = [B+ (cid:0) ][u(y) y]: (3) (cid:0) 2 (cid:0) Finally, I need to express the value functions for a defecting banker. What is of interest when it comes to expressing the incentive constraints for bankers is the initial-defector expected discounted utility, whichIcalculaterecursively. Ifabankerchoosestobecomeadefector, she agrees to consume in all single-coincidence meetings in which it is possible for her to consume (and so continues to issue inside money), but she disagrees to produce whenever production is possible (and so chooses not to redeem inside money). Let vb denote the expected (cid:28) discounted utility of an undiscovered defector at time (cid:28) 1;T 1 . e 2 f (cid:0) g Then 1 1 B vb = [B+ (cid:0) ][u(y)]+(cid:12)vb (4) (cid:28) S 2 (cid:28)+1 e e with the terminal condition that vb = 0: (5) T e The terminal condition incorporates the fact that once discovered, a 14

defecting banker is punished with autarky forever. The expected discountedutilityforabankerfrom aninitialdefectiongiventhatnoone else defects, (cid:12)vb, is what is relevant for banker incentive constraints. 1 This is obtained by solving vb recursively from the terminal condition e (cid:28) vb. Forthisallocation, theexpecteddiscountedutilityfromaninitial T e defection is e vb = 0 if T = 0 1 T 1 1 1 B (cid:0) e = [B+ (cid:0) ][u(y)] (cid:12)(cid:28) if T 1: (6) S 2 (cid:21) (cid:28)=0 X It is immediately obvious that vb is increasing in T. 1 There are two relevant incentive constraints. One is a participae tion constraint for nonbanker producers, y+(cid:12)vn (cid:12)vn; (7) 1 0 (cid:0) (cid:21) and the other is a participation constraint for banker producers, y+(cid:12)vb (cid:12)vb: (8) 1 (cid:0) (cid:21) e Welfare from the inside money example, denoted W(I), is (1+B)2 W(I) = [u(y) y] (9) 4S(1 (cid:12)) (cid:0) (cid:0) and is maximized by the production of y for all single-coincidence (cid:3) meetings for which production takes place. 15

CW(a,b) has shown that y is implementable when T = 0 for (cid:3) su(cid:14)ciently high (cid:12). This is because the right-hand side of (8) is zero, so that the only relevant participation constraint to satisfy is (7), which is nonbinding at y for high enough (cid:12). (cid:3) I now present the main result of this section. PROPOSITION 1. W(I) is weakly decreasing in T. Proof. Note that W(I) is increasing in y for the range y (0;y ]. (cid:3) 2 Substitute (6) into (8) to get T 1 (1 (cid:12))(cid:12)(B+1) 1 (cid:0) y (cid:0) [ (cid:12)(cid:28)]u(y): (10) (cid:20) 2S(1 (cid:12))+(cid:12)(B+1) 1 (cid:12) (cid:0) (cid:0) (cid:0) (cid:28)=0 X Because (6) is weakly increasing in T, the maximum value of y that satis(cid:12)es (10) is weakly decreasing in T.5 Next, I demonstrate that at some point (8) replaces (7) as the relevant constraint for implementation, and that y cannot satisfy (8) (cid:3) for all T. Given that T 1 (cid:0) u(y) lim (cid:12)(cid:28)u(y) = ; T 1 (cid:12) !1(cid:28)=0 (cid:0) X the only y that can satisfy (10) as T is zero, which is less than ! 1 the y that satis(cid:12)es the nonbanker participation constraint (7). This (cid:3) combinedwiththefactsthatW(I)isincreasinginyandthemaximum valueofythatsatis(cid:12)es(10)isweaklydecreasinginT provesthatW(I) is weakly decreasing in T. (cid:4) 5The maximum value of y that satis(cid:12)es (10) is the value of y at equality. 16

The intuition for Proposition 1 is that increasing the updating lag for banker histories tightens the banker participation constraint (8) su(cid:14)ciently so that the maximum value of y that can be implemented decreases. As T increases, the short-term incentive to agree to the trades that permit banker consumption (some of which involves note issue) and to defect from the trades that require banker production (some of which involves note redemption) increases. Therefore, to implement the above type of allocation, the common level of output must be decreasing in T and must approach 0 as T goes to in(cid:12)nity. Although the proof of Proposition 1 is based on a particular type of inside money allocation, the result is more general. This particular inside money allocation provides the least disutility for bankers for a giveny. Forexample,theallocationexcludesgift-givingfrombankers to nonbankers (see footnote 4). If an allocation included gift-giving, itwouldlowerthevaluefunctionforanondefectingbankerbyincreasing the probability that a banker has to produce y in equation (3), but would not reduce the expected discounted utility from an initial defection, leaving (6) unchanged. This has the e(cid:11)ect of tightening the participation constraint for banker producers, (8). As a result, the critical T for which y cannot satisfy banker participation would (cid:3) be higher than that for an allocation without gift-giving by bankers. 17

3.1. Comparison with Outside Money I can compare the above result to that obtained when there is only one outside money and no inside money. This example also has asset holdings limited to the set 0;1 and allocations are restricted to be f g both stationary and symmetric. Now consider an allocation in which the same output level, y 2 (0;y ]; is produced in all single-coincidence meetings in which the (cid:3) consumerinthemeetinghasaunitofoutsidemoneyandtheproducer does not. I assume that half of all agents within each specialization type start with a unit of outside money, whereas the other half do not. As with the inside money scheme, it is easy to verify that the distribution of money is a steady state distribution. In contrast to the example involving inside money, trade takes place in strictly fewer single-coincidence meetings. This is because banker histories are completely ignored, so that there are no credit opportunities.6 The bene(cid:12)t of such an allocation, however, is that it is independent of the updating lag. The value functions for agents without money and with money 6For example, banker consumers in meetings with nonbanker producers must have a unit of outside money while their trading partner must not. All that was required in the inside money example was that their trading partners not have a unit of inside money because bankers could issue new notes. 18

are7 S(1 (cid:12))v = 1[ y+(cid:12)(v v )] (11) (cid:0) 0 2 (cid:0) 1 (cid:0) 0 S(1 (cid:12))v = 1[u(y) (cid:12)(v v )]: (12) (cid:0) 1 2 (cid:0) 1 (cid:0) 0 As is well known and easy to verify, these equations have a unique solution with S(1 (cid:12))+(cid:12) [v v ] = 1[u(y)+y] > 0: f (cid:0) g 1 (cid:0) 0 2 Thereisonlyonerelevantconstraint, aparticipationconstraintfor producers: y+(cid:12)v (cid:12)v : (13) 1 0 (cid:0) (cid:21) Welfare for the outside money example, denoted W(O), is 1 W(O) = [u(y) y] (14) 4S(1 (cid:12)) (cid:0) (cid:0) and, as with the previous example, is maximized by the production of y for all single-coincidence meetings in which trade takes place. (cid:3) It is well known that for (cid:12) su(cid:14)ciently high, (13) is not binding and y is implementable. Because trade occurs less frequently in (cid:3) the example with outside money than it does in the example with inside money, W(I) > W(0) if y is implementable (i.e., when T is (cid:3) small enough). This is essentially the result of CW(b) that inside money is strictly better than outside money under perfect monitoring of bankers. Inside money gives bankers liquidity when needed that 7Because banker identity is ignored in this example, I do not keep track of agents’ information types. 19

increases the frequency of trades. When T = 0, monitoring is enough incentive to prevent overissue and have y implementable. While (cid:3) outside money can duplicate the y when (cid:12) is su(cid:14)ciently high, it (cid:3) cannot duplicate the frequency of trades because agents cannot issue outside money. AtsomeT, however, y cannotbeimplementedwithinsidemoney. (cid:3) This reduces the maximum attainable welfare using inside money. Outside money does not depend on the lag, so y can be implemented (cid:3) using outside money, no matter how large T is. Thus, there may be a range of updating lags for which it is optimal to use both outside and inside money in order to take advantage of each type’s unique feature: inside money’s ability to provide liquidity and outside money’s ability to maintain a higher level of consumption. 4. ESSENTIALITY OF OUTSIDE AND INSIDE MONEY In this section, I set up three monetary arrangements|one that uses two distinct inside monies, one that uses two distinct outside monies, and one that uses one inside and one outside money|and provide an example where the use of both inside and outside money is essential to implement certain allocations. As in the previous examples, this one maintains the assumption that asset holdings are limited to the set 0;1 and allocations are restricted to be both stationary and f g 20

symmetric. The outside-money arrangement and the inside-money arrangement use two assets, to provide comparable alternatives to the arrangement that uses both. Aiyagari et al. (1996) show that having two distinct assets may improve welfare in environments with indivisibility of assets and the upper bound on holdings because it increases thefrequencyoftradesbyallowingagentstoexchangeahigher-valued asset for a lower-valued asset and production. Modeling two distinct assets in the outside-money and inside-money arrangements removes the possibility that the essentiality of both types of money is tied to this feature. Fortheexample,thereisanimplicitfunctionthatmapsassetholdings and histories into states, which are members of a three-element set A 0;1;2 . Because nonbankers have unobservable histories, (cid:17) f g their states can only represent asset holdings, where state 0 indicates no asset holdings, state 1 indicates holdings of a unit of asset 1; and state 2 indicates holdings of a unit of asset 2. The state interpretation for bankers, however, is contingent on the monetaryarrangement. Intheoutside-moneyarrangement,abanker’s state describes asset holdings. Because each element of A is needed to representassetholdings,thestatesdonotrepresentabanker’shistory. For the inside-money arrangement, the states do not represent asset holdings of bankers, because bankers do not hold assets, due to the symmetric treatment of notes. Thus each state in A is tied to history. For the arrangement that uses both types of money, refer to asset 1 21

as the inside money and asset 2 as the outside money. If a banker is in state 2, that implies that she has a unit of outside money. If the banker is in either state 0 or 1, then she does not have a unit of outside money and the state can carry some history dependence. In general, allocations for each monetary arrangement can be described in the following way. Let xk denote the fraction of each i specialization type with information type k in state i: Let the set of informationtypesbe b;n ;wherebindicatesthatanagentisabanker f g and n indicates that he is a nonbanker. Let ykl be output when ij 2 < + a producer of information type k announces state i and a consumer of information type l announces state j. Similarly, let pkl(h) = 1 and ij qkl(g) = 1 indicate that the next period’s states for the producer and ij the consumer are h A and g A, respectively. For no-coincidence 2 2 meetings, let rkl(h) = 1 indicate that h A is the next period’s state ij 2 for the agent of information type k who announces state i in a no-coincidence meeting with an agent of information type l who announces state j. For a given list of fractions of agents in states, an allocation is outcomes in single-coincidence meetings and outcomes in no-coincidence meetings, [ykl;pkl(h);qkl(h);rkl(h)]; ij ij ij ij for all i;j A and k;l b;n . 2 2 f g Recall that the alternative monetary arrangements are di(cid:11)erentiated by the constraints that need to be satis(cid:12)ed for implementability. 22

The intuition for this follows. A formal derivation of these di(cid:11)erences in the general problem is in the Appendix. An environment that uses only inside money will have the fewest feasibilityconstraintsbecausetheissueandredemptionofinsidemoney adds liquidity that can lead to more trading.8 In contrast, an environment that uses only outside money will have the most feasibility constraints because no one can issue or redeem outside money. An environment that uses both inside and outside money has an intermediate number of feasibility constraints. For incentive constraints, the ranking of monetary arrangements in terms of number of constraints from most to least restrictive is reversed. Thisisduetothefactthatabanker’soutside-moneyholdings are observable, but that banker states tied to histories are observed only with a lag. In the inside-money arrangement, all banker states are tied to history, whereas in the outside-money arrangement, all banker states are tied to money holdings. Thus, bankers have the most opportunities to misrepresent their states in the inside-money arrangement, and so have the most truth-telling constraints, whereas bankers have the fewest opportunities to misrepresent their states in the outside-money arrangement, and so have no truth-telling constraints. Thegreaternumberoftruth-tellingconstraintsalsoincreases the expected utility from making an initial defection, making par- 8Recall that the feasibility constraints pertain to agents’ abilities to issue and redeem notes. 23

ticipation constraints tightest in the inside-money arrangement and weakest in the outside-money arrangement. A comparison of various monetary arrangements involves comparing the sets of allocations that are weakly implementable via each monetary arrangement. This comparison is explicitly set out in Definitions 3{5 in the Appendix. These permit comparison of the types of allocations implementable under each arrangement. DEFINITION 2. The use of both inside and outside money is essential if there exists an allocation that is weakly implementable according to De(cid:12)nition 1 for a monetary arrangement that uses both inside and outside money, but is not implementable under a monetary arrangement that uses only inside money or only outside money. The de(cid:12)nition of the essentiality of inside and outside money is weak in that it only requires that an allocation be weakly implementable exclusively in a monetary arrangement that uses both inside and outside money. A stronger de(cid:12)nition would require that such an allocation be a good allocation, in the sense that it achieves a higher level of welfare than can be achieved by any other allocation that can be weakly implemented by the other monetary arrangements. Such a de(cid:12)nition is not used here because the general problem makes it di(cid:14)cult to demonstrate the satisfaction of this stricter requirement. Nonetheless, the results of the previous section suggest that for somebackgroundmodelsthereexistgoodallocations thatcanonlybe implemented via a monetary arrangement that uses both inside and 24

outside money. These background models are ones in which the lag is neither too short|otherwise the incentive constraints for bankers are not much of an issue and the economy would bene(cid:12)t from the exclusiveuseofinsidemoneyasinCW(a,b)|nortoolong|otherwise the incentive constraints devalue inside money su(cid:14)ciently so that the economy would bene(cid:12)t from the exclusive use of outside money. PROPOSITION 2. The set of allocations that satisfy De(cid:12)nition 2 is nonempty. The proof of Proposition 2 requires an example of an allocation that satis(cid:12)es De(cid:12)nition 2. The following section describes such an example. 4.1. Proof of Proposition 2 Consider an allocation with three output levels, yO;yI and yS, where yO > yI > yS and yO (0;y ]. The output level yO trades in sin- (cid:3) 2 gle-coincidence meetings whenever the consumer has a unit of outside money and the producer does not have a unit of either money. Also, the consumer gives the producer the unit of outside money. The output level yI is produced in several di(cid:11)erent types of single-coincidence meetings. It is produced in all single-coincidence meetings between bankers (with no assets changing hands) except for meetings involving the transfer of outside money mentioned above. It is also produced whenever inside money is exchanged. This includes meetings between nonbankers where the consumer has a unit of inside 25

money but the producer does not. It also includes meetings between nonbanker producers without an asset and banker consumers without outsidemoney(insidemoneyisissuedtothenonbanker)andmeetings between banker producers regardless of asset holdings and nonbanker consumers with a unit of inside money (inside money is redeemed by the banker). The output level yS is exchanged in meetings between nonbankers where the producer has a unit of inside money and the consumer has a unit of outside money. In such a meeting, the agents swap assets as well. The suggested actions are summarized in Table 1. The (cid:12)rst element of each triplet in the box represents output in the meeting. The second element is the end-of-period state for the producer, and the third element is the end-of-period state for the consumer. Boxes with \nt" indicate that there is no trade. A distribution of states that satisfy the steady-state conditions contained in the Appendix is9 xn = 0:36;xn = 0:18;xn = 0:36;xb = 0:0333;xb = 0:0167;xb = 0:05: 0 1 2 0 1 2 To satisfy De(cid:12)nition 2 such an allocation must be implementable using inside and outside money, but must not be implementable using only inside money or only outside money. It is immediate that such anallocationcannotbeimplementedwithonlyoutsidemoneybecause there are meetings in which (inside) money is issued and redeemed. 9See conditions (A.1){(A.4). 26

For example, nonbanker producers in state 0 (no asset holdings) can produce for a banker consumer in state 0 and leave the meeting in state 1. This violates feasibility constraints that must be satis(cid:12)ed for outside money; outside money cannot be issued. What remains to be shown is that such an allocation is weakly implementable using both types of money, but is not implementable usingonlyinsidemoney. Idemonstratethisnumericallyforthemodel speci(cid:12)cation 1 T;S;B;(cid:12);u(x) = 65;3;0:1;0:99;x2 f g f g with output levels yO = 0:25 yI = 0:20 yS = 0:10: Table 2 provides the relevant expected discounted utilities for the example, where vk denotes the no-defection expected discounted utili ity of an agent of information type k who is in state i at the start of a period, and vb (I) and vb (M) denote the expected utility of an initial i1 i1 defection by a banker in state i under the inside money arrangement, e e I, and the arrangement that uses both types of money, M, respectively. First, notice that the expected utilities for nonbankers are increasing in states. Asset 2, which typically trades for higher levels 27

of output, is valued more than asset 1, which is more valued than holding no asset. The expected utilities for bankers are weakly increasing in states. Being in state 2 is preferable to being in state 1 or 0. Consider the expected utilities for defecting bankers with the inside money arrangement. They are the same for all states, because defecting bankers have the (cid:13)exibility to represent the state that provides the highest period utility. For the arrangement using both types of money, the expected utility is the same for states 0 and 1 and less than state 2 because defecting bankers have the freedom to misrepresent themselves only when their true states are 0 and 1. These expectedutilitiesarelessthanthatforstate2becausestate2typically commands higher levels of consumption relative to the others. One can now use Table 2 to verify that the example allocation satis(cid:12)es all of the constraints set forth in De(cid:12)nition 5 in the Appendix, and so is implementable under the monetary arrangement that uses both inside and outside money. Further, one can verify that the inside-moneyarrangementcannotimplementtheallocationbecausecertain banker-producer participation constraints are violated. Speci(cid:12)cally, in meetings with nonbanker consumers who enter with a unit of asset 1, banker producers in states 0 and 1 are not willing to redeem inside money for the amount of output called for by the allocation. Also, in meetings with banker-consumers, bankers in states 0 and 1 are not willing to produce for other bankers in states 0 or 1. 28

5. CONCLUSION I present features of an environment for which both outside and inside money are essential as means of payment. The key model feature is that there is imperfect monitoring of issuers of inside money. In deriving the results, I make use of the assumption that agents can only hold one unit of one asset at a time. I conjecture that this assumption is not crucial. This is because what makes both types of money essential is the aforementioned trade-o(cid:11) between outside and inside money. This trade-o(cid:11) would still exist if the assumption about the unit upper bound on money holdings were dropped. An important issue not addressed in this paper is whether both typesofmoneyareessentialinthestrongersenseofimprovingwelfare. Establishing this is di(cid:14)cult because of the large dimensionality of an allocation (there are 36 possibly distinct single-coincidence meetings) and the large number of complicated constraints. REFERENCES Aiyagari, S. R., N. Wallace, and R. Wright (1996) Coexistence of money and interest-bearing securities. Journal of Monetary Economics 37, 397{420. Cavalcanti, R. and N. Wallace (1999a) A model of private bank-note issue. Review of Economic Dynamics 2, 104{136. 29

Cavalcanti, R. and N. Wallace (1999b) Inside and outside money as alternativemediaofexchange.JournalofMoney, CreditandBanking 31 Part 2, 443{57. Friedman, M. (1959) A Program for Monetary Stability. New York: Fordham University Press. Kehoe,T.andD.Levine(1993)Debt-constrainedassetmarkets.The Review of Economic Studies 60, 865{888. Kocherlakota, N. (1998) Money is memory. Journal of Economic Theory 81, 232{251. Kocherlakota, N. and N. Wallace (1998) Incomplete record-keeping andoptimalpaymentarrangements.Journal of Economic Theory 81, 272{289. Ostroy, J. (1973) The informational e(cid:14)ciency of monetary exchange. The American Economic Review 63, 597{610. Shi, S. (1995) Money and prices: A model of search and bargaining. Journal of Economic Theory 67, 467{498. Townsend, R. (1989) Currency and credit in a private information economy. Journal of Political Economy 97, 1323{1344. Trejos, A. and R. Wright (1995) Search, bargaining, money and prices. Journal of Political Economy 103, 118{141 30

APPENDIX: FORMAL CHARACTER- IZATION OF ALTERNATIVE MON- ETARY ARRANGEMENTS A.1. STEADY-STATE AND FEASIBILITY CON- STRAINTS In describing the steady-state and feasibility requirements imposed on state transitions, I anticipate the satisfaction of one of the constraints described later: no free disposal of assets. Because each person must be in one of the states, the fractions of each specialization type in each state must satisfy xb = B and xn = 1 B: (A.1) i i (cid:0) i i X X Additionally, because each person can be in only one state at a particular point in time, state transitions must satisfy pkl(h) = 1 if and only if pkl(g) = 0 for all g = h; ij ij 6 qkl(h) = 1 if and only if qkl(g) = 0 for all g = h; ij ij 6 rkl(h) = 1 if and only if rkl(g) = 0 for all g = h: (A.2) ij ij 6 A steady-state distribution of agents over states requires that the fraction of bankers in each state and the fraction of nonbankers in 31

each state be constant. This can be expressed by equating the in(cid:13)ow and out(cid:13)ow of each state for nonbankers and bankers. These are, for each i A, 2 xn xb[pnb(i)+qbn(i)+(S 2)rnb(i)] = xn xb[ pnb(h)+qbn(h)+(S 2)rnb(h)] h j hj jh hj i j ij ji ij (cid:0) (cid:0) h=i j j h=i X6 X X X6 (A.3) for nonbankers and xb xn[pbn(i)+qnb(i)+(S 2)rbn(i)] = xb xn[ pbn(h)+qnb(h)+(S 2)rbn(h)] h j hj jh hj i j ij ji ij (cid:0) (cid:0) h=i j j h=i X6 X X X6 (A.4) for bankers. There are also feasibility constraints implied by the preservation of asset holdings in meetings. The general conditions can be written as pkl(i) = qkl(i) = rkl(i) = 1; (A.5) ii ii ii pkl(j) = 1 if and only if qkl(i) = 1; (A.6) ij ij rkl(j) = 1 if and only if rlk(i) = 1: (A.7) ij ji Condition (A.5) says that if both agents have the same asset holdings, then they leave with the same asset holdings. Conditions (A.6) and (A.7)saythatifagentA’snext-periodstateisagentB’scurrentstate, then B’s next-period state is agent A’s current state (they swap asset holdings). Somewhat weaker conditions than (A.6) and (A.7) that 32

will become relevant for the mixed mechanism are pkl(j) = 1 if and only if qkl(j) = 0; ij ij qkl(j) = 1 if and only if pkl(j) = 0; ji ji rkl(j) = 1 if and only if rlk(j) = 0: (A.8) ij ji These are weaker than (A.6) and (A.7) because for agent A to enter agentB’sstate,allthatisrequiredisthatagentBleavehisstate(they do not have to swap states). A.2. VALUE FUNCTIONS AND INCENTIVE CONSTRAINTS I now describe a general set of participation, truth-telling, and freedisposal constraints. I begin by describing the expected discounted utilities of agents. These are all expressed given that no one else defects. Let vk denote the no-defection expected discounted utility of an i agent of information type k who is in state i at the start of a period. For a single-coincidence meeting in which the producer is of information type k in state i and the consumer is of information type l in state j, let Pkl and Qkl be producer and consumer payo(cid:11)s, respecij ij tively, from following the suggested outcome in the second stage of a 33

meeting. Then Pkl = ykl +(cid:12) pkl(h)vk (A.9) ij ij ij h (cid:0) h X and Qkl = u(ykl)+(cid:12) qkl(h)vk: (A.10) ij ij ij h h X For no-coincidence meetings, let Rkl be the payo(cid:11) to the agent of ij information type k in state i when his partner is of information type l in state j from following the suggested outcome in the second stage of a meeting. Then Rkl = (cid:12) rkl(h)vk: (A.11) ij ij h h X Given these de(cid:12)nitions, vk is i xl vk = j [Pkl +Qlk +(S 2)Rkl]: (A.12) i S ij ji (cid:0) ij l b;n j 2Xf gX Icalculaterecursivelytheinitial-defectorexpecteddiscountedutility. Let Pbl and Qlb be producer and consumer payo(cid:11)s, respecmj(cid:28) jm(cid:28) tively,toanundiscovereddefectingbankerwho(cid:12)rstdefected(cid:28) periods e e ago and announced state m in the (cid:12)rst stage of a single-coincidence meeting with a trading partner of information type l in state j from following the suggested outcome in the second stage. Then Pbl = ybl +(cid:12) pbl (h)vb (A.13) mj(cid:28) mj mj h;(cid:28)+1 (cid:0) h X e e 34

and Qlb = u(ylb )+(cid:12) qlb (h)vb : (A.14) jm(cid:28) jm jm h;(cid:28)+1 h X e e Similarly, de(cid:12)ne Rbl for no-coincidence meetings as mj(cid:28) e Rbl = (cid:12) rbl (h)vb : (A.15) mj(cid:28) mj h;(cid:28)+1 h X e e Now consider a banker in the (cid:12)rst stage of a meeting whose true state is i. The announced state of an undiscovered defecting banker depends on the ability of that banker to misrepresent her state, what is known about the state of her trading partner, whether she is in a single-coincidence or no-coincidence meeting, and in the case of a single-coincidence meeting, whether she is a producer or consumer. Let A A be the set of states over which a banker can misrepre- (cid:26) sent. For the outside-money mechanism A = , for the inside-money e ; mechanism, A = A, and for the mixed mechanism, A = 0;1 . Let e f g I be an indicator variable that equals 1 if i A and is 0 otherwise. i e e 2 Thus, if I = 1, then a banker can announce a state in A, whereas i e if I = 0, then she is constrained to report truthfully. Similarly, let i e Jl be an indicator variable that equals 1 if j A and is 0 otherwise, j 2 where j is the true state of the banker’s trading partner and l is her e trading partner’s information type. Let (cid:22) p;q;r denote the type of meeting a defecting banker 2 f g is in where p indicates that she is a producer in a single-coincidence meeting, q indicates that she is a consumer in a single-coincidence 35

meeting and r indicates that she is in a no-coincidence meeting. Then de(cid:12)ne m (cid:22)l (I ;Jl) to be the optimal message of a defecting banker who ij i j is in meeting type (cid:22), whose true state is i, and whose trading partner is of information type l in state j. There are four types of optimal messages for a banker in a meeting oftype(cid:22): m (cid:22)l (0;0);m (cid:22)l (0;1);m (cid:22)l (1;0);andm (cid:22)l (1;1). LetM =Pbl ij ij ij ij mj(cid:28) if (cid:22) = p, M = Qlb if (cid:22) = q, and M = Rbl if (cid:22) = r. Then the jm(cid:28) mj(cid:28) f e optimal messages are f e f e (cid:22)l (cid:22)l m (0;0) = m (0;1) = i; ij ij (cid:22)l m (1;0) = argmaxM; ij m xl (cid:22)l f j m (1;1) = argmax M. (A.16) ij m j A xl j Xi A 2 2 P f e e The(cid:12)rstmessagere(cid:13)ectsthefactthatifabankercannotmisrepresent her type (I = 0) then she reports truthfully. The (cid:12)nal two messages i say that when given the freedom to misrepresent, a defecting banker chooses the state that gives the highest expected discounted utility; (cid:22)l m (1;0) indicates that the state of the trading partner is known with ij (cid:22)l certainty, whereas m (1;1) indicates that what is known is that the ij state of the trading partner is an element of A. Finally, consider a defecting banker in the second stage. The ane nouncements concerning states have been revealed and now a defecting banker chooses whether to agree to the suggested outcome. Let Pbl[m pl (I ;Jl);j] = Pbl such that m = m pl (I ;Jl). Similarly, de(cid:12)ne (cid:28) ij i j mj(cid:28) ij i j b e 36

Qlb[j;m ql (I ;Jl)] and Rbl[mrl(I ;Jl);j]. (cid:28) ij i j (cid:28) ij i j Then for i A and (cid:28) 1;2;:::;T 1 , the expected discounted b b 2 2 f (cid:0) g utility of an undiscovered defecting banker is xl vb = j max[Pbl(m pl (I ;Jl);j);(cid:12)vb ] i(cid:28) S f (cid:28) ij i j i;(cid:28)+1 l b;n j 2Xf gX e +max[Qlb(j;m ql (I b ;Jl));(cid:12)vb ] e (cid:28) ij i j i;(cid:28)+1 +(S 2b)max[R (cid:28) bl(mr ij l(I i ;J ej l);j);(cid:12)v i b ;(cid:28)+1 ] (A.17) (cid:0) g b e with the terminal condition that vb = 0 (A.18) iT e for all i A. 2 Nowconsidertheconstraintsthatarerelevantforimplementation. Participation constraints require that agents are ex post sequentially rational. This is equivalent to the requirement that they receive nonnegative gains from trade. For nonbankers, the participation constraints are min Pnl;Qln;Rnl (cid:12)vn (A.19) ij ji ij i f g (cid:21) for all i;j A and l b;n . The right-hand side of (A.19) is due 2 2 f g to the fact that defecting nonbankers can only be punished with no trade at that date because they will never be discovered. 37

For bankers, the participation constraints are min Pbl;Qlb;Rbl (cid:12)vb (A.20) ij ji ij i1 f g (cid:21) e for all i;j A and l b;n . The right-hand side of (A.20) re(cid:13)ects 2 2 f g the fact that if a banker does not agree to the suggested outcome, she does not trade at that date and becomes an initial defector. Now consider truth-telling constraints on bankers. Bankers must report their true state in the (cid:12)rst stage of a meeting for all possible meetings: m (cid:22)l (I ;Jl) = i (A.21) ij i j for all i;j A, l b;n and (cid:22) p;q;r . 2 2 f g 2 f g Finally, the free-disposal constraints for nonbankers and nondefecting bankers are vk vk (A.22) i 0 (cid:21) For undiscovered defecting bankers, the free-disposal constraints are vb vb (A.23) i(cid:28) 0(cid:28) (cid:21) e e for all (cid:28) 1;2;:::;T 1 . 2 f (cid:0) g A.3. DEFINITIONS The outside-money mechanism has the largest number of feasibility restrictions because outside-money holdings must be preserved in all 38

meetings, whereastheinside-moneymechanismhasthesmallestnumber of feasibility constraints because inside-money holdings must be preserved only in meetings between nonbankers. The incentive constraints are most restrictive for the inside-money mechanism, and least restrictive for the outside-money mechanism. There are two reasons for the di(cid:11)erence. The (cid:12)rst is that the outsidemoneymechanismhas,ine(cid:11)ect,notruth-tellingconstraints,themixed mechanism has truth-telling constraints only for some bankers, and the inside-money mechanism has truth-telling constraints for every banker. The second is that, because the di(cid:11)erent mechanisms have di(cid:11)erent numbers of truth-telling constraints, the value to a banker from making an initial defection, vb , may also vary with the mechai1 nisms. This leads to stricter participation constraints for bankers in e mechanisms where vb is higher. i1 The following de(cid:12)nitions formalize the conditions necessary for an e allocation to be implementable via each mechanism. DEFINITION3. Anallocationisimplementableviaoutsidemoney if it satis(cid:12)es (A.1){(A.7) for i;j A, k;l b;n and (A.9){(A.23) 2 2 f g for i;j A, k;l b;n , (cid:22) p;q;r , (cid:28) 1;2;:::;T , where 2 2 f g 2 f g 2 f g I = Jl = 0 for all i;j A. i j 2 DEFINITION4. Anallocationisimplementableviainsidemoney if it satis(cid:12)es (A.1){(A.4) for i;j A, k;l b;n , (A.5){(A.7) for 2 2 f g i;j A; k = l = n, and (A.9){(A.21) for i;j A, k;l b;n , 2 2 2 f g (cid:22) p;q;r , (cid:28) 1;2;:::;T , where Jn = 0 and I = Jb = 1 for all 2 f g 2 f g j i j 39

i;j A, and (A.22) for i A;k = n. 2 2 DEFINITION 5. An allocation is implementable via outside and insidemoneyifitsatis(cid:12)es (A.1){(A.4)fori;j A;k;l b;n , (A.5) 2 2 f g for i A;k = l = n and k and/or l = b with i = 2, (A.6) and (A.7) 2 for i;j A;k = l = n, (A.8) for j = 2;i = 2, and k and/or l = b, 2 6 (A.9){(A.21) for i;j A, k;l b;n , (cid:22) p;q;r , (cid:28) 1;2;:::;T , 2 2 f g 2 f g 2 f g where I = Jn = Jb = 0 and I = Jb = 1 for i;j 0;1 , and (A.22) 2 j 2 i j 2 f g and (A.23) for i A;k = n and k = b with i = 2. 2 40

noitacolla elpmaxe eht ni sedarT .1 elbaT recudorP etatS reknabnoN etatS reknaB 2 1 0 2 1 0 tn tn )0;1;Iy( )1;2;Iy( )0;1;Iy( )0;0;Iy( 0 etatS reknaB remusnoC tn tn )1;1;Iy( )1;2;Iy( )1;1;Iy( )1;0;Iy( 1 tn tn )0;2;Oy( )2;2;Iy( )0;2;Oy( )0;2;Oy( 2 tn tn tn tn tn tn 0 etatS reknabnoN tn tn )0;1;Iy( )0;2;Iy( )0;1;Iy( )0;1;Iy( 1 tn )1;2;Sy( )0;2;Oy( tn )0;2;Oy( )0;2;Oy( 2 41

Table 2. Value functions for the example allocation State vn vb vb (I) vb (M) i i i1 i1 0 2.432 3.742 3.595 3.253 1 2.712 3.742 3.595 3.253 e e 2 2.821 3.877 3.595 3.303 42