Alternative Central Bank Credit Policies for Liquidity Provision in a Model of Payments

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Alternative Central Bank Credit Policies for Liquidity Provision in a Model of Payments David C. Mills, Jr. 2005-55 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.

Alternative Central Bank Credit Policies for Liquidity Provision in a Model of Payments David C. Mills, Jr.1 2 Federal Reserve Board Version: March 2005 I explore alternative central bank policies for liquidity provision in a model of payments. I use a mechanism design approach so that agents(cid:146)incentives to default are explicit and contingent on the credit policy designed. In the (cid:133)rst policy, the central bank investsincostlyenforcementandchargesaninterestratetorecovercosts. Ishowthatthe second best solution is not distortionary. In the second policy, the central bank requires collateral. If collateral does not bear an opportunity cost, then the solution is (cid:133)rst best. Otherwise, the second best is distortionary because collateral serves as a binding credit constraint. JEL Classi(cid:133)cation: E40; E58; C73 Key Words: Payments systems, central banking, liquidity, collateral 1Correspondence: Mail Stop 188, 20th & C Streets, NW, Washington, DC 20551. Email: david.c.mills@frb.gov. O¢ ce: 202-530-6265. Fax: 202-872-7533. 2The author thanks Charles Plosser, Robert Reed, an anonymous referee, and seminar participants at the University of Kentucky, Federal Reserve Banks of New York and Richmond, the Federal Reserve Board, the European Central Bank, 2004 Money, Banking and Payments Workshop at the Federal Reserve Bank of Cleveland, the 2004 Summer Meeting of the Econometric Society,the2004MidwestMacroeconomicsMeetingsandtheFall2003MidwestEconomicTheory Conference. The views in this paper are solely the responsibility of the author and should not beinterpretedasre(cid:135)ectingtheviewsoftheBoardofGovernorsoftheFederalReserveSystem or any otherperson associated with the FederalReserve System. Allerrors are my own. 1

1. INTRODUCTION A primary role of a central bank is to facilitate a safe and e¢ cient payments system. One source of ine¢ ciency in payment systems is a potential shortage of liquidity. Centralbanksoftenrespondbyprovidingliquiditythroughtheextension ofcredit. Becauseofthisrole,acentralbankmustmanageitsexposuretotherisk that an agent does not repay. Some central banks, such as the European Central Bank, manage this risk by requiring borrowers to post collateral. Others, such as the Federal Reserve in the U.S., charge an explicit interest rate on credit and limit the amount any particular agent can borrow. In this paper, I explore these alternative credit policies in a theoretical model of payments and o⁄er a rationale for why some central banks may choose one credit policy over another. I do this in a mechanism design framework, paying particular attention to the moral hazard issues associated with the repayment of debt that alternative credit policies aim to mitigate. The payment systems most relevant to this paper are large-value payment systems which are mainly intraday, interbank payment systems. Many large-value payment systems are operated by central banks and are often real-time gross settlement (RTGS) systems. In an RTGS system, payments are made one at a time, with (cid:133)nality, during the day. Examples of RTGS systems include Fedwire operated by the Federal Reserve in the U.S. and TARGET, operated by the European CentralBankintheEMU.3 Becausepaymentsaremadeoneatatime, liquidityis needed to complete each transaction. If participants do not have enough liquidity to make a payment at a particular point in time, they can typically borrow funds 3TARGET is the collection of inter-connected domestic payment systems of the EMU that settle cross-borderpayments denominated in euros. 2

from the central bank by overdrawing on an account with the central bank, which they then pay back by the end of the day. The central bank faces a trade-o⁄between supplying this intraday liquidity at little or no cost to enhance the e¢ ciency ofthesystemandaccountingformoralhazardissuesassociatedwiththeextension of credit. Of fundamental interest in this paper is how a central bank should design a credit policy for the provision of liquidity in an RTGS system to improve e¢ ciency while dealing with moral hazard associated with debt repayment. The main contribution of this paper is a framework with which to study the alternative credit policies of central banks. The key features of the framework are (i)defaultdecisionsofagentsareendogenous,and(ii)mechanismdesign. The(cid:133)rst is important to rigorously introduce a moral hazard problem that arises when debt is extended. The second is a useful approach to evaluate what good outcomes are achievable under alternative credit policies taking into account agents(cid:146)incentives to default. This framework is applied to a model of payments that is similar to that of Freeman (1996). Such a model captures some key features of large-value payment systems. These features are i) (cid:133)at money is necessary as a means of payment, ii) there is a need to acquire liquidity (in the form of (cid:133)at money) during the day to make such payments, and iii) money is also necessary to repay debts by the end of the day. These three features provide an endogenous role for an institution such as a central bank to provide liquidity to facilitate payments. AnimportantabstractioninFreeman(cid:146)soriginalmodel, however, isthatthereis costless enforcement that exogenously guarantees that debts are repaid. Such an abstraction has led to conclusions by Freeman (1996), Green (1997), Zhou (2000), Kahn and Roberds (2001) and Martin (2003) that a credit policy of free liquidity 3

provisionisoptimal. Theseconclusionsareimmediategiventhatthereisnoexplicit moral hazard problem in most of these models4. As a result, these models do not fully capture the trade-o⁄ between providing liquidity to facilitate payments and minimizing the exposure of credit risk associated with that provision. Moreover, Mills (2004) endogenizes the repayment decision of agents under costless enforcement in Freeman(cid:146)s model and shows that money is not necessary to repay debts if enforcement is too strong and so the need for liquidity in the model is questioned. As in Mills (2004), I shall depart from this abstraction so that the default decision of agents is not trivial. In the context of the background environment, I look at two alternative credit policies that resemble some of the features of such policies in actual large-value payment systems. The (cid:133)rst such policy is that of costly enforcement and pricing. Thecentralbankinvestsinacostlyenforcementtechnologythatallowsittopunish defaulters by con(cid:133)scating some consumption goods. The second policy is that of requiring those who borrow from the central bank to post collateral. Under this policy,thecentralbankdoesnotchargeanexplicitinterestrateondebt. Collateral, however, may have an opportunity cost in that it cannot earn a return that it otherwise would have. I use a mechanism design approach to see if the credit policies can achieve good allocations, which I de(cid:133)ne to be Pareto-optimal allocations. It is possible for both types of credit policies to implement these good allocations. In the case of the pricing policy, I (cid:133)nd an example of where the optimal intraday interest rate is positive because of a requirement for the central bank to recover its costs of enforcement. This di⁄ers from the aforementioned literature and supports a 4Martin (2003) is an exception. See below. 4

suggestion made by Rochet and Tirole (1996) that the intraday interest rate be positive because monitoring and enforcement is costly. In the case of collateral, if itdoesnothaveanopportunitycost,suchapolicycanimplementagoodallocation that is (cid:133)rst-best. If, on the other hand, there is a positive opportunity cost of collateral, requiring collateral adds binding incentive constraints that distort the allocation away from Pareto optimality. Collateral serves as an endogenous credit constraint. ThispaperismostcloselyrelatedtoMartin(2003). Bothpapersareinterested inevaluatinghowalternativecreditpoliciesaddressparticipants(cid:146)moralhazardina general equilibrium model where money is necessary. Martin (2003) models moral hazard by endogenizing some agents(cid:146)choice of risk arising from a central bank(cid:146)s free provision of liquidity. Agents can choose a safe production technology or a risky one that exogenously leads to some default and central banks cannot enforce a choice of the safe asset. In his model, agents cannot strategically default. In this paper, agents do not have an opportunity to engage in risky behavior, but rather have the choice to strategically default. The central bank can enforce some repayment only after investing in a costly enforcement technology. Martin (2003) compares alternative central bank credit policies and concludes that a collateral policy with a zero intraday interest rate is preferred to debt limits in mitigating the credit risk. The collateral in his model is debt issued by private agentswhoexogenouslycommittorepayment(i.e. thereisnochoicetostrategically default) and does not bear an opportunity cost. In this paper, the issuers of collateral do have the opportunity to strategically default and collateral does bear anopportunitycost. Moreover,Martin(2003)doesnotconsidercostlymonitoring ofagentsreceivingcentralbankliquidityandhowthatmightcomparetoacollateral 5

policy. Finally, in this paper the default decision of agents is endogenous, but the liquidityshortageisexogenous. Thisiscomplementarytoanareaintheliterature by Bech and Garratt (2003), Angelini (1998) and Kobayakawa (1997). These papers endogenize the liquidity shortage by focusing on the incentives agents have to coordinate the timing of payments given alternative credit policies, but do not endogenize the need for such credit policies. The paper is organized as follows. Section 2 presents the environment while Section3providesabenchmarkofoptimalallocations. Sections4and5containthe mainresultsaspertainstothecreditpolicywithpricingandcollateral,respectively. Section6extendstheanalysistoincludeexogenousdefaultandcentralbanklosses. Section 7 concludes. 2. THE ENVIRONMENT The model is a variation of both Freeman (1996) and Mills (2004). It is a pure exchange endowment model of two-period-lived overlapping generations with two goods at each date, good 1 and good 2. The economy starts at date t = 1. There is a [0;1] continuum of each of two types of agents, called creditors and debtors, born at every date.5 These two types are distinguished by their endowments and preferences. Each creditor is endowed with y units of good 1 when young and nothing when old. Each debtor is endowed with x units of good 2 when young and nothing when old. Let ct denote consumption of good z 1;2 at date t by a creditor of zt 0 2 f g 0 5Thenamegiventocreditorsisabitmisleadingbecausetheseagentsneverlendinequilibrium. 6

generation t. The utility of a creditor is u(ct ;ct ); where u : 2 . Notice 1t 2;t+1 <+ ! < that a creditor wishes to consume good 1 when young and good 2 when old. The functionuisstrictlyincreasingandconcaveineachargument,isC1,andu(0)= 0 1 and u( )=0. 0 1 Letdt denoteconsumptionofgoodz 1;2 atdatet byadebtorofgenerazt 0 2f g 0 tiont. Theutilityofadebtorbornatdatetisv(dt ;dt )wherev : 2 . Hence, 1t 2t <+ !< a debtor wishes to consume both good 1 and good 2 when young. A debtor does not wish to consume either good when old. The function v is strictly increasing and concave in each argument, is C1, and v (0)= and v ( )=0. 0 0 1 1 At date t=1, there is a [0,1] continuum of initial old creditors. These creditors are each endowed with M divisible units of (cid:133)at money. It is assumed that agents cannot commit to trades and that there is no public memory of trading histories. It is also assumed that agents do not consume any goods until the end of the period. There is also an institution called a central bank that has three technologies uniquetoit.6 The(cid:133)rsttechnologyistheabilitytoprint(cid:133)atmoney. Thesecondis arecord-keepingtechnologythatenablesthecentralbanktokeeptrackofindividual balancesofbothmoneyandgoodsthataprivateagentmayhavewithit. Thethird technologyisanenforcementtechnologythatcanbeacquiredatarealresourcecost (cid:13) >0 per period.7 The enforcement technology allows the central bank to punish defaultersbycon(cid:133)scatinggoods. Theresourcecost(cid:13) canbethoughtofasthecost of monitoring and the use of channels to con(cid:133)scate goods to satisfy repayment. 6WhatIcallacentralbankmayalsobeinterpretedasaprivateclearinghousethatisseparate from theotheragents. Asnoted in Green (1997),itremainsan open question asto whetherthe liquidity-providing institution in the model should be a public or private one and is beyond the scope ofthis paper. 7In Freeman (1996),(cid:13)=0. 7

There are four stages within a period. At the (cid:133)rst stage, young debtors meet the central bank. As we shall see, young debtors may seek liquidity from the central bank at this time. At the second stage, young debtors and young creditors meet. This is the only opportunity for young debtors to acquire good 1. At the third stage, young debtors and old creditors meet. This is the only opportunity for old creditors to acquire good 2. Finally, at the fourth stage, young debtors are reunited with the central bank. At this time, young debtors have an opportunity to repay the central bank for any liquidity provided by it at the (cid:133)rst stage. Debtors are endowed with an investment technology that allows them to invest some of their endowment (I x) at the end of the (cid:133)rst stage, that yields with (cid:20) certainty, RI units of good 2 at the beginning of the third stage, where R 1. (cid:21) The sequence of events for each date is summarized in Figure 1. The setup captures some key elements of large-value RTGS payment systems. In such systems, banks use funds in central bank accounts to make payments. In themodel,(cid:133)atmoney(intheformofcurrency)isnecessaryasameansofpayment if trade is to take place because of the timing of trading opportunities within a period and the fact that there is no commitment and no public memory.8 In actual RTGS systems, banks may face a liquidity problem during the day because ofthemismatchbetweenpaymentsreceivedandpaymentsmade. Incentralbankoperated systems, banks may borrow funds from the central bank by overdrawing on their accounts to make payments. The liquidity problem is approximated in the model via the timing of events within a period; because young debtors are not endowedwith(cid:133)atmoneytheymust(cid:133)rstacquiresomeviaacreditrelationshipwith the central bank. In actual central-bank operated RTGS systems, overdrafts are 8See Kocherlakota (1998) for a general discussion and Mills (2004) for one in the context of this type ofmodel. 8

repaid via deposits into central bank accounts. This is approximated in the model by requiring young debtors to repay the central bank at the (cid:133)nal stage within a period so that it may retire the same amount of money that it injected into the economy at the beginning of the period. Moreover, as in Townsend (1989) money servesasacommunicationdevicethatsignalstothecentralbankthepastbehavior of debtors. Thus, money is essential for the repayment of debt in the model. Themodeldoesabstractfromexplicitlymodellingbanksasintermediariesthat make payments on behalf of customers. What is important for the analysis, however, is that the model contain a liquidityproblem in the payment system that can bealleviatedbythecentralbank. Bynotmodelingbanksexplicitly,Iamassuming that ine¢ ciencies arising from an interbank payment system have real e⁄ects on the economy. 3. BENCHMARK: OPTIMAL ALLOCATIONS Beforedescribingthealternativecreditpolicies,I(cid:133)rstde(cid:133)nesomeoptimalallocations. A(cid:133)rst-bestallocationisonethatmaximizesex-anteexpectedsteady-state utility of debtors and creditors subject to a limited set of feasibility constraints. This limited set abstracts from incentive constraints which will be important for implementation. Denote the steady state levels of consumption of both good 1 and good 2 by d z for a debtor and c for a creditor for z 1;2 . The problem is then to maximize z 2f g u(c ;c )+v(d ;d ) (1) 1 2 1 2 9

with respect to I;d ;d ;c ;c and subject to the following feasibility constraints: 1 2 1 2 x I (2) (cid:21) y d +c (3) 1 1 (cid:21) RI+(x I) d +c : (4) 2 2 (cid:0) (cid:21) De(cid:133)ne u as the partial derivative of creditor utility with respect to good z z and v as the partial derivative of debtor utility with respect to good z for z z 2 1;2 . Optimal allocations require that (2)-(4) are satis(cid:133)ed at equality. The (cid:133)rst f g order conditions, then, which satisfy the Kuhn-Tucker conditions for necessity and su¢ ciency simplify to: u v 1 = 1 (5) u v 2 2 Condition (5) states that optimal allocations are those that are Pareto optimal. Thus, in what follows, I shall look for implementable allocations (ones that take into account the incentives of agents) that satisfy (5). 4. LIQUIDITY PROVISION WITH COSTLY ENFORCEMENT AND PRICING In this section, I provide an example of a payment mechanism where a central bank provides liquidity with a credit policy of paying a real cost (cid:13) >0 for the enforcementtechnologyandcharginganintradayinterestrate(orprice)forliquidity. I characterize a set of implementable allocations via the mechanism as those that satisfyasetofincentiveconstraints. Allocationsareimplementableiftheyaresubgame perfect equilibrium allocations. Finally, I show that the second-best optimal allocation that is implementable via the pricing mechanism is Pareto-optimal. 10

Recall that investment in the enforcement technology enables the central bank to con(cid:133)scate goods from a defaulting debtor. The central bank can e⁄ectively choose some combination of goods 1 and 2 to con(cid:133)scate so that, in equilibrium, debtorswillchoosenottodefault.9 Thiscostlyenforcementismeanttomodelthe opportunities a central bank may have when it monitors the behavior of payment system participants. The exogenous parameter, (cid:13), is a proxy for the real cost of monitoring agents and the costs associated with the potential liquidation of assets in the event of a default. The central bank charges an intraday interest rate, r 0 proportional to the (cid:21) amount borrowed. The interest payment is payable in units of good 2. This is convenient so as to provide an easy comparison with the collateral policy of the next section. Some central banks, such as the Federal Reserve, have a mandate to fully recover costs of the operation of their payment services. Such an assumption in the context of the model is that the central bank charge an intraday interest rate such that the nominal value of the interest payment be at least as large as the nominal value of the cost of enforcement. I shall describe the payment game that agentsplayundertheassumptionthatthecentralbankmustfullyrecoveritscosts. Thegameisasfollowsforanydatet. Atthe(cid:133)rststageofaperiod,generationt debtors choose whether or not to seek liquidity from the central bank. Those that seek such credit acquire M units. Let (cid:14)t [0;1] be the fraction of debtors 1 2 who seek credit from the central bank. The generation-t debtor then invests the entire amount of good 2 (I = x). At the second stage, the mechanism suggests that generation-t creditors who want to participate in exchange each o⁄er d of 1 good 1 and that generation-t debtors who want to participate o⁄er M units of (cid:133)at 9In Section 6,Ishallconsiderthe possibility that some debtors may exogenously default. 11

money. The creditors and debtors simultaneously choose whether to participate in exchange or not. Let (cid:20)t [0;1] be the fraction of generation-t creditors who 2 2 agree to o⁄er d of good 1 and (cid:14)t [0;(cid:14)t] be the fraction of generation-t debtors 1 2 2 1 who agree to exchange M units of money for some consumption of good 1. Each debtor who agrees to trade M units of money for consumption receives (cid:20)t 2d units (cid:14)t 1 2 of good 1. Each creditor that agrees then receives (cid:14)t 2M units of money and has (cid:20)t 2 ct = y d units of good 1 left for consumption. Those that disagree leave with 1 (cid:0) 1 autarky. At the third stage of date t, the generation-t debtors(cid:146)investments pay o⁄and each now has Rx units of good 2. The mechanism then suggests that generationt debtors who want to participate in exchange each o⁄er c of good 2 and that 2 generation-t 1 creditors who have (cid:14)t 2(cid:0) 1 M units of money and who want to par- (cid:0) (cid:20)t 2(cid:0) 1 ticipate in exchange o⁄er up (cid:14) 2 t (cid:0) 1 M units of money. Generation-t 1 creditors (cid:20) 2 t (cid:0) 1 (cid:0) with no money are not able to participate in exchange. The debtors and those creditors able to participate in exchange simultaneously choose to participate or not. Let(cid:14)t [0;(cid:14)t]bethefractionofgeneration-tdebtorswhoagreetoo⁄erc of 3 2 2 2 good 2 and (cid:20)t 1 [0;(cid:20)t 1] be the fraction of generation-t 1 creditors who agree 3(cid:0) 2 2(cid:0) (cid:0) to exchange money for some consumption of good 2. Each creditor who agrees to trade money for consumption receives (cid:14)t 3 c units of good 2. Each debtor that (cid:20)t 3(cid:0) 1 2 agrees then receives (cid:20) 3 t (cid:0) 1 (cid:14) 2 t (cid:0) 1 M units of money and has Rx c units of good 2 (cid:14)t 3 (cid:20) 2 t (cid:0) 1 (cid:0) 2 left over. Those that disagree leave with autarky. At the (cid:133)nal stage, if a generation-t debtor who has borrowed money at the (cid:133)rst stagenowhas (cid:20) 3 t (cid:0) 1 (cid:14) 2 t (cid:0) 1 M unitsofmoney,thenthedebtormaychoosetorepaythe (cid:14)t 3 (cid:20) 2 t (cid:0) 1 central bank (cid:20) 3 t (cid:0) 1 (cid:14) 2 t (cid:0) 1 M units of money plus an interest payment, i ; in units of (cid:14)t 3 (cid:20) 2 t (cid:0) 1 2 good 2: The nominal value of the interest payment is determined by the nominal 12

value of the amount borrowed. Speci(cid:133)cally, let p and p represent the prices 1 2 of goods 1 and 2 in terms of money, respectively. Then p i = rM = rp (cid:20)t 2d 2 2 1(cid:14)t 1 2 or i = rp1 (cid:20)t 2d . I can express i in terms of good 2, i = r (cid:14)t 3 (cid:20)t 2(cid:0) 1 c by 2 p2 (cid:14)t 2 1 2 2 (cid:20)t 3(cid:0) 1 (cid:14)t 2(cid:0) 1 2 noting that the nominal value of good 1 acquired by generation-t 1 creditors is (cid:0) p (cid:14)t 3 c = (cid:14) 2 t (cid:0) 1 M so that p (cid:14)t 3 (cid:20) 2 t (cid:0) 1 c = M = p (cid:20)t 2d where the latter equality 2 (cid:20)t 3(cid:0) 1 2 (cid:20) 2 t (cid:0) 1 2 (cid:20) 3 t (cid:0) 1 (cid:14) 2 t (cid:0) 1 2 1(cid:14)t 2 1 represents the nominal value of good 1 acquired by the generation-t debtors. The central bank then removes the (cid:20) 3 t (cid:0) 1 (cid:14)t 2(cid:0) 1 M units of money from circulation and (cid:14)t 3 (cid:20)t 2(cid:0) 1 the debtor has dt = Rx c r (cid:14)t 3 (cid:20)t 2(cid:0) 1 c units of good 2 for consumption. If 2 (cid:0) 2 (cid:0) (cid:20) 3 t (cid:0) 1 (cid:14)t 2(cid:0) 1 2 the young debtor does not have (cid:20) 3 t (cid:0) 1 (cid:14)t 2(cid:0) 1 M units of money and does not o⁄er (cid:14)t 3 (cid:20)t 2(cid:0) 1 i =r (cid:14)t 3 (cid:20) 2 t (cid:0) 1 c units of good 2 to the central bank, then that debtor is punished 2 (cid:20) 3 t (cid:0) 1 (cid:14) 2 t (cid:0) 1 2 by surrendering the amount of good 1 he acquired at stage 2. Notice that equations (2)-(3) are satis(cid:133)ed at equality by the mechanism but that (4) is not because i = r (cid:14)t 3 (cid:20) 2 t (cid:0) 1 c (cid:13) > 0 represents a real resource cost 2 (cid:20) 3 t (cid:0) 1 (cid:14) 2 t (cid:0) 1 2 (cid:21) borne by the private agents. I (cid:133)rst characterize the set of allocations that are implementable via the payment mechanism with pricing. Proposition 1. A steady-state allocation is implementable if it satis(cid:133)es the following participation constraints: v[y c ;Rx (1+r)c ] v[0;Rx] (6) 1 2 (cid:0) (cid:0) (cid:21) for debtors and u[c ;c ] u[y;0] (7) 1 2 (cid:21) for creditors. The proof of Proposition 1 is in the appendix. Proposition 1 gives two simple conditions that an allocation must meet for it to be implementable. They essen- 13

tially ensure that both debtors and creditors wish to participate in exchange. The next proposition characterizes the second-best optimal allocation via the payment mechanism with pricing and shows that it is Pareto-optimal. The optimal allocation is always second-best because the enforcement technology combined with the cost-recovery constraint reduces the amount of good 2 available to the agents. Proposition 2. The optimal allocation implementable via the payment mechanism with pricing satis(cid:133)es (5). Proof. Theoptimizationproblemcanbewrittenasmaximizing(1)withrespect to I;d ;d ;c ;c ;r and subject to (2)-(3), (6)-(7), and 1 2 1 2 RI+(x I) d +(1+r)c (8) 2 2 (cid:0) (cid:21) rc (cid:13) (9) 2 (cid:21) where (8) replaces (4) and (9) is the cost recovery constraint of the central bank. Giventhat(2)-(3)and(8)-(9)holdatequality,andsubstitutingtheserelationships into the optimization problem, the (cid:133)rst order conditions, which satisfy the Kuhn- Tucker conditions for necessity and su¢ ciency simplify to (1+(cid:21) )v (1+(cid:21) )u d 1 = c 1 (10) (1+(cid:21) )v (1+(cid:21) )u d 2 c 2 (cid:21) u[c ;c ] u[y;0] = 0 (11) c 1 2 f (cid:0) g (cid:21) v[y c ;Rx (cid:13) c ] v[0;Rx] = 0 (12) d 1 2 f (cid:0) (cid:0) (cid:0) (cid:0) g (cid:21) ;(cid:21) 0 (13) c d (cid:21) where (cid:21) and (cid:21) are the multipliers for the creditor and debtor participation conc d straints, respectively. Inspection of (10) reveals that (5) is satis(cid:133)ed regardless of 14

whether the participation constraints (6) and (7) bind or not. Whiletheintradayinterestratemayin(cid:135)uencewhetherornotdebtorsparticipate in trade (for trade to take place at all under this credit policy, it is important that r = (cid:13) is not too high that constraint 6 is violated), it does not create a wedge c2 betweentheratiosofmarginalratesofsubstitutionandsoisPareto-optimal. This isbecausedebtorsdonothavetopaytheinterestrateuntilstage4sothatthecost in terms of good 2 can be shared among debtors and creditors. One interpretation of r = (cid:13) > 0 is that it is the optimal risk-free intraday inc2 terestrate. Thisisbecause(i)apositiveinterestrateisnecessaryforcentralbank liquidity provision (because of the cost recovery constraint) and (ii) investment in the enforcement technology eliminates the risk that a debtor defaults. This is a departurefromthecasewherer =0(freeintradayliquidity),whichhasbeenfound to be optimal in papers such as Freeman (1996), Green (1997), Zhou (2000) Kahn and Roberds (2001) and Martin (2003). In each of those cases, it is implicitly assumed that (cid:13) =0 so that there was no social cost attached to providing intraday liquidity. The positive optimal risk-free interest rate found here supports a recommendation suggested by Rochet-Tirole (1996) that costly monitoring of agents is necessary and liquidity providers should be compensated. 5. LIQUIDITY PROVISION WITH COLLATERAL In this section, I provide an example of a payment mechanism where a central bank provides liquidity with a credit policy of requiring collateral. As in the previoussection,Icharacterizeasetofimplementableallocationsviathemechanism as those that satisfy a set of constraints. Finally, I show that the second-best optimal allocation implementable via the collateral policy is not Parteo optimal if 15

collateral bears an opportunity cost. The (cid:133)rst-best optimal allocation is achieved, however, if there is no opportunity cost to posting collateral. Theyoungdebtors,whentheyseekliquidityfromthecentralbank,pledgesome of their endowment of good 2 as collateral which they will then buy back from the central bank at the end of the period (during the fourth stage). Recall that young debtorscaninvesttheirendowmentofgood2andreceiveacertainreturnofR 1. (cid:21) Because the amount of good 2 they pledge is transferred to the central bank, there isanopportunitycostinthatthecollateralisnolongeravailabletoinvestwhenever R>1. Intermsofactuallarge-valuepaymentsystems,onecanthinkoftheopportunity cost of collateral in the following way.10 Suppose that participants of the system canpostonlyalimitedsetofassetsascollateral. Theseassetsaregenerallyviewed as safe from the point of view of the liquidity-provider. Typically, such safe assets have lower (expected) returns. To the extent that participants seeking intraday liquidityholdmoreofthesesafeassetsthantheyotherwisewouldwithouttheneed to post them as collateral, one could argue that there is an opportunity cost to pledging collateral. Thegameisasfollowsforanydatet. Atthe(cid:133)rststageofaperiod,generation-t debtors choose whether or not to seek liquidity from the central bank. Those that seek such credit acquire M units and deposit (cid:27) x units of good 2 at the central (cid:20) bank as collateral. Generation-t debtors then invest their remaining supply of good 2 (I =x (cid:27)). Let (cid:14)t [0;1] be the fraction of debtors who seek credit from (cid:0) 1 2 the central bank. At the second stage, the mechanism suggests that generation-t creditors who want to participate in exchange each o⁄er d of good 1 and that 1 10Zhou (2000) also makes this argument. 16

generation-t debtors who want to participate o⁄er M units of (cid:133)at money. The creditors and debtors simultaneously choose whether to participate in exchange or not. Let (cid:20)t [0;1] be the fraction of generation-t creditors who agree to o⁄er 2 2 d of good 1 and (cid:14)t [0;(cid:14)t] be the fraction of generation-t debtors who agree to 1 2 2 1 exchange M units of money for some consumption of good 1. Each debtor who agrees to trade M units of money for consumption receives (cid:20)t 2d units of good 1. (cid:14)t 1 2 Each creditor that agrees then receives (cid:14)t 2M units of money and has ct = y d (cid:20)t 2 1 (cid:0) 1 units of good 1 left for consumption. Those that disagree leave with autarky. At the third stage of date t, the generation-t debtors(cid:146)investments pay o⁄and eachnowhasR(x (cid:27))unitsofgood2availableatthisstage. Themechanismthen (cid:0) suggests that generation-t debtors who want to participate in exchange each o⁄er c R(x (cid:27)) of good 2 and that generation-t 1 creditors who have (cid:14)t 2(cid:0) 1 M units 2 (cid:20) (cid:0) (cid:0) (cid:20)t 2(cid:0) 1 of money and who want to participate in exchange o⁄er up (cid:14)t 2(cid:0) 1 M units of money. (cid:20)t 2(cid:0) 1 Generation-t 1 creditors with no money are not able to participate in exchange. (cid:0) The debtors and those creditors able to participate in exchange simultaneously choosetoparticipateornot. Let(cid:14)t [0;(cid:14)t]bethefractionofgeneration-tdebtors 3 2 2 who agree to o⁄er c of good 2 and (cid:20)t 1 [0;(cid:20)t 1] be the fraction of generation- 2 3(cid:0) 2 2(cid:0) t 1 creditors who agree to exchange money for some consumption of good 2. (cid:0) Each creditor who agrees to trade money for consumption receives (cid:14)t 3 c units of (cid:20)t 3(cid:0) 1 2 good 2. Each debtor that agrees then receives (cid:20)t 3(cid:0) 1 (cid:14)t 2(cid:0) 1 M units of money and has (cid:14)t 3 (cid:20)t 2(cid:0) 1 R(x (cid:27)) c units of good 2 left. Those that disagree leave with autarky. 2 (cid:0) (cid:0) Atthe(cid:133)nalstage,ifageneration-tdebtorwhohasborrowedM unitsatthe(cid:133)rst stage,nowhas (cid:20) 3 t (cid:0) 1 (cid:14) 2 t (cid:0) 1 M unitsofmoney,thenthedebtormaychoosetorepaythe (cid:14)t 3 (cid:20) 2 t (cid:0) 1 central bank (cid:20) 3 t (cid:0) 1 (cid:14) 2 t (cid:0) 1 M units of money in exchange for the return of the (cid:27) units (cid:14)t 3 (cid:20) 2 t (cid:0) 1 ofgood2thatservedascollateral. Thecentralbankthenremovesthe (cid:20)t 3(cid:0) 1 (cid:14)t 2(cid:0) 1 M (cid:14)t 3 (cid:20)t 2(cid:0) 1 17

units of money from circulation and the debtor has dt = R(x (cid:27)) c +(cid:27) units 2 (cid:0) (cid:0) 2 of good 2 for consumption. If the young debtor does not have (cid:20)t 3(cid:0) 1 (cid:14)t 2(cid:0) 1 M units of (cid:14)t 3 (cid:20)t 2(cid:0) 1 money, then the central bank does not return the collateral. Notice that (3) and (4) are satis(cid:133)ed at equality by the mechanism but that (2) is not when there is an opportunity cost of collateral (R >1). Rather I =x (cid:27). (cid:0) The opportunity cost of collateral is then (R 1)(cid:27) which is the di⁄erence between (cid:0) RxandR(x (cid:27))+(cid:27): Thereisalsoanadditionalfeasibilityconstraintthatrequires (cid:0) c R(x (cid:27)). This constraint re(cid:135)ects the fact that the amount of good 2 that 2 (cid:20) (cid:0) generation-t 1creditorscanconsumemustbelessthanthetotalamountavailable (cid:0) at the third stage. Inowcharacterizethesetofallocationsthatareimplementableviathepayment mechanism with collateral. Proposition 3. A steady-state allocation is implementable if it satis(cid:133)es the following incentive constraints: v[y c ;R(x (cid:27)) c +(cid:27))] v[0;Rx] (14) 1 2 (cid:0) (cid:0) (cid:0) (cid:21) v[y c ;R(x (cid:27)) c +(cid:27))] v[y c ;R(x (cid:27))] (15) 1 2 1 (cid:0) (cid:0) (cid:0) (cid:21) (cid:0) (cid:0) for debtors and u[c ;c ] u[y;0] (16) 1 2 (cid:21) for creditors. The proof of Proposition 3 is in the appendix. Compared with Proposition 1, Proposition 3 has an additional incentive constraint beyond participation. This constraint, (15), essentially requires that the amount of collateral that a debtor 18

buys back from the central bank must be at least as much as the amount of good 2 a creditor is expected to receive ((cid:27) c ). Otherwise, a debtor, after acquiring 2 (cid:21) some of good 1, would prefer not to exchange with old creditors to acquire money and so default on his debt to the central bank. The following proposition states that the payment mechanism under a credit policy with collateral cannot achieve Pareto-optimal allocations when there is an opportunity cost of collateral. Proposition 4. When R > 1, the optimal allocation implementable via the payment mechanism with collateral does not satisfy (5). Proof. Theoptimizationproblemcanbewrittenasmaximizing(1)withrespect to I;d ;d ;c ;c ;(cid:27) and subject to (3)-(4), (14)-(16), and 1 2 1 2 x (cid:27) I (17) (cid:0) (cid:21) RI+(x (cid:27) I) c : (18) 2 (cid:0) (cid:0) (cid:21) where (17) replaces (2) from the benchmark problem and (18) is an additional feasibilityconstraintforstage3. Giventhat(3),(4), and(17)willholdatequality, and substituting these relationships into the optimization problem, the (cid:133)rst order conditions, which satisfy the Kuhn-Tucker conditions for necessity and su¢ ciency 19

simplify to (1+(cid:21) )v (1+(cid:21) )u d 1 = c 1 (19) (1+(cid:21) +(cid:21) )v +(cid:21) (1+(cid:21) )u 2 d 2 1 c 2 (1+(cid:21) )v (R 1)+(cid:21) R = (cid:21) v (20) d 2 1 2 2 (cid:0) (cid:21) R(x (cid:27)) c = 0 (21) 1 2 f (cid:0) (cid:0) g (cid:21) v[d ;R(x (cid:27)) c +(cid:27))] v[d ;R(x (cid:27))] = 0 (22) 2 1 2 1 f (cid:0) (cid:0) (cid:0) (cid:0) g (cid:21) u[c ;c ] u[y;0] = 0 (23) c 1 2 f (cid:0) g (cid:21) v[y c ;R(x (cid:27)) c +(cid:27)] v[0;Rx] = 0 (24) d 1 2 f (cid:0) (cid:0) (cid:0) (cid:0) g (cid:21) ;(cid:21) ;(cid:21) ;(cid:21) 0 (25) 1 2 c d (cid:21) where (cid:21) is the multiplier for (18), (cid:21) is the multiplier for (15), and (cid:21) and (cid:21) are 1 2 c d the multipliers for the creditor and debtor participation constraints, (14) and (16), respectively. Condition (20) is the (cid:133)rst-order condition with respect to (cid:27). For (19) to equal (5), it must be the case that (cid:21) =(cid:21) =0 which is the case if 1 2 (15)and(17)donotbind. If(cid:21) =(cid:21) =0,then(20)reducesto(1+(cid:21) )v (R 1)=0 1 2 d 2 (cid:0) implying that v = 0, which violates the assumptions about debtor preferences.11 2 Therefore, a solution to this optimization problem cannot have both (cid:21) and (cid:21) be 1 2 equal to 0. The intuition for Proposition 4 is as follows. Because there is an opportunity cost to pledging collateral, a solution to the optimization problem should minimize the amount of collateral required. For such an allocation to be incentive feasible for debtors, (cid:27) c . Thus, an optimal allocation should have (cid:27) = c so that (15) 2 2 (cid:21) binds. But if (15) binds, then it turns out that debtors are credit constrained. 11v2=0ifand only ifd2= which is not feasible. 1 20

That is to say they cannot borrow "enough" from the central bank to acquire the desired amount of good 1 from young creditors. Thus, when collateral bears an opportunity cost, it serves as an endogenous credit constraint. This result is consistent with other papers on the use of collateral, such as Lacker (2001). Given that the constraint (15) binds, the central bank credit is fully collateralized. Toseethis,notethatp d =M,i.e.,thenominalvalueofdebtorconsumption 1 1 of good 1 equals the money supply. This is also true of the nominal value of creditor consumption of good 2, p c = M. Because (cid:27) = c , we have p d = p (cid:27), or 2 2 2 1 1 2 thenominalvalueoftheamountofgood1(cid:133)nancedbycreditfromthecentralbank equals the nominal value of the collateral. Finally, it is worth exploring the case when there is no opportunity cost of collateral. This may be the case in actual large-value payment systems when the central bank accepts a wide range of assets as collateral, mitigating the need to have an asset portfolio with a heavier than optimal weight on safe assets. Proposition 5. When R = 1, the optimal allocation implementable via the payment mechanism with collateral satis(cid:133)es (5) if the allocation has c < x. 2 2 21

Proof. When R=1, the (cid:133)rst order conditions from Proposition 4 simplify to (1+(cid:21) )v (1+(cid:21) )u d 1 = c 1 (26) (1+(cid:21) +(cid:21) )v +(cid:21) (1+(cid:21) )u 2 d 2 1 c 2 (cid:21) = (cid:21) v (27) 1 2 2 (cid:21) x (cid:27) c = 0 (28) 1 2 f (cid:0) (cid:0) g (cid:21) v[d ;x c ] v[d ;x (cid:27)] = 0 (29) 2 1 2 1 f (cid:0) (cid:0) (cid:0) g (cid:21) u[c ;c ] u[y;0] = 0 (30) c 1 2 f (cid:0) g (cid:21) v[y c ;x c ] v[0;x] = 0 (31) d 1 2 f (cid:0) (cid:0) (cid:0) g (cid:21) ;(cid:21) ;(cid:21) ;(cid:21) 0 (32) 1 2 c d (cid:21) As before, I need (cid:21) =(cid:21) =0 which is the case if (15) and (17) do not bind. Such 1 2 a condition does not violate (27) and is met when c <(cid:27) <x c or c < x. 2 (cid:0) 2 2 2 This gives su¢ cient conditions for which the debtor incentive constraint (15) does not bind. In this case, because there is no opportunity cost of collateral, the optimumdoesnotrequire(cid:27) tobesmall. Thusitispossibletochoosefromarange of (cid:27) that does not lead to any credit constraints. Notice that when there is no opportunity cost of collateral, the Pareto-optimal allocations are (cid:133)rst-best. This is because the use of collateral in this case does not add any additional social cost. Only a subset of such allocations, however, are achievable because of the need to satisfy the incentive constraint of debtors. 6. EXOGENOUS DEFAULT Up to this point in the analysis, the only type of default that is possible is strategic default. As a result, the central bank is assured of no equilibrium credit 22

losses under either policy because both e⁄ectively address the moral hazard issues associated with the repayment of debt. The banks that use central bank liquidity, however, are typically complex (cid:133)nancial institutions12. Although credit policies maybedesignedsothatapaymentsystemdoesnotprovideanincentivefordefault, theremaybeotherfactorsexogenoustothesystemthatcouldleadabanktofailto repayitsdebt,potentiallyleadingtocentralbanklosses. Forexample,abankcould become insolvent prior to repaying the central bank. In addition, the monitoring andenforcementtechnologythatthecentralbankemploysinthepricingpolicymay be less e⁄ective than has been assumed here. It may not be able to completely identify institutions that are more likely to become insolvent, and in the event of a liquidation, the central bank may face uncertainty about its claims. In this section, I extend the model to capture these concerns by introducing exogenous default. Speci(cid:133)cally, assume that between stages 3 and 4 within a period, a debtor receives a shock with probability " that he loses all of the money and goods that he has in his possession. With probability 1 ", the debtor enters (cid:0) stage4aspreviouslyassumed,witheverythinghehadattheendofstage3. Neither theagentsnorthecentralbankknowwhichdebtorswillreceivetheshockuntilafter it is realized. If the debtor receives the shock, he is unable to repay the central bank. Under the pricing policy, the central bank is unable to recover anything from an exogenously defaulting debtor. Under the collateral policy, the central bank keeps the collateral. IchoosetomodelexogenousdefaultthiswaybecauseIwishtofocusexclusively on the prospect of central bank losses. As noted earlier, the extension of central 12See Bliss (2003) for a discussion of the complexities of bankruptcy procedures for (cid:133)nancial institutions. 23

bankliquidityarisesinlarge-valuepaymentsystemsthatarereal-timegrosssettlementsystemswithpayment(cid:133)nality. Payment(cid:133)nalityisaguaranteebythecentral bankthatonceapaymentismadefromonepartytoanother,itcannotbeundone. In the context of this paper, this means that a debtor defaults only on the central bank at stage 4 and not on old creditors at stage 3. Thus, I model exogenous default so that the incentive constraints that must be satis(cid:133)ed for implementation undereitherpolicyremainsunchanged. Toseethisnotethatatstage4,withprobability 1 ", a debtor faces the same decision as to whether to repay the central (cid:0) bankasbefore,whilewithprobability"hereceivesnothingandsodefaults. Thus, in each preceding stage, a debtor(cid:146)s incentive constraint takes the following form (1 ")v[accepting proposed trade]+"v[0;0] (1 ")v[rejecting proposed trade]+"v[0;0] (cid:0) (cid:21) (cid:0) which reduces to the debtor incentive constraints in Propositions 1 and 3 and their proofs. Under the collateral policy, the central bank is protected fully from losses because, as shown in Section 5, borrowing is fullycollateralized.13 Underthe pricing policy,however,thecentralbankmustrecoveritscosts. Thepresenceofexogenous default changes the cost recovery constraint of the central bank, (9), to (1 ")rc (cid:13) (33) 2 (cid:0) (cid:21) 13This is a bit contrived because it is assumed that there is no real cost to the central bank of su⁄ering a default under the collateral policy. This may not be true in practice, for example, because a default could lead to an unanticipated increase in the money supply. The selling of collateralcanhelpundothisunanticipatedincreaseandsoreduceoreliminatethecostofdefault. 24

which, given that the constraint will bind, translates to an intraday interest rate (cid:13) r = : (34) (1 ")c 2 (cid:0) Exogenousdefault,therefore,addsariskpremiumtotherisk-freeintradayinterest rate for a given level of creditor consumption of good 2. Alternatively, should the centralbankwishtocontinuetochargetherisk-freerate,implementableallocations would haveahigherlevelof creditorconsumption ofgood 2and, by(5)would lead toalowerlevelofdebtorconsumptionofgood1. Inotherwords,exogenousdefault leads to "less" borrowing by debtors in terms of the amount of good 1 they can buy. Moreover, exogenous default tightens the debtor incentive constraint, (6). If the probability of default is high enough then the pricing policy could lead to a distortion that is more serious than that generated by the collateral policy when collateral has an opportunity cost; access to central bank liquidity would be shut downandtherewouldbenotrade. Whilethisresultmayseemextreme, thepoint to take from the analysis is that less e⁄ective monitoring and enforcement powers ofthecentralbankmayleadtoarationingofaccesstocentralbankliquidityunder thepricingpolicywhichmaybedistortionaryandperhapsevenmoredistortionary thanwhatisgeneratedbythecollateralpolicywhencollateralbearsanopportunity cost.14 14Ithank an annoymous referee forpointing this out. 25

7. CONCLUSION The above analysis sheds some light on why di⁄erent central banks may have di⁄erent credit policies for RTGS systems. Collateral is preferred if there is no opportunity cost of collateral, such as may be the case when a wide range of assets are accepted as collateral. This is because it can achieve (cid:133)rst-best allocations. If collateraldoeshaveanopportunitycost,comparisonoftherelativecost(intermsof good 2 in the model) is important. For example, the European Central Bank does not have monitoring authority over participating banks. Thus, it may be di¢ cult tocoordinatemonitoringandenforcementauthorities. Inthecontextofthemodel, this is a high enough (cid:13) so that collateral may be the preferred option. On the other hand, the Federal Reserve already has supervisory authority over depository institutions it serves over Fedwire, so that economies of scope are likely to yield a low (cid:13) so that pricing may be the preferred option. In the case where the cost of both policies would be the same ((cid:13) = (R 1)(cid:27)), the pricing policy would clearly (cid:0) be preferred due to the result that collateral adds a binding endogenous borrowing constraint that does not permit a Pareto-optimal allocation. Another credit policy tool that has not been modeled here is that of setting limits or caps to the amount a debtor can borrow. The Federal Reserve, for example, sets net debit caps that limit the amount that Fedwire participants can borrowtolimittheFed(cid:146)sexposuretocreditrisk. Inthecontextofthemodel,such binding constraints could have a similar e⁄ect on the pricing policy as what takes place under exogenous default when the central bank held the intraday interest rate (cid:133)xed at the risk-free rate. That is, it would reduce a debtor(cid:146)s consumption of good 1, but Pareto-optimal allocations may still be achievable. This would simply reducethesetofPareto-optimaloutcomesthatwouldbeachievableattheexpense 26

of debtors and for the gain of creditors. The results of the paper suggest that the existence of an opportunity cost of collateraliskeytothattypeofcreditpolicyleadingtoine¢ cientallocations. Thus, it is important to empirically understand whether or not there is an e⁄ective opportunity cost of intraday collateral. Collateral in this model is riskless to the central bank. An extension might involve the introduction of a range of collateral that varies according to riskiness. This complicates matters in that the central bank may have to decide what types ofriskyassetsareacceptableascollateral. Theconjecturehereisthatasacentral bank accepts a wider range of assets, the opportunity cost to the participant of posting collateral is less, but collateral provides less protection to the central bank intheeventofdefaultsunlessthevalueofthecollateralisdiscountedappropriately. Finally, this paper restricts itself only to two credit policies designed to replicate actual central bank policies. A more generalized study may reveal that a third policy may be more appropriate especially when some of the aforementioned complications are present in the model. APPENDIX A: PROOFS Proof of Proposition 1. The proof solves for subgame perfect equilibria of the game via backwards induction. The equilibria are those where every agent agrees at every stage. Begin with stage 4 within a period at date t. Generation-t debtors who have agreeduptothisstagehave (cid:20) 3 t (cid:0) 1 (cid:14) 2 t (cid:0) 1 M unitsofmoney. Theywillchoosetoreturn (cid:14)t 3 (cid:20) 2 t (cid:0) 1 27

the money and pay i units of good 2 if 2 (cid:20)t (cid:14)t (cid:20)t 1 v[ 2d ;Rx c r 3 2(cid:0) c ] v[0;Rx c ]: (35) (cid:14)t 1 (cid:0) 2 (cid:0) (cid:20)t 1 (cid:14)t 1 2 (cid:21) (cid:0) 2 2 3(cid:0) 2(cid:0) Note that the right-hand side of (35) represents utility after the central bank con- (cid:133)scates the debtor(cid:146)s amount of good 1 he previously acquired. Now turn to stage 3. A creditor from generation t 1 enters this stage with (cid:0) either (cid:14) 2 t (cid:0) 1 M or 0 units of money which is private information. Suppose that all (cid:20) 2 t (cid:0) 1 otheragentsagreeinthethirdstage. Ifthecreditordoesnothaveanymoneythen she cannot trade. If she has (cid:14) 2 t (cid:0) 1 M units of money then it is trivial that she will (cid:20) 2 t (cid:0) 1 want to agree to trade as well because (cid:14)t u[y d ; 3 c ] u[y d ;0]: (36) (cid:0) 1 (cid:20)t 1 2 (cid:21) (cid:0) 1 3(cid:0) Thus, (cid:20)t 1 =(cid:20)t 1. 3(cid:0) 2(cid:0) A generation-t debtor enters the third stage with either (cid:20)t 2d or 0 units of good (cid:14)t 1 2 1, which is private information. Suppose that allotheragents that can participate in trade will agree in the third stage. If the debtor has (cid:20)t 2d units of good 1, he (cid:14)t 1 2 will also agree if (cid:20)t (cid:14)t (cid:20)t 1 max v[ 2d ;Rx c r 3 2(cid:0) c ];v[0;Rx c ] v[0;Rx]: (37) f (cid:14)t 1 (cid:0) 2 (cid:0) (cid:20)t 1 (cid:14)t 1 2 (cid:0) 2 g(cid:21) 2 3(cid:0) 2(cid:0) The left hand side of the expression represents a debtor(cid:146)s stage 4 decision. The right hand side takes into account the fact that if a debtor disagrees, he will not receive money and will then be punished by losing (cid:20)t 2d at stage 4. Because (cid:14)t 1 2 28

v[0;Rx c ] <v[0;Rx] (37) reduces to 2 (cid:0) g (cid:20)t (cid:14)t (cid:20)t 1 v[ 2d ;Rx c r 3 2(cid:0) c ] v[0;Rx] (38) (cid:14)t 1 (cid:0) 2 (cid:0) (cid:20)t 1 (cid:14)t 1 2 (cid:21) 2 3(cid:0) 2(cid:0) and(35)issatis(cid:133)edif(38)issatis(cid:133)ed. Ifthedebtorhasnoneofgood1,itistrivial that he chooses not to agree to trade. Nowconsideranarbitrarygeneration-tdebtoratstage2. Ifthedebtordisagrees at this stage, he enters the third stage with 0 units of good 1 and will disagree in the third stage as well. Thus, he will receive only autarkic utility, v[0;Rx]. If the debtor agrees when everyone else does, then his second-stage participation constraintisidenticaltohisthird-stageparticipationconstraint,(38). Thisimplies that all of the generation-t debtors who agree at stage 2 will also agree at stage 3, that is, (cid:14)t =(cid:14)t. 3 2 Now, consider an arbitrary generation-t 1 creditor at the second stage of date (cid:0) t 1. If the generation t 1 creditor disagrees at the second stage when young, (cid:0) (cid:0) sheentersthethirdstagewhenoldwithnomoneyand,therefore,receivesautarkic utility, u[y;0]. She also knows that (cid:14)t = (cid:14)t. If she agrees at the second stage of 3 2 datet 1, shewillenterthethirdstageofdatetwithmoneyandagreesothatshe (cid:0) receives u[y d ; (cid:14)t 2 c ]. She will agree if (cid:0) 1 (cid:20) 3 t (cid:0) 1 2 (cid:14)t u[y d ; 2 c ] u[y;0] (39) (cid:0) 1 (cid:20)t 1 2 (cid:21) 3(cid:0) where (cid:14)t is substituted for (cid:14)t. 2 3 Finally, consider generation-t debtors at stage 1 of date t. Here, if all other debtors agree to borrowing from the central bank, then (cid:14)t = (cid:14)t = (cid:14)t = 1 and an 1 2 3 29

arbitrary debtor also agrees if (cid:20)t 1 v[(cid:20)td ;Rx c r 2(cid:0) c ] v[0;Rx] (40) 2 1 (cid:0) 2 (cid:0) (cid:20)t 1 2 (cid:21) 3(cid:0) which turns out to be the debtor participation constraint for stage 2 and stage 3. As a result, (cid:20)t 1 = (cid:20)t = (cid:20)t 1 = 1 because (39) is now satis(cid:133)ed because (7) is 2(cid:0) 2 3(cid:0) satis(cid:133)ed by hypothesis. Thus, generation-t debtor constraints at stages 2 and 3 reduce to (6), the stage 1 generation-t constraint is then trivially satis(cid:133)ed so that (cid:14)t = 1, and all nontrivial creditor constraints reduce to (7), both of which are 4 satis(cid:133)ed. Proof of Proposition 3. The proof solves for subgame perfect equilibria of the game via backwards induction. The equilibria are those where every agent agrees at every stage. Begin with stage 4 within a period at date t. Generation-t debtors who have agreeduptothisstagehave (cid:20) 3 t (cid:0) 1 (cid:14) 2 t (cid:0) 1 M unitsofmoney. Theywillchoosetoreturn (cid:14)t 3 (cid:20) 2 t (cid:0) 1 the money in exchange for collateral if (cid:20)t (cid:20)t v[ 2d ;R(x (cid:27)) c +(cid:27)] v[ 2d ;R(x (cid:27)) c ] (41) (cid:14)t 1 (cid:0) (cid:0) 2 (cid:21) (cid:14)t 1 (cid:0) (cid:0) 2 2 2 which trivially holds. Now turn to stage 3. A creditor from generation t 1 enters this stage with (cid:0) either (cid:14) 2 t (cid:0) 1 M or 0 units of money which is private information. Suppose that all (cid:20) 2 t (cid:0) 1 otheragentsagreeinthethirdstage. Ifthecreditordoesnothaveanymoneythen she cannot trade. If she has (cid:14) 2 t (cid:0) 1 M units of money then it is trivial that she will (cid:20) 2 t (cid:0) 1 30

want to agree to trade as well because (cid:14)t u[y d ; 3 c ] u[y d ;0]: (42) (cid:0) 1 (cid:20)t 1 2 (cid:21) (cid:0) 1 3(cid:0) Thus, (cid:20)t 1 =(cid:20)t 1. 3(cid:0) 2(cid:0) A generation-t debtor enters the third stage with either (cid:20)t 2d or 0 units of good (cid:14)t 1 2 1, which is private information. Suppose that allotheragents that can participate in trade will agree in the third stage. If the debtor has (cid:20)t 2d units of good 1, he (cid:14)t 1 2 will also agree if (cid:20)t (cid:20)t v[ 2d ;R(x (cid:27)) c +(cid:27)] v[ 2d ;R(x (cid:27))]: (43) (cid:14)t 1 (cid:0) (cid:0) 2 (cid:21) (cid:14)t 1 (cid:0) 2 2 The right hand side of the expression takes into account the fact that if a debtor disagrees,hewillnotreceivemoneyandwillthennotbeabletoreclaimhiscollateral at stage 4. If the debtor has none of good 1, it is trivial that he chooses not to agree to trade. Nowconsideranarbitrarygeneration-tdebtoratstage2whohasborrowedfrom the central bank. If the debtor disagrees at this stage, he enters the third stage with 0 units of good 1 and will disagree in the third stage as well. Thus, he will receive only autarkic utility, v[0;R(x (cid:27))]. If the debtor agrees when everyone (cid:0) else does, then his second-stage participation constraint is (cid:20)t (cid:20)t max v[ 2d ;R(x (cid:27)) c +(cid:27)];v[ 2d ;R(x (cid:27))] v[0;R(x (cid:27))] (44) f (cid:14)t 1 (cid:0) (cid:0) 2 (cid:14)t 1 (cid:0) g(cid:21) (cid:0) 2 2 which is trivially satis(cid:133)ed. Thus, (cid:14)t = (cid:14)t. Those that have not borrowed from 2 1 the central bank will not be able to agree to trade. 31

Now, consider an arbitrary generation-t 1 creditor at the second stage of date (cid:0) t 1. If the generation t 1 creditor disagrees at the second stage when young, (cid:0) (cid:0) sheentersthethirdstagewhenoldwithnomoneyand,therefore,receivesautarkic utility, u[y;0]. If she agrees at the second stage of date t 1, she will enter the (cid:0) third stage of date t with money and agree so that she receives u[y d ; (cid:14)t 2 c ]. (cid:0) 1 (cid:20)t 3(cid:0) 1 2 She will agree if (cid:14)t u[y d ; 3 c ] u[y;0] (45) (cid:0) 1 (cid:20)t 1 2 (cid:21) 2(cid:0) where (cid:20)t 1 is substituted for (cid:20)t 1. 2(cid:0) 3(cid:0) Finally, consider generation-t debtors at stage 1 of date t. Here, if all other debtorsagreetoborrowingfromthecentralbank,then(cid:14)t =(cid:14)t =1andanarbitrary 1 2 debtor also agrees if max v[(cid:20)td ;R(x (cid:27)) c +(cid:27)];v[(cid:20)td ;R(x (cid:27))] v[0;Rx]: (46) f 2 1 (cid:0) (cid:0) 2 2 1 (cid:0) g(cid:21) Now it remains to be shown that (cid:20)t 1 = (cid:20)t = (cid:14)t = 1 is supported in an 2(cid:0) 2 3 equilibrium. If (cid:20)t 1 = (cid:20)t = 1, then (46) trivially holds given that (14) holds by 2(cid:0) 2 hypothesis. Thus, (cid:14)t = (cid:14)t = 1 and (43) reduces to (15) which is satis(cid:133)ed and 1 2 (cid:14)t =1. If (cid:14)t =(cid:14)t =(cid:14)t =1, then it is obvious that (cid:20)t 1 =(cid:20)t =1 because (45) is 3 1 2 3 2(cid:0) 2 satis(cid:133)ed given that (16) is. REFERENCES [1] Angelini, P. (1998), "An Analysis of Competitive Externalities in Gross Settlement Systems," Journal of Banking & Finance, 22 (1), 1-18. [2] Bech, M. and Garratt, R. (2003), "The Intraday Liquidity Management Game," Journal of Economic Theory, 109 (2), 198-219. 32

[3] Bliss,R.(2003),"BankruptcyLawandLargeComplexFinancialObligations," Federal Reserve Bank of Chicago Economic Perspectives, 27 (1), 48-58. [4] Freeman, S. (1996), (cid:148)The Payments System, Liquidity, and Rediscounting,(cid:148) The American Economic Review, 86 (5), 1126-1138. [5] Freeman, S. (1999), (cid:148)Rediscounting Under Aggregate Risk,(cid:148)Journal of Monetary Economics, 43 (1), 197-216. [6] Green, E. (1997), (cid:148)Money and Debt in the Structure of Payments,(cid:148)Bank of Japan Monetary and Economic Studies, 215, 63-87. Reprinted in Federal Reserve Bank of Minneapolis Quarterly Review, Spring 1999, 23 (2), 13-29. [7] Kahn,C.andRoberds,W.(2001),"Real-timeGrossSettlementandtheCosts of Immediacy," Journal of Monetary Economics, 47 (2), 299-319. [8] Kobayakawa, S. (1997), "The Comparative Analysis of Settlement Systems," Center for Economic Policy Research Discussion Paper No. 1667. [9] Kocherlakota, N. (1998), "Money is Memory," Journal of Economic Theory, 81 (1), 232-251. [10] Lacker, J. (2001), "Collateralized Debt as the Optimal Contract," Review of Economic Dynamics, 4 (4), 842-859. [11] Martin, A. (2003), Optimal Pricing of Intraday Liquidity," Journal of Monetary Economics, forthcoming. [12] Mills,D.(2004),"MechanismDesignandtheRoleofEnforcementinFreeman(cid:146)s Model of Payments," Review of Economic Dynamics, 7 (1) 219-236. [13] Rochet, J. and Tirole, J. (1996), "Controlling Risk in Payment Systems," Journal of Money, Credit, and Banking, 28 (4), 832-862. 33

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FIG. 1 Sequence of Events in a Period Stage 1 Young Money Central Debtors Bank Stage2 Good 1 Young Young Debtors Creditors Money Stage3 (Investment Realized) Good 2 Young Old Debtors Creditors Money Stage 4 Money Young Central Debtors Bank 35