LIQUIDITY_ INTERBANK MARKET_AND THE SUPERVISORY ROLE OF THE CENTRAL BANK

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					Macroeconomic Dynamics, 7, 2003, 192–211. Printed in the United States of America. DOI: 10.1017.S136510050101015X

LIQUIDITY, INTERBANK MARKET, AND THE SUPERVISORY ROLE OF THE CENTRAL BANK
YOUNG SIK KIM
Kyung Hee University

This paper provides an explanation for the supervisory role of the central bank in a monetary general equilibrium model of bank liquidity provision. Under incomplete information on the individual banks’ liquidity needs, individual banks find it optimal to invest solely in bank loans holding no cash reserves, and rely on the interbank market for their withdrawal demands. Using the costly state verification approach under uncertainty in aggregate liquidity demands, the supervisory role of the central bank as a large intermediary arises as an incentive-compatible arrangement by which banks hold the correct level of cash reserves. First, it takes up a delegated monitoring role for the banking system. Second, it engages in discount-window lending at a penalty rate, where the discount margin covers exactly the monitoring cost incurred. Finally, under the central banking mechanism, currency premium no longer exists in the sense that currency is worth the same as deposits having an equal face value. Keywords: Liquidity, Interbank Market, Central Bank

1. INTRODUCTION Why do banks need a central bank? This paper provides an explanation for the supervisory role of a central bank in the context of a general equilibrium model with fiat money where banks underinvest in the cash reserves. Given a positive nominal interest rate and incomplete information on individual banks’ liquidity needs, banks invest solely in illiquid assets—bank loans—which yield higher returns than cash reserves, and rely on the interbank market for their liquidity needs. Thus, there is a market failure in which no liquidity is available in the aggregate at all. The central bank improves upon the interbank coordination problem by not only taking up the delegated monitoring role that induces individual banks to reveal truthfully their liquidity needs, but also by providing discount-window loans at a penalty rate.

This paper is dedicated to the memory of Tae Won Kim, my mentor and my father. I would like to thank Stephen Burnell, Steve Williamson, Hyun Park, and seminar participants at Kyung Hee University and a money-credit session at the 1998 North American Econometric Society Summer Meeting in Montreal for their valuable comments and discussions. I also wish to thank a referee and the Associate Editor whose comments and suggestions have improved the paper substantially. Any errors are my own. Address correspondence to: Young Sik Kim, School of Economics and International Trade, Kyung Hee University, 1 Hoegi-dong, Dongdaemun-ku, Seoul 130-701, Korea. c 2003 Cambridge University Press

1365-1005/03 $12.00

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Bhattacharya and Gale (1987) considered a Diamond–Dybvig (1983) framework to characterize the role of a central bank for risk sharing across banks.1 They argued that under privately observed liquidity shocks, setting up a Walrasian interbank market creates a free-rider problem in the sense that individual banks will rely on the ex post market to provide them with liquidity and will underinvest in the liquid asset. The solution to the second-best risk-sharing problem requires the role of a central bank as a nonmarket entity that implements borrowing/lending through a discount window and imposes reserve requirements.2 Chari (1989) also constructed a Diamond–Dybvig type model with uncertainty about local liquidity needs [referred to as “community risk” by Chari (1989)] to capture two key features of the U.S. National Banking System: the limits on bank branching and the ability of banks to circumvent reserve requirements. He showed that an appropriate choice of reserve requirements and a well-functioning interbank market can generate efficient outcomes. It was further argued that an effective central bank is necessary to delegate the task of changing the proper level of reserves over time and engaging in discount-window policy. However, as Bhattacharya and Gale (1987) pointed out, without an explicit incorporation of fiat money into models of banking, it is clearly difficult to provide detailed monetary interpretations of results on underinvestment in liquid assets by banks at a positive nominal interest rate. Further, both Bhattacharya and Gale (1987) and Chari (1989) considered only idiosyncratic shocks to liquidity demands at individual bank levels, abstracting from uncertainty in aggregate liquidity demands. This paper reconsiders the ideas of Bhattacharya and Gale (1987) and Chari (1989) by introducing fiat money and aggregate uncertainty. Champ et al. (1996) recently constructed a general equilibrium model of bank liquidity provision in which fiat currency and the provision of liquidity by banks are closely related. Their model is extended here by introducing the interbank market where individual banks’ liquidity needs are private information in the presence of uncertainty in aggregate liquidity demands. More specifically, as in Wallace (1988), aggregate liquidity demands are not known with certainty in a given period.3 Agents are initially assigned to a location and face relocation risk. Limited communication prevents claims on specific agents from being traded across locations and only paper liabilities (e.g., fiat currency) can be transported between locations. This generates a transactions role for currency in the sense that in the event of relocation an agent needs liquidity through banks in the form of currency.4 Banks can be regarded as being distinguished by geographical location and depositors attach themselves to particular banks by locational proximity. Then, local economic conditions in the area where the bank operates will have a marked impact on the demand for liquidity. For example, when unemployment is high, agents are forced to move to another location and the need for liquidity becomes high. Under the “inelastic currency” regime where banks are prohibited from issuing notes, the nominal interest rate is positive and there is a positive probability of bank panics in which the banking system will have to suspend cash payments

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following the exhaustion of its cash reserves. The possibility of bank panics also implies that agents in need of liquidity have to suffer relative to those not in need. The imperfectly correlated liquidity shocks across banks imply an incentive to form an interbank market by which they can collectively provide extra liquidity by borrowing and lending after their liquidity needs are known. However, as in Bhattacharya and Gale (1987), there arises a moral-hazard problem due to the incomplete information on the proportion of depositors wishing to make early withdrawals. In an equilibrium with a positive amount of bank loans at a positive nominal interest rate, individual banks find it optimal to invest all the deposits in illiquid assets—bank loans—which yield higher returns than cash reserves, and rely on the interbank market for their liquidity needs. That is, the interbank rates do not create the incentives for individual banks to hold a correct amount of cash reserves. Given the market liquidity limited by the individual banks’ reserve holdings, no agents would write deposit contracts with banks unless a mechanism is designed by which banks have no incentive to underinvest in the cash reserves. In the presence of uncertainty in aggregate liquidity demands, Townsend’s (1979) costly state verification approach yields the incentive-compatible, secondbest optimal arrangement that resembles the unique supervisory roles of a central bank in the banking system. The central bank, as a large intermediary, improves upon the interbank coordination problem by taking up the delegated monitoring role which not only induces individual banks to reveal truthfully their liquidity needs, but also eliminates the needs for monitoring by its depositors (i.e., lending banks). Further, it engages in discount-window loans to illiquid banks at penalty rate. Under the central banking mechanism, agents in need of liquidity (or currency) do not suffer relative to the others and hence no currency premium exists. Finally, the central bank’s zero-profit condition implies that the penalty rate or discount margin is equal to the monitoring cost incurred by the central bank. The higher discount margin implies the higher optimal reserve/deposit ratio and hence the smaller probability that the representative bank will exhaust cash reserves. The paper is organized as follows. Section 2 describes the basic model of bank liquidity provision with fiat currency and aggregate uncertainty. In Section 3, an interbank market is introduced to examine the potential interbank coordination problem due to the incomplete information on individual banks’ liquidity needs. It follows how an institution such as a central bank can arise to mitigate the coordination problem. Finally, Section 4 summarizes the paper with a few remarks. 2. A MODEL OF BANK LIQUIDITY PROVISION Champ et al. (1996) develop an overlapping generations model in which banks exist to insure agents against random needs for liquidity and to intermediate between lenders and borrowers. Time is discrete and indexed by t = 1, 2, . . . . In each period, a continuum of two-period-lived agents with unit mass is born in a location. Half of these agents are “lenders” and the remaining half are “borrowers.” All agents born at t have preferences given by

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E[u(ct , ct+1 )] = E(ln ct + β ln ct+1 ), where E is the expectation operator, ct > 0 is consumption in period t, and β ∈ (0, 1) is a discount factor. Noting that the period-t state of the economy is not public information at t, agents maximize expected utility conditioned on information at t − 1 (instead of t). However, the i.i.d. feature of the aggregate state implies the equivalence between conditional and unconditional expectations. Each lender born at t has endowment x > 0 when young and no endowment when old. A borrower has no endowment when young and y > 0 when old. Assume that βx < y. At t = 1 there is a continuum of old agents with unit mass in each location. These agents are each endowed with M > 0 units of fiat currency and there are no subsequent injections or withdrawals of currency. There are a finite number of infinitely lived banks in each location. The following summarizes the sequence of events within a period in the model economy:
(1) At the begining of period t, agents in each location receive their endowment and young lenders can deposit resources with a bank and trade with old agents. (2) Young lenders consume and then cannot contact other agents until they learn whether or not they are to be relocated. (In particular, young lenders and young borrowers never meet.) (3) Young borrowers contact banks to take loans and then young lenders learn whether they are to be relocated. In period t, a random fraction πt of young lenders (“movers”) must relocate where πt ∈ [πtl , πtu ], 0 ≤ πtl < πtu ≤ 1. (4) Movers can contact their banks and withdraw their deposits plus any promised interest. Since all period-t resources have been consumed, these agents receive claims to future consumption in the form of fiat currency. (5) At t + 1, these liabilities can be used to purchase goods when old agents contact young lenders and/or a bank in their new location.

As in Townsend (1987), limited communication prevents the quality of checks written by movers from the other location, or of claims on borrowers in the other location, from being verified. However, fiat currency can be universally verified and hence lenders demand currency in the event of relocation. For any given period t, the relocation probabilities are uniformly distributed over the interval [πtl , πtu ] where θt ≡ (πtl , πtu ) is random, so that uncertainty enters the model in an aggregate sense as well as in an idiosyncratic (or location-specific) sense. It is assumed that θt is i.i.d. over time and its realization is not public information. The feature of unobservable aggregate state of the economy implies that the aggregate liquidity demands in a given period are not known with certainty. For a given θt ≡ (πtl , πtu ), let the distribution function of the random variable πt ∈ [πtl , πtu ] be denoted as F(πt ) and its continuously differentiable density function as f (πt ). After agents receive endowments, banks announce payoff schedules that specify gross real returns to depositors contingent on the realization of idiosyncratic relocation risk πt ∈ [πtl , πtu ]. Let rtm (πt ) and rt (πt ) denote, respectively, the oneperiod return per unit deposited for movers and nonmovers contingent on πt . Each

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bank simply accepts all deposits offered and makes loans charging the competitively determined gross real loan rate Rt . Note that Rt is independent of πt . This comes from the model environment where borrowers are physically separated from lenders when πt is realized. Borrowers therefore do not observe πt , making it natural not to condition repayments on it.5 2.1. The Agent’s Problem Having observed announced repayment schedules at t, each lender chooses a deposit dt and a bank where dt is chosen to maximize the lender’s expected utility: E ln(x − dt ) + β +β
πtu πtl πtu πtl

π ln rtm (π ) dt f (π ) dπ

(1 − π) ln[rt (π ) dt ] f (π ) dπ . βx , 1+β

The solution is dt =

and each lender chooses the bank whose return schedules maximize her expected utility. Borrowers choose a loan quantity lt to maximize expected utility, taking Rt as given6 : E[ln lt + β ln(y − Rt lt )]. The solution to this problem sets lt = y . (1 + β)Rt

2.2. The Bank’s Problem Against dt , deposits per depositor, a bank holds per-depositor cash reserves with a real value of z t , and makes loans (per depositor) with a real value of dt − z t . Let γt ≡ z t /dt be a bank’s reserve/deposit ratio. Reserves earn the one-period real gross return pt+1 / pt , where pt is the period-t inverse price level. Banks take Rt and pt+1 / pt as given. After γt is chosen and loans are made, πt ∈ [πtl , πtu ] is realized for a given state θt ≡ (πtl , πtu ). Then the bank faces real per-depositor withdrawal demand equal to dt πt rtm (πt ) pt , pt+1

which is paid by the bank in the form of currency. The presence of the term pt / pt+1 reflects the fact that the bank gives currency to movers at t, who will take the currency to their new location to make purchases at t + 1. These earn the real

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rate of return pt+1 / pt between t and t + 1. Thus, a payment of rtm (πt ) pt / pt+1 to a mover at t results in a perceived return to the mover of rtm (πt ). Note that since borrowers have no resources and have already consumed at the end of t, there is no ability to liquidate loans.7 Let αt (πt ) denote the fraction of cash reserves that the bank pays out to movers at t (as a function of πt ). Then, payments to movers at t must satisfy the constraint pt+1 πt rtm (πt ) ≤ αt (πt )γt (1) pt and payments to nonmovers at t + 1 must satisfy (1 − πt )rt (πt ) ≤ γt [1 − αt (πt )] pt+1 + (1 − γt )Rt . pt (2)

In equilibrium, rtm (πt ), rt (πt ), αt (πt ), and γt must be chosen to maximize the expected utility of depositors, taking their savings behavior as given: E ln x 1+β
πtu πtl

+β

πtu πtl

π ln rtm (π )

βx f (π ) dπ 1+β

+β

(1 − π) ln rt (π )

βx f (π ) dπ , 1+β

subject to (1) and (2). Competition for deposits will force banks to earn zero profits in equilibrium, and hence (1) and (2) hold with equality in equilibrium. Further, the reserve payout contingent on πt can be expressed as αt (πt ) ≤ πt rtm (πt ) pt = πt 1 + γt pt+1 1 − γt γt Rt pt , pt+1 (3)

where the equality holds if αt (πt ) < 1 in which rtm (πt ) = rt (πt ) and, hence, from (1) and (2), pt+1 rtm (πt ) = γt + (1 − γt )Rt . pt Now the optimal reserve/deposit ratio can be expressed as8 γt = 1 −
πtu πt∗

F(π ) dπ,

(4)

where πt∗ is defined to be the value of πt that satisfies (3) as an equality with αt (πt∗ ) = 1; that is, πt∗ = 1 + 1 − γt γt Rt pt pt+1
−1

≡ g(γt , It )

(5)

and It ≡ Rt pt / pt+1 denotes the gross nominal interest rate. When πt ≥ πt∗ holds, a “banking panic” occurs in the sense that the banking system exhausts its cash

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reserves.9 Moreover, in a panic, rtm (πt ) ≤ rt (πt )—i.e., those agents needing liquidity (movers) suffer relative to nonmovers.10 Noting that the ratio of returns, rt (πt )/rtm (πt ), can be interpreted as the currency premium, the bank panic (which occurs when πt ≥ πt∗ ) implies a positive currency premium in the sense that rt (πt )/rtm (πt ) ≥ 1.11 Given πt∗ as defined in (5), the bank’s optimal reserveliquidation strategy, (3), can be written as αt (πt ) = min πt ,1 . πt∗ (6)

The optimal choice of the reserve/deposit ratio can also be characterized by defining the function H : [0, 1] → [0, 1] as H (x) =
x πtu

F(π ) dπ.

Then (4) can be written as 1 − γt = H [g(γt , It )]. (7)

Equation (7) is depicted graphically in Figure 1. It can be shown that H [g(γ , I )] is a decreasing function of γ , and is concave in γ for I ≥ 1. In addition, H [g(0, I )] = H (0) < 1 and H [g(1, I )] = H (1) = 0, for all I . The slope of H [g(γ , I )] exceeds one in absolute value at γ = 1 for all I > 1, and is equal to one in absolute value if I = 1. It follows that, if It = 1, (7) is satisfied only by γt = 1.

FIGURE 1. Optimal reserve/deposit ratio.

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If It > 1 (the nominal interest rate is positive), then (7) has two solutions. It can be verified that the interior solution solves the bank’s optimization problem. From (5) and the definition of H (·), an increase in I shifts H [g(γ , I )] upward, as depicted in Figure 1, where 1 < I1 < I2 . In summary, the optimal reserve/deposit ratio is given by γt = γt (It ), where γt (1) = 1 and γt < 0. The fractional reserves (γt < 1) therefore imply a positive nominal interest rate and a positive probability of bank panics (πt∗ < 1) from (5). 2.3. General Equilibrium Note first that, in equilibrium, loans must be made in positive quantities. Then, since γt (1) = 1 and γt < 0, it follows that It > 1 must hold, or Rt > pt+1 pt (8)

for all t ≥ 1; that is, in equilibrium nominal interest rates are always positive. Moreover, the net per-capita savings of each young generation must equal the per-capita supply of real balances in each period: βx y = pt M. − 1+β (1 + β)Rt Finally, all real balances are held as cash reserves by banks in equilibrium: γt dt ≡ γt βx = pt M. 1+β (10) (9)

Notice that pt M > 0 (a positive stock of valued fiat currency) in equation (9) implies y Rt > > 1. βx Hence, as long as y is sufficiently large relative to x for given β ∈ (0, 1), Rt > pt+1 / pt , and therefore nominal interest rate is positive in equilibrium. 3. INTERBANK MARKET UNDER INCOMPLETE INFORMATION Now suppose that banks can participate in the Walrasian interbank market where they can collectively provide extra liquidity by borrowing and lending reserves after their liquidity needs are known. In a given period t where θt ≡ (πtl , πtu ), banks across locations are assumed to be uniformly distributed in their liquidity needs over the interval [πtl , πtu ]. For a given state of the world θt ≡ (πtl , πtu ), banks are assumed to be identical ex ante, discovering ex post their πt ’s. More importantly, to capture the information asymmetries in the banking system, an individual bank’s πt is assumed to be private information.

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Let bt (πt ) denote the amount of borrowing (per depositor) in the interbank market by an “illiquid” bank and rtc be the interbank-market interest rate per unit amount of borrowing or lending. To guarantee the nontrivial presence of the interbank market, I consider a situation in which rtc ≥ pt+1 / pt . Further, noting that borrowers rely on bank loans for their consumption in youth, I focus on an equilibrium in which Rt ≥ rtc , and hence bank loans are made in positive quantities. 3.1. The Interbank Coordination Problem Now, an individual bank’s payments to movers and nonmovers must satisfy, respectively, the following feasibility conditions: πt rtm (πt ) ≤ αt (πt )γt pt+1 bt (πt ) pt+1 + , pt pt dt bt (πt ) c r , dt t (11) (12)

(1 − πt )rt (πt ) ≤ γt [1 − αt (πt )]rtc + (1 − γt )Rt −

where bt (πt ) = 0 if αt (πt ) < 1 and bt (πt ) > 0 if αt (πt ) = 1. In equilibrium, banks must earn zero profits and choose payment schedules along with reserve/deposit ratio and payout strategy to maximize the expected utility of depositors, taking their savings or deposit demand schedules as given, subject to (11) and (12). Note that banks take rtc as well as Rt and pt+1 / pt as given. PROPOSITION 1. In the presence of the interbank market with incomplete information on individual banks’ liquidity needs, γt = 0 is a strictly dominant strategy for an individual bank. Proof. Competition for deposits forces banks to earn zero profits in equilibrium, and hence the feasibility conditions (11) and (12) holding with equality imply the following payment schedules: rt (πt ) = rtm (πt )  γt αt (πt ) pt+1 + γt [1 − αt (πt )]r c + (1 − γt )Rt  t  pt =  pt+1 bt (πt ) c pt+1 γ  t rt − + (1 − γt )Rt − pt dt pt

if if

αt (πt ) < 1 , αt (πt ) = 1

where γt ∈ [0, 1] and bt (πt ) > 0 if αt (πt ) = 1 or γt = 0. With γt = 0, the above payment schedules become rt (πt ) = rtm (πt ) = Rt − bt (πt ) c pt+1 rt − . dt pt (13)

Now, the amount of interbank borrowing (per depositor) can be obtained after substituting (13) into (11):

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201

bt (πt ) = dt

πt 1 − πt

pt Rt pt+1 pt 1 − rtc pt+1 pt Rt pt+1 pt 1 − rtc pt+1

.

(14)

Substituting this back into (13) yields the following: πt rt (πt ) = rtm (πt ) = Rt − 1 − πt rtc − pt+1 . pt

(15)

With γt > 0, the payout of cash reserves contingent on πt can be expressed as αt (πt ) ≤ πt rtm (πt ) pt γt pt+1 pt + pt+1 1 − γt γt Rt pt pt+1

= πt αt (πt ) + [1 − αt (πt )]rtc

where equality obtains if αt (πt ) < 1, in which case bt (πt ) = 0. Further, this equality can be rewritten as pt pt 1 − γt Rt + πt rtc pt+1 γt pt+1 αt (πt ) = . pt c 1 − πt 1 − r t pt+1 Substituting this into the payment schedule with αt (πt ) < 1 yields the following: πt rt (πt ) = rtm (πt ) = (1−γt )Rt +γt rtc − pt pt+1 Rt − γt Rt − rtc pt pt+1 rtc − pt+1 . pt (16) Subtracting (16) from (15) yields   γt Rt − rtc 1 −  πt rtc 1 + πt  pt −1  pt+1  ≥ 0,  pt c rt −1 pt+1

1 − πt 1 − rtc

for all πt ∈ [πtl , πtu ] since Rt ≥ rtc and πt (rtc ppt − 1)/[1 + πt (rtc ppt − 1)] < 1. Fit+1 t+1 nally, if αt (πt ) = 1 with γt > 0, then bt (πt ) > 0 and (13) implies that in an equilibrium with Rt > pt+1 / pt (i.e., a positive nominal interest rate) the payment schedule with αt (πt ) = 1 and γt > 0 is lower than the one with γt = 0 for all πt ∈ [πtl , πtu ]. Therefore, from an individual bank’s point of view, γt = 0 is a strictly dominant strategy.

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In equilibrium with rtc ≤ Rt where borrowers have a positive consumption in their youth, prices do not create the correct incentives for an individual bank to hold a positive amount of cash reserves. That is, in the presence of the interbank market where the interbank rate is below the bank loan rate, individual banks with private information on their liquidity needs would adopt the following profitmaximizing strategy: invest all the deposits in bank loans that yield higher returns and obtain liquidity as required ex post from the interbank market. Then, given the market liquidity limited by the individual banks’ holdings of cash reserves, no agents would make deposits with banks unless there exists an incentive-compatible mechanism by which banks do not underinvest in the liquid asset or cash reserves. 3.2. Delegated Monitoring and the Central Bank An incentive-compatible mechanism under the information asymmetries in the banking system yields the “second-best” optimal arrangement. An individual bank with liquidity needs πt chooses its payment schedule, [rt (πt ), rtm (πt )], as well as its reserve/deposit ratio and payout strategy to maximize the expected utility of depositors subject to the feasibility constraints (11) and (12), and the following incentive-compatibility constraint: E[u(ct , ct+1 ; πt )] ≥ E[u(ct , ct+1 ; πt )] for πt = πt . (17)

Notice that, with complete information on individual πt ’s, the interbank lending would be made contingent on the individual bank’s liquidity needs. Hence, the strategy of zero cash reserves (γt = 0) and interbank borrowing as given by (14) would not be feasible for an individual bank; that is, the incentive constraint (17) would not be binding under complete information. With the incentive constraint (17) binding under incomplete information, I take the costly state verification approach, which has the effect of relaxing the incentive constraint at a cost. Suppose that an individual bank’s πt (or liquidity needs) can be observed by any other banks at a monitoring cost of φ units of the consumption good. Thus, there is costly state verification as in Townsend (1979). The interbank coordination problem as discussed above implies that, for deposits to be made, contracts must be written to provide for monitoring under some contingencies. They will be written in a way that gives an individual bank the incentive to truthfully report its liquidity needs while allowing the other banks to economize on monitoring costs. When attention is restricted to pure strategy contracts with nonstochastic monitoring, arguments similar to those of Williamson (1986) imply that an optimal arrangement is for all lending to be delegated to a large intermediary that borrows from and lends to many banks and monitors borrowing banks only if they report shortages of cash reserves. For instance, suppose that k banks, indexed by j = 1, 2, . . . , k, fund a particular bank’s liquidity shortages in period t. The bank can make either of two declarations: to meet its depositors’ withdrawals, it needs extra cash or it does not. At least one of these declarations must induce bank j to monitor, since otherwise the bank would

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always declare that it is short of liquidity (see Proposition 1). The bank j’s optimal strategy is to monitor only if the borrowing bank declares liquidity shortages. Given this monitoring strategy, the bank always reports the true outcome to the lending banks. Moreover, when each lending bank lends directly to a borrowing bank, the total expected monitoring cost is πt φk, where πt ≥ πt∗ . It is clear that there must be a way to improve on this arrangement, since in the case of a bank panic all the lending banks monitor and acquire the same information. Suppose, then, that the group of k lending banks delegates the task of monitoring to one group member. If this monitoring bank always reports the true outcome to the group, the group needs to incur the expected monitoring cost πt φ, which is less than or equal to πt φk for k ≥ 1. However, the moral-hazard problem crops up again because the monitoring bank may not have an incentive to report the truth. If the liquidity shortage is due to the underinvestment in cash reserves, the monitoring bank could declare the outcome was “bad”—i.e., πt ≥ πt∗ —and collude with the (monitored) bank to split its extra profits. However, an arrangement that optimally delegates the monitoring role does exist and it resembles the special supervisory roles associated with central banking. In this arrangement, the central bank is an individual monitoring bank that intermediates between many lending and borrowing banks. The central bank diversifies by lending to and borrowing from a large number of banks, which makes it possible to exploit the law of large numbers. Thus, for a given aggregate state of the model economy, a central bank in this model can perfectly predict the fractions of “liquid” and “illiquid” banks. The central bank can therefore commit to making fixed payments to the lending banks, without making the payments hinge on the liquidity outcomes of individual banks. Hence, diversification eliminates delegated monitoring costs since the lending banks need never monitor the central bank. Suppose that the central bank contracts to fund m banks, indexed by i = 1, 2, . . . , m. Without loss of generality, assume that the central bank writes identical contracts with each of the (ex ante) identical banks. Let btc (πi ) denote the amount of discount-window lending (per depositor), in the event that the central bank monitors bank i following its report of a high πi (so that it becomes illiquid), and let ωt be the “penalty rate” per unit borrowed from the central bank. Further, let rtc ∈ [ pt+1 / pt , Rt ] be the market return from loans in the interbank market. The total return to the central bank from the m loan contracts is
m m

= ωt
i=1

max

btc (πi ) ,0 . dt

By the strong law of large numbers, it follows that 1 m→∞ m plim = ωt
B

m

btc (πi ) f (πi ) dπi , dt

where B = {πi |btc (πi )/dt > 0}. Now, let N denote the number of “illiquid” or borrowing banks with the central bank (and hence get monitored). Then, N φ is the

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cost to the central bank of monitoring these banks. Since N is a binomial random variable with parameters (m, B f (πi ) dπi ), by the strong law of large numbers, plim
m→∞

Nφ =φ m

f (πi ) dπi .
B

Therefore, if the following weak inequality holds per unit amount of loan to illiquid banks,     ωt − φ  
B

f (πi ) dπi
B btc (πi )

dt

f (πi ) dπi

   ≥ rtc , 

then as the central bank grows large (m → ∞), it can guarantee the return of rtc to the lending banks without making it contingent on the liquidity needs of the individual borrowing banks. Given the contract [btc (πi ), ωt ], a finite-sized central bank must write contracts with its depositors (i.e., lending banks) that involve monitoring and, given the market return (rtc ), lending banks must be compensated for these monitoring costs by the central bank. This compensation lowers expected profit for the central bank, so that for given [btc (πi ), ωt ], expected profit is higher for a large (i.e., infinitesized) central bank, which lending banks need not monitor, than for a central bank of finite size. As in Diamond (1984) and Williamson (1986), the costs of delegated monitoring go to zero in the limit as the intermediary grows large.12 Now, it remains to determine the contract [btc (πi ), ωt ] for the large central bank. 3.3. The Equilibrium Contract Now, an individual bank chooses rtm (πt ), rt (πt ), αt (πt ), and γt to maximize depositors’ expected utility subject to the following feasibility conditions: πt rtm (πt ) ≤ αt (πt )γt pt+1 bc (πt ) pt+1 + t , pt pt dt btc (πt ) ωt , dt (18) (19)

(1 − πt )rt (πt ) ≤ γt [1 − αt (πt )]rtc + (1 − γt )Rt −

where btc (πt ) = 0 if αt (πt ) < 1, btc (πt ) > 0 if αt (πt ) = 1, and rtc ∈ [ pt+1 / pt , Rt ] can be interpreted as the interest rate on reserves held at the central bank. PROPOSITION 2. The optimal payment schedule for an individual bank is  γt αt (πt ) pt+1 + γt [1 − αt (πt )]r c + (1 − γt )Rt if αt (πt ) < 1  t  pt rt (πt ) = rtm (πt ) = bc (πt ) ˆ  pt+1 γt φt + (1 − γt )Rt − t if αt (πt ) = 1  pt dt ˆ if φ t ≡ ωt − pt+1 / pt > 0.

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205

Proof. Following the realization of liquidity shocks, a borrowing bank reports πt to the central bank. Define S ⊂ [πtl , πtu ] such that, if πt ∈ S, then monitoring occurs, whereas if πt ∈ S, then monitoring does not occur. At first, competition for deposits forces banks to earn zero profits in equilibrium and hence the feasibility conditions (18) and (19) hold with equality. The optimal payment schedule as given above is feasible in the sense that it is obtained from (18) and (19) when they hold with equality. Further, the optimal payment schedule is incentive compatible since rtc ≥ pt+1 / pt implies lower payment schedule at the illiquid banks than at the liquid banks: that is, γt αt (πt ) = γt > γt pt+1 + γt [1 − αt (πt )]rtc + (1 − γt )Rt pt

pt+1 pt+1 + (1 − γt )Rt + γt [1 − αt (πt )] rtc − pt pt bc (πt ) ˆ pt+1 + (1 − γt )Rt − t φt . pt dt

Therefore, πt ∈ S if and only if αt (πt ) = 1. Notice that, unlike the situation without the interbank market, rtm (πt ) = rt (πt ); that is, those agents needing liquidity (movers) need not suffer relative to nonmovers. However, in the event of borrowing from the discount window following ˆ the liquidity shock, the positive “discount margin” (φ t > 0) leads to a lower payment schedule to depositors at the illiquid banks than those at the liquid banks. This induces only the illiquid banks to report truthfully their liquidity shortages for monitoring purposes. As is verified below, the equilibrium discount margin amounts to the monitoring cost per unit borrowed from the central bank. Noting that it is the positive discount margin that induces “truth-telling” of individual banks, the costly verification of individual banks’ liquidity needs has the effect of replacing the incentive constraint (17) faced by an individual bank in choosing its payment schedule. As in the model economy with no access to the interbank market, the payout of cash reserves contingent on πt can be expressed as αt (πt ) ≤ πt rtm (πt ) pt γt pt+1 pt + pt+1 1 − γt γt Rt pt , pt+1 (20)

= πt αt (πt ) + [1 − αt (πt )]rtc

where equality obtains if αt (πt ) < 1, in which case btc (πt ) = 0 and hence from the optimal payment schedule,

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rtm (πt ) = γt αt (πt )

pt+1 + γt [1 − αt (πt )]rtc + (1 − γt )Rt . pt

Further, the reserve/deposit ratio can be obtained from the first-order condition with respect to γt after substituting the optimal payment schedules into the depositor’s expected utility function. In particular, when rtc = pt+1 / pt , the optimal reserve/ deposit ratio becomes13 γt = 1+ ˆ φ t πtc∗ R t − ωt 1 , 1 − It−1 (21)

where πtc∗ is defined to be the value of πt that satisfies (20) as an equality with αt (πtc∗ ) = 1; that is, πtc∗ = 1 + 1 − γt γt Rt pt pt+1
−1

.

(22)

Notice that an individual bank’s optimal reserve/deposit ratio increases with the ˆ discount margin, φ t , and decreases with the nominal interest rate, It . It can be also verified from (22) that the higher reserve/deposit ratio associated with the higher discount margin increases the cutoff point, πtc∗ , for discount-window borrowing from the central bank. Now, the individual bank’s optimal reserve-liquidation strategy, (20), can be rewritten as αt (πt ) = min αt πtc∗ , 1 , where πt rtc pt + pt+1 1 − πt 1 1 − γt γt − rtc Rt pt pt+1 (23)

αt πtc∗ =

pt pt+1

,

which can be expressed as αt (πtc∗ ) = πt /πtc∗ in an equilibrium with rtc = pt+1 / pt . When a bank has πt ≥ πtc∗ , the bank exhausts its cash reserves and has to obtain the additional liquidity required from the discount window at a penalty rate. The amount of borrowing (per depositor) from the central bank can be obtained after substituting the optimal payment schedule into (18): btc (πt ) = dt πt (1 − γt )Rt − (1 − πt )γt πt ωt + (1 − πt ) pt+1 pt pt+1 pt ,

(24)

where πt ≥ πtc∗ . This implies that the per-depositor borrowing from the central bank decreases with the reserve/deposit ratio. Finally, the central bank’s expected profit (per depositor), denoted as II, becomes the following:

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207

= E ωt − rtc
πtc∗ πtl

πtu πtc∗

btc (π ) f (π ) dπ − φ dt

πtu πtc∗

f (π ) dπ (25)

γt 1 − αt πtc∗

f (π ) dπ ,

where rtc ∈ [ pt+1 / pt , Rt ]. Note that this is the expected profit from banking intermediation (through discount window) between liquid and illiquid banks. PROPOSITION 3. The equilibrium discount-window rate is  ωt = rtc   +φ 
πtu

 f (π )dπ f (π ) dπ   .  (26)

πtc∗ πtu c bt (π ) πtc∗

dt

Proof. The market-clearing condition in the interbank market is
πtu πtc∗

btc (π ) f (π ) dπ = dt

πtc∗ πtl

γt 1 − αt πtc∗

f (π ) dπ.

(27)

Substituting this back into (25), the zero-profit condition of the central bank implies the equilibrium discount-window rate given by (26). That is, the equilibrium penalty rate charged by the central bank for the discountwindow lending is higher than the interest rate on reserves where the difference or discount margin is precisely the amount of the monitoring cost per unit borrowed from the central bank. 3.4. General Equilibrium In a monetary equilibrium where bank loans are made in positive quantities, the equilibrium values of πtc∗ , γt , pt , and Rt are determined jointly by (21)—or (A.3) in the Appendix—and (22), along with the market-clearing conditions (9) and (10). The equilibrium amount of discount-window lending can be determined by integrating (24) over πt ∈ [πtc∗ , πtu ] and the equilibrium penalty rate, ωt , by (26). Finally, rtc ∈ [ pt+1 / pt , Rt ] is determined by the interbank market-clearing condition (27). A couple of remarks are in order. First, the determinations of all the equilibrium values as explained above depend on the realization of aggregate uncertainty θt ≡ (πtl , πtu ). For example, in the case of two possible realizations of θt such that θi ≡ (πil , πiu ) for i = 1, 2, a monetary equilibrium at t consists of πic∗ , γi , pi , Ri , ωi , ric , and discount-window lending for i = 1, 2. Second, for a sufficiently high monitoring cost due to a severe information problem, there may not exist a monetary equilibrium with a central bank.

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3.5. “Suspension Mechanism” and Aggregate Uncertainty Is it possible to achieve, with a simpler mechanism, welfare as high as or higher than the central banking mechanism as proposed in the paper? Consider an alternative mechanism that does not involve costly delegated monitoring. Without aggregate uncertainty, the aggregate liquidity demands are known with certainty as in Chari (1989). Suppose that the banks are asked to simultaneously report their liquidity needs. If they add up to what they are supposed to, then the appropriate transfers are implemented. If not (i.e., if at least one bank misrepresents), then permanent autarky prevails. This is similar to the “suspension mechanism” of Diamond and Dybvig (1983). As in their setup, it implements truth telling. However, in the presence of aggregate uncertainty as introduced in the paper, the suspension mechanism will not necessarily induce truth telling. Now, under the suspension mechanism, individual banks would have an incentive to misreport their true liquidity needs by exploiting the uncertain nature of aggregate liquidity demands. Therefore, as long as the monitoring costs are not sufficiently large, the central banking mechanism would do at least as well as the suspension mechanism.

4. CONCLUDING REMARKS Williamson (1987) classifies models of financial intermediation into the two groups. One group, represented by Diamond and Dybvig (1983), focuses specifically on liquidity service by banks, deposit contracts, and bank runs to justify the view that banking is inherently unstable (and hence requires special regulatory intervention). The other group focuses more on how financial intermediation serves to economize on the information costs in an environment where borrowers know more about the realized outcomes of their investment than do lenders. For example, Diamond (1984) and Williamson (1986) show that an arrangement with a diversified financial intermediary permits individual lenders to delegate the responsibility for monitoring borrowers to the intermediary and hence to economize on the costs of monitoring investment outcomes. This paper has built on both groups of models to explain how the supervisory role of a central bank can arise endogenously when the banking system has an inherent problem of liquidity underprovision by individual banks whose asset portfolios consist largely of illiquid loans with only fractional cash reserves. In the presence of the interbank market with incomplete information on the individual banks’ liquidity needs, there is a market failure in the sense that the interbank rate does not provide the correct incentives for individual banks to hold a positive amount of cash reserves. The central bank provides an incentive-compatible mechanism by which banks hold the correct level of liquid assets. It takes up a delegated monitoring role in the banking system. Further, it engages in discountwindow loans to illiquid banks at a penalty rate, where the discount margin covers exactly the monitoring cost incurred by the central bank. The higher discount margin leads to the higher reserve/deposit ratio for a bank, and hence the less chance

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209

of exhausting its cash reserves following the liquidity shock. Finally, under the central banking mechanism as proposed in the paper, no currency premium exists in the sense that currency is worth the same as deposits having an equal face value.
NOTES
1. Diamond and Dybvig (1983) consider a role for central bank policy, in the case of aggregate preference shock, arising from the central bank’s assumed superior ability to make payoffs to depositors contingent on realized aggregate outcome. 2. Holmstrom and Tirole (1998) show the benefit of a centralizing supply of liquidity and the potential for free riding on liquidity in the production sector. 3. The significance of this assumption is discussed later in the paper. 4. The nature of interlocation exchange that introduces a role for a medium of exchange is also discussed by Townsend (1987), Mitsui and Watanabe (1989), and Hornstein and Krusell (1993). 5. In general, competition would not preclude banks from charging a loan rate contingent on πt . Whether the contingent loan rate matters for the main results of the paper requires further investigation. 6. Note that borrowers do not face relocation risk. 7. The feature of the model that there is a completely illiquid asset and a liquid asset held as reserve by the bank resembles that of Jacklin and Bhattacharya (1988). 8. It is obtained from the first-order condition with respect to γt after substituting the following payment schedules into the depositor’s expected utility function: rt (πt ) = rtm (πt ) = γt rt (πt ) = 1 − γt 1 − πt pt+1 + (1 − γt )Rt pt and rtm (πt ) = if γt πt πt < πt∗ ; pt+1 pt if πt ≥ πt∗ .

Rt

9. The banking panics in the model are not “belief-based” ones; instead they are caused by fractional reserve banking under the “inelastic currency” regime in which banks are prohibited from issuing notes. 10. The payment schedules as given in note 8 imply that rt (πt ) = rtm (πt ) πt 1 − πt 1 − γt γt It ≥ 1,

if and only if πt ≥ πt∗ , where πt∗ is given by (5). Notice also that banks respond optimally to this panic, treating all movers alike. In particular, there is no “sequential service” constraint in effect, of the type discussed by Diamond and Dybvig (1983). 11. A currency premium exists if currency is worth more than deposits having an equal face value. Suppose that either movers or nonmovers can withdraw their deposits at t, receiving rtm (πt ) in the form of currency. However, nonmovers have no use for currency and hence they will exchange any currency they obtain for claims to the deposits of movers. They will be indifferent between doing so and leaving their deposit until t + 1 as long as they obtain a real value of rt (πt ) per unit withdrawn. Similarly, movers will be indifferent between obtaining currency from their bank or from nonmovers if and only if they can obtain currency with a real value of rtm (πt ) at t + 1 in exchange for deposits with a real value of rt (πt ). Thus, after πt is realized, currency must exchange for deposits at the rate rt (πt )/rtm (πt ) and a currency premium exists if [rt (πt )/rtm (πt )] − 1 is positive. 12. According to the mechanism by which a large intermediary takes up the central banking role, there can arise many central banks in general, instead of a single central bank as in reality. It can be further assumed that, in addition to the given monitoring costs, a lending bank in the interbank market must expend ψ units of effort in making loans to another bank. The lender-specific transactions cost ψ can be interpreted as the time spent in writing contracts and collecting payments from borrowing

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banks, and so forth. Then the central bank will be the one with a transactions cost of zero. That is, if any lending bank with a positive transactions cost acts as a bank intermediary (i.e., a central bank) and offers contracts to banks that earn nonnegative profits, a bank with lower transactions cost could enter and offer contracts preferred by these banks and still earn positive profits. 13. When rtc > pt+1 / pt , the reserve/deposit ratio cannot be expressed as in (21). See the Appendix for the set of equations that determine γt along with other endogenous variables.

REFERENCES
Bhattacharya, S. & D. Gale (1987) Preference shocks, liquidity, and central bank policy. In W.A. Barnett & K. Singleton (eds.), New Approaches to Monetary Economics, pp. 69–88. New York: Cambridge University Press. Champ, B., B.D. Smith & S.D. Williamson (1996) Currency elasticity and banking panics: Theory and evidence. Canadian Journal of Economics 29, 828–864. Chari, V.V. (1989) Banking without deposit insurance or bank panics: Lessons from a model of the U.S. National Banking System. Federal Reserve Bank of Minneapolis Quarterly Review 13, 3–19. Diamond, D.W. (1984) Financial intermediation and delegated monitoring. Review of Economic Studies 51, 393–414. Diamond, D.W. & P.H. Dybvig (1983) Bank runs, liquidity, and deposit insurance. Journal of Political Economy 91, 401–419. Holmstrom, B. & J. Tirole (1998) Private and public supply of liquidity. Journal of Political Economy 106, 1–40. Hornstein, A. & P. Krusell (1993) Money and insurance in a turnpike environment. Economic Theory 3, 19–34. Jacklin, C. & S. Bhattacharya (1988) Distinguishing panics and information-based bank runs: Welfare and policy implications. Journal of Political Economy 96, 568–592. Mitsui, T. & S. Watanabe (1989) Monetary growth in a turnpike environment. Journal of Monetary Economics 24, 123–137. Townsend, R.M. (1979) Optimal contracts and competitive markets with costly state verification. Journal of Economic Theory 21, 265–293. Townsend, R.M. (1987) Economic organization with limited communication. American Economic Review 77, 954–971. Wallace, N. (1988) Another attempt to explain an illiquid banking system: The Diamond-Dybvig model with sequential service taken seriously. Federal Reserve Bank of Minneapolis Quarterly Review 12, 3–16. Williamson, S.D. (1986) Costly monitoring, financial intermediation, and equilibrium credit rationing. Journal of Monetary Economics 18, 159–179. Williamson, S.D. (1987) Recent developments in modeling financial intermediation. Federal Reserve Bank of Minneapolis Quarterly Review 11, 19–29.

APPENDIX
When rtc = pt+1 / pt , the equilibrium values of πtc∗ , γt , ωt , and Rt are determined jointly by (9), (21), (22), and (27), where
πtu πtc∗

f (π) dπ = 1 − πic∗

(A.1)

INTERBANK MARKET AND THE CENTRAL BANK and
πtu πtc∗

211

γt ( pt+1 / pt ) + (1 − γt )Rt btc (π ) f (π ) dπ = 1 − πtc∗ dt ωt − ( pt+1 / pt ) ωt Zt ln ωt − ( pt+1 / pt ) πtc∗ [ωt − ( pt+1 / pt )] + pt+1 / pt Zt ≡ , (A.2)

− where

( pt+1 / pt )[γt ( pt+1 / pt ) + (1 − γt )Rt ][γt ωt + (1 − γt )Rt ] . [ωt − ( pt+1 / pt )][γt ( pt+1 / pt ) + (1 − γt )Rt ]

When rtc > pt+1 / pt , the equilibrium values of πtc∗ , γt , ωt , Rt , and rtc are determined jointly by (9), (22), (27), and the following two equations: πtc∗ = γt +
πtu πtc∗

Rt γt

πtc∗ γt rtc − Rt + Rt 1 − πtc∗ πtc∗

1−

πtc∗ 2

1−

rtc pt+1 / pt (A.3)

ωt − Rt γt ωt + (1 − γt )Rt

btc (π ) pt+1 / pt f (π ) dπ = γt c∗ dt πt ( pt+1 / pt ) − rtc pt+1 / pt − πtc∗ ln 1 − ( pt+1 / pt ) − rtc 1− rtc pt+1 / pt πtc∗ , (A.4)

× πtc∗ + where
πtu πtc∗

[btc (π )/dt ] f (π) dπ is as given by (A.2).


				
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