Optimal Policy Response to Systemic Bank Insolvency
Keiichiro Kobayashi∗
October 16, 2002

Abstract
In some cases of recent financial crises, most of the domestic banks or the banking
sector as a whole becomes insolvent (systemic bank insolvency). We analyze thewelfare eﬀects of policy responses to the systemic bank insolvency by examining
the Diamond-Dybvig type model. The source of ineﬃciency in banking crisis of ourmodel is the premature liquidation of assets that is caused by fixed liability of banks,
which is not changeable contingent on the state of bank asset.
We assume that the systemic bank insolvency is caused by an exogenous macroeconomic shock that destroys a part of bank asset. We analyze the consequences ofdiﬀerent policy responses to the insolvency: (1) laissez faire, (2) deposit guarantee(without immediate recapitalization), (3) immediate recapitalization, and (4) control of inflation. We show that the policy of immediate recapitalization is optimalin our model. Our findings imply that the protracted recession (and the asset-pricedeflation) in Japan for over a decade may be caused by inappropriate policy responseto the systemic bank insolvency.

JEL Classification: E5, E6, G2, H3.
Keywords: Financial crises, deposit guarantee, recapitalization.

∗

The Research Institute of Economy, Trade and Industry, e-mail: kobayashi-keiichiro@rieti.go.jp

11

Introduction

In this paper, we analyze and compare the consequences of diﬀerent policy responses tofinancial crises. In the recent theoretical research, there are considerably rich literatureson the mechanism how financial crises occur (Diamond and Dybvig [1983], Postlewaite
and Vives [1987], and Allen and Gale [1998, 2000a, 2001]). But there are not so many
theories that explain the diﬀerence of recovery paths from the crises in accordance with
diﬀerent policy responses. For example, Diamond and Dybvig (1983), Freixas, Parigi,
and Rochet (1999), and Martin (2001) argue the policies to prevent bank runs, notto respond to systemic insolvency. Few authors like Diamond and Rajan (2002), andBergoeing, et al. (2002) argue about ex post policy responses to financial crises.
Meanwhile, the world has experienced a sizable number of banking crises in the lasttwenty five years. Caprio and Klingebiel (1999) identified 113 systemic banking crisesthat have occurred in 93 countries since the late 1970s, and 50 borderline and smallerbanking crises in 44 countries during that period. In these experiences, researchers havebecome to pay more attention to bank insolvency rather than to bank runs. Recent crisis
episodes show that bank insolvency is the central problem that should be rectified andthat a rush of bank runs or temporary shortage of liquidity is rather a symptom. Several
stylized facts about the recovery paths from banking crises have been found by casestudies and empirical researches of recent financial crises (See, for example, Claessens etal. [2001], Alexander et al. [1997], Caprio and Klingebiel [1996]).
Honohan and Klingebiel (2000) find that open-ended liquidity supports, regulatory
forbearance and an unlimited depositor guarantee are all significant contributors to thefiscal cost of banking crisis (See also Leggett[1994] for forbearance policy). They alsofind that liquidity support significantly increases the output loss or delays the economic
recovery (This result is confirmed by Bordo et al. [2001] who used a broader data set).
On the other hand, Claessens et al. (2001) show that the liquidity support and depositor
guarantee may be eﬀective for recovery of corporate sector when they are implemented
with the measures to restore solvency of banks (by disposing of nonperforming assets).
In short, these empirical analyses seem to indicate that open-ended liquidity support2without resolving systemic insolvency may hinder economic recovery and magnify thefiscal cost.
The researchers emphasized on moral hazard or managerial or political distortions
as the causes why liquidity supports tend to increase the output loss and fiscal cost. In
this paper we neglect all these microeconomic distortions, and emphasize the excessive
liquidation of premature assets made by banks that try to fulfill the commitment to
repay when they are insolvent and do not have enough aggregate liquidity.
Among the recent financial crises around the world, Japan’s case is quite peculiar.
As Caprio and Klingebiel (1999) point out, the banking sector of Japan has fallen into
the crisis at the beginning of the 1990s as a result of the burst of asset-price bubble,
and the Japanese banks have stayed in the crisis situation for over a decade until today.
The characteristics of recent Japanese economy are (a) continuation of low economic
growth, (b) continuation of asset-price decline, i.e., asset-price deflation, (c) increase in
the liquidity of bank assets and of corporate liability. Figure 1 shows the proportion ofliquid assets and of loans and bills discounted in bank assets of all Japanese banks during
the period of 1986-2002. The liquidity in bank asset surged in 1994 when the financial
crisis was widely recognized for the first time, and it has continued to grow after thesecond crisis of 1997-1998.
Figure 1: Assets of All Banks in Japan
Figure 2 shows the liquid liability of non-financial corporations. Although liquidliability decreases over time, the liquidity has been consistently above the trend since
mid-1990s.
Figure 2: Liquid Liability of Non-Financial Corporations
Our aim in this paper is to analyze the eﬀects of diﬀerent policy responses to bankinsolvency, to find the optimal policy response, and eventually to construct a consistent
explanation of the above features of Japan’s protracted recession.
In order to analyze the banking crises, we use a variant of the Diamond-Dybvig
model (Diamond and Dybvig [1983]). In our model, we do not describe how financial
3

crises occur, but we just set that all banks become insolvent as a result of an unspecified
macroeconomic shock (e.g., a burst of asset-price bubble, fall in currency exchange rate)that suddenly decrease the value of bank asset. Taking this systemic bank insolvency
as given, we focus our analysis on welfare properties of the following policy responses:(1) laissez faire, (2) deposit guarantee, (3) immediate recapitalization, and (4) control of
inflation.
If the government chooses no action (laissez faire) to systemic bank insolvency, allhouseholds would withdraw their deposits immediately since they know that bank assets
are smaller than bank liabilities. In this case, all banks are run on, premature liquidationof all assets occurs, and welfare of households becomes worse.
If the government guarantees the depositors (but does not recapitalize banks immediately), the use of assets becomes ineﬃcient. Since the government guarantees deposits,
depositors do not run on banks. But banks are forced to partially liquidate their premature assets to meet the demand of withdrawals, since the government does not supply
additional liquidity into the economy. Private agents rationally believe that in the future
the government must recapitalize insolvent banks to fulfill the commitment to guarantee
deposits. In this case as a result of ineﬃcient liquidation of premature assets, the welfare of households becomes suboptimal, and the fiscal cost imposed on households in the
future becomes large.
If the government immediately recapitalizes insolvent banks by issuing the government bonds, there are several tradeoﬀs for private agents. Since the government bonds
provide the economy with additional liquidity, households can withdraw their deposits
in full. But they must pay tax to redeem the government bonds in the future. Banks
do not need to liquidate premature assets since they are given liquidity, i.e., the government bonds. Since assets can be used eﬃciently and the output can be maximized
if the government recapitalizes banks appropriately, households can attain the optimal
consumption in this case.
If the government can control the inflation rate immediately and accurately, it iseasily shown that the optimality can be attained. But in the case where it takes time4

for the government to change the inflation rate, households run on banks before inflation
occurs, anticipating that the real value of their deposit will be small due to the inflation.
In this case, the outcome is similar to the case of laissez faire.
The rest of the paper is organized as follows. Section 2 describes the basic model.
Section 3 analyzes the consequences of several policy options. Section 4 concludes.

2

Model

The economy consists of a continuum of households and of banks. The measure of eachcontinuum is normalized to 1. The economy continues two periods with three dates:t = 0, 1, 2. There is one type of good (consumer good ) in this economy that can be
consumed by households.

2.1

AssetsThere are two investment technologies in this economy. The following two assets represent these investment technologies.
(1) The short-term asset: one unit of the consumer good invested in the short-term asset
at date t yields R1 units at date t + 1, for t = 0, 1. Although we will assume that R1 = 1
for simplicity in Section 3, we describe our model for general values of R1 in this section.
(2) The long-term asset: one unit of the consumer good invested in the long-term asset
at date 0 produces R2 units at date 2, where R2 > R1 . It is assumed that partial liquidation of the long-term asset is available to banks. One unit of long-term asset yields
xl units of consumer good at date 1 and (1 − l)R2 units at date 2, where the bank canchoose the degree of liquidation l ∈ [0, 1] at date 1.
We assume that the short-term asset is available to both households and banks, andthe long-term asset is available only to banks. The parameters satisfy
x < 1 ≤ R1 ≤ R12 < R2 .

5

(1)

2.2

Households

Each household is endowed with E units of consumer goods at date 0 and nothing at
subsequent dates. A household can either deposit E in bank account, or invest all of
the endowment in short-term asset by itself. We analyze the case where expected payoﬀ
for households is larger when they make a deposit than when they hold their assets bythemselves.
We assume there are two types of households, early households, who only consume
the good at date 1, and late households, who only consumes at date 2. The type of eachhousehold is revealed at date 1, and all households are identical at date 0. The type of ahousehold is private information that cannot be observed by banks or other households.
Each household has a probability η of being an early household and it is assumed that thelaw of large number holds so that the proportion of early households in the populationis η. Let c1 (c2 ) denote the consumption of an early (late) household at date 1 (date 2).
The household’s ex ante utility is
ηU (c1 ) + (1 − η)U (c2 ).
The utility function U (·) satisfies the usual neoclassical properties, U 0 (c) > 0, U 00 (c) < 0,and limc↓0 U 0 (c) = ∞. In the following, we assume for simplicity that
U (c) =c1−θ, θ > 1.
1−θThe assumption: θ > 1 is necessary to guarantee that the liquidity insurance given bybanks can be Pareto superior to the market allocation attained in the case where assets
are directly owned by households and households can trade their assets and the goods
in a competitive market (See von Thadden [1999], Diamond and Dybvig [1983]).
Possibility of Bank Run We focus our attention to the case where households makea deposit at date 0. Since late households can invest in short-term asset at date 1,
they can pretend to be early households and withdraw their deposit at date 1, invest
it in short-term asset, and consume the proceeds at date 2. Therefore, if households

6believe that all late households withdraw early, a bank run can occur as a result of selffulfilling expectations even when banks have enough resources to meet their commitment
to early households at date 1 and make an equal or greater payment to late households
at date 2, as Diamond and Dybvig (1983) describe (Allen and Gale [2001] call this type
unnecessary bank runs). Since our analysis focus on the eﬀect of policy responses tofinancial crises, we ignore the self-fulfilling prophecies, which make the equilibrium pathunnecessarily complicated. Thus following Allen and Gale (2001), we assume that inequilibrium late households wait until date 2 to withdraw from the bank as long as it isincentive-compatible to do so.

2.3

Systemic Bank Insolvency

We formalize a financial crisis as destruction of assets due to an exogenous macroeconomic
shock at a time between date 0 and date 1. At date 0, households make a deposit of E
in banks, and banks make investments (S, L) and commitments (W1 , W2 ), where S is
the amount invested in short-term asset, L ≡ E − S is the long-term asset, W1 is theamount to be repaid when a depositor withdraws at date 1, and W2 is the amount to be
repaid at date 2.
Assumption 1 The commitments (W1 , W2 ) and investments (S, L) are observable andverifiable for all agents in the economy.
We assume the following for contractual technology of demand deposit:
Assumption 2 A bank’s commitment to repay W1 is not renegotiable at date 1.
This assumption implies that a bank must repay W1 to all withdrawer at date 1 unlessthe bank runs out of asset, even if it becomes unable to repay W2 at date 2 as a resultof the repayment at date 1. The specific reason why banks cannot renegotiate theamount of repayment with depositors is out of the scope of this paper. One justificationof Assumption 2 may be given by the agency problem that is thoroughly analyzed by
Diamond and Rajan (2001).

7

After these investments (S, L) and commitments (W1 , W2 ) are made, a macroeconomic shock hits the economy at, say, date 12 , and destroys the proportion 1 − λ of allbanks’ short-term assets and 1 − µ of all banks’ long-term assets where λ and µ are
random variables that satisfy (λ, µ) ∈ [0, 1]2 and (λ, µ) ∼ F (λ, µ) where F (·, ·) is thejoint distribution function over [0, 1]2 .
Assumption 3 The random shock λ and µ are both observable but not verifiable. Thus
agents cannot write a contract contingent on (λ, µ). When the government takes policy
action at date 1, the action can be contingent on (λ, µ).
Thus the short-term asset of a bank becomes S 0 = λS, and the long-term asset
becomes L0 = λL at date 1 whereas banks cannot change the commitment Wt (t = 1, 2)
that is made at date 0. 1
Note that the macroeconomic shock (λ, µ) is formalized as a convenience to createa situation where banks’ deposit liability exceeds their asset (systemic bank insolvency).
We formalize the macroeconomic shock that brings about insolvency of the banking
system as a physical destruction of productive asset, although its counterpart in reality
is a burst of asset-price bubble or a fall of currency exchange rate, which typically does
not involve physical destruction of productive assets.2

2.4

Structure of the GameAt date 1, the government chooses one policy among the policy options that are specified
in Section 3, given (W1 , W2 , S) and (λ, µ). Note that each bank can choose diﬀerent valuesof (W1 , W2 , S), and the government can apply diﬀerent policy to each bank contingent on

1

If the bank insolvency is an idiosyncratic events, then banks can insure themselves by forming thedeposit insurance. But the deposit insurance among banks is useless if the shock is macroeconomic andall banks become insolvent as in the model of this paper.
2
We may regard our model as a variant of the model by Allen and Gale (1998) in which business cycle
fluctuation is formalized as a random return on long-term asset. In their model, one unit of long-term
asset generates R units of consumer goods where R is a random variable the realized value of which iscommon for all banks.

8the values of (W1 , W2 , S) since these variables are observable and verifiable (Assumption
1).
We denote the government’s choice of policy by P : P = P (W1 , W2 , S, λ, µ). The government’s objective is to maximize the average utility of households, given (W1 , W2 , S, λ, µ).
The government chooses P (W1 , W2 , S, λ, µ), anticipating banks’ reaction (m, l) where mis the fraction of consumer goods generated from short-term asset that is to be reinvested in short-term asset, and l is the fraction of long-term asset that is to be liquidated
prematurely at date 1. At date 1 after the government chooses the policy, banks choose(m, l). Thus a bank pays out (1 − m)R1 λS + xlµL units of consumer goods at date 1
to withdrawers. Banks’ choice m (or l) is a mapping from (W1 , W2 , S, λ, µ, P ) to a real
number in [0, 1]. Thusm = m(W1 , W2 , S, λ, µ, P ), and l = l(W1 , W2 , S, λ, µ, P ).
Therefore, the bank’s problem at date 0 is written as follows:
max E[ηU (c1 ) + (1 − η)U (c2 )]W1 ,W2 ,Ssubject to P = P (W1 , W2 , S, λ, µ),m = m(W1 , W2 , S, λ, µ, P ), l = l(W , W , S, λ, µ, P ).
1

2

The expectation E[·] is taken over λ and µ. As we will see in Section 3, the consumptions
c1 and c2 are uniquely determined by the choice of (P, m, l).
Since all banks solve the same problem at date 0, the solution (W1 , W2 , S) should be
identical for all banks. Thus we can focus our attention to the symmetric equilibrium
where all banks choose the same values of (W1 , W2 , S) when we examine the policy
response of the government at date 1.
Note that we have assumed time consistency in the government’s choice of policy: The
government cannot commit, before (λ, µ) is revealed, to any policy that is not optimal
ex post. Under the assumption of time consistent government, we can compare the eﬀect9

of diﬀerent policy chosen at date 1 on the premise that (W1 , W2 , S, λ, µ) are all fixed andidentical for all banks.

3

Policy Responses to Systemic Bank Insolvency

In sections 3.1 – 3.5, we compare the policy options at date 1 for fixed values of(W1 , W2 , S, λ, µ). In Section 3.6, we see the date 0 problem for banks, in which they
determine (W1 , W2 , S) anticipating which policy to be chosen at date 1 by the government for each (λ, µ).

3.1

Optimal Allocation

The values of (W1 , W2 , S, λ, µ) are all revealed at date 1. The optimal allocation is definedas (m, l) that maximizes the average utility of household, given the resource constraints
only. Thus the optimal allocation is the solution to the following problem:(PO) max ηU (c1 ) + (1 − η)U (c2 )m,lsubject toηc1 ≤ (1 − m)λR1 S + xlµL,







(1 − η)c2 ≤ (1 − l)R2 µL + mλR21 S,




L=E−S






m≥0





 l ≥ 0.

If the solution satisfies l ≥ 0 and m = 0, the first order condition isU 0 (c1 ) −R2 0
U (c2 ) ≥ 0,
x

(2)

1−l
R2 µL. If the solution satisfies m ≥ 0 andwhere c1 = η1 (λR1 S + xlµL) and c2 = 1−η

l = 0, the first order condition isU 0 (c1 ) − R1 U 0 (c2 ) ≤ 0,

(3)
1−θ

1
c
−1
2
where c1 = 1−mη λR1 S and c2 = 1−η (mλR1 S + µR2 L). Since U (c) =1−θ , the firstorder conditions (2) and (3) imply that the solution (l, m) to (PO) satisfies that l > 0
10and m = 0 if (λ, µ) falls in Region I, that l = m = 0 if (λ, µ) falls in Region II, and thatl = 0 and m > 0 if (λ, µ) falls in Region III, where Regions I, II, and III are defined asfollows.
½

2

Region I = (λ, µ)| (λ, µ) ∈ [0, 1] , µ >½Region II = (λ, µ)| (λ, µ) ∈ [0, 1]2 , µ ≤
½

³

³

R2
x

R2
x
1

´1

´1
θ

¾

(1−η)R1 S
ηR2 L λθ,1¾(1−η)R1 S
θ (1−η)R1 S
ηR2 L λ, and µ ≥ R1
ηR2 L λ¾

1SRegion III = (λ, µ)| (λ, µ) ∈ [0, 1]2 , µ < R1θ (1−η)RηR2 L λ .

,These regions I, II, and III are shown in Figure 3.
Figure 3: Optimal Allocation
If (λ, µ) ∈ Region I, the optimal consumptions ct (t = 1, 2) and l are uniquely determinedby (2) in equality. If (λ, µ) ∈ Region II, the optimal consumptions ct (t = 1, 2) are
uniquely determined by l = m = 0. If (λ, µ) ∈ Region III, the optimal consumptions ct(t = 1, 2) and m are uniquely determined by (3) in equality.
In problem (PO), we implicitly assume that the social planner observes the type ofeach household and assigns c1 (c2 ) to an early (late) household. Since the type of a household is private information, we should impose a restriction that the government cannot
observe types of households. In this case, the consumption allocation must satisfy thefollowing incentive compatibility constraint to prevent late households from mimicking
early households:
R1 c1 ≤ c2 .
When we add this constraint to (PO), this constraint becomes redundant only if R1 ≤ 1.
Therefore in this section we assume for simplicity of analysis that
R1 = 1.

(4)

If R1 = 1, the solution to (PO) always satisfies the incentive compatibility constraint(c2 ≥ R1 c1 = c1 ) that the government faces when it chooses policy action. Before moving
on to other policies, we examine two policy options that are excluded from available
options in this paper: Deposit Cut and Monetary Easing.
11

Deposit Cut

Deposit cut forced by the government is the most straightforward pol-icy to realize the optimal allocation. If the government can order arbitrary change of
repayments (W1 and W2 ) at date 1, then the optimal allocation is obviously attained.
In some episodes of banking crises, the governments coercively cut the deposit liabilityof banks without causing major macroeconomic impacts (Baer and Klingebiel [1995]).
But those episodes show that deposit cut in which the government order overrides private contracts and property rights can be implemented only when the country falls into
the emergency situation: for example, the occupation by foreign army (Japan 1946) or
the great depression (the United States 1933). Since it seems plausible to assume that
deposit cut policy incurs prohibitively high political costs, we exclude deposit cut fromthe available policy options in this paper.
Monetary Easing Since ineﬃciency in our model is caused by early liquidation of
long-term asset, liquidity provision by monetary policy seems eﬀective to mitigate theineﬃciency. But careful consideration shows that the monetary policy is ineﬀective in
our model. Monetary easing can be defined as the following policy: The government
supplies “cash” to banks at date 1 in exchange for the banks’ commitment to repay
the prespecified amount of “cash” at date 2. “Cash” is something exchangeable with
consumer good. Suppose that early households withdraw W1 = Wc + Wm from banks
at date 1, where Wc is consumer goods and Wm is cash. Since in monetary policy thegovernment does not impose any tax on early households, all early households try toconvert all amount of cash (Wm ) to consumer goods. As long as the aggregate supply of
consumer goods at date 1 ((1 − m)λR1 S + xlµL) is less than ηW1 , the price of consumer
good in terms of cash is bid up to infinity and the cash becomes worthless. Thus themonetary easing cannot prevent early liquidation of long-term asset.
This analysis is based on the implicit assumption that deposit contracts are madein terms of consumer good. If we allow deposit contract be nominal, the monetary
authority may control the price level to induce the optimal allocation. We analyze thiscase in Section 3.5.

12

3.2

Laissez Faire

If the government does not take any policy action to the systemic bank insolvency, allbanks are run on at date 1 for small λ and µ. It is shown as follows.
At date 1, the values of (W1 , W2 , S, λ, µ) are revealed. We assumed that late households wait to withdraw if and only if it is incentive compatible to do so. Thus bank run
occur iﬀ c2 satisfies c2 < R1 c1 = c1 where c2 is what a late household can get by waiting
to withdraw until date 2. The condition for occurrence of bank runs is thus rewritten as
min{W2 , c2 (l, m)} < min{W1 , (1 − m)λS + xlµL} where (l, m, c2 (l, m)) is determined by(PLF) max ηU (min{W1 , (1 − m)λS + xlµL}) + (1 − η)U (c2 (l, m))m,lsubject to ηW1 ≤ (1 − m)λS + xlµL, or l = 1 − m = 1,
 c2 (l, m) ≡ mλS + (1 − l)R2 µL.

(5)

If min{W2 , c2 (l, m)} < min{W1 , (1 − m)λS + xlµL}, late households try to withdraw
W1 at date 1, invest it in short-term asset, and consume W1 at date 2. That all latehouseholds run on banks in this case is guaranteed by Assumption 2 and the technological constraint that premature liquidation of long-term asset generates some amount ofconsumer goods at date 1.
Assuming that W2 is suﬃciently large, we can rewrite the condition for bank runs as
ηW1
R1 W1 − λR12 S
and µ <
R1 S
R2 L

(6)

(1 − η)R1 W1 + η Rx2 W1 − Rx2 λR1 S
ηW1and µ <
,
R1 S
R2 L

(7)

λ≥
or
λ≤

where R1 = 1. We define a region BR(W1 , S) in [0, 1]2 byBR(W1 , S) = {(λ, µ)| λ and µ satisfy conditions (6)and (7).}Region BR(W1 , S) for fixed values of W1 and S is shown in Figure 4.
Figure 4: Region of Bank Runs (Case of Laissez Faire)
13(8)

Therefore, the consumption allocation in the case of laissez faire is summarized as follows:
If (λ, µ) ∈ BR(W1 , S), bank runs occur and each household consumes λS + µxL units ofconsumer goods3 ; Otherwise early households consume W1 , and late households consumemin{c2 (l, m), W2 } where c2 (l, m) is the solution to (PLF).4
If bank runs occur, banks are forced to liquidate all asset. Therefore (l, m) = (1, 0)
if (λ, µ) ∈ BR(W1 , S). Comparing Figures 3 and 4, it is easily shown that laissez fairepolicy cannot realize the optimal allocation for given (W1 , W2 , S) without measure-zerocoincidence in which (λ, µ) happens to satisfy either (2) or (3) in equality with c1 = W1
and c2 = c2 (l, m).

3.3

Deposit Guarantee

In order to prevent bank runs, the government can declare unlimited guarantee of depositors. We assume that to initiate this policy incurs the social cost CD that is a deadweight loss for the economy. Since banks are insolvent if λ and µ are small, the government must use fiscal expenditure to fulfill the commitment of deposit guarantee. In
order to contrast with recapitalization policy in Section 3.4, we assume the following forthis deposit guarantee policy.
Assumption 4 When the government implements the deposit guarantee policy, it restricts itself to use public money only after banks run out of all assets.
To subsidize individual banks or the deposit insurance company in order to guarantee
depositors is politically controversial in reality. This assumption formalize this politicaldiﬃculty in a simple form (We will assume in Section 3.4 that the government can
subsidize banks before they run out of asset if the government pays higher politicalcost). The government must collect tax to cover the fiscal cost of deposit guarantee. We
assume the following for the government ability of tax collection:3

We assume as Allen and Gale (1998) that all proceeds of liquidation of bank asset are equally divided

by all depositors at date 1 when bank runs occur.
4
We assume for simplicity that if late households wait until date 2, all remaining proceeds of bankassets are equally devided by remaining depositors at date 2.

14

Assumption 5 Although the government does not observe type of each household, it cancollect tax from household contingent on the timing of its withdrawal. The government
can impose tax (T11 , T12 , T2 ) where an early withdrawing household must pay T11 at date
1 and T12 at date 2, and a late withdrawing household must pay T2 at date 2.
Note that an early withdrawing household must save TR121 = T12 and invest it in shortterm asset at date 1 in order to pay tax T12 at date 2. Given (W1 , W2 , S, λ, µ), thegovernment problem is written as follows:
(PDG)

max

l,m,T11 ,T12 ,T2 ,F1 ,F2

ηU (W1 − T11 − T12 ) + (1 − η)U (W2 − T2 )subject to  ηW1 = λS + xµL + F1 and l = 1 − m = 1,
 







 ηW1 = (1 − m)λS + lxµL,








ηT11 ≥ F1 ,
if λS + xµL < ηW1 ,otherwise,(1 − η)W2 = mλS + (1 − l)R2 µL + F2 ,





 ηT12 + (1 − η)T2 ≥ F2 ,







ηT11 + ηT12 + (1 − η)T2 ≥ F1 + F2 + CD ,





 W1 − T11 − T12 ≤ W2 − T2 .

(9)

The last constraint of (9) is the incentive compatibility constraint to prevent bank runs.
As long as this constraint is satisfied, early households withdraw at date 1, and latehouseholds withdraw at date 2. This fact justifies the other constraints in (9). Define
F (λ, µ), F ∗ (λ, µ), F ∗∗ (λ, µ), and F ∗∗∗ (λ, µ) by

F (λ, µ) =F ∗ (λ, µ)



∗∗

F (λ, µ)
 F ∗∗∗ (λ, µ)1if λ ≥ ηWS ,1
if λ ≤ ηW
S and λS + µxL ≥ ηW1 ,if λS + µxL ≤ ηW1 .

F ∗ (λ, µ) ≡ (1 − η)W2 − R2 µL + ηW1 − λS + CD ,F ∗∗ (λ, µ) ≡ (1 − η)W2 − R2 µL + Rx2 (ηW1 − λS) + CD ,F ∗∗∗ (λ, µ) ≡ (1 − η)W2 − xµL + ηW1 − λS + CD .
Problem (PDG) can be simplified to(PDG’)max

T11 ,T12 ,T2

ηU (W1 − T11 − T12 ) + (1 − η)U (W2 − T2 )
15

(10)

subject to ηT11 + ηT12 + (1 − η)T2 ≥ F (λ, µ),T11 ≥ max{0, ηW1 − λS − xµL}, W −T −T ≤W −T .
1

11

12

2

(11)

2

We can specify when this policy attains the optimal allocation. In order to simplify1the comparison, we assume CD = 0 for now. If λ < ηW
S , banks must liquidate long-term asset prematurely at date 1, and the degree of liquidation l is uniquely determinedby l = minnoηW1 −λS
xµL , 1. Given (W1 , W2 , S), the value of l cannot be optimal without

1 −λSmeasure-zero coincidence in which (λ, µ) satisfies equation (2) in equality for l = ηWxµL.
1
Therefore, if λ < ηW
S , ineﬃcient liquidation of long-term asset occurs almost always, andthus the government cannot attain the optimal by choosing any values for (T11 , T12 , T2 ).
1
If λ ≥ ηW
S , no ineﬃcient liquidation of long-term asset occurs. In this case it iseasily shown that the government can attain the optimal consumption by appropriately
choosing (T11 , T12 , T2 ) if the optimal value of l equals 0. The incentive compatibility
constraint is automatically satisfied by the optimal consumption since R1 = 1. Therefore,
we have shown the following proposition for the optimality of deposit guarantee policy.
Proposition 1 If CD = 0, the deposit guarantee policy can attain the optimal only when1
(λ, µ) satisfies λ ≥ ηW
S and µ ≤

³

R2
x

´1
θ

(1−η)R1 S
ηR2 L λ.

See Figure 5 for the region where the deposit guarantee policy can attain the optimal.
Figure 5: Region of Optimality (Case of Deposit Guarantee)

3.4

Recapitalization

In the previous subsection, we made a restriction that the government does not give
subsidy to banks unless they run out of all assets. We assume that the government can
subsidize or recapitalize banks before they run out of asset if it pays higher politicalcost than in the case of deposit guarantee policy. Thus we assume that to initiate therecapitalization policy incurs social cost CR , where CR ≥ CD .
We may consider that the government subsidizes banks by issuing the government
bonds and giving the government bonds to the banks. We make a crucial assumption:
16

Assumption 6 Households accept the government bonds as liquidity. Banks can repay
deposits in the form of the government bonds.
Thus the recapitalization policy in our paper means that the government produces andsupplies additional liquidity (i.e., the government bonds) to prevent excessive liquidationof long-term asset. Given (W1 , W2 , S, λ, µ), the government implements recapitalization
policy by solving the following problem:
(PR)

max

l,m,T11 ,T12 ,T2 ,B

subject toηU (W1 − T11 − T12 ) + (1 − η)U (c2 − T2 )
ηW1 = (1 − m)λS + lxµL + B,







η(W1 − T11 − T12 ) ≤ (1 − m)λS + lxµL,





 (1 − η)c2 (l, m) = mλS + (1 − l)R2 µL,
c2 = min{W2 , c2 (l, m)},







ηT11 + ηT12 + (1 − η)T2 ≥ B + CR ,





 W −T −T ≤ c −T .
1

11

12

2

(12)

2

The essential diﬀerence between problems (PDG) and (PR) is that in (PR) the government can determine B, the subsidy for recapitalization of banks5 , before the valueof (l, m) is fixed whereas in (PDG) the government must determine F1 and F2 only after (l, m) is determined by banks’ obligation to repay (Assumption 2). Therefore, thegovernment can set B as a function of (l, m).
In the case where CR = 0 and W2 is suﬃciently large, problem (PR) can be degenerated to (PO) by setting B = B(l, m) = ηW1 − (1 − m)λS − lxµL, ηT11 + ηT12 = B, andT2 = 0. Thus the optimal value of (l, m) is attained by recapitalization policy. We haveshown the following:
Proposition 2 If CR = 0, the recapitalization policy can always attain the optimal.
This result appears to be quite diﬀerent from Diamond and Rajan (2002) who claim that
recapitalization of failing banks during a financial crisis may worsen the crisis. Diamond5

We assume the subsidy B can take a negative value. In this case B is interpreted as tax on banks.

17and Rajan seem to use the term “recapitalization” to represent the policy that transfers
liquidity to an insolvent bank from the other banks. Thus in their model, the amountof aggregate liquidity does not increase at date 1, and therefore bank bailouts result in
more ineﬃcient liquidation of long-term assets. Therefore, “recapitalization” in Diamondand Rajan (2002) is similar policy as deposit guarantee without creation of additional
aggregate liquidity, just like the policy in our Section 3.3.
In (PR) of our model however, the government can create aggregate liquidity when
it recapitalizes insolvent banks by issuing the government bonds, which are accepted asliquidity by early households at date 1. Thus in our model, recapitalization attains the
optimal allocation, while deposit guarantee (without recapitalization) results in ineﬃcientliquidation of long-term asset.
We need to emphasize, however, that increase in aggregate liquidity alone is not suﬃcient to obtain the optimal. Let us compare recapitalization policy and monetary easing
by the central bank. The increase of money supply by the central bank may increase
the amount of aggregate liquidity but it does not rectify insolvency of banks since thecentral bank gives them cash only in exchange for their assets. The premature liquidation occurs because banks have too much obligation to repay at date 1. Thus monetaryeasing cannot prevent premature liquidation that is caused by bank insolvency, unlessbanks are given de facto subsidy by the central bank operation. Supply of aggregateliquidity and filling the gap of insolvency are both necessary to prevent ineﬃcient asset
liquidation.

3.5

Inflation

Inflation may be one of the policy options that can be used to address bank insolvency. In
our simplified model, it is diﬃcult to formalize inflation in self-consistent way. Instead ofmodeling the mechanism that generates inflation, we simply assume that the government
can control inflation rates (πt ) at date t = 1, 2 by unspecified monetary policy which incursocial cost of CM . Crucial point is that the government can change the inflation rate
without aﬀecting the real value of bank asset. In this case, the government problem

18

becomes(PI)
subject tomax ηUl,m,π1 ,π2

µ

W1
π1

¶

+ (1 − η)U

µ

W2
π2

¶




η W1 ≤ (1 − m)λS + lxµL,


 π1
W2

(1 − η) π2 ≤ mλS + (1 − l)R2 µL − CM ,
 W1 ≤ W2 .
π1

(13)

π2

This problem is identical to (PO) if the cost CM = 0 and the government can freely
control both π1 and π2 . But if it takes time to change inflation rate, the consequence
becomes diﬀerent. Suppose that the government cannot generate inflation at date 1:
π1 = 1, while it can change π2 freely. In this case, problem (PI) become identical to(PLF) if we assume W2 is suﬃciently large. Therefore, we have shown the following:
Proposition 3 If CM = 0, and if the government can freely change π1 and π2 , inflation
policy attains the optimal. If π1 is fixed at 1 and the government can change π2 only,
the outcome of inflation policy is identical to that of laissez faire.

3.6

Optimal Policy Response

Welfare Eﬀect

We have analyzed welfare eﬀects of diﬀerent policies in Sections 3.1– 3.5. If political cost is zero: i.e., CD = CR = CM = 0, the optimal policy response atdate 1 is clear. The recapitalization of insolvent banks by issuing the government bonds
is the optimal policy. Controlling the inflation rate is also optimal if the government or
the central bank can immediately and accurately change the inflation rate of date 1. If
CD = CR ≥ 0, it is obvious that for any given (λ, µ), the recapitalization at date 1 issuperior to the deposit guarantee.
Fiscal Cost Honohan and Klingebiel (2000) found that liquidity provision and deposit
guarantee tend to increase the fiscal cost. This is confirmed in our model by comparing
deposit guarantee policy and recapitalization policy. In order to compare the fiscal costsof policy actions, we need to measure the costs in the same condition. Thus we use
B + (1 − η)(W2 − c2 (l, m)) + CR as the fiscal cost of recapitalization policy instead of19B + CR , since the cost (1 − η)(W2 − c2 (l, m)) is directly borne by late households in(PR) while this cost is once borne by the government and is transferred to households bytaxation in (PDG). The fiscal cost of deposit guarantee is F1 + F2 + CD . Let us assume
CD = CR = 0 for simplicity. It is easily confirmed that the fiscal costs defined above isrewritten as ηW1 + (1 − η)W2 − λS − {lx + (1 − l)R2 }µL, where l is chosen in response
to deposit guarantee policy or recapitalization policy. The value of l may be larger orsmaller in the case of deposit guarantee than in the case of recapitalization. So do thefiscal costs. But we can easily confirm the following:1
Proposition 4 Suppose that CD = CR = 0. If λ < ηW
S and the optimal consumption

c1 satisfies c1 < W1 , the value of l and the fiscal cost are larger in the case of deposit
guarantee than in the case of recapitalization.
Banks’ Problem at Date 0

If the values of (CD , CR , CM ) are fixed, the optimal

policy response that maximizes ηU (c1 ) + (1 − η)U (c2 ) is uniquely determined for each
(W1 , W2 , S, λ, µ). Thus we can define a function P (W1 , W2 , S, λ, µ) which maps (W1 , W2 , S, λ, µ)to the optimal policy. If we make restriction that the inflation rate at date 1 is notchangeable (π1 = 1), the optimal policy is chosen from the following set:
P (W1 , W2 , S, λ, µ) ∈ {Laissez Faire, Deposit Guarantee, Recapitalization}
Banks determine (W1 , W2 , S) at date 0 by solving the following problem:(P)
subject tomax E[ηU (c1 ) + (1 − η)U (c2 )]W1 ,W2 ,S
P = P (W1 , W2 , S, λ, µ),







m = m(W1 , W2 , S, λ, µ, P ),l = l(W1 , W2 , S, λ, µ, P ),






c1 = (1 − m)λS + lxµL,





 c2 = mλS + (1 − l)R2 µL.

Note that the solution to (P) can be the equilibrium only if the government can choose
the diﬀerent policy to each bank that is contingent on the bank’s choice of (W1 , W2 , S). If
20not, it is easily shown as follows that a bank has an incentive to deviate from the solution
to (P). For example, suppose that the government chooses the policy that is contingenton (W1 , W2 , S), the social average of (W1 , W2 , S). In this case a bank can make its own
depositors better oﬀ by setting W1 at arbitrarily large value, because W1 is guaranteed
and the tax contingent on W1 is limited when the government chooses deposit guaranteeor recapitalization.

4

Conclusion

When the banking system falls into the situation of systemic insolvency, diﬀerent policy
responses have quite diﬀerent welfare eﬀects. (1) Laissez faire policy may brings aboutbank runs and ineﬃcient liquidation of long-term asset. (2) Deposit guarantee without
immediate recapitalization results in excessive liquidation of long-term asset, suboptimal
consumption, and larger fiscal cost than in the case of recapitalization. (3) Immediate
recapitalization by the government bond issuance can attain the optimal. (4) Control of
the inflation rate generates the same outcome as laissez faire if the government cannotchange the inflation rate immediately and accurately.
These results are quite consistent with the stylized facts of recent financial crisesthat we refer to in Section 1. Although open-ended liquidity support and blanket depositor guarantee are distinguished in empirical research (Honohan and Klingebiel [2000],
Claessens et al [2001]), both tools serve for orderly deposit withdrawals. Therefore thatthe liquidity support without resolving systemic insolvency prolongs the crisis can beinterpreted as that the deposit guarantee without recapitalization decreases the total
welfare in our model.
Let us examine the 1990s of the Japanese economy in our theoretical framework.6
Although Japan experienced a full-scaled crush of asset-price bubble at the beginning of
the 1990s, the government began to recapitalize major banks only since 1998, while the6

We can easily generalize this three-period model into a multi-period model preserving the intuition

so that the basic implication of the model can be applied to the decade-long recession of the Japanese
economy.

21

government had insisted during the 1990s that they would never allow the occurrenceof any bank closure. This attitude of the Japanese government can be interpreted as
employing the policy of deposit guarantee without recapitalization. The observations in
the Japanese economy seem consistent with the prediction of the model: low production,
increase in liquidity of bank assets, and remarkable increase in (potential) fiscal cost ofbank bailouts.
Another characteristic of the Japanese economy, i.e., the continuous decline of assetprices seem to be consistent with the model too. In our model, the productivity oflong-term asset falls from R2 to (1 − l)R2 as a result of deposit guarantee without
recapitalization. Since we do not explicitly formulate the asset market in the model, we
cannot directly conclude that asset-price decline is caused by this productivity decline oflong-term asset. But since it seems plausible to interpret asset-price fall as productivity
decline of the asset in reality, we can argue that our model provides a possible explanation
for the continuous decline of asset prices in Japan.7
In summary, our model implies that Japan’s prolonged recession since the beginningof the 1990s may have been caused by inappropriate policy response to systemic bankinsolvency: deposit guarantee8 with “too little and too late” recapitalization.

References
Alexander, W.E., J.M. Davis, L.P. Ebrill, and C.-J. Lindgren eds. (1997). Systemic
Bank Restructuring and Macroeconomic Policy, International Monetary Fund.

7

Several authors incorporate the Diamond-Dybvig banking model with asset market. See Allen andGale (2000b, 2001), Diamond (1997), and von Thadden (1999).
8
Although Japan established the deposit insurance system in the 1970s, that guarantees up to 10
million yen for one depositor, there have been a tacit understanding that the government never let any
bank close and thus deposits are guaranteed unlimitedly. Since 1995, the government declared explicitlythat it guarantees all deposits unlimitedly for the time being. This unlimited deposit guarantee isscheduled to continue until 2005.

22

Allen, F. and D. Gale (1998).

“Optimal Financial Crises,” Journal of Finance 53(4):1245-1284.
– and – (2000a). “Bubbles and Crises,” The Economic Journal 110:236-256.

– and – (2000b). “Banking and Markets,” Wharton Financial Institutions Center
Working Paper 00-44-B, University of Pennsylvania.

– and – (2001). “Financial Fragility,” Wharton Financial Institutions Center Working
Paper 01-37, University of Pennsylvania.

Baer, H. and D. Klingebiel (1995). “Systematic Risk When Depositors Bear Losses: Five
Case Studies,” Research in Financial Services: Private and Public Policy 7:195-302

Bergoeing, R., P.J. Kehoe, T.J. Kehoe, and R. Soto (2002). “Policy-Driven Productivity
in Chile and Mexico in the 1980’s and 1990’s,” American Economic Review (Papers andProceedings) 92 (5): 16-21.

Bordo, M., B. Eichengreen, D. Klingebiel, and M. S. Martinez-Peria (2001). “Is The
Crisis Problem Growing More Severe?,” Working Paper, World Bank.

Caprio, G., Jr., and D. Klingebiel (1996). “Bank Insolvency: Bad Luck, Bad Policy,
or Bad Banking?,” Paper presented at Annual World Bank Conference on Development
Economics 1996.

– and – (1999). “Episodes of Systemic and Borderline Financial Crises,” World Bank,
mimeo.

Claessens, S., D. Klingebiel, and L. Laeven (2001). “Financial Restructuring in Banking
and Corporate Sector Crises: What Policies to Pursue?,” NBER Working Paper 8386.

23

Diamond, D. (1997). “Liquidity, Banks, and Markets,” Journal of Political Economy
105: 928-956.

– and P. Dybvig (1983). “Bank Runs, Deposit Insurance, and Liquidity,” Journal of
Political Economy 91: 401-419.

– and R. G. Rajan (2001). “Liquidity Risk, Liquidity Creation, and Financial Fragility:
A Theory of Banking,” Journal of Political Economy, 109(2): 287-327.

– and – (2002). “Bank Bailouts and Aggregate Liquidity,” American Economic Review(Papers and Proceedings) 92 (5): 38-41.

Freixas, X., B. Parigi, and J.-C. Rochet (1999). “Systemic Risk, Interbank Relationsand Liquidity Provision by the Central Bank,” mimeo.

Honohan, P. and D. Klingebiel (2000). “Controlling Fiscal Costs of Banking Crises,”
Policy Research Working Paper 2441, World Bank.

Leggett, K.J. (1994). “The Consequences of a Policy of Necessity: Bank Regulatory
Forbearance, 1986-1989,” Journal of Economics and Finance 18(1): 1-12.

Martin, A. (2001). “Liquidity Provision vs. Deposit Insurance: Preventing Banking
Panics without Moral Hazard?, ” Working Paper, Research Division, Federal Reserve
Bank of Kansas City.

Postlewaite, A. and X. Vives (1987). “Bank Runs as an Equilibrium Phenomenon,”
Journal of Political Economy, 95(3): 485-491.

24von Thadden, E.-L. (1999). “Liquidity creation through banks and markets: Multiple
insurance and limited market access,” European Economic Review 43: 991-1006.

25

Asset of All Banks in Japan

Liquid Assets
Loans and Bills Discounted

70.0%
65.8%
60.0%

60.7%

60.8%

60.8%

64.3%

64.7%

64.7%

64.6%
61.9%

61.7%
59.6%

58.4%

61.9%

63.4%

59.7%

58.4%

59.2%

23.0%

23.3%

2001

2002

50.0%

40.0%

30.0%

20.0%

22.4%

22.1%

21.8%

21.4%
19.3%

17.8%

19.6%
16.0%

15.3%

1992

1993

19.1%

17.2%

17.9%
15.9%

19.6%
16.4%

10.0%

0.0%
1986

1987

1988

1989

1990

1991

1994

1995

1996

1997

1998

1999

2000

Note: Liquid Assets consist of (1) Cash and Due from Banks, (2) Call Loans, (3) Receivables under Resale agreements, (4) Bills Bought, (5) Monetary Claims Bought, (6) Trading Assets,(7) Trading Account Securities, (8) Money Held in Trust, (9) Government Bonds, (10) Local Government Bonds, and (11) Corporate Bonds.
Source: Bank of Japan

All Industries
Manufacturingnon-manufacturing

Figure 2: Liquid Liability of Non-Financial Corporations
65.0%

trend (All Industries 1985Q1-1994Q2)

60.0%

55.0%

50.0%

45.0%

Source: Financial Statements Statistics of Corporations by Industry (Houjin Kigyou Toukei), Ministry of FinanceNote1: Q1:January-March quarter, Q2: April-June quarter, Q3: July-September quarter, Q4: October-December quarter
Note2: liquid liability = bills and accounts payable + short-term borrowings + short-term allowance + bonds

2002 Q1

2001 Q3

2001 Q1

2000 Q3

2000 Q1

1999 Q3

1999 Q1

1998 Q3

1998 Q1

1997 Q3

1997 Q1

1996 Q3

1996 Q1

1995 Q3

1995 Q1

1994 Q3

1994 Q1

1993 Q3

1993 Q1

1992 Q3

1992 Q1

1991 Q3

1991 Q1

1990 Q3

1990 Q1

1989 Q3

1989 Q1

1988 Q3

1988 Q1

1987 Q3

1987 Q1

1986 Q3

1986 Q1

1985 Q3

1985 Q1

40.0%

Parameters:
R1=1, R2=2, x=0.7,θ=2,η=1/2, S=5, L=4, W1=8