A Lender of Last Resort for Public Banks? Theory and an
Application to Japan Post Bank∗

Achim Hauck†

Uwe Vollmer‡

August 7, 2009

Abstract
This paper analyzes different regulatory arrangements for public banks in a liquidity crisis
and applies the analysis to the current privatization process of the Japan Post Bank (JPB).
We show that with suitably designed recapitalization rules, social welfare is higher thanwhen complete discretion is allowed. For the latter case, we derive conditions under which
the central bank or deposit insurer should act as lender of last resort (LLR). Moreover,
we report details of the JPB privatization and show that in the case of a liquidity crisisat the JPB, the current allocation of the LLR function to the Bank of Japan is inefficient.
Keywords: Public banking; lender of last resort, central bank, deposit insurance, forbearance, Japan Post Bank.
JEL-Codes: G21, G28, L32

∗

†

‡

Parts of this research were conducted while Uwe Vollmer was on leave at Kobe University. He wishes
to thank Kobe University (especially Ralf Bebenroth) for its hospitality and the Fritz Thyssen Stiftung
for its financial support. For their instructive comments we thank Tobias Hiller and the participantsof the 42nd Radein Research Seminar. Of course, the usual disclaimer applies.
Corresponding author: Heinrich-Heine-University Düsseldorf, Department of Economics, Universitätsstraße 1, 40225 Düsseldorf, Germany, Tel.: +49/(0)211/81-15342, Fax.: +49/(0)211/81-12196,
email: achim.hauck@uni-duesseldorf.de
University of Leipzig, Department of Economics, Grimmaische Straße 12, 04109 Leipzig, Germany,
Tel.: +49/(0)341/97-33571, Fax: +49/(0)341/97-33579, email: vollmer@wifa.uni-leipzig.de

1Introduction

Although there has been a tendency towards the privatization of financial institutions until
recently, public ownership of banks remains common. In 1995, the mean of government
ownership of banks around the globe was about 41.6 percent (La Porta et al., 2002).
Furthermore, several large banks have been partially or fully nationalized in the course of
the recent financial crisis. There are two opposing views on the welfare effects of publicbanks. On the one hand, several papers argue that public banks enhance social welfare
as they improve financial stability or provide funding for socially beneficial projects that,for instance due to externalities, could otherwise not be realized (Stiglitz, 1994; Allen andGale, 2000; Hakenes and Schnabel, 2004; 2006). On the other hand, it is claimed thatpublic banks lead to inefficiencies due to political pressures (Kornai, 1979; Shleifer and
Vishny, 1994; Boycko, Shleifer and Vishny, 1996).1
Unlike these contributions, our paper focuses on the liability side of public banks. In
particular, we analyze the optimal design of financial assistance for a public bank in aliquidity crisis. With this focus, our paper is closely related to Repullo (2000) and Kahn
and Santos (2005, 2006), who consider whether the central bank or the deposit insurer
should act as lender of last resort (LLR) or as a bank supervisor when information about
the solvency of banks is non-verifiable. They show that the optimal allocation of the
LLR function depends on the extent of a bank’s liquidity shortage and that, in a multiregulator arrangement, it is optimal to give supervisory power to the deposit insurer. A
central assumption of these papers is that, in the event of a liquidity crisis, a bank will beforced into bankruptcy unless it obtains financial assistance from the LLR. Accordingly,
1

The empirical literature provides more support for the hypothesis that public banks have negative
welfare effects, see, e.g., Dinç (2005), Faccio (2006), Faccio, Masulis and McConnel (2006) and Micco,
Panizza and Yanez (2007).

2

their results can be applied to private banks with dispersed ownership, which are unable
to raise fresh capital at short notice.2 In contrast, our model allows for a recapitalization
by bank owners when a liquidity crisis occurs, which is a plausible assumption in the caseof a public bank.
Our theoretical analysis focuses on two aspects. First, we investigate the incentivesof a public bank’s owner to provide fresh capital when a liquidity crisis has occurred.
We find that under discretion, recapitalization decisions are not socially optimal (i) if theowner places a different weight on social bank failure costs than society, or (ii) if deposits
are guaranteed by a deposit insurer, so that the owner does not bear all losses if thebank fails. In both cases, however, there is an optimal recapitalization rule that aligns
the bank owner’s preferences with social preferences, either by subsidizing (if the capital
injections are too small) or taxing (in the case of excessively large capital injections)recapitalization. Second, we discuss the implications of adding a LLR - the central bankor deposit insurer - to the discretionary case. We find that the allocation of the LLR
function will be relevant only if the deposit insurer guarantees deposits. Then, unlike thecentral bank, the deposit insurer provides funds as the LLR only if potential long-termlosses from guaranteeing deposits are not excessively large. Moreover, we derive conditions
under which the implementation of a LLR enhances social welfare. A LLR, however, does
not generally ensure a socially optimal recapitalization policy.
While these theoretical results can be applied to any public bank, we use them specifically to assess the regulatory framework of the Japan Post Bank (JPB), which - as partof Japan Post Holdings Co. - is one of the largest banks in the world. At present, Japan
2

Other papers that analyze the bail-out of private banks are Aghion, Bolton and Fries (1999), Diamond(2001a, 2001b) and Osano (2002, 2005). These papers focus on the incentive effects of public bail-outs
on bank managers.

3

Post Holdings Co. is owned completely by the government (the Ministry of Finance). It
is, however, undergoing a privatization process which commenced in October 2007. Overa ten year transition period, the government is obliged to reduce its stake in the Holding
Company to 33 percent. Moreover, the JPB is expected to start going public in 2010. At
the end of the transition period in September 2017, all of its shares must be traded on thestock market (Japan Post, 2008a).
JPB is an interesting case because during the privatization process, it has a hybrid
liability structure as contracts for postal savings have been differentiated between thoseconcluded before privatization (”old deposits”) and those concluded since October 1, 2007(”new deposits”). Old deposits together with corresponding asset accounts were assigned
to a separate corporate body that is legally independent of the JPB. These deposits areguaranteed by the government but the administration and investment of assets and liabilities are commissioned to the JPB (CEFP, 2004: 7-9; Office of the United States Trade
Representative, 2007). All new contracts, however, are insured through the Deposit Insurance Company of Japan (DICJ). These agencies and the Bank of Japan (BoJ), which
serves as lender of last resort for Japanese banks, may pursue different goals. Accordingly,
we assess how the current regulatory framework for the JPB affects the recapitalizationof the JPB in the event of a liquidity crisis.
With the assessment of the regulatory framework for JPB, our paper contributes tothe existing literature on JPB privatization. Porges and Leong (2006) describe the JPB
privatization and consider whether privatization creates a level playing field for postal
services. Amyx et al. (2005) and Maclachlan (2004, 2006) describe the political power
struggle before the decision to privatize the Japanese postal service. Imai (2007) analyzes
politician’s attitudes towards post bank privatization. Imai (2008) examines whether

4JPB helped raising untapped savings from savers who did not have access to private
depository services. We add to this literature by analyzing the likely outcomes of liquidity
assistance to JPB during the ten years transition period in which JPB is still owned by
the government.
The paper is organized as follows: Section 2 presents the model and derives optimalliquidity assistance in a benchmark case with no informational asymmetries. Section 3
analyzes the government’s incentives to recapitalize a public bank under discretion andproposes a rule which ensures optimal liquidity assistance. Section 4 assumes that sucha rule does not exist and analyzes the behavior of the government and the LLR under
discretion. Section 5 assesses the regulatory framework for JPB during privatization.
Section 6 concludes.

2

Model set-up and benchmark case

We consider an economy at date t with three types of agents: a government (the Ministryof Finance) that owns a public bank and two regulatory authorities, the central bank(CB) and the deposit insurer (DI), which may act as LLR. The public bank consists of
two divisions, i = o, n, where o stands for the ”old” division and n for the ”new” division.
As is the case for JPB, all deposits in the old division are guaranteed by the government,
whereas deposits with the new division are guaranteed by the deposit insurer.3
Inspired by Repullo (2000), we assume that at a previous date t − 1, each division
i of the bank has invested 1 USD in an indivisible, long-term project which matures at
date t + 1. If the division pursues the project until this date, it yields either a return
Ri (with probability ui ) or nothing at all. At the investment date t − 1 , the probability
3

To simplify the exposition, we normalize the insurance premium to zero.

5

of success ui is a random variable with density function f (ui ) satisfying E [ui ] Ri > 1.
Thus, when making the investment, the project will have a positive expected net present
value if held until maturity. At the current date t, the government as the division’s owner
privately observes the realization of ui . Moreover, the project is illiquid. If it is terminatedprematurely at the current date t, it will have only a small liquidation value Li ∈ (0, 1).
The long-term project is the only asset that division i possesses. That is, it has no liquid
funds.
Each bank-division i has financed its investment at date t − 1 through demandable
deposits Di = 1. They can either be withdrawn early at date t or late at date t + 1. A
publicly observable fraction vi ∈ (0, 1) of deposits is withdrawn early at the current date t.
As there are no liquid funds at this date, vi can be interpreted as a liquidity shock sufferedby division i. If it obtains funds of at least vi from the government as its owner or the LLR
in t, the division can pay out early depositors without terminating the long-term project.
Project returns at date t + 1 will then be either Ri or zero. Otherwise, the division will
be forced to liquidate its project. The liquidation value Li then materializes and there are
no returns in t + 1.
There is no legal relationship between the two divisions. Accordingly, the returns
earned by one division cannot be used for paying out depositors from the other. Therefore,a bankruptcy of either division can occur at date t or at date t + 1. It will occur at t if
the division does not receive sufficient liquidity from the owner or from the LLR, because
the liquidation proceeds Li do not suffice to pay out all depositors. A Bankruptcy will
occur at t + 1 if the division has obtained financial assistance at t and the project fails
at t + 1, so that there are no funds available for paying out late depositors. If division igoes bankrupt, there is a social cost Di ci = ci due to, e.g., systemic risk and contagion to6other banks. In addition, bankruptcy implies that the agent who has guaranteed deposits
is obliged to pay out those depositors who receive nothing from the bankrupt division. In
the event of bankruptcy of the old division, this obligation devolves on the government.
If, in contrast, the new division is bankrupt, the DI will pay out the remaining depositors.
All agents in the economy are risk-neutral. The preferences of agents are represented
by an additive-separable utility function defined over private profits and the social costsof bankruptcy. That is, as the owner of the bank, the government cares about expected
net profits of the bank’s divisions and the social bankruptcy costs ci . It weights thelatter with a factor α > 0. The two regulatory authorities, the CB and the DI, bothweight social bankruptcy costs with a factor β > 0. In addition, they care about expected
losses (or profits) when acting as LLR and, in case of the DI, about the expected costs ofguaranteeing deposits with the new division. In what follows, we restrict attention to thecase α + β ≥ 1.4 Finally, we assume that the regulatory authorities supervise the bank.
Hence, they find out the realization of the success probabilities uo and un at date t. Thisis, however, assumed to be non-verifiable by the public.
Before we analyze government and regulator behavior in a liquidity crisis, it is useful
to examine a benchmark case in which the probabilities of success uo and un are publicly
observable and verifiable. In this first-best case, from the perspective of social welfare,
it is optimal to provide financial assistance to division i at date t if the expected returns
less the social bankruptcy costs of continuing the project are (weakly) higher than the
4This assumption simplifies the formal analysis without affecting our qualitative results.

7liquidation proceeds, net of ci . Division i should thus obtain sufficient liquidity at date t
if ui Ri − (1 − ui ) ci ≥ Li − ci . Accordingly, as in Repullo (2000), we obtain:

ui ≥

Li
=: u∗i .
Ri + ci(1)

Hence, in order to maximize social welfare, each bank division should be closed if and only
if the success probability of its long-term project is below a certain threshold u∗i . The
threshold is higher, the higher is the liquidation value Li of the project and the lower are
the potential long-term project returns Ri and the bankruptcy costs ci . Moreover, u∗i does
not depend on the liquidity shock at t, as neither the potential liquidation proceeds nor
the long-term project returns or the social bankruptcy costs depend on vi .

3

Recapitalization by the government: Discretion vs. rules

When the success probabilities uo and un of the divisions’ projects are not publicly observable or verifiable, it is impossible to directly implement the first-best recapitalization rule(1). This raises the question of whether there is a suitable regulatory framework which
induces a first-best recapitalization. This section concentrates on government incentives torecapitalize the bank’s divisions after the liquidity shocks have materialized. We proceed
in two steps. First, we analyze the government’s behavior assuming that it can make discretionary recapitalization decisions. In this situation, the first-best will not be achieved
unless some rather specific conditions are met. Second, we show that recapitalization rules
can ensure that the government always chooses the first-best recapitalization policy.

8

3.1

Recapitalization under discretion

If no LLR provides liquidity assistance at date t, the government can choose between two
options after a liquidity shock vi has emerged. It can either inject its own liquid funds
KiGOV = vi into division i so that bankruptcy is avoided, or it can refuse to provide
funds implying that division i goes bankrupt at date t.5 The government’s incentives torecapitalize will differ between the divisions as it guarantees only the deposits of the olddivision but not the deposits of the new one.
For a given liquidity shock vo , the government will provide a liquidity injection KoGOV =vo to prevent the old division going bankrupt at date t if:−vo − (1 − vo ) + uo Ro − (1 − uo ) αco ≥ Lo − 1 − αco .

(2)

The left-hand side of (2) represents the government’s expected payoff if it recapitalizes theold division so that it continues to exist until t + 1. The government must then inject vo
into the division at date t. This amount is used to pay out early depositors. Moreover,at date t + 1, a total of 1 − vo must be paid to late depositors, irrespective of the successof the long-term project. At this date, the project yields a return Ro with probability uo ,while it fails with probability 1 − uo , in which case the government has to bear bankruptcy
costs co which are weighted by α. The right-hand side of (2) is the government’s payoff
when the division is closed at date t. Liquidation then yields Lo , the government mustpay 1 to all depositors and bear the weighted bankruptcy costs.
5

When recapitalizing a division, the government could, of course, also decide to provide more liquid
funds than needed by choosing KiGOV > vi . It will become apparent from the subsequent analysis,
however, that the government never prefers a recapitalization with KiGOV > vi over a recapitalization
with KiGOV = vi . Therefore, restricting attention to the latter case does not sacrifice any generality.

9

It follows from (2) that the government will recapitalize the old division only if:

uo ≥

Lo
=: uo .
Ro + αco(3)Thus, if the probability of success uo is not below the threshold uo , the government will
recapitalize the old division after the liquidity shock has emerged. As was the case in
the first-best scenario, the threshold does not depend on the magnitude of the liquidity
shock. This is because the government guarantees all deposits, implying that the timing ofdeposit withdrawals is irrelevant for its recapitalization decision. However, the thresholduo can differ from the first-best threshold u∗o . On the one hand, if α < 1 holds true, uo will
be higher than u∗o . In this case, the government is too tough. That is, for any uo ∈ [u∗o , uo ),it refuses to provide liquid funds to the old division, although this would be optimal froma first-best perspective. This overly tough behavior stems from the too small weight thatthe government places on bankruptcy cost avoidance when recapitalizing the division. If,
on the other hand, α > 1 and thus uo < u∗o holds true, the government will be too soft sothat it recapitalizes the division in some instances, although the division would be closed
in a first-best world.
At the new division, deposits are guaranteed by the DI. Therefore, the government
will inject liquidity KnGOV = vn into the new division if:−vn + un [Rn − (1 − vn )] − (1 − un ) αcn ≥ −αcn .

(4)The left-hand side of (4) is the government’s expected payoff in the case of a recapitalization of the new division; the right hand side is the expected payoff of liquidation.
In the former case, the liquidity injection vn at date t goes to early depositors. If the
10

project is successful at t + 1, returns Rn materialize and late depositors receive 1 − vn . If it
fails, the division pays nothing to late depositors and the government bears the weighted
bankruptcy costs. In the latter case, the liquidation proceeds Ln < 1 are used entirely
to repay depositors so that the government receives nothing but suffers from bankruptcy
costs.
Condition (4) results in:

un ≥

vn
=: un (vn ) ,Rn − (1 − vn ) + αcn(5)so that the government will, as was the case with the old division, recapitalize the newdivision at date t only if the probability of success is sufficiently high. Now, however, thethreshold un does depend on the magnitude of the liquidity shock with un being higher, thehigher is the level of vn . The intuition for this result is straightforward. The larger is theliquidity shock of the new division at t, the larger will be the amount that the government
loses if it keeps the division in operation and the project ultimately fails. Relative to afirst-best scenario, the government will thus be too soft in recapitalizing the new division
Rn −1+αcn
if the shock is small, vn < R
Ln =: vn∗ , while, at least as long as α is not too large,n −Ln +cn

it will be too tough if vn is large with vn > vn∗ .

3.2

Recapitalization under a rule

We have seen that discretion almost never leads to a first-best government recapitalization
policy. A recapitalization rule might, therefore, be a useful instrument for improving
governmental decisions. We will now determine whether this is indeed the case.
As a starting point, assume that, at date t − 1, the following rule is stipulated: If theobservable liquidity shock of division i turns out to be vi at date t, the government may
11recapitalize the division only if it injects KiGOV into it. The remainder KiLLR = vi −KiGOV
will then be provided by the LLR. Three remarks useful in clarifying this rule. First, we do
not place any restriction on KiGOV . That is, a rule can, for example, stipulate KiGOV > vi ,implying that KiLLR is negative. This could be interpreted as a tax that the government
must pay when recapitalizing a division. Second, we treat KiLLR as a one-off payment fromthe LLR to the division (in case of KiLLR > 0) or vice versa (in case of KiLLR < 0) thatis not repaid.6 Third, while the rule can be made contingent on the observable liquidity
shock vi , it cannot be made contingent on ui as the success probabilities of the projects
are not verifiable.
Given a rule KoGOV ; KoLLR , the government will prefer recapitalization of the olddivision over liquidation if:−KoGOV − (1 − vo ) + uo Ro − (1 − uo ) αco ≥ Lo − 1 − αco .

(6)Note that (6) differs from the discretionary case (2) in only one respect. Under the rule,
the government must pay KoGOV for a recapitalization instead of vo . From (6), it follows
that the government will recapitalize the old division if:

Lo − vo − KoGOV
uo ≥Ro + αco=: uruleo .

then leads directly to:
A comparison of u∗o and uruleo6

Alternatively, we could assume that, for instance, in the case of KiLLR > 0, the LLR provides a credit
which must be repaid with a positive probability. Then, we could reinterpret KiLLR as the difference
between the credit amount provided at date t and the expected credit repayment at date t + 1. This,
however, would complicate the analysis without yielding much additional insight.

12

Proposition 1 A recapitalization rule:

KoGOV = vo −1−α1−α
co Lo ; KoLLR =co Lo
Ro + coRo + co

ensures that the government will recapitalize the old division at date t if uo ≥ u∗o .
The essence of the proposition is that a rule can induce the government to recapitalize
the old division as in the first-best case. This is achieved by a money transfer KoLLR , which
does not depend on the liquidity shock vo . Recall from (3) that under discretion, the government will be too tough if α < 1. The optimal rule then stipulates a payment from the
LLR to the government, provided that the latter recapitalizes the division. This payment
makes the government softer so that a first-best recapitalization policy is achieved. If,
however, α > 1 holds true, the government tends to be too soft so that the optimal rule
induces first-best behaviour by forcing the government to pay a fixed amount to the LLR
when recapitalizing the division. Thus, in case of the old division, the optimal recapitalization rule will involve a subsidy if discretion is associated with too tough behaviour
while it will involve a tax if discretion is associated with too soft behaviour. Moreover,the subsidy or tax is independent of vo as the behavior of the government under discretion
does not depend on vo either. This is due to the government’s obligation to guarantee
deposits fully.
The implications of a rule KnGOV ; KnLLR for the new division are somewhat similar.
Here, the government is willing to recapitalize the division at date t if:−KnGOV + un [Rn − (1 − vn )] − (1 − un ) αcn ≥ −αcn .

13

(7)

Again, the only difference between (7) and the discretionary case (4) is the different
payment that the government must make if it wishes to avoid liquidation. Accordingly,
the threshold for a recapitalization of the new division is given by:

un ≥

KnGOV
=: urulen ,Rn − (1 − vn ) + αcn

and we thus obtain:

Proposition 2 A recapitalization rule:Rn − (1 − vn ) + αcn
Rn − (1 − vn ) + αcn
KnGOV =
Ln ; KnLLR = vn −Ln
Rn + cn
Rn + cnensures that the government will recapitalize the new division at date t if un ≥ u∗n .
The proposition shows that under an optimal rule, the government’s decision on recapitalizing the new division will always be the first-best. Contrary to the case of the olddivision, the optimal rule for the new division depends on the realization of the liquidity
LLR

shock vn . If it is relatively small, vn < vn∗ , the rule stipulates KnLLR < 0 with ∂K∂vnn> 0.

It thus punishes recapitalization with a tax that decreases in the liquidity shock vn of the
new division. This leads to first-best recapitalization as the government would be too softunder discretion with the degree of overly soft behaviour being lower, the higher is thelevel of vn . For large shocks vn > vn∗ , the rule implies a subsidy KnLLR > 0, which increases
in vn since under discretion, the government tends to tougher, the higher is the liquidity
shock.
To sum up this section, first-best recapitalization decisions can be achieved by suitably
designed recapitalization rules, even though the realization of the success probabilities of14

the divisions’ projects is not publicly observable. The optimal rule either subsidizes (whenthe government tends to be too tough) recapitalization or taxes it (when the government
tends to be too soft). For the old division, the optimal subsidy or tax depends on the weight
α that government places on bankruptcy costs while it is independent of the liquidity
shock. For the new division, recapitalization is taxed for small shocks and it is subsidized
for large shocks. Note that the optimal rule leaves no room for self interested discretionary
behavior on the part of the LLR. Therefore, under a recapitalization rule, it is actuallyirrelevant which institution serves as LLR.

4

The interaction of the government and the LLR

One might argue that optimal recapitalization rules, as derived in the preceding section,
are somewhat hard to implement. Because the government usually has at least somede facto or de jure legislative power, it could simply change the rules after a liquidity
shock has emerged. Consequently, this section analyzes the implications of discretionary
behavior of both the government, as the owner of the public bank, and the LLR for therecapitalization of the bank’s divisions. For either division i, we proceed in three steps.
First, we determine the maximum amount of liquidity that the government is willing to
inject in order to prevent bankruptcy. We then determine the maximum amount that
the LLR would be willing to provide in order to save the division. In the last step, we
determine the conditions under which the division will survive until date t + 1 and derive
the optimal institutional arrangement under discretion.

15

4.1

Old division

Under what conditions will the old division of the public bank obtain sufficient funds
at date t to prevent its bankruptcy when the government and the LLR jointly makediscretionary recapitalization decisions? The government prefers a recapitalization of thedivision over liquidation at date t if (6) is met. This condition can be rearranged to:b oGOV ,KoGOV ≤ vo − Lo + uo (Ro + αco ) =: Kb oGOV denoting the government’s maximum willingness to pay for a recapitalization
with Kb GOV decreases in the potential liquidation proceeds
at date t. As one would expect, Ko
Lo while it increases in the success probability uo of the project, the potential return
Ro , and social bankruptcy costs co . In the next step, we derive the behavior of the LLR
under discretion. For the old division, it is not necessary to differentiate between thecase of the CB acting as LLR and the DI as LLR. There are two reasons for this. First,
neither institution is responsible for paying out remaining depositors when the division
goes bankrupt. Second, both institutions attach the same weight β to social bankruptcy
costs co . Therefore, the preferences of the CB as LLR and the DI as LLR are identical. In
either case, the LLR cares only about the payoff of subsidizing or taxing recapitalization
at date t and about the weighted expected bankruptcy costs. That is, the LLR prefers to
pay KoLLR for a recapitalization of the old division rather than to liquidate it at date t if:−KoLLR − (1 − uo ) βco ≥ −βco16b oLLR . The LLR is thus willing to provide more funds for recapor if KoLLR ≤ uo βco =: K
italization, the higher is the success probability of the project and the higher are social
bankruptcy costs.
b oGOV of the government and the maximum
Based on the maximum willingness to pay K
b LLR of the LLR, we can now assess whether or not the old division will
willingness to pay Kob oGOV and K
b oLLR covers the liquidity shock vo ,be recapitalized at date t. If the sum of K
the government and the LLR will agree to recapitalize the old division which then survives
b GOV + K
b LLR falls short of vo , no arrangement exists which isuntil t + 1. If, however, Koo
acceptable for both parties and prevents liquidation. Accordingly, the government andthe LLR will recapitalize the division under discretion only if:

uo ≥

Lo
=: u
eo .
Ro + (α + β) co(8)

It is evident from (8) that the interaction of the government and the LLR leads to arecapitalization policy for the old division that is independent of the liquidity shock vo .
This is because in any circumstances the responsibility for paying out depositors devolves
on one of the two parties, namely the government. In addition, due to u
eo < uo , adding a
LLR to the discretionary case unambiguously leads to a softer recapitalization policy. As
the LLR, be it the CB or the DI, derives utility from a recapitalization of the old divisionin form of a possible avoidance of bankruptcy costs, it is always willing to provide someliquid funds for recapitalization so that the threshold for the success probability decreases.
Put differently, the implementation of a LLR leads to an internalization of the utility ofrecapitalization that would otherwise not be accounted for.

17

Unless α + β = 1, the threshold u
eo differs from the first-best threshold u∗o so that thecoexistence of the government and the LLR does not lead to a first-best recapitalization
policy under discretion. Instead, if α + β > 1, their recapitalization decisions will be toosoft. However, given that there is no recapitalization rule, should there be a LLR for theold division from the perspective of social welfare? In order to answer this question, we
need to compare the social loss under discretion with and without an LLR. This leads to:

Proposition 3 From a social welfare perspective, the existence of a LLR for the old division under discretion:• is never beneficial if α ≥ 1;
• is beneficial if α < 1 and β < β crit , where β crit is defined implicitly by:Z uo(uo (Ro + co ) − Lo ) f (uo ) duo = 0.
ueoProof. See Appendix.
The proposition suggests to distinguish between two cases under discretion. First, theintroduction of a LLR that can act in its own self interest will never lead to a recapitalization policy that is closer to first-best if the government places a high weight α ≥ 1 onsocial bankruptcy costs. In this case the government will already be too soft if there is no
LLR. If a LLR then is implemented, the policy will become even softer so that there are
more instances in which the old division is not liquidated, although this would raise social
welfare. Second, if the government places a relatively small weight α < 1 on bankruptcy
costs, the existence of a LLR may lead to higher social welfare. In this case, the government would be too tough if there were no LLR. The introduction of a LLR then again

18

leads to a softer recapitalization policy which will be more in line with first-best recapitalization as long as β does not exceed a threshold β crit . Otherwise, the LLR would induce
the government to recapitalize the old division, even for very small success probabilities,
so that social welfare will be higher if there is no LLR.

4.2

New division

In case of bankruptcy of the new division, depositor claims are not guaranteed by thegovernment. Therefore, a recapitalization of the new division that costs KnGOV will makethe government (weakly) better off than liquidation at date t if condition (7) is met.
b GOV := un (Rn − (1 − vn ) + αcn ) for a
The government is thus willing to pay, at most, Knrecapitalization at date t.
Let us now determine the LLR’s willingness to pay for a recapitalization of the newdivision at date t. Here, in contrast to the old division, we need to differentiate between
the CB and the DI acting as LLR. While the former institution suffers only from theweighted social costs if the division goes bankrupt, the latter must make an additional
payment to the remaining depositors, because it has guaranteed the deposits.
Assume for the moment that the CB acts as LLR. If it provides liquidity KnCB to thenew division in order to prevent liquidation, the CB will lose this amount and suffer fromthe bankruptcy costs with a probability of 1 − un . As liquidation leads to bankruptcy
costs with certainty, the CB prefers the former alternative over the latter if:−KnCB − (1 − un ) βcn ≥ −βcn .

b nCB := un βcn .
Consequently, its maximum willingness to pay for recapitalization satisfies K

19

The government and the CB as LLR will agree on a recapitalization of the new divisionb GOV + K
b CB covers the division’s liquidity needs vn . Under discretion, they will
only if Knnthus recapitalize the new division at date t only if:

un ≥

vn
=: u
eCB
n (vn ) .
Rn − (1 − vn ) + (α + β) cn(9)The threshold ueCB
increases in vn since, from the perspective of the government and thenCB, higher liquidity shocks are associated with a lower value of the DI’s deposit guarantee.
Moreover, a comparison of ueCBn and un reveals that adding the CB as LLR to the scenario
leads to a softer recapitalization policy, ueCBn < un . As was the case for the old division, thisis due to the possible internalization of the utility gained by the CB from not liquidating
and un imply:
the new division. With respect to social welfare, ueCBn
Proposition 4 From a social welfare perspective, the existence of the CB as LLR for the
new division under discretion:n +cn )• is never beneficial if α ≥ 1 + (1−Lncn)(RLn

n +cn )• is beneficial if α < 1 + (1−Lncn)(Rand vn > venCB , where venCB is implicitly defined
Ln

by:
Z un
(un (Rn + cn ) − Ln ) f (un ) dun = 0.
ueCBnProof. See Appendix.
If the choice is between no LLR or the CB as LLR, it is necessary two differentiate
between two cases. First, if α is sufficiently large, it will never pay to appoint the CB as
LLR. In this case the government is already too soft in the event of a liquidity shock if
there is no LLR. Accordingly, it makes no sense from a social perspective to implement an
20

LLR that makes the recapitalization policy even softer. In the case of a relatively small
α, however, appointing the CB as LLR will improve welfare if the liquidity shock is large
so that without a LLR, the government would recapitalize the new division in too few
instances.
When appointed as LLR, the DI will behave differently than the CB, because the
DI is obliged to pay out the remaining depositors when the new division goes bankrupt.
Therefore, if it acts as LLR and must pay KnDI for rescuing the division, it is ready to do
so only if:−KnDI − (1 − un ) (1 − vn + βcn ) ≥ − (1 − Ln ) − βcn .

(10)The left hand side of (10) reflects the expected payoff of the DI if the division survives
at date t. Then, the insurer pays KnDI at date t. If the project fails at date t + 1 andyields no return, the DI must pay 1 − vn to late depositors and incurs the bankruptcy
costs. According to the right hand side of (10), liquidating the division at date t leads to apayment obligation 1 − Ln of the DI and to inevitable bankruptcy costs. As the maximumb nDI := (1 − un ) vn − Ln +willingness to pay for a recapitalization, we therefore obtain: Kun (1 + βcn ). In contrast to all previous cases, the maximum willingness to pay of the LLR
may be negative. This will occur if the liquidity shock vn and the success probability unof the project are sufficiently small. In the case of a recapitalization, the DI then regards
the potential avoidance of bankruptcy costs as less important than the higher long-termcosts due to the deposit guarantee.

21

b nGOV of the government does not depend on the allocation of
The willingness to pay K
the LLR function. Consequently, the government and the DI would agree to recapitalizeb GOV + K
b DI ≥ vn . This leads to:
the new division in negotiations if K
n
n

un ≥

Ln
=: u
eDI
n .
Rn + (α + β) cn(11)

The threshold ueDIn does not depend on the liquidity shock. This is because in any circumstances, one of the two negotiating parties, the government or the DI, is responsible forpaying out depositors so that the share of early withdrawals. Furthermore, for sufficiently
small liquidity shocks, ueDIn > un holds true. When the shock vn is small, the DI incurs
an expected loss due to the deposits guarantee if the division survives. Accordingly, as
the LLR, it agrees to a recapitalization only if it obtains a payment from the government
so that the latter tends to be tougher than in the case without a LLR. However, such apayment can only be enforced if the LLR has the right to close the division at date t even
if the government wishes to continue operations. Otherwise, if the LLR had no unilateralright to close the division, the relevant threshold would be min{un , ueDIn }. Then, in thecase of ueDIn > un , the government could simply recapitalize the new division and ignore
the losses of the deposit insurer as if there were no LLR.
Assuming that the LLR does indeed have the unilateral right to close the new division,
a comparison of the expected social loss with and without the DI as LLR leads to:22

Proposition 5 From a social welfare perspective, the existence of the DI as LLR for
the new division under discretion is beneficial if either vn < ṽ˜nDI or vn > venDI , with
Rn −1+αcn
Ln , and where venDI > ṽ˜nDI is defined implicitly by:ṽ˜nDI := Rn −L
n +(α+β)cn

Z un(un (Rn + cn ) − Ln ) f (un ) dun = 0.
ueDInProof. See Appendix.
The proposition states that the social value of the DI as LLR for the new division
depends on the extent of the liquidity shock vn . If the shock is small, vn < ṽ˜nDI , thegovernment would pursue a very soft recapitalization policy. The DI should then be
appointed as LLR in order to toughen the behaviour of the government. This will indeed
be achieved, because for small liquidity shocks, the DI is unwilling to provide funds forrecapitalization. Instead, the DI will impose a tax on recapitalization. If the shock is rather
large, vn > venDI , the existence of the DI as LLR will be beneficial as well. The government
would then be too tough without a LLR and the DI has a positive willingness to pay forrecapitalization, so that recapitalization will be softer and closer to the first-best if the DIserves as LLR. In the case of intermediate liquidity shocks vn ∈ ṽ˜nDI , venDI , adding the
DI as LLR to the scenario does not enhance social welfare. That is, the recapitalization
decisions of the government will then be more in line with the first-best if the DI does notserve as LLR.
Form propositions 4 and 5, we can infer two essential properties of the optimal institutional arrangement for the new division in a liquidity crisis. First, if the liquidity
shock is sufficiently moderate, from a social point of view, it will never pay to appointa LLR which is willing to provide funds for recapitalization. Then, there should either23

be no LLR at all, or the DI as LLR should tax recapitalization. Second, if the liquidity
shock is sufficiently large, there should be a LLR which is willing to provide funds forrecapitalization. Whether this role should be allocated to the CB or to the LLR depends
on the extent of the shock and on the respective preferences of the involved agents.

5

Assessment of JPB privatization

The scenario introduced in section 2 of a bank divided into two separate divisions with arisky project and subject to substantial deposit withdrawals is appropriate for JPB for thefollowing reasons. As of March 31, 2008, JPB held 73.9% of its total assets in the form ofJapan Government Bonds (JGB) that are subject to substantial interest rate risk. Sinceinterest rates were abnormally low during the last decade, they are more likely to increase
than to fall in the near future; even a small increase of the interest rate for 10 years JGBs
of 50 basis points for five years is likely to result in a substantial decrease in JPB’s cash
flow and profit in fiscal year 2011. Moreover, JPB is rapidly losing market share to otherJapanese banks. While the balance of postal savings amounted to 204 trillion YEN at theend of March 2006, they decreased to 183 trillion YEN at the end of February 2008 (a lossof 10.3%; Japan Post, 2008b).
In Japan at present, there are no recapitalization rules for public banks. This is also
true for JPB during the privatization process. That is, there is no formal agreement
between the Ministry of Finance and either DICJ or BoJ on how to recapitalize an illiquid
division of JPB after a liquidity shock has occured. In the light of our theoretical model,
the involved parties thus forego the opportunity to ensure a first-best recapitalization
policy by stipulating rules like those suggested in propositions 1 and 2. Instead, they24

have complete discretion with respect to a recapitalization of a JPB division with urgent
liquidity needs.
Taking the non-existence of recapitalization rules as given, the question arises ofwhether the current regulatory framework is at least the socially optimal arrangement
under discretion. As mentioned in the introduction to this paper, the LLR function in
Japan is allocated to the BoJ. It can decide freely whether or not to serve as LLR whena bank is in trouble. In particular, there is no formal regulation that prevents liquidity
provision by the BoJ in the case of relatively small liquidity shocks. Moreover, there are
no formal restrictions on the extent to which the BoJ can provide liquidity assistance.
According to our results in section 4, there should be a LLR for the old division in
the discretionary case only if two prerequisites are met. First, the government places asufficiently small weight α on social bankruptcy costs co and, second, the LLR’s weight βof co is not too large. Only then, having no LLR would result in too tough recapitalization
decisions and the implementation of a LLR, by leading to a softer recapitalization policy,
would improve social welfare. It is reasonable to suspect that neither of the two prerequisites is fulfilled in the Japanese case because the decision makers had painful experiences
during the financial crisis in the 1990s.7 Accordingly, the current regulatory framework
for the old division of JPB is likely to be suboptimal even if only the discretionary case isconsidered. However, since the importance of the old division will decline in the future as
old savings expire, this problem may resolve itself in due course.
With regard to the new division of JPB, the analysis in section 4 shows that the
impact of a LLR on social welfare depends on the size of the liquidity shock and the7

According to Nakaso (2001: 28) there ”...may well be cases in which the central bank will still decide
to extend emergency support for the sake of financial stability, even knowing that it may result in aloss with a certain probability.”25

weight attached by the government and the respective LLR to the social bankruptcy
costs. Here, assigning the LLR function to BoJ is socially beneficial under discretion if
the government’s weight α is smaller than the benchmark value given in proposition 4 andif the liquidity shock is not too small. As argued in the preceding paragraph, the former
condition may be violated while the latter condition will definitely be violated whenever
the liquidity crisis is not too severe. In such a situation, it would be better to prohibit
liquidity assistance by the BoJ and either to have no LLR at all or to appoint the DIJC
as LLR.
To conclude this section, we have demonstrated that the current regulatory framework
for the divisions of JPB in a liquidity crisis is likely to be suboptimal from a social welfare
perspective. The Japanese government would be well advised to introduce recapitalization
rules like those defined in propositions 1 and 2. Alternatively, if explicit rules are unenforceable, social welfare could at least be improved by limiting the BoJ’s rights to provide
liquidity assistance. A reform of the regulatory framework is of particular importance forthe new division of JPB as this division will gain in size relative to the old division overthe next years.

6

Summary

The purpose of this paper was twofold. First, we considered the consequences of different
regulatory arrangements for public banks in a liquidity crisis. Second, we applied our
results to the case of the Japan Post Bank (JPB) and showed that the current allocationof regulatory powers over JPB, which is still owned by the Japanese government, is suboptimal. We have shown that discretionary recapitalization decisions by the government
result in either excessively small or excessively large capital injections in a wide variety
26

of circumstances while a suitably designed recapitalization rule could ensure a first-best
recapitalization policy. We have also shown that there should be a LLR that is willing toprovide funds for recapitalization only if the liquidity shock is sufficiently large. Whether
this role should be allocated to the CB or to the DI depends on the size of the liquidity
shock and on the respective preferences of the involved agents.

Appendix
Proof of proposition 3
In a first-best world, (1) implies that the expected social payoff of division o at date t is:πof b =






∗
 uo Ro − (1 − uo ) co if uo ≥ uo Lo − coif.

uo < u∗o Accordingly, as uo is unobservable, it follows from (3) that at date t the expected social  Ruloss under discretion without a LLR for the old division is E lo = u∗o φo (uo ) f (uo ) duoowith φo (uo ) := uo (Ro + co ) − Lo while it follows from (8) that the expected social loss
h i R
u
e
under discretion with a LLR for the old division is E e
lo = u∗o φo (uo ) f (uo ) duo . Underodiscretion, the existence of a LLR for the old division thus reduces the expected social loss
only if:Z uo
φo (uo ) f (uo ) duo > 0.

(12)

u
eo

Note that φo (uo ) is increasing in uo and equal to zero for uo = u∗o . Now, we can distinguish
two cases. First, suppose that α ≥ 1 (and thus u
eo < uo ≤ u∗o ). Then, φo (uo ) ≤ 0 for all
uo ∈ [euo , uo ], so that (12) is violated. Second, suppose that α < 1 (and thus ueo ≤ u∗o < uo ).
27

Then, the left-hand side (LHS) of (12) is positive for β = 1 − α (in which case u
eo = u∗oholds true) and decreasing in β (as ueo is decreasing in β). Therefore, (12) is met only if
β < β crit , where β crit is implicitly defined by

R uoueo φo (uo ) duo = 0.

Proof of proposition 4
By parallel arguments as in the proof of proposition 3, it can be shown that under discretion, having the CB as LLR for the new division reduces the expected social loss only
if:

Z un
φn (un ) f (un ) dun > 0.

(13)

u
eCB
n

Note that φn (un ) := un (Rn + cn ) − Ln is increasing in un and equal to zero for un = u∗n .
n +cn )
Now, we can distinguish two cases. First, suppose that α ≥ 1 + (1−Lncn)(R(and thus
Ln

 CB

u
eCB
< un ≤ u∗n for all vn ). Then, φn (un ) ≤ 0 for all un ∈ u
en , un so that (13) isnn +cn )
nviolated. Second, suppose that α < 1 + (1−Lncn)(Rand define vn∗∗ := RnR−1+(α+β)cLn .
Ln
n −Ln +cn

Then, there are three subcases. First, if vn < vn∗ , it follows that u
eCB< un < u∗n so thatn CB
φn (un ) ≤ 0 for all un ∈ u
en , un implying that (13) is violated. Second, if vn ∈ [vn∗ , vn∗∗ ],it follows that ueCB≤ u∗n ≤ un so that the LHS of (13) is (i) not positive if vn = vn∗n(and thus ueCB< u∗n = un ), positive if vn = vn∗∗ (and thus ueCB= u∗n < un ), and (iii)nnincreasing in vn . Therefore, (13) is met if vn > venCB , where venCB is defined implicitly by
R un

φn (un ) f (un ) dun = 0.
ueCBnThird, if vn > vn∗∗ , it follows that u∗n < ueCB≤ un so that
n

 CB

φn (un ) > 0 for all un ∈ u
en , un implying that (13) is met.

28

Proof of proposition 5
By parallel arguments as in the proof of proposition 3, it can be shown that under discretion, having the DI as LLR for the new division reduces the expected social loss only
if:

Z un
φn (un ) f (un ) dun > 0.

(14)

u
eDI
n

Rn −1+αcn
Define ṽ˜nDI := Rn −L
Ln . Now, we can distinguish three cases. First, suppose thatn +(α+β)cn



∗
eDI
eDI
(with strict
vn < ṽ˜nDI (and thus un < un ≤ un ). Then, φn (un ) ≤ 0 for all un ∈ un , uninequality for some un ) implying that (14) is met. Second, suppose that vn ∈ ṽ˜nDI , vn∗
 DI
≤ un < u∗n ). Then, φn (un ) ≤ 0 for all un ∈ u
en , un implying that(and thus ueDIn∗(14) is violated. Third, suppose that vn ≥ vn∗ (and thus ueDIn ≤ un < un ). Then, the∗LHS of (14) is (i) non-positive if vn = vn∗ (and thus ueDIn ≤ un = un ) and (ii) increasing

in vn . Therefore, (14) is met if vn > venDI , where venDI > ṽ˜nDI is defined implicitly by
R un

φn (un ) f (un ) dun = 0.
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