The Seeds of a Crisis: A Theory of Bank Liquidity

and Risk-Taking over the Business Cycle1

Viral Acharya2

New York University Stern School of Business, NBER, CEPR and ECGI.

Hassan Naqvi3

National University of Singapore and NUS Risk Management Institute.

March 10, 2010

1We thank Anjan Thakor for useful suggestions. Comments by conference and

seminar participants at the 2010 AEA Annual Meeting, CEMFI, Lancaster University Management School, and National University of Singapore are also appreciated.

We also acknowledge Hanh Le and Michelle Zemel for research assistance.

2Contact: Department of Finance, Stern School of Business, New York University, 44 West 4th Street, Room 9-84, New York, NY 10012. Tel: +1 212 998 0354.

Fax: +1 212 995 4256. e-mail: vacharya@stern.nyu.edu

3Contact: Department of Finance, NUS Business School, Mochtar Riady Building, BIZ 1 #7-62, 15 Kent Ridge Drive, National University of Singapore, Singapore

119245. Tel: +65 6516 5552. Fax: +65 6779 2083. e-mail: naqvi@nus.edu.sg
Abstract

We examine how the banking sector may ignite the formation of asset price

bubbles when there is access to abundant liquidity. Inside banks, given

limited liability and lack of observability of effort, loan officers (or risk takers)

are compensated based on the volume of loans. Outside banks, when there is

heightened macroeconomic risk, investors reduce direct investment and hold

more bank deposits. This ‘flight to quality’ leaves banks flush with liquidity,

lowering the sensitivity of bankers’ payoffs to downside risks of loans and

inducing excessive credit growth and asset price bubbles. The seeds of a

crisis are thus sown. We show that the optimal monetary policy involves

a “leaning against liquidity” approach: A Central Bank should adopt a

contractionary monetary policy in times of excessive bank liquidity in order

to curb risk-taking incentives at banks. Conversely, a Central Bank should

follow an expansionary monetary policy in times of scarce liquidity so as to

boost investment.

JEL Classifications: E32, E52, E58, G21

Keywords: Bubbles, flight to quality, Greenspan put, leaning against

liquidity, liquidity, monetary policy, moral hazard
“For too long, the debate has got sidetracked. Into whether we can

rely on monetary policy ‘mopping up’ after bubbles burst. Or into whether

monetary policy could be used to control asset prices as well as doing its

orthodox job of steering nominal trends in the economy...” - Paul Tucker,

Executive Director for Markets and Monetary Policy Committee (MPC)

member at the Bank of England.1

1 Introduction

It is clear that in the period leading up to the global financial crisis of 2007-

2009, credit and asset prices were growing at a ferocious pace.2 In the United

States, for example, in the five-year period from 2002 to 2007, the ratio of

debt to national income went up from 3.75 to one, to 4.75 to one. It had

taken the prior full decade to accomplish this feat, and fifteen years prior

to that. During this same period, house prices grew at an unprecedented

rate of 11% per year while there was no evidence of appreciating borrower

quality over this period. The credit growth was across board, in mortgages,

especially sub-prime ones, in financing of leveraged buy-out transactions

(LBOs), and through increasing issuance of low-rated bonds. This rapid

rise in asset volume and prices met with a precipitous fall. For instance, the

median house price divided by rent in the United States3 over the 1975 to

2003 period varied within a relatively tight band around its long-run mean.

Yet starting in late 2003, this ratio increased at an alarming rate. In mid

2006, however, the ratio flattened and kept falling sharply until 2009 (See

Figure 1).

What caused this tremendous asset growth and the subsequent puncture

is likely to intrigue economists for years. Some have argued that the global

economy was in a relatively benign low-volatility environment in the decade

leading up to the ongoing crisis (the so-called “Great Moderation", see Stock

1Bank of England’s Quarterly Bulletin 2008 Q2, Volume 48 No. 2.

2The series of facts to follow are borrowed from Acharya and Richardson (2009a).

3 In particular, this is the ratio of the Office of Federal Housing Enterprise Oversight

(OFHEO) repeat-sale house price index to the Bureau of Labor Statistics (BLS) shelter

index (i.e., gross rent plus utilities components of the CPI)

1
Figure 1: House Price to Rent Ratio. The Figure graphs the demeaned

value of the ratio of the Office of Federal Housing Enterprise Oversight

(OFHEO) repeat-sale house price index to the Bureau of Labor Statistics

(BLS) shelter index (i.e., gross rent plus utilities components of the CPI).

Because of demeaning, the average value of this ratio is zero.

2
and Watson, 2002). In departure from this view, we argue that it is likely

not a coincidence that the phase of remarkable asset growth described above

started at the turn of the global recession of 2001—2002. In fact, in response

to the unprecedented rate of corporate defaults and heightened macroeconomic risk during that recession, the Federal Reserve lowered interest rates

to 1%, the lowest level since 1958. A period of abundant availability of

liquidity to the financial sector ensued, bank balance-sheets grew two-fold

within four years, and when the “bubble burst”, a number of agency problems within banks in those years came to the fore. Such problems were

primarily concentrated in centers that were in charge of undertaking large

risks and took the form of risk-takers being paid out huge bonuses based on

the volume of assets they created rather than on (long-term) profits they

generated.4

In this paper, we develop a theoretical model that combines these ingredients and explains why access to abundant liquidity aggravates the risktaking moral hazard at banks, giving rise to asset price bubbles. Somewhat

perversely, the seeds of crisis are sown when the macroeconomic risk is high

and investors in the economy switch from investments to savings in the

form of bank deposits. We argue that these bubbles can be counteracted

by Central Banks with a contractionary monetary policy in such times, and

conversely are exacerbated by expansionary monetary policy. Expansionary

monetary policy may be tempting in such times, but this may flush banks

with (even more) liquidity, which can lead to credit booms and asset price

bubbles.

We first develop a benchmark model in Section 2.1 wherein the representative bank collects deposits from investors and then allocates a fraction

of these deposits to investment projects. The bank faces random deposit

withdrawals and in case of liquidity shortfalls suffers a penalty cost. The

penalty cost could be interpreted as the cost of fire sales or alternatively

4See Rajan (2008) and Chapter 8 of Acharya and Richardson (2009b) for a discussion

of bank-level principal-agent problem — the “fake alpha” problem when performance is

measured based on short-term returns — and the role that this problem played in causing

the financial crisis of 2007—2009.

3
the cost of raising finance from markets. In order to avoid such costs the

bank has an incentive to set aside some reserves (cash and marketable assets

or other forms of ready liquidity). The rest of the deposits are invested in

projects (e.g. houses) depending on the demand for loans (e.g. mortgages).

The problem of the bank is to choose the optimal lending rate that maximizes its expected profits subject to the depositors’ participation constraint.

We show in this benchmark model that the bank lending rate appropriately

reflects the underlying risk of the project.

In Section 2.2 we enrich the model to study how agency problems within

the bank affect the pricing of loans. In practice, bankers and loan officers

(“bank managers”) often have incentives to give out excessive loans since

their payoffs are proportional to the amount of loans that are advanced.5

We show that such incentives can arise as part of an optimal contracting

outcome of a principal-agent problem when managerial action or effort is

unobservable. Further, under the optimal contract, bank managers may

have an incentive to underprice the risk of the investments. However, we

show that such mispricing of risk only occurs when the bank is awash with

liquidity (deposits). This is because in presence of excessive liquidity the

manager attaches too little weight to the scenario where the bank might

later face liquidity shortfalls. In other words, excessive liquidity encourages

managers to disregard downside risk, overinvest and underprice the risks of

projects.

We then show in Section 2.2.3 that such behavior ultimately has an

impact on asset prices. We assume that the demand for loans arises from

investments by the household or corporate sector in underlying assets of the

economy. To show how asset price “bubbles” are formed we first define the

“fundamental” asset prices as those that arise in the absence of any agency

frictions within banks. We construct the optimal demand function for assets

by bank borrowers and then solve for the underlying asset price given the

5The Bureau of Labor Statistics reports that “Most (loan officers) are paid

a commission based on the number of loans they originate.” (See the Bureau

of Labor Statistics’ Occupational Outlook Handbook, 2008-09 Edition available at

http://www.bls.gov/oco/ocos018.htm#earnings.)

4
market clearing condition that the aggregate demand for assets should equal

their finite supply. If the bank lending rate underprices risks, then there

is an increase in aggregate borrowing from banks. This in turn fuels an

excessive demand for assets in the real sector which leads to prices rising

above their fundamental values. We interpret this asset price inflation as a

“bubble”. Importantly, such bubbles are formed only when bank liquidity

is high enough as only then do bank managers underprice risk.

In Section 3 we study when asset price bubbles are most likely to be

formed. We show that this is the case when the macroeconomic risk in

the economy is high. When macroeconomic risk increases, depositors avoid

direct entrepreneurial investments and prefer to save their money in bank

deposits which are perceived to be safer. Gatev and Strahan (2006) offer

direct empirical evidence consistent with this effect. In our model, such

“flight to quality” results in excessive bank liquidity and induces bubble

formation in line with our earlier results.

Finally, we study the implications of this result for optimal monetary

policy. We show that if the Central Bank adopts a contractionary monetary policy in times of excessive bank liquidity, then it can counter the

flight to quality by drawing out the increases in bank liquidity and avoiding

the emergence of bubbles. On the contrary, if the Central Bank adopts an

expansionary monetary policy then this accentuates the formation of bubbles. Intuitively, an increase in the money supply only serves to increase

bank liquidity further when there is already a flight to quality of deposits.

Our model can thus explain how lax monetary policy by the Scandinavian

Central Banks in 1980’s, Bank of Japan during 1986-1987, and the Federal

Reserve in the United States during most of the Greenspan era culminated

in housing and real estate bubbles in these countries.

However, in times of scarce bank liquidity, banks raise lending rates

which adversely affects aggregate investment. We show that during these

times if the Central Bank adopts an expansionary monetary policy then

it can boost aggregate investment by effectively injecting liquidity into the

banking system. We thus argue in Section 4 that the optimal monetary

policy involves a “leaning against liquidity” approach.

5
Proponents of the ‘Greenspan camp’ argue that monetary policy should

not be geared towards avoiding the emergence of bubbles and should focus

on targeting the natural interest rate and the natural rate of employment

as has traditionally been the case. This is justified on the basis that central

banks cannot pinpoint an asset price bubble. Nevertheless, we prove that

targeting bank liquidity is optimal even if central banks are not aware of

where the economy is in the business cycle. Since the asset price bubble is

intuitively tied to bank liquidity, we believe that the central banks’ task in

identifying times for employing a contractionary policy is not as onerous as

is often suggested.6

While Jensen and Meckling (1976) showed that leverage induces equityholders to prefer excessive risk, our point is concerned with risk-taking

incentives inside banks as a function of liquidity. On this front, our result is

similar to Myers and Rajan (1998) wherein access to liquidity allows financial firms to switch to riskier assets, and in anticipation of such behavior,

renders them illiquid ex ante. The channel in our model is somewhat different though: When banks are flush with liquidity, managers are hedged from

the downside risks they undertake, and this induces risk-taking incentives.

Allen and Gale (2000) show in a model of risk-shifting that uncertainty

in monetary policy acts to exacerbate the risk-taking incentives ex ante and

fosters an asset price bubble. Diamond and Rajan (2008) show that lowering interest rates ex post may be desirable for a central bank in order

to avoid bank runs and fire sales, but that this can induce moral hazard

and incentivize banks to hold more illiquid assets. It may thus be desirable for the Central Bank to commit to raising interest rates when they are

6 In fact, a number of economists, including those who traditionally believed that monetary policy should not react to asset price bubbles, have revised their priors on its conduct.

Some examples include: (i) “Given the events of the last eight months, it would be foolish

not to reconsider the Greenspan doctrine,” by Kenneth Rogoff, Financial Times, 16 May

2008; (ii) “I think I am still with the orthodoxy but I have to admit that recent events

are sowing seeds of doubt,” by Alan Blinder, Financial Times, 16 May 2008; and, (iii) “A

Central Bank should bear in mind those long-run consequences of asset price bubbles and

financial imbalances in the setting of current interest rates,” by Charles Bean, Financial

Times, 16 May 2008.

6
low. In our model, the focus is also on the ex-ante effects of central bank

intervention. The difference arises from the fact that our model embeds

the principal-agent problem between dispersed shareholders of banks and

the loan officers or the risk takers. The severity of the agency problem is

affected by the availability of liquidity, and thus by central bank interventions. Importantly, in the absence of bank-level agency problems, there is no

excess in credit growth and central bank interventions do not distort credit

outcomes.

In another related paper Farhi and Tirole (2009) study how monetary

policy has an affect on the collective behavior of banks. They argue that

banks have an incentive to correlate their risk exposures because if everyone

engages in maturity transformation, authorities have little choice ex post

other than facilitating refinancing. They thus argue that the optimal monetary policy is time inconsistent. Acharya and Yorulmazer (2007) also study

the incentives for banks to correlate their risks when regulatory forbearance

(bailout policy) suffers from a time-consistency problem. In a model with

two banks they show that banks have an incentive to reduce their correlations given the possibility that one bank may acquire the other. However, if

the rents obtained from such an acquisition are outweighed by the bailout

guarantee when both banks fail, then banks herd and seek to maximize

their correlation. In contrast to these papers, our paper studies whether or

not monetary policy should target asset prices given the agency problems

inherent in banks.

Thakor (2005) argues that bank over-lending is due to them permitting

higher loan commitments and not invoking the MAC clause during booms

given reputational concerns.7 The alternative view provided by our paper

is that over-lending occurs mainly due to higher spot lending by banks subsequent to a deposit influx.

7Material Adverse Change (MAC) clause permits the bank to decline to lend under

the commitment if the borrower’s financial condition has declined significantly since the

commitment was sold.

7
2 Benchmark model

2.1 Base case

We consider a three-period model of a bank wherein the bank at t = 0

receives deposits D from risk-neutral investors. Each investor deposits 1

unit of his endowment in the bank. The reservation utility of depositors is

given by u ̄. Hence in order to secure deposits the bank needs to set the rate

of return on deposits, rD, such that the depositors earn an expected payoff

of at least u ̄.

The bank subsequently makes investments in projects while holding a

fraction of the deposits as liquid reserves, R. The bank-funded projects

either succeed or fail at t = 2. The probability of success of bank projects is

given by θ and in the event the project is successful it pays off at t = 2. The

project is illiquid in the sense that if it were to be liquidated prematurely

at t = 1 the bank faces a penalty or a liquidation cost. The bank observes

θ and sets rL which is the (gross) rate of return on loans. When choosing

the lending rate, the bank takes into account the demand function for loans

(say, by the corporate sector) which is given by L(rL) where L0 (rL) < 0.

Bank reserves are thus given by:

R = D − L(rL)

The bank may experience withdrawals at t = 1 and for simplicity we

assume that the fraction of depositors who experience a liquidity shock is a

random variable given by x ̃, where x ∈ [0, 1].

8 The cumulative distribution

function of x ̃ is given by F (x) while the probability distribution function is

denoted by f (x). Each depositor who withdraws early receives 1 unit of his

endowment back at t = 1. Thus the total amount of withdrawals at t = 1

is given by xD ̃ . If the realization of xD ̃ is greater than R, then the bank

8As in Allen and Gale (1998) and Naqvi (2007) we could have assumed that x ̃ is

correlated with asset quality news in the sense that depositors receive a noisy signal of θ

on which they base their decision on whether or not to run. While this is more realistic,

it does not affect our qualitative result but highly complicates the analysis. Hence similar

to Diamond and Dybvig (1983) and Prisman, Slovin and Sushka (1986) we assume that

x ̃ is random.

8
faces a liquidity shortage, and it incurs a penalty cost, given by rp (xD − R),

which is proportional to the liquidity shortage, where rp > rL > 1.

The penalty cost can be justified in a number of ways. The bank may

be forced to cover the shortfall in a costly manner by selling some of its

assets prematurely at fire-sale prices. This is particularly likely when firms

in other industries are also facing difficulties.9 Alternatively the bank can

raise external financing via capital markets. However, this is also costly

because raising equity leads to dilution of existing shareholders due to the

debt overhang problem (Myers, 1977). Furthermore, raising external finance

entails a price impact due to the adverse selection problem a la Myers and

Majluf (1984). Capital-raising can also entail costs related to monitoring

that the new financier must undertake. Finally, if the bank attempts to

cover the shortfall by emergency borrowing from the central bank, this can

also be costly as the central bank may charge a penalty rate. Finally, apart

from pecuniary costs, the bank may also suffer non-pecuniary costs such

as a reputational cost, e.g., the stigma associated with borrowing from the

central bank’s emergency facilities.

If the projects financed by bank borrowings are successful, then the bank

is solvent and is able to repay the patient depositors the promised rate of

return of rD at t = 2, whilst the equityholders consume the residual returns

from bank investments after these depositors have been paid off. However,

in case of the failure of bank-funded projects, the surplus reserves, R − xD ̃ ,

if any, are divided amongst the patient depositors whilst the equityholders

9Shleifer and Vishny (1992) argue that the price that distressed firms receive for their

assets is based on industry conditions. In particular, the distressed firm is forced to sell

assets for less than full value to industry outsiders when other industry firms are also

experiencing difficulties. There is strong empirical support for this idea in the corporatefinance literature, as shown, for example, by Pulvino (1998) for the airline industry, and by

Acharya, Bharath, and Srinivasan (2006) for the entire universe of defaulted firms in the

US over the period 1981 to 1999 (see also Berger, Ofek, and Swary, 1996, and Stromberg,

2000). In the evidence of such specificity for banks and financial institutions, James (1991)

shows that the liquidation value of a bank is typically lower than its market value as a

going concern. In particular, his empirical analysis of the determinants of the losses from

bank failures reveals a significant difference in the value of assets that are liquidated and

similar assets that are assumed by acquiring banks.

9
t = 0

• Bank raises deposits

• Bank observes success

 probabilit L • Investments made

 and bank sets aside

reserves R

t = 1

• Bank suffers early

 withdrawals, xD

• Bank incurs a penalty

cost if xD>R

t = 2

• Bank projects

 either succeed

 or fail

• Payoffs divided

 among parties

Figure 2: Benchmark model: Timeline of events

consume zero. The sequence of events is summarized in the timeline depicted

in Figure 2.

Given this setup the risk neutral bank’s problem is as follows:

max r∗

L,r∗D,R∗ Π = π − rpE [max ( ̃xD − R, 0)] (1)

subject to

E ( ̃x) + (1 − E ( ̃x)) ∙θrD + (1 − θ)

E [max (R − xD,  ̃ 0)]

(1 − E ( ̃x)) D

 ̧

≥ u ̄ (2)

where π is given by:

π = θ {rLL(rL) − rDD (1 − E ( ̃x)) + E [max (R − xD,  ̃ 0)]} . (3)

The above program says that the bank chooses deposit and lending rates as

well as the level of bank reserves so as to maximize its expected profits, π,

net of any penalty costs incurred in the case of liquidity shortage subject

to the participation constraint of the depositors. Expression (2) represents

the participation constraint of depositors. A depositor withdraws his funds

early with a probability of E ( ̃x) in which case he receives a payoff of 1. With

a probability of 1−E ( ̃x) the depositor does not experience a liquidity shock

in which case he receives a promised payment of rD if the bank projects

succeed (which is with probability θ). In case of the failure of bank investments (which happens with probability 1 −θ) any surplus bank reserves are

10
divided amongst the patient depositors. Thus expression (2) states that the

depositors must on average receive at least their reservation utility. Equation (3) represents the expected profit of the bank exclusive of the penalty

costs. With probability (1 − θ) bank profits are zero since the investment

projects fail. With probability θ the bank projects succeed in which case

the bank’s expected profit is given by the expected return from the loans

(rLL(rL)) minus the expected cost of deposits (rDD [1 − E ( ̃x)]) plus the

expected value of net reserve holdings at the end of the period (which is

given by the last term of the equation).

Note that for simplicity we have considered a setup with a given penalty

cost. In the appendix we consider a setup wherein the penalty costs are

explicitly calculated in an environment where the bank finances the shortfall

by selling its assets at fire-sale prices. We show that in this three-period

environment, the objective function of the bank is analogous to equation

(1) and is given by π minus a cost term which is proportional to the bank’s

liquidity shortfall. Hence our qualitative results are unchanged and thus we

use the simpler setup given its parsimony and tractability.

We next solve the bank’s optimization problem and derive the first-best

lending and deposit rates and the optimal level of bank reserves. Subsequently we study how the riskiness of bank projects as well as bank liquidity affects the loan rate offered by the bank. The results are summarized in

Proposition 1.

Proposition 1 1. The optimal gross lending rate is given by

r∗

L = 1+(rp − 1) Pr ( ̃xD ≥ R∗)

θ

3

1 − 1

ηL

 ́ (4)

where ηL = −rLL0 (rL) /L > 0 is the elasticity of the demand for loans.

The optimal gross deposit rate is given by

r∗

D = ( ̄u − E ( ̃x)) D − (1 − θ) Pr ( ̃xD < R∗) (R∗ − E ( ̃xD|xD < R  ̃ ∗)

θ (1 − E ( ̃x)) D .

(5)

And, the optimal level of reserves is given by:

R∗ = D − L(r∗

L)

11
2. (Risk effect) ∂r∗

L

∂θ < 0, i.e., an increase in risk (1 − θ), ceteris paribus,

increases the equilibrium lending rate.

3. (Liquidity effect) ∂r∗

L

∂D < 0, i.e., an increase in bank liquidity, ceteris

paribus, decreases the equilibrium lending rate.

Proof. See Appendix.

It is interesting to note that as the elasticity of demand for loans decreases, the lending rate increases and hence the spread between the loan

rate and deposit rate increases. This result is consistent with the MontiKlein (Klein, 1971 and Monti, 1972) model. As expected, the greater the

market power of the bank, the higher will be the lending rate that it can

charge and vice versa.

The second and third parts of the proposition are also intuitive. The

lending rate prices both risk and liquidity. An increase in liquidity lowers

the expected cost of liquidity shortage and the bank passes some of this

benefit to the borrowers via a lower loan rate.

2.2 Agency problem, over-lending and asset price bubbles

2.2.1 Setting of the problem

So far we have abstracted from any agency issues between the bank equityholders and the bank manager. Let us now consider a setting where such

issues are relevant.

A study by OCC (1988) found that “Management-driven weaknesses

played a significant role in the decline of 90 percent of the failed and problem

banks the OCC evaluated... directors’ or managements’ overly aggressive behavior resulted in imprudent lending practices and excessive loan growth.”

They also found that 73% of the failed banks had indulged in over-lending.

This indicates that principal-agent problems within banks have been one of

the key reasons for bank failures and that bank managers tend to engage in

‘overly aggressive risk-taking behavior’.10 Perhaps even more striking evidence is presented by the financial crisis of 2007-2009 which has revealed that

1 0The OCC’s study is based on an analysis of banks that failed, became problems and

recovered, or remained healthy during the period 1979-1987. The study analysed 171 failed

12
in the period preceding the crisis, traders and large profit/risk centers at a

large number of financial institutions were paying themselves bonuses based

on the size of their risky positions rather than their long-run profitability.

Moreover, in many cases, it was a conscious choice of senior management

to silence the risk management groups that had spotted weaknesses in the

portfolio of building risks.11

To study how such managerial agency problems can have an effect on

bank policies, we model the agency problem within banks explicitly. Let e

denote the unobservable effort level of the manager, such that e ∈ {eL, eH},

where eH > eL. We assume that although the loans are affected by effort,

they are not fully determined by it. The stochastic relationship is necessary

to ensure that effort level remains unobservable. We assume that the distribution of loan demand L(rL) conditional on eH first-order stochastically

dominates the distribution conditional on eL. In other words, for a given

level of lending rate, the manager on average makes a higher volume of loans

when he exerts high effort relative to the case where he exerts lower effort,

i.e. E [L(rL)|eH] > E [L(rL)|eL]. In addition to L(rL), both the principal

and the agent can observe the realization of bank’s net profit, Π.

The manager is risk averse and his utility is increasing in wages, w, but

decreasing in effort. Furthermore, we suppose that E [Π|eH] − E [Π|eL] >

E [w|eH] − E [w|eL]. This means that the incremental increase in the expected profit from implementing a high effort is greater than the increase in

the expected wage costs from implementing the high effort. In other words

the principal has an incentive to implement the high effort level since the

gains from doing so are greater than the associated costs.

We assume that bank liquidity is non-verifiable. This is plausible given

that liquidity is not even well-defined as it can take several forms. Furthermore, bank liquidity may be lent out to other banks via the interbank market

or conversely it may be the excess liquidity of other banks that makes it way

banks to identify characteristics and conditions present when the banks deteriorated. 1 1See Chapter 8 of Acharya and Richardson (2009b), which contains a detailed account

of governance and management failures at a large number of financial institutions. The

most detailed evidence is for UBS based on its “Shareholder Report on UBS’s Write

Downs” prepared for the Swiss Federal Banking Commission.

13
t = 0

• Bank raises deposits

• Principal offers contract

to manager

• Manager sets rL

• Manager chooses e

L(rL) realized

t = 0.5 t = 1

A fraction x of

depositors

withdraw early

t = 2

Payoffs realized

and divided among

parties

Figure 3: Timeline of events.

to the bank in question. It is also particularly difficult to verify off-balance

sheet liquidity which may take the form of unused loan commitments.

The time line is as follows: At t = 0 the bank raises deposits, the

principal offers a contract to the manager (such that eH is chosen) and the

manager chooses rL. At t = 0.5, for a given level of rL the volume of loans

will be realized. As before, at t = 1 there may be early withdrawals and

finally at t = 2 the payoffs are realized and divided between the parties given

the contractual terms. The timeline is depicted in Figure 3.

In the presence of symmetric information, there is no agency problem

and the bank’s problem is analogous to that of Section 2.1 with the bank

maximizing

Π = π − rpE [max ( ̃xD − R, 0)|e = eH] (6)

subject to the following participation constraint

E ( ̃x) + (1 − E ( ̃x)) ∙θrD + (1 − θ)

E [max (R − xD,  ̃ 0)|e = eH]

(1 − E ( ̃x)) D

 ̧

≥ u ̄ (7)

where π is given by

π = θ {rLE [L(rL)|eH] − rDD (1 − E ( ̃x)) + E [max (R − xD,  ̃ 0)|e = eH]}

(8)

The first-best lending rate analogous to equation (4) is given by

r

f

L = 1+(rp − 1) Pr £¡xD ̃ ≥ R ̄f

¢

|e = eH

¤

θ

3

1 − 1

η ̄L

 ́ (9)

14
where η ̄L = −rL

∂E[L(rL)|eH]/∂rL

E[L(rL)|eH] > 0 and R ̄f = D − E [L(rL)].

Next, we introduce the agency problem. We assume that the manager

can observe the quality of the project (θ) and distribution of depositor withdrawals or its liquidity needs (F (x)) but this information is not observable to

the principal and hence the principal cannot infer the first-best loan rate.12

Consequently, the principal will be unable to enforce the first-best rate and

there will be a deviation from the first-best even if the principal is able to

implement eH via the incentive compatibility condition. We assume, however, that the risk premium required on deposits (defined as κ∗ ≡ r∗

D − u ̄)

is public information and hence the participation constraint (7) still needs

to be satisfied.13

Given this setup with asymmetric information, we prove in the appendix

the following important proposition on the manager’s compensation contract.14

Proposition 2 The optimal contract between the principal and the manager

entails that the manager’s payoffs be monotonically increasing in the amount

of loans as well as bank profits net of penalty costs.

Proof. See Appendix.

Such contracts are ubiquitous in practice whereby the bonuses of managers are increasing in the volume of loans and profits. For simplicity we

limit the analysis to linear contracts such that the manager’s payoff is a

1 2 If θ and F (x) are observable then the principal would be able to ascertain rf

L given (9)

and hence enforce the first-best rate, for instance by penalizing the manager if the actual

rate diverges from the first best. Thus this assumption ensures that there is asymmetry

of information regarding the optimal loan rate. 1 3As before, the participation constraint (7) will hold with equality and the optimal deposit rate can be written in generic form as r∗

D =  ̄u + κ∗ where κ∗ =

κ (θ, F (x), E [L|eH] , D). The assumption that κ∗ is public information implies that the

participation constraint needs to be satisfied and at the same time we remain in an environment with asymmetric information. 1 4 In order to ensure monotonocity we use the monotone likelihood ratio property as is

usual in contract theory literature. MLRP ensures that both β and δ in expression (10)

are positive.

15
proportion β of loans and a proportion δ of profits net of penalty costs.15

Hence the bank manager maximizes the following:

Πm = βE [L(rL)|eH] + δ (π − rpE [max ( ̃xD − R, 0)|e = eH]) (10)

subject to the participation constraint (7) and where π is given by (8). We

can now prove the following proposition.

Proposition 3 In the presence of an agency problem between the bank manager and equityholders, the loan rate is lower (compared to the case where

there is no agency problem) and consequently the volume of credit in the

economy is higher.

Proof. See Appendix.

The intuition behind the above proposition is straightforward. The bank

manager has an additional incentive to lower lending rates as this allows him

to make more loans and his payoff is increasing in the amount of lending.

2.2.2 Liquidity-induced agency problem

The intuition as to why the optimal contract in the preceding section depends on both the volume of loans and net profit is as follows: Managerial

compensation solely dependent on profits would be suboptimal from a risksharing standpoint since profits are affected not just by loans but also by

random withdrawals by depositors. In other words, net profit is not a sufficient statistic for loans with respect to effort. On the other hand, if managerial compensation only depends on the volume of loans, then the manager

will be incentivized to lower lending rates as much as possible because he

will not be penalized when low lending rates adversely impact net profits.

In this case, the volume of loans will be excessive, reserves will be too low

1 5The assumption of linear contracts is for simplicity and has no bearing on our results. Furthermore, Holmstrom and Milgrom (1987) argue that real-world compensation

schemes involve linear contracts as more complex contracts are subject to costly “gaming”

by agents. Holmstrom and Milgrom show that in a dynamic environment with CARA

preferences and (binomial) i.i.d. increments in output, optimal linear contracts can be

obtained.

16
and hence liquidity shortages very likely. Hence the optimal contract is a

function of both loans and net profits which are observable to the principal.

Ideally the principal would want the manager to exert high effort as well

as set the first-best lending rate. But given asymmetric information the

manager cannot be explicitly penalized for setting a lending rate lower than

the first-best. Now let us suppose the principal can conduct a costly audit

which verifies whether or not the manager acted overly aggressively. In particular, the audit reveals the quality of the project (θ) and the distribution

of liquidity needs (F (x)). This enables the principal to infer the first-best

loan rate, as given by (9), and thus the principal is able to infer whether

or not the manager had acted aggressively by setting a lending rate below

the first-best. More specifically the contracting problem involves the following stages: The principal offers a contract to the manager. The manager

optimizes his payoff under this contract by choosing the effort level and setting the lending rate. The principal conducts an audit following a liquidity

shortfall and if it is inferred that the manager had acted overly aggressively

by setting a lending rate below the first-best then the manager is penalized

a proportion γ of the penalty costs, where γ ∈ (0, 1].

16 It is assumed that

the manager has limited liability and hence there is an upper bound on the

punishment that can be imposed. Given that γ ≤ 1, this implies that at

most the manager incurs the entire penalty costs.

Optimal risk sharing now dictates that the manager’s payoff only be

a function of loan volume as long as there is no liquidity shortfall. This

is because in the absence of an audit, the manager’s payoff is conditioned

on bank profits so as to constrain his risk-taking appetite. However, in

the presence of an audit, conditioning on bank profits is superfluous since

managerial behavior can be constrained directly by imposing a penalty on

the manager following a liquidity shortfall if it is inferred that he had set

a lending rate below the first-best. Hence the manager’s payoff will be a

1 6This is a realistic setup given that in practice audits are carried out following liquidity

shortages to ascertain the cause of the banking crisis. In the event that it is inferred that

bank managers were responsible due to excessive risk-taking, then the managers are likely

to suffer reputational costs, job losses, congressional hearing costs, etc.

17
function of loan volume as long as there are no liquidity shortfalls but he

will be penalized in the event of a liquidity shortfall if it is inferred that

he had acted over-aggressively. The manager now faces a trade-off. If he

behaves overly aggressively, his utility is higher as long as there are no

liquidity shortfalls and subsequently no penalties. However, if a liquidity

shortfall is realized then the manager is penalized and would be worse off.

We can then prove the following proposition.

Proposition 4 The manager will engage in overly-aggressive behavior if

and only if bank liquidity, D, is sufficiently high.

Proof. See Appendix.

The above proposition says that for high enough bank liquidity the manager has an incentive to engage in overly-aggressive behavior by mispricing

in the loan rate the underlying risk of loans. It should be noted that the

agency problem is only actuated when bank liquidity is high enough. This is

because even though the manager bears a proportion of the penalty costs, in

the presence of excessive liquidity, the probability of experiencing a liquidity shortage is low and this encourages the manager to indulge in excessive

lending. Because of this ‘insurance effect’ of liquidity, the manager’s compensation becomes more sensitive to loan volume. The manager will thus

indulge in excessive lending as long as his payoffs from doing so exceed his

payoffs in the first-best world. On the other hand, for low enough liquidity

the agency problem is immaterial since in this case the manager’s expected

payoff in the first-best world exceeds his expected compensation from acting over-aggressively. Hence for low enough liquidity the manager does not

sanction excessive loans for the fear of incurring a penalty in the event of a

liquidity shortfall.

2.2.3 Asset pricing

Next we consider the asset pricing implications of our model. We define

the fundamental asset price as the price that holds in the absence of any

distortions created by agency problems. We then compare the fundamental

asset price with the actual asset price which may or may not be distorted

18
depending on whether or not agency problems have been actuated within

the banking system. To facilitate this comparison we first model the asset

demand by bank borrowers which was so far taken as given. We assume that

there exists a continuum, n, of risk-neutral borrowers who have no wealth

and hence need to borrow in order to finance investments. Since there is a

continuum of borrowers and loans cannot be conditioned on their size, they

can borrow as much as they like at the going rate of interest.

Although there is a continuum of borrowers, we shall analyze the behavior of a representative borrower. This implies that the equilibrium is

symmetric and that all borrowers choose the same portfolio. The fact that

all borrowers are identical ex post means that the bank cannot discriminate

between borrowers by conditioning the terms of the loan on the amount

borrowed or any other characteristic.

Let Xd denote the number of units of the asset demanded by the representative borrower and Xs denote the total supply of the risky asset. We

assume for simplicity that the supply of the asset is finite and fixed. The

asset returns a cash flow of C per unit with a probability of θ. Let P denote

the price of one unit of the asset.

As in Allen and Gale (2000) we assume the borrowers face a non-pecuniary

cost of investing in the risky asset b (Xd) such that it satisfies the usual neoclassical properties: b (0) = b0 (0), b0 (Xd) > 0 and b00 (Xd) > 0 for all Xd > 0.

The purpose of the investment cost is to restrict the size of the individual

portfolios and to ensure the concavity of the borrower’s objective function.

The optimization problem faced by the representative borrower is to

choose the amount of borrowing so as to maximize expected profits. That

is

max

Xd

θ [CXd − rLPXd] − b (Xd). (11)

Note that the borrower has to pay an interest of rL on its borrowing. The

market-clearing condition for the asset is:

nXd = Xs. (12)

The first-order condition of the above problem is as follows:

θ [C − rLP] − b0 (Xd)=0

19
Setting Xd = 1 in the first order condition and letting τ (Xd) = b0 (Xd)

denote the marginal investment cost, the per unit asset price can be written

as:

P = θC − τ (1)

θrL

. (13)

As expected, the asset price is the discounted value of the expected cash

flows net of the investment cost. It is also clear that there is a one-to-one

mapping from the (gross) lending rate, rL, to the asset price, P. Let r

f

L

denote the fundamental (gross) lending rate which is the rate obtained in

the absence of any distortions such as agency problems. Recall that r

f

L is

the rate which maximizes the equity value of the bank. Since depositors

always earn u ̄ on average, r

f

L is also the rate which maximizes the value of

the bank. Hence r

f

L is given by expression (9). Then the fundamental asset

price is given by:

Pf = θC − τ (1)

θrf

L

. (14)

Since the total amount of loans for a given rL is L(rL) and the amount

borrowed by an individual borrower is PXd, it has to be the case that

nPXd = L(rL). Hence, using the market-clearing condition (12) the asset

price can also be written as:

P = L(rL)

Xs

. (15)

Using the above formulation it is again clear that there is a one-to-one

mapping from the lending rate to the asset price.17 Thus the fundamental

asset price can also be rewritten as:

Pf =

L

3

r

f

L

 ́

Xs

. (16)

Having derived the fundamental asset price we can next define an asset

price bubble. An asset price bubble is formed whenever P >Pf since the

asset is overpriced. Note that P >Pf as long as rL < rf

L. Intuitively, a

1 7Note that θC−τ(1)

θrL = L(rL)Xs and hence rL = (θC−τ(1))XsθL(rL) . Since nPXd = L(rL) and

nXd = Xs we again get P = θC−τ(1)

θrL .

20
lending rate lower than the fundamental rate creates a high demand for the

asset which in turn leads to an increase in asset prices over and above the

fundamental values.

From Proposition 3 we know that the loan rate in the presence of an

agency problem is lower than the fundamental rate. Furthermore, from

Proposition 4 we know that this agency problem is actuated for high enough

bank liquidity. Given that the asset price is given by expression (13) and the

fundamental asset price is defined by (14) we immediately have the following

corollary to Propositions 3 and 4.

Corollary 1 In the presence of an agency problem between the bank manager and the equityholders, an asset price bubble is formed for high enough

bank liquidity.

To better understand the mechanics behind the formation of a bubble,

the four-quadrant diagram in figure 4 is useful. Quadrant I in the figure

depicts the relationship between the risk of project failure, (1 − θ), and the

loan rate, rL, charged by the bank. In general the higher the riskiness the

higher would be the equilibrium lending rate as is captured by the line AA.

The loan rate in turn determines the demand for loans and the volume of

credit in the economy. For any given lending rate, the amount of bank loans

is given by L(rL). Since L0 (rL) < 0 we know that the lower the loan rate

the higher is the amount of investment in the economy as is captured by

the line BB in quadrant II. From the expression derived in equation (15) we

know that the market clearing condition is PXs = L(rL). In other words,

the price of the assets multiplied by the supply of the asset has to be equal to

the total demand for the asset. Hence the demand for the asset determines

the asset price and this relationship is captured by the line Y Y in quadrant

III. Notice that the higher the demand for the asset, the higher is its price.

Finally quadrant IV derives the relationship between the asset price and

risk. In general, the higher is the underlying risk the lower will be the asset

price as is depicted by the line ZZ. The equilibrium relationship between

asset price and risk is derived by tracing the effect of risk on the loan rate,

which in turn has an effect on the amount of investment which subsequently

21
Risk

Loan rate, rL

Price, P

Investment, L A

A

A1

A1

B

B

Y

Y

Z

Z

Z

1

Z

1

I II

III IV

Figure 4: The mechanics of the formation of asset price bubbles.

determines the asset price.

Let the line AA represent the fundamental relationship between risk

and the bank loan rate, i.e. the relationship that would be obtained in the

absence of agency issues. Then for any given level of risk, the fundamental

asset price would be represented by the line ZZ. However, as we showed

in Proposition 4 an agency problem is actuated for sufficiently high bank

liquidity levels where the bank loan rate is lower for any given level of risk.

This in turn shifts the AA line to A1A1. From quadrant II we know that

the volume of credit in the economy increases following lower loan rates.

Consequently asset prices increase as a result of market-clearing as is shown

in quadrant III. The final relationship between asset prices and risk is shown

22
in quadrant IV and it is obvious that the actuation of the principal-agent

problem shifts the ZZ line to Z1Z1. In the end we see that the asset price

is higher for the same level of risk leading to the formation of a bubble.

It is also interesting to note that our model implies that the size of the

bubble is monotonic in the leverage of bank borrowers. This is because bank

borrowers in the model borrow more the lower the lending rates offered by

the banks. The greater the severity of the agency problems, the lower are

the lending rates, which encourages excessive bank borrowing. This increase

in leverage in turn increases asset demand which pushes up asset prices.

Our model helps explain how agency problems in the banking sector can

induce the formation of asset price bubbles. In terms of the four-quadrant

diagram we would be reducing our attention to quadrant IV alone if we fail

to consider the role of the banking sector. Embedding of the banking sector

in a pricing framework gives us a fuller picture of how the banking sector

contributes to equilibrium asset or investment demand in the economy and

to the formation of asset price bubbles.

3 When are bubbles likely to be formed?

3.1 High macroeconomic risk

We have established so far that asset price bubbles are formed when bank

liquidity is substantially high. So the question that arises is when are banks

most likely to be flushed with liquidity. In an empirical study, Gatev and

Strahan (2006) find that as spreads in the commercial paper market increase,

bank deposits increase while bank asset (loan) growth also increases. The

spreads on commercial paper are a measure of the investors’ perception of

risk in the real economy. Intuitively, when investors are apprehensive of

the risk in the entrepreneurial sector they are more likely to deposit their

investments in banks rather than make direct investments.18

1 8The flight of depositors to banks may be due to banks having greater expertise in

screening borrowers during stress times, inducing a natural negative correlation between

the usage of lines of credit and deposit withdrawals as argued by Kashyap, Rajan and

Stein (2002). Alternatively, the flight may simply be due to the fact that bank deposits

23
To formalize the above intuition we integrate with the model the entrepreneurial sector, endogenize the quantity of deposits in the economy and

show that bank deposits will increase at a time when the underlying risk

increases. Consider an economy where entrepreneurs have access to projects

that yield a terminal cash flow C if it succeeds and 0 otherwise. The probability of success depends partly on the realization of the state variable,  ̃θ,

and partly on the entrepreneurs’ effort decision, e, which identifies whether

the entrepreneur is diligent (e = 1) or shirks (e = 0) in which case, entrepreneurs extract a private benefit of B. If the entrepreneur is diligent, the

probability of success is θ but in the presence of shirking the probability of

success is φθ, where φ ∈ (0, 1). The realization of the state variable θ is

observable to the entrepreneurs.

Entrepreneurs promise to pay the risk-neutral investors who invest directly in their projects a face value of d. Thus the expected payoff of the

risk-neutral entrepreneurs whose projects are financed by direct investment

is given by:

E (p) = (

θ (C − d)

φθ (C − d) + B

if e = 1

if e = 0 . (17)

One can then prove the following proposition.

Proposition 5 Entrepreneurs will shirk (i.e. e = 0) if and only if θ<θ∗

where θ∗ = B

(1−φ)(C−d) .

Proof. See Appendix.

The above proposition says that entrepreneurs are more likely to shirk

when macroeconomic risk is high. Intuitively, an increase in macroeconomic

risk lowers the success probability of entrepreneurial projects which encourages entrepreneurs to shirk so as to consume their private benefits.

We know that investors on average earn u ̄ from bank investment. Let

E (ue|θ) denote the expected utility of the investor from entrepreneurial

are insured (up to a threshold) by the Federal Deposit Insurance Corporation (FDIC)

whereas commercial paper and money market funds are uninsured, at least until the

extraordinary actions taken by the Federal Reserve during 2008 and 2009. Pennacchi

(2006) finds evidence supportive of this latter hypothesis by examining lending behavior

of banks during crises prior to the creation of the FDIC.

24
investment. We assume there exists θ∗ such that:

E (ue|θ<θ∗) < u<E  ̄ (ue|θ ≥ θ∗).

That is, the expected utility of investors from direct investment will be

higher than that from bank investment as long as  ̃θ ≥ θ∗ and conversely the

expected utility from bank investment will be higher than that from direct

investment as long as  ̃θ<θ∗. This will be the case in the presence of deposit

insurance since investors would then perceive bank deposits to be safer and

hence in bad times would prefer to invest in bank deposits rather than make

direct entrepreneurial investments. Alternatively it can be argued that in

bad times banks are better monitors of entrepreneurial activity and thus

entrepreneurs cannot shirk if financed by banks.

If investors perfectly observe θ before deciding whether to invest directly

in projects or in banks, all investments will either be channeled directly into

projects or into banks depending on the realization of θ. Hence in order

to make a more realistic distribution of investments between entrepreneurs

and deposits we assume that depositors do not observe θ but each depositor

receives an imperfect signal, s, on the basis of which they decide whether

to allocate their endowments to direct entrepreneurial investment or bank

deposits. A signal sj = g received by investor j is a good signal which

implies that θ ≥ θ∗ but a signal sj = b is a bad signal which would be an

indication to the investor that θ<θ∗. Thus an investor allocates his capital

to entrepreneurial projects only if he receives a good signal.19

The probability distribution of the signals is assumed to be identical and

independent across depositors and given as: Pr (s = g) = νθ and Pr (s = b) =

1 − νθ, where ν ∈ (0, 1). Investors only observe their own signals and are

not aware of the probability distribution of the signals. The simple formulation of the probability distribution implies that a proportion ν ̃θ of the

1 9This will have implications for the participation constraint (2). In the constraint, θi

would need to be replaced by a Bayesian expectation of θ which will slightly alter the

equations in Proposition 1. However since all qualitative results are unchanged we can

assume for simplicity that when offered a banking contract depositors are able to ascertain

whether rD is high enough to satisfy their reservation utility. This will be the case if the

required risk premium on deposits, defined as κ∗ ≡ r∗

D − u ̄, is public information.

25
investors will allocate their endowments to entrepreneurial projects while a

proportion 1 − ν ̃θ will allocate their endowments to bank deposits. This

formulation implies that the better the macroeconomic state θ, the higher

would be the amount of direct investments. However a low realization of θ

would imply that a larger amount of endowments will be tunneled to bank

deposits. Hence, bank deposits are given by D = (1 − νθ) I assuming that

there are a total of I risk neutral investors.

It is clear from the formulation of D that bank liquidity increases when

the underlying macroeconomic risk is high. This is consistent with the findings of Gatev and Strahan (2006) cited earlier. However, we hypothesize

that for extremely high macroeconomic risk (very low θ) bank liquidity may

be adversely affected. This is because casual observation of crises suggests

that once macroeconomic risk crosses a certain threshold the bubble bursts

subsequently, adversely affecting banks’ liquidity. Nevertheless since this

paper focuses on the formation of bubbles rather than the burst of bubbles

we do not model this phenomenon but simply capture this by assuming

that D = (1 − νθ) I ∀θ ∈ [θ, 1] where θ is the threshold below which bank

liquidity decreases as θ decreases.

We can then prove the following proposition.

Proposition 6 A bubble is formed in the economy when the macroeconomic

risk is high enough. More formally, there exists a threshold θc such that

P >Pf if θ<θc where θc ∈ [θ, 1].

Proof. See Appendix.

As macroeconomic risk increases, there is a flight to quality whereby

investors prefer to invest in bank deposits rather than engage in direct lending. Subsequently, banks find themselves flushed with liquidity during times

when spreads in the commercial paper market (i.e. the direct funding costs

of entrepreneurs by investors) are high. Excessive liquidity encourages bank

managers to increase the volume of credit in the economy by mispricing of

underlying risk. This in turn fuels a bubble in asset prices.

26
3.2 Loose monetary policy

Before we turn to the implications for central bank’s monetary policy, we

briefly discuss how monetary policy has a direct effect on bank’s liquidity.

When embarking on an expansionary monetary policy via open market operations, central banks buy government securities from primary dealers who

have accounts with depository institutions. The way this transaction works

in practice is that the central bank directly credits the reserves which commercial banks have with the central bank, hence effectively increasing the

deposit base of the bank. On the other hand, in order to implement a contractionary monetary policy, the central banks sell government securities to

primary dealers and at the same time debit their accounts which effectively

reduces the deposit base of banks. Hence bank deposits are a function of

both macroeconomic risk (θ) as well as monetary policy (M):

D = D (θ; M) (18)

The above relationship is depicted in Figure 5. As discussed in the previous

section, as macroeconomic risk increases there is a flight to quality whereby

bank deposits increase and this continues until risk crosses the threshold 1−θ

after which more and more investors prefer to just consume their reservation utility. In the absence of an active monetary policy, the relationship

between bank liquidity and risk is given by DD. However, following an expansionary monetary policy, bank liquidity increases for the same level of

risk and the DD line shifts upwards to D+D+. Conversely, subsequent to a

contractionary monetary policy, bank liquidity decreases for the same level

of risk and consequently the DD line moves downwards to D−D−.

In the figure, D∗ = (1 − νθc

) I is the liquidity threshold above which

asset price bubbles are formed. When macroeconomic risk increases above

1−θc to say 1−θ1, bank liquidity crosses the threshold D∗ to D1 leading to

the formation of a bubble. However the central bank can offset this effect via

a contractionary monetary policy which will shift the DD line downwards.

The magnitude of the contractionary monetary policy should be such that

the DD line moves downwards to at least D−D−. As can be seen from the

figure this is the minimum shift that is required to ensure that for the new

27
1- Risk

D*

c 1

D1

Liquidity, D

D

D

D_

D_

D+

D+

Figure 5: The interplay between bank liquidity, macroeconomic risk and

monetary policy.

level of risk 1 − θ1, bank liquidity is at or below D∗.

It is interesting to note from Figure 5 that even if the macroeconomic

risk level is below 1 − θc

, central banks can fuel asset price bubbles by

adopting loose monetary policies thereby shifting the DD line upwards such

that the liquidity level crosses the threshold D∗. Indeed Kindleberger (2005)

in his study on the history of financial crises notes that: “Speculative manias

gather speed through expansion of money and credit.”

4 Optimal monetary policy

We next formalize the argument in Section 3.2 and study implications for

optimal monetary policy in the presence of the following trade-off faced by

the central bank: An increase in money supply increases aggregate investment. However, an increase in money supply also increases bank liquidity

and we know from our earlier results that excessive bank liquidity can induce

bubbles in asset prices. As discussed formally in the extension (section A.2

28
of the Appendix) bubbles are costly given that aggressive behavior of managers and underpriced loan rates result in a deterioration in the quality of

bank investments which in turn increases the average default risk and hence

the expected default costs. Let the expected cost of the bubble, conditional

on P >Pf , be denoted by Ω (∆), where ∆ ≡ E £¡P − Pf ¢

|P >Pf ¤ denotes the expected size of the bubble. It is plausible to argue that the cost

of the bubble increases with the size of the bubble, i.e. Ω0 (∆) > 0. (This is

also shown formally in extension A.2 of the Appendix).

The trade-off faced by the central bank can be expressed by the following

expected welfare (objective) function:

max

M∗ W = L ̄H (rL) − [Pr (D (θ) > D∗) Ω (∆)] (19)

where L ̄H (rL) ≡ E [L(rL)|eH] represents the expected demand for bank

loans conditional on high effort exerted by bank managers. Since entrepreneurs have zero wealth, L ̄H also represents the expected investment made

by entrepreneurs. The second term denotes the expected cost of a bubble

since a bubble is formed when deposits cross the threshold D∗.

Taking the FOC of (19) with respect to M we get the following:

L ̄0

H

drL

dM = Ω (∆)

∂ Pr (D>D∗)

∂M + Pr (D>D∗)

∂Ω (∆)

∂M (20)

where L ̄0

H = ∂L ̄0H/∂rL. The LHS in (20) represents the marginal benefits

(MB) of expansion. The RHS represents the marginal costs (MC) of expansion. Thus the central bank expands money supply up to the point where

the marginal benefits of expansion (in terms of increased investment) just

equal the marginal costs of expansion (in terms of a higher likelihood of a

bubble and the associated costs).

We assume that the SOC is satisfied, namely: ∂MB

∂M < ∂MC∂M . This will

be the case if there exist diminishing returns on investment and if the marginal cost of a bubble is a non-decreasing function of the money supply.

The assumption of diminishing returns to investment implies that as money

supply increases, the marginal benefits (in terms of higher investment) increase but at a decreasing rate. As discussed in section A.2 of the appendix,

managers progressively make worse quality loans, which can also explain the

29
diminishing returns on investment. The assumption that the marginal cost

of a bubble is a non-decreasing function of money supply implies that as

bank liquidity and subsequently the expected size of the bubble increases,

the incremental cost of the bubble does not decrease. This is also plausible because if anything we expect the marginal cost of a bubble to be an

increasing function of the size of the bubble.

We can now prove the following proposition.

Proposition 7 The optimal monetary policy dictates that the central bank

decrease the money supply as macroeconomic risk, (1 − θ), increases as long

as d∆/dθ < 0. However, if d∆/dθ > 0, then the optimal monetary policy

dictates that the central bank increase the money supply as macroeconomic

risk increases. More formally,

dM∗

dθ (

> 0

< 0

if d∆

dθ < 0

if d∆

dθ > 0 .

Proof. See Appendix.

The intuition behind the above proposition is as follows: If the expected

size of the bubble is increasing in macroeconomic risk, i.e. d∆

dθ < 0, say

for instance due to a flight to quality effect which increases bank liquidity,

then this raises the cost of bubbles. The central bank can counter this

effect by decreasing the money supply and hence draining out liquidity from

the banking system. If, on the other hand, bank liquidity is decreasing

in macroeconomic risk and consequently the expected size of the bubble

decreases as the underlying risk increases, i.e. d∆

dθ > 0, say for instance, due

to investors’ loss of confidence in times of a crisis, then the central bank can

offset this effect by increasing the money supply. In other words, the central

bank should lean against macroeconomic risk as long as the expected cost of

a bubble is increasing with risk, but should lean with macroeconomic risk as

long as the expected cost of a bubble is decreasing with risk.

Proponents of the Greenspan camp may argue that the central bank

may not be aware where we are in the business cycle and hence whether

bank liquidity is increasing or decreasing in macroeconomic risk. Nevertheless, it can be shown that a much simpler policy recommendation is to lean

30
against bank liquidity regardless of where we are in the business cycle. The

optimality of this policy is stated in the following proposition.

Proposition 8 The optimal monetary policy implies a leaning against liquidity approach, i.e. tightening monetary policy in times of excessive bank

liquidity and loosening monetary policy in times of falling bank liquidity.

More formally, dM∗

dD < 0 ∀θ.

Proof. See Appendix.

The above proposition is intuitive. In times of excessive bank liquidity,

bubbles are likely to be formed and the central bank can avoid the formation

of bubbles by tightening monetary policy. On the other hand, in times of

scarce liquidity, banks raise loan rates and hence aggregate investment is

adversely affected. The central bank can prevent the fall in investment by

loosening monetary policy.

We thus argue that the ‘Greenspan put’ should be employed in times of

falling bank liquidity. However, in times when banks are flush with liquidity, a loose monetary policy only enhances the liquidity insurance enjoyed

by banks, and thus aggravates their risk-taking incentives. This in turn

increases the likelihood of bubbles in asset prices.

4.1 Discussion

Traditionally, as suggested by the Taylor rule, monetary policy has targeted interest rates and employment. However, in the light of our results,

we argue that monetary policy should also target asset prices. Our results

suggest that asset prices can be targeted if the monetary authorities adopt

a “leaning against liquidity” approach. In fact we show that a “leaning

against liquidity” policy performs a twofold purpose: In times of abundant

liquidity it counters the surge in asset prices, whilst in times of scarce liquidity it performs the role of quantitative easing and subsequently encourages

investment.

Allen and Gale in their book “Understanding financial crises” document

the following: “In Finland an expansionary budget in 1987 resulted in massive credit expansion. The ratio of bank loans to nominal GDP increased

31
from 55 percent in 1984 to 90 percent in 1990. Housing prices rose by a

total of 68 percent in 1987 and 1988... In Sweden a steady credit expansion

through the late 1980’s led to a property boom.” These observations are

perfectly in line with our model. Loose monetary policies can potentially

lead to excessive liquidity in the banking system which in turn encourages

bank mangers to underprice the underlying risk and thereby increase the

volume of credit in the economy. This in turn creates an asset price bubble.

Our model can thus explain how lax monetary policy in Japan during

the mid 1980s led to asset price inflation. Bank of Japan (BOJ) reduced the

official discount rate five times between January, 1986 and February, 1987,

leaving it finally at 2.5 percent. It is widely accepted that the easy credit

policies adopted by BOJ created excess liquidity in the Japanese economy,

as also acknowledged by Goyal and Yamada (2004). The sequence of events

started with the Plaza Accord (1985), in which the G5 countries agreed

on a stronger yen so as to lower the U.S. trade deficit. However, BOJ’s

intervention in foreign exchange markets appreciated the yen rapidly. Responding to the strengthening yen and seeking to avert deflationary effects

in the domestic economy, Bank of Japan lowered interest rates and consequently increased liquidity in the economy. In the subsequent years a large

real estate bubble was formed.

One of the causes of the current subprime crisis has been suggested to

be the loose monetary policy adopted by the Federal Reserve in the United

States. In 2003, the Fed lowered the federal funds rate to 1% - a level that

was last seen only in 1958. Subsequently banks mispriced risk and indulged

in over-lending which finally culminated in the subprime crisis. In fact the

world was awash with liquidity prior to the crisis. We would thus argue that

this excess of liquidity contributed significantly in causing the crisis.

Finally, Naqvi (2007) shows that the central bank’s lender of last resort

operations need to be complemented ex ante by an efficient supervisory

framework so as to avoid the moral hazard repercussions of bail-outs. What

we learn from our paper is that such supervision is even more essential

during times when the banking system is flushed with liquidity. This is

because during such times bank managers are more likely to under-price

32
risk and hence over-invest. Thus adequate supervision in times of abundant

liquidity can mitigate the risk-taking appetites of banks.

5 Conclusion

We develop a theory of banking explaining how the seeds of a crisis may be

sown when banks are flush with liquidity. The main empirical implication

of our model is that excessive liquidity induces risk-taking behavior on the

part of bank managers. In summary, we obtain the following results: (a)

bank managers will behave in an overly-aggressive manner by mispricing risk

when bank liquidity is sufficiently high; (b) asset price bubbles are formed

for high enough bank liquidity; (c) bubbles are more likely to be formed

when the underlying macroeconomic risk is high inducing investors to save

with banks rather than make direct entrepreneurial investments; and, finally

(d) bubbles are more likely to be formed following loose monetary policies

adopted by the central bank.

We also show that the optimal monetary policy involves a “leaning

against liquidity” approach, i.e. a central bank should adopt a contractionary monetary policy at times when banks are awash with liquidity so as

to draw out their reserves; and it should adopt an expansionary monetary

policy at times when banks have scarce liquidity so as to boost investment.

Some, most notably Alan Greenspan, have argued that “we are never

certain where we are in the cycle”20 and hence monetary policy should not

be used to target asset prices. Nevertheless, our model showed that even

if this is the case a “leaning against liquidity” policy can be rationalized.

Thus we argue that monetary policy should target not just interest rates

and employment but also asset prices.

2 0Alan Greenspan, Financial Times, 27 May 08.

33
Appendix

A.1 Extension: Bank’s objective function in the presence of

fire-sale prices for assets liquidated prematurely. In this appendix

we will derive the bank’s objective function in a setup where instead of taking

the penalty cost term as exogenous we derive the term in an environment

where the bank can finance the liquidity shortfall by liquidating assets at

fire-sale prices. We show that the objective function in this setup has the

same generic form as that in the main body of the paper and hence the

qualitative results are unaffected.

The timing of the game is as follows. At time t = 0 the bank receives

deposits and then invests a fraction of these deposits in investment projects

via loans to entrepreneurs. A proportion of the total loans, L, are of very

good quality such that they repay early with probability 1 at t = 1. These

loans are indexed by Le and the rate of return on these loans is re

L. However

such investments are scarce so the bank cannot make its entire investments

in such assets. The rest of the investments are made in assets which repay

at t = 2 with a probability of θ. These investments are indexed by L and the

rate of return on these investments is given by rL. Thus the total investment

is given by L= L+ Le. The rest of the deposits are retained as reserves, R.

At time t = 1 the bank distributes a fraction of its ‘early returns’, re

LLe,

to its equityholders in the form of dividends where the dividend payout ratio is given by d. For simplicity we normalize d = 1, which implies that

the early returns are accrued by the equityholders. Furthermore, at t = 1

the bank experiences a random liquidity shock such that there are withdrawals of xD ̃ . If the bank reserves at t = 1 are insufficient to service

these withdrawals then the bank will have to inefficiently liquidate a fraction, ξ, of its portfolio. The value of the bank’s portfolio at t = 1 is given by

re

LLe+rLθL. However, the fire-sale value of the portfolio is λ (reLLe + rLθL),

where λ ∈ (0, 1). Thus the fraction of premature liquidation is given by

ξ = (xD − R) / [λ (re

LLe + rLθL)]. We assume for simplicity that the value

of the bank’s portfolio is high enough so that it can service its withdrawals

without going bankrupt. This is just for simplicity and allowing for bankruptcy does not change the results.

34
Finally at time t = 2 the returns from bank investments, if any, are divided amongst the depositors and the bank equityholders. With probability

θ, the return from the ‘late’ projects will be realized and the equityholders

will consume rLL−rDD (1 − x ̃)+(R − xD ̃ )+ where (R − xD ̃ )+ = R−xD ̃ if

xD < R  ̃ and 0 otherwise. Also note that the equityholders need to make the

promised payment rDD to the patient depositors. With probability 1−θ the

late projects fail and the value of reserves is divided amongst the depositors.

In this case the equityholders only consume their dividends at t = 1.

Given this setup the expected profits of the bank’s equityholders is given

by:

Π1 = Pr ( ̃xD < R) [θ {rLL − rDD (1 − E ( ̃x)) + E (R − xD ̃ )} + re

LLe

]

+ Pr ( ̃xD ≥ R) [θ {rL (1 − ξ)L − rDD (1 − E ( ̃x))} + re

L (1 − ξ)Le

] .

Since Pr ( ̃xD < R)=1 − Pr ( ̃xD ≥ R) and defining

πNF = θ {rLL − rDD (1 − E ( ̃x)) + E (R − xD ̃ )} and

πF = θ {rL (1 − ξ)L − rDD (1 − E ( ̃x))} we can rewrite the above as

follows:

Π1 = (πNF + re

LLe

) − Pr ( ̃xD ≥ R) [(πNF − πF ) + ξre

LLe

]

where (πNF − πF ) denotes the difference in the profits from the late projects

with and without fire sales while ξre

LLe denotes the loss in value from the

early projects due to fire sales. Thus the expected profit of the bank is given

by the expected profit in the absence of any fire sales, (πNF + re

LLe), minus

the probability of a liquidity shortage times the loss in value due to fire sales.

To show the qualitative equivalence of the above dynamic setup with

that of the static setup, we first note that πNF = π as defined in equation

(3) given that E (R − xD ̃ ) = Pr ( ̃xD < R) [R − E ( ̃xD|xD < R  ̃ )]. Thus,

Π1 = (π + re

LLe

) − Pr ( ̃x ≥ R) [(πNF − πF ) + ξre

LLe

]

Note that re

LLe is a constant and hence maximizing Π1 w.r.t. rL and rD is

equivalent to maximizing the following

Π2 = π − Pr ( ̃xD ≥ R) [(πNF − πF ) + ξre

LLe

] (21)

35
Also note that [(πNF − πF ) + ξre

LLe] is increasing in ( ̃xD − R) given that

∂πF

∂ξ < 0 and ∂ξ

∂( ̃xD−R) > 0. Thus [(πNF − πF ) + ξre

LLe] is increasing in

( ̃xD − R). Note the stark similarity of maximizing Π as in equation (1)

with that of equation (21) above. In both cases, the expected profit of the

bank is given by π minus the probability of a liquidity shortfall times a cost

term that is proportional to the shortfall. Thus in the main body of the

paper, we use the simpler setup with an exogenous penalty cost given that

the qualitative results are unaffected.

A.2 Extension: Cost of a bubble

Suppose that bank borrowers have an outside option given by u ̄B. In

other words, if they do not borrow from banks to invest in projects they can

consume their outside option. Furthermore, bank borrowers are heterogenous in the sense that the project of borrower i succeeds with probability

θi

. Banks cannot observe the success probability of individual entrepreneurs

and hence under a pooling equilibrium they set a loan rate which cannot

be conditioned on the individual θi

’s but is based on the average risk of the

participating entrepreneurs. The projects require a capital of k and if the

project succeeds it generates a cash flow Ck, where C > 1. For simplicity

we assume that borrowers have limited wealth and hence need to borrow k

from banks in order to become entrepreneurs. Given limited liability and

a loan rate, rL, charged by banks, borrower i will invest in his project and

hence borrow from the bank if and only if

θi (Ck − rLk) ≥ u ̄B

or if and only if

rL ≤ r ̄i

L = Ck − u ̄B/θi

k .

Hence as long as rL > r ̄i

L the agent would prefer to consume his outside

option and would invest in risky projects only if the loan rate is low enough.

Suppose that there are many entrepreneurs each with a unique success

probability θi such that θ is uniformly distributed between [0, 1]. As long as

there are no agency problems between bank equityholders and bank managers, the loan rate is given by r

f

L. Thus in the no-agency world, all entrepreneurs with r ̄i

L > rf

L will invest whilst the infra-marginal entrepreneur

36
will be indifferent between investing and consuming his outside option. Thus

the marginal entrepreneur’s success probability, denoted by θm, satisfies the

following:

rf

L = Ck − u ̄B/θm

k

or

θm = u ̄B

k

3

C − r

f

L

 ́.

Hence the average success probability of the active entrepreneurs in the

no-agency world is given by:

 ̄θ = 1 + θm

2 .

Next we introduce a cost of default in the economy denoted by Ψ. If

the projects of the (active) entrepreneurs fail then a default occurs and the

economy suffers a cost Ψ. This cost can take several forms. It could be a

cost suffered by the taxpayers who eventually foot the bill for bank bailouts.

It could be a cost suffered by the banking system: if banks are not bailed

out it would be the cost of bank failures; it can represent a reputational cost

for banks; banks can also face a cost in terms of tougher regulations (for

instance limits on the size of banks, limits on proprietary trading and/or

higher regulatory taxes). Finally it could represent a political cost borne by

regulators. Hence in the absence of agency problems, the average default

cost of an active entrepreneur is given by

Ψ ̄ = ¡1 −  ̄θ

¢

Ψ.

We have shown that in the presence of an agency problem, bank managers act over-aggressively and set a loan rate, ra

L, such that raL < rf

L. When

the loan rate is lowered, this encourages over-investment and some agents

who were previously consuming their outside options will have an incentive

to indulge in bank borrowing. In the presence of an agency problem, the

marginal entrepreneur’s success probability is given by:

θm0 = u ̄B

k

¡

C − ra

L

¢

37
where θm0 < θm since ra

L < rf

L. Hence the average success probability of the

active entrepreneurs in the presence of an agency problem is given by:

 ̄θ

0

= 1 + θm0

2 .

It is clear that the average success probability of projects falls in the

presence of an agency problem. In other words, the average quality of loans

deteriorates when managers behave over-aggressively. The average default

cost is now given by

Ψ ̄ 0 =

3

1 −  ̄θ

0

 ́

Ψ

where Ψ ̄ 0 > Ψ ̄ .

Intuitively when managers act over-aggressively they set a loan rate

which encourages excessive borrowing. This leads to a deterioration of the

average quality of loans and hence increases the economy’s cost of default.

Thus in bubble periods there is over-investment due to a worsening of the

quality of loans. This in turn is conducive to higher default costs on average.

Thus the cost of the bubble can be defined as

Ω ≡ Ψ ̄ 0 − Ψ ̄ .

Let ∆ ≡ P − Pf denote the size of the bubble. Then note that Ω0 (∆) > 0.

This is because a reduction in lending rates (which implies an increase in

asset prices given equation (13)) lowers θm0 which in turn lowers  ̄θ

0

and hence

increases the average default cost Ψ ̄ 0

. In other words, the cost of the bubble

is increasing in the magnitude of the bubble.

A.3 Proofs

Proof of Proposition 1. The participation constraint of the bank will

be binding because otherwise the bank can increase its expected profits by

slightly reducing rD. Thus, r∗

D is given by the solution to the following:

E ( ̃x) + (1 − E ( ̃x)) ∙θrD + (1 − θ)

E [max (R − xD,  ̃ 0)]

D

 ̧

=  ̄u

where E [max (R − xD,  ̃ 0)] = Pr ( ̃xD < R) [R − E ( ̃xD|xD < R  ̃ )]. Solving

for r∗

D we get (5).

38
We can then substitute r∗

D in the bank’s objective function and hence r∗L

will be the solution to the following unconstrained maximization problem:

max

r∗

L

Π = θ {rLL(rL) − r∗

DD (1 − E ( ̃x)) + Pr ( ̃xD < R) [R − E ( ̃xD|xD < R  ̃ )]}

−rpE [max ( ̃xD − R, 0)] .

Assuming that Π is quasi-concave in rL and noting that R = D − L, the

maximum is characterized by the following first order condition:

∂Π

∂rL

= θL(rL) − θ Pr [ ̃xD < R]L0 (rL) + θrLL0 (rL)

−rp Pr [ ̃xD ≥ R]L0 (rL) − θD (1 − E ( ̃x)) ∂r∗

D

∂rL

= 0. (22)

Noting that ∂r∗

D/∂rL = (1 − θ) Pr [ ̃xD < R]L0 (rL) /θD and solving for rL

after some simplification we get (4). Thus the optimal reserve level is given

by R∗ = D − L(r∗

L) which proves the first part of the proposition.

From the FOC (22), if we solve for r∗

L directly without exploiting the

definition of ηL we get the following expression for the return on loans:

r∗

L = 1

θ − LL0 + (rp − 1) Pr ( ̃xD ≥ R∗)θ (23)

Taking the partial derivative of the above expression w.r.t. θ we get:

∂r∗

L

∂θ = −1+(rp − 1) Pr ( ̃xD ≥ R∗)

θ2 < 0 (24)

since rp > rL > 1, which proves the second part of the proposition.

Next note that ∂ Pr ( ̃xD ≥ R) /∂D < 0, i.e. an increase in bank liquidity

(deposits) lowers the probability of liquidity shortfalls since R = D−L. Then

taking the partial derivative of (23) w.r.t. 1 − F (R) = Pr ( ̃xD ≥ R) we get:

∂r∗

L

∂ [1 − F (R)] = rp − 1θ > 0 (25)

Hence ∂r∗

L

∂D = ∂r∗

L

∂[1−F(R)]

∂[1−F(R)]

∂D < 0, which proves the third part of the

proposition. Q.E.D.

Proof of Proposition 2. Both the principal and the agent can observe

the realized amount of loans and the bank’s net profit. The joint density of

39
loans and net profits conditional on e is given by g

3

L(rL), Π ̃|e

 ́

whilst the

distribution function is given by G

3

L(rL), Π ̃|e

 ́

, where Π ̃ is the realized

value of net profits.

In our setup L is not a sufficient statistic for Π ̃ w.r.t. effort and vice

versa. Hence given the results of Holmstrom (1979) the optimal compensation schedule should be a function of both L(rL) and Π ̃.

The manager is an expected utility maximizer with a Bernoulli utility

function u (w, e) over his payoffs or wage w and effort e. The utility function

satisfies uw (w, e) > 0, uww (w, e) < 0 and ue (w, e) < 0 (where the subscripts

denote the partial derivatives). This implies that the manager prefers more

wealth to less, he is risk averse and dislikes high effort. More specifically

we assume that the utility function is given by u (w, e) = v (w) − e, where

v0 (w) > 0 and v00 (w) < 0.

We can now write down the optimal contract. Since bank profits are

decreasing in managerial wages, the optimal compensation scheme solves

the following:

M in

w(L,π)

Z

L(rL)

Z

Π ̃

w (L, π) dG 3L(rL), Π ̃|e

 ́

subject to

Z

L(rL)

Z

Π ̃

v (w (L, π)) dG 3L(rL), Π ̃|e

 ́

− e ≥ uo (26)

Z

L(rL)

Z

Π ̃

v (w (L, π)) dG 3L(rL), Π ̃|eH

 ́

−eH ≥

Z

L(rL)

Z

Π ̃

v (w (L, π)) dG 3L(rL), Π ̃|eL

 ́

−eL

(27)

where uo is the manager’s reservation utility.Hence, the second best contract

minimizes the expected wages subject to the participation constraint of the

manager and subject to the incentive compatibility constraint.

Letting μ1 and μ2 denote the multipliers for constraints (26) and (27)

respectively, then optimization of the Lagrangian yields the following characterization for the optimal compensation schedule:

1

v0 (w (L, π)) = μ1 + μ2

⎡

⎣1 −

g

3

L(rL), Π ̃|eL

 ́

g

3

L(rL), Π ̃|eH

 ́

⎤

⎦

40
We know from standard contract theory that in any solution to the

optimization problem with e = eH, both μ1 and μ1 must be strictly positive.

(See for instance, Holmstrom (1979)). This implies that both constraints

(26) and (27) are binding. Hence unlike the first best solution, the optimal

compensation scheme is not a constant but varies with L and Π ̃.

As is common in the literature, we then invoke the monotone likelihood ratio property (MLRP), i.e. h

g

3

L(rL), Π ̃|eL

 ́

/g 3L(rL), Π ̃|eH

 ́i is

decreasing in L and Π ̃. In words, this means that as bank loans and bank

profits increase, the likelihood of getting a given level of loans and profits if

effort is eH, relative to the likelihood if effort is eL must increase. Given that

v0 > 0, this implies that the manager’s wages are monotonically increasing

in L and Π ̃. Q.E.D.

Proof of Proposition 3. The proof is along the same lines as the proof

to Proposition (1). Define α = β/δ. Thus maximizing (10) is equivalent to

maximizing the following objective function:

Πm = αE [L(rL)|eH]+(π − rpE [max ( ̃xD − R, 0)|e = eH])

Let h (L(rL)|e) denote the density function of loans conditional on effort.

Define L ̄H ≡ E [L(rL)|eH] and L ̄0

H ≡ ∂E [L(rL)|eH] /∂rL = R L0 (rL) h (L(rL)|eH) dL.

Then taking the partial derivative of Πm w.r.t. rL we get the following FOC:

αL ̄0

H + θL ̄H − θ Pr [( ̃xD < R)|e = eH]L ̄0H + θrLL ̄0H

−rp Pr [( ̃xD ≥ R)|e = eH]L ̄0

H − θD (1 − E ( ̃x)) ∂r∗

D

∂rL

= 0.

The participation constraint is given by (7) and as before it holds with

equality. Solving for r∗

D and taking the partial derivative w.r.t. rL we get

∂r∗

D/∂rL = (1 − θ) Pr [( ̃xD < R)|e = eH]L ̄0H/ [θD (1 − E ( ̃x))]. As before

inserting ∂r∗

D

∂rL in the FOC and solving for r∗

L we get:

r∗a

L = 1 − α + (rp − 1) Pr [( ̃xD ≥ R)|e = eH]

θ

3

1 − 1

η ̄L

 ́ (28)

where r∗a

L denotes the lending rate in the presence of an agency problem and

η ̄L = −rL

∂E[L(rL)|eH]/∂rL

E[L(rL)|eH] = −rL

L ̄0

H

L ̄H . Comparing the above expression with

41
(9) it is clear that the loan rate is lower, i.e. r∗a

L < rf

L. Since L0 (rL) < 0,

the volume of bank credit increases. Q.E.D.

Proof of Proposition 4. If the manager engages in overly-aggressive

behavior, his expected payoff is given by the following expression:

βL ̄a

H − γrpE [max ( ̃xD − Ra, 0)|e = eH]

where Ra = D − La and L ̄a

H = E [La|eH] which denotes the expected loan

volume in the presence of an agency problem conditional on the manager

exerting high effort. (Note that choosing an expected loan volume is equivalent to choosing a lending rate since there is a one-one mapping from the

lending rate to the expected loan volume).

However, in the absence of agency problems, the expected loan volume

is given by L ̄f

H = E £

Lf |eH

¤

which denotes the expected loan volume in the

first best world conditional on the manager exerting high effort. Thus the

expected payoff of the manager in the absence of agency problem is simply

given by Πna

m = βL ̄f

H. Since the manager can always earn this expected payoff in the absence of agency problems, Πna

m can be treated as an opportunity

cost in the agency world. In other words, the manager has to earn at least

Πna

m in the agency world as otherwise the manager would be better off by

not acting over-aggressively.

Thus the problem of the manager is to maximize

max

L ̄a

H

Πa

m = βL ̄aH − γrpE [max ( ̃xD − Ra, 0)|e = eH] − βL ̄f

H (29)

The first order condition is given by:

β − γrp

∂E [max ( ̃xD − Ra, 0)|e = eH]

∂L ̄a

H

(30)

We assume that the second order condition is satisfied, i.e.

∂2E [max ( ̃xD − Ra, 0)|e = eH]

∂L ̄a2

H

< 0.

Differentiating the total derivative of the FOC wrt D we get:

∂2Πa

m

∂L ̄a2

H

dL ̄a∗

H

dD +

∂2Πa

m

∂L ̄a

H∂D = 0.

42
where L ̄a∗

H denotes the equilibrium value of the expected loan volume in the

agency world.

Since ∂2Πa

m

∂L ̄a2

H

< 0 given the SOC, it follows that

sign μdL ̄a∗

H

dD ¶

= sign μ ∂2Πa

m

∂L ̄a

H∂D¶

Thus one needs to prove that ∂2Πa

m

∂L ̄a

H∂D > 0 as this would imply that an

increase in deposits would increase the equilibrium loan volume (in expected

terms) and hence overall the manager would be better off.

Taking the partial derivative of (30) wrt D we obtain:

−γrp

∂E [max ( ̃xD − Ra, 0)|e = eH]

∂L ̄a

H∂D (31)

where E [max ( ̃xD − Ra, 0)|e = eH] = Pr (xD > Ra)

£

xD − D + L ̄a

H

¤

. Thus

∂E[max( ̃xD−Ra,0)|e=eH]

∂L ̄a

H

= Pr (xD > Ra). It follows that ∂E[max( ̃xD−Ra,0)|e=eH]

∂L ̄a

H∂D <

0 since ∂ Pr(xD>R)

∂D < 0. Hence the sign of expression (31) must be positive.

Hence as deposits increase, managers have an incentive to act overaggressively and increase the expected loan volume (via setting a lower

lending rate) and they will indulge in this practice as long as their payoffs from doing so increase their payoffs in the default first best world (i.e.

their opportunity cost). Conversely, for low enough liquidity there will be no

agency problems as in this case the manager would be better off by setting

the first best lending rate. Q.E.D.

Proof of Proposition 5. We know from (17) that entrepreneurs will

choose e = 0 if and only if φθ (C − d) + B>θ (C − d). This implies that

entrepreneurs will shirk if and only if θ < B

(1−φ)(C−d) . Q.E.D.

Proof of Proposition 6. Comparing (13) with (14) we know that

P >Pf if and only if rL < rf

L. From the proof to Proposition 4 we know

that rL < rf

L for sufficiently high D. Let D∗ denote the threshold below

which rL < rf

L and assume the plausible that the number of investors I is

big enough so that D∗ exists. Hence all we need to show is that dD

dθ < 0

∀θ ∈ [θ, 1]. Since D = (1 − νθ) I ∀θ ∈ [θ, 1] it follows that dD

dθ = −νI < 0

∀θ ∈ [θ, 1]. Since D is monotonically decreasing in θ for all θ ∈ [θ, 1] it

43
follows that there exists a threshold θc below which D>D∗ and hence

P >Pf , where θc is such that it solves D∗ = (1 − νθc

) I. Q.E.D.

Proof of Proposition 7. Differentiating the total derivative of the

FOC (20) wrt θ we get:

∂2W

∂M2

dM∗

dθ +

∂2W

∂M∂θ = 0.

Since ∂2W

∂M2 < 0 given the SOC, it follows that

sign μdM∗

dθ ¶

= sign μ ∂2W

∂M∂θ¶

.

Taking the partial derivative of (20) wrt θ we obtain after some simplification:

∂2W

∂M∂θ = −∂ Pr (D>D∗)∂M

∂Ω (∆)

∂θ − ∂Ω (∆)∂M

∂ Pr (D>D∗)

∂θ . (32)

We know that ∂ Pr(D>D∗)

∂M > 0 since an increase in the money supply increases

bank liquidity. Also,

∂Ω (∆)

∂θ = Ω0 (∆)

d∆

dθ

Given that Ω0 (∆) > 0, it follows that the first term in (32) is positive if

d∆

dθ < 0 and vice versa. Similarly,

∂Ω (∆)

∂M = Ω0 (∆) d∆dM

where d∆

dM > 0 since an increase in money supply increases bank liquidity,

lowers the loan rate and thus increases ∆. This implies that the expected

marginal cost of monetary expansion is positive.

Next note that ∂D

∂θ < 0 is a necessary condition for d∆dθ < 0 since the

expected size of the bubble increases as liquidity increases. Conversely,

∂D

∂θ > 0 is a necessary condition for d∆dθ > 0. It follows that,

∂ Pr (D>D∗)

∂θ (

< 0

> 0

if d∆

dθ < 0

if d∆

dθ > 0

Thus, the second term in (32) is positive if d∆

dθ < 0 and vice versa. Q.E.D.

44
Proof of Proposition 8. Differentiating the total derivative of the

FOC (20) wrt D we get:

∂2W

∂M2

dM∗

dD +

∂2W

∂M∂D = 0.

Since ∂2W

∂M2 < 0 given the SOC, it follows that

sign μdM∗

dD ¶

= sign μ ∂2W

∂M∂D¶

.

Taking the partial derivative of (20) wrt D we obtain after some simplification:

∂2W

∂M∂D = −∂ Pr (D>D∗)∂M

∂Ω (∆)

∂D − ∂Ω (∆)∂M

∂ Pr (D>D∗)

∂D (33)

We know that ∂ Pr(D>D∗)

∂M > 0. Also ∂Ω(∆)∂D > 0 since an increase in bank

liquidity lowers the loan rate, increases expected asset prices and thus increases ∆. Thus the first term is negative. Similarly, we know ∂Ω(∆)

∂M > 0

and ∂ Pr(D>D∗)

∂D > 0. Thus the second term is also negative. Q.E.D.

45
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