Modello bank run

T

his need for liquidity, because the need is private information and, thus, is not

due to some observable event such as a hurricane. However, contracts can

be designed that indirectly provide this insurance. Liquid assets that offer a

D. W. Diamond: Banks and Liquidity Creation 191

smaller loss when liquidated early can provide indirect insurance. As a result,

for some risks, investors can save and liquidate the liquid assets as needed. I

=

will call an investor a “type 1” if he needs to liquidate at 1 and a “type

T

=

2” if he can wait until 2. For this example, this means he will consume

T

= =

only at 1 if of type 1, and only at 2 if of type 2 (or he can store date

T T

1 consumption goods and consume them at date 2).

As of date 0, an investor does not know which type he will be, but each

−

investor has a probability of being of type 1 and 1 of being of type 2.

t t

There is no aggregate uncertainty, and there will be a fraction of investors of

t

= 1

type 1. To be concrete, suppose that and that there are 100 investors.

t 4

As a result, 25 will be of type 1 and 75 will be of type 2, but it is not known

at date 0 which investors will be of each type. =

A type 1 investor with utility function who consumes at 1 has

U (c) c T

1

=

utility A type 2 investor who consumes at 2 (this includes

U (c ). c T c

1 2 2

any stored date 1 consumption goods) has utility The utility function is

U (c ).

2

the same for both types, but the date on which an investor wishes to consume

depends on his type. An investor who holds the asset (r , ), which gives a

r

1 2

=

choice of at date 1 or at date 2, consumes if of type 1 (with

r r > r c r

1 2 1 1 1

= −

probability or if of type 2 (with probability 1 The investor’s

t) c r t).

2 2

expected utility is given by + −

tU (r ) (1 t)U (r ).

1 2 −1

= .

I assume that the investor has the risk-averse utility function of U (c) c

To simplify exposition, I add a constant of 1 to the utility (with no effect on

= − 1

any decisions) and use the utility function 1 This allows the

U (c) ( ).

c

expected utility calculations to yield positive numbers.

Comparing More and Less Liquid Assets

Consider the following two assets, both of which cost 1 at date 0. The illiquid

= =

asset has 1, and a more liquid asset has 1, In-

(r r R) (r > r < R).

1 2 1 2

vestors have access only to the illiquid asset. Later, I will show how banks can

create the more liquid asset, although there is no physical asset with its payoffs,

but for now I simply illustrate the demand for liquidity with the following nu-

= 1

merical example for the case where the probability of being of type 1 is t 4

= = =

and the illiquid asset has 1, 2). As a comparison, consider

(r r R

1 2 = =

a hypothetical more liquid asset that has 1.28, 1.813). Section 2

(r r

1 2

explains why these particular numerical values are used. The expected utility

from holding the illiquid asset is

1 3

+ = 0.375.

U (1) U (2)

4 4

192 Federal Reserve Bank of Richmond Economic Quarterly

The expected utility from holding the more liquid asset is

3

1 + = 0.391 0.375.

U (1.28) U (1.813) >

4 4

Each investor prefers the more liquid asset. A risk-averse investor prefers

this smoother pattern of returns; holding the illiquid asset is risky because it

delivers a low amount when liquidated early, on date 1.

Note that if investors were not risk averse and had constant marginal utility

of consumption, they would not prefer this particular liquid asset. That is, if

= then the expected utility of holding any asset is equal to its expected

U (c) c,

payoff given the policy of liquidating when of type 1. For the illiquid asset,

the expected payoff is 1 3

+ = 1.75.

(1) (2)

4 4

The more liquid asset gives an expected payoff of:

3

1 + = 1.68 1.75.

(1.28) (1.813) <

4 4

The more liquid asset has a lower expected rate of return. Sufficiently risk-

averse investors, but not risk-neutral investors, are willing to give up some

expected return to get a more liquid asset.

Investors choose to liquidate assets when consumption is highly valuable

to them. In particular, a type 1 investor liquidates the asset at a time when his

marginal utility of consumption is high. An investor’s demand for liquidity

is greater the higher his (relative) risk aversion is, because liquidating early

implies low consumption and, thus, high marginal utility of consumption.

Entrepreneurial Liquidity Demand

An alternative motivation for a large demand for liquid assets comes from an

entrepreneur who may have a sudden need to fund a very high return project

at date 1 (which cannot be funded elsewhere). The entrepreneur wishes only

to consume on date 2 but may choose to liquidate assets on date 1 to fund this

high return project. As a result, the entrepreneur places an especially high

value on date 1 liquidation proceeds in the circumstances where he wants to

liquidate early. Suppose that with probability the entrepreneur will be able

t,

to fund the high return investment project and that it returns per unit

> R

−

invested. With probability 1 he does not get this opportunity and has

t,

access only to storage (storing one unit of goods at date 1 returns one unit at

date 2). The availability of the high return is private information. Consider

an asset that costs 1 at date 0 and offers either at date 1 or , at date

r r > r

1 2 1

2. When the entrepreneur has access only to storage, he will not liquidate the

asset. However, when he needs to fund the high return project he will liquidate

r

2

it if the project’s return exceeds , the rate of return from continuing to hold

r

1

D. W. Diamond: Banks and Liquidity Creation 193

the asset. As of date 0, the entrepreneur values an asset that can be liquidated

+ − r

2

for at date 1 or at date 2, as follows: if , and

r r tr (1 t)r , >

1 2 1 2 r

1

+ − ≤ r

2

if . This is qualitatively similar to the risk-averse

as (1 t)r ,

tr 1 2 r

1

consumer, because the entrepreneur liquidates when the value of the proceeds

= = = 1

is very high. Suppose that if 2.5, 2, and , the entrepreneur

R t 4

= = + =

1 3

1, 2) as 2.125, and

then values the illiquid asset r (1) (2)

(r 1 2 4 4

= = + =

1 3

1.28, 1.813) as 2.160.

the liquid asset r (1.28) (1.813)

(r

1 2 4 4

The entrepreneur prefers the more liquid asset.

The entrepreneurial demand for liquidity will be even more similar to the

investor/consumer demand for liquidity if the high return project has decreas-

ing returns to scale.

I do not continue to analyze the entrepreneurial demand for liquidity here,

but refer the reader to Diamond and Rajan (2001) and Holmström and Tirole

(1998). I now return to the consumer demand for liquidity.

2. BANK LIQUIDITY CREATION

I now show that a bank can provide the more liquid asset by offering demand

= =

1, 2).

deposits, even though the bank invests in the illiquid asset r

(r

1 2

I assume a mutual bank without equity (purely for expositional simplicity).

=

Suppose that in return for a deposit of 1 at date 0, the bank offers to pay

T

= = =

1.28 to those who withdraw at 1 or to pay 1.813 to those who

r T r

1 2

=

withdraw at 2.

T

If the bank receives $1 from each of the 100 investors, it receives $100

=

in deposits on date 0. If the bank invests in the illiquid asset, it will

T =

need to liquidate some of the illiquid asset at 1 to pay 1.28 to those who

T

withdraw.

=

At 1, the bank’s entire portfolio is worth $100. Suppose 25 depositors

T =

withdraw 1.28 each, then 25(1.28) 32 assets must be liquidated: 32 percent

of the portfolio must be liquidated. If 32 assets are liquidated, then 68 will

= =

remain until 2, when they will be worth 2 each. On date 2, there

T R

remain 75 depositors, each will receive

−

[100 32]2 [68]2

= = 1.813.

75 75

Depositors prefer the more liquid asset to the illiquid asset. A bank can

provide the more liquid deposit which has a smaller loss from early liquida-

tion than is available from holding the illiquid assets directly. This liquidity

transformation service is one of the most important functions of banks. If the

bank offers the more liquid deposits and invests in the illiquid assets, it can

create liquidity. It is an equilibrium (a Nash equilibrium) for 25 depositors to

= =

withdraw at 1, because if all depositors expect 25 to withdraw at 1,

T T

only type 1 depositors will withdraw because the 75 type 2 depositors prefer

= =

the 1.813 available at 2 to the 1.28 available at 1.

T T

194 Federal Reserve Bank of Richmond Economic Quarterly

When assets are illiquid and risk-averse investors do not know when they

will need to liquidate, the bank can create a more liquid asset that allows

investors to share the risk of liquidation losses. The bank can give a fraction

− = [1−tr ]R

1

of investors at date 1 and a fraction 1 of investors at date

t r t r

1 2 1−t

=

2, because if a fraction of the depositors get in period 1, this will

t r T

1

−

leave a fraction [1 ] of the assets unliquidated and in place until date 2.

tr 1 − = [1−tc ]R

1

Each of the remaining fraction of depositors can receive

(1 t) r 2 1−t

= =

in period 2. Note that for the illiquid asset, 1 and

r r R.

1 2

The Optimal Amount of Liquidity

This section derives the optimal amount of liquidity for the bank to create.

The optimal levels of and will maximize the ex ante expected utility

r r

1 2

of each investor at date 0. The optimum is to provide a choice between

= =

1.28 at date 1 or 1.813 at date 2. This derivation is not essential to

r r

1 2

understanding the balance of the article. The optimal amount of liquidity to

create is the amount that maximizes each investor’s ex ante expected utility,

= = + −

choosing to maximize subject to

c r , c r tU (r ) (1 t)U (r ),

1 1 2 2 1 2

≤ ≥ ≥

[1−tr ]R

1 0, 0. For an interior optimum, the optimal values

r , r r

2 1 2

1−t

=

satisfy so the marginal utility is in line with the marginal

U (r ) RU (r ),

1 2 = [1−tr ]R

1

cost of liquidity, and , because no liquidity is wasted. For the

r 2 1−t =

= − 1 1

, marginal utility is

case used in the example where 1 U (c)

U (c)

√ 2

c c

2

r = =

r 1

2

2

and the c