Money and Banking Lecture 3a

The DD model of ‘liquidity insurance’

Marcus Miller

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Review: Buying insurance - competition and

monopoly

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F M

E

FO CB

CG

0

Endow

ment

FO

45°

Fair Odds

OC

W

W -L

EU

Monopoly

Price 1

p

p

NB: CG plotted on vertical axes; CB on horizontal. Results same.

Diamond Dybvig created a Model

of Liquidity Insurance

• Banks as socially efficient, but fragile institutions –

because there’s a coordination problem among

depositors

• And what to do about it

• Before coming to banking, we will examine

non-banking solutions first.

• We begin with a solitary person on a desert island!

This is Robinson Crusoe Island , off Chile, where

Robinson Crusoe was marooned alone, 1704-1708

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.

(It now has pop of 700 - but still no bank!)

Robinson Crusoe on the beach

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Choice of what to do with endowment

• Assume the consumer has an endowment of one

unit of good at date 0; and no endowment at future

dates.

• Think of a bunch of coconuts.

• Technology

• One possible strategy (storage) is to keep access

to this endowment in either period 1 or period 2.

• Think of this as leaving coconuts on the beach.

• This is convenient; but not very productive. (Like

keeping all your money on deposit.)

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Alternative: long-term investment

• The alternative strategy (investment) is to give up

access to endowment in period 1 so as to get

higher return in period 2 (i.e. to hold a longer term

asset that allows one unit of the good at date 0 to

be converted into R>1 units of the good at time 2).

• Think of burying coconuts under the ground so

they will grow into trees in period 2 (productive, but

illiquid)

• What should Robinson Crusoe do? Will surely

depend on his tastes. These are rather special.

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Crusoe is uncertain as to whether he will want

to consume early, or later

• Tastes

• Consumption is to take place in the future, at dates

1 and 2, but the consumer is uncertain at which

date. Let be the probability of being an early

consumer with preference and 1- the

probability of being a late consumer with

preference.

• So the consumer is uncertain not about the world,

but about his/her own preferences. This is one way

of modelling the demand for liquidity.

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Expected Utility of consumer

• Expected Utility =

• EU is a weighted average of preferences for early

consumption , with weight λ, and

• late consumption, with weight (1- λ)

• Can represent these in diagram as we did

yesterday, see below.

• Can also see the technology .

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R

c1

c2

A

Most likely autarky equilibrium: storage,

no long term investment

1

Unlikely to get tangency solution

here R

Expected Utility Curve

L

Autarky Equilibrium: most likely (1,1)

.10

Point A is like leaving all coconuts on the beach; L is like putting them

all underground. Can choose to split between the two. Shading shows

all the Consumption Possibilities. Looks like Crusoe will go for A.

Now assume that there are a number of people

on the island; but no bank.

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• They can do better than Crusoe on his own, as follows.

• Each can make investments in line with the population

parameters λ, 1-λ (e.g. all put half their coconuts

underground if there’s a 50% chance of being a late

consumer) . But if there’s a market in which consumers

can switch between the short and long assets at one to

one when they learn their type, then earlier consumers

can get 1 and late consumers get R by trading on

learning their type. Hence EU is higher, namely

• EU=λU(1)+(1-λ)U(R)

.12

How trading at the end of period 1, improves

on autarky: see Market equilibrium at M

M R

c1 1

c2

Feasible

Autarky

Allocations

A Expected Utility

Curve

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The banking solution

• Finally we come to the banking solution.

• The bank takes one unit from each agent at time 0 and

invests it in a portfolio of y units of the short asset and x

units of the long asset, and by exploiting the law of large

numbers offers each consumer a consumption profile

(c1,c2), which can be interpreted as a deposit contract,

where the depositor has the right to withdraw either c1

at date 1 or c2 at date 2, but not both.

• Assume that, with free entry and competition, banks will

maximise the EU of the typical depositor subject to a

zero-profit constraint.

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Banking Solution: competitive banking

(and the demand for liquidity shown by OC)

M

C Consumers’ Offer Curve

R

1

Constant Expected Utility

Banks’ No-Profit Constraint

c1

c2

1/

B

What the bank can offer

Note that the set of possible contracts, shown as the

Banks No-profit Constraint, is obtained as:

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so

The competitive equilibrium will be where the EU

curves are tangent to this constraint.

As yesterday, we can draw an Offer Curve showing

what depositors would like at different prices for

liquidity. The CE is the demand at the fair odds

price.

Formal summary p. 1

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Max

s.t.

so

i.e

Typically we assume , where measures risk

aversion.

Banking:

increasing Risk Aversion (RA) in the DD model

• Low RA log: 1

𝑐1=

𝑅

𝑐2 i.e. 𝑐2 = 𝑅 𝑐1

• More RA, rho =2, 1

𝑐1 =

𝑅

𝑐2 or 𝑐2 = 𝑅 𝑐1

• Extreme RA, as for rho = n 𝑐2 = 𝑅𝑛

𝑐1 then

as rho tends to infinity we find 𝑐2 = 𝑐1.

• This is the solution for Leontief preferences.

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R

c1 1/

c2

R/(1-)

Leontief

preferences

(c1 = c2) L

1 c1

R

= 1/2

P

M

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Bank runs

• Suppose there exists a liquidation process, so

that if the long-term asset is liquidated

prematurely at date 1, one unit of the long asset

yields r < 1 units of the good.

• Then there exists another equilibrium if the bank

is required to liquidate whatever assets it has in

order to meet the demands of consumers who

decide to withdraw at time 1.

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Bank Runs (cont’d)

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To see this suppose that all depositors decide to

withdraw at date 1. As the banks assets will be ,

the bank cannot pay all its depositors more than 1

at date 1. In fact, if liquidated, it can only pay less

than 1 on average.

Because all assets will be used up trying to pay

those who withdraw at period 1, anyone who

waits for second period will get nothing.

Given a late consumer thinks all others will

withdraw, it pays him to withdraw. There are two

equilibria of this coordination game.

Bank runs (cont.)

Pay-off to the one late consumer who does not run,

average payoff to all others

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Run

(by all other late

consumers)

No Run

(by all other late

consumers)

Run (by him) (rx + y, rx+y)

No Run (by

him)

(0,rx + y)

1 2( , )c c

2 2( , )c c

Avoiding Bank Runs

a) Lender of Last Resort (as implemented by the Bank

of England). In 1873 Walter Bagehot laid out three

principles for central bank intervention:

i. Lend freely at a penalty rate against good collateral

ii. Value assets at between panic and pre-panic values

iii. Institutions without good collateral should be allowed to fail.

Allen and Gale wrote: “The last true panic in the UK was

the Overend and Gurney crisis of 1886.”

But that was before Northern Rock in 2007!

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Avoiding Bank Runs (continued)

b) Deposit Insurance

The was no central bank in the US from 1836 until – after a

succession of bank panics – the Federal Reserve System

was established in 1914.

The banking panic of 1933 led to the Glass-Steagall Act of

that year, which introduced Deposit Insurance and required

the separation of commercial and investment banking

operations. (The FDIC was set up in 1934.)

c) Suspension of convertibility (‘Bank holiday’)

In the 1930s some US States ordered the closure of banks

until panic was over. 23

Questions for Seminar

• What is the Autarky solution for Robinson Crusoe,

alone on the island?

• How can Crusoe and his friends improve on this by

trading at the end of period 1, even with no bank?

• How can a bank improve on this no-banking

solution?

• Why is the banking equilibrium fragile?

• How to rescue a bank facing a ‘bank run’?

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