week 8 diamond dybvig model and financial crisis · 2008-12-02 · model • consumers don ......

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Week 8 Diamond Dybvig Model and Financial Crisis

• Diamond-Dybvig Model of Banking Runs

• Lender of Last Resort or Deposit Insurance

• Subprime Crisis and The Entire Financial System

• Summary of Causes of Financial Crisis


The Diamond-Dybvig Banking Model

• Explain the emergence of banks.

• Show that a bank run is a possible equilibrium

• Show the usefulness of deposit insurance

• Banks emerge to pool liquidity risk: an individual does not know when he needs money



• Three periods, 0, 1, and 2.

• Every agent has 1 endowment in period 0

• It can invest in a technology giving a payoff 1+r in period 2

• The investment can be interrupted in period 1. You get your money back and get 1

• There are two types of consumers: early (consume in period 1) and late (consume in period 2)



• Consumers don’t know beforehand whether they are early or late consumers. As in the real world, you could face shocks such as a hospital bill

• The outcome without a bank is that each agent gets 1 in period 1 if she interrupts investment and gets 1+r in period 2 if she does not interrupt

• But the economy can do better: set up a bank to share risk,

The liquidity risk that you need early money, i.e. in period 1



• Why can the economy do better with a bank?

• A bank can pool liquidity risks and give each consumer the average rate of return

• This implies that a consumer gets more than 1 in period 1 and less than 1+r in period 2

• The consumer is better off, because he likes to smooth consumption


Figure 15.6 The Utility Function For a Consumer in the Diamond–Dybvig Model

Decreasing marginal utility implies a desire to smooth consumption


Expected Utility Function

Expected utility over the two periods is the sum of the utilities in both periods, multiplied (weighted) by the probability that you consume in one of the two periods

( ) ( ) ( )21 1 CUtCtUEU −+=

With t the probability that you want to consume in period 1


Equation 15.12

The marginal rate of substitution of early consumption for late consumption is


Figure 15.7 The Preferences of a Diamond–Dybvig Consumer


A Bank with a Deposit Contract

• There are N consumers paying N amounts of resources to the bank.

• First constraint that a deposit contract must satisfy is:

• So, the number of consumers who want to consume in period 1, Nt times their consumption C1, has to be equal to the fraction of total resources invested that is interrupted, xN


Equation 15.14

Second constraint that a deposit contract must satisfy is

The fraction of consumers that does not interrupt, N(1-t) times their consumption C2 has to be equal to resources available in period 2


Equation 15.15

Combine the two constraints to get one:


Equation 15.16

Re-write the constraint:


Equilibrium with Bank

There is one (representative) bank making zero profit, assuming free entry

This implies that an efficient outcome is the equilibrium (remember chapter 5, second welfare theorem)

So, we search for the efficient outcome, maximize utility subject to budget constraint of bank

This determines what consumer gets in the two periods


Figure 15.8 The Equilibrium Deposit Contract Offered by the Diamond–Dybvig Bank

The equilibrium A determines the fraction of investments interrupted x.

15-16

Two Characteristics of Equilibrium

D is the outcome without a bank: 1 in period 1 and 1+r in period 2

Characteristic 1: in equilibrium A with a bank there is more consumption in period 1 than in D without a bank. This reflects first, the desire to smooth consumption and second, the possibility to smooth consumption, because there is a bank

15-17

Two Characteristics of Equilibrium

B would be the outcome with equal consumption in both periods

Characteristic 2: in A consumption in period 1 is smaller, but in period 2 larger than with equal consumption in BThis reflects the technological payoff in the economy that keeping the investment until period 2 generates a rate of return of 1+r.

Hence you want to make it optimal for consumers who don’t face a shock to wait with consumption till period 2, paying them more

The MRS is equal to

–t/(1+t)


Good and Bad Equilibrium

• Consumers who don’t face a shock, a fraction

1-t, will wait with consumption until period 2, because they get more in period 2

• Consumers who face a shock, do consume in period 1, and forego the rate of return from waiting

• This is the good equilibrium


Good and Bad Equilibrium

• Suppose that the bank works with a first come first serve system: people first in the line for the bank get their deposits first

• When a period 2 consumer thinks that all other period 2 consumers want to consume in period 1 already…

• Then it is optimal to also queue in period 1. This gives the probability of at least some payoff, whereas with waiting till period 2, there is for sure nothing left


Bad Equilibrium: Bank Run

• There is nothing left when all other consumers want to consume in period 1, because the bank only has Nx resources, whereas (N-1)C1 is the amount of consumption N-1 agents want to withdraw

• Remember that C1 is larger than 1, because the bank could use risk pooling

• Given that all others want to consume in period 1, I will also queue up

• The result is a bank run with everybody queuing to get quick money


Bank Run: Intuition

• The reason the bank has not enough resources in period 1, is that part of the money from the depositers is invested in the technology, assuming that only a fraction t wants to consume in period 1

• If people behave as expected this runs fine, but when everybody wants to withdraw its money in period 1, there is a problem

• The bank run is an equilibrium, because it is optimal to queue up given that others do the same


Possible Solution: Deposit Insurance

• If the government guarantees the value of all deposits, the bad equilibrium disappears

• It is not optimal anymore for an individual period 2 consumer to queue up in period 1, also if all others queue up

• She will get her money in period 2, as it is guaranteed by the government. And it is more than what she could get in period 1

• So, she does not queue up and the bank run does not take place


Problem of Deposit Insurance: Moral Hazard

• If banks know that deposits are insured, it becomes attractive to take excessive risk (moral hazard):

• The government guarantees the deposits anyway, so deposit holders have no incentive to look for a careful bank and will only look for the bank with the highest rate of return

• Therefore, deposit insurance has to go along with regulation and supervision to prevent excessive risk taking


Optimality of Deposit Insurance: Too Big To Fail Doctrine

• The government has a direct interest in protecting deposit holders, because bank runs lead to losses for the deposit holders

• But a wider reason for deposit insurance is that a bank run might be the onset for further problems:

• If a bank collapses, other banks that invested in that bank may also collapse, because the value of their assets becomes too small

• Hence, many financial institutions are ‘too big to fail’


Lack of Regultation and Too Big To Fail Doctrine

• Part of the problem in present crisis is that non-bank financial intermediaries like investment banks and hedge were too big to fail

• But still are hardly regulated

• Hence, they took excessive risk and still the government had to step in to prevent their collapse

week 8 diamond dybvig model and financial crisis · 2008-12-02 · model • consumers don ......

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