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
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10-1
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
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15-2
The Diamond-Dybvig Banking
Model
• 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)
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15-3
The Diamond-Dybvig Banking
Model
• 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
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15-4
The Diamond-Dybvig Banking
Model
• 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
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15-5
Figure 15.6 The Utility Function For a
Consumer in the Diamond–Dybvig Model
Decreasing
marginal utility
implies a desire to
smooth
consumption
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15-6
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
EU = tU (C1 ) + (1 − t )U (C2 )
With t the probability that you want to consume in
period 1
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15-7
Equation 15.12
The marginal rate of substitution of early
consumption for late consumption is
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15-8
Figure 15.7 The Preferences of a
Diamond–Dybvig Consumer
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15-9
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
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15-10
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
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15-11
Equation 15.15
Combine the two constraints to get one:
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15-12
Equation 15.16
Re-write the constraint:
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15-13
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
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15-14
Figure 15.8 The Equilibrium Deposit
Contract Offered by the Diamond–Dybvig
Bank
The
equilibrium A
determines the
fraction of
investments
interrupted x.
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15-15
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
15-16
Two Characteristics of
Equilibrium
B would be the outcome with
equal consumption in both
periods
The MRS is equal to
–t/(1+t)
Characteristic 2: in A consumption in period 1 is smaller, but in
period 2 larger than with equal consumption in B
This 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 15-17
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
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15-18
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
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15-19
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
15-20
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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
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15-21
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
15-22
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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
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15-23
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’
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15-24
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
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15-25