EU ( c , c ) U ( c ) ( 1 ) U ( c ) 1 2 - - PDF document

SLIDE 1

1

Banking Crises – Model by Diamond und Dybvig (1983) Three periods: t = 0, 1, 2, infinitely many households One-good-economy, good is storable and may be interpreted as liquidity. Depositors (private households) have 1 unit of the commodity in t=0. They may

• store their good,
• invest directly in a technology,
• deposit their good in a bank.

With probability γ a household needs liquidity (goods) in t=1 („early consumer“) With probability 1 – γ a household needs liquidity only in t=2 („late consumer“)

2

In t=0, a household does not yet know when he needs liquidity (his “type”). Utility from consumption in the appropriate period is given by U(ct). Assume U’ > 0 and U’’ < 0 Objective: maximize expected utility

) ( ) 1 ( ) ( ) , (

2 1 2 1

c U c U c c EU      

1

c consumption in t=1

2

c consumption in t=2

1  

discount factor

SLIDE 2

3

Technology: there is an infinite number of potential projects. A project requires input of 1 unit of the good in t=0. The project can be liquidated in t=1. Then, it delivers λ ≤ 1 units of the good. (Here, we follow Diamond/Dybwig and assume λ = 1.) If the project runs until t=2, it yields payoff R > 1 (in units of the good). Optimal behavior without bank If λ = 1, a household cannot lose by investing. If he needs liquidity in t=1, he must liquidate his project and receives 1 unit of goods (storage would yield the same return). => c1 = 1 If he needs liquidiuty only in t=2, he receives R>1, which is better than just storing his good. => c2 = R His expected utility is

) ( ) 1 ( ) 1 ( ) , (

2 1

R U U c c EU      

Note: for λ < 1 a risk averse household should invest only a part of his endowment.

4

Efficient allocation by introducing a bank If liquidity need („type“) can be verified, each household can engage in contingent contract with a bank, guaranteeing a repayment

* 1

c in t=1, if he is an early consumer, and repayment

* 2

c in t=2, if he is a late consumer. How big should

* 1

c and

* 2

c be?

SLIDE 3

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Derive the efficient allocation infinitely many depositors (households), law of large numbers

 a fraction γ needs liquidity in t=1 and receives c1 each.  a fraction 1 – γ needs liquidity in t=2 and receives c2 each.

Allocation (c1, c2). The efficient allocation maximizes expected utility subject to feasibility Feasibility constraint: deposits in t=0 1 withdrawels in t=1 γ c1 A fraction of project 1 – γc1 will be continued to t=2. In total, these projects yields a return R(1 – γc1). The total return is distributed among the remaining depositors (late consumers) whose share is 1–γ. Result: If early consumess receive c1, late consumers can at most receive (per capita)

     1 ) 1 ( ) (

1 1 2

c R c c

. (1) Note that R c c   ) 1 ( 1

2

.

λ=1: The bank can invest a share 1 – γc1 of deposits projects, or she invests all deposits and liquidates a share γc1 of projects in t=1. Both strategies result in the same returns. For λ<1 the bank should invest only a fraction 1 – γc1 in projects and avoid the loss resulting from premature liquidation.

6

Maximize EU(c1,c2) s.t.

) 1 ( ) 1 (

1 2

c R c     

)] 1 ( ) 1 [( ) ( ) 1 ( ) ( max

1 2 2 1 , 2

1

c R c c U c U

c c

           

FOCs: (c1)

) ( '

1

     R c U

(2) (c2)

) 1 ( ) ( ' ) 1 (

2

        c U

(3) (2) <=>

R c U   ) ( '

1

(3) <=>

   ) ( '

2

c U

=> Optimality condition:

R c U c U ) ( ' ) ( '

2 1

 

(4)

SOC holds, because utility function is concave (U’’<0) and restriction is linear. Result: The efficient allocation (

* 1

c ,

* 2

c ) is characterized by optimality condition (4) and feasibility (1). To yield the solution explicitly, we may insert (1) in (4) and solve the resulting equation for c1:

(1)

     1 ) 1 ( ) (

1 1 2

c R c c

=>

              1 ) 1 ( ' ) ( '

1 1

c R U R c U

(5)

(4)

SLIDE 4

7

Consider equation (5)

              1 ) 1 ( ' ) ( '

1 1

c R U R c U

Since U’’<0, the left-hand side is decreasing in c1. The right-hand side is increasing in c1. c1 c1

*

feasible and optimal R ρ U’(c2(c1)) U’(c1) U’ U’(c1

*)

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Compare the optimal allocation (

* 1

c ,

* 2

c ) with the best allocation attaineable without bank (1, R) .

• 1. Allocation (1, R) is feasible without bank.
• 2. If in the optimal allokation c1

* > 1, then it is not attainable without bank.

Under which condition is c1

* > 1?

The figure shows that

) ( ' ) 1 ( ' 1

* 1

R U R U c    

. c1 c1

*

feasible and optimal R ρ U’(c2(c1)) U’(c1) U’ 1 U’(1) U’(c1

*)

R ρ U’(R)

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Lemma: if relative risk aversion (RRA) exceeds 1, then

1

* 1 

c

. Proof: 1. View ) ( ' R U R  as a function on R. Then

) ( ' ' ) ( ' ) ( ' R U R R U R U R dR d     

.

• 2. This derivative is negative, if and only if RRA(R) > 1:

. 1 ) ( ' ) ( ' ' ) ( ' ' ) ( '      R U R U R R U R R U

• 3. If

) ( '  R U R dR d 

, then

) ( ' ) 1 ( ' R U R U   

, because R>1. This implies

) ( ' ) 1 ( ' R U R U  

, because

1  

. QED

RRA(R)

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Result:

• 1. If consumers are sufficiently risk averse, a bank can improve upon the risk allocation by offering

a conditional contract, allowing early consumers to consume more than 1, while late consumers receive less than R.

• 2. Consumers are better of with this contingent contract. Therefore, they have an incentive to

depose their money in the bank instead of investing directly.

• 3. If the need for liquidity can be verified, the contingent contract solves the problem of attaining

the optimal allocation. The bank functions as an insurance company:

• The „damage“ is an early need for liquidity. Without insurance, this event would leave a consumer

with a low level of consumption.

• With insurance the damage will be mitigated (c1>1).

For this, the consumer pays an insurance premium, lowering his payoff in case the “damage” does not occur (c2<R). => Early consumers take a share in the returns of technology.

SLIDE 6

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Model with bank, if need for liquidity cannot be verified If a consumer’s type cannot be verified, a contingent contract is impossible. Instead, the bank may offer a deposit contract that allows each consumer (independent of his type) to withdraw money in t=1. Now incentive constraints must prevent that late consumers withdraw liquidity in t=1. Bank offers a deposit contract: consumer deposes funds in t=0 at the bank. Bank promises repayment 1

r ≥ 1 in t=1 oder 2 r in t=2.

• 1. Contract must be feasible

     1 ) 1 (

1 2

r R r

If

     1 ) 1 (

1 2

r R r

, the bank yields a profit. We concentrate on contracts with

     1 ) 1 (

1 2

r R r

. (6)

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• 2. Late consumers must have an incentive to leave their funds in the bank:

Incentive constraint:

1 2

r r 

Using (6), this is equivalent to

) 1 ( 1 ) 1 ( ) 1 ( ) 1 ( 1 ) 1 (

1 1 1 1 1 1

               R R r R r R r r R r r R       

If

* 1 1

c r 

and

* 2 2

c r 

, the deposit contract can implement the optimal allocation. This requires

* 2 * 1

c c 

• r equivalently:

) 1 ( 1

* 1

   R R c 

.

SLIDE 7

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BUT: If some positive mass of late consumers withdraw in t=1 and

1

1 

r

, then the bank cannot keep its promise. We can see this from the feasibility constraint:

     1 ) 1 (

1 2

r R r

If a share γ’ > γ of all consumers withdraws deposits in t=1, remaining funds are too small to pay

• ut r2 to the remaining 1 – γ’ consumers, because

) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( 1 ) 1 (

2 1 2 1 1 1 2 1 1 1

                                    r R r R R Rr Rr r R Rr d r R d

Intuition: late consumers subsidise consumption of early consumers, provided that

1

1 

r

and

R r 

2

. If a positive fraction of late consumers wants to receive a subsidy instead of providing it, there are more receivers than needed and fewer net-payers. Hence, the bank cannot pay the promised amount r2 to the remaining late guys.

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What happens if the bank cannot keep its promises? → Bankruptcy rule, subject to regulation Assume: „first come, first serve“ depositors who want to withdraw money in some period t arrive randomly and queue up in front of the bank. The bank pays them their promised return rt , until the bank is empty. Given this rule, the model has two equilibria in pure strategies for all interesting values of r:

• 1. if

) 1 ( 1

1

   R R r 

and thereby

1 2

r r 

, then there is an equilibrium, in which late consumers wait for t=2. Proof: If all late consumers wait, the bank can keep its promise and a late consumer has no incentive to withdraw early.

• 2. If

1

1 

r

, there is an equilibrium, in which all consumers withdraw in t=1 already (bank run). Proof: If all want to withdraw early and

1

1 

r

, then the bank must liquidate all projects and can

• nly serve a fraction

1

/ 1 r of all depositors.

When a depositor runs the bank, (s)he does not know whether (s)he ends up in front or in the back of the queue. The ex- ante probability of being served is

1

/ 1 r . If a late consumer waits

instead, he will receive zero for sure, because in t=2 the bank is already closed.

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Corollary: for

) 1 ( 1 1

1

    R R r 

there are two equilibria in pure strategies: one in which all depositors run the bank in t=1 and one in which all late consumers wait until t=2. If

) 1 ( 1 1

* 1

    R R c 

, there exists an equilibrium, in which the optimal allocation can be implemented by a deposit contract with

* 1 1

c r 

and

* 2 2

c r 

. But, there is another equilibrium in which the bank is run. For

) 1 ( 1

* 1

   R R c 

, there is no equilibrium implementing the optimal allocation by a deposit contract. r1 ) 1 ( 1   R R  1

Unique equilibrium: bank run Unique equilibrium without bank run Multiple equilibria For r1<1 a bank is not needed.

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How to solve the problem that the economy may coordinate on an inefficient equilibrium although it could as well coordinate on the efficient one? 1st approach: change the bankruptcy rules: suspension of convertibility. 2nd approach: deposit insurance Both approaches change the rules of the game A deposit contract usually allows each depositor to withdraw money anytime. Suspension of convertibility changes that.

SLIDE 9

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Suspension of convertibility As soon as a fraction

1

/ 1 ˆ r f 

has withdrawn deposits in t=1, the right to withdraw money from the bank is suspended. Now, remaining depositors (fraction

f ˆ 1

) have to wait until t=2. How does this change incentives? Agents withdrawing in t=1, recveive either 1

r or zero.

(depending on whether they are in the front or in the back of the queue and on suspension of convertibility.) Agents withdrawing in t=2, receive their share of the remaining value of the bank (at most

2

r ).

Let f be the fraction who may withdraw in t=1 (

f f ˆ 

). Remaining funds in the bank are

1

1 r f 

. They are invested in projects, earn interest and are distributed among remaining depositors in t=2. Hence, a depositor who did not (or could not) withdraw in t=1, receives in t=2

f R r f   1 ) 1 (

1

. Note that this amount is decreasing in f . Proof:

) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) )( 1 ( 1 ) 1 (

2 1 2 1 1 1 2 1 1 1

                           f r R f R r f R R r f R r f R r f R r f f R r f df d

QED

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Since

f f ˆ 

, the amount available for a depositor in t=2 is at least

f R r f ˆ 1 ) ˆ 1 (

1

 

. Late consumers have an incentive to wait, if ) 1 ( ˆ ) 1 ( ˆ ˆ ˆ ) ˆ 1 ( ) ˆ 1 ( ˆ 1 ) ˆ 1 (

1 1 1 1 1 1 1 1 1 1 1

                   R r r R f r R R r f R r f R r f r R r f r f f R r f r A deposit contract with

1

* 1 1

  c r

and

* 2 2

c r  , that suspends convertibility after withdrawel of f

ˆ ,

implements the optimal allocation with a unique Nash equilibrium, if and only if

) 1 ( ˆ

* 1 * 1

    R c c R f 

. Proof: If the condition holds, late consumers have an incentive to wait => uUnique equilibrium. If f

ˆ is smaller than  , some early consumers are not served in t=1, which is not optimal.

If f ˆ >

) 1 (

* 1 * 1

  R c c R

, there exists an equilibrium, in which late consumers run the bank in t=1. QED

SLIDE 10

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Condition ) 1 ( ˆ

* 1 * 1

    R c c R f  requires that

) 1 (

* 1 * 1

   R c c R 

. . 1 ) 1 ( ) 1 ) 1 ( ( ) 1 (

* 1 * 1 * 1 * 1

            R R c R R c c R R c    Result: If 1 ) 1 (

* 1

   R R c  , the optimal allocation can be implemented by a unique Nash equilibrium. BUT: the bank needs to know the fraction of early consumers and must be sure that this fraction does not exceed f

ˆ .

If  is a random variable, then it may happen that convertibility is suspended before all early consumers are served. The bank would need to set f ˆ sufficiently high to guarantee that all early consumers are served. It also needs to know  in order to promise optimal repayments, since the allocation (

* 1

c ,

* 2

c ) depends

• n  . Assume next that  is unknown. Denote the ex-post optimal allocation by (

) (

* 1 

c , ) (

* 2 

c ).

20

Deposit insurance The government offers a deposit insurance that guarantees depositors a certain return.

 is random with ] , [

max min 

  

Deposit contract

) , (

2 1 r

r

. Deposit insurance steps in if (and only if) there is a bank run in t=1. To finance the insurance, the government taxes commodities (liquidity) in t=1. Thus, we consider a tax on households in t=1. tax rate:

           

max 1 max 1 * 1

1 1 ) ( 1 ) (    f if r f if r f c f

Remaining goods for consumption in t=1:

       

max max * 1 1 1

1 ) ( ) 1 (    f if f if f c r c

Tax revenue is given to the bank who pays consumers and, thus, can avoid premature liquidation of investment projects.

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Now, consumers who withdraw in t=2, receive

          

max max * 2 * 1 2

) ( 1 )) ( 1 (   f if R f if f c f f c f R c

Since

) ( ) (

* 1 * 2

f c f c 

and 1  R , the amount a depositor receives in t=2 exceeds the net amount available for consumption, if a depositor withdraws in t=1. Thereby, a late consumer has no incentive to run the bank.. Only early consumers withdraw in t=1 and   f . Hence, early consumers receive

) (

* 1 

c

in t=1, while late consumers receive

) (

* 2 

c

in t=2. the optimal allocation is implemented by a unique equilibrium. If  is known and ) (

* 1 1

 c r  , then the deposit insurance will never be have to pay. Hence, taxed will never be paid. The existence of an insurance is sufficient to avoid a bank run and to avoid that the insurance case will ever occur.

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Critique:

• 1. Premature liquidation of projects is costless,

1  

. If

1  

, an investment may be harmful. If the bank invests too much, to pay out all early depositors, ressources are wasted and the efficient allocation cannot be achieved. (unless the government gives the bank additional ressources, financed by a tax).

• 2. Taxes are neutral. If taxes are distortive, there is a welfare loss associated with charging taxes

in t=1.

• 3. There is just a single type of projects. Thereby, we exclude the possibility that the bank

engages in „moral hazard“. Banking crises usually occur if banks have too risky portfolios or lost money due to overly risky investments. (destabilizing speculation).