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

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“) 1 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(c t ). Assume U’ > 0 and U’’ < 0 Objective: maximize expected utility       EU ( c , c ) U ( c ) ( 1 ) U ( c ) 1 2 1 2 c consumption in t=1 1 c consumption in t=2 2   discount factor 1 2

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). => c 1 = 1 If he needs liquidiuty only in t=2, he receives R>1, which is better than just storing his good. => c 2 = R His expected utility is       EU ( c , c ) U ( 1 ) ( 1 ) U ( R ) 1 2 Note: for λ < 1 a risk averse household should invest only a part of his endowment. 3 Efficient allocation by introducing a bank If liquidity need („type“) can be verified, each household can engage in contingent contract with a * * c in t=1, if he is an early consumer, and repayment c in t=2, if bank, guaranteeing a repayment 1 2 he is a late consumer. * * c and c be? How big should 1 2 4

Derive the efficient allocation infinitely many depositors (households), law of large numbers  a fraction γ needs liquidity in t=1 and receives c 1 each.  a fraction 1 – γ needs liquidity in t=2 and receives c 2 each. Allocation ( c 1 , c 2 ). The efficient allocation maximizes expected utility subject to feasibility λ =1: The bank can invest a share 1 – γ c 1 of deposits Feasibility constraint: projects, or she invests all deposits and liquidates a share deposits in t=0 1 γ c 1 of projects in t=1. Both strategies result in the same returns. withdrawels in t=1 γ c 1 For λ <1 the bank should invest only a fraction 1 – γ c 1 in A fraction of project 1 – γ c 1 will be continued to t=2. projects and avoid the loss resulting from premature liquidation. In total, these projects yields a return R (1 – γ c 1 ). The total return is distributed among the remaining depositors (late consumers) whose share is 1– γ . Result: If early consumess receive c 1 , late consumers can at most receive (per capita)   R ( 1 c )  c ( c ) 1 . (1) 2 1   1  )  c ( 1 c 1 R Note that . 2 5      ( 1 ) c R ( 1 c ) Maximize EU(c 1 ,c 2 ) s.t. 2 1             max U ( c ) ( 1 ) U ( c ) [( 1 ) c R ( 1 c )] 1 2 2 1 c , 2 c 1      U ' ( c ) R 0 FOCs: (c 1 ) (2) 1         ( 1 ) U ' ( c ) ( 1 ) 0 (c 2 ) (3) 2   U ' ( c ) R (2) <=> 1    U ' ( c ) (3) <=> 2   U ' ( c ) U ' ( c ) R => Optimality condition: (4) 1 2 SOC holds, because utility function is concave (U’’<0) and restriction is linear. * * c , c ) is characterized by optimality condition (4) and feasibility (1). Result: The efficient allocation ( 1 2 To yield the solution explicitly, we may insert (1) in (4) and solve the resulting equation for c 1 :     R ( 1 c )   R ( 1 c ) (4)     1 U ' ( c ) R U '  c ( c ) 1   (1) => 1   (5) 2 1   1   1 6

    R ( 1 c )     U ' ( c ) R U ' 1   Consider equation (5) 1   1   Since U’’<0, the left-hand side is decreasing in c 1 . The right-hand side is increasing in c 1 . U’ R ρ U’(c 2 (c 1 )) feasible and optimal * ) U’(c 1 U’(c 1 ) c 1 * c 1 7 * * c , c ) with the best allocation attaineable without bank (1, R ) . Compare the optimal allocation ( 1 2 1. Allocation (1, R) is feasible without bank. * > 1, then it is not attainable without bank. 2. If in the optimal allokation c 1 * > 1? Under which condition is c 1 U’ R ρ U’(c 2 (c 1 )) U’( 1 ) feasible and optimal * ) U’(c 1 R ρ U’(R) U’(c 1 ) c 1 * c 1 1     * c 1 U ' ( 1 ) R U ' ( R ) The figure shows that . 1 8

1  * c 1 Lemma: if relative risk aversion (RRA) exceeds 1, then . d      R  R U ' ( R ) U ' ( R ) R U ' ' ( R ) U ' R ( ) Proof: 1. View as a function on R. Then . dR 2. This derivative is negative, if and only if RRA( R ) > 1:  R U ' ' ( R )     U ' ( R ) R U ' ' ( R ) 1 . U ' ( R ) RRA( R ) d   R U ' ( R ) 0      U ' ( 1 ) R U ' ( R ) U ' ( 1 ) R U ' ( R ) 3. If , then , because R>1. This implies , dR   1 because . QED 9 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 (c 1 >1). For this, the consumer pays an insurance premium, lowering his payoff in case the “damage” does not occur (c 2 <R). => Early consumers take a share in the returns of technology. 10

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 r ≥ 1 in t=1 oder 2 r in t=2. repayment 1 1. Contract must be feasible   R ( 1 r )  r 1 2   1   R ( 1 r )  r 1 If , the bank yields a profit. 2   1   R ( 1 r )  r 1 We concentrate on contracts with . (6) 2   1 11 2. Late consumers must have an incentive to leave their funds in the bank: r  r Incentive constraint: 2 1 Using (6), this is equivalent to   R ( 1 r )        1 r R ( 1 r ) r ( 1 ) 1 1 1   1 R         R r ( 1 R ) r 1 1    1 ( R 1 ) r  r  * * c c If and , the deposit contract can implement the optimal allocation. 1 1 2 2 R c  * * c  * This requires or equivalently: c . 1 2 1    1 ( R 1 ) 12