1. ECON7003 Money and Banking. Hugh Goodacre. Lectures 1-2. BANK RUNS Bank deposits and uncertain liquidity demand. The Diamond and Dybvig 1983 model, Spencer, ch. 10 version. 1. Trading risk in a two-individual society. 2. The bank deposit contract. Preview: 3. Measures to prevent bank runs.

2. Withdrawal of deposits on demand normally no problem, despite: low deposit : asset ratio high gearing in bank sector (a) Scale economies: withdrawal demands unlikely to be correlated. For banking system as a whole, likely to be inversely correlated: Debits-credits net out! (b) Tradable money market instruments: e.g. Certificates of Deposit (CDs). To meet fluctuations in liquidity needs.

3. These advantages are basic to banks profit through intermediation: i.e. Asset transformation: Short-term / instantly withdrawable deposits long-term / illiquid assets Maturity transformation Small-size deposits large-size assets: Size transformation Low-risk instrument, i.e. deposit, high-risk.: Risk transformation In each case: Interest on asset > interest on liability bank profit.

4. BUT: Loss of confidence in bank withdrawals not motivated by genuine liquidity requirement / transactions motive. May be contagious and panic. In panic, those at end of queue may not be paid in full: Even if bank is solvent and all its assets are liquidated

5. Costs of liquidation Loss of: customer relationships confidential information, etc. i.e. Destruction of informational capital / intangible assets. Inevitably undervalued in fire sale conditions. Net value > 0 when functioning may < 0 if sold off hurriedly.

6. Asymmetric information problem facing bank: Bank unable to distinguish between: withdrawals for genuine / transactions purposes withdrawals through panic cannot pay in sequence: Gain time avoid fire sale liquidate assets at better price.

7. 3-period model of bank runs and measures to prevent them. Assumption: Bank liabilities all consist of deposits withdrawable on demand. Each individual has a primary investment of 1 in period 0 yields 1 if liquidated and consumed in period 1 yields R > 1 if liquidated and consumed in period 2. i.e. R 1 + r

8. Individuals are of 2 types: Type 1s die in period 1 having first liquidated their investment and consumed its entire value. Type 2s survive period 1 but die in period 2 having by that time liquidated their investment and consumed its entire value. The overall proportion (p) of type 1s is publicly known in period 0 i.e. There is no aggregate uncertainy. but individuals do not find out which type they are until period 1, and this information is private. i.e. There is individual uncertainy.

9. i.e. Requirement for liquidation of investment in period 1 drives the demand for liquidity. Cost of early death is R – 1. Because R > 1, type 2s optimally set C 1 = 0.

10. Individuals expected utility E [U] in period 0: E [U] = p.U(C 1 1 + C 2 1 ) + (1 – p).U(C 1 2 + C 2 2 ) Type 1s: Expectation of a constant is a constant E[C 1 1 ] = C 1 1 = 1 E[C 2 1 ] = C 2 1 = 0 Type 2s: Expectation that they optimise E[C 1 2 ] = 0 E[C 2 2 ] = R Substituting: E [U] = p.U(1 + 0) + (1 – p).U(0 + R) E [U] = p.U(1) + (1 – p).U(R)

11. Society of two individuals where p = ½ Learning own type revelation of type of other ! i.e. Full state verification / no informational asymmetry. Socially optimal risk-sharing contract possible in period 0: Type 2 will pay fixed sum (π) to type 1 in period 1. Individual 1 consumes C 1 = 1 + π in period 1. Individual 2 consumes C 2 = R(1 – π) in period 2. Only requirement: Mechanism for enforcing contract.

12. Deriving optimal scale of transfer (π): We need to find the value of π which maximises total social utility (SU) U(C 1 ) + U(C 2 ) Express period 2 budget constraint i.t.o. C 1 : C 1 = 1 + π π = C 1 - 1 Substituting into C 2 = R(1 – π) we have: C 2 = R[1 – (C 1 – 1)] = R(2 – C 1 ) = 2R – RC 1 Substituting into expression for total social utility, we have: SU = U(C 1 ) + U(2R – RC 1 )

13. Differentiating SU and setting to zero to maximise, we have: SU = U(C 1 ) + U(2R – RC 1 ) dSU / dC 1 = MU 1 – R.MU 2 = 0 MU 1 / MU 2 = R = 1 + r i.e. MRS (in consumption) = MRT (through investment) We define the values which solve these equations as: C 1 *, C 2 *, and π*

14. C2C2 2C1C1 2R Vertical intercept: Period 2 social budget constraint: C 2 = R(2 – C 1 ) Solving for C 1 = 0: C 2 = 2R Horizontal intercept: Maximum possible consumption by both types (social consumption) is 2. Social budget line

15. C2C2 R 2C1C1 1 2R Allocation point under autarchy / no trading of risk i.e. Social level of consumption under autarchy is: 1 + R A

16. C2C2 2C1C1 2R 45 0 45 0 line indicates complete absence of risk between states / outcomes

17. C2C2 R 2C1C1 1 2R 45 0 With trading in risk / contract to pay π, social IC reaches tangency with BC at A' A' is closer to the 45 0 line, indicating a reduction in risk With no trading in risk, social indifference curve cuts BC at A A'A' A It is on a higher social IC curve, showing that trading in risk results in a socially preferable outcome to autarchy.

18. C2C2 C2*C2* 2 A C1C1 1 2R 45 0 A'A' Rπ*Rπ* π*π* At A', individual 1 consumes C 1 * due to receiving π* At A', individual 2 consumes C 2 * due to loss of Rπ* C1*C1* R Note: C 2 * > C 1 *

19. BUT: Society of more than two individuals: Information on own type remains private in period 1: life expectancy and liquidity requirements no longer publicly revealed. asymmetric information problem in designing contract for trading risk.

20. An intermediary / bank now offers a deposit contract capable of achieving same degree of insurance as in the two-individual case. i.e. : All type 1s will consume C 1 * = 1 + π in period 1. All type 2s will consume C 2 * = R(1 – π) in period 2. C 2 * > C 1 * type 2s still have motive to set C 1 = 0

21. BUT: Bank can only fulfil this contract if only type 1s withdraw their deposits in period 1. i.e. for genuine liquidity requirement. Fragility of this result: In period 1 liabilities > assets bank relies on type 2s not withdrawing.

22. Period 1 liabilities > assets: Recall the assumption: All the banks assets / funds are sourced from its depositors. Let there be N depositors, then the funds available to the bank for distribution to depositors in period 1 are: N.1 = N The banks liabilities to depositors in period 1 are: N.C 1 * And N.C 1 * > N !

23. Let p = ½ Good outcome period 1: Type 2s will optimise by setting C 1 2 = 0 Only type 1s withdraw deposits in period 1. Liquidity demand in period 1 is: pNC 1 * + (1 – p)N.0 = ½NC 1 * < N i.e. Banks liabilities do not exceed its assets. All deposit withdrawal demands can be met.

24. Bad outcome period 1: Type 2s fear a bank run / begin to withdraw deposits in period 1. If all do so (bank panic), type 2 liquidity demand in period 1 is: (1-p).NC 1 * = ½NC 1 *. Total liquidity demand: ½NC 1 * + ½NC 1 * = NC 1 * > N i.e. Banks assets insufficient to meet liabilities. Some depositors get 0.

25. Deposit : liability ratio of banks in period 1: N : N.C 1 * i.e. 1 : C 1 * Assumption: No deposit insurance arrangements are in place. Maximum proportion of depositors who can withdraw their deposits in period 1 in the presence of a run: Deposits divided by liabilities: N / NC* i. e. deposits : liabilities ratio (1 : C 1 *) expressed as a fraction: f = 1 / C 1 * C 1 * > 1 f < 1

26. Fraction of depositors who get nothing through being last in the queue: 1 – f = 1 - 1 / C 1 * = (C 1 * - 1) / C 1 * We have: C 1 * = 1 + π Substituting:1 – f = (1 + π – 1) / C 1 * = π / C 1 * i.e. Fraction who receive nothing is π / C 1 *

27. i.e. Intermediation / bank deposits offer solution to informational problems of trading in risk of early death. BUT That solution is not robust to fear of banks insolvency: Such fear may self-fulfilling prophecy / fear becomes general (panic). Sequential service constraint / bank cannot meet all withdrawal demands / last in queue get nothing. Expectations of run may actual run, with no change in fundamentals. Banks are inherently fragile. If fear is contagious, may threaten whole banking system.

28. Preview: The good and bad outcomes will be defined as Nash equilibria. Measures to prevent bank runs. Influence expectations / provide confidence. Make good Nash equilibrium unique. 3 possible solutions: Action by banks themselves: Suspend convertibility Government actions: Government-backed deposit insurance Lender of last resort facility