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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.

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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.

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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.

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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

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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.

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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.

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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

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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.

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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.

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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)

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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.

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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 )

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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*

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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

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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

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

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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.

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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 *

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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.

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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

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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.

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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 !

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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.

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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.

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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

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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 *

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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.

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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

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