ECON7003, 2008-9. Money and Banking. Hugh Goodacre. Lectures 1-2. LIQUIDITY DEMAND AND B ANK RUNS

1. ECON7003, 2008-9. Money and Banking. Hugh Goodacre.Lectures 1-2.LIQUIDITY DEMAND AND BANK RUNS Bank deposits and uncertain liquidity demand. • Trading risk in a two-individual ‘society’. • Intermediation: the bank deposit contract. Ranking outcomes. • ‘Autarchy’. • Intermediation with no run. • Intermediation with run and without deposit insurance. • Intermediation with run with deposit insurance. Measures to prevent bank runs. • Suspension of convertibility. • Government-backed deposit insurance. • Lender of last resort facility.

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 bank’s 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 • 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 C1 = 0.

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 knownin 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. Individual’s expected utility E [U] in period 0: • E [U] = p.U(C11 + C21) + (1 – p).U(C12 + C22) • Type 1s: Expectation of a constant is a constant → • E[C11] = C11 = 1 • E[C21] = C21 = 0 • Type 2s: Expectation that they optimise → • E[C12] = 0 • E[C22] = R • → Substituting: • E [U] = p.U(1 + 0) + (1 – p).U(0 + R) • → E [U] = p.U(1) + (1 – p).U(R)

10. ‘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 C1 = 1 + π in period 1. • Individual 2 consumes C2 = R(1 – π) in period 2. • Only requirement: • Mechanism for enforcing contract.

11. ‘Society’ of two individuals, contd • Deriving optimal scale of transfer (π): • We need to find the value of π which maximises total social utility (SU) ≡ U(C1) + U(C2) • Express period 2 budget constraint i.t.o. C1: • C1 = 1 + π • → π = C1 - 1 • Substituting into C2 = R(1 – π) we have: • C2 = R[1 – (C1 – 1)] • = R(2 – C1) = 2R – RC1 • Substituting into expression for total social utility, we have: • SU = U(C1) + U(2R – RC1)

12. ‘Society’ of two individuals, contd • Differentiating SU and setting to zero to maximise, we have: • SU = U(C1) + U(2R – RC1) • → dSU / dC1 = MU1 – R.MU2 = 0 • → MU1 / MU2 = R • = 1 + r • i.e. MRS (in consumption) = MRT (through investment) • We define the values which solve these equations as: • C1*, C2*, and π*

13. C2 ‘Society’ of two individuals, contd 2R • ← Vertical intercept: • Period 2 social budget constraint: • C2 = R(2 – C1) • Solving for C1 = 0: • C2 = 2R Horizontal intercept: Maximum possible consumption by both types (‘social’ consumption) is 2. ↓ Social budget line 2 C1

14. A C2 ‘Society’ of two individuals, contd 2R Allocation point under autarchy / no trading of risk R • i.e. Social level of consumption under autarchy is: • 1 + R 1 2 C1

15. C2 ‘Society’ of two individuals, contd 2R 450 line indicates complete absence of risk between ‘states’ / outcomes 450 2 C1

16. A A' ‘Society’ of two individuals, contd C2 With trading in risk / contract to pay π, ‘social IC’ reaches tangency with BC at A' With no trading in risk, ‘social indifference curve’ cuts BC at A 2R A' is closer to the 450 line, indicating a reduction in risk R It is on a higher social IC curve, showing that trading in risk results in a socially preferable outcome to autarchy. 450 1 2 C1

17. C2 ‘Society’ of two individuals, contd 2R At A', individual 2 consumes C2* due to loss of Rπ* A R Rπ* A' C2* At A', individual 1 consumes C1* due to receiving π* Note: C2* > C1* π* 450 C1* 1 2 C1

18. 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.

19. 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 C1* = 1 + π in period 1. • All type 2s will consume C2* = R(1 – π) in period 2. • C2* > C1* → type 2s still have motive to set C1 = 0

20. 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.

21. Period 1 liabilities > assets: • Recall the assumption: All the bank’s 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 bank’s liabilities to depositors in period 1 are: • N.C1* • And N.C1* > N !

22. Let p = ½ • ‘Good’ outcome period 1: • Type 2s will optimise by setting C12 = 0 • Only type 1s withdraw deposits in period 1. • → Liquidity demand in period 1 is: • pNC1* + (1 – p)N.0 • = ½NC1* < N • i.e. Bank’s liabilities do not exceed its assets. • All deposit withdrawal demands can be met.

23. ‘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).NC1* = ½NC1*. • → Total liquidity demand: • ½NC1* + ½NC1* • = NC1* > N • i.e. Bank’s assets insufficient to meet liabilities. • Some depositors get 0.

24. Deposit : liability ratio of banks in period 1: • N : N.C1* • i.e. 1 : C1* • 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 : C1*) expressed as a fraction: • f = 1 / C1* • C1* > 1 • → f < 1

25. Fraction of depositors who get nothing through being last in the queue: • 1 – f • = 1 - 1 / C1* • = (C1* - 1) / C1* • We have: C1* = 1 + π • Substituting: 1 – f = (1 + π – 1) / C1* = π / C1* • i.e. Fraction who receive nothing is π / C1*

26. 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 bank’s 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.

27. 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

28. ‘AUTARCHY’ / no trading in risk – review: • E [U] = ½ [U(1) + U(R)] • ‘Society’ of two individuals where p = ½ • E [U] = ½[U(C1*) + U(C2*)] • ≡ What is offered by intermediary. • The ‘good’ Nash equilibrium

29. ‘Society’ of two individuals where p = ½ -- review. • Scale of π* through simple calculus by maximising • SU ≡ U(C1) + U(C2) • Substitute into this the period 2 BC, i.e. 2R – RC1 • Differentiate w.r.t. C1 and set to 0, etc. • i.e. MRS (in consumption) = MRT (through investment) • Values which solve these equations ≡ C1*, C2*, π* • Giving us: • E [U] = ½[U(C1*) + U(C2*)]

30. Intermediary / bank offers deposit contract providing same degree of insurance as in two-individual case -- review. • C2* > C1* → type 2s still have motive to set C12 = 0 • Fragility of this result: • In period 1 liabilities > assets • N.C1* > N ! • → bank relies on type 2s not withdrawing.

31. ‘Good’ outcome period 1 -- review. • Type 2s optimise by setting C12 = 0 • → Liquidity demand in period 1 is: • pNC1* + (1 – p)N.0 • = ½NC1* < N • ‘Bad’ outcome period 1: • All Type 2s fear run → Total liquidity demand: • ½NC1* + ½NC1* • = NC1* > N • Maximum proportion of depositors who can withdraw : • f = 1 / C1* • We have: C1* = 1 + π* • Substituting: 1 – f = (1 + π* – 1) / C1* = π* / C1* • i.e. Fraction who receive nothing is π* / C1*

32. Intermediation and autarchy compared. • With and without a bank run. • Total expected utility of those who manage to withdraw deposits in presence of run: • N individuals. • Fraction who are paid is 1/C1*. • → E[U] = N.{1/C1*}.U(C1*)}

33. Those who receive nothing have U[0], so total expected social utility is the same! • But can be expanded thus: • Fraction who receive nothing is π*/C1* • → E[SU] = N.{1/C1*.[U(C1*)] + π*/C1*.[U(0)]} • Note again: • We here have the utility of ALL individuals, lucky and unlucky: • 1/C1* + π*/C1* • = (1 + π*) / C1* • = C1* / C1* • = 1

34. Individual’s expected utility / assumption of no bank run: • E[U] = ½[U(C1*) + U(C2*)] • Note: C2* > C1* → E[U] > U(C1*) • Comparing expected utility with run, we have: • With run: E[U] = 1/C1*.U(C1*) [ + 0 ] • 1/C1* < 1 • → E[U] < U(C1*) • From above, we have: • Without run: E[U] > U(C1*)

35. Measures to prevent bank runs. • The ‘good’ and ‘bad’ outcomes: Nash equilibria: • Agents optimise autonomously. • Disregard effect of actions at aggregate level. • Strategy: Make ‘good’ Nash equilibrium unique. • Influence expectations / provide confidence. 3 possible solutions: • Action by banks themselves: Suspend convertibility • Government actions: Government-backed deposit insurance Lender of last resort facility

36. Strategies to eliminate the ‘bad’ Nash equilibrium • (1) Suspension of convertibility. • Problem: Fragility of the contract: • Liquidation value of assets in period 1 is 1. • BUT: C1 = 1, and consequently C2 = R, is same as autarchy would provide anyway! • Thus: • Necessary condition for intermediary to offer improvement on autarchy is: • Must offer to pay out more than liquidation value of its assets in period 1!

37. Advantage of suspension of liquidity: • Time for assets to be liquidated in orderly way • → better price than in ‘fire-sale’ conditions. • Bank announces: • Will suspend convertibility if pays out fraction (p = ½ ) of its deposit liabilities. • Consequences for Type 1s • Some left with 0 • Consequences for Type 2s • All assured of C2* in period 2

38. Type 2s all assured of C2* in period 2 → gives them incentive to keep deposits till period 2. i.e. strategy for eliminating ‘bad’ equilibrium / threat of bank panic. → Counters risk of run. . .

39. Strategies to eliminate the ‘bad’ Nash equilibrium • (2) Deposit insurance. • We now assume: • Government has imposed system of deposit insurance. • Equalises amount Type 1s and Type 2s can withdraw in period 1in the presence of run. • → None left with 0 through being at the back of the queue. • Advantage of government over bank in arranging deposit insurance: • Can use tax system to claw back funds from depositors after withdrawal of deposits.

40. Number of Type 1s = number of Type 2s = ½ N • → Total number of desired withdrawals: • N (assuming all Type 2s panic → wish to set C22 = 0). • Total amount deposited = N • → Each can receive N/N = 1. • → E[U] = ½[U(1) + U(1)] • i.e. E[U] = U(1)

41. Deposit insurance with run and autarchy ranked. • With bank run and deposit insurance, we have: • E[U] = ½[U(1) + U(1)] (i.e. U(1).) • Under autarchy, we have: • E[U] = ½[U(1) + U(R)] • R > 1 → Autarchy ‘ranks’ better in terms of social welfare. • Discrepancy: due to ‘premature liquidation effect’. • i.e. loss of return on the early withdrawals by Type 2s.

42. Ranking E[U] with and without deposit insurance. • Question: Amount paid out same in each case, so SU same??? • Without insurance: • 1/C1* withdraw C1* • 1/π* are left with nothing. • ‘Unfair’! • With insurance: • No one gets more than 1. • At least this is ‘fair’!

43. Answer: • Added utility to the successful of receiving C1* rather than 1 • outweighed by • Loss in utility by the unsuccessful • of receiving 0 rather than 1. • i.e. ‘Fairer’ distribution through insurance maximises social welfare subject to available resources. • i.e. Deposit insurance has reversed the ‘distribution effect’.

44. If bank run develops, deposit insurance can motivate Type 2s not to withdraw. • Even though expected utility in period 2 no more than in period 1 • (1) E[U] in period 2 no worse than in period 1. • (2) Chance that run will not develop into a panic • → might get C2 > 1 after all.