Bank deposits and uncertain liquidity demand, contd.Diamond and Dybvig 1983, Spencer, ch. 10 version.

Measures to prevent bank runs.

Ranking outcomes: ‘Autarchy’.Intermediation -- run.Intermediation with run: without deposit insurance.Intermediation with run: with deposit insurance.

Suspension of convertibility.

ECON7003 Money and Banking. Hugh GoodacreLecture 9.

BANK RUNS, contd.

Motives for excessive risk-taking, review:

Pressure on banks’ profits.Due to increased competition.

Motive to keep E low.E is denominator of ROE.Effect multiplied; E is denom. of EM as well.

Convexity of return to holders of equity. Risk transfers value from debt to E.Bigger the spread, bigger the transfer.

The rewards structure in the financial sector.Bonus payments and excess returns.Real and ‘apparent’ ER.

More on bonus structure – current debates and proposals:

Convex function of convex function!

Same asymmetry as in bondholder-shareholder case modelled above:

Maximum bonus to employee is unlimited.

Minimum bonus zero.

→ Loss / lack of profits suffered by shareholders / shhldr-mngt coalition, not employee.

A further exacerbation in much current practice:

‘Guaranteed bonuses’.

Contradiction in terms!

Bonus structure – Convex function of convex function!, contd.

Problem for cost-cutting:Rigid bonus structure / built into employee remuneration system / contracts.

→ Prompt action over write-downs / cost-cutting, etc, more difficult.

Proposal: delay in payments:– Hedge funds already have a system:– Only part of bonus paid in profitable years. – Rest held as ‘reserve’ for future losses.

‘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

‘Society’ of two individuals where p = ½

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*)]

C2

C2*

2

A

C1

1

2R

450

A'Rπ*

π*

At A', individual 1 consumes C1*

due to receiving π*

At A', individual 2 consumes C2*

due to loss of Rπ*

C1*

R

Note: C2* > C1*

Intermediary / bank offers deposit contract providing same degree of insurance as in two-individual case.

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.

‘Good’ outcome period 1:

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*

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*)}

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

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*)

RegimeRun

Utility of withdrawers in period 1

Utility of withdrawers in period 2

Expected utility

Autarchy -- U(1) U(R) ½[U(1) + U(R)]

Intermediation No run

U(C1*) U(C2*) ½[U(C1*) + U(C2*)]

Intermediation Run U(C1*) --1/C1*.[U(C1*)]

{ + π*/C1*.U(0) }

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 insuranceLender of last resort facility

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!

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 1sSome left with 0

Consequences for Type 2sAll assured of C2* in period 2

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.

.

.

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.

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)

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.

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’!

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’.

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.

Advantage of government over bank in arranging deposit insurance:

Can use tax system to claw back funds from depositors after withdrawal of deposits.

Regime Run DepositInsrnce

U of w’dwrs per 1

U of w’dwrs per 2

Expected utility Order i.t.o.

soc W

Aut-archy

-- -- U(1) U(R) ½[U(1) + U(R)] 2

Inter-medtn

No run

-- U(C1*) U(C2*) ½[U(C1*) + U(C2*)] 1

Inter-medtn

RunWithout

insuranceU(C1*) --

1/C1*.[U(C1*)]

+ π*/C1*.U(0)4

Inter-medtn

RunWith

insuranceU(1) U(1) U(1) 3