9. Banks & Liquidity Demand

Question 1

Summarise the assumptions in the Diamond & Dybvig model (1983)

Answer 1

Time: 3 dates (t=0,1,2)
Consumer preferences: consumers are ex ante identical
Endowment: 1
State contingent preferences
U(c1) if impatient (prob lambda)
U(c2) if patient (prob 1-lambda)
Where the function u is increasing and strictly concave
Types are private info at t=1. No aggregate uncertainty

Question 2

When do short term and long term projects yield returns?

Answer 2

-Short term projects yield r1=1 at date 1 per unit of date-0 investment and r2 at date 2 per unit of date-1 investment
-Long term projects yield R>1 at date 2 per unit of date-0 investment and 0 at date 1.

Question 3

What assumption do we make about autarky which simplifies things?

Answer 3

That it is impossible to liquidate the long term technology. L=0

Question 4

What is consumption in each period given by in the autarky model?

Answer 4

C1= i1
C2= r2i2+ Ri2= R- c1(R-r2)

Question 5

What does the representative consumer maximise?

Answer 5

(Lambda)u(c1) + (1- lambda)u(R-c1(R-r2))

Question 6

Why does the consumer face uncertainty about her liquidity needs?

Answer 6

Because she solves the problem before she knows her type

Question 7

What are the two possible solutions from the optimisation of the representative consumers maximisation?

Answer 7

An interior or a corner solution

Question 8

Why can’t i1=0 be a solution?

Answer 8

The fact that the marginal utility of 0 consumption approaches infinity

Question 9

What does the social planner maximise and what are her constraints?

Answer 9

The same utility as the representative consumer but has different constraints. In particular the social planner only has to meet the aggregate (economy wide) constraints

Question 10

What is a simple way to solve the model of the social planner?

Answer 10

Normalising the aggregate endowment to unity and then interpret lambda as the proportion of impatient consumers

Question 11

Describe the meaning of the three constraints in the social planner problem.

Answer 11

The first constraint requires that whatever has been invested in the short term technology will be equally divided among impatient consumers and the second constraint states that the proceeds of the long term technology will be equally divided among patient consumers. The third constraint ensures that the social planner doesn’t violate the resource constraint

Question 12

Why is the proportion of impatient consumers (lambda) not there in the social planner solution?

Answer 12

Because the social planner can always adjust the amount invested in the short term technology by that proportion.

Question 13

Why is the return of the short term technology not there in the social planner solution?

Answer 13

Because the social planner will never have to liquidate the long term technology.

Question 14

Is the autarky solution efficient?

Answer 14

No the social planner problem is

Question 15

Proposition 1

Answer 15

If the coefficient of relative risk aversion exceeds 1 for all c then at the optimum
1<c1<c2<R

Question 16

When can the optimal allocation be implemented by a bank deposit contract?

Answer 16

When the rate of interest that the consumer receives depends on the date of withdrawal. Namely the rates of interest r(short term) and r(long term) on deposits withdrawn at dates 1 and 2 such that 1+ r(short term)=c1 and (1+ r(long term))^2= c2

Question 17

In what case is it optimal to invest all endowments in long term technology?

Answer 17

L=r1=r2=1 and R>1

Question 18

Why is the solution to bank run the same as before in the social planner problem?

Answer 18

Because in both cases early consumers withdraw the same amount lambda(c1*)

Question 19

What does lambda hat denote?

Answer 19

The fraction of consumers who withdraw at date 1

Lambda hat= lambda + (1- lambda)x
Where c is the fraction of patient consumers who run to the bank

Question 20

As lambda hat increases what happens to liquidation?

Answer 20

There is more liquidation

Question 21

If 1/lambda hat >c1* what happens to the bank?

Answer 21

The bank can’t meet its obligations. It fully liquidated the long term technology and divides the proceeds equally among those who withdraw early

Question 22

What does strategic complementarities refer to?

Answer 22

The fact that the incentive to run increases with the number of other consumers who withdraw

Question 23

What equilibria do strategic complementarities yield?

Answer 23

They often yield multiple equilibria and indeed in this model there is a Pareto inferior equilibrium where everybody withdraws at date 1