9. Banks & Liquidity Demand Flashcards
Summarise the assumptions in the Diamond & Dybvig model (1983)
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
When do short term and long term projects yield returns?
-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.
What assumption do we make about autarky which simplifies things?
That it is impossible to liquidate the long term technology. L=0
What is consumption in each period given by in the autarky model?
C1= i1
C2= r2i2+ Ri2= R- c1(R-r2)
What does the representative consumer maximise?
(Lambda)u(c1) + (1- lambda)u(R-c1(R-r2))
Why does the consumer face uncertainty about her liquidity needs?
Because she solves the problem before she knows her type
What are the two possible solutions from the optimisation of the representative consumers maximisation?
An interior or a corner solution
Why can’t i1=0 be a solution?
The fact that the marginal utility of 0 consumption approaches infinity
What does the social planner maximise and what are her constraints?
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
What is a simple way to solve the model of the social planner?
Normalising the aggregate endowment to unity and then interpret lambda as the proportion of impatient consumers
Describe the meaning of the three constraints in the social planner problem.
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
Why is the proportion of impatient consumers (lambda) not there in the social planner solution?
Because the social planner can always adjust the amount invested in the short term technology by that proportion.
Why is the return of the short term technology not there in the social planner solution?
Because the social planner will never have to liquidate the long term technology.
Is the autarky solution efficient?
No the social planner problem is
Proposition 1
If the coefficient of relative risk aversion exceeds 1 for all c then at the optimum
1<c1<c2<R
When can the optimal allocation be implemented by a bank deposit contract?
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
In what case is it optimal to invest all endowments in long term technology?
L=r1=r2=1 and R>1
Why is the solution to bank run the same as before in the social planner problem?
Because in both cases early consumers withdraw the same amount lambda(c1*)
What does lambda hat denote?
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
As lambda hat increases what happens to liquidation?
There is more liquidation
If 1/lambda hat >c1* what happens to the bank?
The bank can’t meet its obligations. It fully liquidated the long term technology and divides the proceeds equally among those who withdraw early
What does strategic complementarities refer to?
The fact that the incentive to run increases with the number of other consumers who withdraw
What equilibria do strategic complementarities yield?
They often yield multiple equilibria and indeed in this model there is a Pareto inferior equilibrium where everybody withdraws at date 1
