The Banking Problem

The set-up of this problem is exactly the same as the banking setup presented in Lecture Notes 4 and 5. Suppose that there are three dates, indexed by t = 0; 1; 2. There is a single good which can be used either for consumption or investment.

There is a unit-mass continuum of ex-ante identical agents. Each consumer has an endowment of one unit at date 0 and nothing at the future dates. All consumption however takes place at dates 1 and 2. Assets. There are two assets that consumers can use in order to provide for future consumption: there is a short-term liquid asset and a long-term illiquid asset.

The liquid, short asset is represented by a storage technology that allows one unit of the good at date t to be converted into one unit of the good at date t + 1, for either t = 0; 1.

The illiquid, long asset is represented by an investment technology that allows one unit of the good invested at date 0 to be converted into 1 + r units of the good at date 2, with r > 0. If the long asset is liquidated prematurely at date 1, then it pays l where 0 < l < 1.

Preferences. At date 1, each consumer learns his or her type. There are two possible types: early consumers who only want to consume at date 1 and late consumers who only want to consume at date 2. Initially, in period 0, each consumer does not know his own typeñhe only knows the probability of being an early or a late consumer. Let λ denote the probability of being an early consumer and 1-λ be the probability of being a late consumer. The consumer only learns whether he is an early or late consumer at the beginning of date 1. More speciÖcally, the agentsíutility is given by

u (c₁) with probability λ
u (c₂) with probability 1-λ

(a) Consider the consumerís problem under autarky. Derive the conditions that characterize the autarkic allocation of (c1; c2).

(b) Suppose there exists a market at date 1 in which agents can buy and sell holdings of the long asset after learning their true types. Thus, early types will sell their holdings of the long asset and late types will buy these assets. Let p be the equilibrium price in this market. Solve for the market allocation (c1; c2) when this market exists.

(c) Consider the plannerís problem. Derive the conditions that characterize the plannerís solution for the e¢ cient allocation.

(d) In one graph with c₁ on the x axis and c₂ on the y-axis, plot three things: (i) the feasible consumption set under autarky, (ii) the plannerís feasible consumption set, and (iii) the market allocation.

(e) Show that if u (c) = ln c, the e¢ cient allocation is the same as the market solution. In addition, in a graph as in part (d), plot the e¢ cient allocation and show how it coincides with the market solution. (You do not need to plot the feasible set under autarky.)

(f) If utility is CRRA,

u(c) = c^1-γ/1-γ

with γ > 1, then the e¢ cient allocation has greater early consumption c1 and lower late consumption c2 than in the market allocation. Show this in a graph as in part (e).

(g) If γ > 1 does the planner provide more or less liquidity insurance compared to the market allocation? Give intuition for your answer.