Assignment Task

Question

There are three dates t = 0, 1, 2 and a continuum of consumers with measure one, each endowed with wealth one. There are two types of consumers: a fraction a is impatient, i.e. their utility is 1 – (1/c1), where c1 is consumption at t = 1, and a fraction 1 a are patient whose utility is 1 – (1/c2), where where c2 is consumption at t = 2. At t O consumers don’t know their type, i.e. it is unknown whether the consumer wishes to consume at date t 1 or t = 2. At t observe their own type, but cannot observe others’ types (i.e. type is private information). There are two technologies to manage liquidity. Consumers can invest for one period only (short term technology), in which case the investment will yield r≥ 1. This technology is available at both t O and t = 1. Alternatively, wealth can be invested for two periods (long term technology) at date t 0, which yields 0 at t 1, but R > r2 at t 2. The investment in the long-term technology can be liquidated at t 1, which then yields L < 1>
Suppose there is a bank to manage liquidity risk, all consumers deposit their wealth in the bank and the bank maximizes total welfare of its depositors when it chooses c_1^B, c_2^B, the promised payouts for early (t = 1) and late (t = 2) withdrawals, respectively.

i. Set up the bank’s optimisation problem and derive the first order condition. What does the first order condition tell you about the optimal allocation?

ii. Calculate the optimal investment in the long-term technology IB and the optimal allocation c_1^B, c_2^B.

iii. Does the allocation c, c always constitute an equilibrium? If not, derive the necessary condition for c_1^B, c_2^B to be an equilibrium and carefully explain your finding.

iv. Now, suppose the condition that you derived in iii)above does not hold. Derive the feasible (incentive compatible) allocation c that maximizes total welfare and can always be supported in an equilibrium.