Y5

1. Consider the basic setup of the Diamond-Dybvig (1983) model. Specifically, there are three periods, denoted t = 0, 1, 2, a single consumption good, and an illiquid investment opportunity that pays gross return 1 if liquidated at t = 1, or gross return 3 if liquidated at t = 2. There are 7 people in the economy, each endowed with 1 unit of the consumption good at t = 0. At t = 1, exactly 1 will randomly realize that they need to consume at t = 1 (the early consumer ), and the remaining 6 people will need to consume at t = 2 (the late consumers). The utility derived from consumption is 1(1/c1)2 for the early consumer, 1(1/c2)2 for late consumers, where the subscript denotes the time of consumption.

(i) Calculate the expected return (from a t = 0 perspective) of direct investing.

(ii) Calculate the expected utility (from a t = 0 perspective) derived from direct investing.

Suppose a bank can offer an asset that is more liquid, with gross returns Rd1 = 2.25 and Rd2 = 2.25 (i.e., the asset pays the same return, Rd = 2.25 to all depositors regardless of the time of liquidation).

(iii) Calculate the expected utility (from a t = 0 perspective) derived from depositing with the bank. Do you conclude that people would prefer banking to direct investing at t = 0?

(iv) Calculate the bank's profit after t = 2. In other words, what amount of funds remains at the bank once all depositors have withdrawn?

(v) What is the highest gross return Rd that the bank can offer all depositors? [NOTE: Rd is still constrained to be the same for all depositors, regardless of the time of liquidation.]

(vi) What is the gross return Rd that earns the highest profits for the bank?