Consider an economy with n identical investors each endowed with one dollar to deposit in a bank. There are three periods t = 0; 1; 2: At t = 0; the investors decide on whether to deposit their endowment in a bank or not. If the investors deposit their money in the bank, they can withdraw the money at either t = 1 or t = 2:

The bank has total asset of n in period t = 1 and nR in period t = 2; provided that all the investors invested in t = 0: Investors can withdraw from the bank at any period, which yields the investors a payoff of r > 1: However, if the number of withdrawals exceeds the bank's total assets, the asset is divided evenly among the withdrawing investors.

The investors have a preference for late withdrawals. In essence, if they choose to withdraw at t = 1; each dollar amount they receive is worth only 1 -E < 1 for them.

We will assume that R > r > 1; and that all depositors have no choice but to save their endowments in the bank at t = 0.

a. What is an individual depositor's payoff when he withdraws at t = 1 as a function of the number of remaining depositors who withdraw at t = 1?

b. What is an individual depositor's payoff when he withdraws at t = 2 as a function of the number of remaining depositors who withdrew at t = 1?

c. What are the pure strategy Nash equilibria? Hint: There are three of them, but only two of them are significant.