Consider an economy with n identical investors each endowed with one dollar to d
ID: 1177506 • Letter: C
Question
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.
Comments: This is a Diamond Dybvig model on bank runs. Notice that in this particular model, the multiple Nash equilibria is very useful for us to explain possible bad outcomes without any changes to the economic environment. Instead, the bad outcome is entirely driven by a self-fulfilling prophecy borne from the investors' beliefs. In essence, investors believe there will be a bank run, so they withdraw as soon as possible, which then causes a bank run!
Explanation / Answer
1)let x be the investers withdrawing at t=1,
at t=1 total assest of bank is n
if x people withdraw they should get x+x(1-E)=2x-xE
if 2x-xE<n then each get
2-E
therefore payoff is 2-E-1=1-E
else each get n/x
2)so if x withdraw at t=1
total asset of bank taken back is 2x-xe
at t=2 total asset is nR
nR-2x-xE -(n-x)//beacuse it is invested intially is the paid/n-x//(total members withdrawn at t=2) is th payoff for individual