Suppose we have the kind of Diamond-Dybvig model of a bank discussed near the end of chapter 18. . There are a large number N of consumers, each with one dollar in the bank. So the bank start with Ndollars in deposits. . The bank can invest in an asset with interest rate r = 1. This means that for the bank has at the end of period 1, they'll have 1 + r = 2 dollars at the start of period 2. (In this example, there's no inflation, so real interest rate is equal to nominal interest rate.) . The bank offers consumers a contract where they get c, dollars if they withdraw their money in period 1, and ca dollars if they withdraw in period 2. . Half of the consumers will want to consume in period 1 (early consumers", and the other half will want to consume in period 2 ("late consumers"). When the contract is put in place, consumers don't yet know when it is that they will need their money. . After knowing which type of consumer they are, the consumer's utility is u(c) = 1 - 1, where c is their consumption in the period of their choice. . Before knowing which type of consumer they are, a consumer's expected utility from withdrawing when they want to consume is: . A late consumer also has the option of withdrawing early, and storing the money for later. But they don't get any returns on that stored money. Problems: First consider a contract where (c, () = (1, 1 + r) = (1,2) (a) Half of the consumers withdraw early, half leave their money in the bank. How much money will the bank have at the beginning of period 2? Will they be able to pay back the 1 + r dollars promised to each consumer who left their money in the bank? (b) Now suppose that all but 1 of the late consumers panics and decides to withdraw their money early. What should that last consumer do? Will the bank be able to pay them the 1 + r dollars in period 2 as promised? Now consider a contract where (ci, ca) = (1.25, 1.5) (c) Half of the consumers withdraw early, half leave their money in the bank. How much money will the bank have at the beginning of period ?? Will they be able to pay back the 1.5 dollars promised to each consumer who left their money in the bank? (Verify that this contract satisfies the bank's resource constraint.) (d) In expectation, are consumers better off with this contract than they are with the previous one? (e) Now suppose that all but 1 of the late consumers panics and decides to withdraw their money early. What should that last consumer do? Will the bank be able to pay them the 1.5 dollars in period 2 as promised