Question: Consider the following model of bank runs, which is due to Diamond and Dybvig (1983). There are three periods (t = 0, 1, 2). There are many consumers a continuum of them, for simplicity. All consumers arc ex ante identical. At date 0, they deposit their entire wealth, $1, in a bank. The bank invests in projects that yield $R each if the money is invested for two periods, where R > 1. However, if a project is interrupted after one period, it yields only $1 (it breaks even). Each consumer "dies" (or "needs money immediately") at the end of date 1 with probability x, and lives for two periods with probability 1 - x. He learns which one obtains at the beginning of date I. A consumer's utility is u(c1) if he dies in period 1 and (c1 + c2) if he dies in period 2, where u' > 0, u" < 0,="" and="">1 and c2 are the consumptions in periods 1 and 2.

An optimal insurance contract (, ) maximizes a consumer's ex ante or expected utility. The consumer receives if he dies at date 1, and otherwise consumes nothing at date 1 and receives at date 2. The contract satisfies x+ (1 - x)/R = 1 (the bank breaks even)and(equality between the marginal rates of substitution). Note that 1 <> <>. The issue is whether the bank can implement this optimal insurance scheme if it is unable to observe who needs money at the end of the first period. Suppose that the bank offers to pay r1 = to consumers who want to withdraw their money in period 1. If f ? [0,1] is the fraction of consumers who withdraw at date 1, each withdrawing consumer gets r1 if fr1 ≤ 1, and gets 1/f if fr1 > 1. Similarly, consumers who do not withdraw at date 1 receive max {0, R( 1 - r1 f)/(1 - f)} in period 2.

(a) Show that it is a Nash equilibrium for each consumer to withdraw at date I if and only if he "dies" at that date.

(b) Show that another Nash equilibrium exhibits a bank run (f = 1).

(c) Compare with the stag hunt.