Consider a three-period overlapping generations economy with N

Consider a three-period overlapping generations economy with N young people born each period. Each young person receives y goods, but nothing when middle-aged or when old. People can access a storage technology that yields one good next period for every good put in storage in the current period. Alternatively, there is a capital good, with one unit of the consumption good acquiring one unit of capital. For each unit of capital acquired in date t, 1:1 units of the consumption good will be received at date t+ 2. If the person liquidates the capital after one period, though, then only:9 units of the consumption good can be obtained at date t+ 1. Assume that agents are risk neutral, and there is a 60% probability that a person will want to consume only when middle-aged and a 40% probability that a person will want to consume only when old. Whether someone is an "early" or "late" consumer will be revealed only at time t + 1.

a) Derive the equilibrium in this economy without an intermediary bank. How much would people invest in the storage technology?

b) Derive the equilibrium with a bank, assuming the bank operates under perfect competition (no proÖts). Assume also that the bank does not expect a bank run to occur. What is the advantage of pooling risk?

c) What is the outcome in case of an (unexpected) bank run, assuming that the bank equally splits the value of its portfolio among all depositors?

d) Compare and rank outcomes from points (a), (b) and (c), both in terms of ex-ante expected consumption (before agents are revealed whether they are early or late consumers) and ex-post actual consumption for early and late consumers