Assume there are many individuals (aggregate number if normalized to 1) that are each endowed with 1 unit of currency notes. There are three time periods: t = 0,1, and 2. At t =0, individuals have two options with regards to how they can invest their money. They can either stuff in their mattress, where it gets a return equal to 1, or they can invest it in a long-term project that yields a return R = 4 in period two. For example, in individual that invests an amount x will receive 4x in period two, and have 1-x stuffed under the mattress. However, individuals always have the option of withdrawing their money from the long-term project early in period one at a penalty. If they withdraw early, they only receive a return L = in period 1, rather than the return R = 4 in period 2.

At time t = 1, a fraction 0.5 of the individuals receive a liquidity shock. These individuals are impatient and only value consumption in period one. The remaining people, fraction 0.5 of individuals, that do not receive a liquidity shock are patient and only value consumption in period two. At time t = 0, each individual has an equal chance of being hit by the liquidity shock. Assume that individuals do not discount the future, so that their ex-ante expected utility is given by, U = 0.5 * u(c1)+ 0.5 * u(c2), where c1 and c2 is the consumption period 1 and 2 respectively, and u(c) = -1/c.

Let us assume that a bank exists and replicates the decision of the social planner. How much of the 1 unit of endowment will the bank invest in risky asset?

1. 0
3. 1
4. 1/3
5. 2/3