Consider three dates: t=0,1,2. Consumers receive one unit of wealth which can be invested or consumed. Of those consumers, a are impatient and consume at t=1 with a utility of u(c1), leaving 1-a patient consumers who consume at t=2 with utility. The utility is given as a function of:

Where ρ is the level of risk affinity. Consumers don’t know their affinity at t=0, such that they don’t know if they will want to consume at t=1 or t=2. At t=1, consumers will know their own type but cannot observe other consumer’s types. Consumers discount the future with a factor of β.

Consider two different technologies that manage liquidity. Consumers can invest in short term technology over a single period, available at t=0 & 1 and yielding r>1. Consumers can also invest in long run technology over two periods, starting at t=0, yielding 0 at t=1 and R>r at t=2. This investment can be liquidated at t=1, but yields only L<1.

Suppose all consumers deposit their money with a particular bank that manages liquidity appropriately. Set up the optimal allocation problem and calculate c1*, c2*. Explain how these allocations change with the discount factor and risk affinity (β and ρ, respectively).

What is the optimal investment in long run technology?

Consider the situation where R>r².

How much should the bank invest in long run technology?

Now also assume that . Is the allocation found in part (a) an equilibrium?