Model: 3 periods T = 0,1,2

If one invests one unit at date 0, it will be worth r2 at date 2 but only r1 < r2 at date 1.

r1/r2 corresponds to the degree of liquidity

Show that if the consumers were risk-neutral, they would prefer the illiquid asset to the liquid asset. However, if they have the utility function U(c) = 1-1/c, which is strictly concave, i.e. they are riskaverse, they prefer the liquid asset to the illiquid asset.

b. Show that the liquid asset is the optimal amount of liquidity if the bank maximizes the expected ex-ante utility of the consumers.

c. Solve for the optimal amount of liquidity if the utility function is U(c) = 1 1/c^2 d. Going back to the original example, show that type-2 consumers would run at T = 1 if they expect that at least 60% of other consumers run at T = 1.