Consider an economy that extends over three periods, \(t=0,1,2\), with a continuum of banks whose population is normalized to be 1. The banks engage in maturity transformation that finance long-term assets with short-term debts.

In \(t=0\), banks start business, each of them with one unit of capital. Bank capital yields an interim dividend in \(t=1\), which is a stochastic variable that is uniformly distributed over an interval \([0, K]\).

A bank's problem is to choose investment \(I\) in illiquid assets in \(t=0\). Illiquid assets only return in \(t=2\), with expected payoff being \(R I\). The bank finances a fraction \(\alpha\) of the illiquid assets with shortterm debt that is due in \(t=1\), and a fraction \(1-\alpha\) of the illiquid assets with long-term debt that is due in \(t=2 . \alpha\) is constant and exogenously given. The funding cost of short-term debt is lower than long-term debt: the gross two-period interest rate on long term debt is \(r\), and the cost of rolling over short-term debt is \(r-\Delta\), with \(0<\DeltaWith probability \(p\), the economy is in a crisis state. A bank is distressed in the crisis state if its interim dividend is not large enough to service the short-term debts. Assume that stressed banks are not bankrupted, but rather, each of them has to incur a loss, \(c \bar{I}\) with \(c>0\), and \(\bar{I}\) is the average of all banks' investment \(I . \bar{I}\) thus reflects the illiquidity of the economy.

(a) Suppose a benevolent social planner makes the decision of \(I\) in \(t=0\) for all banks to maximize banks' profit. Compute the social planner's choice on \(I\).

(b) Suppose each bank makes a decision of \(I\) individually in \(t=0\) to maximize its profit. Compute a bank's choice on \(I\). Explain why it is different from the social planner's solution.

(c) In question (b), suppose a regulator is able to impose a tax rate \(\tau\) on short-term debt. Compute the optimal \(\tau\) that equalizes the bank's solution with the social planner's solution.