## Debt Deflation and the Great Depression

Problem 1: Debt Deflation and the Great Depression (50 points)

This problem is meant to walk you through the Fisher Debt-Deflation theory of how the Great Depression started in the U.S.

i.) Consider the AS-AD model with money, where money is neutral. Suppose that half of the banks in the economy magically disappear. What will happen to the supply of credit? Suppose that the central bank responds to this by keeping the supply of money fixed. What will happen to the price level P?

ii.) The net worth of a firm is equal to the value of the firm’s assets (things the firm owns) minus the value of the firm’s liabilities (things the firm owes). In our model, the representative firm owns capital, K. One way to value capital is at replacement cost. Simply put, this means the value of capital is the value of the investment goods the firm would have to buy to replace its entire capital stock. To replace the capital stock, the firm would need I = K units of investment goods. This has a nominal value of P × K. We have not discussed liabilities of the firm in class. A substantial amount of the money owed by firms is in the form of loans owed to banks and bonds. Both loans and bonds are promises by the firm to pay a fixed amount of dollars. Assume the total fixed amount of dollars the firm has promised to pay is B. This means that the net worth of the firm is given by NW = P × K − B 1 What happens to the net worth of firms in response to the bank closures in i.)?

iii.) Suppose that the representative firm needs to use its net worth as collateral to finance its investment projects. When net worth is high, the firm has more collateral and can invest more. When net worth is low, the firm has less collateral and must decrease investment. Using the AS and AD curves, explain why GDP will fall.

iv.) In part i.), what could the central bank have done to avoid the drop in output in part iii.)?

Problem 2: How to Pick a Central Banker (50 points)

Consider the following alternative version of the money surprise Phillips curve we discussed in class Y − YT = a(π − πe) + bπ where Y is real GDP, YT is trend real GDP, π is inflation, πe is the private sector’s expectation of inflation, and 0 < a, b. The coefficient a captures the “money surprise” effect of inflation we discussed in class. The coefficient b is meant to capture the idea that there is some small growth benefit to having positive inflation. Suppose that society’s preferences over inflation and GDP can be represented by a utility function

U(π, Y ) = − 1 /2 cπ^2 + Y where c > 0.

A central banker has been appointed to control monetary policy for this society, which here just means setting the inflation rate. The central banker also has preferences over inflation and GDP which don’t necessarily match society’s.

In particular, the central banker’s utility function is Ucb(π, Y ) = − 1/ 2 cCBπ^2 + Y where, again, cCB > 0, but we allow c ̸= cCB.

In what follows, you should  assume that the central banker will always act to maximize his own utility function, not society’s.

i.) Solve for the inflation rate and GDP when the central bank can and cannot commit to monetary policy in terms of the parameters a, b, cCB, YT .

ii.) Use your solution from i.) to compute the utility of society in the case where the central banker can commit to monetary policy. Find the value of cCB that maximizes society’s welfare in this case.

iii.) Use your solution from i.) to compute the utility of society in the case where the central banker cannot commit to monetary policy. Find the value of cCB that maximizes society’s welfare in this case. How does this compare to your answer in part ii.) (i.e. is it larger or smaller, what is its relation to society’s parameter c)? What does this tell you about how central bankers should be chosen if there is a commitment problem?