Department of Economics Columbia University Spring 2023 Instructor: Martin Uribe Homework 9 Economics UN3213 Intermediate Macroeconomics Due April 12, before class on Gradescope Exercise 1 (A Speculative Attack on the Central Bank). Consider the Cagan model with rational expectations. The demand for real balances is given by: Y 0.04 + it where MY denotes the demand for nominal money balances in period t, A denotes the price level in period t, Y = 100 denotes output, and i denotes the nominal interest rate for assets held between periods { and { + 1. The real interest rate, denoted ", is nil(r = 0). Suppose that the monetary policy consists in pegging the price level. Specifically, the central bank pegs the price of one unit of consumption goods at one unit of currency, that is, P = 1 for $ 2 0. Under this policy the money supply, denoted Me, is endogenous, because the central bank stands ready to exchange money for goods at the price P = 1. 1. Find the equilibrium inflation rate, at = P/ P-1 - 1, for t 2 1. 2. Find the equilibrium nominal interest rate, 4, for t 2 0. 3. Find the equilibrium level of real money balances, M/ P, for $ 2 0. 4. Find the equilibrium level of money supply, M, for t 2 0. 5. Find the equilibrium growth rate of the money supply, at = Mr/Mt-1 - 1, for $ 2 1. Now consider a change in monetary policy. Specifically, in period 7 > 0 the central bank unexpectedly announces that beginning in period 7 + 1, the price level peg is abandoned and that the central bank will switch to a money growth rate peg at the rate , = 0.12 (or 12 percent per period), so that My+1/My, My+2/My+1 = ...= 1+ ji = 1.12. 6. Find the equilibrium inflation rate that will prevail from period 7 + 1 on, that is find 7. Find the equilibrium interest rate, i, for t 2 T. 8. Find the equilibrium level of real balances, M / P, for t 2 T.9. Discuss intuitively what happens to real balances in period T relative to the value that would have obtained in the absence of the announcement of the policy change. 10. Find the nominal money supply in period T, MT {recall that the central bank still pegs the price level in period T). 11. Find the money growth rate in period T, p.31 E MT/MT_1. Given your answer to this question, explain the meaning of the title of this exercise, \"A Speculative Attack on the Central Bank.\" Exercise 2 (A Variant of the Baumol-Tobin Model). In class, we derived the BaumolTobin money demand function under the assumption that at the beginning of each period, the individual receives her income in the form of an interest-bearing bank deposit. Suppose instead that the individual receives her income in her hand. The individual can still make trips to the bank and deposit money in an interest-bearing account. The only difference with the setting we discussed in class is that here at the beginning of each period she receives her income in his hand rather than into the bank. The individual spends all of her income by the end of the period. Suppose that Y = 1000, 2' = 0.01, and C = 0.2, where Y, 1', and 0 denote, respectively, income, the interest rate, and the cost of each trip to the bank. 1. Explain why it would not be optimal for the individual to make only one trip to the bank. 2. Write the formulas (using symbols, Y, i, C) for the total cost of going to the bank 0 times, 2 times, 3 times, and in general n times, where n is any integer other than 1. 3. Using a calculator or a spreadsheet, calculate the total cost when n takes the values 0, 2, 3, 4, 5, 6, 7, 8, 9, and 10. What is the number of trips that minimizes the cost? 4. Write the formula for the average money holdings for 0 trips and for 10. trips. 5. What are the average money holdings when the individual makes the optimal number of trips to the bank? Exercise 3 (Elasticities and Semi-Elasticities). Compute the income elasticity and the interest semi- elasticity for the following three specications of the demand for money. In all cases, assume that Y = 1, and 1': 0.1. 1. L(i,Y) = 732' + 21/ 2. L(2',Y) = Y{l +2')-3 3. L(i,Y) = 0.5Y1/2'1/2