Macroeconomic Analysis ECON 6022A Fall 2011 Problem Set 4

Transcription

1 Macroeconomic Analysis ECON 6022A Fall 2011 Problem Set 4 November 2, The price level and money demand Suppose the price level in the economy is P. Real money demand L(Y, i) is the same as we ve discussed in Lecture 6. So nominal money demand is as follows, M d = P L(Y, i) where Y is the real income and i nominal interest rate. 1. Given P is the price level, interpret the economic meaning of a new variable p M = 1 P. 2. Given Y and i, plot the relation between M d and p M, where p M is on the vertical axis. 3. Adding the money supply curve, M s to the graph above. And show the determination of equilibrium p M. 4. Explain how p M responds to an increase in income Y, with the aid of the graph in (Optional) Plot the relation between M d and interest rate, i, where interest rate is on the vertical axis. 6. (Optional) Adding money supply curve to the above graph. And show the determination of equilibrium i. 7. (Optional) Given Y and we assume that price, P, is fixed in the short run. Suppose the nominal money supply increases. Show the determination of new equilibrium interest rate, i. Is it lower than i? Explain why. Solution: 1. Price of money. The number of baskets needed to purchase one unit of money. It is a real variable. 2. See Figure 2. 1

2 3. See Figure When income Y increases, people s demand for money would increase. So the M d curve shifts to the right. The price of money increases. 2

3 5. See Figure See Figure When the nominal money supply increases, without changes in prices in the short-run, the new equilibrium interest rate i would be lower than i. The return on non-monetary asset has to be lower, so that households would hold more money in their portfolio. The mechanism is that households try to get rid of the excess money in their portfolio. Therefore, the price for non-monetary asset (bonds) increases and the return to non-monetary asset decreases. 3

4 2 Money growth and inflation Suppose the money demand function takes the following form M d /P = L(Y, i) = Y ηy ψ(i) where, ψ (i) < Show the income elasticity of money demand is η Y. 2. Suppose that the growth rate of nominal money supply is µ, and inflation rate π and income growth Solution: rate g. The inflation rate is fully anticipated (π e = π). The real interest rate is r and we assume that it is constant. Show the following: π = µ η Y g [hint: In a growing economy, what s the relation between inflation and money supply growth? You may find the following helpful: ln x t+1 ln x t = ln( x t+1 x t 1. The income elasticity of money demand is ) = ln(1 + x t ) x t x t x t dm d /M d dy/y = dm d dy Y M d = P η Y Y η Y 1 ψ(i) Y P Y η Y ψ(i) = η Y. 2. Solution: Asset market equilibrium gives M s t = M d t and therefore, M s t /P t = Y η Y t ψ(i t ) 4

5 Given i t = r t + π e t and r t is constant, we have i t = r + π e t Given π t = π e t, we have, i t = r + π t = r + π And therefore we know that i t = i = r + π is a constant. Given the money demand equation and the equilibrium condition, we have the following, ln(m s t ) ln(p t ) = η y ln(y t ) + ln(ψ(i t )) and ln(m s t+1) ln(p t+1 ) = η y ln(y t+1 ) + ln(ψ(i t+1 )) since i t = i t+1 = r + π, [ln(m s t+1) ln(m s t )] [ln(p t+1 ) ln(p t )] = η y [ln(y t+1 ) ln(y t )] or, ln(m s ) ln(p ) = η y ln(y ) Using the approximation method provided in the hint, M s M s P P = η y Y Y therefore, π = µ η y g 3 Growth rate in money supply ABC, 7th edition: Numerical Problem 6, Page 272. Solutions: (a) π e = M/M = 10%. i = r + π e = 15%. M/P = L = /0.15 = 10. P = 300/10 = 30. (b) π e = M/M = 5%. i = r + π e = 10%. M/P = L = /0.10 = 15. P = 300/15 = 20. The slowdown in money growth reduces expected inflation, increasing real money demand, thus lowering the price level. 4 Nominal and real interest rates (optional) Suppose the nominal interest rate at Period t is i t and real interest rate is r t. Let π t be the inflation rate. The inflation rate from year t to year t + 1, π t, is the ratio of the change in the price level (P t+1 P t ) to the initial price level, P t. We assume that π t r t 0. Show that the following holds: i t = r t + π t 5

6 Solution: The nominal value of asset in year t is x t and in year t + 1 is x t+1. The following relation is satisfied: Since p t+1 = p t (1 + π), we have x t+1 p t+1 Rearrange it, we have the following, and x t p t x t+1 = x t (1 + i t ) x t+1 x t = p t+1 p t (1 + π t ) (1 + i t) x t+1 p t+1 = x t p t (1 + i t) (1 + π t ) are real value of the asset in year t + 1 and t, respectively. Therefore, 1 + r t = (1 + i t) (1 + π t ) Using the assumption that π t r t 0, the result is obtained. Or alternatively, we can approximate the equation by using, and ln(1 + r t ) = ln(1 + i t ) ln(1 + π t ) r t i t π t 6