Answers to Problem Set 4. Homework 4 Economics 301

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1 Answers to Problem Set 4 Homework 4 Economics 301 Dividend Problem: For the questions below, assume that the asset in question is a bond with a two year maturity which will pay $100 at the end of the first year and $100 at the end of the second year. 1) Assume that the current interest rate is 5.25% and that it is expected to rise to 5.5% next year. Additionally assume that agents do not care about risk. What is the price today of this two-year bond? $ 100 $100 P = + = $ (1.0525)(1.055) 2) Calculate the yield to maturity and graph the yield curve. Is the slope positive or negative? Explain why. The yield to maturity is approximately the average of the two interest rates and equals 5.375%. The yield curve therefore shows a one-year bond paying 5.25% and a twoyear bond paying 5.375%. It is upward sloping because interest rates are expected to rise. yield time 3) Calculate the price of the bond if the Fed raises the current interest rate to 5.5% and the rate is expected to remain there for the coming year. Does the increase in the interest rate increase of decrease the price of the bond? $ 100 $100 P = (1.055)(1.055) = $ The increase in the interest rate reduces the price of the bond.

2 4) Return to the assumptions in 1) about interest rates over time, but now assume that agents do care about risk. In fact they want a risk premium of 1%. Recalculate the price of the bond and explain how risk affects it. Think of the asset as the combination of a one period bond with a face value of 100 and a two year bond with a face value of 100. There is no risk on the one year bond so its discount factor does not change. For the two year bond, the risk premium will raise the discount factor in the first period by 1% from 5.25% to 6.25%. The price of the bond becomes $100 $100 P = + = $ Therefore, the aversion to risk reduces the price of (1.0625)(1.055) the bond. Recall that after one year has passed, the two-year bond has become a oneyear bond and will not command a risk premium over other one year bonds. 5) Calculate the yield to maturity under the new assumption about risk and graph the yield curve. How does the slope compare to the slope of your original yield curve? The yield to maturity is the average of the first-period interest rate with the risk premium, 6.25% and the second period interest rate of 5.5%, yielding 5.875% for the two-year bond. The yield curve is steeper. yield time

3 Analytical Problems: Chapter 9 1. (a) The increase in desired investment shifts the IS curve up and to the right, as shown in Figure The price level rises, shifting the LM curve up and to the left to restore equilibrium. Since the real interest rate rises, consumption declines. In summary, there is no change in the real wage, employment, or output; there is a rise in the real interest rate, the price level, and investment; and there is a decline in consumption. Figure 9.21

4 (b) The rise in expected inflation shifts the LM curve down and to the right, as shown in Figure The price level rises, shifting the LM curve up and to the left to restore equilibrium. Since the real interest rate is unchanged, consumption and investment are unchanged. In summary, there is no change in the real wage, employment, output, the real interest rate, consumption, or investment; and there is a rise in the price level. Figure 9.22 (c) The increase in labor supply is shown as a shift in the labor supply curve in Figure 9.23 (a). This leads to a decline in the real wage rate and an increase in employment. The rise in employment causes an increase in output, shifting the FE line to the right in Figure 9.23 (b). To restore equilibrium, the price level must decline, shifting the LM curve down and to the right. Since output increases and the real interest rate declines, consumption and investment increase. In summary, the real wage, the real interest rate, and the price level decline; and employment, output, consumption, and investment rise. Figure 9.23 (d) The reduction in the demand for money gives results identical to those in part (b).

5 2. The increase in the price of oil reduces the marginal product of labor, causing the labor demand curve to shift to the left from ND 1 to ND 2 in Figure Since households expected future incomes decline, labor supply increases, shifting the labor supply curve from NS 1 to NS 2 (but by assumption, the shift to the left in labor demand is larger than the shift to the right in labor supply). At equilibrium, there is a reduced real wage and lower employment. The productivity shock results in a shift to the left of the full-employment line from FE 1 to FE 2 in Figure 9.25, as both employment and productivity decline. Because the shock is permanent, it reduces future output and reduces the future marginal product of capital, both of which result in a downward shift of the IS curve. The new equilibrium is located at the intersection of the new IS curve and the new FE line. If, as shown in the figure, this intersection lies above and to the left of the original LM curve, the price level will increase and shift the LM curve upward (from LM 1 to LM 2 ) to pass through the new equilibrium point. The result is an increase in the price level, but an ambiguous effect on the real interest rate. Since output is lower, consumption is lower. Since the effect on the real interest rate is ambiguous, the effect on saving and investment are ambiguous as well, though the fall in the future marginal product of capital would tend to reduce investment. Figure 9.24 Figure 9.25

6 The result is different from that of a temporary supply shock; when the shock is temporary there is no impact on future output or the marginal product of capital, so the IS curve does not shift. In that case the price level increases to shift the LM curve up and to the left from LM 1 to LM 2 in Figure 9.26 to restore equilibrium. In that case, the real interest rate unambiguously increases. Under a permanent shock, the IS curve shifts down and to the left, so the rise in the real interest rate is less than in the case of a temporary shock, and the real interest rate can even decline. Figure (a) The decrease in expected inflation increases real money demand, shifting the LM curve up, as shown in Figure The real interest rate rises and output declines. Figure 9.27

7 (b) The increase in desired consumption shifts the IS curve up and to the right, as shown in Figure This causes the real interest rate and output to rise. Figure 9.28 (c) The increase in government purchases shifts the IS curve up and to the right, with the same result as in part (b). (d) If Ricardian equivalence holds, the increase in taxes has no effect on either the IS or LM curves, so there is no change in either the real interest rate or output. If Ricardian equivalence doesn t hold, so that the increase in taxes reduces consumption spending, the IS curve shifts down and to the left, as shown in Figure Both the real interest rate and output decline. Figure 9.29 (e) An increase in the expected future marginal productivity of capital shifts the IS curve up and to the right, with the same result as in part (b).