ECON 2123 Problem Set 2 Instructor: Prof. Wenwen Zhang TA: Mr. Ding Dong Due at 15:00 on Monday, April 9th, 2018 Question 1: The natural rate of unemployment Suppose that the markup of goods prices over marginal cost is 5%, and that the wage-setting equation is W = P (1 u) where u is the unemployment rate. a. What is the real wage, as determined by the price-setting equation? b. What is the natural rate of unemployment? c. Suppose that the markup of prices over costs increases to 10%. What happens to the natural rate of unemployment? Explain the logic behind your answer. a. Price setting equation: P = (1 + m)w, m = 0.05. W/P = 1/1.05 = 0.952 b. From wage setting: u n = 1 W/P = 1 0.952 = 4.8%. c. W/P=1/1.1=.91; u=1-.91=9%. The increase in the markup lowers the real wage. Algebraically, from the wage-setting equation, the unemployment rate must rise for the real wage to fall. So the natural rate increases. Intuitively, an increase in the markup implies more market power for firms, and therefore less production, since firms will use their market power to increase the price of goods by reducing supply. Less production implies less demand for labor, so the natural rate rises. Question 2: Demand shocks and demand management Assume that the economy starts at the natural level of output. Now suppose there is a decline in business confidence, so that investment demand falls for any interest rate. a. In an AS-AD diagram, show what happens to output and the price level in the short run and the medium run. 1
b. What happens to the unemployment rate in the short run? in the medium run? Suppose that the Federal Reserve decides to respond immediately to the decline in business confidence in the short run. In particular, suppose that the Fed wants to prevent the unemployment rate from changing in the short run after the decline in business confidence. c. What should the Fed do? Show how the Feds action, combined with the decline in business confidence, affects the AS-AD diagram in the short run and the medium run. d. How do short-run output and the short-run price level compare to your answers from part (a)? e. How do the short-run and medium-run unemployment rates compare to your answers from part (b)? a. The AD curve shifts left in the short run. Output and the price level fall in the short run. In the medium run, the expected price level falls, and AS shifts right, returning the economy to the original natural level of output, but at a lower price level. b. The unemployment rate rises in the short run, but returns to its original level (the natural rate, which is unchanged) in the medium run. c. The Fed should increase the money supply, which shifts the AD curve right. A monetary expansion of the proper size exactly offsets the effect of the decline in business confidence on the AD curve. The net effect is that the AD curve does not move in the short run or medium run, and neither does the AS curve. d.under the policy option in part (c), output and the price level are higher in the short run. In the medium run, output is the same in parts (a) and (c), but the price level is higher in part (c). e. The unemployment rate is lower in the short run in part (c). In the medium run, the unemployment rate is the same in parts (b) and (c). Question 3. Suppose that the interest rate has no effect on investment. 2
a. Can you think of a situation in which this may happen? b. What does this imply for the slope of the IS curve? c. What does this imply for the slope of the LM curve? d. What does this imply for the slope of the AD curve? Continue to assume that the interest rate has no effect on investment. Assume that the economy starts at the natural level of output. Suppose there is a shock to the variable z, so that the AS curve shifts up. e. What is the short-run effect on output and the price level? Explain in words. f. What happens to output and the price level over time? Explain in words. a. (The solution is not unique) Firms may be pessimistic about sales and the economy, therefore they do not want to borrow at any interest rate. b. IS curve is vertical. Interest rate is not affected by the interest rate. c. The LM curve is normal upward sloping. d. The AD curve is vertical. Price level does not affect equilibrium output. e. The natural level of output changes due to the change in variable z. The equilibrium output doesnt change. Therefore, current output is greater than the natural level of output. Price level will increase. f. Output remains unchanged, natural level of output remains below current output. AS curve will shift up forever, thus price level will continue to increase forever. Question 4. Consider the production function = K + 2N a. Suppose that K = 10 and N = 20. What is the value of? b. Suppose that K and N triple. What happens to? c. How would you qualify the returns to scale for this function? d. Write this function as a relation between output per worker and capital per worker. e. Suppose K/N = 2. What is the value of /N? What happens to /N if K/N is doubled? 3
f. Does the relation between output per worker and capital per worker exhibit the same returns to scale as the above production function? g. What is the difference between the two functions? h. Plot the relation between output per worker and capital per worker. How does the shape of the relationship between the two compare with Figure 10.4? a. =50. b. triples. c. Constant returns to scale. d. /N = K/N + 2. e. K/N=2 implies /N=4. K/N=4 implies /N=6. Output less than doubles. f. No. g. In part (e), we are essentially looking at what happens to output when we increase capital only, not capital and labor in equal proportion. There are decreasing returns to capital. h. The curves are not the same. One is linear and this other is not. Question 5: The Cobb-Douglas production function and the steady state This problem is based on the material in the chapter appendix. Suppose that the economy s production function is given by: = K α N 1 α and assume that α = 1/3. a. Is this production function characterized by constant returns to scale? Explain. b. Are there decreasing returns to capital? c. Are there decreasing returns to labor? d. Transform the production function into a relation between output per worker and capital per worker. e. For a given saving rate, s, and deprecation rate, δ, give an expression for capital per worker in the steady state. f. Give an expression for output per worker in the steady state. g. Solve for the steady-state level of output per worker when s = 0.32 and δ = 0.08. 4
h. Suppose that the depreciation rate remains constant at δ = 0.08, while the saving rate is reduced by half, to s = 0.16. What is the new steady-state output per worker? a. es. Doubling inputs K and N will double output. b. es. Keep labor constant and only increasing capital, output doesnt increase with constant returns to scale. c. es. Keep capital constant and only increasing labor, output doesnt increase with constant returns to scale. d. = Kα N 1 α = ( K N N N )α. e. s = δ K N N s( K N )α = δ K N K N = ( s δ )1.5 f. N = ( s δ )0.5 g. K N = (4)1.5 = 8 = N (4)0.5 = 2 h. K = N (2)1.5 N = (2)0.5 THIS IS THE END OF PROBLEM SET 2. 5