Homework Assignment 1 Economics 215 Intermediate Macroeconomics 1. Production Function. Assume that an economy has a Cobb-Douglas production function with a = 12 . For simplicity, we will assume that the technology level is constant (i.e. T = 1). 1 1 Q F ( K , L) K 2 L2 K L a. Demonstrate that this production function demonstrates constant returns to capital. Calculate the production level when K = 1 and L = 1. Calculate the level of output after the inputs double, K = 2 and L= 2. Calculate the level of output when inputs triple, K = 3, and L = 3. If F ( K , L) K L , then F (K , L) K L K L K L When K = 1 and L = 1, Q=1. When K = 2 and L = 2, Q=2 When K = 3 and L = 3, Q=3. Double all your inputs, double your output Q b. Show that the average productivity of labor, APL = , is L decreasing in the amount of labor used if we hold the level of capital constant. Assume that capital is constant at K = 1. Calculate the APL when L = 1, L =4, L = 9, L = 16, L = 25. Is this decreasing as L increases? L Q Q L 1 1 1 4 2 .5 9 3 1/3 16 4 .25 25 5 .2 The larger is L, the larger is Q, but the smaller is Q/L. c. Calculate the marginal product of labor for this production function in two ways. Continue to assume that K = 1 throughout. i. Calculate Q F ( K 1, L L) F ( K 1, L) L L L L L L when L = 1 and L = 3 (i.e. Calculate the extra production per extra worker as we move from 1 worker to Q 4 workers). Calculate , when L = 4 and L = 5. L Calculate the marginal product of labor when we move from 9 workers to 16 workers and when we move from 16 workers to 25 workers. Is this amount decreasing as L increases? ii. Calculus tells us that the marginal change in output from an infinitessimal change in labor gives us the formula: L L 1 4 9 16 25 3 5 7 9 Q 1 Q when L = 1, 4, 9, 16, and 25. Is this amount L 2 L decreasing as L increases? Q 1 Q Q L L L 2 L L L L .5 1/3 .25 .2 1/6 1/7 .125 1/9=.111 .1 2. In our economy, we find that the income and substitution effects of real wage changes always cancel out. Thus, the labor supply curve showing the W relationship between the real wage rate , and labor supply is perfectly P vertical at L* = 4. The production function of firms is given by the CobbDouglas function Q K L . Assume that the capital stock, K = 1. a. How much output will be produced when L = L* = 16. Q = 4 Q 1 Q 1 1 b. Calculate the marginal product of labor, when L L 2 L 2 L = L*. Graph the labor market. Show, using the graph, that the equilibrium real wage, w*, equals the marginal product of labor calculated at L*. What is w* in numbers? LS w w* MPL=LD L L* = 4 w* = .25 c. The workers must sign a dollar wage contract that specifies a wage rate W and allows firms to choose the number of workers they want to hire at that wage rate. Assume that workers specify a wage rate that they expect will net them a real wage rate of w*. If workers expect a price level of PE = 4, what wage rate, W , will they choose in order to achieve the real wage rate, w*in section 2.a. Workers would choose a wage rate W =1, to set the expected real wage = .25. d. The labor contracts have been signed at the wage rate solved for in section 2.c. The workers have overestimated the price level. It actually turns out that the wage rate will by P = 2. What is the ex post real wage? How much labor will firms want to hire at the ex post real wage? Answer this second question, by solving for the ex post real wage, then solving for the level of labor L, that sets the marginal product of labor equal to the ex post real wage. Is there a worker shortage or a worker surplus? The expost real wage turns out to be .5, which is larger than the equilibrium real wage. At a real wage rate of .5, the firm will hire a level of labor equal to L=.5. There is a worker surplus, since at the given real wage, workers would choose a labor supply of L = 4 3. Real Consumption [Computer Assignment]. You are an analyst in the department of statistics. You gather data on the level of consumer expenditure in 2001 and are assigned to calculate the real consumption level for 2001. The base year used by the department is 1990. You also have comparable data for 1990. Calculate consumption expenditure, PCC for 1990 and 2001. There are N = 10 types of goods. Calculate weights w1…wN for each type of good based on their share of overall consumption expenditure in the base year. Use these weights to calculate the weighted average price level of consumer goods in 2001 (relative to the base year), P2001. Calculate real consumption spending in 2001 in 1990 dollars, C. [Download an Excel file from the Web Page]. Table 1 Consumer Goods Hamburgers Electricity Cigarettes Televisions Shirts Bus Tickets Automobiles Shoes Haircuts Doctor Visits Total Prices 10 4 40 1600 400 8 12000 250 100 400 Consumption Spending 1990 Consumption Spending 2001 Weighted Average of Relative Prices Real Consumption Spending 2001 1990 Mil. HK$ Spending 5500 12000 8000 4000 6000 3000 17500 2000 9000 7000 74000 74000 90000 1.299775 69242.77 Weights 0.074324 0.162162 0.108108 0.054054 0.081081 0.040541 0.236486 0.027027 0.121622 0.094595 Prices 8 5 50 2000 500 10 16000 400 150 600 2001 Mil. HK$ Spending 8500 14500 6000 8000 7000 4500 22500 2500 9000 7500 90000 69242.77 Relative Prices 0.8 1.25 1.25 1.25 1.25 1.25 1.333333 1.6 1.5 1.5 1.299775

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