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2 The Theory of the Firm (50 pts. ) Suppose Jason starts a firm making high-end basketball shoes. He has a production function Q(L, K) = AL K where Q is the number of units produced. He has MPL = A() and MPK = A(;). Let A = 1 for now. a. Suppose the firm hires L = 16 and rents capital units K = 81. How many units will it produce? b. How does the marginal product of each input behave as & and K increase? c. In a figure, plot the firm's production function as Q(L) keeping A constant. What "Law" describes it's shape? d. Now, as a result of an employee training program hosted by the Chicago Bulls training facility, the overall technological level becomes A = 4. For the same L = 16 and rental of capital K = 81, calculate how many units will it produce? Show the effect of this change in A in the figure you drew above, plotting the output for L = 16. e. Suppose the wage w = 9 and rental rate of capital r = 16. The firm gets a constant P = $12 per unit when sold. For L = 16 and K = 81, what is the value of the marginal product of labor, MP, in dollar terms (the value of marginal benefit of the 16th worker)? f. Now suppose the wage w = 16 and rental rate of capital r = 16. You have 64 units of capital, K = 64. The firm gets a constant P = $12 when it sells it's good. Solve for the optimal number of workers L. g. Assume Jason's firm wants to produce Q = 120 units with wage w = 9 and rental rate of capital r = 16. What combination of L and A minimizes costs in the long run using the production function Q(L, K) = 1LK ? Remember, use the tangency condition. h. Suppose the market (inverse) demand curve in the shoe market is given by P = 100 - 0.50 and suppose Jason's firm has constant MC = 10. Write the firm's total revenue function TR(Q), marginal revenue function MR(Q), and total cost function TC(Q). i. Write the firm's profit function #(Q) and plot it as we did in class. j. Using the firm's production rule (functions from from g), solve for the optimal output level Q"?1. Suppose the production function for the output produced by a rm is f (L,K) = 40143\"r 4K \"4, where L is labor and K is capital. 1. What is the marginal product of labor? What is the marginal product of capital? 2. Suppose the rental rate of capital in the market is r=2, the wage of labor is w=10, the price of output is p=1f3 and the amount of capital is xed at 16 units (i.e., the rm is maximizing a function of a single variable, L, since K is xed). Assuming that the rm acts to maximize prot (note that prot is given by H = p'fJQ w*L r*K), how many units of labor should the rm hire? Show that both the rst and second order conditions for a maximum are satised. What is the associated optimal output level? 11. Suppose now that the capital stock is variable, but the rm is constrained (by its lenders) to spend no more than a total of C=400. Assume the rm chooses inputs to maximize production and all prices are as indicated above, how much capital and labor will it hire? How much will it produce and what are its prots? EHNT: Maximize output, L, K), subject to the constraint on total costs (C), where total costs equal the sum of the cost of employing capital and labor (i.e., C=400=10*L + 2*K). 46. A firm produces according to the following production function: F(L, K) = L K. The price of capital is $3 and the wage rate is $7. What is the optimal combination of labor and capital to produce 100 units of output? a. [' =50, K' =50 b. D' = 100, K* =50 C. [' = 100. K* =100 d. [' = 10, K' = 10 47. You are an efficiency expert hired by a manufacturing firm that uses K and L as inputs. The firm produces and sells a given output. If w =$40 , r = $100 , MPP. =4 , and MPP = 40 : a. the firm is cost minimizing. b. the firm should use less L and more K to cost minimize. c. the firm should use more K and less L to cost minimize. d. the firm is profit maximizing but not cost minimizing. 48. A firm is producing in the short run with output given by: =102 -0.254. The firm hires labor at wage rate of $16 and sells the good for P = $8 . How many workers does the firm hire? a. L = 24 b. L=96 C. L=16 d. L=820. (Figure: Understanding Surplus and Efficiency) In the graph above, efficiency In this market is achieved at a price of: 18 8 1 12 1O Prue Manna $0. $10. $14. $4. 000 (:1 21. If bad weather persists in a summer resort town, we can expect demand to shift left and the equilibrium price for hotels to rise. L) False 0 True 22. Which factor would NOT cause the demand curve for green tea to shift to the right? C) All of these will shift the demand for green tea to the right. 0 new research showing that dn'nking green tea reduces the risk of illnesses O a decrease in the price of lemons, a complement to green tea 0 a decrease in the price of green tea 1, What structure do mod scientific articles follow/ (1pt) 2. What purpose does the introduction section serve? (1 pts) 3. What are 3 features al an anicle that you can use to help identify the author's main points? (3 pts) 4. What scops can you take to improve your understanding of an article? (2 pt) 5. What are 3 key types of information that you should include when taking notes on an article/ (3 pts)
