1 DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE INSTRUCTED TO BEGIN Econ 30010 Intermediate Microeconomic Theory Exam 2, Spring 2014 Instructions: 1) You may use a pen or pencil, a hand-held nonprogrammable, non-graphing calculator, and a ruler or straightedge. No other materials may be at or near your desk. 2) Once you are instructed to begin, check that your exam has 8 numbered pages. 3) Be sure to show all of your work. Answers without supporting calculations will receive zero credit. You will receive credit only for the answers and supporting calculations that appear in this test booklet. 4) You have 120 minutes from the beginning of the exam period to complete this exam. No extensions will be granted. 5) The times listed below are only estimates that are intended to help you manage your time. NAME __Answer Key_______________________________________ Question 1 - 10 minutes: _______ (15 points) Question 2 - 20 minutes: _______ (31 points) Question 3 - 20 minutes: _______ (27 points) Question 4 - 20 minutes: _______ (20 points) Total - _______ (93 points) 70 minutes: 2 1. Define each of the following terms without using equations. (3 points each) a. Isocost curve All combinations of inputs whose cost to the firm is the same. b. Substitution effect The change in a firm's cost-minimizing combination of inputs with respect to a change in relative input prices. c. Economies of scale A characteristic of a firm's cost function in which average total cost decreases as output increases. d. Marginal cost The rate at which cost changes with changes in output. e. Economic profit A firm's revenues less its economic cost. 3 2. Short answer problems. a. Write down the equations needed to solve for a firm's cost-minimizing input combination. Provide an economic interpretation for each equation. (6 points) Technological Efficiency Equation: f ( K , L) q This equation identifies all the combinations of inputs that allow the firm to meet its output target with the absence of waste. Marginal Value Equation: MPK ( K , L) / r MPL ( K , L) / w This equation identifies all the combinations of inputs for which the last dollar spent on each input increases output equally. b. List the properties of a long-run equilibrium in a perfectly competitive market. (3 points) i. Profit-maximizing firms (firms choose their output levels to maximize profit). ii. At the market price, the quantity demanded equals the quantity supplied (market-clearing price). iii. Zero economic profit. c. List the market structure characteristics of a perfectly competitive industry. (4 points) i. Many firms, all with insignificant market share ii. Homogeneous goods iii. Free entry and exit iv. Perfect information 4 d. What properties must all conditional factor demand functions satisfy? Provide an example of a conditional demand function that satisfies all the properties and an example of a conditional demand function that violates each property. (5 points) i. The conditional quantity demanded of a factor must be decreasing in that factor's price. ii. The conditional quantity demanded of each factor can change only if there is a change in the factor's relative price. Example satisfying both properties: K ( q) ( w / r )q Example violating (i): K ( q) ( r / w)q Example violating (ii): K ( q) ( w2 / r ) q e. Suppose there are two perfectly competitive markets. In market 1, every firm has the long-run total cost function, TC1 ( q) 100 q 2q2 . In market 2, every firm has the long-run total cost function, TC2 ( q) 200 .5q q 2 . In which market will the long-run equilibrium output of each firm be larger? Explain. (8 points) In a long-run equilibrium, each firm will produce at its efficient operating scale (the quantity at which average cost is minimized) because this is the only quantity consistent with zero economic profit. We can find this quantity by equating average cost and marginal cost. Market 1: AC1 ( q) 100 / q 1 2q and MC1 ( q) 1 4q so AC1 ( q ) MC1 ( q) implies q 50 7.07 . Market 2: AC2 ( q) 200 / q .5 q and MC2 ( q) .5 2 q so AC2 ( q) MC2 ( q) implies q 200 14.14 . Thus, the long-run equilibrium firm output will be larger in market 2. f. A consulting firm has just finished a study for a manufacturer of wine. It has determined that an additional man-hour of labor would increase wine output by 500 gallons per day. Adding another machine-hour of fermentation capacity would increase output by 300 gallons per day. The price of a man-hour of labor is $5 per hour. The price of a machine-hour of fermentation capacity is $6/hour. Is there a way for the wine manufacturer to lower its total costs of production and yet keep its output constant? If so, what is it? Explain. (5 points) To answer this question, one needs to compare the marginal value ratios of labor and fermentation capacity. If the ratios are not equal, the manufacturer can lower its production costs by substituting towards the input for which the last dollar spent on that input increases output more and away from the other input. Since MPL / w 500 / 5 100 and MPK / r 300 / 6 50 , the manufacturer can lower its total production costs by hiring more labor and employing less fermentation capacity. 5 3. Consider a firm with production function f ( K , L, M ) K 1/4 L1/4 M 1/4 . The rental rate is equal to 5, the wage rate is equal to 2, and the price of materials is equal to 1. a. Calculate the firm's long-run total cost function, average cost function, and marginal cost function. (8 points) Step 1: Calculate the conditional factor demands. This involves solving 3 equations: (TE) K 1/4 L1/4 M 1/4 q , (MVKL) K 3/4 L1/4 M 1/4 / 20 K 1/4 L3/4 M 1/4 / 8 , and (MVLM) K 1/4 L3/4 M 1/4 / 8 K 1/4 L1/4 M 3/4 / 4 . (MVKL) implies L = 2.5K and (MVLM) implies M = 2L. Substituting these equations into (TE) implies (12.5)1/4 K 3/4 q . Thus, K * ( q) .43q 4/3 , L* ( q) 1.08q 4/3 , and M *( q) 2.15q 4/3 . Step 2: Calculate the total cost function and then calculate the average and marginal cost functions. TC ( q) (5)(.43q 4/3 ) (2)(1.08q 4/3 ) (1)(2.15q 4/3 ) 6.46q 4/3 AC ( q) 6.46q1/3 MC ( q) 8.62 q1/3 b. Now suppose the firm's amount of capital is fixed at 10. Calculate the firm's short-run total cost function, average total cost function, and marginal cost function. (6 points) The TE equation becomes 101/4 L1/4 M 1/4 q and (MVLM) still implies M = 2L. Solving these two equations simultaneously implies L* ( q ) .22q 2 and M * ( q) .44 q 2 . Thus, STC ( q) (5)(10) (2)(.22q 2 ) (1)(.44q 2 ) 50 .88q 2 , SAC ( q) 50 / q .88q , and SMC ( q) 1.76q . 6 c. Calculate the short-run equilibrium in a perfectly competitive market in which there are 50 firms and market demand is Qd 1000 2 p . Each firm's capital is fixed at 10 (even if it shuts down) and each firm has the above production function. Each firm faces the same factor prices given above. (11 points) Step 1. Calculate the firm and market supply functions. i. Calculate the minimum value of ANSC. Since ANSC ( q) .88q , its minimum value is zero and a firm will be willing to produce at any positive price. ii. Calculate the profit-maximizing quantity. SMC ( q) p implies that q p / 1.76 .57 p . Each firm's supply function is qs ( p) .57 p and the market supply function is Qs ( p) 28.4 p . Step 2. Calculate the market-clearing price. Qd ( p ) Qs ( p ) implies 1000 2 p 28.4 p or p* 32.9 . Step 3. Describe the equilibrium. p* 32.9 , q* (.57)(32.9) 18.75 , and Q * (50)(18.75) 937.5 . Because SAC (18.75) 19.17 p* , each firm is earning positive economic profit. d. How would you expect the number of firms to change in the long run? Why? (2 points) The number of firms will increase due to entry since firms are earning positive economic profit in the short-run equilibrium. 7 4. Suppose the long-run conditional factor demands for each firm in a perfectly competitive market are K ( q) w 2 q and L( q ) r r 9r q . w w a. If the wage rate, w, increases while the rental rate, r, stays the same, how will each firm's conditional quantity demanded of labor change? What is the economic reason for this change? (3 points) The conditional quantity demanded will decrease due to the substitution effect caused by an increase in the relative price of labor. b. If the wage rate, w, increases while the rental rate stays the same, how will each firm's total labor expenditures change for a fixed amount of output? Explain your answer in light of your answer to part (a). (3 points) Total labor expenditures for a firm equals wL* ( q) rwq 9r . It is increasing in w, so a higher wage rate increases total labor expenditures even though the firm will employ less labor. This means the demand for labor is wage inelastic. c. Calculate the long-run equilibrium in this market when r = 1 and market demand is Qd ( p ) 10 p . Your answers will be functions of w. (8 points) Step 1. Calculate the minimum value of average cost. TC ( q) wq2 wq 9 , AC ( q) wq w 9 / q , and MC ( q) 2 wq w . Average cost is minimized where AC ( q) MC ( q) which is at q 3 / w . The minimum value of average cost is 7 w . Thus, p* 7 w and q* 3 / w . Step 2. Calculate the quantity demanded at the long-run equilibrium price. Q * Qd (7 w ) 10 7 w Step 3. Calculate the long-run equilibrium number of firms. 8 N * Q * / q* 10 7 w 10 w 7 w 3 3/ w d. If the wage rate, w, increases while the rental rate stays the same (at r = 1), how will each firm's total long-run equilibrium labor expenditures change? Why? Compare your answer to your answer in part (b) and explain any differences. (3 points) wL* ( q* ) 12 so each firm's long-run labor expenditures do not change as w changes. This occurs because as w increases, the long-run response of each firm is to reduce the quantity it produces which in turn reduces its long-run quantity demanded of labor. e. If N*equals the long-run equilibrium number of firms, then the total long-run equilibrium labor income for workers in this market is N times the total long-run equilibrium labor expenditures for a single firm. Suppose w is currently .6 and r = 1. If the government passes a law to increase the wage rate, what will happen to total long-run equilibrium labor income for workers in this market? Explain. (3 points) N * / w 5 7 57 w 0 at w = .6. This means that N* is decreasing in w. As a result 3 w 3 3 w there will be fewer firms in the market after an increase in w. Since an increase in w does not change the long-run equilibrium amount each firm spends on labor, a decrease in the number of firms means that long-run equilibrium labor income also decreases.

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