Name Davidson College Mark C. Foley Department of Economics Fall 2001 Intermediate Microeconomic Theory Final Examination DUE 5:00 P.M. THURSDAY, DECEMBER 13, 2001 in Chambers 202 Structure: There are 200 points on the exam. Each multiple-choice question is worth 2 points. The short answers in Section II are worth 3 points each. The problems in Section III are 35, 35, and 30 points, respectively. The problems in Section IV are worth 25 points each. Directions: You must show all your work to receive full credit. Any assumptions you make and intermediate steps should be clearly indicated. Do not simply write down a final answer to the problems without an explanation. This review is to be taken under the honor code. It is closed-book, closed-notes, untimed, and you may use a calculator. I do require that you take it in one sitting. Sign the honor pledge. Carpe diem. Honor Pledge Section 1: Multiple Choice Indicate your answer in the space provided to the right of each question. 1. After some level of output, marginal cost begins to rise because (a) total costs always increase. (b) marginal product eventually decreases. (c) poorer quality inputs are hired as output expands. (d) average variable costs eventually increase. 2. If raising the price of a good results in less total revenue, ceteris paribus, (a) the demand for the good must be elastic. (b) the demand for the good must be inelastic. (c) the demand for the good must be unit-elastic. (d) the demand for the good must be perfectly inelastic. 3. If, at a given consumption bundle on the budget line, the marginal utility of yams is three times larger than the marginal utility of xylophones, and the price of yams is four times the price of xylophones, the utility-maximizing consumer should (a) make no change in her consumption bundle. (b) consume more xylophones and fewer yams. (c) consume more yams and fewer xylophones. (d) consume more yams and the same number of xylophones. 4. If a perfectly competitive firm is producing at an output where price is below average total cost but above average variable cost, (a) it is making normal profits. (b) it is not covering variable cost. (c) it should shut down. (d) it is not covering fixed cost. 5. Consider the production function Q = 4L½K½ where Q is output, L is the number of workers, and K is the number of machines. If the cost of labor is $25 per unit and the cost of machines is $64 per unit, then the total cost of producing 6 units of output will be (a) $120 (b) $240 (c) $150 (d) None of the above are correct. 6. Assume that as new firms enter a competitive market in response to positive economic profits, input prices tend to rise. Under these circumstances, we would expect the long-run market supply curve to be (a) upward-sloping. (b) horizontal. (c) downward-sloping. (d) insufficient information 7. Kristie thinks apples and oranges are perfect substitutes, one for one. If apples currently cost $5 per unit and oranges cost $6 per unit, and if the price of apples increases to $9 per unit, (a) the income effect of the change in demand for apples will be bigger than the substitution effect. (b) there will be no change in demand for oranges. (c) the entire change in demand for apples will be due to the substitution effect. (d) one-quarter of the change in demand for apples will be due to the income effect. 8. The following production function, Q = K½L½, is characterized by (a) constant returns to scale, upward-sloping TC, and increasing MC (b) decreasing returns to scale, upward-sloping TC, and increasing MC (c) constant returns to scale, constant TC, and constant MC (d) constant returns to scale, upward-sloping TC, constant AC 9. The marginal revenue curve of a perfectly price discriminating monopolist (a) is horizontal. (b) coincides with the demand curve. (c) lies below the demand curve. (d) None of the above are correct. 10. An excise tax is imposed on a product in an increasing-cost competitive industry. The price elasticity of demand is less (in absolute value) than is the price elasticity of supply. As a result of the tax, (a) consumers bear a greater proportion of the tax than do sellers. (b) consumers and sellers bear an equal burden of the tax. (c) sellers bear a greater burden of the tax than do consumers. (d) Cannot be determined from the given information. Section II: Short Answers On these pages, respond to the following. 1. Define an inferior good, mathematically and in words, and give an example, explaining why your good is inferior. 2. What is a natural monopoly? Draw and explain a graph illustrating a sustainable natural monopoly. What is the welfare-best (socially optimal) pricing scheme? 3. Define dominant strategy equilibrium, Nash equilibrium, and subgame perfect Nash equilibrium. 4. Define price discrimination, give an example, and explain which degree it is. 5. Define risk aversion. 6. List and explain the characteristics of a monopolistically competitive industry. Describe sources of inefficiency under monopolistic competition and explain why government intervention is not common in such industries despite their inefficiency. 7. Explain why firms might experience increasing returns to scale at lower level of production and then face decreasing returns to scale at higher levels of production. 8. Define zero economic profit. 9. Define and show graphically consumer surplus, producer surplus, and deadweight loss. 10. Define peak-load pricing and give an example. Section III: Problems Do these 3 questions (No choice ). 1. Consider the utility function U ( X , Y ) X Y and budget constraint I PX X PY Y , where I is income, X indicates the quantity of good X, Y indicates the quantity of good Y, PX is the price of good X, and PY is the price of good Y. (a) Show that the percent of income devoted to good X is and to good Y is . Assume + = 1. (b) Calculate the own-price elasticity of demand, cross-price elasticity of demand, and income elasticity of demand for X from part (a). (c) Consider the following CES utility function, U ( X , Y ) X 1 Y 1 . Are the preferences represented by this utility function homothetic? Why or why not? (d) Using the Lagrangian method of optimization, derive the demand functions for X and Y from part (c). (e) If the consumer’s I = $100, PX = $1, and PY = $1, what are the utility-maximizing consumption bundle and optimal level of utility? (f) What is the value of the Lagrangian multiplier, , at the optimum? What is the economic interpretation of ? Show mathematically that this holds (approximately). (g) Write down and explain intuitively what the second order conditions for a maximum are. Why do we care about second order conditions? 2. Assume two firms face a market demand curve of P = a – b Q, where a and b are positive constants, and marginal cost equals c for both firms. Fill in the following table, identify the respective equilibria on the graph below, and explain the relevant characteristics of each model as you work through them. Show all your work. Be organized and neat. Assume “collusion” means the firms split equally the optimal monopoly quantity and profit. Comparison of Oligopoly Models Firm Firm Total Market Consumer Firm Firm Total Model 1’s 2’s Output Price Surplus 1’s 2’s Profit quantity quantity (Q) (P) (CS) profit profit () (q1) (q2) (1) (2) Competitive Cournot duopoly Collusion Bertrand Stackelberg with Cournot follower (1 = leader) q1 q2 2. (work space) (b) Referring to your answers above, explain intuitively how output and price vary by market structure. 3. A competitive firm producing hockey sticks has a production function given by Q = 2K½L½. In the short-run, the firm’s amount of capital equipment is fixed at K = 100. The rental rate for K is r = $1 and the wage rate for L is w = $4. Recall that TC = wL + rK. (a) Calculate and graph the firm’s short-run average total cost, average variable cost, and marginal cost curves. (b) What is the firm’s shutdown price and at what price will it earn zero economic profits? (c) The production function for a firm in the business of calculator assembly is given by Q = 2L½, where Q is finished calculator output and L is hours of labor input (the firm does everything by hand). The firm is a price-taker for both calculators (which sell for $P) and workers (which can be hired at a wage $w per hour). Calculate the supply function for assembled calculators, q f ( P, w) ? Show that the supply function is homogeneous of degree zero in P and w and that profits are homogeneous of degree one in P and w. Explain how changes in w affect the supply of calculators. Section IV: Problems Do two (2) of the following problems (Choice ). If you start more than one, be sure to indicate (by circling the question #) which you want graded. 4. Each day, Jack, a precocious third-grader, eats lunch at school. He only likes Twinkies (T) and orange slices (S), which provide him utility according to U (T , S ) T 2 S 2 . 1 1 (a) If Twinkies cost $0.10 each and a slice is $0.25 per unit, how should Jack spend the $1 his mother gives him in order to maximize his utility? (b) If the school tries to discourage Twinkie consumption by raising the price to $0.40, by how much will Jack’s mother have to increase his lunch allowance to provide him with the same level of utility he received in part (a)? How many Twinkies and slices will he buy now? Assume fractional units are possible. 5. (a) We Make Stuff, Inc. produces high quality stuff for sale throughout the world. The cost function for total stuff production (Q) is given by Total Cost = ¼Q 2. Stuff is only demanded in Australia, where the demand curve is QA = 100 – 2P and Canada , where demand is QC = 100 – 4P. If We Make Stuff, Inc. can control the quantities supplied to each market, how many should it sell in each location in order to maximize total profits? What price will be charged in each location? What are total profits? (b) Assume that a very large number of firms in an industry all have access to the same production technology. The total cost function associated with this technology is TC(q) = 40q – 24q2 + 4q3. If the demand function for the industry’s product is Q = 19 – P, how many firms will produce positive amounts of output at a long-run competitive equilibrium? 6. (a) On the graph below, draw a diagram of an individual’s optimal consumption decision between xylophones and yams. Now assume the price of a xylophone decreases. Graphically show the income and substitution effects such that xylophones are an inferior but not Giffen good. Be sure to explain the movements between your initial consumption bundle (call it point a), the intermediate/hypothetical (call it point b), and the final consumption bundle (call it point c). Yams Xylophones (b) As you’ve drawn it, are xylophones and yams complements or substitutes? Why? (c) Plot the compensated and uncompensated demand curves for xylophones, identifying points “a,” “b” and “c” in your graph. Be sure to label your axes and curves appropriately. Which curve would you use to measure the change in consumer surplus from a price decrease? Why? 7. The mowing of lawns requires only labor (gardeners) and capital (lawn mowers). These inputs must be used in a fixed proportion of one worker to one lawn mower, and production exhibits constant returns to scale. Suppose that the wage rate of gardeners is $2 per hour, that lawn mowers rent for $5 per hour, and that the price elasticity of demand for mowed lawns is –2. (a) What is the wage elasticity of demand for gardeners? (b) What is the elasticity of demand for lawn mowers with respect to their rental rate? (c) What is the cross-price elasticity of demand for lawn mowers with respect to the wage rate? 8. Suppose a government wanted to provide universal health care, at some minimum level, for its citizens. It proposes doing so by offering an earmarked grant of $H for health care. This $H must be spent on health care, and not other goods. Assume that people cannot resell their health care voucher. (a) Show how such an earmarked grant can result in lower utility for some consumers compared to an unrestricted cash grant of the same $H. Draw the original budget line and optimal indifference curve as well as the budget line and optimal indifference curves under both government provision of an earmarked grant and an unrestricted cash grant from the government. CCG Health care (b) Explain how both types of grants exhibit a “crowding-out” of the private amount spent on health care (the subsidized good). Indicate the crowding out on your diagram above. (c) Using graphical analysis, show that a consumer will be worse-off with an excise subsidy for a given good compared to a cash grant which costs the government the same amount. 9. (a) Explain why a monopolist will never produce in the inelastic portion of its demand curve, P = a – bQ, where a and b are positive constants. Draw a graph to support your argument, identifying the elastic and inelastic portions of the (linear) demand curve. (b) What is the elasticity of demand when marginal revenue equals 0? Prove it. (c) Derive and explain MR = P (1+ 1/), where MR is marginal revenue, P is the price of the output, and is the own-price elasticity of demand. 10. Barney, Baxter, and Benjamin are law partners. A client wants them to sue a chemical plant that has been dumping toxins into the river near her home. After listening to the situation, the three partners meet to decide if they should take the case. Decisions are majority rule. They all agree that if they take the case, there is a 30% chance they will win big and make $4,000,000 for the firm, a 30% chance they will win small and make $1,000,000, and a 40% chance they will lose and win nothing. If they take the case, they must forgo another case in which they are certain they will win $1,210,000. Barney has a utility function U(I) = I2, Benjamin has U(I) = I, and Baxter has ½ U(I) = I , where I is the total amount of income for the partnership. (a) Are the partners risk-averse, risk-neutral, or risk-loving? Support your answers mathematically and explain. (b) Will the firm take the case? Who will vote for taking the case? Support your answers mathematically.