eco200y-m08 - UNIVERSITY OF TORONTO Faculty of Arts and...

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Unformatted text preview: UNIVERSITY OF TORONTO Faculty of Arts and Science APRIL/MAY 2008 EXAMINATIONS ECO 200 Y Duration - 2 hours Examination Aids: Non-Programmable Calculators Instructions: The exam is divided into three parts. Students will have to answer a total of six questions in the following way: PART 1: Students have to answer two out of two questions. PART II: Students have to answer two out of three questions. PART III: Students have to answer two out of three questions. PART I Answer the following two questions from Part I Question 1.1 [18 points] The Mall Street Journal is considering offering a new service which will send news articles to readers by e-mail. Their market research indicates that there are two types of potential users: impecunious undergraduate students and higher—level executives. Let x. be the number of articles that a user requests per year. The executives have an inverse demand fimction Pe(x) = lOO-x and the students have an inverse demand function Pu(x) = 80—x. (Prices are measured in cents). The journal has a zero marginal cost of sending articles via e-mail. Draw these demand functions on a graph and label them. (2 points) (a) Suppose that the Journal can identify which users are students and which are executives. It offers each type of user a different all-or—nothing deal. A student can either buy access to 80 articles per year or none at all. What is the maximum price an undergraduate student will be willing to pay for access to 80 articles? An executive can either buy access to 100 articles per year or none at all. What is the maximum price an executive would be willing to pay for access to 100 articles? Explain. (3 points) (b) Suppose that the Journal can’t tell which users are executives and which are undergraduates. Thus it can’t be sure that the executives wouldn’t buy the student package if they found it to be a better deal for them. In this case, the Journal can still offer two packages, but it will have to let the users self-select the one that is optimal for them. Suppose that it offers two packages: one that allows up to 80 articles per year the other that allows up to 100 articles per year. What is the highest price that the undergraduates will pay for the 80 article subscription? Explain. (4 points) (c) What is the total value to the executive of reading 80 articles per year? (3 points) ((1) What is the maximum price that the journal can charge for 100 articles per year if it wants executives to prefer this deal to buying 80 articles a year at the highest price the undergraduates are willing to pay for 80 articles? Explain. (6 points) confd.h Question 1.2 [18 points] Let the utility function of an individual consumer be U(C,H) = 0.5\/C + 3‘1(24-H), where C is consumption and H<24 is hours worked. Assume that the consumer gets paid a wage of w per hour. The budget constraint of the consumer is p*C = w*H + 10, where p is the price of the consumption good and Io>0 is the initial income (independent of the level of hours worked). aU(c,H)/ac = 0.25Nc aU(c,H)/aH = -3/(2*\/(24-H)) (a) Write the utility maximization consumer problem. (2 points) (b) Derive the labor supply function of the individual consumer, i.e., H*(p,w,Io)? (5 points) (0) Suppose that the government decide to set a tax on consumption t>0. Assume that this tax enters the budget constraint in the following way. The new budget constraint is (l+t)*p*C = w*h + 10, where p is the price of the consumption good before tax. (c.l) Derive the labor supply function of the individual consumer, i.e., H*(p,w,Io,t), under the taxation system. (7 points) (c.2) Prove or disprove whether the labor supply of the individual consumer is increasing, decreasing or not affected by the level of taxation. Furthermore, suppose that leisure is defined as L=24-H. What is the impact of a consumption tax in the optimal choice of leisure, i.e., will leisure increase, decrease or not be affected with the tax? (4 points) cont’d . . . PART 11 Answer the two out of the following three questions from Part II cont’d . .. Question 11.1 [16 points] Consider a profit-maximizing firm with the production function q=F(K,L)=20*\/(L*K), facing output price p and factor (i.e., input) prices r and w for capital and labor. Suppose that in addition to the costs of the inputs, the firm is taxed according to the total cost of labor, i.e., tax=twL. Consequently, assume that the total costs of the firm are C(K,L)=(1+t)*w*L + r*K. aF(K,L)/aL = (10*K) / (V(L*K)) aF(K,L)/aK = (10*L) / (\/(L*K)) (a) Write the profit maximization problem that the firm solves. Similarly write the cost minimization problem that the firm solves. (3 points) (b) Given r and w, what the marginal rate of technical substitution of labor for capital, i.e., MRTS? (1 point) (b) Derive the firm factor demand function, i.e., the demand for K and L. (7 points) (c) Prove or disprove that if the tax rate, i.e., t, increases then the firm will use less of factor 2 for any level of production q. (6 points) confd.“ Question 112 [16 points] Let the cost function of a profit-maximizing firm in a perfectly competitive market be C(q) = 1 + qz, where the fixed cost to operate is 1. Assume that the firm has no sunk costs. Let the price of the homogenous good be p. Recall that 8C(q)/0q = 2*q (a) Write the profit-maximization problem that the firm solves. (2 points) (b) Derive the firm supply function. Please clearly specify for what ranges of prices the firm will produce zero quantity and for What ranges of prices the firm will produce a positive amount. (7 points) (c) Derive the maximum firm profit as a function of the price p. That is, if the profit fianction is 7: (p,q), the maximum firm profits as a function of the price is it (p,q*(p)), where q*(p) is the optimal quantity that the firm will produce for any given price p. Please clearly specify the maximum firm profit function for all ranges of prices p>0 allowing the firm to shutdown whenever it is optimal. (4 points) ((1) Consider that the level of price is uncertain. Suppose that the price p can be either pH=IO, pM=5, or pL=l, where the probability Pr(p=pH) = 0.2 and the probabilities Pr(p:pM) = Pr(p=pL) = 0.4. Calculate the expected profit of the firm allowing the firm to shutdown whenever it is optimal. (3 points) cont’d . .. Question 11.3 [16 points] Suppose Shannon has the following utility function: U(X,Y) = (X"‘Y)°'5 where X is her consumption of candy bars and Y is the consumption of espressos. Let Px be the price of candy bars and Py be the price of espressos. aU(X,Y)/ax = 0.5*(X'°'5)*Y°'5 aU(X,Y)/aY = 0.5*(Y'°'5)*X°‘5 (a) Write the utility maximization problem that Shannon solves. (2 points) (b) Given Px and Py, write the marginal rate of substitution between of good X for Y, i.e., the MRS? (1 point) (c) Derive Shannon’s demand functions for candy bars and espressos. (5 points) ((1) Suppose that the price of candy bars is Px=1 and that the price of espressos is Py=3. Assume that Shannon’s income is 1:300. Calculate Shannon’s optimal consumption bundle (X*,Y*)? Prove or disprove that if the income of Shannon increases by u>1 then Shannon’s expenditures in each product will increase by a? (4 points) (e) Samantha is a friend of Shannon. Suppose that Samantha has the following utility function: U(X,Y) =1n(X) + ln(Y). Prove or disprove that Shannon’s and Samantha’s utility functions generate identical demand functions for candy and espressos. Recall that 61n(X)/6X = l/X. (4 points) cont’d . . . PART III Answer two out of the following three questions from Part III confd.“ Question 111.] [16 points] Let the marginal cost of producing a product be MCp(q) = 30+q, where q is the firm’s output of the product. Unfortunately, the production of q results in pollution. The marginal social cost of production is MCs(q) = 30 + 2*q. The inverse market demand for the product is P(Q) = 300 — Q, where Q is the market output for the product. (a) Draw a detailed diagram. How does the competitive equilibrium for this market differ from the social optimum? Explain (3 points) (b) Define and represent in a diagram the total surplus at the social optimum and at the competitive equilibrium. Does a deadweight loss occur at the competitive equilibrium? (4 points) (0) Is the socially optimal amount of pollution zero or positive? Why? (3 points) (d) Illustrate how a per-unit tax on output can be used to achieve the socially optimal amount of pollution. Compare this tax to a quota on total output that achieves the same socially optimal point. What are the disadvantages of these two methods? (6 points) cont’d . . . Question 111.2 [16 points] This problem is an illustration of the Hawk dove game. The game was first used by biologist John Maynard-Smith to illustrate the uses of game theory in the theory of evolution. Males of a certain Species frequently come into conflict with other males over the opportunity to mate with females. If a male runs into a situation of conflict, he has two alternative strategies. If he plays Hawk, he will fight the other male until he either wins or is badly hurt. If he plays Dove, he males a bold display but retreats if his opponent starts to fight. If two Hawk players meet, they are both seriously injured in battle. If a Hawk meets a Dove, the Hawk gets to mate with the female and the Dove slinks off to celibate contemplation. If a Dove meets another Dove, neither chases the other away. Eventually the female may select one of them at random or leave. The expected payoffs to each male are shown in the box below. The Hawk Dove Game Animal B ' Animal A Hawk Dove Hawk -5, —5 10, 0 Dove 0, 10 4, 4 (a) Now while wandering in the forest 3 male will encounter many situations of this type. Suppose that he cannot tell in advance whether another animal that he meets is a Hawk or a Dove. The payoff to adopting either strategy for himself depends on the proportions of Hawks and Doves in the population at large. For example, that there is one Hawk in the forest and all of the other males are Doves. The Hawk would find that his rival always retreated and would therefore enjoy a payoff of [---] on every encounter. Given that all other males are Doves, if the remaining male is a Dove, his payoff on each encounter would be [——-]. Briefly explain. (3 points) (b) If strategies that are more profitable tend to be chosen over strategies that are less profitable, explain why there cannot be an equilibrium in which all males act like Doves. Briefly explain. (2 points) (c) If all the other males are Hawks, then a male who adopts the Hawk strategy is sure to encounter another Hawk and would get a payoff of [—--]. If instead, this male adopted the Dove strategy, he would again be sure to encounter a Hawk, but his payoff would be [--—] Briefly explain. (3 points) ((1) Explain why there could not be an equilibrium where all of the animals act like Hawks. (2 points) (e) Since there is not an equilibrium in which everybody chooses the same strategy, we look for an equilibrium in which some fraction of the males are Hawks and the rest are Doves. Suppose that there is a large male population and the fraction p are Hawks. Then the fraction of any player’s encounters that are with Hawks is about p and the fraction cont’d . . . that are with Doves is about l-p. Therefore with a probability p a Hawk meets another Hawk and gets a payoff of -5 and with probability l-p he meets a Dove and gets 10. It follows that the payoff to a Hawk when the fraction of Hawks in the population is p, is p*(-5) + (1-p)*10 = 10 — 15p. Similar calculations show that the average payoff of being a Dove when the proportion of Hawks in the population is p will be [-—-]. (3 points) (f) Write an equation that states that when the proportion of Hawks in the population is p, the payoff to Hawks is the same as the payoffs to Doves. Solve the equation for the value of p such that at this value Hawks do exactly as well as Doves. This requires that p = [---].(3 points) cont’d .. . Question 111.3 [16 points] You have been hired by the competition authority of your country to explain the following puzzle. Two separate industries in the country produce a homo genous good. In each industry, there are two price setting firms with constant and average marginal costs. Demand in each industry appears to be downward sloping and approximately linear. In one industry, however, price~cost margins are very small. On the other industry, price- cost margins are very large. The only difference that you can identify is that, in the low price cost margin industry the tenure of the chief executive officers of the two companies has been very short. In fact, over the last ten years, each of the firms has changed chief executive officer three times. In the other industry, there has been no change in tenure of the chief executives. Using the material on oligopoly and game theory, analyze the industry as follows: (a) Write down the elements of the “game” that is being played in this industry, as you see it. (5 points) (b) Suggest an explanation for the differences between these two industries. (5 points) (c) Now, suppose that the chief executives’ tenure has been the same in both industries: there have been no changes in the last ten years. On the other hand, in the low price—cost margin industry you observe that sales of the product are infrequent and conducted privately between buyer and seller. The exact terms of the sale are not known in the industry until months after the sale has been concluded, in fact. Suggest an explanation for why the price—cost margin in this industry is low. (6 points) ...
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This note was uploaded on 04/10/2010 for the course ECO ECO200 taught by Professor Carlosserrano during the Spring '10 term at University of Toronto.

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eco200y-m08 - UNIVERSITY OF TORONTO Faculty of Arts and...

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