ECONOMICS 336Y5Y Fall/Winter 2011/12. PUBLIC ECONOMICS Spring Term Test Sample Solutions Feb. 28, 2014

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1 UNIVERSITY OF TORONTO MISSISSAUGA DEPARTMENT OF ECONOMICS ECONOMICS 336Y5Y Fall/Winter 2011/12 PUBLIC ECONOMICS Spring Term Test Sample Solutions Feb. 28, 2014 Please fill in your full name and student number in the spaces below. NAME: Robert McMillan Question 1 (worth 7 points) a) Take an allocation of goods to consumers in a pure exchange economy. At that allocation, suppose that the marginal rates of substitution across two goods (A and B) are not equalized across two consumers. Can that allocation be Pareto-efficient? Please explain, illustrating your answer (if necessary) using an Edgeworth box. (4 points) A necessary condition for Pareto efficiency is that the slopes of everyone s indifference curves at the optimal allocation are equal. Mathematically, the marginal rates of transformation of good A for good B (for any pair of goods A and B) should be equal across any agents i and j: MRS AB i = MRS AB j for all agents i and j. If those rates are not equalized, then a Pareto-improving gain can be made. Specifically, moving along one of the consumer s indifference curves, the other consumer can be made better off (so moving onto a higher indifference curve) while keeping the first consumer at the initial level of utility. b) State the First Welfare Theorem in the context of the pure exchange economy (one without any producers). (3 points) The First Welfare Theorem states that, in the absence of externalities, every competitive equilibrium must be Pareto-efficient. 1

2 Question 2 (worth 7 points) a) What is true of all points on the utility possibilities frontier? [One sentence.] (2 points) All those points are Pareto-efficient. For the next part, consider a simple two-person economy. The curved line in the diagram below represents the utility possibilities frontier, with person 1 s utility plotted on the horizontal axis and person 2's utility plotted on the vertical axis. Suppose that the underlying social welfare function is given by S = w 1 U 1 + w 2 U 2. At point A, the slope of the utility possibilities frontier is -1/10. U 2 A B Slope at B = -6 C U 1 b) Given that the slope of the curve at point B is equal to 6, if the two people have equal welfare weights, what portion of the utility possibilities frontier would the socially efficient point 2

3 lie on (AB or BC)? Please explain, justifying your answer carefully. [Hint: your answer should involve consideration of the (equal) welfare weights.] (5 points) The socially efficient point would fall on portion AB. The socially efficient point is found where the highest social indifference curve forms a point of tangency with the utility possibilities frontier. The set of social indifference curves derived from the give social welfare function each have a slope given by (-w 1/ /w 2 ). In the case where the welfare weights are equal, the relevant slope is -1. Thus the point of tangency must occur to the left of point B, on portion AB of the utility possibilities frontier where the slope becomes less negative. Question 3 (worth 15 points) A monopolist faces a downward-sloping demand curve (this is implied by having market power), which we can write in general as P = P(Q), with P (Q) < 0, meaning that demand, Q, increases as price, P, falls. Let the total cost of producing output Q be given by C(Q), where the marginal cost C (Q) is positive. a) In setting Q efficiently - that is, in a socially efficient way - what condition needs to hold, and why? [Two sentences.] (3 points) The quantity Q should be set such that P(Q) = MC(Q). Doing so equates the social value of an extra unit of output with the additional cost to society of producing one more unit. b) Prove in the general case (above) of demand given by P = P(Q) that the monopolist will under-provide the good relative to the efficient level given in part a). Please show your derivations clearly. [Hint: start by writing down the monopolist s profit function etc.] (4 points) First, following the hint, the profit function of the monopolist is given by Π(Q) = P(Q) Q C(Q). To maximize profits, differentiate with respect to Q to yield the first-order condition P + Q dp/dq C (Q) = 0. (Given that total revenue is given by P(Q) Q, note that marginal revenue MR(Q) = P + Q dp/dq, 3

4 the first term on the left-hand side of the first-order condition.) Thus we have that the monopolist will choose Q M to solve MR(Q) = MC(Q). Given that AR(Q) > MR(Q) for all Q>0, we must have that P = AR > MR = MC. This arises when the monopolist holds back Q below the efficient level in order to raise prices and increase profits. For the remainder of this question, consider an industry characterized by continually downwardsloping average costs (over the relevant output range). Specifically, assume that AC(Q) = 10-1/4 Q, for Q 40. Further, the demand curve faced by the industry is given by P(Q) = 15 - Q. c) Given the average cost curve given above, derive the industry marginal cost curve, and show your steps. [Hint: AC(Q) = C(Q)/Q. And marginal cost can be derived from C(Q) using calculus.] (3 points) Given that AC(Q) = 10-1/4 Q, then multiplying through by Q, total costs are given by C(Q) = 10Q - 1/4 Q 2. So by differentiating with respect to Q, MC(Q) = C (Q) = 10 ½ Q. d) How many firms would it be cost-efficient to have operate in this industry, and why? [Two sentences.] (2 points) It would be cost-efficient to have just one firm operate in this industry. The declining AC curve implies that two firms (or more) would always produce a given output at greater cost than one firm. e) What would the socially efficient level of output be? Please show your calculations. (3 points) The socially efficient output level Q* solves P(Q) = MC(Q). Here, equating the two, we have that 15 - Q = 10 ½ Q. This equation is solved by Q* = 10, in turn implying P(Q*) = 5. f) If the government regulator chose to run a competition for a single firm to operate in the industry, and stipulated that the winning firm should produce the socially efficient level of output, how much profit would that firm make in total, exactly? Please show your calculations. (5 points) 4

5 At Q* = 10, AC = = 7.5. Therefore the loss per unit = P(Q) AC(Q) = 2.5, and the total loss on all ten units = 2.5 x 10 = 25. Question 4 (worth 12 points) a) A firm produces output Q and damages the environment in the process, where the damages are given by the firm s marginal damage (MD) schedule. Please explain precisely how a government regulator would set a Pigouvian subsidy for this firm. [Note: you need to explain the setting of both the level of the subsidy per unit of Q and the overall amount paid to the firm.] (4 points) A Pigouvian subsidy s = MD(Q*), where Q* is the efficient level of production. The subsidy is paid for each unit of output the firm cuts back on below the profit-maximizing level, Q 1. Thus the total amount of the subsidy paid to the firm = (Q 1 Q*) s. b) Calculate the total amount of the subsidy paid to the firm in the following case. The firm s MPC schedule is given by: MPC(Q) = 4 + 1/2 Q. The firm s MD schedule is given by: MD(Q) = 1/2 Q. And the benefit schedule is given by: MB(Q) = 8 Q. Please show your calculations clearly. [Hint: you will need to calculate the firm s quantity choice in the absence of regulation as part of your answer.] (6 points) The firm will set Q 1 = 2 2/3 in the absence of regulation. That is, Q 1 solves MPC(Q) = MB(Q), or 4 + 1/2 Q = 8 Q. In contrast, Q* solves MPC(Q) + MD(Q) = MSC(Q) = MB(Q), or 4 + Q = 8 Q, so Q* = 2. MD(Q*) = 1, so s = 1, the Pigouvian subsidy. The total subsidy amount is 1 x (2 2/3 2) = 2/3. 5

6 c) Please give one argument in favour of Pigouvian taxes versus Pigouvian subsidies that relates to firm entry. (2 points) Taxes make the industry less profitable and also raise revenue, while subsidies do not make the industry less profitable, and also lead to a revenue requirement, which will distort the economy elsewhere though the need to raise taxes. Question 5 (worth 15 points) Canada has extensive reserves of oil, in the form of the Alberta tar sands. Oil extraction in this context is very costly. Let the private costs of a major oil firm associated with extraction level Q be given by MPC(Q) = 4 + Q. Especially in light of the recent sharp increases in the price of oil, oil extraction from the tar sands is highly profitable for the firms involved. Suppose the benefit associated with oil extraction is MB(Q) = 12 Q. a) In an unregulated world, what level of extraction would the firm choose? Please solve for this level explicitly. (2 points) The firm would choose the extraction level that satisfied MPC(Q) = MB(Q), or 4 + Q = 12 Q. This is solved by Q = 4. b) Extracting oil from tar generates externalities. Let these be captured by the curve MD(Q) = 4. If the government were willing to regulate oil extraction, what would the efficient level (Q*) be? Please derive this in full. (3 points) The efficient level Q* solves MPC(Q) + MD(Q) = MSC(Q) = MB(Q), or 8 + Q = 12 Q, solved by Q* = 2. c) What is the extent of the inefficiency arising when there is no regulation? Here, please solve for the relevant amount explicitly, and set out your calculations. (3 points) The inefficiency is represented by the triangular area whose base = 2 and height =4. So the efficiency loss is given by = 4. d) Suppose a second firm wins a concession to extract oil from tar sands. It faces the same conditions given above, but its oil extraction is somewhat less environmentally damaging, given by: MD(Q) = 2. 6

7 What firm-specific quantity standard would be efficient for this second firm? (2 points) For this firm, MSC(Q) = 6 + Q, so we have 6 + Q = 12 Q, which is solved by Q 2 * = 3. e) If the government were constrained (in this two-firm case) to set just one standard, what level would you recommend, and why? Please explain. (5 points) The standard should be set exactly in the middle of the two firm-specific standards, at a quantity of 2.5. In this instance, given the slopes of the relevant curves, equalizing the base of the two efficiency loss triangles minimizes the total efficiency loss. (This is intuitively clear if you draw the associated diagram.) Question 6 (worth 15 points) Consider the interactions between two parties on adjacent properties: a rancher, who raises cattle, and a wheat farmer. Let the marginal benefit curve of the rancher associated with C cattle be MB R = 6 C, while the marginal cost curve of the farmer is given by MC F = 2 + C. a) What is the efficient (surplus-maximizing) number of cattle on the land, C*? Please solve for this level explicitly. (2 points) The efficient level C* is solved where MB R (C) = MC F (C), or where 6 C = 2 + C, implying C* = 2. b) Assuming that property rights can be defined, and have been defined in favour of the rancher, what deal would the farmer propose to the rancher? Please be precise, and spell out your reasoning. (5 points) The deal would involve specifying a quantity equal to the efficient output level, combined with a mutually beneficial transfer. Thus the farmer would propose C= 2, and provide a transfer T, no less than 8 and no greater than 24. (The lower limit of 8 would leave the rancher indifferent between the proposed deal and doing nothing.) c) Would it matter for efficiency if property rights were defined differently? Please explain. [One sentence.] (3 points) It would not make any difference for efficiency.. This is implied by the Coase Theorem. 7

8 d) Suppose that, instead of bargaining, the farmer could build a fence that would prevent the cattle roaming. What is the maximum that the farmer would be willing to pay? Please set out your reasoning clearly. [Think about this one carefully.] (5 points) The loss to the farmer under the proposed Coasian bargain (where C* = 2) is 6, which is the area under the MC curve between 0 and 2. So as long as the fence is less costly than that, the farmer would install it, as the fence would eliminate the negative externality entirely. 8