371_practice_3_sol

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UBC Vancouver School of Economics ECON 371 Question #1 Suppose there are two people in the society who enjoy a pure public good. The demand for the good of these individuals are: Q 1 = 100 – P and Q 2 = 200 – P. a) What is the optimal level of this public good if it can be produced at a constant marginal cost of $150 per unit? Total MB from the public good is the(vertical) sum of the MB of the two individuals (that is add P) = 100 - Q +200 - Q = 300 -2Q For optimal solution, ∑MB = MC 300 - 2Q = 150 Q =75. b) If it were left to the private market how much of it might be produced? In the private market, each will provide the level where its MB = MC. So first individual: MB = 100 - Q = MC = 150 Q 1 = - 50 (consider 0). Second individual: 200 - Q = 150 Q 2 =50 So total private provision = Q 1 + Q 2 =50. Thus private provision under- provides the public good (due to free riding problem). 200 100 100 200 Q P MC = 150 D 1 D 2 75 total D 125 25 At the optimal level of Q =75, MB 1 = 25 and MB 2 =125 for the total MB = 150.

Question #2 Fireworks in Kitsilano on July 1 is a public good as it is non-excludable and non- rival. There are two types of people who enjoy this fireworks with the following demand functions: High income: Ph = 40 – Q Low income: PL = 20 – Q There are a total of 3 people in the “high income” group and a total of 5 individuals in the “low income” group. Assume that the marginal cost of producing the good is: MC = 3Q. a) What is the efficient output level, Q*, of this public good? b) Calculate the total cost of providing Q* (assuming fixed cost =0). c) Calculate the total surplus to the society if this efficient amount is provided. d) Obviously, no private entrepreneur would be willing to provide this public good as it is non-excludable. Why do you think, the government should provide this public good? e) If government provides the good and finance it by taxing the consumers of the good (according to their willingness to pay), how much should it charge to each high and low income individual? Question #3 . A monopolist is a friend of a conservationist. Explain. Hint: A monopoly of a natural resource uses/extracts less than competitive amount and in that sense conserves the resource. Question 4: In a perfectly competitive market for gasoline in Vancouver, total demand is given by Q d = 60 – 2P + 2P n , where Q = quantity of gasoline in gallons, P = price of gasoline per gallon and P n = price of a unit of natural gas which, at present, is equal to $10. Total market supply of gasoline is given by Qs = 6P. (a) Calculate the equilibrium price and quantity of gasoline sold/consumed in Vancouver. Q d = 60 – 2P + 2P n = 80 – 2P At a competitive market equilibrium: Q d = Q s 80 – 2P = 6P P = $10 Q = 60 gallon

(b) Calculate at the equilibrium point the price elasticity of demand and the cross-price elasticity of demand for gasoline with respect to the price of natural gas. Explain how your results can be used to determine if gasoline and natural gas are substitutes or complements? Elasticity of demand E d = (dQ/dP) *(P/Q) = - 2 *10/60 = - 1/3 (inelastic). Cross-price elasticity of demand E = (dQ/dP n )* (P n /Q) = 2*10/60 = 1/3 Since E > 0, gasoline and natural gas are substitutes (when the price of natural gas goes up, the demand for gasoline increases). (c) If the city of Vancouver sets a price ceiling equal to $4 per gallon, what would be the quantity exchanged in the market? Also, calculate the gain or loss in consumer surplus caused by this price ceiling. With P c = $4, the quantity that is exchanged in the market Q s = 6*4 = 24 gallons. Demand P (at Q = 24) = 40 – 0.5*24 = 28. CS 1 = 0.5(40 – 10)*60 = $900 CS 2 = 0.5{(28 - 4)+(40 – 4) }*24= $720 Loss in CS = $180 D $ Gasoline 6 60 24 P c = 4 40 40 28 MC 10 Question #5 A energy company produces power according to the following production function: Q = 5LK 2 , where K = tons of coal and L = labor in person hours. The market wage rate is w = $20 per hour and the price of coal is r = $50/ton.

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a) Determine (numerically) the firm’s marginal product of labor (MP L ) and Marginal product of coal (MP K ) when the firm is using 5 tons of coal and 10 units of labor. MP L = 5K 2 = 5*5 2 = 125 MP K = 10LK = 10*10*5 = 500. b) Does this function exhibit the property of diminishing marginal returns in labor? MP L is independent of L or does not change with L (note that dMP L /dL = 0). That means this production function does not exhibit the property of diminishing marginal returns in labor. (c) Does the firm's production function exhibit increasing returns to scale? Increase both inputs by t (>1) times, then new Q = 5(tL) (tK) 2 = t 3 5LK 2 > Q, which implies increasing returns to scale in production. d) Does this firm enjoy economies of scale when it employs twice as much of both the inputs? Demonstrate this by comparing the percentage change in quantity with the percentage change in total cost. When the firms employs twice as much of both the inputs, cost of production doubles or increases by 100%. That is %  C = 100% From (b), we see that the production increases by 2 3 = 8 times; that is %  Q = (8Q - Q)/Q = 700%. Economies of scale ES = %  C/%  Q = 100/700 = 0.14 < 1 (when Q increases by 1%, C increases only by 0.14%, less than 1%). Therefore the firm enjoys economies of scale. e) If this firm’s production technology, for som e reason, changes to Q = K + 2L, calculate the marginal product of labor and Average product of labor. What can you say about the relationship between MP and AP of labor and the nature of AP of labor? MP L = 2 AP L = K/L + 2. MP L < AP L for all values of L (given K > 0). That means, AP L is decreasing for all levels of labor.

f) Now assume that the firm's cost function is given by C = 200 + 4Q. Is this firm a natural monopoly? MC = 4 and AC = 200/Q +4, which implies that MC < AC for all levels of Q. Thus this firm's AC declines as it increases Q, implying that the firm enjoys economies of scale and is a natural monopoly. Question #6 BC Hydro is a highly profitable monopoly firm in Vancouver's hydro-power market. BC Hydro's MC of supply is constant and equal to $5. Its annual fixed cost is F. The annual demand for power in BC is given by Q d = 100,000 – 5000P. a) Is BC Hydro a natural monopoly? BC Hydro's C = F + 5Q ATC = F/Q + 5. MC = $5, Since MC < ATC for all values of Q, implying that ATC falls with Q and hence BC Hydro is a natural monopoly. (Note that in this case, the industry naturally supports only one firm because if one firm supplies the entire market, ATC per firm is lower than if two or more firms supplied the market.) b) Calculate the profit maximizing output and price for BC Hydro. Illustrate your results on a carefully labeled graph. A monopolist produces Q where MR = MC. Rearranging the demand function gives P = 20 – Q/5000. Therefore, R = (20 – Q/5000) Q, and MR = 20 – Q/2500. Since MR = MC, 20 – Q/2500 = 5 → Q = 37,500 From demand function: P = 20 – 37,500/5000 = $12.5

c) The provincial government is considering regulating BC Hydro by requiring it to set price equal to average total cost (i.e., earn zero profits). Let N  denote BC Hydro's profits without the regulation, and let R  denote its profits with the regulation. Similarly, let N CS and R CS denote consumer surpluses without and with the regulation, respectively. Which is likely to be larger: N R   − or R N CS CS − ? Using a carefully labeled diagram, briefly explain your answer. g U P 1 MC P 0 MR D ATC f e d c b a Q 0 Q 1 $5 $12.5 P MR MC D Q 37,500

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With the regulation, industry price decreases from P 0 to P 1 , consumer surplus increases by area P 1 P 0 ab, and profit decreases from P 0 agU to 0. The diagram clearly shows that N R   − < R N CS CS − . Or P 0 agU < P 1 P 0 ab. (note also that producer surplus decreases by area P 1 P 0 ac less area fcbe.) Q 7. Thousands of people enjoy BC provincial parks every year. Assume there are two types of people, students and non-students. Adults inverse demand is given by P = 20 – 0.2Q; the demand for senor citizens is given by P = 50 – 0.2Q. The marginal cost of maintaining the parks MC = $5 per visitor. a. If the provincial park is a profit maximizer, how much show it charge to each group? b. Is there a possibility of arbitrage (resale of tickets from one group to the other)? What can the park administration do to prevent this arbitrage? Students : MR =MC 20 - 0.4Q = 5 Q = 37.5 P = 20 – 0.2*37.5 = $12.50 Others: 50 - 0.4Q = 5 Q = 112.5 P = $27.50 Since price charged to students is lower, they can resell tickets to others at any price between 12.5 and 27.5 and make profits. To prevent this arbitrage (resale), the park administration can require to produce student ID at the park entrance . Q 8. Suppose the daily demand for electricity in summer is P = 100 – Q and in winter is P = 140 – Q, where Q = energy in kwh and P = price in cents. The MC of energy supply is MC = 10 cents. The capacity of energy supply is 60 kwh. a)) What are the profit-maximizing quantity and price in each season? b) if the Hydro can increase its capacity at an extra MC of 2 cents, would it expand it s capacity. If so by how much? c) If capacity can be expanded only with extra MC of 12 cents, would the hydro expand its capacity?

a) Summer MR = MC 100 – 2Q = 10 Q = 45 and P = 100 - 45 = P = 55 cents. Winter MR = MC 140 – 2Q = 10 Q = 65 > 60 the capacity So Q = 60 and P = 140 – 60 = 80 cents b) If capacity can be expanded with MC =12, profit maximization occurs at Q given by 140 – 2Q = 22 Q = 64 So the Hydro would expand the capacity to 64, its profit maximizing output level. 140 -2Q = 22 Q = 59 The profit maximizing quantity is less than its current capacity. Hence it would NOT expand the capacity. Question #9 Suppose the government of BC is contemplating to regulate natural monopolies on an efficiency ground. As a CEO of one of the natural monopolists, how will you argue against such a regulation? Explain with the help of a diagram. Hint: To achieve efficiency regulated P = competitive P = MC. In this case, the company will suffer loss (and the company may not be able to provide the essential utility services at all). Question #10 Consider 3 neighbors (Ben, Jill and Lisa) deciding whether or not to have a streetlight in the neighborhood. The light costs $600. They agree in advance that if the light is provided they would share the costs equally. Ben is willing to pay $175, Jill $100 and Lisa $450 for the light. a) Determine if each of Ben, Jill or Lisa is pivotal in the sense that her/his presence or absence will affect the (optimal) outcome. Explain the reasons briefly. With Jill, the total Benefit = 175+100+450 = 725 > 600 (that is cost). Without Jill, the total benefit 175 + 450 = $625 > cost = $600)

Thus whether Jill is present or absent total benefit is higher than costs (or her presence or absence does not change the (optimal) outcome ). So Jill in non-pivotal . With Ben, the total Benefit = 175+100+450 = 725 > 600 (that is cost). Without Ben, the total benefit = 100+450 = $550 < cost = $600 Ben is pivotal! By the same token Lisa is pivotal . (Pivotal part only one calculation/reason is OK. Alternatively the calculation can be for Lisa: without Lisa B = 175+150 = 325 < 600, with Lisa B > 600 ) (Alternative: One might calculate in terms of Net Value. Without Ben, the net benefit 250 – 25 = $225 >0; but with his presence, net value = 250-25-50 = $175 >0. So with or without Ben net value is still positive and so he is non-pivotal. By the same token Jill is also not pivotal. However, Lisa is pivotal as without her net value = - 50 – 25 = - 75 < 0 (Public good is not worth providing). But with her net value = 250 -25 – 50 = 175 > 0 (worth providing)). b) To what extent Ben must lie to change the outcome? If Ben lies to this extent, what would be the Clarke tax that must be imposed upon her? (6 pts) With Jill and Lisa, total benefits = 100+450 = 550. The cost = 600. So Bill needs to lies such that his willingness to pay < 50. (Alternative: Net benefits to Jill and Lisa = (100 – 200) + (450 – 200) = 150. So Ben must lie such that his net benefits is less than – 150.) With this lying, the good will not be provided and as a result, Jill gains 200 – 100 = 100 but Lisa suffers $450 - 200 = 250 Net negative pecuniary externality = 250 -100 = $ 150. So Clarke tax on Ben = net negative pecuniary externality = $150 for lying! Question #11 Suppose you manage a private natural park of its own kind where hundreds of visitors come every year. You are trying to decide how to maximize your profits. The market or total demand for the park is given by Q = 100 - P; the marginal cost of managing the park per visitor is $10 (there are no other costs). If visitors are identical and you implement two-part pricing (membership fee and usage fee), calculate your revenue

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from this pricing scheme, clearly distinguishing the revenue from the two sources. (You may draw a diagram) R = membership fee + usage fee = (100 – 10)*90/2 + 10*90 = $4,950 Question #12 Consider a Hotelling's location game with two hot dog vendors (X and Y) who are deciding to locate (at the same time) along a 400 m stretch of a beach. If there are 9 beach goers who are equally spaced along that stretch, find the Nash equilibrium location(s) and the total (one-way) travel distance for the beach goers under this equilibrium. Assume each buys only one hotdog from the nearest vendor. Nash equilibrium location is both locate at the center. Travel distance = 2*(200+150+100+50) = 1000 m. The Nash equilibrium is not efficient in that the travel distance is not minimized (Travel distance is minimized when the two vendors are located 200 m apart). Question #13 A common pool resource (that is rival but non-excludable) may be overexploited due to its non- excludability nature, leading to what Garett Hardin calls “tragedy of the commons”. Explain by relating it to PD game. (refer the PD game explained in class and how it relates to tragedy of the commons) Question #14 Consider a town in BC with 90 households and an electricity supplier with ATC = 90/Q + 4, where Q = number of households. How much (total) costs saving is possible if this town is served by only this service provider versus two identical service providers each with the above given ATC (sharing the market equally)? ( Try it based on natural monopoly example covered in class )

371_practice_3_sol

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