Fall 2011 Economics 431 Final Exam Name Question 1. (30 points) The Coase Theorem A firm pollutes a local river and causes damage to a swim club downstream. The line MB represents the firms Marginal Benefit from polluting and the line MC represents the Marginal Cost of the pollution damages to swimmers. State your answers to each of the following questions in terms of the areas I, II, III, IV, V, and VI. Firm Swim Club MB (polluter) MC (pollutee) I II III IV V VI X* Zero pollution Water Pollution Full pollution a) If the firm has the property right to pollute the water, what area represents the minimum total payment the firm would be willing to accept to reduce water pollution to the efficient level? Area VI b) If the firm has the property right to pollute the water, what area represents the maximum total payment the swim club would be willing to pay to reduce water pollution to the efficient level? Area II + IV + VI c) If the firm has the property right to pollute the water, what area represents the swim club s gain from trading water rights with the firm at the fair bargaining price? Area II d) If the swim club has the property right for the river, what area represents the maximum total payment that the firm would be willing to pay to pollute the water to the efficient level? Area I + III + V e) If the swim club has the property right for the river, what area represents the total payment that the firm would pay to the swim club at the fair bargaining price? Area III + V
2. Public Goods (40 points) It s going to be a cold winter night on the shore of Lake Tahoe, and you and your two ski pals are deciding how much wood to buy for a fire. Adding logs to the fire will warm up the whole cabin and benefit all three of you. The marginal private benefit from the number of logs burned is different for you, Angela, and Clyde the Canadian, and each can be written as: MPB(you) = 15 0.75Q MPB(Angela) = 10 0.5Q MPB(Clyde) = 5 0.25Q, where Q is the number of logs. Logs can be purchased for $3.00 each. A. (10 points). On a graph, draw the marginal private cost, the marginal social benefit and the MPB curves. How many logs would the three of you burn to be socially efficient? Label this quantity Q* on your graph. $/log MSB MPB you MPB Angela MPB Clde MPC = $3.00 Q*=18 20 Q (logs) The marginal social benefit is MSB = MPB you + MPB Angela + MPB Clyde = (15 0.75Q) + (10 0.5Q) + (5-0.25Q) MSB = 30 1.5Q The optimal socially efficient level of logs is derived by setting MSB = MPC. We have 30 1.5 Q = 3.00 Q* = 18 (logs) B. (10 points) In Lindahl tax system, each person pays a unit price equal to their MPB at Q*, that is, for each individual, P i = MPB (at Q*), where i = You, Angela, Clyde. So each person pays the following Lindahl tax per log: You: P You = MPB You (at Q*) = 15 0.75Q* = 15 0.75 (18) = $1.50; Angela: P Angela = MPB Angela (at Q*) = 10 0.5Q* = 10 0.5(18) = $1.00; Clyde: P Clyde = MPB Clyde (at Q*) 5 0.25Q* = 5 0.25(18) = $0.50; C. (20 points). If you impose Lindahl taxes, would Clyde be better off reporting his true MPB or free-riding by reporting that his MPB is zero? Calculate her consumer surplus in either case to earn full credit. First, suppose Clyde reports his true Marginal Private Benefit. From the graph I below, we could see that consumer surplus is, CS = 1/2(Base) (Height) = 1/2 (18) ( 5-0.5) = (9)(4.5) = $40.5
$/logs $/logs MSB Graph I MSB^ Graph II $5 MPB Clyde $5 MPB Clyde $0.5 MPC = $3.00 $0.6 MPC = $3.00 Q* = 18 Q (logs) Q^= 17.6 Q (logs) Next, suppose Clyde reports zero marginal private benefit. In this scenario, only you and Angela express willingness to pay for logs. The Marginal Social Benefit becomes: MSB^ = MPB You + MPB Angela = (15 0.75Q) + (10 0.5Q) = 25 1.25Q By MSB = MCP = P, we have 25 1.25 Q = 3 Q^ = 22/ 1.25 = 17.6 logs, Clyde s valuation of the last unit is: P^Clyde = 5 0.25Q^ = 5 0.25(17.6) = $ 0.60. Since Clyde reports zero marginal private benefit, he actually does not pay anything for his (indirect) use of the 17.6 logs, but free rides instead. His consumer s surplus is the whole area underneath his MPB line and to the left of Q^, which is a triangle plus a rectangle in graph II. From Graph II, we could see that her consumer s surplus is, CS = 1/2(Base)(Height of triangle) + (Base)(Height of rectangle) = 1/2(17.6) (5 0.60) + (17.6) (0.60) = $49.28 Clyde would be better off if he reports zero marginal private benefit. Question 3. (50 points) Transferable Permits The U.S. Clean Air Act Amendments in 1990 introduced a permit trading system for SO 2 emissions at electric utilities in the U.S. This regulation largely impacts coal-fired electric plants. Among coal-fired electric plants subject to regulation, there are two types of electric utilities: (1) Table A plants, which are the dirtiest electric utilities that burn high-sulfur coal from the Appalachian region; and (2) type B plants, which use low-sulfur coal from the Powder River Basin of Wyoming and Montana. Suppose a representative Table A plant has total benefit from SO 2 emissions given by TB A = 600S A ½S 2 A, and a representative type B plant has total benefit from SO 2 emissions given by TB B = 600S B S 2 B. Total emissions of SO 2 (in tons) for the two plants combined is S = S A + S B, and the total social cost of SO 2 emissions is given by C(S) = 50S + ½S 2. A. (10 points) Calculate the marginal benefit of SO 2 emissions for each electric plant (MB A and MB B ). How much SO 2 do the plants emit in an unregulated market (S C A and S C B )? Marginal benefit is the derivative of total benefit with respect to emissions: for firm A: MB A (S A ) = 600 - S A for firm B: MB B (S B ) = 600-2S B In an unregulated market, each firm pollutes until MB(pollution) = 0, so S C A : 600 S A = 0 => S C A = 600. S C B : 600 2S B = 0 => S C B = 300.
B. (10 points) Draw a diagram that shows the marginal benefit of air pollution for each firm, the combined marginal benefit of air pollution for both firms, the marginal social cost of pollution and the social optimum. $/S P 2 MSC P* P 1 MB T = Total pollution demand S B * S A * S* S*/2 Pollution C. (15 points). What is the socially optimal level of air pollution for each firm, S A * and S B *, and what is the total amount of SO 2 emissions at the social optimum, S*? Total Social Cost of pollution is C(S) = 50S + ½S 2. Marginal social cost is C (S) = 50 + S = 50 + S A + S B To calculate the socially optimal standard on total emissions, maximize welfare: W = TB 1 + TB 2 C(S) = 600S A ½S 2 A + 600S B S 2 B 50(S A + S B ) ½(S A + S B ) 2 Conditions for an optimum (FOC): (1) W/ S A = 600 S A 50 (S A + S B ) = 0 (2) W/ S B = 600 2S B 50 (S A + S B ) = 0 Note: Optimality condition equate MB A = MSC and MB B = MSC Rearranging (1): 550 2S A = S B Rearranging (2): 550 3S B = S A Plugging (2) into (1): 550 2(550 3S B ) = S B => 5S B = 550 => S B * = 110 Solving for S A in (2): S A * = 550 3(110) = 220 => S A * = 220 S* = S A * + S B * = 110 + 220 = 330 D. (15 points) Suppose transferable permits are distributed to firms, where each firm is given an equal share of ½S* as an initial allocation of permits. Derive the equilibrium permit price. Show the gains to permit trading for each firm from the initial transferable permit allocation on your diagram for part C. See diagram above. The permit price equates MB A = MB B = MSC => p* = 50 + S* = $380
Question 3. Deposit-Refund System for Do-It-Yourself Oil Changers (40 points) Suppose consumers derive total benefit of B(Q) = 40Q Q 2 from consuming motor oil (Q), where Q denotes gallons of motor oil. The total cost of producing motor oil is given by c(q) = 10Q + 0.5Q 2. After consumption, motor oil can be recycled at a certified collection center (R) or disposed down the drain (D), so that Q = R+D. Recycling used motor oil requires that consumers who change their oil at home travel to a collection center (e.g., Jiffy Lube) to return their used oil, where the total cost of recycling for consumers is C(R) = 0.5R 2. Disposing used oil down the drain is costless (D); however, used oil is hazardous waste that contaminates local water resources when poured down the drain. Suppose the environmental damage of fluorescents disposed in landfills is given by E(D) = 1.5D 2. A. (20 points) What is the welfare-maximizing mix of recycled and disposed motor oil (R*, D*)? B(Q) = 40Q Q 2 = 40(D + R) (D + R) 2 c(q) = 10Q + 0.5Q 2 = 10(D + R) + 0.5(D + R) 2 W = 40(D + R) (D + R) 2-10(D + R) - 0.5(D + R) 2 0.5R 2 1.5D 2 FOC: (1) dw/dr = 40 2(D + R) 10 (D + R) R = 0 => (1) 30 3(D + R) = R (2) dw/dd = 40 2(D + R) 10 (D + R) 3D = 0 => (2) 30 3(D + R) = 3D Solving (1) and (2) together => R = 3D Plugging this value into (1): 30 3(D+3D) = 3D => 30 = 15D => D* = 2 Noting that R = 3D => R* = 6 B. (10 points) What is the outcome in an unregulated, competitive market for fluorescents? CP: Max CS = B(Q) pq 0.5R 2, where Q = D + R CS = 40(D + R) (D + R) 2 p(d + R) 0.5R 2 FOC: (1) dcs/dr = 40 2(D + R) p R = 0 => p = 40 2(D + R) - R (2) dcs/dd = 40 2(D + R) p = 0 => p = 40 2(D + R) Notice that these 2 conditions can only be met when R c = 0. The equilibrium price equates supply and demand in the motor oil market, where Q = D. Formally, the firm s problem (FP) is: Max = pq - 10Q - 0.5Q 2 FOC: d /dq = p 10 Q = 0 => p = 10 + Q Market Equilibrium: p = MB = MC => 40 2Q = 10 + Q => Q c = D c = 10 P c = $20.00 and R c = 0 C. (10 points) Suppose it is impossible to monitor consumers and tax them for disposing fluorescents in garbage bins. Derive a deposit-refund system to achieve (R*, D*) with a combination of a tax on fluorescent bulbs and a refund on fluorescent bulbs returned to recycling centers.
The policy levies a tax of $t per unit on all fluorescent bulbs and returns $s back to consumers on recycled fluorescents. The consumer s problem under the deposit-refund system is: CP: Max CS = B(Q) pq 0.5R 2 tq + sr, where Q = D + R CS = 40(D + R) (D + R) 2 p(d + R) 0.5R 2 t(d + R) + sr FOC: (1a) dcs/dr = 40 2(D + R) p R t + s = 0 => p = 40 2(D + R) R t + s (2a) dcs/dd = 40 2(D + R) p t = 0 => p = 40 2(D + R) t Substitution in from the firm s problem, p = 10 + (D+R), gives: (1a) 30 3(D + R) = R + t s (2a) 30 3(D + R) = t The goal is to line this solution up with the optimal outcome in equations (1) and (2) from part A. Equating (1) and (1a): From (1), 30 3(D* + R*) = R*, so t* = s* in (1a) Equating (2) and (2a): From (2), 30 3(D* + R*) = 3D*, so t = 3D* or t* = $6.00 Check: Plugging t* = 6 into equation (2a) and factoring gives: D + R = 8. Plugging t* = s* = 6 and D+R = 8 into (1a) gives: R c = R* = 6. Since D + R = 8 and R c = 6, D c = 2. 4. Abatement Investment (45 points) Producing clean energy is a major challenge for society in the 21 st century. Suppose the total benefit of pollution as an input in electricity production is given by B(Q) = 300Q 0.75Q 2, where Q is the amount of electricity produced (in kilowat hours). Suppose the environmental damage associated with the pollution is given by E(Q) = 120Q. A. (5 points) Derive the private market equilibrium (Q c ) and social optimum (Q * ). Q c : MB = 0 => 300 1.5Q = 0 => Q c = 200 Q * : MB = MEC => 300 1.5Q = 120 => Q * = 120 B. (10 points) Calculate the tax on pollution that achieves Q* in the private market equilibrium. For full credit, verify that Q* maximizes after-tax returns to pollution. After-tax returns ( ): = B(Q) tq FOC: d /dq = 300 1.5Q t = 0 t* that attains Q* = 120 solves, 300 1.5(120) = t* or t* =120 (recall: t* = MEC) To verify, the FOC (when t*=120) is d /dq = 300 1.5Q 120= 0 => Q c = Q* = 120 C. (10 points) Calculate the marginal abatement cost function of electricity producers. Draw a diagram that depicts MAC. Label your function MAC 0.
Marginal abatement cost is the lost value incurred by an electricity producer for each unit of abatement. Thus, MAC is zero beyond Q c = 200 and rising at rate 1.5A back along the MB line, where A = 200 Q is abatement Formally: Substituting Q = 200 A into MB, MAC = 300 1.5(200 A) = 1.5A Plugging this into MB: MAC = 200 2(100 A) = 2A or MAC = 1.5A. MAC MAC 0 = 1.5A MAC 1 = 0.8A t*=120 64 Extra return to innovation under a tax A 0 * = 80 A 1 * = 150 Abatement (A) D. (5 points) What is the socially optimal standard on the amount of pollution abatement electricity producers should select (A 0 *)? What is the associated tax on pollution that attains A 0 *? Show A 0 * and the optimal tax t* on your diagram. The socially optimal abatement level is: A 0 * = 200 Q* = 80 units The optimal tax on pollution is the same as in part B, t* = E (Q) = $120 E. (15 points) Suppose a technology exists to scrub pollution as it leaves the smokestack of an electricity producer and that this technology reduces marginal abatement cost to MAC 1 = 0.8A. If the fixed cost of installing scrubbers is $1,000, show that producers will adopt the modern technology under a tax on pollution but not under an abatement standard at A 0 *. Draw a diagram that shows MAC 0, MAC 1, A 0 *, t*, and the return to investment under a standard and a tax. This technology (scrubbers) does exist and have been adopted by many coal-fired electric plants facing regulation under the U.S. Clean Air Act (a transferable permit approach that provides incentives much like a tax). The representative electricity producer will invest in the modern technology if the excess return from operating the new technology exceeds the fixed cost that must be paid to adopt it. That is, the fixed cost of adopting scrubber technology is $1,000, then electricity producers will adopt the technology if it raises their return by $1,000 or more and otherwise not adopt it. Under an abatement standard of A 0 * = 80, the return to adopting the modern technology is the reduction in total abatement cost. This is the cross-hatched area of the figure (the area between the two lines MAC 0 and MAC 1 ). We can find this area by integration, or equivalently, by comparing abatement costs prior to and after adoption
of the modern technology. Old technology: TAC 0 = ½(80)(120) = $4,800 Under the new technology, evaluating MAC 1 at A 0 * = 80 => MAC 1 = 0.8(80) = $64 New technology: TAC 1 = ½(80)(64) = $2,560 The savings in abatement cost with modern technology is: Return = TAC 0 TAC 1 = $2,240. This is not enough to cover the $3,500 fixed cost, so electricity producers do not adopt modern technology. Under an abatement tax of t* = $120, adopting the modern technology allows producers to capture the $2,240 return on the first 80 units abatement (the cross-hatched area), while also allowing producers to make further reductions in pollution to defray tax payments. The producer will abate each unit for which incurring the abatement cost is cheaper than paying the $120 tax, which implies that MAC 1 = 120 in equilibrium, or an abatement level of 150 units rather than 80 units. (0.8A = 120 => A 0 * = 150.) The extra return to abating these additional units is the shaded region of the figure, so that the return to adopting the modern technology sums the cross-hatched and shaded regions: Return = $2,240 + ½ (150-80)(120-64) = $4,200. This is more than enough to cover the $3,500 investment, so producers adopt scrubber technology under a tax, but not under a standard.