Answer each of the questions below. Round off values to one decimal place where necessary.

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1 Fall 2012 Economics 431 Mid-Term Exam Name KEY Answer each of the questions below. Round off values to one decimal place where necessary. Question 1. A firm uses two inputs to produce widgets, labor (L) and capital (K). Capital pollutes and labor does not. Suppose the output level of the firm at an optimal allocation is 20 percent lower than the current (unregulated) level. Draw a diagram that shows a standard on output that requires the firm to produce 20 percent less output is suboptimal. Show the optimal solution on your diagram, explain how it is different than a standard requiring a 20 percent reduction of output, and describe 2 types of alternative policies that could attain the optimum. L L* L 0 L 1 K* K 1 K 0 C/r* C 1 /r C 0 /r K The initial production level is shown as the isoquant Q 0. The cost of producing Q 0 is C 0, where C 0 = wl + rk is the isocost line. The K-intercept is therefore C 0 /r. If a production standard is implemented requiring a 20% reduction in output to Q 1, the firm scales back production proportionately by selecting (roughly) 20% less labor (L 1 L 0 ) and 20% less capital (K 1 K 0 ). This is not optimal, since input prices in the economy do not fully reflect negative external costs. Specifically, since K polluted and L does not, the rental rate on capital is not high enough (r < r*). The social price that reflects the marginal social cost of capital is r* and the optimal mix of inputs that achieves the isoquant line Q 1 at the social price level is L* and K*, which involves less capital and an additional input substitution effect away from capital and towards labor due to the relative price change. The optimal policy involves L* and K* at the optimal production level Q 1. Policies that can achieve this result: (1) a standard on K* and L*; (2) a tax of t*= E (K*) on polluting capital; (3) a standard or tax on pollution.

2 Question 2. Transferable Permits (20 points). Two firms contribute to air pollution in the San Luis Obispo air basin. The total benefit of pollution for each firm are TB 1 = 60X 1 X 1 2 for firm 1 and TB 2 = 60X 2 1/2X 2 2 for firm 2. Neither firm pays any cost of pollution in the private market. The total social cost of air pollution in the Orlando air basin is TSC = 12, where = X 1 + X 2 is the combined air pollution of the two firms. A. In an unregulated market (without taxes or standards), how much air pollution does each firm produce? (5 points) MB 1 =60-2X 1 MB 2 =60-X 2 MB 1 =MB 2 =0 60-2X 1 =0 60-X 2 =0 X 1 =30 X 2 =60 B. What is the socially optimal level of air pollution for each firm, X 1 * and X 2 *, and what is the total air pollution ( *) at the social optimum? (5 points). TSC=12 MSC=12 MB 1 = X 1 =12 X 1 =24 MB 2 =12 60-X 2 =12 X 2 =48 = X 1 + X 2 = = 72 C. Suppose transferable permits are distributed to firms, where each firm is given an equal share of (1/2) * in permits. Draw a graph that indicates the gains to permit trading for each firm from the initial transferable permit allocation. (10 points).

3 D. If the regulator gives out (1/2) * permits to each firm, what is the fair bargaining price? How many permits will each firm sell (buy) at this price? (5 points). 1 1 (72) =36 permits to each firm 2 2 Firm 1 needs 24 permits so it will sell the extra 12. Firm 2 needs 48 permits so it will buy the 12permits from firm 1. Price=MB (for X 1 or X 2 ), and 60-2(24)=60-(48)=12, so fair bargaining price is $12. Question 3. (30 points) Uncertainty and the Choice of Standards or Taxes Suppose the State of California wants to reduce air pollution by regulating emissions at electric utilities in the State. Suppose the environmental damage from emissions on local bird, fish and wildlife populations is given by D(X) = 20X + ½X 2, where X is pollution. The state regulators are uncertain about the benefit of pollution to electric utilities, but their best guess is that the total benefit of pollution is B(X) = 200X 2.5X 2. A. Based on the information available on pollution benefits and costs, what pollution tax would the state regulator set to obtain the (expected) social optimum? What pollution standard would the regulator set? The regulator sets the tax or standard to equate estimated marginal benefit of pollution to the utility industry with marginal damage from pollution, MB est = MD. To find this point, set B (X) = 200 5X = 20 + X ==>180 =6X ==> Xs=30 At this level of X, we can find the pollution tax by plugging Xs = 30 back into marginal damage. That is, t* = MEC(X*) = MD(X*), where the regulator s guess of X* is Xs = 30. So, t* = 20 + Xs = => t* =$50 Suppose the true benefit of pollution to electric utilities is B T (X) = 128X 2.5X 2. B. What is the true social optimum for pollution, X*. The regulator s guess of the marginal benefit to firms of pollution turned out to be way off. The true social optimum satisfies MB true = MD, or B (X) = 128 5X = 20 + X ==>108 =6X ==> X* = 18 So, regulation under the standard misses the mark, by allowing too much pollution under a standard of Xs = 30. Tax regulation also misses the mark by setting the tax higher than optimal. To see this, consider the optimal tax: t* = 20 + X* = => t* =$38 Setting a tax that is too high results in less than optimal pollution.

4 C. Provide a graph that shows the true marginal benefit of pollution to electric utilities, the true social optimum, the outcome under a pollution tax set at the level you computed in part A and the outcome under a pollution standard set at the level you computed in part A. Identify deadweight loss under each policy. $/X DWLtax DWLstandard MD t =$50 MB est (X) MB True (X) Xt=15.6 X*=18 Xs=30 X Under the (incorrect) tax, the firm reduces pollution too much, leading to the gray DWL triangle, whereas, under a subsidy, the firm does not reduce pollution enough, which leads to the black DWL triangle. D. Based on your calculations and graph above, briefly describe how uncertainty over the electric utilities pollution benefit function can inform the regulator on whether taxes or standards are likely to perform better for regulating pollution in this market. If you were hired as a consultant to help set up pollution policy based on the state s estimate of pollution benefits, would you recommend taxes or standards? The DWL from the tax is smaller than DWL the standard, which suggests the regulator should choose taxes as the better form of regulation. The reason is not because the regulator knows true pollution benefits, but based strictly on the fact that the estimated pollution benefits results in a relatively inelastic marginal benefit of pollution function.

5 Question 4. Deposit-Refund Systems Suppose consumers derive total benefit of B(Q) = 10Q 0.25Q 2 from consuming fluorescent lights (Q), where Q denotes fluorescent bulbs (in hundred millions). The total cost of producing fluorescent lights is given by c(q) = Q Q 2. After consumption, fluorescent lights can be recycled at a recycling center (R) or disposed in a landfill (D), so that Q = R+D. Recycling fluorescent lights requires that consumers travel to a recycling center (e.g., Home Depot), which is a hassle, and the total cost of recycling for consumers is C(R) = 0.5R 2. It is much easier for consumers to simply discard fluorescent lights in the garbage (D); however, fluorescents contain toxic chemicals such as argon and mercury that contaminate the environment through their accumulation in landfill waste. According to the Environmental Protection Agency (EPA), approximately 800 million fluorescent bulbs are improperly disposed of every year. Suppose the environmental damage of fluorescents disposed in landfills is given by E(D) = 2D 2. A. What is the welfare-maximizing mix of recycled and disposed fluorescents (R*, D*)? B(Q) = 10Q 0.25Q 2 = 10(D + R) 0.25(D + R) 2 c(q) = Q Q 2 = (D + R) (D + R) 2 W = 10(D + R) 0.25(D + R) 2 (D + R) 0.25(D + R) 2 0.5R 2 2D 2 FOC: (1) dw/dr = (D + R) 1 0.5(D + R) R = 0 => (1) 9 (D + R) = R (2) dw/dd = (D + R) 1 0.5(D + R) 4D = 0 => (2) 9 (D + R) = 4D Solving (1) and (2) together => R = 4D Plugging this value into (1): 9 (D+4D) = 4D => 9 = 9D => D* = 1 Noting that R = 4D => R* = 4 B. What is the outcome (R c, D c ) in an unregulated, competitive market for fluorescents? CP: Max CS = B(Q) pq 0.5R 2, where Q = D + R CS = 10(D + R) 0.25(D + R) 2 p(d + R) 0.5R 2 FOC: (1) dcs/dr = (D + R) p R = 0 => p = (D + R) - R (2) dcs/dd = (D + R) p = 0 => p = (D + R) Notice that these 2 conditions can only be met when R c = 0. The equilibrium price equates supply and demand in the fluorescent bulb market, where Q = D. Formally, the firm s problem (FP) is: Max = pq - Q Q 2 FOC: d /dq = p 1 0.5Q = 0 => p = Q Market Equilibrium: p = MB = MC => Q = Q => Q c = D c = 9 P c = $5.50 and R c = 0 There are 2 dimensions of the market inefficiency. The private market produces too many fluorescent bulbs from the social perspective (Q* = R* + D* = 5 vs. Qc = 9) and consumers have an incentive to save the transaction cost per unit by not recycling them, leading to 100% waste disposal and 0% recycling. A depositrefund system can correct both problems.

6 C. 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 recycling. A deposit-refund system levies a tax of $t per unit on all fluorescent bulbs and returns $s back to consumers on recycled fluorescents, where the optimal values of t and s are what we seek to solve for in our model. Consider the effect of the deposit and refund on consumer behavior. The firm s problem is the same as the outcome listed above, where (inverse) supply is given by: p = Q = (R + D). Notice that the values of R and D are not identified in supply, only the total number of fluorescent bulbs (Q = R + D). 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 = 10(D + R) 0.25(D + R) 2 p(d + R) 0.5R 2 t(d + R) + sr FOC: (1a) dcs/dr = (D + R) p R t + s = 0 => p = (D + R) R t + s (2a) dcs/dd = (D + R) p t = 0 => p = (D + R) t Substitution in from the firm s problem, p = (D+R), gives: (1a) 9 (D + R) = R + t s (2a) 9 (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), 9 (D* + R*) = R*, so t* = s* in (1a) Equating (2) and (2a): From (2), 9 (D* + R*) = 4D*, so t = 4D* or t* = $4 Check: Plugging t* = 4 into equation (2a) and factoring gives: D + R = 5. Plugging t* = s* = 4 and D+R = 5 into (1a) gives: R c = R* = 4. Since D + R = 5 and R c = 4, D c = 1.