Environmental Economics: Exam December 2011 Answer to the short questions and two Problems. You have 3 hours. Please read carefully, be brief and precise. Good luck! Short Questions (20/60 points): Answer each of the four questions with few sentences. (a) Provide one argument against allowing banking of tradable emissions permits (i.e. transferring some unused permits from one phase to the next one). With banking, firms can transferred all or part of its unused permits from one phase to the next one. It might lead firms to use more permits than the target emission level in one phase which is a problem if pollution concentration at one period of time is an issue such as for SO2 emissions. (b) When firm s abatement costs are private information, explain why a more complex regulation than a flat tax rate or a standard on emissions can improve welfare. How does such a regulation look like? Direct revelation mechanism does better. For instance, an emission standard and a bonus for emitting less than the standard up to some level is better because it leads to emission levels closer to the first-best. Indeed, high cost firms will just comply with the standard while low cost firms with abate more to obtain the bonus. The later firms enjoy an informational rent. (see lecture notes). (c) When a common-pool resource is exploited under open-access, all users obtain no rent from its extraction. True or False? Explain. False. If users have heterogeneous extraction costs, then only the user with the higher cost obtains no rent form extraction. Users with lower costs enjoy a positive rent. (See lecture notes). (d) When firm s emissions are taxed but costly to monitor, explain why a firm that opportunistically underreports emissions to pay less taxes might nevertheless emit the efficient level of emissions. see lecture notes Problem 1 (20/60 points) A firm is producing sugar from sugarcane in Candy Island. It is using a polluting technology: for each kilogram of sugar produced, 1 liter of a chemical pollutant ends up in the aquifer which is the only source of fresh water on Candy Island. The technology is used only for two periods. According to the scientists, the pollutant has a negligible impact on water quality as long as the aquifer contains no more than Ē liters. Above the threshold Ē, the water becomes so toxic that it cannot be consumed anymore. Let us denote production q i, production emissions e i for periods 1
i = 1, 2. The inverse demand for sugar is the same both periods: P (q i ) = A q i for i = 1, 2 with A > 0. Marginal production costs are constant equal to c i for i = 1, 2. Welfare and profits are discounted at rate one. (a) What is the shape of the damage function? An horizontal line from 0 to Ē and then a vertical line at Ē. (b) Find the efficient productions qi under the constraint that pollution concentration should not exceed Ē. Explain how the pollutant concentration threshold Ē and costs c i impact the production choices. Find the tax on emissions that implements the first-best assuming that the firm is price-taker. Explain why the tax rate should be the same both periods. The optimal production allocation (q1, q 2 ) maximizes the discounted sum of consumers surplus under the constraint that pollution concentration does not exceed Ē. It thus solves max (q 1,q 2 ) q1 0 (P (x) c 1 )xdx + q2 0 (P (x) c 2 )xdx subject to the pollution constraint q 1 + q 2 Ē. Denoting λ the multiplier associated to the constraint, it leads to the following first-order conditions: P (q i ) = c i + λ for i = 1, 2. That is the price should be equal to the social cost of production that is the marginal cost and the shadow cost of pollution. If binding, the pollution constraint implies that price net of marginal cost should be equalized: P (q 1) c 1 = P (q 2) c 2. Substituting for the demand function in the above equality and using the biding pollution constraint, we obtain q1 = (c 2 c 1 + Ē)/2 and q 2 = (c 1 c 2 + Ē)/2. Unsurprisingly, production in period i is decreasing with marginal cost in the same period and decreasing on the marginal cost for the next period. It is also increasing with the emission level Ē. Using the first order condition, we can find the Pigouvian tax level λ = A (c 1 +c 2 +Ē)/2. It is the same each period because it reflect the marginal damage of emissions on welfare which is the same both periods (the discounted rate is one and emissions impact pollution concentration the same way each period). (c) The firm acts as a monopolist. What are the productions qi m for both periods? Illustrate the efficient and monopoly solution in the same graph. Each period the monopoly chooses qi m that maximizes its profit (P (q i ) c i )q i. The first-order condition leads to P (q m i )q m i + P (q m i ) = c i, 2
which is the standard marginal revenue equals to marginal cost condition. Using the definition of P (q i ) we obtain qi m = (A c i )/2. See the lecture notes for the graph. (d) The council of economic advisors in Candy Island disagree on the regulation that should be applied to the monopolist sugar producer. Some economists advice to tax emissions while others are in favor a cap on sugar price. Who is right? How to choose among the two regulations? The choice among the two instruments depends on wether monopoly production qi m is higher or lower than the first-best production qi. If qm i > qi, the monopoly produces more than it should because it ignores the impact of pollution. Production should be taxed at the Pigou level (see (b)). If qi m < qi, the monopoly reduces production from the first-best to exploit market power. Price is higher than the first-best, i.e. P (qi m) > P (q i ), so a regulation that cap price to P (q i ) at period i implements the first-best. (e) Suppose now that part of the pollution is absorbed from one period to the next one: for every liter emitted in period 1, only γ liters are remaining in the aquifer in period 2. How does that change the tax on emissions found in (b)? Explain why. The tax should be lower in period 1 than period two because emissions (or production) in the first period has a lower impact on pollution concentration in second period when the pollution constraint is binding. Indeed the binding pollution constraint becomes γe 1 + e 2 = γq 1 + q 2 Ē. The first-order condition in period 1 is then P (q 1 ) = c 1 + γλ. The tax rate should then be γλ in the first period and λ in the second period. Problem 2 (20/60 points) A set of n > 1 cities are sending their dangerous wastes to a storing and treatment plant. Wastes must be sorted out and labeled properly before being sent to the plant. Some of the chemical are explosive if mixed. Therefore waste sorting and conditioning is essential to avoid an accident at the treatment facility. Let us denote by e i the safety effort (in terms of sorting and labeling of waste containers) of city i and its cost by C i (e i ). It is assumed increasing and convex with safety C i (e i) > 0 and C i (e i) > 0. The probability that the waste treatment plant explodes depends on the total safety effort E = n i=1 e i according to the decreasing convex function p(e) (with p (E) < 0 and p (E) > 0, p(0) = 1, p( ) < 1). The damage for city i is evaluated to d i so that the overall damage to society is D = n i=1 d i. Cities are risk neutral: each city minimizes its expected costs and damages. 1. Find the conditions that characterize the first-best safety efforts e = (e 1,..., e n) and the equilibrium ones e e = (e e 1,..., ee n). Comment those conditions and explain why cities underinvest in safety efforts in equilibrium. The first-best safety efforts e minimize the sum of costs and expected damages 3
for all cities that is n i=1 (p(e)d i + C i (e i )) that is p(e)d + n i=1 C i(e i ), i.e. the expected overall damage plus the total costs. The first-order conditions yield: C i(e i ) = p (E )D. for i = 1,..., n with E = n i=1 e i. The left-hand side the marginal cost of reducing effort in terms of safety costs for a city i while the right-hand side is the marginal reduction of expected damage for all cities. The first-order conditions imply that the safety effort e i for city i should equalize its marginal cost to the overall expected marginal benefit in term of reduction of risk of accident for all cities i = 1,..., n. The equilibrium efforts e e is such that each city i minimizes the sum of its effort cost and the expected damage to itself C i (e i )+p i (e i )d i. The first-order conditions yield: C (e e i ) = p (E e )d i for i = 1,..., n with E e = n i=1 ee i. The marginal cost of the safety effort should be equal to its marginal benefit in term of expected damage reduction for the city. When choosing its safety effort, a city takes into account only the reduction of expected damage on itself d i and not on others j i d j. Therefore each city tends to under-invest on safety: e e i < e i. Indeed, the prevention of accident is a public good. Each city under contributes to its provision. 2. Suppose each city i has to pay a fine F i in case of accident for i = 1,..., n. Find the minimal fines that implement the first-best safety efforts e in equilibrium. Interpret this fines in terms of liability. With the fine F in case of accident, city i now choose the safety effort e f i minimizes its expected cost with the fine C i (e i ) + P (E)(d i + F ) for i = 1,..., n. The first-order conditions are: C i(e f i ) = p (E f i )(D i + F i ), for i = 1,..., n with E f i = n i=1 ef i. The city equalizes the marginal cost of safety effort with its marginal benefit which is marginal reduction of the probability of accident times the cost of an accident including the fine. Comparing the above condition and the efficient ones for a given i shows that the efficient safety efforts e are implemented for F i = j i d j, i.e. the damage for all other cities. A city should pay the overall damage of an accident to internalize the impact of its safety effort one society so the fine should be the damage for other cities. that 4
3. For legal reasons, the fine has to be the same F for all cities. Moreover, it should be lower than the overall damage: F < D. The regulator is considering replacing the fine by a minimal safety standard e applied to all cities or using both regulations. Suppose that costs are the same C i (e i ) = e 2 i /2 for all cities i = 1,..., n and that damages are ranked as follows d 1 < d 2 <... < d n. Discuss the performance of (i) a uniform safety standard e, (ii) a uniform fine F (iii) a mix of both regulations. Since costs are the same, the first-best is the same safety effort e = p (ne )D. So a uniform standard e = e can implement the first-best. With a fine F the first-order conditions find in 2. leads to e f i = p (E f )(d i +F ) with E f = n i=1 eu f. The first-best cannot be implement because agents with lower damage under-invests while those with higher damage overinvest. Mixing both instrument is not needed wince a uniform standard is enough to implement the first-best. 4. Suppose that the waste treatment facility is shared by n = 3 cities. The economy is unregulated (no liability or safety standard). The cities are identical in terms of costs and damages: C i (e i ) = C(e i ) and d i = d for i = 1, 2, 3. The cities are considering coordinating their safety efforts through a voluntary safety standard. Each city is free to adhere or not to this standard. Explain why the adoption of a common standard by all cities depends on a city s expectation about the behavior of the two other s cities concerning the adoption of their own standard. Show that under pessimistic expectations, all cities accept a common safety standard. If all cities agree on a standard, the standard e 3 minimizes the total expected costs 3C(e)+p(3e)D. If two cities agree on a standard, the standard e 2 minimizes the sum of their expected cost 2C(e) + p(2e + e )D given the safety effort of the other city e which minimizes its own cost C(e)+p(2e 2 +e)d. To be accepted, each city should be better-off if it implements the standard than if it does not. Under optimistic expectations, a city expects the others to agree on a standard e 2 if it does not adhere. So a country agrees if its expected cost is lower in the sense: C(e 3 ) + p(3e 3 )d C(e ) + p(2e 2 + e )d. Under pessimistic expectations, a city expects that the other will not agree on a standard so the status-quo is the equilibrium effort e e. So a country agrees if: C(e 3 ) + p(3e 3 )d C(e e ) + p(3e e )d. By definition of e 3, we have 3C(e 3 ) + p(3e 3 )D < 3C(e e ) + p(3e e )D which, divided by 3, implies C(e 3 ) + p(3e 3 )d < C(e e ) + p(3e e )d since D = 3d. 5