MPC MPC. Under a Pigouvian subsidy, s, the firm s MPC curve shifts up to MPC = MPC + s. This is less
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1 Old Exam Solutions Question 1 (worth 17 points) Consider a firm producing a quantity Q of a product that generates a negative externality. The firm has a downward-sloping marginal benefit (MB) curve associated with output Q, and an upward-sloping marginal private cost (MPC) curve. The externality is represented by an upward-sloping marginal damage (MD) curve. These curves are illustrated in the diagram. 1a) Referring to the diagram below, please explain why a firm would not set Q = 0 under a Pigouvian subsidy. [Hint: To gain full marks, you will need to explain how the subsidy is set and how it affects the firm s choice, with the aid of the diagram.] (6 points) $ MSC MPC MPC MD MB 0 Q* Q Under a Pigouvian subsidy, s, the firm s MPC curve shifts up to MPC = MPC + s. This is less than MB evaluated at Q = 0, and also at Q slightly greater than zero. So the firm would have an incentive to increase output until MB = MPC. In doing so, it would forego the subsidy on each unit of output it produced above zero, but the greater MB would more-than compensate. 1
2 2
3 Now consider some more specific structure on the problem. Let the firm s marginal benefit schedule be given by MB ( Q) = 8 2 * Q, the firm s marginal private cost of production be MPC(Q) = 6 + (1/4)*Q, and the marginal damage function be given by MD Q = (3/ 4) *. ( ) Q 1b) At what level should the Pigouvian subsidy be set? Please show your calculations in full, making clear how the subsidy should be set. (5 points) The Pigouvian subsidy, s, should be set equal to the marginal damage at the efficient level of output. The efficient level of output, Q*, is found where MSC = MB. We know that MSC = MPC + MD. Here, MSC = Q Q = 6 + Q, and MB = 8 2Q. Thus Q* solves 6 + Q = 8 2Q, implying that 3Q = 2, so Q* = 2/3. Therefore, s = MD(Q*) = 0.75 (2/3) = c) Consider the claim that the total subsidy amount paid to the firm will be less than 0.4. Is this true? [Note: no marks for guessing.] Please explain precisely, showing any relevant calculations. (6 points) The subsidy is paid on each unit of output the firm cuts back on below its initial output choice Q 1. Thus the total subsidy paid = s (Q 1 Q*). Now Q 1 solves MB = MPC, or 8 2Q = Q. This implies Q 1 = 8/9. Thus (Q 1 Q*) = 8/9 6/9 = 2/9, and the total subsidy = 0.5 2/9 = 1/9. This is clearly less than 0.4. Thus the claim is true. Question 2 (worth 7 points) To improve environmental quality, a regulated firm is forced to adopt a quantity standard of e T (in the diagram below) as a means of reducing its pollution emissions below e 0. The firm is 3
4 deciding whether to adopt some new technology that yields the same marginal benefit as before but at a lower level of emissions. In the diagram below, the adoption of the new technology can be represented as a shift in the MB schedule from MB OLD to MB NEW. $ MB OLD MB NEW A e T e 0 e The firm faces an installation cost of $I. Suppose that its time horizon is three periods long and that there is no discounting. Please state the conditions under which the firm would adopt the new technology, making appropriate reference to the diagram. Be sure to explain your reasoning. Adopting the new technology allows the firm to save on abatement costs of A per period. Over the 3-period horizon, the firm saves 3A. The installation cost is $I. Thus the firm would adopt the new technology if I < 3A. Question 3 (worth12 points) 3a) Give an example of an industrial activity that generates positive externalities. (2 points) Research and development by firms is an example of an activity that generates spillover benefits (a type of positive externality) in the form of new knowledge. 3b) Explain clearly the sense in which a permit trading scheme is optimal. [One sentence.] (5 points) A permit trading scheme is optimal in the sense that it attains a given target emissions level at the least abatement cost to society. 4
5 3c) If the marginal benefits of emissions are not equalized across firms in a given permit allocation, is it possible that total abatement costs could be minimized? Please explain clearly. [More than one sentence.] (5 points) Equalization of marginal benefits is a necessary condition of minimizing total abatement costs. This is straightforward to show. Suppose the overall target abatement level has been attained, but let two firms have different marginal abatement costs. Then take the firm with the higher marginal abatement cost. Increasing its emissions by a small amount ε and reducing the other firm s emissions by the same amount ε will allow the same level of emissions to be attained, but now at lower total abatement costs (higher marginal abatement has been replaced by lower marginal abatement). Thus is marginal abatement costs are not equalized, it must be possible to lower total abatement costs hence the necessity of the condition in the first sentence. Question 4 (worth 16 points) The electricity generation industry produces pollution emissions. Suppose it has N firms with low marginal benefit of emissions (MB L ) schedules, and N firms with high marginal benefit of emissions (MB H ) schedules. To regulate emissions, a permit trading scheme operates in this industry. Throughout this question, take the supply of permits to be fixed. 4a) Conditions in other sectors affect entry into and exit from the industry. Consider a situation in which N/2 low-valuation firms exit the electricity generating industry. Qualitative speaking, what do you predict will happen to the equilibrium price? (Will it rise or fall?) Explain your answer. [Two sentences.] (4 points) The price will fall. This is because the supply of permits is fixed but exit causes the industry marginal benefit of emissions schedule to shift in. Before any firms exit the electricity generating industry, consider the operation of the permit trading market in more detail (note that there are N firms of each type). In particular, suppose that the marginal benefit of emissions schedule of the low -type firms is given by MB L = e and the marginal benefit of each high -type firm is given by MB H = 6 e, where e is the emissions level. Suppose also that the initial permit allocation is 4 permits per firm. 4b) Which type of firm will choose to buy more permits at the initial permit allocation? [One sentence.] (1 point) At the initial permit allocation of 4, the high-valuation firms will buy more permits. (In contrast, the low-valuation firms will wish to sell.) 5
6 4c) What emissions level for each type of firm will emerge in equilibrium? Please show your calculations. (8 points) In equilibrium, the total number of permits bought must equal the total sold. Given equal numbers of each type, we know that each high-valuation firm must buy what each low-valuation firm sells. Suppose each low-type firm sells α permits in equilibrium. Then the equilibrium emissions levels must be e L = 4 α and e H = 4 + α for the two types respectively. We also know that marginal benefits of emissions must be equalized across both types of firm. Thus MB L (e L ) = MB H (e H ). Now, substituting from above, MB L (e L ) = 3 1/2(4 α) and MB H (e H ) = 6 (4 + α). This implies that 3 1/2(4 α) = 6 (4 + α) so 1 + 1/2α = 2 α implying 3/2 α = 1. Therefore α = 2/3, implying that e L = 3 1/3 and e H = 4 2/3. 4d) What will the equilibrium permit price be? (3 points) Building on the reasoning in part c), P* = MB H (e H ) = 6 4 2/3 = 1 1/3. Question 5 (worth 12 points) The town of Carthage has two types of residential neighborhood: those in the suburbs and those downtown. In order to assess the benefits of environmental improvements, researchers wish to estimate household valuation of clean air using a hedonic price regression approach. 6
7 Air quality in location i is measured by Ai, which increases as the air gets cleaner, and air quality is cleaner on average in the suburbs than downtown. The researchers plan to estimate the following hedonic model, with data on house prices (Pi), house size (Si), neighborhood attributes (Ni), and air quality (Ai): Pi = a_0 + a_1 Si + a_2ni + a_3ai + v_i where v_i represents factors unobserved to the researcher. The true model is Pi = α_0 + α_1 Si + α_2ni + α_3ai + α_4xi + e_i, where some variable Xi is omitted from the model that is actually estimated. Suppose that Xi measures local traffic noise (with Xi being higher, the greater is the traffic). Suppose also that Xi is higher downtown. 5a) What relation would you expect in the true model between Xi and house prices, and why? [Two sentences.] (3 points) We would expect Xi and house prices to be negatively related (so α_4 < 0). As traffic noise increased, so demand for housing would tend to fall, other things equal, leading to lower prices. 5b) What relation would you expect between traffic noise and air quality, and why? Please explain. [Two sentences.] (2 points) We would expect traffic noise to rise with the volume of traffic, and more traffic to make air quality worse. Therefore, we would expect a negative relation. 5c) If the traffic noise measure is omitted from the estimated model, what impact will this have on the estimate of a_3, and why? (7 points) In the true model, we would expect there to be a positive relation between house prices and air quality. (That is, the slope of the line with Ai on the horizontal axis and Pi on the vertical axis would be positive (α_3 > 0). In the suburbs, Ai is higher, on average, and traffic noise is lower. Both factors lead to an increase in house prices. So we would expect the estimated coefficient to be biased up the estimated curve to be steeper than it should be because Ai picks up itself and an additional good thing. Question 6 (worth 16 points) Consider an industry that generates atmospheric pollution, consisting of two firms, one located in a rural area with a relatively flat marginal benefit of emissions schedule (MB L ), the second one located in a city, with a steeper marginal benefit schedule (MB H ). (See diagram.) Environmental damages differ across the two locations. In the rural area, they are represented by the relatively low marginal damage schedule, MD L. In the city, they are represented by the much higher schedule, MD H (again, see diagram). 7
8 MB H B A MD H MB L C MD L emission e * L = x et y * e H Note that y - e T = e T x. Note also that y should be located directly below the intersection of MD L and MB H (and so is not drawn completely accurately in the diagram above). An environmental regulator is trying to decide whether a uniform standard would be more or less efficient (in an overall sense) than a tradeable permit scheme. Under a uniform standard, both firms would produce at e T. Under the permit-trading scheme, suppose that the initial allocation of permits to each firm is given by e T. 6a) In the diagram, draw the permit trading equilibrium, showing clearly the equilibrium permit price, and the emissions levels of each firm. Also, provide a clear verbal explanation of your answer, giving the conditions needed for equilibrium. (4 points) Equilibrium is found where the total number of permits bought equals the total sold, and the marginal benefits of emissions are equalized across the two firms. This occurs where P = MB L, and e L = x and e L = y, and the amount bought equals the amount sold = e T x. 6b) Which of the two regulatory approaches would be more efficient in an overall sense? Please explain your answer precisely, making reference to the diagram. [Hint: make clear what the overall efficient outcome would be.] (8 points) 8
9 The overall efficient level of emissions for the low-type firm is found where e H = x, and for the high-type firm, where e H = e H *, found at the intersection of MB H and MD H. The inefficiency of the permit scheme is due entirely to the high-type firm, as the low-type firm is efficient. The inefficiency of the high-type firm = A + B. The inefficiency associated with the uniform standard is B for the high types and C for the low types. As drawn, A > C, so the uniform standard is more efficient. 6c) Suppose the population in the city rises, shifting up the MD H schedule there. Would this affect which regulatory instrument would be more efficient? Please explain. (4 points) As MD H rises, both areas A and B get bigger, making the permit even less efficient relative to the uniform standard. 9
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