Homework 2:: Homework 2

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1 Homework 2:: Hendrik Wolff Environmental Economics ECON436 Homework 2 Kolstad, Chapter 3 Answer Question 6, part a) and part b). Use Excel Kolstad, Chapter Consider the market for electricity. Suppose demand (in megawatt hours) is given by QQ = 50 PP and that the marginal private cost of generating electricity is $10 per megawatt hour (P is the same units). Suppose further that smoke is generated in the production of electricity in direct proportion to the amount of electricity generated. The health damage from the smoke is $15 per megawatt hour generated. a. Suppose the electricity is produced by competitive producers. What price will be charged, and how much electricity will be produced? The profit maximizing firms in the perfectly competitive market each set marginal revenue equal to marginal cost, where each of their marginal revenue curves are MMMM = PP. As represented by point A on the figure below, qq cccccccc = 4444 and pp cccccccc = 1111.

2 b. How would your answer in part a change if electricity were produced by an unregulated monopolist? The monopolist sets marginal revenue to marginal cost, where their marginal revenue is MMMM = Thus, as represented by point B in the figure below, qq mmmmmm = 2222 and pp mmmmmm = 3333.

3 c. In parts (a) and (b), what is the consumer surplus from the electricity generation? What is the net surplus, taking into account the pollution damage? i. The net surplus is 200. As presented in the figure below, the net surplus in part (a) of the question is the sum of consumer surplus (CS) and producer surplus (PS) minus the pollution damage (Dmg). CS = Area(AYZ) = = PS = Area(AY) = 0 Dmg = Area(AWXY) = = ii. The net surplus is 300. As presented in the figure below, the net surplus in part (b) of the question is the sum of consumer surplus and the monopolist producer surplus minus the pollution damage. CS = Area(BTZ) = = PS = Area(BTVY) = = Dmg = Area(UVWY) = =

4 Remark: In a market with complete property rights (without pollution externalities), typically a) In a competitive equilibrium there is no DWL (Dead weight loss), whereas b) the monopoly generates a DWL because the firm restricts the quantity and charges a higher price. In the above example, however, we do not have complete property rights and each unit of production of q produces marginal damage MD = 15. So what economy system is better, competitive or monopoly? Here in the above example the net surplus of the monopoly is HIGHER (300) compared to the competitive market (net surplus = 200)., so the monopoly is to be preferred. Is this always the case? No, it really depends on the amount of MD produced by each unit of q. If MD is small relative to the marginal cost, then the competitive market can be still be the preferred option. Make sure that you understand this by moving around supply, MD and demand functions and convince yourself that the above statement is correct.

5 2. In a photocopy of Figure 4.1, find and label the following points: a. A point D such that Z and Y are Pareto preferred to D but S is not. b. A point E such that the arc BZ is the portion of the Pareto frontier, which are Pareto improvements on E. Example:

6 4. Combine Figure 4.2 and 4.5 into a single figure showing how much garbage disposal and wine will be produced in an economy consisting of Ivorytower Land Services (ILS the producer) and Brewster, our consumer. Show the relative prices of wine and garbage disposal services in the figure. Assume Brewster s income is fixed. The amount of garbage disposal and wine is shown as qq GG and qq WW, respectively, on the figure below. These values are given by the point of tangency between Brewsters highest indifference curve and the production possibilities frontier. The slope at this point of tangency represents the relative price ration of the two goods P w/p G. 7. Suppose Humphrey and Matilda live together. Humphrey currently smokes 20 packs of cigarettes per month; Matilda hates the smoke. They currently have no agreement restricting smoking. Their only joint expense is monthly rent, which they split 50:50. Draw an Edgeworth box with two goods smoke and rental payments. Make up some reasonable indifference curves. Show the initial endowment. What Pareto efficient points might result from bargaining to restrict smoke? How does the graph show what price per pack Matilda might pay to buy down Humphrey s smoking (i.e., show the relative prices on your figure)? How would your answer change if the status quo is that the two have an agreement for no smoking and Humphrey would like to smoke as much as 20 packs per month? He must seek Matilda s permission to do so. (Hint: For Matilda, redefine Humphrey s smoking as smoke reduction.) The figure below depicts an Edgeworth Box with indifference curves for Humphrey and Matilda. First consider Humphrey: his utility depends on packs of cigarettes (a good) and rental payments (a bad). He therefore has upward sloping indifference curves of the type introduced in this chapter of the text. Their concave (downward) shape suggests diminishing marginal utility of smoking. Matilda's utility depends on her rental payments and Humphrey 's

7 smoking, both bads. As suggested by the hint in the problem, we can redefine the leftward direction on the horizontal axis for Matilda as smoke reduction (a good - for Matilda). Consider the interpretation of the upper-left and bottom-right corners of the box. Point C represents an arrangement where Humphrey pays 100% of the rent and smokes zero packs of cigarettes. This is the point of highest utility in the box for Matilda, and the point of lowest utility in the box for Humphrey. Point D is the opposite: Matilda pays all the rent and Humphrey smokes 20 packs of cigarettes. This is the feasible point of highest utility for Humphrey and lowest utility for Matilda. The two endowment points considered in the problem are labeled A and B in the figure below. At point A, rent is split 50:50 and Humphrey smokes 20 packs per month. Indifference AA AA curves that pass through A are labeled UU MM and UU HH for Matilda and Humphrey, respectively. These indifference curves are not tangent to one another, suggesting that bargaining over rent and Humphrey's smoking can lead to a Pareto improvement. More specifically, consider the lens-shaped area to the south-west of A. All points inside the lens are preferred to A by both agents; it appears Matilda will be able to "buy down" Humphrey's smoking in an arrangement that makes both of them better off.

8 Bargaining would result in an allocation like point E. At E, their indifference curves are tangent, and no further mutually beneficial trades are possible. At E, the prevailing price of smoke reduction is reflected in the slope of the dashed line connecting points E and A. This is the rate at which Matilda must buy down her roommate's smoking. Point B in the second figure represents the initial allocation where the rent is split 50:50, but a no-smoking arrangement is in place. Here the lens-shaped area of preferred allocations lies to the north-east, suggesting that bargaining will involve Humphrey paying Matilda for the right to smoke. An equilibrium arrangement would be supported by a point like F in the figure. Humphrey could pay his roommate a price per pack given by the slope of the dashed line that connects points Band F. Notice that while the first arrangement would be strongly preferred by Humphrey, and the second strongly preferred by Matilda, both lead through bargaining to allocations that are Pareto optimal. This is the important result regarding the assignment of property rights to which the text returns in Chapter 13.

9 Kolstad, Chapter Consider an air basin with only two consumers, Huck and Matilda. Suppose Huck's demand for air quality is given by qq HH = 1 pp where pp is Huck's marginal willingness to pay for air quality. similarly, Matilda's demand is given by qq MM = 2 2pp. Air quality can be supplied according to qq = pp where pp is the marginal cost of supply. a. Graph the aggregate demand for air quality along with individual demands. Huck and Matilda's demand curves (DD HH, DD MM ) are presented in the figure below along with the aggregate demand (DD AAAAAAAA ). We can characterize the inverse aggregate demand function as PP AAAAAAAA = 22 QQ AAAAAAAA, QQ AAAAAAAA, QQ AAAAAAAA [00, 11) QQ AAAAAAAA [11, 22] b. What is the efficient amount of air quality? The efficient quantity of air quality is 0.8 and is found where Supply = Aggregate Demand as seen in the figure below. In this case, QQ = QQ 22 AAAAAAAA.

10 4. Consider an airport that produces noise that decays as the distance (d), in kilometers, from the airport increases: NN(dd) = 1 dd2. Fritz works at the airport. Fritz's damage from noise is $1 per unit of noise and is associated with where Fritz lives. His costs of commuting are $1 per kilometer (each way). The closest he can live to the airport is d = 0.1 km. a. Write an expression for Fritz's total costs (noise and transportation). We are given that "noise costs" are NN = 11 dd22 and transportation costs are TT = 2222 ($1 per kilometer, each way). Daily total costs are therefore given by Daily TTTT = TT + NN = dd22 b. What is the distance Fritz will live from the airport in the absence of compensation for the noise? What are his total costs? Where would Fritz live? He will choose dd to minimize total costs from noise and transportation. Differentiating TTTT with respect to dd and setting this expression equal to zero, the condition for minimization of the function in part (a) is = 22 dd = 00 dd33 = 11 dd = 11km Daily total costs are minimized where dd = 11 km and Fritz's total cost is 3. c. Suppose Fritz is compensated for his damage, wherever he may live. How close to the airport will he choose to live? How much will he be compensated? (Hint: Solve graphically or using calculus.) Fritz is now told he will be compensated for any noise damage he suffers. Total costs are now given by

11 TTTT = dd = 2222 dd22 The problem is therefore reduced to one of minimizing travel costs. Fritz would move as close as possible to the airport, which in this case is dd = km. Compensation, in turn, is maximized at Compensation = NN = 11 dd 22 = 11 = $ (00. 11) 22 This outcome demonstrates what is known as "moving to the nuisance," and represents a strong argument against compensation based on damages for victims of externalities. 5. Two types of consumers (workers and retirees) share a community with a polluting cheese factory. The pollution is nonrival and nonexcludable. The total damage to workers is pp 2 where pp is the amount of pollution and the total damage to retirees is 3pp 2. Thus marginal damage to workers is 2pp and marginal damage to retirees is 6pp. According to an analysis by consulting engineers, the cheese factory saves 20pp pp 2 by polluting pp, for a marginal savings of 20 2pp. a. Find the aggregate (including both types of consumers) marginal damage for the public bad. We have the individual marginal damage functions for the two types of residents: MMDD WW = 2222 MMDD RR = 6666 Because pollution is a nonrival bad, the aggregate marginal damage function is the vertical sum of these individual damage functions: MMDD TT = = 8888 b. Graph the marginal savings and aggregate marginal damage curves with pollution on the horizontal axis. The graph of marginal savings and aggregate marginal damage is below.

12 c. How much will the cheese factory pollute in the absence of any regulation or bargaining? What is this society's optimal level of pollution? In the absence of any regulation, the firm would pollute as long as the marginal savings from pollution are positive. That is, the uncontrolled level of pollution, pp 00, is found where marginal savings equals zero: = 00 pp 00 = 1111 The socially optimal level of pollution, pp, is defined by the level at which marginal savings from pollution are equal to the marginal damages from pollution: = 8888 pp = 22 d. Starting from the uncontrolled level of pollution calculated in part (c), find the marginal willingness to pay for pollution abatement, A, for each consumer class. (Abatement is reduction is pollution; zero abatement would be associated with the uncontrolled level of pollution.) Find the aggregate marginal willingness to pay for abatement. The uncontrolled level of pollution has been identified as pp 00 = The question asks to turn the problem around and find marginal willingness to pay (demand) for pollution abatement. Let abatement equal A, so that pp + AA = At the uncontrolled level, marginal damages to workers are 20. Marginal willingness-to-pay for pollution reduction at pp 00 is therefore 20. Marginal damages fall by 2 per unit as pollution is decreased (abatement is increased), and are zero where = 00 (AA = 1111). To transform

13 the marginal damage (from pollution) function to a marginal willingness-to-pay (for abatement) function, substitute for pollution using pp = 1111 AA MMMMMMPP WW (AA) = 22(1111 AA) = MMMMMMPP RR (AA) = 66(1111 AA) = MMMMMMPP TT (AA) = 88(1111 AA) = The marginal willingness to pay (aggregate) can also be found by the vertical summation of the Workers' and Retirees' marginal willingness to pay: MMMMMMPP TT (AA) = ( ) + ( ) = e. Again starting from the uncontrolled level of pollution, what is the firm's marginal cost of pollution abatement? What is the optimal level of A? Since we know pp = 1111 AA we can also substitute this identity into the Marginal Savings function to get the Marginal Cost of Abatement for the firm: MMMM(AA) = (1111 AA) = 2222 The socially optimal level of abatement is 8 and is defined as the point at which the marginal cost of abatement is equal to the aggregate marginal willingness to pay: MMMM(AA) = 2222 = = MMMMMMPP TT (AA) AA = 88 f. Are the problems of optimal provision of the public bad (pollution) and the public good (abatement) equivalent? Explain why or why not. The answers to parts (c) and (e) are the equivalent. The problems of optimal provision of public bads (pollution in part (c)) and public goods (pollution abatement in part (e)) are logically the same. Problems of this type are cast in one way or the other for convenience, but the underlying objective in choosing A or p is the same: to maximize social welfare.