Solution to exercise - 795 Externalities in Competitive Markets 21.10 PolicyApplication Pollution that increases firm costs 7 Barney’s Solution

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Unformatted text preview: 795 Externalities in Competitive Markets 21.10 PolicyApplication: Pollution that increases firm costs 7 Barney’s Solution: Consider the same situation as the one described in exercise 21.9. A: Assume again that the only impact of pollution is that it increases firm fixed costs by 6 for every unit ofx that is produced in the industry. (a) Suppose there are N firms in the equilibrium you described in exercise 21.9. What is the pol- lution related cost offirm i producingone more unit ofx? Answer: The pollution related cost of one more unit of output is then 6 N — because the fixed cost of every one of the N firms is increased by 6. (b) How much ofthis pollution related cost does firm i not take into account? Iffirm i is one of a large number of firms, is it a good approximation to say that firm i does not take any ofthe pollution related cost into account? How is this similar to our “price-taking” assumption fi2r competitive firms? Answer: Of the marginal pollution related cost 6N, the firm only takes into account the impact on its own firm — not the (N— 1) other firms. Thus, the firm does not take into account 6 (N — 1) of the 6 N pollution related costs it causes by producing one more output unit. As N gets large, this implies that the percentage of the pollution related costs it causes approaches zero. It is therefore reasonable to simply assume the competitive firm does not take any of the pollution related costs it causes into account — and it simply takes the level of pollution as given. This is similar to our price-taking assumption in that both appeal to the fact that each firm is so small that it makes no sense for it to behave strategically in its price setting or its pollution production. (c) Suppose that our benevolentsocial planner Barney can tell firms what to count as costs. Il- lustrate how Barney’s suggestion foreach firm’s marginal costcurve is related to the marginal costcurve firms would otherwise use (given a fixed number N of firms in the industry)? Answer: This is illustrated in panel (a) of Graph 21.10 — the SM C that Barney would want the firm to use includes the (SN marginal cost of pollution that results from marginal in- creases in output. (M smc m gm Graph 21.10: Private and Social Marginal Costs on the Lake (d) What does your answer imply about the relationship between the firm’s AC curve and Barney’s suggestion for what the firm’s AC curve should be? Answer: This is illustrated in panel (b) of Graph 21.10. The additional marginal cost that Barney would like to include is like a per-unit tax that shifts up the AC curve in a parallel way — leaving the lowest point unchanged. Externalities in Competitive Markets 796 (e) True or False: 1 f firms used Barney’s suggested cost curves, the long run industry supply curve would be upward sloping as you should have concluded in exercise 21.9 it is in the absence of Barney— but now it would lie above where it was in exercise 21.9. Answer: As the industry expands, the AC curves for firms will still be shifting up because of the increased pollution costs — but the “SAC” in panel (b) of the graph will always lie above the private AC curve used by firms. Thus, if Barney caused firms to consider the full cost of their production choices, the lowest point of the AC curve for any size of the industry would be higher — implying that the long run industry supply curve would be higher (and still upward sloping). (f) True or False: Under the efficient outcome, the industry would produce less ata higher price. Answer: This is true — the efficient outcome would have firms take into account the full cost of pollution — which would lead to an increase in cost, an increase in price and a drop in output (assuming downward sloping demand). (g) I f a single corporation acquired all the firms around the lake, would that corporation take the costs of pollution into account more like Barney or more like the individual competitive firms? (In exercise 23.1 1, yo u’ll be asked to revisit this in the context ofsuch a monopoly.) Answer: If a single corporation owned all the firms around the lake, then it would consider the full cost of the pollution since the only impact of the pollution is on the costs of the firms around the lake. Thus, the corporation would take the pollution costs into consideration like Barney. If the corporation still behaved as a price taker, that would fully resolve the externality problem. (But if the corporation establishes a monopoly as it acquires all the firms around the lake, it will cause efficiency problems deriving from the existence of market power, as we will see in Chapter 23). B: Consider the same set-up as in part B of exercise 21.9. In the previous case where we derived the marketequilibrium, we said thati in a model with many firms 7 it was reasonable to model each individual firm as not taking its own impact of pollution into account and to simply model the cost function as c(x) = fixz + (SN? (where the latter entered as a fixed cost). (a) Now consider the cost function that benevolent Barney would use for each firm: From the social planner’s perspective, the firm’s variable costs (captured by ) would still matter; as would the fixed cost from pollution (captured by 5N7 where? is the amount produced by each firm and N is the number of firms in the industry.) B utBarney also cares about thefi2llowing: each unit of x produced by firm i causes an increase in costs of6 for each of the N firms 7 which implies that the pollution cost Barney would consider firm i as imposing on society is 6Nx. This implies that Barney’s costfunctionfor each firm is c3 (x) = + (mm 6Nx. Derive from this the marginal and average cost functions that Barney would use for each firm (beingsure to not treat the last term as a fixed cost.) Answer: We now get 603m =2fix+6N and AC(x) = 030‘) = fix+6N+ E. (21.80) (ix x x (b) Repeat parts (c) through (i) from exercise 21.9 using the cost functions Barney would use for eachfirm to arrive atN*, p* and X*. Answer: Setting MC equal to AC, we get the quantity at the lowest point of the AC curve — i.e. x = (5W/fi)llz. Setting )6 equal to x (since every firm produces the same in equi- librium), we also again get x(N) = (SN/fl. Aggregating across N firms, we still get X (N) = 6N2 lfi — and the inverse N (X) = (fiX /6)1/2. Thus far, nothing has changed from the anal- ysis in the previous problem. But when we calculate p(N) — the zero profit price as a func- tion of how many firms are in the market — we now get a different answer. We derive this by substituting x(N) into either the MC or the AC curves to get MC(x) = p(N) = 36N (21.81) rather than 26 N as in the previous problem. This should make sense: Since firms now take into account their pollution costs for the industry, the zero profit price has to be larger than it was before. Substituting N (X) into p(N), we then get the long run industry supply curve 797 Externalities in Competitive Markets p(X) =3(fi§X)1l2 (21.82) as opposed to 2(fit5X)“2 as in the previous problem. Thus, the industry supply curve is shifted up — again because producers now are taking into account the social cost of the pollution they are producing. Setting demand equal to supply, solving for X and then com- pleting the remaining steps, we end up with 1/2 1I4 YEW, p* =(3A)1’2(fi6)“4, and N*=(§) . (21.83) (c) Compare your answers to those from exercise 21.9. How do they differ? Answer: Comparing these to the solutions from the previous problem, we see that industry output X * and the numb er of firms N * fall while the price p* increases. Put differently, were firms to take into account the externality costs they impose on other firms, there would be fewer of them producing less and selling at a higher price. (d) Suppose, as in part (i) of exercise 21.9 thatfi = 1, 6 = 0.1 and A = 10,580. What are X *, p* and N * ? How much does each individual firm produce? Answer: Plugging these values into our equations above, we get p" = 56.34, X" = 35,267 and N" = 1,878 (21.84) with each firm producing approximately 18.78 units of output. (e) Compare these to your answers in exercise 21.9. Can you give an intuitive explanation for why these answers differ despite the fact that pollution only affects the firms in the indusry? Answer: As already suggested by our formulas above, price has increased while (firm and in- dustry) output as well as the number of firms have fallen. The fact that pollution only affects the costs of the firms in the industry therefore does not mean that the industrywill internal- ize the pollution costs — because no single firm has an incentive to take into account the cost of the pollution it produces. In Chapter 27 we will call this the free-rider problem. (D What is the Pigouvian tax that is required in order fi2r competitive firms to implement the equilibrium you just calculated in (d)? What price does this imply consumers would pay and what price does it imply producers would receive? Answer: The tax would have to cause the MC as perceived by the firms to rise to the MC as desired by our benevolent Barney. This implies a per-unit tax of 6 N — which is simply the marginal social cost of the pollution from a firm increasing output by one unit. But the important part is to get N right — because Pigouvian taxes are determined at the optimum, not at the equilibrium (in the absence of taxes). We just calculated that, at the optimum, there would be 1,878 firms. Thus, the necessaryPigouvian tax is t = 0.01(1,878) = 18.78. This implies a consumer price of pd = 56.34 and a supplier price of p; = 56.34 — 18.78 = 37.56. (g) Verijji that your Pigouvian tax in fact results in prices for consumers and the industry that lead them to demand and supply the output level you calculated in part (d). (Note: You will need to refer back to your answers to exercise 21.9 to do this part.) Answer: We calculated in exercise 21.9 that the industry supply curve (in the absence of Barney's intervention) is p = 2(fi6x)1/2. Substituting [3 = 1 and 6 = 0.01, this means an industry supply curve for our example of p = 0.02x1l2 — or an industry supply function of x; = 25p2. The market demand curve is p = A/xllz. Substituting A = 10,580 and inverting to get the demand function, we get xd = (10580/ p)2. Substituting the supplier price of 37.56 into the supply function, we get x3 = 35,269, while substituting the consumer price of 56.34 into the demand function gives us xd = 35,265. Thus, aside from some rounding error, we get that the quantity demanded is equal to the quantity supplies — which in turn is equal to the industry supply X * we calculated in (b) for the cost curves recommended by Barney. ...
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This note was uploaded on 11/17/2010 for the course ECON 100A taught by Professor Woroch during the Fall '08 term at University of California, Berkeley.

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