Pollution that increases firm costs - 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 δ for every unit of x that is produced in the industry.
(a) Suppose there are N firms in the equilibrium you described in exercise 21.9. What is the pollution related cost of firm i producing one more unit of x?
(b) How much of this pollution related cost does firm i not take into account? If firm i is one of a large number of firms, is it a good approximation to say that firm i does not take any of the pollution related cost into account? How is this similar to our "price-taking" assumption for competitive firms?
(c) Suppose that our benevolent social planner Barney can tell firms what to count as costs. Illustrate how Barney's suggestion for each firm's marginal cost curve is related to the marginal cost curve firms would otherwise use (given a fixed number N of firms in the industry)?
(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?
(e) True or False: If 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.
(f) True or False: Under the efficient outcome, the industry would produce less at a higher price.
(g) If 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 23.11, you'll be asked to revisit this in the context of such monopoly.)
B: Consider the same set-up as in part B of exercise 21.9. In the previous case where we derived the market equilibrium, we said that-in a model with many firms-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) = βx2 +δN (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 βx2) would still matter, as would the fixed cost from pollution (captured by δN where  is the amount produced by each firm and N is the number of firms in the industry.) But Barney also cares about the following: each unit of x produced by firm i causes an increase in costs of δ for each of the N firms - which implies that the pollution cost Barney would consider firm i as imposing on society is δNx. This implies that Barney's cost function for each firm is cB (x) = βx2 +δNx +δNx. Derive from this the marginal and average cost functions that Barney would use for each firm (being sure to not treat the last term as a fixed cost.)
(b) Repeat parts (c) through (i) from exercise 21.9 using the cost functions Barney would use for each firm to arrive at N∗, p∗ and X ∗.
(c) Compare your answers to those from exercise 21.9. How do they differ?
(d) Suppose, as in part (j) of exercise 21.9 that β = 1, δ = 0.1 and A = 10,580. What are X ∗, p∗ and N∗? How much does each individual firm produce?
(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 industry?
(f) What is the Pigouvian tax that is required in order for 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?
(g) Verify 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.)