Pollution that increases firm costs €” the Market Outcome: In the text, we assumed for convenience that the ill effects of pollution are felt by people other than producers and consumers. Consider instead the following case: An entire competitive industry is located around a single lake that contains some vital property needed for the production of x. Each unit of output x that is produced results in pollution that goes into the lake. The only effect of the pollution is that it introduces a chemical into the lake €” a chemical that requires firms to reinforce their pipes to keep them from corroding. The chemical is otherwise harmless to the population as well as to all wildlife in the area.
A. We have now constructed an example in which the only impact of pollution is on the firms that are creating the pollution. Suppose that each unit of x that is produced raises every firm€™s fixed cost by δ.
(a) Suppose all firms have identical decreasing returns to scale production processes, with the only fixed cost created by the pollution. For a given amount of industry production, what is the shape of an individual firm€™s average cost curve?
(b) In our discussion of long run competitive equilibria, we concluded in Chapter 14 that the long run industry supply curve is horizontal when all firms have identical cost curves. Can you recall the reason for this?
(c) Now consider this example here. Why is the long run industry supply curve now upward sloping despite the fact that all firms are identical?
(d) In side-by-side graphs of a firm€™s cost curves and the industry (long run) supply and demand curves, illustrate the firm and industry in long run equilibrium.
(e) Usually we can identify producer surplus€” or firm profit€” as an area in the demand and supply picture. What is producer surplus here? Why is your answer different from the usual?
(f) In chapter 14,we briefly mentioned the term decreasing cost industries€”industries in which the long run industry supply curve is downward sloping despite the fact that all firms might have identical production technologies. Suppose that in our example the pollution causes a

Graph 21.9: Increasing Cost Industry due to Pollution Externality
decrease rather than an increase in fixed costs for firms. Would such a positive externality be another way of giving rise to a decreasing cost industry?
B. Suppose that each firm€™s (long run) cost curve is given by c(x) = βx2 +δX where x is the firm€™s output level and X is the output level of the whole industry. Note that x is contained in X €” and thus we could write the cost function as c(x) = βx2 +δx +δ where is the output produced by all other firms. When each firm is small relative to the industry, however, the impact of a single firm€™s pollution output on its own production cost is negligible €” and it is a good approximation (that makes the problem a lot easier to solve) to simply write a single firm€™s cost curve as c(x) = βx2 +δX . Furthermore, if all firms are identical, it is reasonable to assume that all firms produce the same output level . Letting N denote the number of firms in the industry, we can therefore write X = N and re-write the cost function for an individual firm as c(x) = βx2 +δN.
(a) How is our treatment of a producer€™s contribution to her own costs similar to our €œprice taking€ assumption for competitive firms?
(b) Derive the marginal and average cost functions for a single firm (using the final version of our approximate cost function). (Be careful to realize that the second part of the cost function is, from the firm€™s perspective, simply a fixed cost.)
(c) Assuming the firm is in long run equilibrium, all firms will make zero profit. Use your answer to (b) to derive the output level produced by each firm as a function of δ, β, N and
(d) Since all firms are identical, in equilibrium the single firm we are analyzing will produce the same as each of the other firms€”i.e. x = . Use this to derive a single firm€™s output level x(N) as a function of δ, N, and β. What does this imply about the equilibrium price p(N) (as a function of δ and N) given that firms make zero profit in equilibrium?
(e) Since each firm produces x(N), multiply this by N to get the aggregate output level X (N) €” then invert it to get the number of firms N(X ) as a function of β, δ and X .
(f) Substitute N(X) into p(N) to get a function p(X ). Can you explain why this is the long run industry supply curve with free entry and exit?
(g) Suppose the aggregate demand for X is given by the demand curve pD(X)= A/(X 0.5). Set the industry supply curve equal to the demand curve to get the equilibrium market output X ˆ— (as a function of A, δ and β).3
(h) Use your answer to (g) to determine the equilibrium price level pˆ— (as a function of A, δ and β).
(i) Use your answer to (g) to determine the equilibrium number of firms Nˆ— (as a function of A, δ and β).
(j) Suppose that β = 1, δ = 0.01 and A = 10,580. What are X ˆ—, pˆ— and Nˆ—? How much does each individual firm produce? (Do exercise 21.10 to compare these to what is optimal.)

Transcribed Image Text:

Firm Indvstry MC LR LR