A: Suppose a firm produces x using a technology that emits pollution through smokestacks. The firm must ensure that it has sufficient pollution vouchers v to emit the level of pollution that escapes the smokestacks, but it can reduce the pollution by installing increasingly sophisticated smokestack filters s.
(a) Suppose that the technology for producing x requires capital and labor and, without considering pollution, has constant returns to scale. For a given set of input prices (w, r ), what does the marginal cost curve look like?
(b) Now suppose that relatively little pollution is emitted initially in the production process, but as the factory is used more intensively, pollution per unit of output increases — and thus more pollution vouchers have to be purchased per unit absent any pollution abating smokestack filters. What does this do to the marginal cost curve assuming some price pv per pollution voucher and assuming the firm does not install smokestack filters?
(c) Considering carefully the meaning of “economic cost”, does your answer to (b) depend on whether the government gives the firm a certain amount of vouchers or whether the firm starts out with no vouchers and has to purchase whatever quantity is necessary for its production plan?
(d) Suppose that smokestack filters are such that initial investments in filters yield high reductions in pollution, but as additional filters are added, the marginal reduction in pollution declines. You can now think of the firm as using two additional inputs — pollution vouchers and smokestack filters — to produce output x legally. Does the overall production technology now have increasing, constant or decreasing returns to scale?
(e) Next, consider a graph with “smokestack filters” s on the horizontal and “pollution vouchers” v on the vertical axis. Illustrate an isoquant that shows different ways of reaching a particular output level legally — i.e. without polluting illegally. Then illustrate the least cost way of reaching this output level (not counting the cost of labor and capital) given pv and ps.
(f) If the government imposes additional limits on pollution by removing some of the pollution vouchers from the market, pv will increase. How much will this affect the number of smokestack filters used in any given firm assuming output does not change? What does your answer depend on?
(g) What happens to the overall marginal cost curve for the firm (including all costs of production) as pv increases? Will output increase or decrease?
(h) Can you tell whether the firm will buy more or fewer smokestack filters as pv increases? Do you think it will produce more or less pollution?
(i) True or False: The Cap-and-Trade system reduces overall pollution by getting firms to use smokestack filters more intensively and by causing firms to reduce how much output they produce.
B: Suppose the cost function (not considering pollution) is given by C (w, r, x) = 0.5w 0.5 r 0.5 x, and suppose that the tradeoff between using smokestack filters s and pollution vouchers v to achieve legal production is given by the Cobb-Douglas production technology x = f (s, v) = 50s0.25 v0.25 .
(a) In the absence of cap-and-trade policies, does the production process have increasing, de- creasing or constant returns to scale?
(b) Ignoring for now the cost of capital and labor, derive the cost function for producing different output levels as a function of ps and pv — the price of a smokestack filter and a pollution voucher. (You can derive this directly or use the fact that we know the general form of cost functions for Cobb-Douglas production functions from what is given in problem 12.4).
(c) What is the full cost function C (w, r, ps , pv)? What is the marginal cost function?
(d) For a given output prince p, derive the supply function.
(e) Using Shephard’s lemma, can you derive the conditional smokestack filter demand function?
(f) Using your answers, can you derive the (unconditional) smokestack filter demand function?
(g) Use your answers to illustrate the effect of an increase in pv on the demand for smokestack filters holding output fixed as well as the effect of an increase in pv on the profit maximizing demand for smokestack filters.