Question

3. Consider a perfectly competitive firm that is polluting. The firm produces output y which sells at a per unit price p. The cost of production is given by the cost function C(y,a), where a is pollution abatement. Assume Cy>0, Cyy>0, Ca>0, Caa>0, Cay>0. The pollution production function is given by e(y,a). Assume ey>0, eyy>0, ea<0, eaa>0, eay>0. Finally, the government taxes each unit of emission at a tax rate t. Hence, the profit of the firm is given by π = p y - c(y,a) - t e(y,a) Obviously, in this problem y and a are endogenous, whereas p and t are parameters (exogenous).

i) Write down the first-order conditions for this profit maximization problem. (There is no need to solve for anything.)

ii) Tell me in words what we need in order for the second-order condition to be satisfied. (Note: you do not need to prove how it may hold.)

iii) Assuming that the second-order condition for the problem holds, work out the comparative static results y/p and a/p (i.e., work out the impact of an increase in the price of the firm’s output on its optimal levels of output and abatement). [Note: I need both the expression and the sign for the comparative static results asked for in this question.]

Answer 1

Explanation:-

Please find attached the image which contains solution for both the parts (1) (2)  and (3);

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