Consider a firm whose technology satisfies assumption 5.2. The cost function is c(x)=c·x²/2. The output price is fixed and denoted by:

p. Emissions are proportional to output according to e=α·x. The firm is subject to an emission tax τ.

(a) Calculate the optimal output and determine the firm’s profit function.

(b) Determine the firm’s marginal abatement cost function (as a function of emissions) when output is the only option to reduce emissions.

Data from assumption 5.2

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Assumption 5.2 The cost function C(x) is twice continuously differentiable satisfying C')>0 and C()0, and emissions depend on x directly according to e=(x), where (x)>0. For some discussions we will use a simplification of the emissions technology in which 8(x)=8-x, for >0. Note that under assumption 5.2 the marginal abatement cost function is tied to the firm's profit per unit of output. If p is the output price of x the competitive firm's profit is -px-C(x), and adopting the simplification that e=&x allows us to more readily express profit as a function of emissions, so that -p(e/8)-C(e/6). Differentiating with respect to e we obtain dIIp-C'(el8) 8 de (5.5)