Consider instances when a pollution tax is equivalent to an output tax by examining the following:

(a) Suppose a firm’s cost function satisfies assumption 5.2. Show that in this case, an emissions tax is equivalent to a sales tax.

Determine the optimal sales tax.

(b) Suppose now that a firm’s cost function satisfies assumption 5.1.

Show that it is in general not possible to achieve a first-best allocation by charging a sales tax.

(c) Determine the comparative static effect from increasing the output tax. Derive signs for the change in firm’s output, emissions, and the market price.

(d) Determine the second-best optimal output tax when firms are symmetric.

Data from assumption 5.2

Data from assumption 5.1

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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=ax 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'(el 8) 8 de (5.5)