Consider a firm whose cost function is given by c(x)=c·x²/2.
The output price is fixed and denoted by p. Emissions are proportional to output according to e=α₀·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.

(c) Assume now that the firm can adopt a new technology with emissions coefficient α_I<α₀. Determine the new marginal abatement cost function. Draw a diagram with the marginal abatement cost functions resulting from the old and from the new technology.

(d) Determine the cost savings from adopting the new technology.
How does this change with the tax rate? At which tax rate are the cost savings maximized?

(e) Assume now there is a continuum of firms represented by the unit interval. A firm is denoted by x∈ [0,1]. Adoption costs are identical and denoted by F (alternatively: adoption costs are firm specific and given by F(x)=x²/2). The social damage is given by D(E)=d·E²/2. Determine the socially optimal allocation as a function of the damage slope parameter.

d. What do you observe? In which case can we decentralize the social optimum by an emission tax (or by tradable permits)? Determine the optimal tax rate (the optimal amount of tradable permits).

(f) Proceed as in (e), assuming constant marginal damage, i.e. D(E)=d·E.