Consider a monopolist facing linear demand P(X)=1X and constant marginal costs c>0 (and/or increasing marginal cost C(X)

Consider a monopolist facing linear demand P(X)=1−X and constant marginal costs c>0 (and/or increasing marginal cost Cʹ(X) = cX). The social damage function is quadratic of the form D(E) = dE²/2. Emissions are proportional to output, where E = X.

(a) Determine the socially optimal level of production (emissions) and the optimal tax rate.

(b) For which damage parameters is the tax rate positive (negative)?

(c) For which damage parameters can the social optimum be achieved with an emission cap?

(d) Consider now a cost function C(X,E) = (aX - BE)²/2B and derive the second-best optimal tax.