Consider a firm that produces output at constant marginal cost c>0, and emissions are proportional to output so that e=αx, where x is output, and e is emissions. The output price is normalized to 1. Assume that the firm has a technology with emission coefficient α0 and that a new technology is available that leads to a lower emission coefficient α_I<α0. There is no adoption cost.

(a) Show that there is a switching tax rate where the regulator wants to adopt the new technology for low tax rates, and stays with the old technology for high tax rates.

(b) Assume there is a social damage function D(e)=d·e²/2.
Determine the socially optimal allocation as a function of the damage slope parameter d.

(c) How do the answers in

(a) and

(b) change if there is a positive adoption cost F>0?