Consider J local monopolists with cost functions C^j(x_j,e_j) and facing inverse demand functions P_j(x_j). The pollutant is homogeneous and the total supply of tradable permits is L, with each firm holding a permit endowment of ē_j.

(a) Show that a competitive permit market leads to an efficient allocation of permits among the monopolists.

(b) Assume now there are only J=2 (J=3) such firms. Firms can trade bilaterally with prices being negotiated. Show that firms can do no better than exchanging the permits in a way that produces an efficient allocation.