Assume electricity is generated by three technologies: a base load technology with cost function C_b(x_b) and proportional emission coefficient α_b; gas turbines with cost function C_g(x_g) and emission coefficient α_g< α_b; and an emission-free renewable energy source with cost function C_r(x_r). All cost functions are convex, and there is a total emission cap of L. The inverse demand function for electricity is given by P(X).

(a) Characterize the optimal allocation given the emission cap.

(b) Investigate the comparative static effects of lowering the emission cap on quantities, total output, price, and the percentage of renewable energy, denoted by β=x_r/(x_b+x_g+x_r).

(c) Assume now that in addition to the emission cap there is a quota for renewable energy. Show that a quota raises the total cost of electricity production.