Consider the energy sector consisting of J firms where energy producers are characterized by their cost functions C^j(x_j,e_j). The firms are subject to an emission trading system with a total emission cap E̅.

(a) Assume that an (additional) energy tax on output is introduced. Show that individual and total energy output goes down while total emissions remain constant.

(b) Assume now there are K other firms emitting CO₂. Those firms are characterized by their abatement cost functions C^k (e_k). All J+K firms are subject to the same emission trading system with a total emission cap E̅, but the energy sector is subject to an additional output tax. Investigate the comparative static effects of an energy tax increase on emissions and output. Show that emissions move from the energy to the other sector.