3.) Modeling Carbon Tax and Cap and Trade Assume an electricity generator can produce electricity from two different production technologies, a dirty technology and a clean technology. The quantity of electricity produced from the dirty technology is Qd in megawatt hours (MWh) and the amount from the clean technology is Qc. The dirty technology produces carbon emissions at a constant rate of Bd tons per MWh of electricity generation. The clean technology produces carbon emissions at a constant rate of Bc tons per MWh, where Bc < Bd. The cost functions for the two generation technologies are C(Qd) and C(Qc). The market price of electricity is P. Under these assumptions the firms profit function is: B = PQd + PQc - C(Qd) - C(Qc).

a.) Take the partial derivative of profit with respect to each output (Qd and Qc) and set each equal to zero to find the profit maximizing (first-order) conditions for each type of electricity. Denote marginal costs of each type of electricity as MCd and MCc. Write down and interpret these first-order conditions. (Hint: Are these expressions consistent with the behavior of a profit-maximizing firm?)

b.) To fight climate change, policy makers implement a carbon tax where the firm is charged dollars per ton of carbon emissions released into the atmosphere. Under this policy the firms profit function is: B = PQd + PQc - C(Qd) - C(Qc) -B(BdQd + BcQc), where the last term is the amount of carbon taxes paid (hint: check the units on these terms). Write down the firms first order conditions for each type of electricity under the carbon tax policy. How do these expressions differ from the expressions in part 1 above?

c.) In instead of a carbon tax, imagine policymakers adopt a cap on carbon emissions equal to Qp. For simplicity, assume the firm above is the only generator in the market and produces all the regulated carbon emissions. In this case, the firms emissions must not exceed the regulated cap, specifically: dQd + cQc Qp. Here, the firm faces a two-part problem. First, to maximize profit, i.e. the firms usual problem. Second, to satisfy the carbon constraint dQd + cQc Qp. Mathematically, we can write this two-part problem as: maxQd,Qc = PQd + PQc - C(Qd) - C(Qc) subject to: dQd + cQc Qp. In words, the firm chooses how much to produce in order to maximize profits subject to meeting the constraint imposed by the carbon cap. We can solve this problem for the firms first order conditions just like we did in parts 1 and 2 above. However to do this we first need to use a mathematical trick. Define the Lagrangian of the firms constrained profit maximization problem as: L = PQd + PQc - C(Qd) - C(Qc) - PpQp (dQd + cQc Qp), where is the shadow price of the constraint. The shadow price here is the marginal effect on the firms profit of increasing or decreasing the carbon cap. Under a cap and trade system, the firm purchases Qp permits (to cover their emissions) at price Pp. Therefore, PpQp is the firms expenditure on emissions permits. Now, find the three first order conditions by taking partial derivatives with respect to Qd Qc and Qp and setting each equal to zero. In the first-order condition for Qp, solve for as a function of Pp (simple) and substitute into the remaining two first-order conditions. You should have two expressions analogous to the first order conditions in part 2 above. Interpret these two relationships.

d.) Discuss how the first-order conditions for Qd and Qc compare with emissions taxes discussed in class? Can these expressions attain the Pigovian (tax) ideal?