Clean Car Discount / Clean Car Standard

    There are two types of car, distinguished by how fuel efficient they are. Type 0 is the less fuel efficient type, and type 1 is the more fuel efficient. The inverse demand curves for the two types of car are: P0 = 250 − Q0 − Q1/2, P1 = 120 − Q1 − Q0/2. (1) Cost functions are C0(Q0) = 50Q0, C1(Q1) = 20Q1 (2) respectively. 1. Until question 5, we consider a “feebate” or “Clean Car Discount”. That generally means there would be a subsidy on the purchase of some cars, and a tax on others, but in the following analysis it will be possible to have taxes on both or subsidies on both. In the current question, assume that there are two monopolies, one for type 0 cars and one for type 1 cars. Mathematically, this is equivalent to a Cournot duopoly with differentiated goods. (a) Let type 0 cars be taxed at τ0 = 20 per car sold, and type 1 cars be subsidised at 20 per car. To keep the notation consistent between the two types, this subsidy will be represented as a negative tax: τ1 = −20. The profits of the monopolist for type 0 cars are (250 − 50 − 20 − Q0 − Q1/2)Q0. Write down an expression for profits of the monopolist selling type 1 cars. (b) Take first-order conditions for the two monopolists. (c) Simultaneously solve your first-order conditions to find the equilibrium quantities sold of the two types of car. (d) What would τ0 and τ1 have to be set to, for the equilibrium quantitites to be Q0 = 60, Q1 = 60? Note that while this is a bit different conceptually from what you have done before, it is simpler mathematically. Instead of having to simultantaneously solve the two conditions, you should be able to solve them one-by-one. Remember to replace −20 with −τ0 in the expression for profits from type 0, and +20 with −τ1 in the profits for type 1.