Please answer q5 (q1 is below for reference)

5. Fifth and finally, we return to the framework of question 2 (with a single firm providing both types of car) to consider Clean car standards instead of a Clean car discount. Assume that 0 = 0, 1 = 0. That is, a firm does not have to pay any taxes, but it must keep average CO2 emissions per kilometer travelled of the cars it sells down to some required level, x. Let CO2 per kilometre of the two types of car be fixed (per vehicle) at 120 for type 0 cars and 80 for type 1 cars. This means that the only way a firm can reduce its average is to sell a higher proportion of the more fuel efficient type. The regulation requires a firm to choose Q0, Q1 to satisfy the following constraint:

120Q0 + 80Q1/Q0 + Q1 x

although it might be more convenient to express it as:

120Q0 + 80Q1 (Q0 + Q1) x

Remember that there is a single firm who provides both types of car.

(a) Set up Lagrangian to represent the situation faced by the firm, assuming that it wishes to comply with the standard (i.e., treat the regulation as a constraint). Remember to set both tax rates to zero.

(b) Take first-order conditions.

(c) Is there some value of the standard x that would induce the firm to choose the values of Q0, Q1 proposed in question 1(d) (i.e., Q0 = Q1 = 60)? If so, what is it?

(d) What is the intuition for your answer to question 5(c)?

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.

Cost functions are

C0(Q0) = 50Q0, C1(Q1) = 20Q1 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 0 = 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: 1 = 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 0 and 1 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 simultaneously solve the two conditions, you should be able to solve them one-by-one. Remember to replace 20 with 0 in the expression for profits from type 0, and +20 with 1 in the profits for type 1.