I here are two types of car, distinguished by how fuel emcient they are. Type U is the less fuel efficient type, and type 1 is the more fuel emiclent. The inverse demand curves for the two types of car are: Po = 250 - Q0 - Q/2, P = 120 - Q1-Q0/2. (1) Cost functions are C (Q0) = 50Q0, G(Q) = 200, (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 To = 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: 71 = -20. The profits of the monopolist for type cars are (250 - 50 - 20 - Qo - Q./2,Q. 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. (1) What would and 7, have to be set to, for the equilibrium quantitites to be Qo = 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 To in the expression for profits from type 0, and +20 urth - T1 in the profits for type 1. 2. For the second setting, imagine that there is only a single monopolist that sells both types of car. That is, the same firm chooses both Qo and Q1 (a) Write down the monopolist's profits. It should contain separate terms relating to the two types of car. For questions 2-4, leave general terms To and 7) in your expression as in question 1(d), rather than imposing 70 = 20,51 = -20 as in questions 1(a)-1(c). (b) Take first-order conditions for the single monopolist (e) What would the tax rates To, have to be equal to, in order for the equilibrium quantities to be Qi = Q1 = 60 as in question 1(d)? (a) Compare these tax rates with the rates that were assumed in questions 1(a) (C). What is the intuition? 3. In our third setup, there are four firms. Two firms who are identical with each other produce type O cars. Two other firms that are identical with each other (but not with the first two) produce type 1 cars. Each firm maximises profits given the output levels of the other three firms. The originally specified demand functions, (1), still apply to each type of car, and the originally specified cost functions, (2), still apply to individual firms. For example, if firms A and B produce type 0 cars, and firms C and D provide type 1, then Qo = A + Os and Q = 0c + Op and the cost functions are: CA(94) = 5094, C8(98) = 509, Cc9c) = 209c, Cp (90) = 2090- (a) Write down a profit expression for a representative firm providing type O cars, and the profit expression for a representative firm providing type 1. (b) Take first-order conditions (c) What would the tax rates Tot have to equal, in order for the equi- librium quantities Q0, Q. to be the same as the values you found in question 1(d)? Feel free to assume that two identical firms producing a type of car, will provide the same amount as each other. (d) How does your answer to 3(C) compare with your answer to 1(a)? What is the intuition for this? 4. In the fourth setup there are only two firms, but both of them provide both types of car. You might think of this as the two firms competing against cach other in two markets. Each firm chooses two quantities to provide, given the two quantities chosen by the other firm. The two firms are identical. Demand functions for the total quantity of type 0 cars, and for the total quantity of type 1 cars are unchanged. The cost functions for producing the two types of car are also unchanged. (a) Write down an expression for a representative firm (You might need to introduce some new notation). (b) Take two first-order conditions for that firm. (c) What would the tax rates To have to equal, in order for the equi- librium quantities Q- Q: to be the same as the values you found in question 1(d)? (d) What is the intuition for your answer to question 4(c)? 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 To -0,7 -0. That is, a firm does not have to pay any taxes, but it must keep average CO, emissions per kilometre travelled of the cars it sells down to some required level, i. Let CO, 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 QoQ. to satisfy the following constraint 120Q1 + 80Q Q0 + although it might be more convenient to express it as: 120Q. + 80Q. (Qo + Qu). 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 that would induce the firm to choose the values of Q0, Q. 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)