I. A nn's production function is given by is q = KL 0.8K2 0.2!,2 where q is the quantity of output produced, I. is the amount of labor input used, and K is the amount of capital input used. Suppose the amount of capital is fixed at 10 units of capital in the short run. I. 2. Find this finn's short run (a) total product of labor function, (b) average product of labor function, and (c) marginal product of labor function. At what level of labor input usage does the marginal productivity of labor reach a maximum? How much output is produced at that point? Show and briey discuss how you arrived at your answer. Is the marginal product of labor ever negative for this production function? If so, when. If not why not. Show how you arrived at your answer. At what level of labor input usage does the average productivity of labor reach a maximum? How much output is pmduced at that point? Show and briey discuss how you arrived at your answer. At what level of labor input usage does the total productivity of labor reach a. maximum? How much output is produced at that point? Show and briey discuss how you arrived at your answer. What is the economic region of the short run production function you found in part I? Show and briey explain how you arrived at your answer. Calculate the output elasticity of labor when this rm uses the amount of labor at which its average product of labor is maximum. In a carefully labeled diagram, graphically illustrate this rm's total product of labor in the upper panel and the marginal product of labor and average product of labor in the lower panel using your results in parts 1 to 4. This question is based on the following three supply curves. where P is price and O is the quantity supplied. I P = 20 II P = Q III P = 1Q Which of the following statements is TRUE? A All three curves have a supply elasticity of 1 at all levels of output. B Only Option II has a supply elasticity of 1 at all levels of output. None of them ever has a supply elasticity of 1. D All three have a supply elasticity of 1 at only one output level.There are two types of car, distinguished by how fuel efficient they are. Type 2. For the second setting, imagine that there is only a single monopolist that 0 is the less fuel efficient type, and type 1 is the more fuel efficient. The inverse sells both types of car. That is, the same firm chooses both ( and Q1- demand curves for the two types of car are: (a) Write down the monopolist's profits. It should contain separate terms Po = 250 - Qo - Q1/2, A = 120 -Q1 - Qo/2. (1) relating to the two types of car. For questions 2 4, leave general terms o and 7) in your expression as in question 1(d), rather than imposing Cost functions are 78 = 20, n = -20 as in questions 1(a)-1(c). Co(Qo) - 50Qu, G1(Q1) = 20Q, (2) b) Take first-order conditions for the single monopolist. e) What would the tax rates To, 7) have to be equal to, in order for the respectively. equilibrium quantities to be Qo = Q1 = 60 as in question 1(d)? d) Compare these tax rates with the rates that were assumed in questions 1. Until question 5, we consider a "feebate" or "Clean Car Discount". That 1(a) (c). What is the intuition? 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 3. In our third setup, there are four firms. Two firms who are identical with each other produce type 0 cars. Two other firms that are identical with each other on both or subsidies on both. In the current question, assume that there are "but not with the first two) produce type 1 cars. Each firm maximises profits two monopolies, one for type ( cars and one for type 1 cars. Mathematically; given the output levels of the other three firms. The originally specified this is equivalent to a Cournot duopoly with differentiated goods 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 (A) Let type 0 cars be taxed at To = 20 per car sold, and type 1 cars be firms A and B produce type 0 cars, and firms C and D provide type 1, then subsidised at 20 per car. To keep the notation consistent between the 20 = 44 + 48 and Q1 = de + 90 and the cost functions are two types, this subsidy will be represented as a negative tax: 71 = -20. CA(qA) = 50gA, Ca(48) = 50qs; Ce(qc) = 20gc, Co(4D) = 20qp- The profits of the monopolist for type 0 cars are (250 - 50 -20 -Qo - Q1/2)20- (a) Write down a profit expression for a representative firm providing type Write down an expression for profits of the monopolist selling type 1 0 cars, and the profit expression for a representative firm providng type cars. b) Take first-order conditions (b) Take first-order conditions for the two monopolists. (c) What would the tax rates ro, ") have to equal, in order for the equi- (c) Simultaneously solve your first-order conditions to find the equilibrium librium quantities Qo, Q, to be the same as the values you found in quantities sold of the two types of car. 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) What would 7 and 7, have to be set to, for the equilibrium quantitites to d) How does your answer to 3(c) compare with your answer to 1(d)? What be Qo = 60, Q1 = 60? Note that while this is a bit different conceptually is the intuition for this? from what you have done before, it is simpler mathematically. Instead of having to simullanlaneously solve the two conditions, you should be 1. 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 each able to solve them one-by-one. Remember to replace -20 with -To in other in two markets. Each firm chooses two quantities to provide, given the the expression for profits from type 0, and +20 with -71 in the profits two quantities chosen by the other firm. The two firms are identical. Demand for type 1. 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, n - 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 Qo. Q, to satisfy the following constraint: 120Q0 + 8001