2. Keegan and Cindy share an apartment and need to decide how much to run the air conditioner this summer. For them, air conditioning is a public good: the whole apartment must reach a single temperature and the marginal cost of cooling the apartment is given by MC = 2Q+10 where Q is the total amount of air conditioning. Keegan's marginal benefit from cooling the apartment is MBk = 1502Q and Cindy's is MB. = 20 20. Because this is summer, assume that Q> 0 and that no one's marginal benefits of air conditioning can fall below zero (for example, this means that Q = 11 = MB. = 0). (a) (2) Plot Keegan's and Cindy's marginal benefit curves and the marginal cost curve in a well-labelled graph. Be sure to solve for and label the points of intersection between each marginal benefit curve and marginal cost. (b) (3) Derive an equation for the total marginal benefit curve and add it to your graph Hint: the marginal benefit curve will be piecewise linear and your equation should include two cases that depend on the value of Q. c) (2) Find the socially optimal quantity of air conditioning. (d) (2) If Keegan and Cindy's don't pay any costs of cooling the apartment, what is their combined total benefit if they choose the socially optimal quantity? (e) (2) If Keegan and Cindy's landlord pays all costs associated with cooling the apartment, do you expect them choose a quantity above, below, or equal the socially optimal quantity? Justify your answer. Imagine now that Keegan and Cindy pay their cooling bill by setting up an air con- ditioning fund. At the beginning of the month, they each state the amount of cooling they will contribute (9k and q) and pay the total cost of this contribution. Assume Keegan and Cindy contribute to the fund strategically and choose a contribu- tion that sets their personal marginal benefit equal to marginal cost. They both know each other's strategies and account for the fact that actual cooling will be Q =qk+93. (f) (2) Derive each of Keegan's and Cindy's best responses to the other's choice (i.e. BR is the optimal qr for any given quantity 4s) (g) (2) Is there a free-rider under-provision problem in this case? Justify your answer by describing what you expect to be the Nash equilibrium of this game. Hint: Find Keegan's best response to provision from Cindy and check what Cindy's best response to that would be. It might also be informative to care- fully examine your graph from part (a). 2. Keegan and Cindy share an apartment and need to decide how much to run the air conditioner this summer. For them, air conditioning is a public good: the whole apartment must reach a single temperature and the marginal cost of cooling the apartment is given by MC = 2Q+10 where Q is the total amount of air conditioning. Keegan's marginal benefit from cooling the apartment is MBk = 1502Q and Cindy's is MB. = 20 20. Because this is summer, assume that Q> 0 and that no one's marginal benefits of air conditioning can fall below zero (for example, this means that Q = 11 = MB. = 0). (a) (2) Plot Keegan's and Cindy's marginal benefit curves and the marginal cost curve in a well-labelled graph. Be sure to solve for and label the points of intersection between each marginal benefit curve and marginal cost. (b) (3) Derive an equation for the total marginal benefit curve and add it to your graph Hint: the marginal benefit curve will be piecewise linear and your equation should include two cases that depend on the value of Q. c) (2) Find the socially optimal quantity of air conditioning. (d) (2) If Keegan and Cindy's don't pay any costs of cooling the apartment, what is their combined total benefit if they choose the socially optimal quantity? (e) (2) If Keegan and Cindy's landlord pays all costs associated with cooling the apartment, do you expect them choose a quantity above, below, or equal the socially optimal quantity? Justify your answer. Imagine now that Keegan and Cindy pay their cooling bill by setting up an air con- ditioning fund. At the beginning of the month, they each state the amount of cooling they will contribute (9k and q) and pay the total cost of this contribution. Assume Keegan and Cindy contribute to the fund strategically and choose a contribu- tion that sets their personal marginal benefit equal to marginal cost. They both know each other's strategies and account for the fact that actual cooling will be Q =qk+93. (f) (2) Derive each of Keegan's and Cindy's best responses to the other's choice (i.e. BR is the optimal qr for any given quantity 4s) (g) (2) Is there a free-rider under-provision problem in this case? Justify your answer by describing what you expect to be the Nash equilibrium of this game. Hint: Find Keegan's best response to provision from Cindy and check what Cindy's best response to that would be. It might also be informative to care- fully examine your graph from part (a)
