Anna and Bob are the only residents of a small town. The town currently funds its fire department solely from the individual contributions of these two residents. Each of the two residents has a utility function over private goods x and total number of firemen M, of the form: u(x,M)=2 ln x+ln M?. The total provision of firemen hired, M, is the sum of the number hired by each of the two persons: M=M^A+M^B. Ann and Bob both have income of 200 each, and the price of both the private good and a fireman is 1. They are limited to providing between 0 and 200 firemen. For the purposes of this problem, you can treat the number of firemen as a continuous variable (it could be man-years). Suppose that the government recruits additional ? firemen and taxes Ann and Bob equally to cover the cost. Therefore, the total number of firemen is M^A+M^B+N, where M^A, M^B are appropriate individually-optimal contributions of A and B (i.e., the agents behave optimally, conditional on the policy), and each agent is taxed N/2. How many additional firemen should the government hire in order to guarantee that in equilibrium the total number of firemen is optimal?
Anna and Bob are the only residents of a small town. The town currently funds its fire department solely from the individual contributions of these two residents. Each of the two residents has a utility function over private goods x and total number of firemen M, of the form: u(x,M)=2 ln x+ln M?. The total provision of firemen hired, M, is the sum of the number hired by each of the two persons: M=M^A+M^B. Ann and Bob both have income of 200 each, and the price of both the private good and a fireman is 1. They are limited to providing between 0 and 200 firemen. For the purposes of this problem, you can treat the number of firemen as a continuous variable (it could be man-years).
Suppose that the government recruits additional ? firemen and taxes Ann and Bob equally to cover the cost. Therefore, the total number of firemen is M^A+M^B+N, where M^A, M^B are appropriate individually-optimal contributions of A and B (i.e., the agents behave optimally, conditional on the policy), and each agent is taxed N/2. How many additional firemen should the government hire in order to guarantee that in equilibrium the total number of firemen is optimal?
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