The town of Springfield has two residents: Homer and Bart. The town currently funds its fire department solely from the individual contributions of these residents. Each of the two residents has a utility function over private goods (X) and total firefighters (M) of the form U = X^1/2 M^1/2. The total provision of firefighters hired, M, is the sum of the number hired by each of the two persons: M = M_H + M_B. Homer and Bart both have income of $100, and the price of both the private good and a firefighter is $1. Thus, they are limited to providing between 0 and 100 firefighters.

a. How many firefighters are hired if the government does not intervene? How many are paid for by Homer? By Bart?

b. What is the socially optimal number of firefighters? Is your answer differs from a, why?

can you show me how to solve this problem?