4. A town 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, M) = 6 ln X+ 2 ln M. The total provision of firefighters hired, M, is the sum of the number hired by each of the two persons: M = +. Homer and Bart both have income of $100, the price of the private goods is = $100,000 and the price of hiring a firefighter is = $50,000. NOTE: - The MRS should be setup based on how the utility function is written with "M" on the vertical axis.
a. How many firefighters are hired if the government does not intervene? How many are paid for by Homer and by Bart respectively? Show your work
b. What is the socially optimal number of firefighters?
Show your work.