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 has a utility function over private goods X (donuts) and the number of firemen M: € Ui = 4ln(Xi) + 2ln(M), where € i ∈{B,H}. The total number of firemen hired is the sum of the number hired by each of the two: € M = MH + MB . Homer and Bart both have income of 100, and the price of both a donut and a fireman is 1.
a. Find the number of firemen that each would hire in the private market Nash equilibrium. Characterize this equilibrium on a graph that shows both individuals’ best-response functions.
b. What is the socially optimal number of firemen? If your answer differs from that in part a), explain why.
c. Suppose the government is not happy with the private equilibrium, and it decides to provide 10 more firemen. It taxes Homer and Bart equally to pay for the new hires. What is the new total number of firemen? How does you answer compare to a)? Have we achieved the social optimum? Why or why not?
d. Suppose now that the government is still not happy, so it decides to provide 35 new firemen. It taxes Bart 10 to pay for them, and it taxes Homer 25. What is the new total number of firemen? How many are provided by Homer and how many by Bart? How does this compare to the level of provision in c), and why? (HINT: remember that neither person can consume negative units, which affects the purchase decisions of both parties.)
e. Finally, suppose that both Bart and Homer still have an income of 100 each, but that their utility functions have changed to: € UB = 2ln(XB ) + 2ln(M) and € UH = 6ln(XH) + 2ln(M). There are no taxes that are levied and there is no public provision of firemen. Find the new private contribution Nash equilibrium and relate it to your answer in a).