A community has 3 households, H, M, L. They are deciding how many school teachers to hire in their community elementary school. Each school teacher costs $60,000. The marginal utility for each household is MBH = 100, 000 10, 000N MBM = 80, 000 20, 000N MBL = 40, 000 20, 000N, where MBH, MBM, MBL indicates, respectively, the marginal benefit of an additional teacher for household H, M, and L. N is the number of teachers.

(a) (5 pts) Write down expression for the social marginal benefit, as a function of the number of teachers N. (Hint: express it in equations, not in graph. The equation has kinks.)

(b) (5 pts) Solve for the socially optimal number of teachers. What is each household's willingness-to-pay at the optimal number of teachers?

(c) (5 pts) If the households pay a head tax, what is each household's preferred number of teachers?

(d) (5 pts) If the community decides the number of teachers by pairwise voting, how many teachers will the elementary school have?

(e) (10 pts) Suppose now the community decides to fund the elementary school via a proportional property tax on housing value. Suppose H, M, and L lives in a house that is worth $3, $2, and $1 million, respectively. In order to finance the optimal number of teachers, what should the tax rate be? How much each household pays for each teacher? Do you think this allocation of cost is fair? Which household(s) should be charged more and which household(s) should be charged less? Explain your answer.