1. Only 2 families, the Arrows (A) and the Benthams (B), inhabit Little Society. Each family has onechild. The 2 families pay for the single, private, school in Little Society. School quality (Q) can bebought as a price of 1 per unit and is given by the sum of A and B's spending on it, denoted SA and SB,respectively. Each family has an income of M and gets utility U(x, Q) from consuming the singleprivate good, quantity denoted x, and the quality of education,

Q. A and B have identical utilityfunctions,

(a) Suppose A and B choose the amount they spend on the school independently. What is theNash equilibrium level of the school's quality in Little Society?

(b) Suppose, instead, Little Society's government levies lump-sum taxes on A and B to financethe level of Q that maximises social welfare. What level of tax should it impose on each of Aand B? Why is Q higher in this case than in case (a)?

(c) The Benthams now move abroad to the Big Society, in which there are many schools ofdifferent qualities. B can now enrol their child in a school of quality Q for a fee of Q. B'sincome remains M. What quality of school will B purchase? (15%)(d) The Big Society's government decides to make state schooling of quality Q = M/4 availablefor free. Will the B child be sent to a state school or to a private school?