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Individuals a and b each consume amounts of a private good a and a public good y, with preferences represented by: u" (Ta, y) = La + alny, u (xb, y) = X6 + Bly There is a total endowment of X units of the private good. The public good can be provided at a per-unit cost of c units of the private good, so that: Ta tro = X - cy (a) The Samuelson condition for Pareto-efficiency here is: Given the assumed preferences, show that the Samuelson condition implies: a + B y = C (b) Suppose that the above preferences are public knowledge, but with the values of a and / known only privately, to individuals a and b respectively. An arbitrator asks a and b to report values a and B, and then in accordance with the efficiency condition in (a) sets: at B y = C The remaining private good (X - cy) is then distributed equally between a and b. Given B, show that: dy dxa da and = -0.5 da What is the significance of these derivatives, for individual a in considering what value to report