Let u(x1, x2, x3) = (x1x2x3)1/3. Suppose that p1 = p2 = p3 = 1 and that w = 6 and that good 1 is rationed, so that the consumer can buy no more than one unit of it.
(a) Compute the demands for goods 1, 2, and 3.
(b) Suppose there is a black market for good 1. For what black market prices q would the consumer buy no more than the one unit of good 1 received at price of 1 under rationing?
(c) Suppose there was no black market. For what prices would the consumer be willing to pay for more ration tickets, assuming that the tickets could be sold in arbitrarily small units?