Q. Consider a small exchange economy with two consumers, Ann and Bill, and two commodities, ham and cakes. Ann's initial endowment is 4 units of ham and 1 unit of cake. Bill's initial endowment has no ham and 7 units of cakes. Ann's utility function is U(HA,CA) = HACA. Bill is a more inflexible person. His utility function is U(HB,CB) = minHB,CB. (Here HA and CA are the amounts of ham and cakes for Ann, and HB and CB are amounts of ham and cakes for Bill.)

(a) Draw an Edgeworth box, showing the initial allocation and sketching in a few indifference curves (using black ink for Ann's and blue ink for Bill's). Measure Ann's consumption from the lower left and Bill's from the upper right.

(b) Show the locus of Pareto optimal allocations in red. (Hint: Since Bill is kinky, calculus won't help much here. But notice that because of the rigidity of the proportions in which he demands the two goods, it would be inefficient to give Bill a positive amount of either good if he had less than that amount of the other good. What does that tell you about where the Pareto efficient locus has to be?)

(c) Suppose price of cake pC = 1 and pH denotes the price of hams. Write an expression for the amount of hams that Bill will demand at these prices.

(d) Suppose price of cake pC = 1 and pH denotes the price of hams. What is the value of Ann's initial endowment? How much ham will Ann demand at price pH?

(e) If it is competitive market, when the equilibrium makes, what condition does pH satisfied.