A little exchange economy has just two consumers, named Ken and Barbie, and two commodities, quiche and wine. Ken's initial endowment is 3 units of quiche and 2 units of wine. Barbie's initial endowment is 1 unit of quiche and 6 units of wine. Ken and Barbie have identical utility functions. We write Ken's utility function as U(QK, WK) = QKWKand Barbie's utility function as U(QB, WB) = QBWB, where QK and WKare the amounts of quiche and wine for Ken and QB and WBare amounts of quiche and wine for Barbie. (a) Draw an Edgeworth box below to illustrate this situation. Put quiche on the horizontal axis and wine on the vertical. Measure Ken’s goods from the lower left corner and Barbie’s goods from the upper right corner. (Let the box’s length be the total supply of quiche and the height the total supply of wine.) Locate the initial allocation, and label it W. On the sides, label the two consumers’ initial endowments of quiche and wine.
(b) Use blue ink to draw an indifference curve for Ken that shows allocations in which his utility is 6. Use red ink to draw an indifference curve for Barbie that shows allocations in which her utility is 6.
(c) At any Pareto optimal allocation where both consume some of each good, Ken's marginal rate of substitution between quiche and wine must equal Barbie's. Write an equation that states this condition in terms of the consumptions of each good by each person.
(d) On your graph, show the locus of points that are Pareto efficient. (Hint: If two people must each consume two goods in the same proportions as each other, and if together they must consume twice as much wine as quiche, what must those proportions be?)
(e) At any Pareto efficient allocation in which both persons consume both goods, Ken's indifference-curve slope will be . Therefore, since we know competitive equilibrium is Pareto efficient, we know that Pq /Pw = .
(f) In competitive equilibrium, Ken's consumption bundle must be . How about Barbie's consumption bundle? (Hint: You found competitive equilibrium prices above. You know Ken's initial endowment and you know the equilibrium prices. In equilibrium Ken's income will be the value of his endowment at competitive prices. Knowing his income and the prices, you can compute his demand in competitive equilibrium. Having solved for Ken's consumption and knowing that total consumption by Ken and Barbie equals the sum of their endowments, it should be easy to find Barbie's consumption.)
(g) On Ken and Barbie’s Edgeworth box, show the competitive equilibrium allocation and Ken's competitive budget line (with black ink).