A little exchange economy has just two consumers, named Kenand 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) = QKWK and Barbie’s utility function as U(QB,WB) = QBWB, where QK and WK are the amounts of quiche and wine for Ken and QB and WB are 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 axis. Measure goods for Ken from the lower left corner of the box and goods for Barbie from the upper right corner of the box. (Be sure that you make the length of the box equal to the total supply of quiche and the height equal to the total supply of wine.) Locate the initial allocation in your box, and label it W. On the sides of the box, label the quantities of quiche and wine for each of the two consumers in the initial endowment.
(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.
(e) In this example, at any Pareto efficient allocation, where both persons consume both goods, the slope of Ken’s indifference curve will be ________ Therefore, since we know that competitive equilibrium must be Pareto efficient, we know that at a competitive equilibrium, pQ/pw = ______________
(f) In competitive equilibrium, Ken’s consumption bundle must be ___________ How about Barbie’s consumption bundle? ________________
(g) On the Edgeworth box for Ken and Barbie, draw in the competitive equilibrium allocation and draw Ken’s competitive budget line (with black ink).