Question

Smith and Jones are stranded on a desert island. Each has in their possession slices of ham

(H) and cheese (C). Smith prefers to consume ham and cheese together in a fixed proportion
of 1 slice of cheese to each slice of ham; her utility is U S = min(H, C). Jones, on the other
hand, regards ham and cheese as perfect substitutes. She is always willing to trade 1 slice of
ham for 1 slice of cheese; her utility is U J = H + C. Suppose Smith’s initial endowment is 60
slices of ham and 80 slices of cheese, and Jones’ initial endowment is 40 slices of ham and
120 slices of cheese. Draw an Edgeworth box that shows the endowment point and all the
possible allocations of ham and cheese across Smith and Jones. Illustrate that the initial
endowment point is inefficient and then clearly indicate all the efficient allocations of ham
and cheese to which Smith and Jones would voluntary trade, given their preferences and
endowments

Answer 1

Smith and Jones' preferences and endowments lead to an inefficient initial allocation, but efficient trade can be achieved along the contract curve where Smith has equal slices of ham and cheese, and Jones has the remainder.

Step-by-step explanation:

In the scenario described, we're asked to imagine an Edgeworth box in which Smith and Jones are stranded on a desert island with slices of ham and cheese. Smith's utility function US = min(H, C) indicates a lexicographic preference, desiring ham and cheese in a 1:1 ratio. Meanwhile, Jones views these goods as perfect substitutes with a utility function of UJ = H + C, indicating a willingness to exchange them one-for-one.

Initially, Smith has 60 slices of ham and 80 slices of cheese, while Jones has 40 slices of ham and 120 slices of cheese. To create an Edgeworth box, draw a rectangle where the width represents the total amount of ham (100 slices) and the height represents the total amount of cheese (200 slices). The initial endowment point is found at the coordinate (60,80) for Smith and (40,120) for Jones when plotted from each individual's origin.

The initial allocation is inefficient because Smith would be willing to give up some cheese for ham until she has equal amounts of both, while Jones would be indifferent to such a trade as long as the total quantity of goods remains the same. Efficient allocations would be where Smith has equal slices of ham and cheese, and Jones has the remaining quantities since Smith would reach her optimal utility and Jones would be indifferent to the specific combination. These points would lie along the contract curve, which, in this example, is where Smith's slices of ham and cheese are equal.