Question

Jim and Stacy are the only two consumers on an island. Jim has 40 Apples and Stacy has 40 Pop Tarts.

1. Is the current allocation efficient? If not, use an Edgeworth box to demonstrate why.
2. Assume Jim trades 10 Apples for 10 of Stacy's Pop Tarts. How do I know this trade improves overall welfare?
3. Now, assume that Stacy is health-conscious and refuses to eat Pop Tarts at all. Describe the set of efficient allocations under this assumption.

Answer 1

To provide the background, let's begin by stating what are edge-worth boxes and what do they explain:

Imagine a square/rectangle. Edge-worth boxes comprise two origins, for two individuals, one on the bottom left (O1) and the other on the top right (O2). The two axes associated to an origin represent the two goods. Each point in the edge-worth box, thus, represent the allocation of 2 goods to two individuals. The good 1 consumption by individual 1 is measured horizontally from O1 and the consumption of good 2 vertically from O2, and similarly for individual 2 from O2.

a) In the present case where Jim has 40 Apples and Stacy has 40 Pop Tarts, the two goods are Apples and Pop Tarts. The allocation is not Pareto efficient as demonstrated in the Edge-Worth box below:

In the diagram, A1 refers to the original allocation. where A1 is not Pareto efficient. This is because Jim and Stacy can exchange the goods to make them both better off. Any allocation such as A2 is more Pareto efficient than A1. It is further from both the origins.

b) When Jim trades 10 Apples for 10 of Stacy's Pop Tarts, the endowment point in the box changes to A3 (in the diagram). Both Jim's and Stacy's indifference curves move towards the increasing utility direction. This follows from the Non-satiation property of utility functions. Jim's utility increase towards north-east, while Stacy's utility increases towards the south-west. Hence, both are better-off after the trade.

If Stacy doesn't like Pop-tarts at all, consuming zero Pop tarts would make her the happiest. The optimal allocation for her would be 40 apples and 0 pop tarts while for Jim, the option would be to have only 40 Pop tarts. This allocation takes them both farthest from their respective origins, Therefore efficient allocation point would be (0,40);(40,0) where the first number gives the number of apples and the second number gives Pop-tarts in the order of Jim and Stacy.