It was assumed in the text that George’s indifference curves do not cut the bottom or left-hand side of the box, and that Harriet’s indifference curves do not cut the top or right-hand side of the box. The characterization of Pareto optimal and competitive allocations is a little more complex if they do. The next question deals with these complexities.

Assume that George’s indifference curves are linear and have slope −1. Assume that Harriet’s indifference curves are linear and have slope −2.

a) Let X be an allocation in the interior of the box. Show that all of the allocations that both George and Harriet believe to be at least as good as X lie within a triangular or diamond-shaped area.

b) Show that every allocation on the left-hand side or top of the box is Pareto optimal. Show that no other allocation is Pareto optimal.

c) Let p be the price of ale measured in bread. Find George’s best attainable commodity bundle(s) when p is equal to 1 and when it is greater than 1. Find Harriet’s best attainable commodity bundles when p is equal to 2 and when it is less than 2.