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

d) Show that:

i) If a budget line with slope -1 cuts the top of the box, then 1 is a market-clearing price.

ii) If a budget line with slope -2 cuts the left side of the box, then 2 is a market-clearing price.

iii) If a budget line with slope -p cuts the top left corner of the box, and if p is between 1 and 2, p is a market-clearing price.