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Answer :
Final answer:
To determine how much y Zeke consumes at a Pareto optimal allocation, we compare the ratio of marginal utility of good y to x for Zeke and Maude. Based on their utility functions, y = 1/64 for Zeke.
Explanation:
At a Pareto optimal allocation in which both individuals consume some of each good, we can determine how much y Zeke consumes by comparing the ratio of the marginal utility of good y to the marginal utility of good x for Zeke, with the ratio of the marginal utility of good y to the marginal utility of good x for Maude.
Using their utility functions, we can find that the ratio for Zeke is ((1/2)y^(-1/2))/(1) = 1/(2y^(1/2)), and the ratio for Maude is (4)/(1) = 4.
To achieve a Pareto optimal allocation, these two ratios must be equal. Therefore, 1/(2y^(1/2)) = 4, which implies y = 1/64.
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