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Problem: Consider a consumer with preferences over goods x and y represented by the utility function U(x, y) = a In x + (1 - a) In y where a ∈ (0, 1):
a. Determine whether the consumer's preferences are:
i. Complete
ii. Transitive
iii. Rational
iv. Monotonic
v. Convex
b. Supposing the consumer has income of $100 and faces prices px = 2 and py = 5 and that a = 0.2, set up the utility maximization problem and then solve it to derive the utility maximizing consumption levels x* and y*.
c. For any values of this model's parameters (i.e., exogenous variable), show that the consumer always spends a and 1 - a proportions of his income on goods x and y, respectively, at an optimum.
d. Explain why the answers to the previous parts of this question remain unchanged if the utility function is changed to U(x, y) = xay1-a .
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