Problem consider a consumer with preferences over goods x

Question

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) = x^ay^1-a .