Write down the utility maximization problem for this

Question

Instructions: Answer all questions as completely as possible. If you cannot solve the problem, explaining how you would solve the problem.

1. Consider an individual who faces prices p₁, p₂ and p₃, where p_i > 0 for i = 1, 2, 3 and has the following utility function:

u (x₁, x₂, x₃) = x₁, x₂ + {1/(1 + x₃)}

The individual has income y > 0 and the following indirect utility function:

v (p, y) = (y² / p₁p₂) + 1

a. Based on the utility function u (x₁, x₂, x₃), determine whether or not the preference relation that it represents is convex and monotonic on the consumption set X = ^R3₊. Explain how you know.

b. Verify that the indirect utility function is homogeneous of degree zero in prices and income.

c. Find the individual's Marshallian demand functions, x_i (p, u) for i = 1, 2, 3.

d. Find the individual's expenditure function e (p, u).

e. Find the Hicksian demands x^h_i (p, u) for i = 1, 2, 3.

f. If p₁ = 4, p₂ = 1/4, p₃ = 12 and u = 37, calculate the Slutsky matrix for this consumer.

Are the goods substitutes or complements for each other.

2. Suppose that an individual has a vN-M utility function u (ω) = 7 + ω^1/3. Let ω = 300.

a. Calculate the Arrow-Pratt coefficient of absolute risk aversion, R_a (ω), at the level of wealth ω = 300.

b. Does this individual have increasing, constant, or decreasing absolute risk aversion (LARA, CARA or DARA)? Explain how you know.

c. Calculate the certainty equivalent for the gamble g = (½ o 64, ½ o 27)

3. Consider a worker who consumes one good and has a preference for leisure. She maximizes the utility function:

u (x, L) = xL

where x represents consumption of the good and L represents leisure. The total amount of time that the worker has is 1 unit, so L ∈ [0, 1]. The time that is not spent on leisure is spent on work. In particular, for any L that the worker would choose she will receive income r (1 - L), where r represents the wage rate. Let p denote the price of the consumption good. In addition to her wage income the worker also has a fixed income ω > 0.

a. Write down the utility maximization problem for this consumer. Pay close attention to the budget constraint.

b. Find the Walrasian demands for the consumption good and leisure. Be sure to consider the constraint L ∈ [0, 1].

c. Find the indirect utility as a function of p. r, and ω.