Question

Consider the following three utility functions:
$F = (4x_1 + 2x_2), G = min (4x_1, 2x_2)$ and $H = (\sqrt{x_1} + x_2)$ where, $x_1$ and $x_2$ are two goods available at unit prices $p_{x1}$ and $p_{x2}$ , respectively. Which of the following is/are CORRECT for the above utility functions?

• A. The marginal rate of substitution is given by −1, −2, and $−0.5\sqrt{x_1} $ for the utility functions F, G, and H, respectively
• B. If $p_{x1}$ = $p_{x2} $, then the utility maximisation problem with utility function 𝐹 has a corner solution
• C. If income is 100 and $p_{x1}$ = $p_{x2}$ = 2, then in the utility maximisation problem with utility function G, the sum of the optimal values of $x_1$ and $x_2$ is 50
• D. If income is 100, $p_{x1}$ = 5, and $p_{x2}$ = 5000, then in the utility maximisation problem with the utility function H, the optimal value of $x_2$ is 20

Answer 1

The Correct Option is B, C

Analysis of Utility Functions and Correct Options

Step 1: Utility Function Analysis 

1. Utility Function F = (4x₁ + 2x₂)

• This utility function is linear, meaning the consumer will allocate their entire budget to the good providing the highest utility per unit of cost, leading to a corner solution.
• The Marginal Rate of Substitution (MRS) is given by:

MRS = - (∂F/∂x₁) / (∂F/∂x₂) = - (4/2) = -2

2. Utility Function G = min(4x₁, 2x₂)

• This function represents a perfect complements relationship, meaning optimal consumption occurs when 4x₁ = 2x₂, subject to the budget constraint.

Given income I = 100 and pₓ₁ = pₓ₂ = 2, the budget constraint is:

2x₁ + 2x₂ = 100

Substituting x₂ = 2x₁ into the budget equation:

2x₁ + 2(2x₁) = 100

6x₁ = 100 → x₁ = 50/3

Substituting back:

x₂ = 2(50/3) = 100/3

Sum of optimal values: x₁ + x₂ = (50/3) + (100/3) = 50

3. Utility Function H = (√x₁ + x₂)

• This function represents perfect substitutes in a non-linear form.
• The consumer will allocate most of their budget to the good that provides the highest marginal utility per unit cost.

Given I = 100, pₓ₁ = 5, and pₓ₂ = 5000:

• The price of x₂ is too high, so the consumer will allocate their entire budget to x₁.
• Thus, the optimal value of x₂ = 0.

Step 2: Analyzing the Given Options

Option (A): MRS Values

• The given MRS values are -1, -2, and -0.5√x₁ for functions F, G, and H, respectively.
• Incorrect. The MRS for F is -2, and the MRS for G does not exist (since perfect complements do not allow substitution).
• For H, the MRS depends on x₁ as - (1/2√x₁), which is inconsistent with the given value.

Option (B): Corner Solution for Utility Function F

• If pₓ₁ = pₓ₂, the consumer chooses the good that gives a higher utility per unit price.
• Since function F is linear, it leads to a corner solution.
• Correct.

Option (C): Sum of Optimal Values for Utility Function G

• From the calculations, the sum of x₁ + x₂ = 50.
• Correct.

Option (D): Optimal Value of x₂ in Utility Function H

• Given I = 100, pₓ₁ = 5, and pₓ₂ = 5000, the consumer will allocate all their budget to x₁, making x₂ = 0.
• The statement that x₂ = 20 is incorrect.

Final Answer:

The correct options are:

• (B) Utility function F leads to a corner solution.
• (C) Sum of optimal values for utility function G is 50.