Question

Find the optimal bundle using the following utility functions and budget constraints.

1. U(x₁,x₂) = 3x₁ + 2x₂ and 6x₁ + 2x₂ = 24

2. U(x₁,x₂) = x₁ + 2x₂ and 3x₁ + 6x₂ = 48

3. U(x₁,x₂) = min {x₁,x₂} and 4x₁ + 2x₂ = 48

4. U(x₁,x₂) = min {4x₁,2x₂} and 2x₁ + 3x₂ = 36

Answer 1

To find the optimal bundle, we compare the total utility and the ratio of marginal utility to price for each option. The optimal bundles for the given utility functions and budget constraints are (4, 6), (15, 6), (12, 12), and (6, 6), with option 4 being the correct one.

Step-by-step explanation:

The optimal bundle can be found by comparing the total utility at each point on the budget constraint and choosing the highest total. Another approach is to compare the ratio of the marginal utility to price for each good and ensure that the two ratios are equal at the optimal choice. Let's analyze each utility function and budget constraint to find the optimal bundles.

1. For U(x₁,x₂) = 3x₁ + 2x₂ and 6x₁ + 2x₂ = 24, the ratio of the marginal utility to price is 3/6 for good 1 and 2/2 for good 2. This means that the optimal bundle is (x₁, x₂) = (4, 6).
2. For U(x₁,x₂) = x₁ + 2x₂ and 3x₁ + 6x₂ = 48, the ratio of the marginal utility to price is 1/3 for good 1 and 2/6 for good 2. Therefore, the optimal bundle is (x₁, x₂) = (15, 6).
3. For U(x₁,x₂) = min {x₁,x₂} and 4x₁ + 2x₂ = 48, the ratio of the marginal utility to price is 1/4 for good 1 and 1/2 for good 2. Hence, the optimal bundle is (x₁, x₂) = (12, 12).
4. For U(x₁,x₂) = min {4x₁,2x₂} and 2x₁ + 3x₂ = 36, the ratio of the marginal utility to price is 4/2 for good 1 and 2/3 for good 2. Thus, the optimal bundle is (x₁, x₂) = (6, 6).

Based on the analysis, the correct option is 4. U(x₁,x₂) = min {4x₁,2x₂} and 2x₁ + 3x₂ = 36.