In Problem 2, Ambrose has the utility function U(x₁, x₂) . If Ambrose were initially consuming 49 units of nuts (good 1) and 12 units of berries (good 2), then what is the largest number of berries that he would be willing to give up in return for an additional 32 units of nuts? A) 9 B) 21 C) 8 D) 4 E) 2

Question

Quizplus · Accepted Answer

To determine the largest number of berries Ambrose would give up for 32 additional nuts, we need to find his marginal rate of substitution (MRS) between berries and nuts at his initial consumption bundle (49, 12). This informs us of the rate at which he's willing to substitute berries for nuts.

The MRS is given by the partial derivative of the utility function with respect to berries over the partial derivative of the utility function with respect to nuts.

MRS = (∂U/∂berries)/(∂U/∂nuts)

Taking partial derivatives with respect to each good, we get:

∂U/∂nuts = 11xS1U1B1^(1/3)xS1U1B2^(-2/3)

∂U/∂berries = 22xS1U1B1^(-2/3)xS1U1B2^(1/3)

Plugging in Ambrose's initial consumption (49, 12), we get:

∂U/∂nuts = 11(49)^(1/3)(12)^(-2/3) ≈ 2.36
∂U/∂berries = 22(49)^(-2/3)(12)^(1/3) ≈ 0.76

Therefore, MRS = 0.76/2.36 ≈ 0.322

This means that Ambrose is willing to give up 0.322 berries for each additional nut. To determine the largest number of berries he would give up for 32 additional nuts, we need to solve:

0.322x = 32

x ≈ 99.4

Rounded down, the largest number of berries Ambrose would give up is 8. Hence, the answer is C.

In Problem 2, Ambrose Has the Utility Function U(x1, X2)

In Problem 2, Ambrose Has the Utility Function U(x₁, X₂)