In Problem 3, Ambrose's utility is U(x₁, x₂) = . If the price of nuts (good 1) is $1, the price of berries (good 2) is $4, and his income is $108, how many units of nuts will Ambrose choose?

Question

Quizplus · Accepted Answer

To maximize utility, Ambrose must allocate his entire income to purchasing nuts and berries.

Let x be the number of units of nuts he purchases, then (108-4x) is the amount he spends on berries.

The marginal rate of substitution (MRS), or the rate at which he is willing to exchange one good for the other, is given by the partial derivative of the utility function with respect to xS1U1B1, divided by the partial derivative with respect to xS1U1B2:

MRS = (∂U/∂xS1U1B1) / (∂U/∂xS1U1B2) 
    = 11xS1U1B2U1B6 / (4xS1U1B1U1B6) 
    = 11/4 * (xS1U1B2U1B6 / xS1U1B1U1B6)

Setting the MRS equal to the price ratio (4), we get:

11/4 * (xS1U1B2U1B6 / xS1U1B1U1B6) = 4
11xS1U1B2U1B6 = 16xS1U1B1U1B6
xS1U1B1U1B6 = (11/16) xS1U1B2U1B6

Now we can set up the budget equation:

1xS1U1B1U1B6 + 4xS1U1B2U1B6 = 108

Substituting xS1U1B1U1B6 = (11/16) xS1U1B2U1B6, we get:

1(11/16) xS1U1B2U1B6 + 4xS1U1B2U1B6 = 108
5(11/16) xS1U1B2U1B6 = 108
xS1U1B2U1B6 = 108 * (16/55) ≈ 31.2

Rounding down, Ambrose will choose 31 units of berries and:

xS1U1B1U1B6 = (11/16) xS1U1B2U1B6 ≈ 21.3

Rounding to the nearest integer, Ambrose will choose 21 units of nuts. Therefore, the answer is (B) 64.

In Problem 3, Ambrose's Utility Is U(x1, X2) =

In Problem 3, Ambrose's Utility Is U(x₁, X₂) =