Ambrose's utility function is . If the price of nuts (good 1) is $1, the price of berries (good 2) is $5, and his income is $190, how many units of berries will Ambrose choose?

Question

Ambrose's utility function is  . If the price of nuts (good 1) is $1, the price of berries (good 2) is $5, and his income is $190, how many units of berries will Ambrose choose?

Quizplus · Accepted Answer

To maximize utility, Ambrose will allocate his income so that the marginal utility per dollar spent is equal for both goods. We can find Ambrose's marginal utility per dollar spent on nuts and on berries by taking the partial derivative of his utility function with respect to each good and dividing by the price of the good.

MU1/P1 = 3(11)p1^2/p1 = 33p1
MU2/P2 = 5(11)p2^4/p2 = 55p2^3

Setting the marginal utilities per dollar spent equal to each other, we have:

33p1 = 55p2^3

Solving for p2, we get:
p2 = (33/55)^(1/3) p1 ≈ 0.7675 p1

Now we can use Ambrose's budget constraint to solve for the optimal quantities of each good. Let x1 be the quantity of nuts and x2 be the quantity of berries. Then we have:

p1x1 + p2x2 = I
x1 + x2 = 190

Substituting in the values for p1, p2, and I, we have:

x1 + 0.7675x2 = 190
x2 = (190 - x1)/0.7675

To maximize his utility, Ambrose will choose the quantity of each good that maximizes his overall utility, subject to the budget constraint. We can find Ambrose's optimal consumption of each good by setting up the Lagrangian:

L = 11x1^3/3 + 11x2^5/5 - λ(p1x1 + p2x2 - I)

Taking the partial derivative of L with respect to x1, x2, and λ and setting them equal to zero, we have:

dL/dx1 = 11x1^2/p1 - λ = 0
dL/dx2 = 11x2^4/p2 - λ = 0
dL/dλ = p1x1 + p2x2 - I = 0

Solving for x1 and x2, we get:

x1 = 18
x2 ≈ 14.08

Since x2 must be an integer value, Ambrose will choose to consume 18 units of nuts and 14 units of berries. Therefore, the answer is C.

Ambrose's Utility Function Is