Georgina consumes only grapefruits and pineapples.Her utility function is U(x, y)= x²y⁸, where x is the number of grapefruits consumed and y is the number of pineapples consumed.Georgina's income is $105, and the prices of grapefruits and pineapples are $1 and $3, respectively.How many grapefruits will she consume? A)10.5 B)7 C)63 D)21 E)None of the above.

Question

Quizplus · Accepted Answer

To maximize utility subject to budget constraint, we need to use the Lagrange method:

L(x, y, λ) = U(x, y) + λ(I - Px x -Py y)

where I is the income, Px is the price of grapefruits, Py is the price of pineapples and λ is the Lagrange multiplier.

∂L/∂x = 0 ⇒ 10x-6λ = 0
∂L/∂y = 0 ⇒ 4y-9λ = 0
∂L/∂λ = 0 ⇒ Px x + Py y - I = 0

Solving these equations simultaneously, we get:

x=21, y=14, λ=$10.50

Therefore, Georgina will consume 21 grapefruits.

Georgina Consumes Only Grapefruits and Pineapples