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

Supriya Khatiwada · Accepted Answer

Final Answer : D, Explanation :  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 = 0Solving these equations simultaneously, we get:x=21, y=14, λ=$10.50Therefore, Georgina will consume 21 grapefruits.

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