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?

Question

Quizplus · Accepted Answer

To maximize utility subject to the budget constraint, we need to use the Lagrange method. Our objective function is U(x, y) = x²y⁸, and our constraint is P1x + P2y = M. Taking the partial derivative of the objective function with respect to x and y gives: dU/dx = U1P1^2x^U1-1(yP2)^U2 / (P1P2)^U1 + U1P1^2x^U1(yP2)^U2-1 / (P1P2)^U2 = λP1 dU/dy = U1P1(xP2)^U2 / (P1P2)^U1 + U1(xP2)^U2-1(yP2)^U2 / (P1P2)^U2 = λP2 Solving for x and y gives: x = 21 y = 14 Therefore, Georgina will consume 21 grapefruits.

Georgina Consumes Only Grapefruits and Pineapples