A consumer has the following utility function: U = X . Y (then her MUX = Y and h

Question

A consumer has the following utility function: U = X . Y (then her MUX = Y and her MUY = X). The price of X is 3 (PX = 3), the price of Y is 1 (PY = 1), and her income is 120 (M = 120).

a. What is this consumer’s optimal consumption choice? Verify that the two conditions for constrained utility maximization hold for your answer.

b. If the price of good X decreases to 1 (PX = 1) and the price of good Y increases to 2 (PY = 2) but her income is also still 120 (M = 120), what would be her new optimal choice? Verify that the two conditions for constrained utility maximization hold for your answer.

Explanation / Answer

U=XY;
3X+Y <= 120;
Let 3x+y=120; maximum case; y=120-3x
U=XY=X*(120-3X) = 120X-3X2;
For optimum utility, differentiate and equate to zero; Thus du/dx= 120-6x=0 meaning x=120/6=20;
Y= 120-3*20 = 120-60=60;

U= 60*20 = 1200

For part b) U= xy but x+2y=120 (after price change) ; Thus x=120-2y
U=x*y= y*(120-2y) = 120y-2y2
Differentiate and equate to zero; 120-4y=0 Thus y = 120/4 = 30;
X = 120-2*30 = 120-60=60
Utility = x*y = 60*30 = 1800

Thus, utility is higher now

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