Question

Problem 2

Consider a representative consumer with a budget of $ 400 in a “make-believe” world with two consumer goods X and Y, where the price of X is P_x = $2 and the price of Y is P_y= $6.

His utility function can be expressed as:

U(X,Y) = X³Y²

What are the quantities of commodity X and commodity Y that this consumer must purchase and consume to maximize his/her utility?

Answer 1

Marginal utility function w.r.t X = dU(X,Y)/dX = 3(x^2)*(y^2)

Marginal utility function w.r.t Y = dU(X,Y)/dY = 2(x^3)*(y)

where X is quantity of X purchased and Y is quantity of Y purchased

To maximize utility,

3(x^2)*(y^2)/2(x^3)*(y) = Px/Py = 2/6

=> Y/X = 2/9 or Y=2X/9

and as his budget is $400, so Px*X + Py*Y = 400

2X+6Y = 400

substituting Y = 2X/9, we get

2X + 4X/3 = 400 => X = 120

substituting X = 120 in Y = 2X/9, we get Y = 2*120/9 = 80/3

Quantity of X = 120

Quantity of Y = 80/3

But if quantity can not be in decimal,

new Y = 27 & X= 119