Question

Imagine you consume two goods, X and Y, and your utility function is U = XY. Your budget is $100, the price of Good X is $4, and the price of Good Y is $25. So, the optimal bundle for you to consume is (12.5, 2). Now the price of good X increases to $10. The compensated price bundle is (7.91, 3.16). What is the income effect on X?

Answer 1

Substitution effect on X, SE = 7.91 - 12.5 = -4.59

We find the X after increase in price.
Px' = 10; Py = 25; M = 100

Utility is maximized where MRS = Px'/Py = 10/25 = 0.4
MRS = MUX/MUY = (∂U/∂X)/(∂U/dY) = Y/X
So, Y/X = 0.4
So, Y = 0.4X

Budget line: Px'X + PyY = M
So, 10X + 25(0.4X) = 10X + 10X = 20X = 100
So, X' = 100/20 = 5

TE on X = X' - initial X = 5 - 12.5 = -7.5

Income effect, IE = TE - SE = -7.5 - (-4.59) = -7.5 + 4.59 = -2.91

So, income effect on X is -2.91