Question

1. Kim’s utility function is given by U = XY. For this utility function, MU_x = Y and MU_y = X.

   1. If good X costs $6, and good Y costs $3, what share of Kim’s utility-maximizing bundle is made up of good X? Of good Y?

   2. If the price of good Y rises to $4, what happens to the shares of X and Y in Kim’s utility-maximizing bundle?

Answer 1

Budget line: M = X.Px + Y.Py

Utility is maximized when (MUx/MUy) = Px/Py

Y/X = Px/Py

(1) When Px = 6 and Py = 3,

Y/X = 6/2 = 2

Y = 2X

Substituting in budget line,

M = 6X + 3Y

M = 6X + 3(2X)

M = 6X + 6X = 12X

X = M/12

Y = 2 x (M/12) = M/6

X/Y = (M/12) /(M/6) = 6/12 = 1/2

So good X comprises (1/3)^rd of the bundle and good X comprises (2/3)^rd of the bundle.

(2) When Px = 6 and Py = 4,

Y/X = 6/4 = 3/2

Y = 3X/2

Substituting in budget line,

M = 6X + 4Y

M = 6X + 4(3X/2)

M = 6X + 6X = 12X

X = M/12

Y = [3 x (M/12)/2] = M/8

X/Y = (M/12) /(M/8) = 8/12 = 2/3

So good X comprises (2/5)^th of the bundle and good X comprises (3/5)^th of the bundle.