Marys budget constraint is given by PxXPyY60 and Px5 Py2 Sup

Mary’s budget constraint is given by PxX+PyY=60, and Px=$5, Py=$2. Suppose Mary’s utility function is given by the equation U = XY, where U is the level of utility measured in utils and X and Y refer to good X and good Y respectively. You are also told that the marginal utility of good X can be expressed as MUx = Y; and the marginal utility of good Y can be expressed as MUy = X.

a. What are the X and Y in the market basket (X,Y) that maximize Mary’s utility given her budget constraint?

b. What is the marginal rate of substitution at the utility-maximizing market basket (X,Y).

c. How does your answer to (a) change if the price of good X doubles?

d. Can you identify a utility function for which the chosen bundle would not change when the price of good X doubled?

Solution

U = XY

Budget line: 60 = 5X + 2Y

(a)Utility is maximized when MU(X) / MU(Y) = Px / Py

MU(X) = dU / dX = Y

MU(Y) = dU / dY = X

So,

Y / X = 5 / 2

5X = 2Y

Substituting into budget line,

60 = 5X + 2Y

60 = 5X + 5X = 10X

X = 6 [Demand of X]

Y = 5X/2 = 30/2 = 15 [Demand of Y]

(b)

MRS = MU(X) / MU(Y) = Y / X = 15 / 6 = 2.5

(c)

Px = 10

Budget line: 60 = 10X + 2Y

Since 5X = 2Y, we get

60 = 10X + 5X = 15X

X = 4

Y = 5 x 4 / 2 = 10

(d)

I cannot think of such utility function where demand for X does not depend on its price.