Question

Suppose an individual has the following utility function: U(X,Y) = 2XY. Draw the budget constraint when the price of each unit of X and Y are 5 and 4, and income is 120. Find the choice that maximizes utility.

Answer 1

Budget constraint represents the combination of two goods an individual can buy given his money income. Here X and Y are two goods, and the income is 120. So, the equation showing the combination of goods that can be afforded are:

5X+4Y<=120, where 5X shows the cost of good X and 4Y is the cost of good Y.

U= 2XY

The choice is that the combination of goods where the marginal rate of transformation or the slope of the indifference curve is equal to the slope of the budget line.

Marginal Rate of Transformation= Marginal Utility of X/Marginal utility of Y

Marginal utility of X is the derivative of utility with respect to X.

MUx= U/X= 2Y

MUy= U/Y= 2X

MRT= MUx/MUy= 2Y/2X= Y/X

Slope of budget line= price of X/Price of Y= 5/4

At equilibrium, Y/X= 5/4

5X=4Y

Putting this in the equation of the budget line: 4Y+4Y=120

8Y=120

Y= 15

X= 4Y/5= 4(15)/5= 12

The choice of the combination of goods that would be chosen is (12,15).