Question

An agent has preferences for goods X and Y represented by the utility function U(X,Y) = X +3Y

the price of good X is P_x= 20, the price of good Y is P_y= 40, and her income isI = 400

Choose the quantities of X and Y which, for the given prices and income, maximize her utility.

Answer 1

U = X + 3Y

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

400 = 20X + 40Y

For a linear utility function, indifference curve is a straight line touching both axes, and optimal solution is at one of the corner points. X and Y are substitutes, and only one of them would be consumed.

From Budget line,

When X = 0, Y = 400/40 = 10 and U = 0 + (3 x 10) = 30

When Y = 0, X = 400/20 = 20 and U = 20 + (3 x 0) = 20

Since utility is higher when X = 0 and Y = 10, this is the optimal bundle.