Utility Maximization Subject to the Budget Constraint

Question

Suppose a consumer buys two goods, X and Y, and his utility function is given by U (X,Y) = XY + 10Y. The prices of the 2 goods X and Y are given by P_X = $1 and P_Y = $2. The consumer has an income of $10. What is the consumer's optimum consumption bundle of the goods X and Y? Is the optimum bundle an interior optimum or a corner optimum?

Answer 1

The utility maximizing bundle can be derived by solving the following two conditions:

X + 2Y = 10 ....... budget constraint (1)

MRS_X.Y = P_X/P_Y ... tangency condition (2)

For the given utility function, marginal utility of X, MU_X = Y and marginal utility of Y, MU_Y = X + 10.

Thus, MRS_X,Y = MU_X/MU_Y = Y/(X + 10)

Thus, the tangency condition is Y/(X + 10) = 1/2 as P_X = $1 and P_Y = $2...... (2)

Solving (1) and (2) we get X = 0, Y = 5. We thus have a corner optimum where the consumer only buys 1 good Y.