ASSIGNMENT 8
Utility Maximization

Jon Stewart's utility function is described by u(x,y) = x^1/3y^2/3, where x and y are the amounts of 2 goods. The prices of the goods are $10 (for Good X) and $25 (for Good Y). Jon's income is $300.

1. Does the utility function exhibit diminishing marginal utility w.r.t. Good X? Explain.
2. Describe Jon's utility maximization problem formally.
3. Obtain the Lagrangian function.
4. From the FOCs, obtain the critical values of x and y.
5. Using the bordered Hessian, verify that the SOC for a maximum is met.
6. What is Jon's maximum utility?
7. What is the value of the Lagrange multiplier?
8. Provide an interpretation of the multiplier.
9. Sketch the indifference curve and budget constraint. Indicate Jon's optimal bundle on the graph.
10. Suppose, ceteris paribus, the price of Good X rises by 10%. Obtain the new optimal bundle. Indicate the changes on the graph.
11. Go back to the original values of the parameters. If income rises by 20%, ceteris paribus, what happens to Jon's consumption of Good X and Good Y? What happens to his maximum utility? Provide a sketch, comparing the new equilibrium with the old.