Question

Suppose U (x, y) denotes the utility enjoyed by a person when having x hours of leisure (hours not working) per day (24 hours) and y units per day of other goods. The person gets an hourly wage of w and pays a price of p per unit of other goods. The individual spends all of the money they earn and faces the following budget constraint: py = w(24 - x). Assume a utility function with the following features Ux, Uy > 0; Uxy > 0; and Uxx, Uyy < 0. a) Set up the utility maximization problem using the Lagrangian method and state the first order conditions. b) Let (x, y, ?) denote the solution in the utility maximization problem. Solve for ?x/?w using Cramer’s rule and implicit function theorem. Be sure to state whether the conditions for the use of implicit function theorem hold. Show all your work. c) Sign the comparative static solved for in part b.

          Suppose U (x, y) denotes the utility enjoyed by a person when having x hours of leisure (hours not working) per day (24 hours) and y units per day of other goods. The person gets an hourly wage of w and pays a price of p per unit of other goods. The individual spends all of the money they earn and faces the following budget constraint:
py = w(24 - x). Assume a utility function with the following features
Ux, Uy > 0; Uxy > 0; and Uxx, Uyy < 0.

a) Set up the utility maximization problem using the Lagrangian method and state the first order conditions.

b) Let (x*, y*, ?*) denote the solution in the utility maximization problem. Solve for ?x*/?w using Cramer’s rule and implicit function theorem. Be sure to state whether the conditions for the use of implicit function theorem hold. Show all your work.

c) Sign the comparative static solved for in part b.

Suppose U (x, y) denotes the utility enjoyed by a person when having x hours of leisure (hours not working) per day (24 hours) and y units per day of other goods. The person gets an hourly wage of w and pays a price of p per unit of other goods. The individual spends all of the money they earn and faces the following budget constraint:
py = w(24 - x). Assume a utility function with the following features
Ux, Uy > 0; Uxy > 0; and Uxx, Uyy < 0.

a) Set up the utility maximization problem using the Lagrangian method and state the first order conditions.

b) Let (x*, y*, ?*) denote the solution in the utility maximization problem. Solve for ?x*/?w using Cramer’s rule and implicit function theorem. Be sure to state whether the conditions for the use of implicit function theorem hold. Show all your work.

c) Sign the comparative static solved for in part b.

Added by Krista C.

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