ECON302_Assignment2_F11_Sol_Full

# ECON302_Assignment2_F11_Sol_Full - Page 1 of 8 Pages ECON...

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Page 1 of 8 Pages ECON 302 Macroeconomic Theory 2 Instructor: Sharif F. Khan Department of Economics University of Waterloo Fall 2011 Suggested Solutions to Assignment 2 (Optional) Total Marks: 50 Marks Read each part of the question very carefully. Show all the steps of your calculations to get full marks, unless it is mentioned otherwise. 1. [20 marks] Suppose that the representative consumer’s preferences are given by the utility function ( ) , ln ln , l C l C U α + = where C is consumption and l is leisure. The consumer’s budget constraint is given by ( ) T l h w C- +- = π , where w is the real wage, h is the quantity of time the consumer has available, π is dividend income, and T is the lump-sum tax. a) Set up the consumer’s optimization problem. The representative consumer’s problem is to choose C and l to maximize his or her utility subject to the budget constraint: max ( ) l C l C U ln ln , α + = C, l Subject to ( ) T l h w C- +- = π b) Write down the Lagrangian associated with this consumer’s optimization problem. The Lagrangian associated with this consumer’s optimization problem is: ( ) [ ] C T l h w l C L-- +- + + = π λ α ln ln . c) Assume that there is an interior solution to the consumer’s problem. Determine the consumer’s optimal choice of consumption and leisure. Illustrate your results on a diagram. Page 2 of 8 Pages Given an interior solution to the consumer’s problem, we can characterize the solution by the first-order conditions from the problem of choosing C , l , and λ to maximize L . max ( ) [ ] C T l h w l C L-- +- + + = π λ α ln ln C , l , λ The first-order necessary conditions: 1 =- = ∂ ∂ λ C C L λ = ⇒ C 1 (1) =- = ∂ ∂ w l l L λ α w l λ α = ⇒ (2) ( ) =-- +- = ∂ ∂ C T l h w L π λ ( ) T l h w C- +- = ⇒ π (3) Dividing Equation (2) by Equation (1) we get: λ λ α w C l = 1 w C l = × ⇒ 1 α α wl C = ⇒ (4) Substituting C from Equation (3) using Equation (4) we get: ( ) T l h w C- +- = π T wl wh wl- +- = ⇒ π α T wh wl wl- + = + ⇒ π α w T h l l- + = + ⇒ π α Page 3 of 8 Pages w T h l- + = + ⇒ π α 1 1 w T h l- + = + ⇒ π α α 1 - + + + = ⇒ w T h l π α α α α 1 1 (5) Substituting l from Equation (4) using Equation (5) we get: α wl C = - + + + = ⇒ w T h w C π α α α α α 1 1 +- + + = ⇒ α π α 1 1 T wh C (6) Therefore, the consumer’s optimal choices of consumption and leisure are, respectively, as follows: +- + + = α π α 1 1 * T wh C - + + + = w T h l π α α α α 1 1 * Figure 1 illustrates the consumer’s optimal choices of consumption and leisure where...
View Full Document

{[ snackBarMessage ]}

### Page1 / 12

ECON302_Assignment2_F11_Sol_Full - Page 1 of 8 Pages ECON...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online