ECON302_Assignment2_F11_Sol_Full

ECON302_Assignment2_F11_Sol_Full - Page 1 of 8 Pages ECON...

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

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