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 consumers preferences are given by the utility function ( ) , ln ln , l C l C U + = where C is consumption and l is leisure. The consumers budget constraint is given by ( ) T l h w C- +- = , where w is the real wage, h is the quantity of time the consumer has available, is dividend income, and T is the lump-sum tax. a) Set up the consumers optimization problem. The representative consumers 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 , + = C, l Subject to ( ) T l h w C- +- = b) Write down the Lagrangian associated with this consumers optimization problem. The Lagrangian associated with this consumers optimization problem is: ( ) [ ] C T l h w l C L-- +- + + = ln ln . c) Assume that there is an interior solution to the consumers problem. Determine the consumers optimal choice of consumption and leisure. Illustrate your results on a diagram. Page 2 of 8 Pages Given an interior solution to the consumers problem, we can characterize the solution by the first-order conditions from the problem of choosing C , l , and to maximize L . max ( ) [ ] C T l h w l C L-- +- + + = ln ln C , l , The first-order necessary conditions: 1 =- = C C L = C 1 (1) =- = w l l L w l = (2) ( ) =-- +- = C T l h w L ( ) T l h w C- +- = (3) Dividing Equation (2) by Equation (1) we get: w C l = 1 w C l = 1 wl C = (4) Substituting C from Equation (3) using Equation (4) we get: ( ) T l h w C- +- = T wl wh wl- +- = T wh wl wl- + = + w T h l l- + = + Page 3 of 8 Pages w T h l- + = + 1 1 w T h l- + = + 1 - + + + = w T h l 1 1 (5) Substituting l from Equation (4) using Equation (5) we get: wl C = - + + + = w T h w C 1 1 +- + + = 1 1 T wh C (6) Therefore, the consumers optimal choices of consumption and leisure are, respectively, as follows: +- + + = 1 1 * T wh C - + + + = w T h l 1 1 * Figure 1 illustrates the consumers 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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