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ch06lec - Chapter 6 Demand Econ 306 by Rust Cho and Diaz...

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Chapter 6: Demand Econ 306 by Rust, Cho, and Diaz
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Consumer’s Demand The demanded bundle is the optimal bundle at some set of prices and income. The demand function is a function that relates the optimal bundle to the different levels of prices and income. To find the demand functions for and for the Cobb- Douglas utility function, the problems that we want to solve is Choose , to subject to 1 x 2 x 1 x E D ± x x x x u 1 2 1 ) , ( max m x p x p ² 2 2 1 1 2 x
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Consumer’s Demand (continued) Or by using a monotonic transformation Choose , to subject to There are several ways to solve this problem. First and, 1 x 2 x ± ² x x x x u log log ) , ( max 1 2 1 E D m x p x p ² 2 2 1 1 2 1 2 1 p p x x MRS ³ ³ E D m x p x p ² 2 2 1 1
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Consumer’s Demand (continued) Which are two equations with two unknown that can be solved for the optimal bundle. Thus, substituting: m p x p p x p ² 2 1 1 2 1 1 D E m x p x p ² 1 1 1 1 D E m x p x p D E D ² 1 1 1 1 1 1 p m x E D D ²
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Consumer’s Demand (continued) This is the demand function for good 1. Similarly, the demand function for good 2: The second way is to substitute the budget constraint into the objective function. That is, substituting into the maximization problem at the beginning. That is, 2 2 p m x E D E ² 1 2 1 2 1 ) ln( ln max x p p p m x ³ ² E D
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Consumer’s Demand (continued) The first order condition for this problem is By a little algebra, we do have the followings: 0 2 1 1 1 2 1 1 ³ ³ w w p p x p m p x x E D 2 2 p m x E D E ² 1 1 p m x E D D ²
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