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 •T h e d e m a n d e d b u n d l e i s t h e o p t i m a l b u n d l e a t s o m e s e t o f 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 •C h o o s e , 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) •O r b y u s i n g a m o n o t o n i c t r a n s f o r m a t i o n Choose , to subject to •T h e r e a r e s e v e r a l w a y s t o s o l v e t h i s p r o b l e m . F i r s t 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 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 1 1 2 1 1 D E m x p x p ± 1 1 1 1 m x p x p ± 1 1 1 1 1 1 p m x ±
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Consumer’s Demand (continued) •T h i s i s t h e d e m a n d f u n c t i o n f o r g o o d 1 . S i m i l a r l y , t h e demand function for good 2: h e s e c o n d w a y i s t o s u b s t i t u t e t h e b u d g e t c o n s t r a i n t i n t o the objective function. That is, substituting into the maximization problem at the beginning. That is, 2 2 p m x E D ± 1 2 1 2 1 ) ln( ln max x p p p m x ² ± E
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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 ± ± w w p p x p m p x x E D 2 2 p m x ² 1 1 p m x ²
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ch06lec - Chapter 6: Demand Econ 306 by Rust, Cho, and Diaz...

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