Duality (ii) - Economics 21 Intermediate Microeconomics...

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Economics 21: Intermediate Microeconomics Topic 2: Consumer Theory – Duality Theory (ii) 1 Economics 21: Intermediate Microeconomics Topic 2: Consumer Theory Duality Theory (ii) Reference: Varian, p.93 Outline: I. Introduction II. Cobb-Douglas Preferences I. Introduction We reconsider duality Theory in terms of a Cobb-Douglas form example where u x 1 , x 2 ( ) = Ax 1 b x 2 1 ! b . II. Marshallian Demand Function The consumer’s problem is to: max x 1 , x 2 Ax 1 b x 2 1 ! b st. p 1 x 1 + p 2 x 2 = m (1) We obtain the solution via the following Lagrangian: max x 1 , x 2 , ! { } L g = Ax 1 b x 2 1 " b + ! m " p 1 x 1 " p 2 x 2 ( ) (2) The three first-order conditions are: ! L g ! x 1 = bAx 1 b " 1 x 2 1 " b " # p 1 = 0 (3) . ! L g ! x 2 = 1 " b ( ) Ax 1 b x 2 " b " # p 2 = 0 . (4) ! L g !" = m # p 1 x 1 # p 2 x 2 = 0 (5) Dividing (3) by (4):
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Economics 21: Intermediate Microeconomics Topic 2: Consumer Theory – Duality Theory (ii) 2 bAx 1 b ! 1 x 2 1 ! b 1 ! b ( ) Ax 1 b x 2 ! b = " p 1 " p 2 # bx 1 b ! 1 x 2 1 ! b 1 ! b ( ) x 1 b x 2 ! b = bx 2 1 ! b + b 1 ! b ( ) x 1 b ! b + 1 = bx 2 1 ! b ( ) x 1 = p 1 p 2 # x 1 = b 1 ! b ( ) p 2 p 1 x 2 (6) Rearrange (5): x 2 = m p 2 ! p 1 p 2 x 1 (7) Substitute (7) in (6): x 1 = b 1 ! b " # $ % & p 2 p 1 x 1 = b 1 ! b " # $ % & p 2 p 1 m p 2 ! p 1 p 2 x 1 " # $ % & = b 1 ! b " # $ % & m p 1 ! x 1 " # $ % & ( x 1 + b 1 ! b " # $ % & x 1 = x 1 1 + b 1 ! b " # $ % & = b 1 ! b " # $ % & m p 1 " # $ % & ( x 1 1 ! b + b 1 ! b " # $ % & = x 1 1 1 ! b " # $ % & = b 1 ! b " # $ % & m p 1 " # $ % & ( x 1 = b 1 ! b " # $ % & m p 1 " # $ % & 1 ! b 1 " # $ % &
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