GWLecture17+18

# GWLecture17+18 - Part IIA: Paper 1 Duality Theory Dr Snje...

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1 Part IIA: Paper 1 Duality Theory Dr Sönje Reiche

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2 Outline Utility Maximization Problem (UMP) Marshallian Demand Indirect Utility Function Expenditure Minimization Problem (EMP) Hicksian Demand Expenditure Function Duality
Outline Motivation Slutzky Equation Integrability Problem Welfare effects of price changes Books Cowell: Ch. 4.5 - 4.6 Gravelle and Rees: Ch. 3 A, B, C 3

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The UMP The consumer chooses a bundle that max imizes his utility for a given set of prices and wealth m : subject to This program gives the Marshall ian or uncompensated demand functions Substituting those into the objective function gives the indirect utility function 4 ) , ( max 2 1 , 2 1 x x u x x m x p x p + 2 2 1 1 ) , ( m p x i ) , ( 2 1 p p p = )) , ( ), , ( ( ) , ( 2 1 m p x m p x u m p v =
UMP 5 FOC: The tangency condition! Use the constraint to solve for , and 1 1 p x u λ = 2 2 p x u = 2 1 2 1 / / p p x u x u = m x p x p + 2 2 1 1 ) , ( 1 m p x ) , ( 2 m p x ) , ( m p v

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The Indirect Utility function Maximum level of utility achievable with income m given prices Depends indirectly on prices and money Use v to evaluate the effects of a change in prices and/or income on the consumer’s well being v is increasing in m Lagrange mulitplier of the UMP v is quasi-convex in prices and income 6 0 = λ m v
Roy’s Identity An increase in reduces the consumer’s purchasing power of income First-order effect : A decrease in purchasing power reduces indirect utility by The overall effect on v is the product of those two: 7 i x - i p m m v i i x m v p v - =

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Roy’s Identity 8 + = + = i i i i i p x p p x p p x u p x u p v 2 2 1 1 2 2 1 1 λ 0 2 2 1 1 = + + i i i x p x p p x p i i x p v - = 0 = i dp dm )) , ( ), , ( ( ) , ( 2 1 m p x m p x u m p v = m x p x p = + 2 2 1 1
The Dual Problem: EMP The consumer chooses a bundle that

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## This note was uploaded on 03/07/2012 for the course ECON 201 taught by Professor Cowell during the Spring '10 term at LSE.

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GWLecture17+18 - Part IIA: Paper 1 Duality Theory Dr Snje...

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