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
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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 =
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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
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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
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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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