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# lect4 - Calculating Income and Substitution Eects...

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Calculating Income and Substitution E/ects Intermediate Microeconomics Amy Brown University of California, Los Angeles August 13, 2008 A. Brown (UCLA) Econ 11 Lecture 4 08±13±08 1 ± 35

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Outline of Lecture 1 Review of Income/Substitution E±ects 2 Slutsky Equation 3 4 Duality 5 6 Break 7 Elasticities of Demand 8 Gross and Net Substitutes and Complements 9 Consumer Surplus This should cover the rest of chapter 5 and all we will do of chapter 6 in Nicholson. A. Brown (UCLA) Econ 11 Lecture 4 08/13/08 2 / 35
E/ect of Own Price Changes B is constructed to give the same utility as A, but at the new prices. The income di/erence between B and C measures utility loss due to price change. An increase in p x causes substitution away from A to B. The increase in p x e/ectively reduces income from B to C. A. Brown (UCLA) Econ 11 Lecture 4 08±13±08 3 ± 35

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Slutsky Equation The Slutsky equation is a formal derivation of the income and substitution e/ects within the total change in demand from a change in price: x p x = x c p x x x M income e/ect. A. Brown (UCLA) Econ 11 Lecture 4 08±13±08 4 ± 35
Substitution E/ect a level of utility U . e/ect. A. Brown (UCLA) Econ 11 Lecture 4 08±13±08 5 ± 35

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Derivation of Slutsky Equation Point A can be interpreted as the solution to either the utility maximization problem or the expenditure minimization problem. Thus M = E ( p x , p y , U ) and x c ( p x , p y , M ) = x ( p x , p y , E ( p x , p y , U )) A. Brown (UCLA) Econ 11 Lecture 4 08/13/08 6 / 35
Derivation of Slutsky Equation At point A, x c ( p x , p y , M ) = x ( p x , p y , E ( p x , p y , U )) , and we take derivatives of each side: x c ( p x , p y , M ) p x = x ( p x , p y , E ( p x , p y , U )) p x x c p x = x p x + x E ( p x , p y , U ) E ( p x , p y , U ) p x using the derivative chain rule. A. Brown (UCLA) Econ 11 Lecture 4 08/13/08 7 / 35

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Application of the Envelope Theorem. De±nition price yields the demand function for that good. E p x = L p x = x c A. Brown (UCLA) Econ 11 Lecture 4 08/13/08 8 / 35
Recall the expenditure minimization Lagrangian: L ( p x , p y ; λ ) = p x x + p y y + λ ( U U ( x , y )) (1) We know that the expenditure function is: E ( p x , p y , U ) = p x x c + p y y c (2) Recall that the optimal choice for x and y will guarantee that U = U ( x c , y c ) . Therefore U U ( x c , y c ) = 0, and we add it to ( 2 ) : E ( p x , p y , U ) = p x x c + p y y c + λ ( U

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## This note was uploaded on 09/23/2008 for the course ECON 11 taught by Professor Cunningham during the Summer '08 term at UCLA.

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lect4 - Calculating Income and Substitution Eects...

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