101A_lec9 - Economics 101A (Lecture 9) Stefano DellaVigna...

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Economics 101A (Lecture 9) Stefano DellaVigna February 17, 2009
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Outline 1. Expenditure Minimization II 2. Slutsky Equation 3. Complements and substitutes 4. Do utility functions exist?
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1 Expenditure minimization II Nicholson, Ch. 4, pp. 127-132 (109—113, 9th) + Ch. 5, pp. 151-154 Solve problem EMIN (minimize expenditure): min p 1 x 1 + p 2 x 2 s.t. u ( x 1 ,x 2 ) ¯ u Pick budget set which is tangent to indi f erence curve Optimum coincides with optimum of Utility Maxi- mization! Formally: h i ( p 1 ,p 2 , ¯ u )= x i ( p 1 ,p 2 ,e ( p 1 ,p 2 , ¯ u ))
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Expenditure function is expenditure at optimum e ( p 1 ,p 2 , ¯ u )= p 1 h 1 ( p 1 ,p 2 , ¯ u )+ p 2 h 2 ( p 1 ,p 2 , ¯ u ) h i ( p i ) is Hicksian or compensated demand function Is h i always decreasing in p i ?Y e s ! Graphical proof: moving along a convex indi f erence curve
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Using f rst order conditions: L ( x 1 ,x 2 )= p 1 x 1 + p 2 x 2 λ ( u ( x 1 ,x 2 ) ¯ u ) ∂L ∂x i = p i λu 0 i ( x 1 ,x 2 )=0 Write as ratios: u 0 1 ( x 1 ,x 2 ) u 0 2 ( x 1 ,x 2 ) = p 1 p 2 MRS = ratio of prices as in utility maximization! However: di
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101A_lec9 - Economics 101A (Lecture 9) Stefano DellaVigna...

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