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_Lecture_06 - Lecture 5 1-minute review Quasi-linear Demand...

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Lecture 5 1-minute review Quasi-linear Demand Functions X ( p x , p y , M ) , Y ( p x , p y , M ) Corner solutions Income/Substitution Effects Slutsky Hicks Compensated Demand Functions (Hicks) 1
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Sometimes pictures/graphs are useful. own price demand function : graph X as a function of p x – keeping p y , M constant other price demand function : graph X as a function of p y – keeping p x , M constant Engel curve : graph X as a function of M – keeping p x , p y constant income expansion path trace optimal choice as M – keeping p x , p y constant 2
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Example: Cobb-Douglas utility function U ( x, y ) = x a y b X ( p x , p y , M ) = a a + b M p x Y ( p x , p y , M ) = b a + b M p y 3
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Do these calculations for CES utility U ( x, y ) = 5 x 1 / 5 + 5 y 1 / 5 Demand functions X ( p x , p y , M ) , Y ( p x , p y , M ) Lagrangian L ( x, y, λ ) = 5 x 1 / 5 + 5 y 1 / 5 - λ ( p x x + p y y - M ) = 0 Equations x - 4 / 5 - λp x = 0 y - 4 / 5 - λp y = 0 p x x + p y y - M = 0 8
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x - 4 / 5 p x = y - 4 / 5 p y y - 4 / 5 = p y p x x - 4 / 5 y 4 / 5 = p x p y x 4 / 5 y = p x p y 5 / 4 x 9
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Plug into last equation p x x + p y p x p y 5 / 4 x = M x = M p x + p y p x p y 5 / 4 y = M p y + p x p y p x 5 / 4 10
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