Lecture 7 - Economics 101A (Lecture 7) Stefano DellaVigna...

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Economics 101A (Lecture 7) Stefano DellaVigna September 17, 2009
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Outline 1. Utility maximization — Tricky Cases 2. Indirect Utility Function 3. Comparative Statics (Introduction) 4. Income Changes
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1 Utility maximization — tricky cases First, re-solve CES utility function. max x 1 ,x 2 ³ αx ρ 1 + βx ρ 2 ´ 1 s.t. p 1 x 1 + p 2 x 2 M =0 Solution: x 1 = M p 1 Ã 1+ ³ α β ´ 1 ρ 1 ³ p 2 p 1 ´ ρ ρ 1 ! x 2 = M p 2 Ã ³ β α ´ 1 ρ 1 ³ p 1 p 2 ´ ρ ρ 1 !
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Special case 1: ρ 1 (Perfect Substitutes) lim ρ 1 1 ρ 1 = lim ρ 1 ρ ρ 1 = −∞ (here notice the convergence from the left) If α β p 2 p 1 > 1 (or p 1 /p 2 <α/β ), Ã α β ! 1 ρ 1 Ã p 2 p 1 ! ρ ρ 1 0 x 1 M/p 1 If α β p 2 p 1 < 1 p 1 /p 2 >α/β ), Ã α β ! 1 ρ 1 Ã p 2 p 1 ! ρ ρ 1 →∞ x 1 0
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Solution for Perfect Substitutes Case is x 1 = 0 if p 1 /p 2 >α/β M/p 1 if p 1 /p 2 <α/β any x 1 [0 ,M/p 1 ] if p 1 /p 2 = α/β x
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Lecture 7 - Economics 101A (Lecture 7) Stefano DellaVigna...

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