101A_lec7 - Economics 101A (Lecture 7) Stefano DellaVigna...

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Economics 101A (Lecture 7) Stefano DellaVigna February 10, 2009
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Outline 1. Utility maximization II 2. Utility maximization — Tricky Cases 3. Indirect Utility Function 4. Comparative Statics (Introduction)
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1 Utility Maximization Maximization problem becomes max x 1 ,x 2 u ( x 1 ,x 2 ) s.t. p 1 x 1 + p 2 x 2 M =0 L ( x 1 2 )= u ( x 1 2 ) λ ( p 1 x 1 + p 2 x 2 M ) F.o.c.s: u 0 x i λp i for i =1 , 2 p 1 x 1 + p 2 x 2 M
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Moving the two terms across and dividing, we get: MRS = u 0 x 1 u 0 x 2 = p 1 p 2 Graphical interpretation.
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Second order conditions: H = 0 p 1 p 2 p 1 u 00 x 1 ,x 1 u 00 x 1 ,x 2 p 2 u 00 x 2 ,x 1 u 00 x 2 ,x 2 | H | = p 1 ³ p 1 u 00 x 2 ,x 2 + p 2 u 00 x 2 ,x 1 ´ p 2 ³ p 1 u 00 x 1 ,x 2 + p 2 u 00 x 1 ,x 1 ´ = p 2 1 u 00 x 2 ,x 2 +2 p 1 p 2 u 00 x 1 ,x 2 p 2 2 u 00 x 1 ,x 1 Notice: u 00 x 2 ,x 2 < 0 and u 00 x 1 ,x 1 < 0 usually satis f ed (but check it!). Condition u 00 x 1 ,x 2 > 0 is then su cient
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Example with CES utility function.
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101A_lec7 - Economics 101A (Lecture 7) Stefano DellaVigna...

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