09-22-08 - Econ 302 Intermediate Microeconomic Theory...

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Econ 302: Intermediate Microeconomic Theory Andres Elberg University of Illinois at Urbana-Champaign September 22, 2008 Lecture 8: Comparative Statics September 22, 2008 1 / 53
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Comparative Statics Income changes In our last lecture, we started examining how changes in income a/ect the consumer&s optimal choice. An increase in income can either cause the demand for a good to increase or decrease. The theory does not rule out any of the cases. Empirically it is observed that for most goods their demands change in the same direction as the change in income (case of normal goods). Goods whose demand change in the opposite direction to the change in income are called inferior goods. These are goods with strongly preferred, more expensive, substitutes Lecture 8: Comparative Statics September 22, 2008 2 ± 53
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Changes in Income Income-o/er and Engel curves There are two ways for summarizing how a consumer purchasing decisions vary with changes in income: 1 Income-o/er curve Shows the combinations of good 1 and good 2 demanded at di/erent levels of income 2 Engel curve Shows the demand for one of the goods as a function of income Lecture 8: Comparative Statics September 22, 2008 3 ± 53
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Income O/er Curve x 2 m’ /p 2 m /p 2 x 1 m’ /p 1 m /p 1 Lecture 8: Comparative Statics September 22, 2008 4 ± 53
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Engel Curve m x 1 Lecture 8: Comparative Statics September 22, 2008 5 / 53
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Changes in Income Income-o/er and Engel curves We will now examine the income e/ects for demands generated from di/erent types of preferences 1 Cobb-Douglas 2 Perfect substitutes 3 Perfect complements 4 Quasilinear Lecture 8: Comparative Statics September 22, 2008 6 ± 53
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Income-O/er and Engel Curves Cobb-Douglas Preferences Recall that with Cobb-Douglas preferences ( u ( x 1 , x 2 ) = x a 1 x 1 & a 2 ) the demand functions are x 1 ( p 1 , p 2 , m ) = a m p 1 x 2 ( p 1 , p 2 , m ) = ( 1 & a ) m p 2 Lecture 8: Comparative Statics September 22, 2008 7 ± 53
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Income-O/er and Engel Curves Cobb-Douglas Preferences The equation of the Engel curve for good 1 is m = p 1 a x 1 This is a straight line that goes through the origin with slope dm / dx 1 = p 1 / a . Lecture 8: Comparative Statics September 22, 2008 8 ± 53
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Engel Curve Cobb-Douglas Preferences m Slope = p 1 /a x 1 Lecture 8: Comparative Statics September 22, 2008 9 / 53
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Income-O/er and Engel Curves Cobb-Douglas Preferences Notice that the ratio x 1 / x 2 does not depend on income: x 1 x 2 = a ( 1 & a ) p 2 p 1 (1) Hence, as we vary income, the optimal bundles will lie along a ray from the origin. The income-o/er curve is in this case a straight line with equation: x 2 = ( 1 & a ) a p 1 p 2 x 1 (2) Lecture 8: Comparative Statics September 22, 2008 10 ± 53
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Income-O/er Curve Cobb-Douglas Preferences x 2 m’’ /p 2 m’ /p 2 m /p 2 x / / 1 m’ /p 1 m /p 1 Lecture 8: Comparative Statics September 22, 2008 11 ± 53
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Income-O/er and Engel Curves Perfect Substitutes Recall the demand function for good 1 in the case of perfect substitutes: x 1 ( p 1 , p 2 , m ) = 8 > > > > > > < > > > > > > : m / p 1 if p 1 < p 2 unde&ned if p 1 = p 2 0 if p 1 > p 2 Lecture 8: Comparative Statics September 22, 2008 12 ± 53
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Income-O/er and Engel Curves Perfect Substitutes For the case in which p 1 < p
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This note was uploaded on 02/27/2009 for the course ECON 302 taught by Professor Toossi during the Fall '08 term at University of Illinois at Urbana–Champaign.

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09-22-08 - Econ 302 Intermediate Microeconomic Theory...

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