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

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Econ 302: Intermediate Microeconomic Theory Andres Elberg University of Illinois at Urbana-Champaign September 8, 2008 Lecture 4: Utility Function September 8, 2008 1 / 45
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Summary from Last Lecture In our last lecture we started studying consumer preferences. We represented these preferences using a preference relation & which tells us how the consumer compares two bundles. if x & y then bundle x is at least as good as bundle y if x ± y then bundle x is strictly preferred to bundle y if x ² y then the consumer is indi/erent between bundles x and y Lecture 4: Utility Function September 8, 2008 2 ± 45
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Summary from Last Lecture In order to guarantee that consumer preferences are consistent we made three assumptions on preferences: 1 Completeness. The consumer can compare any two bundles in the consumption set 2 Re&exivity. Any bundle is at least as good as itself 3 Transitivity. For any three bundles x , y and z , if x & y and y & z then x & z Lecture 4: Utility Function September 8, 2008 3 / 45
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Summary from Last Lecture We represented the consumer preferences in an x 1 - x 2 plane using indi/erence curves. An indi/erence curve is the set of all bundles which are equally preferred by the consumer. If preferences are complete, each bundle in the consumption set lies on an indi/erence curve. If preferences are transitive, then indi/erence curves cannot cross. Lecture 4: Utility Function September 8, 2008 4 ± 45
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Summary from Last Lecture We looked at several examples of how di/erent preferences can be represented using indi/erence curves 1 Perfect substitutes 2 Perfect complements 3 Economic "bads" 4 Neutral commodities 5 Discrete goods Lecture 4: Utility Function September 8, 2008 5 ± 45
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Summary from Last Lecture Finally, we made two additional assumptions that de&ne "well-behaved" preferences 1 Monotonicity. 2 Convexity. Lecture 4: Utility Function September 8, 2008 6 / 45
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Monotonicity If x 1 & x 2 , y 1 & y 2 , and either x 1 > y 1 or x 2 > y 2 then ( x 1 , x 2 ) ± ( y 1 , y 2 ) That is, more of a commodity is preferred to less and hence both commodities are economic "goods". Monotonicity implies that indi/erence curves are negatively sloped. Lecture 4: Utility Function September 8, 2008 7 ± 45
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Convexity The weakly preferred set is a convex set. That is, a weighted average of any two bundles in the weakly preferred set must also be in the weakly preferred set. This implies that consumers prefer balanced consumption than extremes. You don&t want a lot of soda and no pizza but a mix Lecture 4: Utility Function September 8, 2008 8 / 45
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Convexity Geometrically, the weighted average between two points lies at some point on the straight line connecting the two points. Thus, in order to &gure out whether preferences are convex we need to check whether a straight line connecting any two points on the indi/erence curve lies entirely in the weakly preferred set.
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09-08-08 - Econ 302: Intermediate Microeconomic Theory...

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