lecture_08.pdf - GARP and Afriats Theorem Production Econ...

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GARP and Afriat°s Theorem Production Econ 2100 Fall 2017 Lecture 8, September 21 Outline 1 Generalized Axiom of Revealed Preferences 2 Afriat°s Theorem 3 Production Sets and Production Functions 4 Pro±ts Maximization, Supply Correspondence, and Pro±t Function 5 Hotelling and Shephard Lemmas 6 Cost Minimization
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From Last Class f x j ; p j ; w j g N j = 1 is a ±nite set of demand data that satisfy p j ° x j ± w j . Directly revealed weak preference : x j % R x k if p j ° x k ± w j Directly revealed strict preference : x j ² R x k if p j ° x k < w j Indirectly revealed weak preference : x j % R x i 1 ; if there exists x i 1 % R x i 2 ; x j % I x k x i 1 ; x i 2 ; : : : ; x i m : : : such that x i m % R x k Indirectly revealed strict preference : x j % R x i 1 ; if there exists x i 1 % R x i 2 ; and one of the relations x j ² I x k x i 1 ; x i 2 ; : : : ; x i m : : : such that x i m % R x k in the chain is strict Using these de±nitions, we can ±nd a conditions that guarantees choices are the result of the maximizing a preference relation or a utility function.
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Generalized Axiom of Revealed Preference Axiom (Generalized Axiom of Revealed Preference - GARP) If x j % R x k , then not ° x k ² I x j ± . If a choice is directly revealed weakly prefered to a bundle, this bundle cannot be indirectly revealed strictly preferred to that choice. Note that if p j ° x j < w j then x j ² R x j and therefore x j ² I x j and therefore GARP cannot hold. Therefore, GARP implies the consumer spends all her money. Now suppose x j % R x k and x k ² I x j . Then there is a chain of weak preference from j to k , and a chain back (from k to j ) that has at least one strict preference. This is a cycle with a strict preference inside; if such a cycle exists one can construct a ²money pump³ to make money o/ the consumer and leave her exactly at the bundle she started from. GARP rules these cycles out. Problem 4, Problem Set 4. Show that GARP is equivalent to the following: If x j % I x k then not ° x k ² I x j ± .
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Utility and Demand Data De±nition A utility function u : R n + ! R rationalizes a ±nite set of demand data f x j ; p j ; w j g N j = 1 if u ( x j ) ³ u ( x ) whenever p j ° x ± w j This de±nition can alternatively be stated by the condition x j 2 arg max x 2 R n + u ( x ) subject to p j ° x ± w j If the observations satisfy the de±nition, the data is consistent with the behavior of an individual who, in each observation, chooses a utility-maximizing bundle subject to the corresponding budget constraint.
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Afriat°s Theorem Theorem (Afriat) Given a °nite set of demand data f ( x j ; p j ; w j ) g N k = 1 , the following are equivalent: 1 There exists a locally nonsatiated utility function which rationalizes the data. 2 The data satisfy the Generalized Axiom of Revealed Preference. 3 There exist numbers v j ; ° j > 0 such that v k ± v j + ° j [ p j ° x k ´ w j ] ; for all j ; k = 1 ; : : : ; N (G) 4 There exists a continuous, strictly increasing, and concave utility function which rationalizes the data.
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