CONSUMER CHOICE 1.1. Unit of Analysis and Preferences. The
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Primal and Dual Approaches to the Analysis of Risk Aversion
378.752 D34 W-01-8 Primal and Dual Approaches To the Analysis of Risk Aversion bi Robert G. Chambers and John Quiggin WP 01-08 Waite Lilorary Dept. Of Appllied Economics Unilwrsfty of Minnesota 1994 Buford Ave - St. 232 CilaOff Paull MN 55108-6040 Department of Agricultural and Resource Economics The University of Maryland, College Park e Primal and dual approaches to the analysis of risk aversion Robert G. Chambers' and John Quiggin2 Preliminary draft for discussion purposes only, please do not cite or quote. July 9, 2001 'Professor of Agricultural and Resource Economics, University of Maryland and Adjunct Pro- fessor of Agricultural and Resource Economics, University of Western Australia 2Australian Research Council Fellow, Australian National University In a classic paper, Peleg and Yaari (1975) observe that the marginal rate of substitu- tion for risk neutral decisionmakers between state-contingent income claims is given by the relative probabilities. Thus, probabilities play the same role as prices in the traditional producer and consumer choice problems. More generally, for risk-averse individuals, the marginal rates of substitution between state-contingent incomes may be interpreted as rel- ative 'risk-neutral' probabilities, a point developed further by Nau (2001). This analogy is particularly important in the case where there are spanning portfolios. In that case, after normalization, the induced Arrow-Debreu contingent claims prices can be interpreted as the equilibrium risk-neutral probabilities for all individuals participating in the market. These observations suggest that preferences under uncertainty can be informatively ex- amined in much the same fashion that one analyzes consumer preferences under certainty, that is, in terms of convex indifference sets and their supporting hyperplanes. -
Notes on Microeconomic Theory
Notes on Microeconomic Theory Nolan H. Miller August 18, 2006 Contents 1 The Economic Approach 1 2ConsumerTheoryBasics 5 2.1CommoditiesandBudgetSets............................... 5 2.2DemandFunctions..................................... 8 2.3ThreeRestrictionsonConsumerChoices......................... 9 2.4AFirstAnalysisofConsumerChoices.......................... 10 2.4.1 ComparativeStatics................................ 11 2.5Requirement1Revisited:Walras’Law.......................... 11 2.5.1 What’stheFunnyEqualsSignAllAbout?.................... 12 2.5.2 Back to Walras’ Law: Choice Response to a Change in Wealth . 13 2.5.3 TestableImplications............................... 14 2.5.4 Summary:HowDidWeGetWhereWeAre?.................. 15 2.5.5 Walras’Law:ChoiceResponsetoaChangeinPrice.............. 15 2.5.6 ComparativeStaticsinTermsofElasticities................... 16 2.5.7 WhyBother?.................................... 17 2.5.8 Walras’ Law and Changes in Wealth: Elasticity Form . 18 2.6 Requirement 2 Revisited: Demand is Homogeneous of Degree Zero. 18 2.6.1 Comparative Statics of Homogeneity of Degree Zero . 19 2.6.2 AMathematicalAside................................ 21 2.7Requirement3Revisited:TheWeakAxiomofRevealedPreference.......... 22 2.7.1 CompensatedChangesandtheSlutskyEquation................ 23 2.7.2 Other Properties of the Substitution Matrix . 27 i Nolan Miller Notes on Microeconomic Theory ver: Aug. 2006 3 The Traditional Approach to Consumer Theory 29 3.1BasicsofPreferenceRelations............................... 30 3.2FromPreferencestoUtility............................... -
Economics 250A Lecture 1: a Very Quick Overview of Consumer Choice
Economics 250a Lecture 1: A very quick overview of consumer choice. 1. Review of basic consumer theory 2. Functional form, aggregation, and separability 3. Discrete choice Some recommended readings: Angus Deaton and John Muellbauer, Economics and Consumer Behavior, Cambridge Press, 1980 Geoffrey Jehle and Philip Reny, Advanced Microeconomic Theory (2nd ed), Addision Wes- ley, 2001 John Chipman, "Aggregation and Estimation in the Theory of Demand" History of Po- litical Economy 38 (annual supplement), pp. 106-125. Kenneth Train. Discrete Choice Methods with Simulation, Cambridge Press 2003. Kenneth A. Small and Harvey A. Rosen "Applied Welfare Economics with Discrete Choice Models." Econometrica, 49 (January 1981), pp. 105-130. 1. Review of basic consumer theory a. Basic assumptions and the demand function. ` We start with a choice set X (closed, bounded below); often (though not always) X=R+, the positive orthant. Preferences over alternatives in X are represented by u : X R, twice continuously differentiable, strictly increasing, strictly quasi concave. (u is s.q.c.! if u(x1) u(x2) u( x1 + (1 )x2) > u(x2) for (0, 1]). ≥ ) 2 ` The demand function x(p, I) maps from prices p R++ and income I R+ to a "prefer- ence maximal choice" x X : 2 2 2 x(p, I) = arg max u(x) s.t. px I. x X ≤ 2 Under the preceding assumptions x(p, I) exists and is well-defined, and is continuous in (p, I) for all (p, I) such that the interior of the "budget set" = x X : px I is non-empty. f 2 ≤ g 0 0 0 x(p, I) is also homogeneous of degree 0 (HD0) in (p, I).