Computational aspects of general equilibrium theory: refutable theories of value PDF
Preview Computational aspects of general equilibrium theory: refutable theories of value
Lecture Notes in Economics and Mathematical Systems 604 FoundingEditors: M.Beckmann H.P.Künzi ManagingEditors: Prof.Dr.G.Fandel FachbereichWirtschaftswissenschaften FernuniversitätHagen Feithstr.140/AVZII,58084Hagen,Germany Prof.Dr.W.Trockel InstitutfürMathematischeWirtschaftsforschung(IMW) UniversitätBielefeld Universitätsstr.25,33615Bielefeld,Germany EditorialBoard: A.Basile,A.Drexl,H.Dawid,K.Inderfurth,W.Kürsten Donald Brown · Felix Kubler Computational Aspects of General Equilibrium Theory Refutable Theories of Value 123 ProfessorDonaldBrown DepartmentofEconomics YaleUniversity 27HillhouseAvenue Room15B NewHaven,CT06520 USA [email protected] ProfessorFelixKubler DepartmentofEconomics UniversityofPennsylvania 3718LocustWalk Philadelphia,PA19104-6297 USA [email protected] ISBN978-3-540-76590-5 e-ISBN978-3-540-76591-2 DOI10.1007/978-3-540-76591-2 LectureNotesinEconomicsandMathematicalSystemsISSN0075-8442 LibraryofCongressControlNumber:2007939284 ©2008 Springer-VerlagBerlinHeidelberg Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerial is concerned, specificallythe rights of translation, reprinting, reuseof illustrations, recitation, broadcasting,reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplication ofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyright LawofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtained fromSpringer.ViolationsareliabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoes notimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Production:LE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig Coverdesign:WMXDesignGmbH,Heidelberg Printedonacid-freepaper 987654321 springer.com To Betty, Vanessa, Barbara and Elizabeth Rose DJB To Bi and He FK Preface This manuscript was typeset in Latex by Mrs. Glena Ames. Mrs. Ames also draftedallthefiguresandeditedtheentiremanuscript.Onlyacademiccustom preventsusfromaskinghertobe aco-author.She hasourheartfeltgratitude for her good humor and her dedication to excellence. New Haven and Philadelphia, Donald J. Brown December 2007 Felix Kubler Contents Refutable Theories of Value Donald J. Brown, Felix Kubler..................................... 1 Testable Restrictions on the Equilibrium Manifold Donald J. Brown, Rosa L. Matzkin................................. 11 Uniqueness, Stability, and Comparative Statics in Rationalizable Walrasian Markets Donald J. Brown, Chris Shannon .................................. 27 The Nonparametric Approach to Applied Welfare Analysis Donald J. Brown, Caterina Calsamiglia............................. 41 Competition, Consumer Welfare, and the Social Cost of Monopoly Yoon-Ho Alex Lee, Donald J. Brown ............................... 47 Two Algorithms for Solving the Walrasian Equilibrium Inequalities Donald J. Brown, Ravi Kannan.................................... 69 Is Intertemporal Choice Theory Testable? Felix Kubler..................................................... 79 Observable Restrictions of General Equilibrium Models with Financial Markets Felix Kubler..................................................... 93 Approximate Generalizations and Computational Experiments Felix Kubler.....................................................109 X Contents Approximate Versus Exact Equilibria in Dynamic Economies Felix Kubler, Karl Schmedders.....................................135 Tame Topology and O-Minimal Structures Charles Steinhorn................................................165 References.....................................................193 List of Contributors Donald J. Brown Rosa L. Matzkin Yale University Northwestern University 27 Hillhouse Avenue 2001 Sheridan Road New Haven, CT 06511 Evanston, IL 60208 [email protected] [email protected] Caterina Calsamiglia Universitat Auto`noma de Barcelona Karl Schmedders Edifici B Northwestern University Bellaterra, Barcelona,Spain 08193 2001 Sheridan Road [email protected] Evanston, IL 60208 Ravi Kannan k-schmedders@kellogg Yale University .northwestern.edu 51 Prospect Street New Haven, CT 06511 [email protected] Chris Shannon University of California at Berkeley Felix Kubler 549 Evans Hall University of Pennsylvania Berkeley, CA 94720 3718 Locust Walk [email protected] Philadelphia, PA 19104-6297 [email protected] Yoon-Ho Alex Lee Charles Steinhorn U.S. Securities & ExchangeCommis- Vassar College sion 124 Raymond Avenue Washington, DC 20549 Poughkeepsie,NY 12604 [email protected] [email protected] Refutable Theories of Value Donald J. Brown1 and Felix Kubler2 1 Yale University,New Haven,CT 06511 [email protected] 2 Universityof Pennsylvania,Philadelphia, PA 19104-6297 [email protected] In the introduction to his classicFoundations of Economic Analysis [Sam47], Paul Samuelson defines meaningful theorems as “hypotheses about empiri- cal data which could conceivably be refuted if only under ideal conditions.” For three decades, the problems of existence, uniqueness and the stability of taˆtonnement were at the core of the general equilibrium research program— see Blaug [Bla92], Ingaro and Israel [II90], and Weintraub [Wei85]. Are the theoremsonexistence,uniquenessandtˆatonnementstabilityrefutablepropo- sitions? To this end, we define the Walrasian hypotheses about competitive mar- kets: H1. Marketdemandis the sumofdemands ofconsumersderivedfromutility maximization subject to budget constraints at market prices. H2. Market prices and consumer demands constitute a unique competitive equilibrium. H3. Market prices are a locally stable equilibrium of the taˆtonnement price adjustment mechanism. The Walrasian model contains both theoretical constructs that cannot be observedsuchasutilityandproductionfunctionsandobservablemarketdata such as market prices, aggregatedemand, expenditures of consumers or indi- vidual endowments. A meaningful theorem must have empirical implications in terms of observable market data. In economic analysis there are two different methodologies for deriving refutable implications of theories. One method, used often in consumer the- ory and the theory of the firm, is marginal, comparative statics, and the other methodology is revealed preference theory. Both methods originated in Samuelson’s Foundations of Economic Analysis. We will follow the revealedpreference approach.The propositionwe shall needisAfriat’sseminaltheoremontherationalizationofindividualconsumer demandincompetitivemarkets[Afr67].Givenafinitenumberofobservations 2 Donald J. Brown and Felix Kubler on market prices and individual consumer demands, his theorem states the equivalence between the following four conditions: (a) The observationsare consistentwith maximizationofa non-satiatedutil- ity function, subject to budget constraints at the market prices, (b) There exists a finite set of utility levels and marginal utilities of income that, jointly with the market data, satisfy a set of inequalities called the Afriat inequalities, (c) Theobservationssatisfyaformofthestrongaxiomofrevealedpreference, involving only market data, (d) The observations are consistent with maximization of a concave, mono- tone, continuous, non-satiated utility function, subject to budget con- straints at the market prices. The striking feature of Afriat’s theorem is the equivalence of these four conditions. In particular, conditions (b) and (c). Moreover, condition (c) ex- hausts all refutable implications of a given data set unlike the necessary, but not sufficient, restrictions derivable from marginal, comparative statics. The Afriat inequalities can be derived from the Kuhn–Tucker first-order conditions for maximizing a concaveutility function subject to a budget con- straint. These inequalities involve two types of variables: parameters and un- knowns. Afriat assumes he can observe not only prices, but also individual demands. The other variables, utility levels, and marginal utilities of income are unknowns. But it follows from Afriat’s theorem that the axiom in (c), a version of the strong axiom of revealed preference, containing only market data:pricesandindividualdemands,isequivalenttotheAfriatinequalitiesin (b) containing the unknowns: utility levels and marginal utilities of income. In going from (b) to (c), Afriat has managed to eliminate all the unknowns. The Afriat inequalities are linear in the unknowns, if individual demands areobserved.Thisisnotthecasewhenrevealedpreferencetheoryisextended from individual demand to market demand, if individual demands are not observed.ThisnonlinearityintheAfriatinequalitiesisthemajorimpediment in generalizing Afriat’s and Samuelson’s program on rationalizing individual demandincompetitivemarketstorationalizingmarketdemandincompetitive markets. There are three general methods for deciding if a system of linear in- equalities is solvable. The first method Fourier–Motzkin elimination, a gen- eralization of the method of substitution taught in high school provides an exponential-time algorithm for solving a system of linear inequalities. In ad- dition there are two types of polynomial-time algorithms for solving systems of linear inequalities: the ellipsoid method and the interior point method. As anillustrationofFourier–Motzkinelimination,supposethat wehavea finitesetoflinearinequalitiesintworealvariablesxandy.Thesolutionsetof thisfamilyofinequalitiesisapolyhedroninR2.ApplyingtheFourier–Motzkin methodtoeliminateyamountstoprojectingthepointsinthepolyhedrononto theX-axis.Indeed,ifxisintheprojection,thenweknowthereexistsaysuch
The list of books you might like
