Table Of ContentLecture 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
