General Equilibrium Theory

379 Pages·2012·1.55 MB·English

by  Ross M. Starr

GENERAL EQUILIBRIUM THEORY
Second Edition
GeneralEquilibriumTheory:AnIntroductionpresentsthemathematicaleconomic
theoryofpricedeterminationandresourceallocationfromelementarytoadvanced
levels, suitable for advanced undergraduates and graduate students of economics.
This Arrow-Debreu model (known for two of its most prominent founders, both
NobelLaureates)isthebasisofmodernpricetheoryandofawiderangeofappli-
cations.Thetextstartswithelementarymodels:RobinsonCrusoe,theEdgeworth
Box, and a two-commodity two-household two-firm model. It gives a brief intro-
ductiontothemathematicsusedinthefield(continuity,convexity,separationtheo-
rems,Brouwerfixed-pointtheorem,point-to-setmappings,andShapley-Folkman
theorem).Itthenpresentsthemathematicalgeneralequilibriummodelinprogres-
sively more general settings, including point-valued, set-valued, and nonconvex
set-valueddemandandsupply.Existenceofgeneralequilibrium,fundamentalthe-
oremsofwelfareeconomics,coreconvergence,andfuturesmarketswithtimeand
uncertainty are treated fully. This new edition updates the discussion throughout
andexpandsthe number andvariety of exercises.It offers a revised andextended
treatment of core convergence, including the case of nonconvex preferences, and
introducestheinvestigationofapproximateequilibriumwithU-shapedcostcurves
andnonconvexpreferences.
RossM.StarrisProfessorofEconomicsattheUniversityofCalifornia,SanDiego,
where he has taught since 1980. He has also served on the faculties of Yale Uni-
versity,theLondonSchoolofEconomics,andtheUniversityofCalifornia,Davis,
and he held a Guggenheim Fellowship at the University of California, Berkeley.
Professor Starr’s research focuses on general equilibrium theory, mathematical
economics,andmonetarytheory.Heistheeditorof GeneralEquilibriumModels
ofMonetaryEconomies(1989)andcoeditorofthethree-volumeEssaysinHonor
ofKennethArrow(CambridgeUniversityPress,1986).Hisarticleshaveappeared
in journals such as Econometrica; Economic Theory; Journal of Economic The-
ory; Journal of Money, Credit, and Banking; Quarterly Journal of Economics;
and Review of Economic Studies. Cambridge University Press published the first
editionofGeneralEquilibriumTheory:AnIntroductionin1997.

GENERAL EQUILIBRIUM THEORY
An Introduction
Second Edition
ross m. starr
UniversityofCalifornia,SanDiego
cambridgeuniversitypress
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,Sa˜oPaulo,Delhi,Tokyo,MexicoCity
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Starr,RossM.
Generalequilibriumtheory:anintroduction/RossM.Starr.–2nded.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-521-82645-7(hardback)
1.Equilibrium(Economics) 2.Economics,Mathematical. I.Title.
HB145.S8 2011
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Contents
Listofillustrations pagexiii
Introductiontothesecondedition xv
Prefacetothesecondedition xxi
Prefacetothefirstedition xxiii
Tableofnotation xxv
Tableofassumptions xxix
A Generalequilibriumtheory:Gettingacquainted 1
1 Conceptandhistoryofgeneralequilibriumtheory 3
1.1 Partialandgeneralequilibrium:Developmentofthefield 3
1.2 Theroleofmathematics 7
1.3 Historyofgeneralequilibriumtheory 8
1.4 Bibliographicnote 10
2 Anelementarygeneralequilibriummodel:TheRobinsonCrusoe
economy 12
2.1 Centralizedallocation 14
2.2 Decentralizedallocation 16
2.3 Paretoefficiencyofthecompetitiveequilibrium
allocation:FirstfundamentalTheoremofWelfareEconomics 23
2.4 Bibliographicnote 24
Exercises 24
3 TheEdgeworthbox 31
3.1 GeometryoftheEdgeworthbox 32
3.2 Calculatinganefficientallocation 35
3.3 AcompetitivemarketsolutionintheEdgeworthbox 37
3.4 Bibliographicnote 40
Exercises 40
vii
viii Contents
4 Integratingproductionandmultipleconsumptiondecisions:
A2×2×2model 44
4.1 A2×2×2model 46
4.2 Technicalefficiency 46
4.3 Paretoefficiency 48
4.4 FirstFundamentalTheoremofWelfareEconomics:
CompetitiveequilibriumisParetoefficient 50
Exercises 52
5 Existenceofgeneralequilibriuminaneconomywithanexcess
demandfunction 58
5.1 Bibliographicnote 64
Exercises 64
B Mathematics 67
6 Logicandsettheory 69
6.1 Quasi-orderings 71
6.2 Functions 73
6.3 Bibliographicnote 73
Exercises 73
7 RN:RealN-dimensionalEuclideanspace 75
7.1 Continuousfunctions 82
7.2 Bibliographicnote 85
Exercises 85
8 Convexsets,separationtheorems,andnonconvexsetsinRN 91
8.1 Separationtheorems 92
8.2 TheShapley-FolkmanTheorem 95
8.3 Bibliographicnote 97
Exercises 98
9 TheBrouwerFixed-PointTheorem 99
9.1 Bibliographicnote 106
Exercises 106
C Aneconomywithboundedproductiontechnologyandsupply
anddemandfunctions 109
10 Markets,prices,commodities,andmathematicaleconomic
theory 112
10.1 Commoditiesandprices 112
10.2 Theformalstructureofpureeconomictheory 112
10.3 Markets,commodities,andprices 113
10.4 Bibliographicnote 114
Exercise 114
Contents ix
11 Productionwithbounded-firmtechnology 115
11.1 Firmsandproductiontechnology 115
11.2 Theformofproductiontechnology 116
11.3 Strictlyconvexproductiontechnology 117
11.4 Aggregatesupply 120
11.5 Attainableproductionplans 120
11.6 Bibliographicnote 121
Exercises 121
12 Households 124
12.1 Thestructureofhouseholdconsumptionsetsandpreferences 124
12.2 Consumptionsets 124
12.3 Representationof(cid:3) :Existenceofacontinuousutilityfunction 129
i
12.4 Choiceandboundednessofbudgetsets,B˜i(p) 131
12.5 Demandbehaviorunderstrictconvexity 134
12.6 Bibliographicnote 137
Exercises 137
13 Amarketeconomy 142
13.1 Firms,profits,andhouseholdincome 142
13.2 ExcessdemandandWalras’sLaw 143
13.3 Bibliographicnote 145
Exercises 145
14 Generalequilibriumofthemarketeconomywithanexcess
demandfunction 147
14.1 Existenceofequilibrium 147
14.2 Bibliographicnote 152
Exercises 152
D Aneconomywithunboundedproductiontechnologyand
supplyanddemandfunctions 161
15 Theoryofproduction:Theunboundedtechnologycase 164
15.1 Unboundedproductiontechnology 164
15.2 Boundednessoftheattainableset 165
15.3 Anartificiallyboundedsupplyfunction 169
15.4 Bibliographicnote 172
Exercises 172
16 Households:Theunboundedtechnologycase 174
16.1 Households 174
16.2 Choiceinanunboundedbudgetset 174
16.3 Demandbehaviorunderstrictconvexity 177
16.4 Bibliographicnote 179
Exercise 179