Competitive equilibrium

Cover
Title page
List of Illustrations
Preface
Acknowledgments
1 Exchange
1.1 Mathematical prerequisites
1.1.1 Sets
1.1.2 Functions
1.1.3 Vector spaces
1.1.4 Linear functionals, hyperplans, and halfspaces
1.2 Walrasian equilibrium
1.2.1 The geometry of exchange
1.2,2 Generalizing the net trade diagram
1.2.3 Walrasian equilibrium and net trades
1.3 Pareto optimality and the core
1.3.1 Pareto optimality
1.3.2 The core
1.4 A numerical example
l.4.1 Walrasian equilibrium, Pareto optima, and the core
1.4.2 Core equivalence
1.5 Commodities and prices
1.5.1 Contingent commodities
1.5.2 Overlapping generations
1.6 Summary
2 Production
2.1 Geometry of vector spaces
2.1.1 Convexity
2.1.2 Linear and affine transformations
2.1.3 Cones
2.1.4 Recession Cones
2.1.5 Polar cones
2.2 CRS production
2,2.1 Activity vectors
2.2.2 Technology sets
2.2.3 Sup and inf
2.2.4 Prices and profit
2.2.5 Consnmption set
2.2.6 Walrasian equilibrium
2.2.7 The net trade diagram
2.2.8 Pareto optimality
2.2.9 The core
2.3 Alternative models of production
2.3.1 The Arrow-Debreu economy
2.3.2 Coalition production economies
2.4 Public goods and joint supply
2.4.1 Marshallian joint supply
2.4.2 Public goods
2.4.3 What is Competition?
2.5 Summary
3 Aumann's model
3.1 Applying Aumann's model
3.1.1 A simple illustration
3.1.2 A type economy
3.2 Dea1ing with non convexity
3.2.1 Nonconvex preferences
3.2,2 A mnrket for automobiles
3.2,3 A market for houses
3.3 Measure and integration
3.3.1 The Riemann integral
3.3.2 The Lebesgue integral
3.3.3 Lebesgue measure
3.3.4 Abstract measure and integration
3.3.5 Nonatomic measure spaces
3.3.6 Formalizing Aumann's model
3.3.7 Economies in distribution form
3.4 Hedonic theory and local public goods
3.4.1 Commodity bundles as measures
3.4.2 Hedonic theory
3.4.3 Tiebout equilibrium
3.5 Summary
4 Topology
4.1 Introduction to topology
4.1.1 Topological spaces
4.1.2 The Eudidean topology
4.1.3 Suhspace and product topologies
4,1.4 Metric topologies
4.1.5 Convergence
4.1.6 Interior, closure, and boundary
4.1.7 Contiuuity
4.1.8 Homeomorphisms
4.1.9 Connectedness and compactness
4.2 Topologies on vector spaces
4.2.1 Topological vector spaces
4.2.2 Topological taxonomy for TVS's
4.2.3 Topological dual spaces
4.2.4 Separation and support by hyperplanes
4.3 Summary
5 Best response
5.1 Preferences
5.1.1 Binary relations and orderings
5.1.2 Neoclassical preferences
5.2 Existence of best response
5.3 Continuity of best response
5.3.1 Graph of a preference relation
5.3.2 Topologies on spaces of subsets
5.3.3 Continuity of correspoudences
5,3.4 Uhc and closed graph
5.3.5 The Maximum Theorem
5.3.6 Applying the Maximum Theorem
5,4 Miscellany
5.5 Summary
6 Clearing markets
6.1 Homogeneity
6.2 Existence of Walrasian equilibrium
6.2.1 The Brouwer Fixed Point Theorem
6.2.2 Free disposal
6.2.3 Simplices
6.2.4 The KKM Theorem
6.3 Computation of equilibria
6.3.1 Simplicial subdivisions and Sperner's Lemma
6.3.2 An algorithm
6.3.3 Choosing labels
6.3.4 Proof of KKM and Brouwer
6.3.5 The Merrill restart algorithm
6.3.6 Some myths about Scarf's algorithm
6.4 The Excess Demand Theorem
6.5 Kakutani Fixed Point Theorem
6.6 Summary
7 Walras meets Nash
7.1 Noncooperative game theory
7.1.1 Nash equilibrium
7.1.2 Cournot oligopoly
7.1.3 Ahstract economies
7.1.4 Existence of equi1ibrium
7.2 Walrasian equilibrium
7.2.1 The Arrow-Debreu model
7.2.2 Elaborations
7.3 External effects
7.3.1 Existence
7.3.2 Beesand blossoms
7.3.3 Tragedy of the commons
7.3.4 A mattcr of optimality
7.4 Nonconvexity
7.4.1 The Shapley-Folkman Theorem
7.4.2 Nonconvexity and equilibrium
7.4.3 The meaning of approximation
7.5 Nonordered preferences
7.5.1 Existence
7.5.2 Drugs and intransitivity
7.5.3 Unmasking a cusp catastrophe
7.6 Summary
8 What is competition?
8.1 The Second fundamental Theorem
8.1.1 Pure exchange
8.1.2 Production
8,2 Core equivalence
8.2.1 The Debreu-Scarf Theorem
8.2.2 Core convergence
8.2.3 The integral of a correspondence
8.2.4 Aumann's Theorem
8.3 Infinite dimensional commodity spaces
8.3.1 Vector spaœs
8.3.2 Linear functionals
8.3.3 Dual pairs and dual topologies
8.3,4 The economic significance of topology
8.3.5 Walrasian equilibrium
8.3.6 Proving existence
8.3.7 Nonempty interior
8.3.8 Properness
8.3.9 A lattice structure for prices
8.4 The large square economy
8.5 Summary
Bibliography
Index