CHAPTER D List of Theorems
In this appendix, I list the theorems presented in various different chapters for reference. Many of these theorems refer to mathematical results used in different parts of the book.
Some of them are economic results that are more general and widely applicable than the results I labeled “propositions”. To conserve space, I do not list additional mathematical results given in Lemmas, Corollaries and Facts.Chapter 2
Theorem 2.1: Euler’s Theorem.
Theorem 2.2: Stability for Systems of Linear Difference Equations.
Theorem 2.3: Stability for Systems of Nonlinear Difference Equations.
Theorem 2.4: Stability for Systems of Linear Differential Equations.
Theorem 2.5: Stability for Systems of Nonlinear Differential Equations.
Chapter 5
Theorem 5.1: Debreu-Mantel-Sonnenschein Theorem.
Theorem 5.2: Gorman’s Aggregation Theorem.
Theorem 5.3: Existence of a Normative Representative Household.
Theorem 5.4: The Representative Firm Theorem.
Theorem 5.5: The First Welfare Theorem for Economies with Finite Number of Commodities.
Theorem 5.6: The First Welfare Theorem for Economies with Infinite Number of Commodities.
Theorem 5.7: The Second Welfare Theorem.
Theorem 5.8: Equivalence of Sequential and Non-Sequential Trading with Arrow Securities.
Chapter 6
Theorem 6.1: Equivalence of Sequential and Recursive Formulations.
Theorem 6.2: Principle of Optimality in Dynamic Programming.
Theorem 6.3: Existence of Solutions in Dynamic Programming.
Theorem 6.4: Concavity of the Value Function.
Theorem 6.5: Monotonicity of the Value Function.
Theorem 6.6: Differentiability of the Value Function.
Theorem 6.7: The Contraction Mapping Theorem.
Theorem 6.8: Applications of the Contraction Mapping Theorem.
Theorem 6.9: Blackwell’s Sufficient Conditions for a Contraction.
Theorem 6.10: Sufficiency of Euler Equations and the Transversality Condition.
Chapter 7
Theorem 7.1: Variational Necessary Conditions for an Interior Optimum with Free End Points.
Theorem 7.2: Variational Necessary Conditions for Interior Optimum with Fixed End Points.
Theorem 7.3: Simplified version of Pontryagin’s Maximum Principle.
Theorem 7.4: Mangasarian’s Sufficient Conditions for an Optimum.
Theorem 7.5: Arrow’s Sufficient Conditions for an Optimum.
Theorem 7.6: Pontyagin’s Maximum Principle for Multivariate Problems.
Theorem 7.7: Mangasarian’s Sufficient Conditions for Multivariate Problems.
Theorem 7.8: Arrow’s Sufficient Conditions for Multivariate Problems.
Theorem 7.9: Pontyagin’s Maximum Principle for Infinite-Horizon Problems.
Theorem 7.10: Hamilton-Jacobi-Bellman Equations.
Theorem 7.11: Mangasarian’s Sufficient Conditions for Infinite-Horizon Problems.
Theorem 7.12: Arrow’s Sufficient Conditions for Infinite-Horizon Problems.
Theorem 7.13: General Transversality Condition for Infinite-Horizon Problems.
Theorem 7.14: The Maximum Principle and the Transversality Conditions for Discounted Infinite-Horizon Problems.
Theorem 7.15: Mangasarian’s Sufficient Conditions for Discounted Infinite-Horizon Problems.
Theorem 7.16: Arrow’s Sufficient Conditions for Discounted Infinite-Horizon Problems.
Theorem 7.17: Existence of Solutions in Optimal Control.
Theorem 7.18: Saddle Path Stability for Systems of Linear Differential Equations.
Theorem 7.19: Saddle Path Stability for Systems of Nonlinear Differential Equations.
Chapter 10
Theorem 10.1: Separation Theorem for Investment in Human Capital.
Chapter 16
Theorem 16.1: Equivalence of Sequential and Recursive Formulations.
Theorem 16.2: Principle of Optimality in Stochastic Dynamic Programming.
Theorem 16.3: Existence of Solutions in Stochastic Dynamic Programming.
Theorem 16.4: Concavity of the Value Function.
Theorem 16.5: Monotonicity of the Value Function in State Variables.
Theorem 16.6: Differentiability of the Value Function.
Theorem 16.7: Monotonicity of the Value Function in Stochastic Variables.
Theorem 16.8: Sufficiency of Euler Equations and the Transversality Condition.
Theorem 16.9: Existence of Solutions with Markov Processes.
Theorem 16.10: Continuity of the Value Function with Markov Processes.
Theorem 16.11: Concavity of the Value Functions with Markov Processes.
Theorem 16.12: Monotonicity of the Value Functions with Markov Processes.
Theorem 16.13: Differentiability of the Value Function with Markov Processes.
Chapter 22
Theorem 22.1: Arrow’s Impossibility Theorem.
Theorem 22.2: The Median Voter Theorem.
Theorem 22.3: The Median Voter Theorem with Strategic Voting.
Theorem 22.4: Downs’s Policy Convergence Theorem.
Theorem 22.5: The Median Voter Theorem without Single Peaked Preferences.
Theorem 22.6: Downs’s Policy Convergence Theorem without Single Peaked References.
Theorem 22.7: The Probabilistic Voting and Preference Aggregation Theorem.
Appendix Chapter A
Theorem A.1: Properties of Open and Closed Sets in Metric Spaces.
Theorem A.2: Continuity and Open Sets in Metric Spaces.
Theorem A.3: The Intermediate Value Theorem.
Theorem A.4: Continuity and Open Sets in Topological Spaces.
Theorem A.5: Convergence of Nets and Continuity in Topological Spaces.
Theorem A.6: The Heine-Borel Theorem.
Theorem A.7 The Bolzano-Weierstrass Theorem.
Theorem A.8: Continuity and Compactness in Topological Spaces.
Theorem A.9: Weierstrass’s Theorem.
Theorem A.10: Continuity of Projection Maps and the Product Topology.
Theorem A.11: Continuity of Discounted Utilities in the Product Topology.
Theorem A.12: Tychonoff’s Theorem.
Theorem A.13: Berge’s Maximum Theorem.
Theorem A.14: Properties of Maximizers under Quasi-Concavity.
Theorem A.15: Properties of Minimizers under Quasi-Convexity.
Theorem A.16: Kakutani’s Fixed Point Theorem.
Theorem A.17: Brouwer’s Fixed Point Theorem.
Theorem A.18: Mean Value Theorems.
Theorem A.19: L’Hospital’s Rule.
Theorem A.20: Taylor’s Theorem and Taylor Approximations.
Theorem A.21: Taylor’s Theorem for Functions of Several Variables.
Theorem A.22: The Inverse Function Theorem.
Theorem A.23: The Implicit Function Theorem.
Theorem A.24: Continuity of Linear Functionals in Normed Vector Spaces.
Theorem A.25: Geometric Form of the Hahn-Banach Theorem.
Theorem A.26: Separating Hyperplane Theorem.
Theorem A.27: The Saddle —Point Theorem.
Theorem A.28: The Kuhn-Tucker Theorem.
Appendix Chapter B
Theorem B.1: Fundamental Theorem of Calculus I.
Theorem B.2: Fundamental Theorem of Calculus II.
Theorem B.3: Integration by Parts.
Theorem B.4: Leibniz’s Rule.
Theorem B.5: Solution to Systems of Linear Differential Equations with Constant Coefficients.
Theorem B.6: Solution to General Systems of Linear Differential Equations.
Theorem B.7: The Grobman-Hartman Theorem on Stability of Nonlinear Systems of Differential Equations.
Theorem B.8: Picard’s Theorem on Existence and Uniqueness for Differential Equations.
Theorem B.9: Existence and Uniqueness for Differential Equations on Compact Domain.
Theorem B.10: Picard’s Theorem on Existence and Uniqueness for Systems of Differential Equations.
Theorem B.11: Existence and Uniqueness for Systems of Differential Equations on Compact domain.
Theorem B.12: Peano’s Theorem of Existence and Uniqueness for Differential Equations.
Theorem B.13: Continuity of Solutions to Differential Equations.
Theorem B.14: Solution to Systems of Linear Difference Equations with Constant Coefficients.
Theorem B.15: Solution to Systems of Linear Difference Equations with Constant Coefficients.
Theorem B.16: Existence and Uniqueness of Solutions to Difference Equations.
Appendix Chapter C
Theorem C.1: One-Stage Deviation Principle.
Theorem C.2: Existence of Markov Perfect Equilibria in Finite Dynamic Games.
Theorem C.3: Existence of Subgame Perfect Equilibria in Finite Dynamic Games.
Theorem C.4: Relationship between Markov and Subgame Perfect Equilibria.
Theorem C.5: Punishment with the Worst Equilibrium.
Theorem C.6: Punishment with the Minmax Continuation Values.
Theorem C.7: The Folk Theorem for Repeated Games.
Theorem C.8: Uniqueness of Markov Perfect Equilibria in Repeated Games.