Mathematical Analysis and Optimization for Economists

Description

In Mathematical Analysis and Optimization for Economists, the author aims to introduce students of economics to the power and versatility of traditional as well as contemporary methodologies in mathematics and optimization theory; and, illustrates how these techniques can be applied in solving microeconomic problems. This book combines the areas of intermediate to advanced mathematics, optimization, and microeconomic decision making, and is suitable for advanced undergraduates and first-year graduate students. This text is highly readable, with all concepts fully defined, and contains numerous detailed example problems in both mathematics and microeconomic applications. Each section contains some standard, as well as more thoughtful and challenging, exercises. Solutions can be downloaded from the CRC Press website. All solutions are detailed and complete. Features Contains a whole spectrum of modern applicable mathematical techniques, many of which are not found in other books of this type. Comprehensive and contains numerous and detailed example problems in both mathematics and economic analysis. Suitable for economists and economics students with only a minimal mathematical background. Classroom-tested over the years when the author was actively teaching at the University of Hartford. Serves as a beginner text in optimization for applied mathematics students. Accompanied by several electronic chapters on linear algebra and matrix theory, nonsmooth optimization, economic efficiency, and distance functions available for free on www.routledge.com/9780367759018. Cover Half Title Title Page Copyright Page Dedication Contents Preface Author Symbols and Abbreviations Chapter 1. Mathematical Foundations 1 1.1 Matrices and Determinants 1.2 Vector Spaces and Subspaces 1.3 Matrix Inversion 1.4 Solution Set of a System of Simultaneous Linear Equations 1.5 Linear Dependence, Dimension, and Rank 1.6 Hyperplanes and Half-Planes (- Spaces) 1.7 Convex and Finite Cones 1.8 Theorems of the Alternative for Linear Systems 1.9 Quadratic Forms 1.9.1 Basic Structure 1.9.2 Symmetric Quadratic Forms 1.9.3 Classification of Quadratic Forms 1.9.4 Necessary Conditions for the Definiteness and Semi-Definiteness of Quadratic Forms 1.9.5 Necessary and Sufficient Conditions for the Definiteness and Semi-Definiteness of Quadratic Forms 1.9.6 Constrained Quadratic Forms 1.10 Linear Transformations 1.10.1 Matrix Transformations 1.10.2 Properties of Linear Transformations 1.10.3 Solvability of Equations 1.10.4 Matrix transformations Revisited Notes Chapter 2. Mathematical Foundations 2 2.1 Real and Extended Real Numbers 2.2 Single-Valued Functions 2.3 Metric Spaces 2.4 Limits of Sequences 2.5 Point-Set Theory 2.6 Continuous Single-Valued Functions 2.7 Operations on Sequences of Sets Notes Chapter 3. Mathematical Foundations 3 3.1 Beyond Single-Valued Functions 3.2 Limits and Continuity of Transformations 3.3 Derivative of a Single-Valued Function 3.4 Derivatives of Vector-Valued Functions 3.5 Derivatives of Quadratic Functions 3.6 Taylor's Formula 3.6.1 A Single Independent Variable 3.6.2 Generalized Taylor's Formula with Remainder Notes Chapter 4. Mathematical Foundations 4 4.1 Implicit Function Theorems 4.2 Chain or Composite Function Rules 4.3 Functional Dependence Chapter 5. Global and Local Extrema of Real-Valued Functions 5.1 Classification of Extrema 5.2 Global Extrema 5.3 Local Extrema Chapter 6. Global Extrema of Real-Valued Functions 6.1 Existence of Global Extrema 6.2 Existence of Global Extrema: Another Look Chapter 7. Local Extrema of Real-Valued Functions 7.1 Functions of a Single Independent Variable 7.2 A Necessary Condition for a Local Extremum 7.3 A Sufficient Condition for a Local Extremum 7.4 A Necessary and Sufficient Condition for a Local Extremum 7.5 Functions of n Independent Variables 7.6 Economic Applications 7.6.1 Elasticity of Demand 7.6.2 Production and Cost 7.6.3 Elasticity and Total Revenue 7.6.4 Profit Maximization 7.6.4.1 Profit Maximization Under Perfect Competition in the Product and Factor (Labor) Markets 7.6.4.2 Monopoly in the Product Market and Perfect Competition in the Factor (Labor) Market 7.6.4.3 Perfect Competition in the Product Market and Monopsony in the Factor (Labor) Market 7.6.4.4 Monopoly in the Product Market and Monopsony in the Factor (Labor) Market Chapter 8. Convex and Concave Real-Valued Functions 8.1 Convex Sets 8.2 Convex and Concave Real-Valued Functions 8.3 Supergradients of Concave and Subgradients of Convex Functions 8.4 Differentiable Convex and Concave Real-Valued Functions 8.5 Extrema of Convex and Concave Real-Valued Functions 8.6 Strongly α-Concave and Strongly α-Convex Functions [Avriel et al. (2010); Vial (1982, 1983)] 8.7 Conjugate Functions (Rockafellar [1970, 1974]; Fenchel [1949]) 8.7.1 Some Preliminary Notions 8.7.2 Conjugacy Defined 8.8 Conjugate Functions in Economics (Lau[1978]; Diewert [1973,1982]; Blume [2008]; Jorgensen and Lau [1974]; Beckman and Kapur [1977]) Appendix 8.A Alternative Proofs of Theorems 8.9 and 8.10 Note Chapter 9. Generalizations of Convexity and Concavity 9.1 Introduction 9.2 Quasiconcavity and Quasiconvexity 9.3 Differentiable Quasiconcave and Quasiconvex Functions 9.4 Strictly Quasiconcave and Quasiconvex Functions 9.5 Strongly Quasiconcave and Strongly Quasiconvex Functions 9.6 Pseudoconcave and Pseudoconvex Functions Appendix 9.A Additional Thoughts on Theorems 9.2, 9.3, and 9.6 (Cambini and Martein [2009]; Mangasarian [1969]; Borwein and Lewis [2000]) Appendix 9.B Additional Thoughts on Differentiable Pseudoconcave and Pseudoconvex Functions (Cambini and Martein [2009]; Mangasarian [1969]; Borwein and Lewis [2000]) Note Chapter 10. Constrained Extrema: Equality Constraints 10.1 Constrained Extrema: A Single Equality Constraint in n Independent Variables 10.2 The Technique of Lagrange 10.3 Interpretation of the Lagrange Multiplier 10.4 Constrained Extrema: m Equality Constraints in n Independent Variables 10.5 The Generalized Technique of Lagrange 10.6 Interpretation of the Lagrange Multipliers λj 10.7 Economic Applications 10.7.1 Household Equilibrium 10.7.2 Marshallian Demand Functions 10.7.3 Hicksian Demand Functions 10.7.4 Constrained Cost Minimization, Constrained Output Maximization, and Long-Run Profit Maximization Appendix 10.7.3.Aƒ The Hicksian Demands Possess the Derivative Property Appendix 10.7.3.B The Income and Substitution Effects Appendix 10.7.4.Aƒ Production-Cost Duality: A Closer Look (McFadden [19781]; Fuss and McFadden [1978]; Diewert [1973, 1982]; Arriel, et al. [2010]; and Jorgenson and Lau [1974]) Notes Chapter 11. Constrained Extrema: Inequality Constraints 11.1 Constrained Extrema: m Inequality Constraints in n Non-Negative Independent Variables 11.2 Necessary Optimality Conditions 11.3 Fritz-John (FJ) Optimality Conditions (Mangasarian [1969]; Mangasarian and Fromovitz [1967]; John [1948]) 11.4 Karush-Kuhn-Tucker (KKT) Optimality Conditions (Kuhn and Tucker [1951]; Tucker [1956]; Arrow et al. [1961]) 11.5 KKT Sufficient Optimality Conditions 11.6 The Optimal Value Function: Lagrange Multipliers Revisited 11.7 Economic Applications 11.7.1 Optimal Resource Allocation 11.7.2 Resource Allocation with Generalized Lagrange Multipliers Notes Chapter 12. Constrained Extrema: Mixed Constraints 12.1 Programs With m Inequality and p Equality Side Relations in n Independent Variables 12.2 KKT Sufficient and Necessary and Sufficient Optimality Conditions 12.3 The Optimal Value Function: Lagrange Multipliers Revisited Notes Chapter 13. Lagrangian Saddle Points and Duality 13.1 Introduction 13.2 Lagrangian Saddle Points (Lasdon [1970]; Kuhn and Tucker [1951]; Arrow et al. [1958]; Uzawa [1958]; Künzi et al. (1966); and Geoffrion [1972]) 13.3 Saddle Points Revisited: Perturbation Functions 13.4 Lagrangian Saddle Points with Mixed Constraints 13.5 Lagrangian Duality with Inequality Constraints (Graves and Wolfe [1963]; Lasdon [1970]; Geoffrion [1972]; Mangasarian [1962]; Wolfe [1961]; Bazaraa et al. [2006]; Minoux [1986]; Fiacco and McCormick [1968]) 13.6 Lagrangian Duality Revisited 13.7 Lagrangian Duality with Mixed Constraints 13.8 Constrained Output Maximization: A Lagrangian Dual Approach Chapter 14. Generalized Concave Optimization 14.1 Introduction 14.2 Quasiconcave Programming 14.3 Extensions of Quasiconcave Programming 14.4 Extensions of Quasiconcave Programming to Mixed Constraints Note Chapter 15. Homogeneous, Homothetic, and Almost Homogeneous Functions 15.1 Homogeneity Defined 15.2 Properties of Homogeneous Functions 15.3 Homothetic Functions 15.4 Almost Homogeneous Functions 15.5 Homogeneity and Concavity (Convexity) 15.6 Homogeneous Programming (LASSERRE and Hiriart-Urruty [2002]; Zhao and Li [2012]) 15.7 Economic Applications 15.7.1 The Long-Run Expansion Path 15.7.2 The Short-Run Cost Functions 15.7.3 The Elasticity of Substitution Between Labor and Capital: Another Look Notes Chapter 16. Envelope Theorems 16.1 Introduction 16.2 Continuous Correspondences 16.2.1 For X=Y=[0,1], let F:X→Y be defined as 16.2.2 Suppose the correspondence F:X→Y is defined as 16.2.3 Let the correspondence F:X→Y be specified as 16.3 The Maximum Theorem (Berge [1963]) 16.4 The Optimal Value or Envelope Function 16.5 Envelope Theorems 16.5.1 y=f(x;α),x∈X⊆R,α∈Ω⊆R. 17.5.2 y=f(x;α),x∈X⊆Rn,α∈Ω⊆Rp 16.5.3 y=f(x;α),G(x;α)=O,x∈X⊆Rn,α∈Ω⊆Rp 16.6 Economic Applications 16.6.1 Long-Run Total Cost: Envelope Results Appendix 16.A A Proof of Berge's Maximum Theorem Notes Chapter 17. The Fixed Point Theorems of Brouwer and Kakutani 17.1 Introduction 17.2 Simplexes 17.3 Simplicial Decomposition and Subdivision 17.4 Simplicial Mappings and Labeling 17.5 The Existence of Fixed Points 17.6 Fixed Points of Compact Point-to-Point Functions 17.7 Fixed Points of Point-to-Set Functions 17.8 Economic Applications: Existence of a Competitive Equilibrium (Debreu [2007]; Takayama [ 1987]; McKenzie [1959, 1961]; Nikaido [1960]; Arrow and Debreu [1954]; Arrow and Hahn [1971]; and Mas-Collel et al. [2007]) 17.8.1 A Pure Exchange Economy 17.8.2 A Private Ownership (Production) Economy Appendix 17.A The Barycentric Subdivision of a k-Simplex (Shapley [1973]; and Scarf [1973]) Notes Chapter 18. Dynamic Optimization: Optimal Control Modeling 18.1 Introduction and Basic Problem 18.2 The Lagrangian and Hamiltonian Functions 18.3 The Maximum Principle (Pontryagin et al. [1962]) 18.4 End-Point and Transversality Conditions 18.5 Sensitivity Analysis: Costate Variables as Shadow Prices 18.6 Autonomous Optimal Control: The Current Value Hamiltonian 18.7 Infinite Time Horizon 18.8 Sufficient Optimality Conditions for the Infinite Horizon Case (Caputo [2005]; Mangasarian [1966]) 18.9 Constraints on the Control Variable 18.10 Economic Applications 18.10.1 The Neoclassical Optimal Growth Model (Solow [1956]; Swan [1956]; Cass [1965]; Koopmans [1965]; and Takayama [1985, 1996]) 18.10.2 A Neoclassical Investment Model With Adjustment Costs (Jorgenson [1967, 1971,1972]; Caputo [2005]; and Takayama [1985, 1996]) Appendix 18. A Ordinary Differential Equations 18.A.1 Introduction 18.A.2 First-Order Differential Equations 18.A.3 Separation of Variables 18.A.4 Linear First-Order Differential Equations 18.A.4a Autonomous Equations 18.A.4b Non-Autonomous Equations 18.A.5 Exact Ordinary Differential Equations 18.A.6 Integrating Factors 18.A.7 Variation of Parameters 18.A.8 Nonlinear Differential Equations of the First Order and First Degree Appendix 18.B Simultaneous Systems of Linear Differential Equations 18.B.1 Introduction 18.B.2 Autonomous Systems 18.B.3 Particular Solution Appendix 18.C Qualitative Analysis of Differential Equations: Phase Diagrams and Stability 18.C.1 Equilibrium Points and Stability for a Single Autonomous Differential Equation 18.C.2 Equilibrium Points and Stability for Autonomous Linear Systems Notes Chapter 19. Comparative Statics Revisited 19.1 Introduction 19.2 The Fundamental Equation of Comparative Statics 19.3 Economic Applications 19.3.1 Constrained Utility Maximization (McKenzie [1957]; Takayama [1977, 1985]; and Silberberg and Suen [2001]) 19.3.2 Constrained Cost Minimization 19.3.3 Long-Run Profit Maximization 19.3.4 The Le Châtelier Effect (Samuelson [1960, 1983]; Currier [2000]; Silberberg [1971]; Silberberg and Suen [2001]) Notes References Index