Mathematical Analysis and Optimization for Economists [1 ed.] 9780367759018, 9780367759025, 9781003164494

Table of contents :
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