European Parliament Library

Mathematical analysis and optimization for economists, Michael J. Panik

Label
Mathematical analysis and optimization for economists, Michael J. Panik
Language
eng
Index
no index present
Literary Form
non fiction
Main title
Mathematical analysis and optimization for economists
Nature of contents
dictionaries
Oclc number
1260347679
Responsibility statement
Michael J. Panik
Summary
"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"--, Provided by publisher
Table Of Contents
Intro -- 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 DependenceChapter 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 -- NoteChapter 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.A The Hicksian Demands Possess the Derivative Property -- Appendix 10.7.3.B The Income and Substitution Effects -- Appendix 10.7.4.A 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 Conditions11.3 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 Defined15.2 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]) -- NotesChapter 18. Dynamic Optimization: Optimal Control Modeling
Classification
Content
Other version
Mapped to

Incoming Resources

Outgoing Resources