Mathematical Economics

by Kelvin Lancaster

Overview

An excellent book which should find wide use. -- Mathematics Reviews.
In this classic volume, a noted economist and teacher has combined a modern text for graduate courses in mathematical economics with a valuable reference book of analytical economics for professional economists.
Unique in its unified and careful presentation of a variety of techniques of economic analysis, the book is divided into two parts: chapters on mathematical economics (i.e. economic models analyzed primarily from the point of view of their mathematical properties) and appropriate mathematical reviews. To keep the exposition as smooth as possible, the economic analysis has been separated from the purely mathematical material -- permitting flexible use of the book as a text. Moreover, the chapters and reviews are designed as a self- contained system, wherein the reviews contain all the mathematics required for the chapters and the chapters illustrate the use of almost all the techniques set out in the reviews. An extensive mathematical background is not required; however, it is assumed the reader has some acquaintance with elementary calculus.
The economic analysis covers linear and nonlinear optimizing techniques, input-output, activity analysis, neoclassical and set- theoretic static economic models, modern general equilibrium theory, the Von Neumann and other models of balanced growth, efficient growth and turnpike theorems, and modern stability analysis.
The mathematical reviews include discussions of set theory, linear algebra, matrices, linear equations and inequalities, convex sets and functions, continuous functions and mappings (including neoclassical calculus methods), topological ideas, differential and difference equations, calculus of variations, and related topics.
Every attempt has been made to give a complete and rigorous exposition (except in topological methods where the approach is descriptive and heuristic) which omits no essential proofs or steps in the argument.

Product Details

ISBN-13:	9780486653914
Publisher:	Dover Publications
Publication date:	11/02/2011
Series:	Dover Books on Computer Science
Edition description:	Revised ed.
Pages:	448
Product dimensions:	5.50(w) x 8.50(h) x 0.91(d)

Table of Contents

1. Introduction
1.1 Mathematical Economics
1.2 Outline of the Book
1.3 Notes on the Literature
Part I: Optimizing Theory
2. The General Optimizing Problem
2.1 Introduction
2.2 The General Structure
2.3 Constraints and the Feasible Set
2.4 The General Optimizing Problem
2.5 The General Solution Principle
2.6 Conditions for a Global Optimum
2.7 Important Special Cases
2.8 Direct Solutions or Optimal Condtions?
Exercises
3. The Theory of Linear Programming
3.1 Introduction
3.2 The Feasible Set
3.3 Duality
3.4 The Optimum Conditions
3.5 Basic Solutions
3.6 The Basis Theorem
3.7 Interpretation of the Dual Variables
Further Reading
Exercises
4. Classical Calculus Methods
4.1 Introduction
4.2 The Lagrangean Function
4.3 Interpretation of the Lagrange Multipliers
4.4 A Geometrical Note
4.5 Second Order Conditions for the Classical Case
4.6 The Substitution Effect of Neoclassical Demand Theory
4.7 Global Optimum Condtions in the Classical Problem
Exercises
5. Advanced Optimizing Theory
5.1 Introduction
5.2 Nonnegative Variables
5.3 Inequality Constraints
5.4 Saddle Points and Duality
5.5 The Dual Variables
5.6 The Minimax Theorem
5.7 Exercises of Optimal Solutions
Further Reading
Exercises
Part II: Static Economic Models
6. Input-Output and Related Models
6.1 Input-Output Models
6.2 The Closed Model
6.3 The Leontief Open Model
6.4 Direct and Indirect Input Requirements
6.5 Factor Intensity in the Leontief Model
6.6 A Labor Theory of Value
6.7 The Substitution Theorem
6.8 Matrix Multipliers
Further Reading
Exercises
7. Linear Optimizing Models
7.1 Activity Analysis of Production
7.2 The Production Set
7.3 Efficient Production
7.4 Constrained Production
7.5 Consumption as an Activity
Further Reading
Exercises
8. Nonlinear Optimizing Models
8.1 Introduction
8.2 Neoclassical Demand Theory
8.3 Convexity Proof of the Substitution Theorem
8.4 The Neoclassical Transformation Surface
8.5 Returns to Scale
8.6 Relative Factor Intensity
8.7 Generalized Production Theory
Further Reading
9. General Equilibrium
9.1 Equilibrium in a Market Economy
9.2 Walras' Law and the Budget Constraint
9.3 The Excess Demand Theorem
9.4 The Walras-Wald Model
9.5 The Arrow-Debreu-McKenzie Model
Further Reading
Part III: Dynamic Economic Models
10. Balanced Growth
10.1 Introduction
10.2 A Leontief-Type Model
10.3 The Von Neumann Growth Model
10.4 The Von Neumann-Leontief Model
10.5 General Balanced Growth Models
Further Reading
11. Efficient and Optimal Growth
11.1 Efficiency and Optimality in Dynamic Models
11.2 The Principle of Optimality
11.3 Efficient Growth
11.4 Properties of Efficient Paths
11.5 A Turnpike Theorem
11.6 An Explicit Turnpike Example
Further Reading
12. Stability
12.1 The Concept of Stability
12.2 Stability Analysis
12.3 Market Stability
12.4 Stability of Decentralized Economic Policy
Part IV: MATHEMATICAL REVIEWS
RI. Fundamental Ideas
R1.1 Sets
R1.2 Ordered and Quasi-Ordered Sets
R1.3 Cartesian Products and Spaces
R1.4 "Functions, Transformations, Mappings, Correspondences"
R1.5 Closedness and Boundedness
R1.6 Complex Numbers
Exercises
R2. Linear Algebra
R2.1 Vectors
R2.2 Fundamental Theorem of Vector Spaces
R2.3 Basis and Rank
R2.4 Sums and Direct Sums
R2.5 Scalar Products
R2.6 Complex Vectors
R2.7 Matrices
R2.8 Matrix Algebra
R2.9 Matrix-Vector Products and Linear Transformations
R2.10 Partitioned Matrices
R2.11 Vector Sets
Exercises
R3. Linear Equations and Inequalities
R3.1 Introduction
R3.2 The Rank of Matrix
R3.3 Homogeneous Equations
R3.4 Nonhomogeneous Equations
R3.5 Nonnegative Vectors and Vector Inequalities
R3.6 Fundamental Theorem on Linear Inequalities
R3.7 Results on Linear Equations and Inequalities
Exercises
R4. Convex Sets and Cones
R4.1 Geometric Ideas
R4.2 Convex Sets
R4.3 Separating and Supporting Hyperplanes
R4.4 Extreme Points
R4.5 Convex Cones
R4.6 Finite Cones and Homogeneous Inequalities
R4.7 The Dual Cone
Exercises
R5. Square Matrices and Characteristic Roots
R5.1 Introduction
R5.2 Determinants and Cramer's Rule
R5.3 The Inverse of a Square Matrix
R5.4 Charateristic Roots and Vectors
R5.5 Diagonalization
R5.6 Convergence of Matrix Series
R5.7 Charateristic Row Vectors
R5.8 Numerical Examples
Exercises
R6. Symmetric Matrices and Quadratic Forms
R6.1 Symmetric Matrices
R6.2 Quadratic Forms
R6.3 Constrained Quadratic Forms
Exercises
R7. Semipositive and Dominant Diagonal Matrices
R7.1 Introduction
R7.2 Indecomposability
R7.3 Properties of Semipositive Square Matrices
R7.4 Properties of Dominant Diagonal Matrices
R7.5 Proofs
Exercises
R8. Continuous Functions
R8.1 Introduction
R8.2 Derivatives and Differentials
R8.3 Some Mapping Relationships
R8.4 Maxima and Minima
R8.5 Convex and Concave Functions
R8.6 Homogeneous and Homothetic Functions
R8.7 The Brouwer Fixed Point Theorem
R8.8 Linear Homogeneous Vector-Valued Functions
Exercises
R9. Point-to-Set Mappings
R9.1 Introduction
R9.2 The Graph of a Mapping
R9.3 Continuity
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