# The Theory of Linear Economic Models

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# The Theory of Linear Economic Models

In the past few decades, methods of linear algebra have become central to economic analysis, replacing older tools such as the calculus. David Gale has provided the first complete and lucid treatment of important topics in mathematical economics which can be analyzed by linear models. This self-contained work requires few mathematical prerequisites and provides all necessary groundwork in the first few chapters. After introducing basic geometric concepts of vectors and vector spaces, Gale proceeds to give the main theorems on linear inequalities—theorems underpinning the theory of games, linear programming, and the Neumann model of growth. He then explores such subjects as linear programming; the theory of two-person games; static and dynamic theories of linear exchange models, including problems of equilibrium prices and dynamic stability; and methods of play, optimal strategies, and solutions of matrix games. This book should prove an invaluable reference source and text for mathematicians, engineers, economists, and those in many related areas.

## Table of Contents

List of Notations

Chapter 1. Linear Programming: Examples, Definitions, and Statements of the Principal Theorems

1. Examples

The diet problem

The transportation problem

Production to meet given demand at minimum cost

Production to maximize income from given resources

2. Duality and prices

3. Further interpretation of duality

4. Price equilibrium

Bibliographical notes

Exercises

Chapter 2. Real Linear Algebra

1. Vector

2. Scalar product, matrices, linear equations

3. Real linear equations and inequalities

4. Basic solutions of equations

5. Geometry of linear inequalities. Convex cones

6. Extreme vectors and extreme solutions

7. Convex sets and polytopes

Bibliographical notes

Exercises

Chapter 3. The Theory of Linear Programming

1. Definitions

2. The duality theorems

3. The equilibrium theorems

4. Basic solutions

5. An application: allocation of resources in a competitive economy

Bibliographical notes

Exercises

Chapter 4. Computation. The Simplex Method

1. Solving simultaneous equations and inverting a matrix

2. The simplex method for linear programming. Discussion

3. Theory of the simplex method

4. Some numerical examples

5. Nonnegative solutions of linear equations

6. Solving linear inequalities

7. Degeneracy. The generalized simplex method

Bibliographical notes

Exercises

Chapter 5. Integral Linear Programming

1. Examples

Transportation problem with indivisible commodity

The optimal-assignment problem

The loading problem

2. Flows in networks

3. The simple-assignment problem

4. The transshipment problem

5. The optimal-assignment problem

6. A problem related to optimal assignment. Price equilibrium

7. The transportation problem

8. Other examples: shortest route; the caterer

9. Concluding remarks and open questions

Bibliographical notes

Exercises

Chapter 6. Two-person Games: Examples, Definitions, and Elementary Theory

1. First examples and definitions

Odds and evens (matching pennies)

Morra

2. Further examples of matrix games

Goofspiel

Bluffing

A, B, C

3. Solutions of games. Mixed strategies

4. Value of a game and optimal strategies

5. Some infinite games

Continuous bluffing

Duels

The oil prospector (a game against nature)

The bomber and the submarine

High number

Low number

6. Saddle points and minimax

7. Symmetric games

8. Proof of the fundamental theorem

Appendix to Chapter 6: A geometric "proof" of the fundamental theorem of game theory

Bibliographical notes

Exercises

Chapter 7. Solutions of Matrix Games

1. Relation between matrix games and linear programming

2. Solving games by the simplex method

3. Optimal strategies

4. Solutions

5. Examples

6. The structure of symmetric games

7. Constructing a game with prescribed solutions

8. Basic optimal strategies

9. A method of "learning" a game

10. Convergence of the learning method

Bibliographical notes

Exercises

Chapter 8. Linear Models of Exchange

1. Examples

The simple exchange model. The price problem

The simple linear model of international trade

2. Equilibrium for the exchange model

3. Dynamic theory

4. Dynamics in the reducible case

5. Price equilibrium for linear exchange models

6. An example of price equilibrium

7. Uniqueness of equilibrium prices

Bibliographical notes

Exercises

Chapter 9. Linear Models of Production

1. The simple linear production model

2. A dynamic property of the simple model

3. The Leontief model

4. The general linear production model. Efficient points

5. Von Neumann’s expanding model

6. Some examples

7. The expanding simple model

Bibliographical notes

Exercises

Bibliography

Index

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