The search for symmetry is part of the fundamental scientific paradigm in mathematics and physics. Can this be valid also for economics? This book represents an attempt to explore this possibility. The behavior of price-taking producers, monopolists, monopsonists, sectoral market equilibria, behavior under risk and uncertainty, and two-person zero- and non-zero-sum games are analyzed and discussed under the unifying structure called the linear complementarity problem. Furthermore, the equilibrium problem allows for the relaxation of often-stated but unnecessary assumptions. This unifying approach offers the advantage of a better understanding of the structure of economic models. It also introduces the simplest and most elegant algorithm for solving a wide class of problems.
Foreword by Michael R. Caputo
Preface
1 Introduction 1
Duality, Symmetry, and the Euler-Legendre Transformation 4
Duality without Constraints 6
Asymmetric Duality with Constraints 9
Symmetric Dual Nonlinear Programs 10
Appendix 1.1 The Euler-Legendre Transformation 12
References 14
2 Lagrangean Theory 15
Unconstrained Maximization 16
Concave and Convex Functions 17
Constrained Maximization 18
Saddle Point Problem 20
Homogeneous Functions 21
A Symmetric Lagrangean Function 22
Exercises 25
Reference 27
3 Karush-Kuhn-Tucker Theory 28
Concave Nonlinear Programming 32
Alternative Specifications of Nonlinear Problems 38
Interpretation of Karush-Kuhn-Tucker Conditions 40
Equilibrium Problem 42
How to Solve Nonlinear Programming Problems 44
Exercises 45
Appendix 3.1 Constraint Qualification 46
References 48
4 Solving Systems of Linear Equations 49
Product Form of the Inverse 52
Summary of the Pivot Method's Rules 54
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