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Preface | |

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Introduction | |

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The Nature of Mathematical Economics | |

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Mathematical versus Nonmathematical Economics | |

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Mathematical Economics versus Econometrics | |

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Economic Models | |

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Ingredients of a Mathematical Model | |

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The Real-Number System | |

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The Concept of Sets | |

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Relations and Functions | |

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Types of Function | |

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Functions of Two or More Independent Variables | |

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Levels of Generality | |

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Static (or Equilibrium) Analysis | |

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Equilibrium Analysis in Economics | |

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The Meaning of Equilibrium | |

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Partial Market Equilibrium--A Linear Model | |

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Partial Market Equilibrium--A Nonlinear Model | |

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General Market Equilibrium | |

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Equilibrium in National-Income Analysis | |

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Linear Models and Matrix Algebra | |

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Matrices and Vectors | |

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Matrix Operations | |

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Notes on Vector Operations | |

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Commutative, Associative, and Distributive Laws | |

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Identity Matrices and Null Matrices | |

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Transposes and Inverses | |

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Linear Models and Matrix Algebra (Continued) | |

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Conditions for Nonsingularity of a Matrix | |

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Test of Nonsingularity by Use of Determinant | |

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Basic Properties of Determinants | |

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Finding the Inverse Matrix | |

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Cramer's Rule | |

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Application to Market and National-Income Models | |

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Leontief Input-Output Models | |

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Limitations of Static Analysis | |

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Comparative-Static Analysis | |

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Comparative Statics and the Concept of Derivative | |

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The Nature of Comparative Statics | |

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Rate of Change and the Derivative | |

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The Derivative and the Slope of a Curve | |

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The Concept of Limit | |

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Digression on Inequalities and Absolute Values | |

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Limit Theorems | |

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Continuity and Differentiability of a Function | |

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Rules of Differentiation and Their Use in Comparative Statics | |

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Rules of Differentiation for a Function of One Variable | |

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Rules of Differentiation Involving Two or More Functions of the Same Variable | |

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Rules of Differentiation Involving Functions of Different Variables | |

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Partial Differentiation | |

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Applications to Comparative-Static Analysis | |

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Note on Jacobian Determinants | |

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Comparative-Static Analysis of General-Function Models | |

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Differentials | |

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Total Differentials | |

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Rules of Differentials | |

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Total Derivatives | |

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Derivatives of Implicit Functions | |

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Comparative Statics of General-Function Models | |

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Limitations of Comparative Statics | |

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Optimization Problems | |

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Optimization: A Special Variety of Equilibrium Analysis | |

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Optimum Values and Extreme Values | |

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Relative Maximum and Minimum: First-Derivative Test | |

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Second and Higher Derivatives | |

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Second-Derivative Test | |

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Digression on Maclaurin and Taylor Series | |

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Nth-Derivative Test for Relative Extremum of a Function of One Variable | |

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Exponential and Logarithmic Functions | |

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The Nature of Exponential Functions | |

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Natural Exponential Functions and the Problem of Growth | |

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Logarithms | |

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Logarithmic Functions | |

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Derivatives of Exponential and Logarithmic Functions | |

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Optimal Timing | |

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Further Applications of Exponential and Logarithmic Derivatives | |

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The Case of More than One Choice Variable | |

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The Differential Version of Optimization Conditions | |

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Extreme Values of a Function of Two Variables | |

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Quadratic Forms--An Excursion | |

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Objective Functions with More than Two Variables | |

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Second-Order Conditions in Relation to Concavity and Convexity | |

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Economic Applications | |

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Comparative-Static Aspects of Optimization | |

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Optimization with Equality Constraints | |

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Effects of a Constraint | |

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Finding the Stationary Values | |

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Second-Order Conditions | |

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Quasiconcavity and Quasiconvexity | |

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Utility Maximization and Consumer Demand | |

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Homogeneous Functions | |

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Least-Cost Combination of Inputs | |

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Some Concluding Remarks | |

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Dynamic Analysis | |

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Economic Dynamics and Integral Calculus | |

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Dynamics and Integration | |

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Indefinite Integrals | |

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Definite Integrals | |

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Improper Integrals | |

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Some Economic Applications of Integrals | |

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Domar Growth Model | |

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Continuous Time: First-Order Differential Equations | |

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First-Order Linear Differential Equations with Constant Coefficient and Constant Term | |

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Dynamics of Market Price | |

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Variable Coefficient and Variable Term | |

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Exact Differential Equations | |

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Nonlinear Differential Equations of the First Order and First Degree | |

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The Qualitative-Graphic Approach | |

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Solow Growth Model | |

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Higher-Order Differential Equations | |

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Second-Order Linear Differential Equations with Constant Coefficients and Constant Term | |

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Complex Numbers and Circular Functions | |

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Analysis of the Complex-Root Case | |

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A Market Model with Price Expectations | |

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The Interaction of Inflation and Unemployment | |

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Differential Equations with a Variable Term | |

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Higher-Order Linear Differential Equations | |

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Discrete Time: First-Order Difference Equations | |

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Discrete Time, Differences, and Difference Equations | |

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Solving a First-Order Difference Equation | |

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The Dynamic Stability of Equilibrium | |

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The Cobweb Model | |

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A Market Model with Inventory | |

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Nonlinear Difference Equations--The Qualitative-Graphic Approach | |

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Higher-Order Difference Equations | |

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Second-Order Linear Difference Equations with Constant Coefficients and Constant Term | |

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Samuelson Multiplier-Acceleration Interaction Model | |

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Inflation and Unemployment in Discrete Time | |

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Generalizations to Variable-Term and Higher-Order Equations | |

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Simultaneous Differential Equations and Difference Equations | |

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The Genesis of Dynamic Systems | |

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Solving Simultaneous Dynamic Equations | |

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Dynamic Input-Output Models | |

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The Inflation-Unemployment Model Once More | |

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Two-Variable Phase Diagrams | |

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Linearization of a Nonlinear Differential-Equation System | |

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Limitations of Dynamic Analysis | |

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Mathematical Programming | |

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Linear Programming | |

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Simple Examples of Linear Programming | |

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General Formulation of Linear Programs | |

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Convex Sets and Linear Programming | |

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Simplex Method: Finding the Extreme Points | |

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Simplex Method: Finding the Optimal Extreme Point | |

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Further Notes on the Simplex Method | |

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Linear Programming (Continued) | |

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Duality | |

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Economic Interpretation of a Dual | |

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Activity Analysis: Micro Level | |

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Activity Analysis: Macro Level | |

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Nonlinear Programming | |

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The Nature of Nonlinear Programming | |

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Kuhn-Tucker Conditions | |

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The Constraint Qualification | |

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Kuhn-Tucker Sufficiency Theorem: Concave Programming | |

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Arrow-Enthoven Sufficiency Theorem: Quasiconcave Programming | |

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Economic Applications | |

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Limitations of Mathematical Programming | |

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The Greek Alphabet | |

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Mathematical Symbols | |

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A Short Reading List | |

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Answers to Selected Exercise Problems | |

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Index | |