# Mathematical methods for economic theory

Martin J. Osborne

## Contents

 Introduction 1. Review of some basic logic, matrix algebra, and calculus 1.1 Logic 1.2 Matrices: determinants, inverses, and rank 1.3 Solving systems of linear equations: Cramer's rule and matrix inversion 1.4 Intervals and functions 1.5 Calculus: one variable 1.6 Calculus: many variables 1.7 Graphical representation of functions 2. Topics in multivariate calculus 2.1 Introduction to multivariate calculus 2.2 The chain rule 2.3 Derivatives of functions defined implicitly 2.4 Differentials and comparative statics 2.5 Homogeneous functions 3. Concavity and convexity 3.1 Concave and convex functions of a single variable 3.2 Quadratic forms 3.2.1 Quadratic forms: definitions 3.2.2 Quadratic forms: conditions for definiteness 3.2.3 Quadratic forms: conditions for semidefiniteness 3.3 Concave and convex functions of many variables 3.4 Quasiconcavity and quasiconvexity 4. Optimization 4.1 Optimization: introduction 4.2 Optimization: definitions 4.3 Existence of an optimum 5. Optimization: interior optima 5.1 Necessary conditions for an interior optimum 5.2 Local optima 5.3 Conditions under which a stationary point is a global optimum 6. Optimization: equality constraints 6.1 Two variables, one constraint 6.1.1 Optimization with an equality constraint: necessary conditions for an optimum for a function of two variables 6.1.2 Optimization with an equality constraint: interpretation of Lagrange multipliers 6.1.3 Optimization with an equality constraint: sufficient conditions for a local optimum for a function of two variables 6.1.4 Optimization with an equality constraint: conditions under which a stationary point is a global optimum 6.2 Optimization with equality constraints: n variables, m constraints 6.3 The envelope theorem 7. Optimization: the Kuhn-Tucker conditions for problems with inequality constraints 7.1 Optimization with inequality constraints: the Kuhn-Tucker conditions 7.2 Optimization with inequality constraints: the necessity of the Kuhn-Tucker conditions 7.3 Optimization with inequality constraints: the sufficiency of the Kuhn-Tucker conditions 7.4 Optimization with inequality constraints: nonnegativity conditions 7.5 Optimization: summary of conditions under which first-order conditions are necessary and sufficient 8. Differential equations 8.1 Differential equations: introduction 8.2 First-order differential equations: existence of a solution 8.3 Separable first-order differential equations 8.4 Linear first-order differential equations 8.5 Differential equations: phase diagrams for autonomous equations 8.6 Second-order differential equations 8.7 Systems of first-order linear differential equations 9. Difference equations 9.1 First-order difference equations 9.2 Second-order difference equations