Table of contents

· Introduction and instructions

· 1. Review of some basic logic, matrix algebra, and calculus

o 1.1 Logic

o 1.2 Matrices and solutions of systems of simultaneous equations

o 1.3 Intervals and functions

o 1.4 Calculus: one variable

o 1.5 Calculus: many variables

o 1.6 Graphical representation of functions

· 2. Topics in multivariate calculus

o 2.1 Introduction

o 2.2 The chain rule

o 2.3 Derivatives of functions defined implicitly

o 2.4 Differentials and comparative statics

o 2.5 Homogeneous functions

· 3. Concavity and convexity

o 3.1 Concave and convex functions of a single variable

o 3.2 Quadratic forms

§ 3.2.1 Definitions

§ 3.2.2 Conditions for definiteness

§ 3.2.3 Conditions for semidefiniteness

o 3.3 Concave and convex functions of many variables

o 3.4 Quasiconcavity and quasiconvexity

· 4. Optimization

o 4.1 Introduction

o 4.2 Definitions

o 4.3 Existence of an optimum

· 5. Optimization: interior optima

o 5.1 Necessary conditions for an interior optimum

o 5.2 Sufficient conditions for a local optimum

o 5.3 Conditions under which a stationary point is a global optimum

· 6. Optimization: equality constraints

o 6.1 Two variables, one constraint

§ 6.1.1 Necessary conditions for an optimum

§ 6.1.2 Interpretation of Lagrange multiplier

§ 6.1.3 Sufficient conditions for a local optimum

§ 6.1.4 Conditions under which a stationary point is a global optimum

o 6.2 n variables, m constraints

o 6.3 Envelope theorem

· 7. Optimization: the Kuhn-Tucker conditions for problems with inequality constraints

o 7.1 The Kuhn-Tucker conditions

o 7.2 When are the Kuhn-Tucker conditions necessary?

o 7.3 When are the Kuhn-Tucker conditions sufficient?

o 7.4 Nonnegativity constraints

o 7.5 Summary of conditions under which first-order conditions are necessary and sufficient

· 8. Differential equations

o 8.1 Introduction

o 8.2 First-order differential equations: existence of a solution

o 8.3 Separable first-order differential equations

o 8.4 Linear first-order differential equations

o 8.5 Phase diagrams for autonomous equations

o 8.6 Second-order differential equations

o 8.7 Systems of first-order linear differential equations

· 9. Difference equations

o 9.1 First-order equations

o 9.2 Second-order equations