Table of contents

• Introduction and instructions
• 1. Review of some basic logic, matrix algebra, and calculus
  • 1.1 Logic
  • 1.2 Matrices and solutions of systems of simultaneous equations
  • 1.3 Intervals and functions
  • 1.4 Calculus: one variable
  • 1.5 Calculus: many variables
  • 1.6 Graphical representation of functions
• 2. Topics in multivariate calculus
  • 2.1 Introduction
  • 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 Definitions
    • 3.2.2 Conditions for definiteness
    • 3.2.3 Conditions for semidefiniteness
  • 3.3 Concave and convex functions of many variables
  • 3.4 Quasiconcavity and quasiconvexity
• 4. Optimization
  • 4.1 Introduction
  • 4.2 Definitions
  • 4.3 Existence of an optimum
• 5. Optimization: interior optima
  • 5.1 Necessary conditions for an interior optimum
  • 5.2 Sufficient conditions for a local optimum
  • 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 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
  • 6.2 n variables, m constraints
  • 6.3 Envelope theorem
• 7. Optimization: the Kuhn-Tucker conditions for problems with inequality constraints
  • 7.1 The Kuhn-Tucker conditions
  • 7.2 When are the Kuhn-Tucker conditions necessary?
  • 7.3 When are the Kuhn-Tucker conditions sufficient?
  • 7.4 Nonnegativity constraints
  • 7.5 Summary of conditions under which first-order conditions are necessary and sufficient
• 8. Differential equations
  • 8.1 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 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 equations
  • 9.2 Second-order equations