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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
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