微观经济学的数学方法(DOC 261)

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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