Mathematical Economics

  1. Optimization Problems
    1. The Concept of Optimization
      1. Objective Functions
        1. Constraints
          1. Types of Optimization Problems
            1. Unconstrained Optimization
              1. Constrained Optimization
                1. Linear vs. Nonlinear Programming
              2. Unconstrained Optimization
                1. Functions of One Variable
                  1. First-Order Conditions for Extrema
                    1. Critical Points
                      1. Second-Order Conditions
                        1. The Second-Derivative Test
                        2. Functions of Several Variables
                          1. First-Order Conditions
                            1. Gradient Vector
                              1. Second-Order Conditions
                                1. The Hessian Matrix
                                  1. Nature of Critical Points
                                    1. Local Maxima
                                      1. Local Minima
                                        1. Saddle Points
                                      2. Concavity and Convexity
                                        1. Definitions and Properties
                                          1. Tests for Concavity and Convexity
                                            1. Implications for Optimization
                                          2. Constrained Optimization with Equality Constraints
                                            1. The Substitution Method
                                              1. Reduction to Unconstrained Problem
                                                1. Advantages and Limitations
                                                2. The Lagrange-Multiplier Method
                                                  1. The Lagrangian Function
                                                    1. First-Order Conditions
                                                      1. Interpretation of the Lagrange Multiplier
                                                        1. Multiple Constraints
                                                        2. Second-Order Conditions
                                                          1. The Bordered Hessian
                                                            1. Sufficient Conditions for Extrema
                                                              1. Classification of Critical Points
                                                            2. Constrained Optimization with Inequality Constraints
                                                              1. The Kuhn-Tucker Conditions
                                                                1. Necessary Conditions
                                                                  1. Sufficient Conditions
                                                                    1. Complementary Slackness
                                                                    2. Constraint Qualification
                                                                      1. Regularity Conditions
                                                                        1. Linear Independence Constraint Qualification
                                                                        2. Economic Interpretation
                                                                          1. Shadow Prices
                                                                            1. Binding and Non-binding Constraints
                                                                          2. Economic Applications of Optimization
                                                                            1. Consumer Theory
                                                                              1. Utility Maximization
                                                                                1. Budget Constraints
                                                                                  1. Derivation of Demand Functions
                                                                                    1. Indirect Utility Function
                                                                                      1. Expenditure Function
                                                                                      2. Producer Theory
                                                                                        1. Profit Maximization
                                                                                          1. Cost Minimization
                                                                                            1. Production Functions
                                                                                              1. Derivation of Supply Functions
                                                                                                1. Input Demand Functions
                                                                                                2. Envelope Theorems
                                                                                                  1. Statement and Proof
                                                                                                    1. Economic Interpretation
                                                                                                      1. Applications in Consumer and Producer Theory
                                                                                                        1. Roy's Identity
                                                                                                          1. Shephard's Lemma

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                                                                                                      4. Comparative Static Analysis

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                                                                                                      6. Dynamic Analysis