Mathematical Economics

6.

Dynamic Analysis

6.1.

The Nature of Dynamic Analysis

6.1.1.

Time as a Variable in Economic Models

6.1.2.

Static vs. Dynamic Models

6.1.3.

Types of Dynamic Models

6.1.3.1.

Discrete Time Models

6.1.3.2.

Continuous Time Models

6.2.

Difference Equations

6.2.1.

Definition and Classification

6.2.2.

First-Order Linear Difference Equations

6.2.2.1.

The Homogeneous Case

6.2.2.2.

The Nonhomogeneous Case

6.2.2.3.

General Solution Methods

6.2.2.4.

Particular Solutions

6.2.3.

Higher-Order Linear Difference Equations

6.2.3.1.

Second-Order Equations

6.2.3.2.

Characteristic Equation Method

6.2.3.3.

Solution Techniques

6.2.3.4.

6.2.4.

Nonlinear Difference Equations

6.2.4.1.

Solution Methods

6.2.4.2.

Linearization Techniques

6.2.5.

Stability Analysis

6.2.5.1.

Criteria for Stability

6.2.5.2.

Economic Interpretation

6.2.5.3.

Phase Diagrams for Discrete Systems

6.2.6.

Economic Applications

6.2.6.1.

The Cobweb Model

6.2.6.1.1.

Assumptions and Structure

6.2.6.1.2.

Stability Analysis

6.2.6.1.3.

Convergent and Divergent Cases

6.2.6.2.

Samuelson Multiplier-Accelerator Model

6.2.6.2.1.

Model Formulation

6.2.6.2.2.

Dynamic Adjustment Process

6.2.6.2.3.

Business Cycle Analysis

6.3.

Differential Equations

6.3.1.

Definition and Classification

6.3.2.

First-Order Linear Differential Equations

6.3.2.1.

The Homogeneous Case

6.3.2.2.

The Nonhomogeneous Case

6.3.2.3.

Solution Methods

6.3.2.4.

Integrating Factors

6.3.3.

Higher-Order Linear Differential Equations

6.3.3.1.

Second-Order Equations

6.3.3.2.

Characteristic Equation Method

6.3.3.3.

Solution Techniques

6.3.3.4.

6.3.4.

Systems of Differential Equations

6.3.4.1.

6.3.4.2.

6.3.4.3.

Eigenvalues and Eigenvectors

6.3.5.

6.3.5.1.

Construction and Interpretation

6.3.5.2.

Phase Portraits

6.3.5.3.

Trajectories and Flow

6.3.6.

Stability of Equilibrium

6.3.6.1.

Local Stability

6.3.6.2.

Global Stability

6.3.6.3.

Lyapunov Stability

6.3.7.

Economic Applications

6.3.7.1.

Solow Growth Model

6.3.7.1.1.

Model Structure

6.3.7.1.2.

Steady-State Analysis

6.3.7.1.3.

Convergence Properties

6.3.7.2.

Market Price Dynamics

6.3.7.2.1.

Price Adjustment Mechanisms

6.3.7.2.2.

Stability of Market Prices

6.3.7.3.

Ramsey Growth Model

6.3.7.3.1.

Optimal Consumption Path

6.3.7.3.2.

Transversality Conditions

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