Course description
The course introduces students to methods of mathematical analysis of dynamic processes, including those that appear in economic settings. The students will learn how to describe and analyze in mathematical terms such processes as equilibrium price setting, economic growth, or population dynamics. The students will master how to determine stability of equilibrium in economic models, and will understand how the starting conditions influence the end result and divide countries into rich and poor. The course covers linear and non-linear equations and systems of equations, the use of phase diagrams in microeconomics and macroeconomics, explaining the nature of the solutions as well as stability analysis of both continuous and discrete dynamical systems.
Topics
Topic 1. Ordinary Differential Equations
- Basic concepts. Special types of differential equations of the first order. Differential equations with separable variables. First order differential equations with homogeneous coefficients. Differential equations with linear coefficients. Exact differential equations. Integrating factors. Linear differential equations of order greater than one. Solutions of the homogeneous and the nonhomogeneous equations of order n with constant coefficients. The method of undetermined coefficients. The method of variation of parameters. Systems of ordinary differential equations.
Topic 2. Stability theory of ordinary differential equations
- Basic definitions. Unperturbed and perturbed motions. Liapunov's definition of stability. Geometric interpretation of stability and asymptotical stability. Equations of perturbed motion. Stability in first approximation. Formulation of the problem. Main theorems of stability in first approximation. The Routh–Hurwitz stability criterion. Stability of linear autonomous systems. Numeric and analytical approaches.
Topic 3. Differential equations in economic dynamics modelling. The stability of equilibria in continuous-time models
- The Evans price adjustment model. The Solow economic growth model. Population dynamics. Malthusian population growth. Logistic growth curve. Competition with no over-crowding. Predatory–prey model with no over-crowding (Lotka–Volterra model). Competitive model with over-crowding. Predatory–prey model with over-crowding. The stability analysis of the second order nonlinear autonomous systems equilibria.
Topic 4. Ordinary Difference Equations
- Basic definitions. Linear ordinary difference equations. The structure of general solution. Casorati determinant. Solutions of the homogeneous and the nonhomogeneous equations of order n with constant coefficients. The method of variation of parameters. Systems of ordinary difference equations.
Topic 5. Stability theory of ordinary difference equations. The stability of equilibria in discrete-time models of economic dynamics
- Basic concepts. Pursuing analogies between difference and differential equations. Stability of linear autonomous systems. The Schur stability criterion. The stability analysis of equilibria in discrete-time models of economic dynamics. The Keynes dynamic model with discrete time. The Samuelson–Hicks discrete-time model. The Baumol–Wolf model.
Literature
- Chiang A. C. Fundamental Methods of Mathematical Economics, 3rd ed. New York: McGraw-Hill, 1984.
- Romer D. Advanced Macroeconomics, 2nd ed. New York: McGraw-Hill, 2001.
- Shone R. Economic Dynamics: Phase Diagrams and their Economic Application. New York: Cambridge University Press, 1997.
- Takayama A. Mathematical Economics, 2nd ed. New York: Cambridge University Press, 1997.