Llicenciatura en Administraciķ i Direcciķ d'Empreses (3323)
Llicenciatura en Economia (3322)
Ālgebra Linial i Sistemes Dināmics(11863)
Tema 1. The Nature of Dynamic Optimization
1.1. Salient Features of Dynamic Optimization Problems.
1.2. Variable Endpoints and Transversality Conditions.
1.3. Alternative Approaches to Dynamic Optimization.
Tema 2. Some Groundwork for Dynamic Analysis
2.1. Continuous Time: Differential Equations.
2.2. Discrete Time: Difference Equations. Introduction to Stability analysis of Discrete Dynamical Systems.
2.2.1. One-Dimensional, Autonomous, First-Order Systems.
2.2.1.1. Linear Systems: The Solution. Existence of Stationary Equilibria. Uniqueness of Stationary Equilibrium. Stability of Stationary Equilibria.
2.2.1.2. Nonlinear Systems: The Solution. Existence, Uniqueness and Multiplicity of Stationary Equilibria. Linearization and Local Stability of Stationary Equilibria. Global Stability.
2.2.2. Multi-Dimensional, Autonomous, First-Order Systems.
2.2.2.1. Linear Systems: The Solution. Existence and Uniqueness of Stationary Equilibria. Examples of a 2-D System: Explicit Solution and Stability Analysis; Phase Diagrams; Stable and Unstable Manifolds. Results from Linear Algebra. The Solution in Terms of the Jordan Matrix. Stability: Distinct Real Eigenvalues; Repeated Real Eigenvalues; Distinct Pairs of Complex Eigenvalues; Repeated Pairs of Complex Eigenvalues.
2.2.2.2. Nonlinear Systems: Local Analysis. Global Analysis.
2.2.3. Higher-Order Autonomous Systems.
2.2.3.1. Linear Systems: Second-Order Systems. Third-Order Systems. N th-Order Systems.
2.2.3.2. Nonlinear Systems.
2.2.4. Non-Autonomous Systems.
Tema 3. The Calculus of Variations
3.1. The Euler Equation.
3.2. Checking Concavity/Convexity.
3.3. Methodological Issues of Infinite Horizon.
Economic Applications: Dynamic Optimization of a Monopolist, Trading Off Inflation and Unemployment, The Optimal Investment Path of a Firm, The Ramsey Model and Optimal Social Saving Behavior.
Tema 4. Optimal Control Theory
4.1. The Maximum Principle.
4.2. The Calculus of Variations and Optimal Control Theory Compared.
4.3. Problems with Several State and Control Variables.
4.4. Infinite Horizon Problems.
4.5. Optimal Control with Constraints.
Economic Applications: The Political Business Cycle, Energy Use and Environmental Quality, Optimal Growth, Exogenous and Endogenous Technological Progress.
Tema 5. Dynamic Programming
5.1. Bellman's Equation.
5.2. Uncertainty.
Economic Applications: Reformulations of Economic Applications, Job Search, Saving under Uncertainty.
Bibliografia
CHIANG, A. C. Elements of Dynamic Optimization. New York: McGraw-Hill Publishers, 1992.
CHIANG, A. C. Fundamental Methods of Mathematical Economics. 2nd. ed. New York: McGraw-Hill, 1974.
GALOR, O. "Introduction to Stability Analysis of Discrete Dynamical Systems". Working Paper 23-96. Brown University, 1996.
KAMIEN, M.; SCHWARTZ, N. Dynamic Optimization, the Calculus of Variations and Optimal Control in Economics and Management. 2nd. ed. Amsterdam: North-Holland cop, 1991.
SIMON, C. P.; BLUME, L. Mathematics for economists. New York: Norton 1994.