Dynamic programming
Table of Contents
The course deals with optimization methods for dynamic problems and with the way they are most used in economics. The course is devided in three parts, the first two deals with varational methods in continuous time and develop both the classical varational approach and the modern control view. The third part delas with recursive methods in discrete time. All methods are presented at a reasonable mathematical deepness, that is, enough to understand the principles they are based on but not much more. Economic applications from industrial, growth and monetary economics are discussed in order to show the use that the economic profession makes of the methods introduced.
Admission requirements
- Parts I and II: a course on differential equations.
- Part III: constrained static optimization.
Evaluation
Students evaluation will be based on two assignment and a written final exam (2h), with weights 1/3 and 2/3 respectively. Correction of both assignment will be provided in two final exam-preparatory meetings (2h each). Whereas the course is condesed in three weeks (due to phd program calendar constraints), the assignment and exam schedule is flexible.
References
- [Ch] A. C. Chiang (1992). Elements of Dynamic Optimization. Waveland Press, Inc. Long Grove Illinois.
- [Su] R. K. Sundaram (1996). A First Course in Optimization Theory. Cambridge University Press. Cambridge.
Program
Part I: The Calculus of Variations
10 hours - Prof. Giulio Bottazzi (Theory) and Dr. Davide Pirino (Tutorials and Applications)
- Euler Equations: Derivation in one dimension and generalization to higher dimensions. [Ch] 2, 2.1, 2.3 (1.5h)
- Transversality Conditions: Derivation of general case and specialization to (truncated) horizontal and vertical line. [Ch] 3, 3.1, 3.2 3.3 (1.5h)
- Application: Dynamic Optimization of a Monopolist. [Ch] 2.4, 3.2 (2h)
- Second Order Conditions: Concavity, Legendre Condition. [Ch] 4, 4.1, 4.2 (1h)
- Infinite Planning Horizon. [Ch] 5, 5.1 (1h)
- Constrained Problems. [Ch] 6, 6.1 (1h)
- Application: The Ramsey Model. [Ch] 5, 5.3 (2h)
Part II: Optimal Control Theory
10 hours - Prof. Giulio Bottazzi (Theory) and Dr. Davide Pirino (Tutorials and Applications)
- Optimal Control: Introduction, Hamiltonian and Costate Variable, A varational view, Transversality conditions. [Ch] 7, 7.1, 7.2, 7.3 , 7.4. (3h)
- More on Otpimal Control: Economic Interpretation, Current Value Hamiltonian, Mangasarian and arrow Succificiency Conditions, Infite Time Horizion, Constraints. [Ch] 8, 8.1, 8.2, 8.3, 9, 9.1, 10.1 (3h)
- Application: Neoclassical Growth Theory, from Cass-Koopmans to Romer. [Ch] 9.3, 9.4 (4h)
Part III: Recursive Methods
10 hours - Dr. Pietro Dindo
- Preliminaries: Correspondences, Maximum Theorem. [Su] 9, 9.1, 9.2, 9.3. (2h)
- Finite Horizon Problems: From Variational Principles to Recursive methods, Bellman Equation and Backward Induction. [Su] Chapter 11. (2h)
- Infinite Horizon Problems: Existence and Characterization of Optimal Strategies, Bellman Equation. [Su] Chapter 12. (2h)
- More on recursive methods: A stochastic example, iteration methods. Lecture notes (1h)
- Applications: Ramsey Model in discrete time, Money in the Utility Function Monetary Theory Models, Lecture notes (1h + 2h)