BEA511 Topics in Dynamic Modeling and Optimal Controls
The objective is to provide students with a capability to formulate and analyze problems in different fields of economics and management science using the tools of optimal control theory. Deterministic and stochastic theory will be presented. Continuous time problems are emphasized. Applied examples (applications) constitute the main part. Classical Hamiltonian formulation with focus on the Maximum Principal (MP) including different constraining relations and their transversality conditions are addressed. Modern formulation applying the value function concept through Dynamic Programming (DP) and its associated Hamilton-Jacobi-Bellman equation (HJB) are introduced to facilitate and bridge the gap to the course BEA514- Topics in numerical optimization.
Stochastic optimal control problems are incorporated in this part.
The relations between MP and DP formulations are discussed. The main focus is put on producing feedback solutions from a classical Hamiltonian formulation. Interpretations of theoretical concepts are emphasized, e.g. that the Hamiltonian is the shadow price on time.
Differential games are introduced.
The ideas of Equivalent Representation and The Principal of Extension are introduced.
Finite and infinite time horizons are treated. Relaxation of the optimality concept is introduced through the notion of "Catching-Up optimality", which may apply if the classical value becomes infinite.
Among others, we study applications such as Ramsey's growth model, production and storage planning, advertising, management of (non-) renewable resources, Pigouvian taxation and pollution control, road planning, maintenance and sale and allocation of private wealth on consumption, secure and risky investments. Real options are presented in a generic setting.
Topics will be lectured in the following sequence:
- Short summary of difference- and differential equations and stability analysis
- Basic concepts and ideas. Global optimality. Shadow prices and theoretical and practical interpretations of basic notions e.g. transversality conditions.
- Introductions to applied control problems. Necessary and sufficient conditions.
- Discounting and present value formulations. Extensions of optimality in infinite horizon problems.
- Constrained problems: State and mixed restrictions. Bang-bang and singular controls.
- Dynamic programming (DP) and the Hamilton-Jacobi-Bellman (HJB) equation.
- Summary of stochastic processes. Stochastic feedback control. Merton´s example
- Differential games - open and closed loop policies
After successfully completing the course, the candidates should be able to:
- analyze and evaluate potential dynamic and stochastic effects on economic quantities and resources depending on policy choices
- identify practical limitations of present day numerical solution approaches
- review, assess and utilize relevant scientific papers addressing dynamic optimization
- formulate and model operational management tasks evolving in time
- identify potential intrinsic deterministic chaos in the formulated model
- take part in and manage interdisciplinary research involving dynamic modelling and decisions tasks in an operational setting
- analyze models with respect to dynamic as well as structural stability
- identify potential or implicit conserved quantities (conservation laws)
- formulate and analyze problems in different fields of economics and management science applying the tools of classical and modern theories of optimal control and the calculus of variation.
Topics/papers will be partly lectured by course responsible and partly presented for discussion in class by students
Knowledge of medium advanced calculus and some familiarity with differential and difference equations and introductory probability theory
Requirements for course approval
Requirements for course approval
- Activity in class
- Exercises during the course
Based on: 1. Final individual home exam/assignments (80%), 2. Short oral examination (20%)
The use of high-level programing in Maple and MatLab will be an integrated part of the course.
Main topics and applications are presented in part two of the textbook "Dynamic Optimization: The theory of Variations and Optimal Control in Economics and Management" by Morton I. Kamien and Nancy L. Schwartz in the series "A series of Volumes in dynamic economics: Theory and applications" volume 4.
Additional topics are given in lecture notes and selected journal articles.
- ECTS Credits
- Teaching language
Leif Kristoffer Sandal