L Karp Course Outline ARE 263, Fall 2004 Methods of Dynamic Analysis and Control Objective: Introduce students to a variety of methods of dynamic analysis and control. Students should learn how to formulate dynamic models, and to develop some understanding of how to solve them - either analytically or numerically. Description of material: Many economic models treat agents as maximizing the present discounted sum of a stream of future payoffs, in situations where current decisions affect future constraints. For example, current extraction of oil or harvest of fish affect the future oil or fish stock, and thus impose limits on future consumption. We will study several ways to analyze these kinds of problems. We will emphasize methods of qualitative analysis for deterministic models. We will discuss a couple of types of stochastic control problems, and mention some numerical methods. The list of topics below contains 13 items. I will cover the first five of them fairly systematically; then I will pick two or three additional topics to cover fairly quickly. Prerequisites: Students should be familiar with differential and integral calculus. They should know how to solve linear ODE's. Course evaluation. There will be at least 6 problem sets which will count for 40% of the grade. A final project (described below) counts for 20% of the grade. The final exam counts for the remaining 40%. This exam will be given during finals weeks at the “official time” -- unless there is unanimous agreement to hold it during some other time. Books: A) (Principal text) Dynamic Optimization, 2nd ed, M Kamien and N Schwartz, North Holland B) Elements of Dynamic Optimization, Alpha Chaing, McGraw Hill. C) Dynamic Programming and Optimal Control, D Bertsekas, Athena Scientific Press (1995) D) Mathematical Bioeconomics, C Clark 2nd ed, Wiley E) Natural Resource Economics, Notes and Problems, Jon Conrad and Colin Clark, Cambridge University Press F) Elementary Differential Equations and Boundary Value Problems, W Boyce and R DiPrima, John Wiley and Sons (or some other ODE book)2 G) Differential Equations, Stability and Chaos, W Brock and A Malliaris, North Holland, 1989 H) Decision and Control in Uncertain Resource Systems, M Mangel, Academic Press, 1985. I) Numerical Methods in Economics, Kenneth Judd, MIT Press 1999 J) Applied Computational Economics and Finance, M Miranda and P Fackler,MIT Press, 2002 Articles and Notes Brander, James and Scott Taylor "The Simple Economics of Easter Island: a Ricardo-Malthus Model of Renewable Resource Use". AER 1998, vol 88 pp 119 - 138. Caputo. Chapter 16 of unpublished Lecture Notes on Dynamics Clarke and Reed. "Consumption/Pollution Tradeoffs..." Journal of Economic Dynamics and Control 1994 vol 18 pp 991 - 1011. Feichtinger et al, "Limit Cycles in intertemporal adjustment models" JEDC 1994 vol 18 pp 353 - 380. Karp. 10 sets of lecture notes Krugman, P. (1991) "History versus Expecation" QJE 651 - 67. LaFrance J and L Barney (1991) "The Envelope Theorem in Dynamic Optimization" J of Econ Dynamics and Control 15: 355-385. Tahvonen and Salo. "Nonconvexities in Optimal Pollution Accumulation", Journal of Environmental Economics and Management (1996) vol 31 pp 160 - 177. Xie, D (1997) "On Time Inconsistency..." JET vol 76 pp 412 - 430. I will cover the first three topics during the first 6-8 weeks. Then I will spend a couple of weeks on one or two other topics (excluding topic numbers 5 and 6). Topic 5 will be expanded, to include material from Judd and from Fackler and Miranda. I will spend the last several weeks of class on this material (numerical methods for dynamic programming). Final project: I would like you to formulate and solve a dynamic optimization problem using numerical methods. (In addition, you can provide qualitative analysis if the model lends itself to that. I want you to emphasize numerical methods.) Provide a verbal description of the problem and its mathematical formulation. Provide a verbal description of the solution algorithm and the computer code. Describe the results. The paper should be no more than a few (4 or 5) pages3 long, excluding computer code. Topics (with readings) 1. Basics of Ordinary Differential Equations (ODEs) and Phase Plane Analysis. D, chapter 6; A, appendix B; F, chapter 9; Karp #1; Krugman, P. (1991) "History versus Expecation" QJE 651 - 67. 2. The Calculus of Variations (COV). A, part I, sections 1 - 11; or B, chapters 1 - 4. Karp #2. 3. The Maximum Principle. A, part II, sections 1 - 9; or B, chapters 7 - 9; Dynamic Envelope Theorem. Caputo; or A, part II section 8; LaFrance and Barney. Karp #3. 4) Uncertainty. A, pp 61 - 63 and 190 - 193; Clarke and Reed; Karp #4. 5) Dynamic Programming, discrete and continuous time. J, Chapters 6 – 9. A, part II, section 21; C (vol 1), chapter 1 and 3; Karp #5. (Applications to “Prices versus Quantities”) 6) Two stochastic control problems. H, pp 50 - 53. C, chapter 6.1 - 6.3; Karp and Pope; Karp #6. 7) Limit Cycles and adjustment cost problems. Two State Variables and Cycles. Feichtinger et al.; Karp #7. 8) Nonconvexities. G pp 159 - 168; Tahvonen and Salo; Karp #8. 9) Linear control problems, deterministic and stochastic. A, part I, section 13 and part II, section 13; D, chapter 2.7; Karp #9; Weitzman’s fish paper. 10) Dynamic Games. Karp #10, Xie. 11) Learning. 12) Hyperbolic discounting 13) Estimating dynamic