STATIC AND DYNAMIC OPTIMIZATION MODELS IN AGRICULTURE

AEB 6533

This course is intended to give the students a background in classical optimization models with emphasis on mathematical programming and practical applications, and to introduce the students to dynamic optimization.

Prerequisites: ESI 4567

Instructor: Charles B. Moss
             1130B McCarty Hall
             Phone:  392-1845 Ext 404
             Email:  cbmoss@ifas.ufl.edu

Two two-hour lectures meet on Tuesday and Thursday. Grades in the course will be assigned based on biweekly homework, two examinations (a mid-term and a final), and a term project. I am willing to meet outside class for consultataion on the students' term projects.

Textbooks: There are three primary textbooks used in this course:

Gill, Philip E. Walter Murray, and Margaret H. Wright. Practical Optimization (New York: Academic Press, 1981).

This book will be used extensively in the first half of the course and will be considered essential. The theoretical development of the calculus of variations and optimal control follows with most of this material as presented in:

Kamien, Morton I., and Nancy L. Schwartz. Dynamic Optimization the Calculus of Variations and Optimal Control Theory in Economics and Management (New York: North-Holland, 1981).

This book is also considered essential.

Tu, Pierre N. V. Dynamic Systems An Introduction with Applications in Economics and Biology Second Edition (New York: Springer-Verlag, 1994).

Will be used to review the mathematics of dynamic systems. Other materials such as lecture notes will be made available on the internet at http://kierkegaard.ifas.ufl.edu/chuck/aeb6533.mathprogramming/syllbus.html.

Computing: The software used in this course is available on my DEC Alpha/OSF workstation, and can be accessed through the network. OSF is a variant of Unix, and can be operated with Xwin, a Windows terminal emulator. GAMS is a standard program on the department network. Further, a student version of GAMS is available with the users guide at a reasonable price. Mathematica and Gauss will be also be used extensively. Both programs are available on the DEC. Gauss is also available over the departmental network. A student version of Mathematica is available at the Technology Hub.

Grades: The grade for this class will be assigned based on two examinations, a term project, class participation, and homework as follows:
Mid - Term Examination
25%
Final Examination
35%
Term Project
25%
Homework
10%
Class Participation
5%

Homework will be due in two week increments. Each assignment will be handed out at the beginning of each increment and will involve topics covered over that time span. For example, the first assignment will involve the derivation of the first and second order necessary conditions for an unconstrained mathematical formulation. The term project consists of two components (1) a written paper of a relevant application of static or dynamic optimization and (2) an oral presentation of the results in class. The intent of this project is to prepare the student for the contributed paper process followed by most academic societies. The 5% assigned to class participation is intended to guarantee that the students are attending classes and keeping up with the readings.

Outline
Class notes will be offered in PDF format. To display these documents on your screen you will require Adobe . The docuements can be printed either with a Adobe PostScript card in a printer or ghostcript.

Overview of Optimality

This material covers the essential elements of unconstrained optimization. We will begin with a discussion of Taylor series exapansion and show how this expansion can be used to derive the traditional first and second order conditons for optimality.

Readings: Gill, Murray and Wright section 2.3 pp. 45-58.

Online Materials
Course Outline
Review of Linear Algebra

My review of linear algebra will focus on the ``null space'' which is used in mathematical programming in the guise of the ``canonical form.'' Other useful topics of review include the discussion of basis and spanning.

Online Materials
Readings: Gill, Murray and Wright section 2.2 pp. 14-45.

Also See Simon, C.P. and L. Blume Mathematics for Economists New York: W.W. Norton & Company, 1994.

Course Outline
Optimality Conditions

Given this broad understanding of optimality and some tools from our discussion of linear algebra, we turn to the development of the optimality conditions for the unconstrained univariate case, the unconstrained multivariate case, the linearly constrained multivariate case and the nonlinearly constrained multivariate case. In addition we discuss the traditional Kuhn-Tucker optimality conditions.

Online Materials
Readings: Gill Murray and Wright Chapter 3, pp. 59-82.

Course Outline

Algorithms

The primary focus of this course is in using mathematical methods to solve economic problems. Given this focus the study of algorithms is very important. Analystical solutions of the first order conditions in applied work are fairly rare. Most applications rely on iterative procedures to solve the problem at hand. This section discusses a number of these iterative procedures.

Online Materials
Readings: Gill Murray and Wright Chapter 4, 83-153.

Course Outline

Optimization on a Computer

The iterative nature of applied mathematical programming necessitates the use of computers in the process. Thus, as a part of learning about mathematical programming we will discuss the merits of several different solvers starting with the OPTMUM routines in Gauss along with similar routines in Matlab. Next, we will discuss the use of ``algebraic engines'' such as Mathematica or MatCad. Finally, we will examine the use of specialized code such as GAMS and Minos.

Online Materials

Course Outline
Sample Midterms
Mathematical Definition of the Dual

Course Outline
A Review of Dynamic Mathematics

The first challenge in studying dynamic systems is to understand its mathematical tools. There are two broad types of tools (1) differential equations which are used in continuous time problems such as fisheries and depletion problems, and (2) difference equations which are used in finance, macroeconomcs and econometrics.

Online Materials

Readings Tu Chapters 2 and 3, and 5 and 6 pp. 5-57 and 83-161.

Course Outline

The Calculus of Variations

A critical building block in developing dynamic optimality is the calculus of variations which yields the Euler conditions for dynamic optimality. In developing this approach the student will learn tools used in the derivation of optimal control.

Online Materials

Readings: Kamien and Schwartz Part I, Sections 1-6, pp 3-42.

Course Outline

Optimal Control

The variational approach has several limiations, but the basic formulation can be used to derive the Pontryagin or optimality conditions for optimal control.

Online Materials
Readings: Kamien and Schwartz Part II. Sections 1-9, pp. 109-169.

Course Outline

Applications of Optimal Control

To complete the study of Optimal Control theory we will look at three applications. The first is the problem of exhaustible resources found in P.S. Dasgupta and G. M. Heal Economic Theory and Exhaustible Resources Chapter 6, pp. 153-92. The second example is the management of renewable resources found in Colin Clark Mathematical Bioeconomics: The optimal management of renewable resources Chapter 4, pp. 88-121, The final example will be taken from the macroeconomic literature.

Online Materials
Course Outline
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