VOLUME 17 APPLICATIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS IN MATHEMATICAL PHYSICS Edited by R. Finn (New York City, April 1964)
VOLUME 18 MAGNETO-FLUID AND PLASMA DYNAMICS Edited by H. Grad (New York City, April 1965)
VOLUME 19 MATHEMATICAL ASPECTS OF COMPUTER SCIENCE Edited by J. T. Schwartz (New York City, April 1966)
VOLUME 20 THE INFLUENCE OF COMPUTING ON MATHEMATICAL RESEARCH AND EDUCATION Edited by J. P. LaSalle (University of Montana, August 1973)
AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures
VOLUME 21 MATHEMATICAL ASPECTS OF PRODUCTION AND DISTRIBUTION OF ENERGY Edited by P. D. Lax (San Antonio, Texas, January 1976)
VOLUME 22 NUMERICAL ANALYSIS Edited by G. H. Golub and J. Oliger (Atlanta, Georgia, January 1978)
VOLUME 23 MODERN STATISTICS: METHODS AND APPLICATIONS Edited by R. V. Hogg (San Antonio, Texas, January 1980)
VOLUME 24 GAME THEORY AND ITS APPLICATIONS Edited by W. F. Lucas (Biloxi, Mississippi, January 1979)
VOLUME 25 OPERATIONS RESEARCH: MATHEMATICS AND MODELS Edited by S. I. Gass (Duluth, Minnesota, August 1979)
VOLUME 26 THE MATHEMATICS OF NETWORKS Edited by S. A. Burr (Pittsburgh, Pennsylvania, August 1981)
VOLUME 27 COMPUTED TOMOGRAPHY Edited by L. A. Shepp (Cincinnati, Ohio, January 1982)
VOLUME 28 STATISTICAL DATA ANALYSIS Edited by R. Gnanadesikan (Toronto, Ontario, August 1982)
VOLUME 29 APPLIED CRYPTOLOGY, CRYPTOGRAPHIC PROTOCOLS, AND COMPUTER SECURITY MODELS By R. A. DeMillo, G. I. Davida, D. P. Dobkin, M. A. Harrison, and R. J. Lip ton (San Francisco, California, January 1981)
VOLUME 30 POPULATION BIOLOGY Edited by Simon A. Levin (Albany, New York, August 1983)
VOLUME 31 COMPUTER COMMUNICATIONS Edited by B. Gopinath (Denver, Colorado, January 198 3)
http://dx.doi.org/10.1090/psapm/032
PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS
VOLUME 1 NON-LINEAR PROBLEMS IN MECHANICS OF CONTINUA Edited by E. Reissner (Brown University, August 1947)
VOLUME 2 ELECTROMAGNETIC THEORY Edited by A. H. Taub (Massachusetts Institute of Technology, July 1948)
VOLUME 3 ELASTICITY Edited by R. V. Churchill (University of Michigan, June 1949)
VOLUME 4 FLUID DYNAMICS Edited by M. H. Martin (University of Maryland, June 1951)
VOLUME 5 WAVE MOTION AND VIBRATION THEORY Edited by A. E. Heins (Carnegie Institute of Technology, June 1952)
VOLUME 6 NUMERICAL ANALYSIS Edited by J. H. Curtiss (Santa Monica City College, August 195 3)
VOLUME 7 APPLIED PROBABILITY Edited by L. A. MacColl (Polytechnic Institute of Brooklyn, April 1955)
VOLUME 8 CALCULUS OF VARIATIONS AND ITS APPLICATIONS Edited by L. M. Graves (University of Chicago, April 1956)
VOLUME 9 ORBIT THEORY Edited by G. Birkhoff and R. E. Longer (New York University, April 1957)
VOLUME 10 COMBINATORIAL ANALYSIS Edited by R. Bellman and M. Hall, Jr. (Columbia University, April 1958)
VOLUME 11 NUCLEAR REACTOR THEORY Edited by G. Birkhoff and E. P. Wigner (New York City, April 1959)
VOLUME 12 STRUCTURE OF LANGUAGE AND ITS MATHEMATICAL ASPECTS Edited by R. Jakobson (New York City, April 1960)
VOLUME 13 HYDRODYNAMIC INSTABILITY Edited by R. Bellman, G. Birkhoff, C. C. Lin (New York City, April 1960)
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AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures published as a subseries of Proceedings of Symposia in Applied Mathematics
PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS
Volume 32
Environmenta l an d Natura l Resourc e
Mathematic s
AMERICAN MATHEMATICAL SOCIETY
PROVIDENCE, RHODE ISLAND
LECTURE NOTES PREPARED FOR THE AMERICAN MATHEMATICAL SOCIETY SHORT COURSE
ENVIRONMENTAL AND NATURAL RESOURCE MATHEMATICS
HELD IN EUGENE, OREGON AUGUST 14-15 , 1984
EDITED BY ROBERT W. McKELVEY
The AMS Short Course Series is sponsored by the Society's Committee on Employment and Education Policy (CEEP). The series is under the direction of the Short Course Advisory Subcommittee of CEEP.
Library of Congress Cataloging in Publication Data Main entry under title:
Environmental and natural resource mathematics. (Proceedings of symposia in applied mathematics; v. 32. AMS short course lecture
notes) Includes bibliographies. 1. Environmental policy—Mathematics—Congresses. 2. Environmental protection-
Mathematics—Congresses. 3. Natural resources—Management—Mathematics—Congresses. I. McXelvey, Robert W., 1929— . I I . American Mathematical Society. HI. Proceedings of symposia in applied mathematics; v. 32. IV. Proceedings of symposia in applied mathematics. AMS short course lecture notes. HC79.E575 1985 33.7'0l'51 85-3917 ISBN 0-8218-0087-6 (alk. paper)
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1980 Mathematics Subject Classification. Primary 49-06, 90-06, 92-06. Copyright © 1985 by the American Mathematical Society.
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CONTENTS
Preface ix
Applications of mathematics in insect pest management
RICHARD E. PLANT 1
Economic incentives for pollution control
MAUREEN L. CROPPER 19
Depletion and discounting: a classical issue in the economics of exhaustible
resources
GEOFFREY HEAL 33
Capital theoretic aspects of renewable resource management
COLIN W. CLARK 45
Applying abstract control theory to concrete models
FRANK H.CLARKE 55
International trade in resources: a general equilibrium analysis
GRACIELA CHICHILNISKY 75
The role of mathematicians in natural resource modeling
PANEL DISCUSSION 127
vii
PREFACE
This volume contains lecture notes, mildly revised from those originally
prepared for the American Mathematical Society's 1984 Summer Short Course, held
in Eugene, Oregon on August 14-15. It also contains an expanded version of a
panel discussion at the Short Course, on the role played by mathematicians in
natural resource modeling.
The term "natural resources" is to be interpreted broadly, encompassing
air and water resources, land and soil, minerals and oil, energy resources, and
such biological resources as fisheries, agricultural crops, forests, and
wildlife. The objective of the Short Course, and of this volume, is to
demonstrate that, despite the great diversity of kinds of natural resources,
there has developed a coherent theory concerning the efficient and conservative
management of resources, and that this theory has a substantial mathematical
component.
The first lecture, by Richard Plant, is a case study, introducing us to
the field of agricultural insect pest management. Several management issues
are investigated, relating to the economically optimal timing of the applica
tion of chemical pesticides, to the maintenance within the insect population of
susceptibility to the pesticides, and to the use of a biological control tech
nique, namely the release of sterile males into an indigenous pest population.
These issues are explored through simple, analytically tractible, dynamic popu
lation models. These, despite their simplifying assumptions, provide insight
into the effects of both environmental and biological factors, some of which
are subject to management control. Stochastic variability is demonstrated to
influence profoundly the behavior of the modeled system, and hence the
appropriate management strategies. While conveying something of the flavor of
a particular applied resource management field, Plant's lecture also
demonstrates the spirit in which analytical resource modeling exercises are
conducted generally.
In Maureen Cropper's lecture we are introduced, in the context of a water
quality problem, to one of the central themes of natural resource theory: the
development of strategies to cope with environmental or resource
"externalities." Here, certain users of a lake or stream pollute its waters,
degrading them for use by others. Technical means exist for abating or miti-
ix
X PREFACE
gating the pollution, at costs below the value to society of the improvements
thus attained. The problem, then, is to work out cooperative arrangements
among the parties to achieve optimal abatement, either through private
bargaining or through government arranged incentives.
What makes the problem difficult is the high cost or even unavailability
of information to achieve optimal abatement: in this case, information about
emission abatement costs, damages from ambient pollution, and the physical
relation between emission and ambient levels. Furthermore, polluting firms
have an incentive to misrepresent their abatement costs and injured parties to
exaggerate damages. The problem of inducing agents to truthfully reveal infor
mation is a general one in economic theory. Professor Cropper describes,
through a simple model applied to the water pollution example, mechanisms that
have been proposed for achieving this aim.
A classical problem associated with the management of an exhaustible
resource, such as an underground pool of oil, is the determination of the
appropriate rate of depletion: that is, of the time profile for the exhaustion
of the resource stock. The aim is to do this in a way that is at the same time
efficient and equitable: i.e., to provide society with in some sense the
greatest possible value from the asset, and at the same time to achieve a fair
distribution of the benefits among the generations.
These considerations are reflected in the choice of an intergenerational
utility function. Very frequently the utility measure incorporates the prin
ciple of the discounting of future returns. But discounting is highly contro
versial, and indeed often is held to be unacceptable ethically.
Geoffrey Heal, in his lecture, demonstrates some of the paradoxical
results of intertemporal utility theory. He shows how, on the basis of reason
able, and seemingly innocuous, postulates concerning intergenerational social
preference orderings, one is led automatically to the discounting of future
returns. He also examines the implications of uncertainty about future tech
nological progress, and its relation to the problem of resource conservation.
For a self-renewing resource, such as a reproducing biological population,
the task of resource conservation is akin to that of determining the optimal
rates of investment in and consumption of an ordinary capital asset. Colin
Clark elaborates on this capital-theoretic perspective on renewable resource
management, through a series of optimal control models of the harvest of a
marine fish stock. These explore, among others, the limitations of myopic
decision rules, the complications that arise in multispecies fisheries, the
effect of irreversible capital investment in fishing vessels and, again, the
profound implications of uncertainty and incomplete information. Indeed much
current research is concentrated on the stochastic fishery, where fish stocks,
PREFACE Xi
market prices and costs fluctuate erratically, and stock levels at any given
time are known only indirectly and very imprecisely.
As several of the lectures demonstrate, the methods and ideas of optimal
control theory play a central role in the theory of management of natural
resources. In his lecture, Frank Clarke undertakes the ambitious task of sur
veying for the applied resource modeler some of the main results, techniques,
and principles involved in analysing specific problems of optimal control. The
illustrative examples are drawn largely from resource management and are
intended to display a gamut of techniques. Clarke's announced intention is to
make several "megapoints":
"(a) It is necessary to have a working knowledge of a variety of
abstract machinery, both of its scope and its limitations.
(b) There are pitfalls associated with incomplete analysis of problems.
(c) Non-trivial optimal control problems are hard. (Yes, this
may be tautological.)"
The final lecture in this volume is that of Graciela Chichilnisky. Her
focus is on the macroeconomic effects of international trade in extractive
resources, for example the current world trade in oil. The discussion is based
on a simple model of "North-South" trade in resources, with an inflow of
foreign capital to the producing country. (The recent transfer of 100 billion
US dollars to Mexico is a relevant example.) The model is a condensed general
equilibrium model, restricted to contain only five markets and four economic
agents or groups. An unusual feature is that this model is mathematically
tractable, and not merely accessible through computer simulation. It can be
used not only to ascertain existence and uniqueness of the equilibrium, but to
make qualitative judgments concerning a wide variety of policy issues, the
results appearing in the form of theorems involving the shapes and global prop
erties of solution manifolds, as the underlying parameters vary. The results
obtained are somewhat surprising, confirming the view that the behavior of
several markets, all interacting with each other, is not readily understood by
the common wisdom usual to a single market. Cross market effects, the very topic of general equilibrium theory, yield the most surprising results and
those which are perhaps the most useful for policy.
The Short Course concluded with a panel discussion, featuring lecturers
and with participation by their audience. The discussion was taped,
transcribed and edited for inclusion in this volume. The discussion explored
the fascinating role that mathematicians and mathematically trained scientists
have played throughout the development of the discipline of natural resource
xii PREFACE
modeling, and in economic theory generally. The discussion then turned to con
sideration of ways in which concepts and techniques of modeling might best be
incorporated into graduate and undergraduate mathematics education.
Robert McKelvey
Department of Mathematical Sciences
University of Montana