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)

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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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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)

COPYING AND REPRINTING. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgement of the source is given.

Republication, systematic copying, or multiple production of any material in this pub­lication (including abstracts) is permitted only under license from the American Mathemat­ical Society. Requests for such permission should be addressed to the Executive Director, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940.

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1980 Mathematics Subject Classification. Primary 49-06, 90-06, 92-06. Copyright © 1985 by the American Mathematical Society.

Printed in the United States of America. This volume was printed directly from copy prepared by the authors. The paper used in this book is acid-free and falls within the guidelines

established to ensure permanence and durability.

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