Natural Resource Economics: Notes and Problems. by Jon M. Conrad; Colin W. ClarkReview by: James KeeslingSIAM Review, Vol. 32, No. 2 (Jun., 1990), pp. 323-325Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2030543 .

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BOOK REVIEWS 323

with the intensity of the mathematics. Our students also found the biological presenta- tions interesting at times. In our opinion, the audience for this book should be mathema- ticians with a decent background in analysis and differential equations and a desire to learn more about dynamical systems and applications of mathematics to biology. It served our seminar study needs well because the survey of some fields of research was useful and the presentation provoked many discussions on the relevance of mathematics to biology.

The authors indicate the hope that this book is (i) an introduction to the theory of dynamical systems based entirely on examples from biology, and (ii) a survey of some recent developments in evolution. They also "hope to point out a few interest- ing sights along the way." They have accom- plished their limited objectives well. The book belongs to a student textbook series of a mathematical society. In my opinion, the uses of the book as a classroom text are rather limited; indeed, as the authors caution, this specialized book is not intended to serve as a text in differential equations. However, the book is well written; it does present a good survey of qualitative behavior of generalized Lotka-Volterra systems; and it deserves to be in the library of mathematicians who enjoy clever applications of mathematics in biological settings.

THOMAS G. HALLAM

University of Tennessee, Knoxville

Natural Resource Economics: Notes and Problems. By Jon M. Conrad and Colin W. Clark. Cambridge University Press, Cam- bridge, UK, 1987. x + 231 pp. $39.50, cloth, ISBN 0-521-53188-9; $14.95, paper, ISBN 0-521-33769-0.

The late Elliott Montroll once applied the theory of natural selection to items of- fered for sale in the Sears Catalogue, and the result was a delectable morsel. Advertise- ments grew as items flourished and waned as sales declined. There was even extinction when an issue finally appeared with a partic- ular article no longer offered for sale. The study was no doubt intended primarily as

entertainment; nevertheless, it stimulates re- flection on a number of other human enter- prises as well. It is interesting to view the intellectual endeavor of a scholarly commu- nity in this way as individuals seek to find the niches where they and their disciples can flourish.

And the authors of this book have found a niche. It is not so much the specific content of the book that gives it survival value (although that surely contributes) as its filling a need in the literature for more books giving students practical applications. Many advanced textbooks, even in areas of applied mathematics, give the student little insight into the difficulties of computing solutions of even the most rudimentary problems.

It will help those who contemplate the use of this book to see precisely where it fits in the scheme of training the practitioners of mathematics. The problem of applying mathematical theory to the real world in any fashion is threefold. The first aspect is con- ceptual. One must learn the art of creating a mathematical grid through which to view some part of the real world. To some this part of the process is natural and subcon- scious just as the wearing of glasses becomes with prolonged use. The model becomes reality itself. One sees an anamorphic view of the world as though it were undistorted. However, in our ideal the applied mathe- matician retains a clear distinction between model and reality.

The second aspect is one of alignment. Parameters must be adjusted, and certain fiduciary marks must match the measure- ments of the real world, so that the quanti- tative aspects of the model are normalized. Parameter estimation and model verification usually use statistical tools. This aspect of applied mathematics may require collabora- tion with one who is expert in the experi- mental techniques in the area of application.

The third aspect of applying mathemat- ical theory is that of determining the features of the model itself. In an earlier era, this meant the application of analytical tech- niques to compute and/or estimate the be- havior of the model. In the age of computers we have a completely new paradigm. One must have a handy toolbox of computer techniques (and, of course, a computer), a ready capability to estimate the errors of various calculations, and the ability to put the results into easily assimilable graphical

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324 BOOK REVIEWS

form. Computers have liberated the model makers. They need not restrict their atten- tion only to models that are analytically tractable; the assumptions made about the real world in constructing the mathematical grid can be much more realistic, though in- timidating. The computer, used intelligently, can supply what is wanting and provide quite useful insights into the behavior of very complicated models. The insights can even lead to new analytical capability by sug- gesting likely behavior based on numerical calculations.

The volume under review mainly pro- vides examples for the student using certain models from natural resource economics. It is more substantial than a typical Schaum's outline, but it is below the level of Halmos's A Hilbert Space Problem Book; the problems are illustrative and use the computer and elementary numerical techniques, but their solution will not qualify for a dissertation. There is a summary of the mathematical techniques together with numerical exam- ples. There are also exercises to be done by the student. The computer is a required tool. Many of the exercises require it, and assorted examples have computer solutions with list- ings of the program code and output. The authors use BASIC. A language more struc- tured than BASIC would have been prefera- ble if the examples had been more compli- cated. However, BASIC is probably the most widely known language, and there is justifi- cation for its use even if it does not provide the most favorable programming environ- ment. After all, one string of spaghetti is not all that confusing. Theory is not given suffi- cient treatment for the book to serve as the main text for a mathematics course; it would be necessary to have a companion volume to provide sufficient substance. However, since so many mathematics texts have a dearth of examples and no insights into the numerical techniques necessary to analyze relevant models, this book would provide a valuable ancillary resource for instructor and student. For a course in economics or envi- ronmental engineering there is a possibility that it could stand alone.

What is the mathematics of natural re- source economics anyway? This is a field of diverse mathematical techniques. It is natu- ral to compare it with operations research. One might in fact consider natural resource

economics to be a new branch of operations research. Operations research itself began as a technical analysis of military tactics and strategy. The applicability of this approach to nonmilitary enterprises soon became apparent. Today most major corporations depend to a considerable extent on linear programming (and lawyers) to maintain their profitability. In natural resource eco- nomics we now have mathematics poten- tially playing a crucial role in husbanding the globe!

Here is a sampling of topics. There is constrained optimization: The method of Lagrange multipliers is explained rather well, and there is some discussion of the Lagrange multiplier, a shadow cost, and a measure of the profitability of varying constraints. Some deterministic and stochastic dynamic pro- gramming problems are presented. The bi- section method and Newton's method for solving equations are covered. The bisection method is intrinsically one-dimensional, but even Newton's method is applied in the book to only one-dimensional problems. The stu- dent is encouraged to use software packages for Newton-Raphson methods of solving systems of nonlinear equations. This is sage advice, but it might have been good experi- ence to have solved a small-dimensional problem to understand the principles in- volved. The standard fourth-order Runge- Kutta method is presented as a numerical method for solving systems of differential equations. Eigenvalues are discussed, but only to show how they are employed to determine the behavior of solutions to sys- tems of ordinary differential equations near an equilibrium. Methods for computing se- paratrices are presented. In the section on growth functions we have an allusion to the chaotic behavior of the logistic population growth x,+ = rx (K - x,) for sufficiently large values of the reproduction rate r. A great deal of theory is hidden in this remark. We also have applications of control theory and numerous optimization problems. We also encounter such topics as discounting and optimal investment strategy, terms fre- quently found in a course in business school. One needs to be concerned with the return on one's investment even in dealing with natural resources and the environment; one cannot ignore the realities of life in the mod- ern world! Here the student can acquire a

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BOOK REVIEWS 325

taste for the actuary's trade on a global scale.

The chapters are entitled "Resource allocation and optimization," "Renewable resources," Nonrenewable resources," "En- vironmental management," and "Stochastic resource models." The focus is on the differ- ent types of analysis necessary for the various resource systems. Sometimes this brings a specific area of mathematics into stark relief. At other times the focus is on the application with the mathematics playing an ad hoc role.

The natural resource systems analyzed include fisheries harvesting, forest harvest- ing, mining of a limited resource, searching for scarce resources, and the economics of residuals (pollutant waste byproducts). By focusing on specific examples we are led to ask the appropriate questions about the sys- tem. Is the system deterministic? Should time be considered continuous or discrete? How significant are the stochastic aspects of the system? What are the economic measures of the resource? What types of optimization techniques are relevant? These questions help the worker decide which mathematical grid is appropriate to properly analyze the natural resource system at hand. Some sys- tems are analyzed in several different ways in order to bring out the usefulness of certain methods. Forest harvesting is analyzed deter- ministically at one point, but when it is analyzed stochastically we can assess the eco- nomic significance of forest fires in the analy- sis. There are also useful comments about the importance of taking into account other matters besides the direct pecuniary measure of the system.

Would this book be useful to you in a course? The answer to that is undoubtedly yes if you are teaching a course in natural resource economics or management of nat- ural resources. Perhaps you are teaching a course in operations research and want to emphasize modelling of natural resource sys- tems. You and your students would find this book a valuable help and it could provide problems for homework assignments. Per- haps there is a seminar in applied mathe- matics whose topic varies from year to year. You could dedicate the course to analysis of resource systems and again get a match! As already mentioned, one would likely assign the book as a supplementary text, particu- larly in a mathematics course. Of course, the

book would be useful for personal study; I hope that some learning is done outside the classroom.

I suppose that Cambridge University Press has estimated the likely demand for the book and the profitability of this venture (perhaps using some of the investment analy- sis covered in the book itself). Or perhaps there is an environmentalist fifth columnist on the Cambridge editorial staff who rec- ommended the book knowing the desperate need in the world for a greater supply of competent people to manage our natural resources. That is a need and this book will certainly help.

JAMES KEESLING University of Florida

at Gainesville

Linear Stochastic Systems. By Peter E. Caines. John Wiley, New York, 1988. xvi + 874 pp. $59.95. ISBN 0-471-08101-9. A vol- ume in the Wiley Series in Probability and Mathematical Statistics.

The stated purpose of this book is "to give a unified account of the main results of the theory of linear discrete-time-parameter stochastic systems" (Preface). This is indeed an ambitious project, but in this book it is carried out in great completeness and detail. It is hard to find a topic of any importance that is not covered in one form or another. As a consequence, this is not a book that anyone is likely to read from cover to cover; rather, it will find its place as a reference work, as I am sure the author has meant it to be.

A couple of nice books on linear sto- chastic systems have already been published. I think of Davis [ 1 ] and Kumar and Varaiya [2]. However, these treat a more selected set of topics. Caines' book is the first of a kind in that the author has spared no effort to present a comprehensive treatment of linear stochastic systems. This is the book's strength and, maybe unavoidably, also its weakness. In fact, its commitment to detail sometimes makes it hard to read. Occasionally, how- ever, this impression is counterbalanced by some very well-written sections.

I think it is fair to say that Caines has done the field a great service by writing this book. Systems and control, or at least the

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