Journal of agricultural economics research

62 Pages·1991·5.5 MB·English

by  United States. Department of Agriculture. Economic Research Service

Historic, archived document
Do not assume content reflects current
scientific knowledge, policies, or practices.
i
Articles
Nonlinear and Chaotic Dynamics: An Economist's Guide
The Changing Structure of the U.S. Flour Milling Industry
Determination of a Variable Price Support Schedule as Applied
to Agricultural Production Control
Nonfarm Prospects Under Agricultural Liberalization
Book Reviews
A A
Taste of the Country: Collection of Calvin Beale's Writings
Three Faces of Power
Imperfect Competition and Political Economy: The New Trade Theory
in Agricultural Trade Research
U.S. Grain Policies and theWorld Market
The Political Economy of U.S, Agriculture: Challenges for the 1990s
A
Risk Analysis: Guide to Principles and Methods for Analyzing Health
and Environmental Risks
Agricultural Risk Management
Vol. 43, No. 3, Summer 1991
—
Editors Contents
Gene Wunderlich
Jim Carlin 1 In This Issue
Gene Wunderlich
Editorial Assistant
Denise M. Rogers Aiiides
Graphics Designer 2 Nonlinear and Chaotic Dynamics: An Economist’s Guide
Susan DeGeorge Michael D. Weiss
Editorial Board 18 The Changing Structure of the U.S. Flour Milling Industry
Ron Fecso C.S. Kim, William Lin, and Mack N. Leath
Roger Hexem
William Kost 26 Determination of a Variable Price Support Schedule as Applied to
Fred Kuchler Agricultural Production Control
Lester Mvers Wen-Yuan Huang and Bengt Hyberg
Kenneth Nelson
Mindy Petrulis 33 Nonfarm Prospects Under Agricultural Liberalization
Gerald Schluter Maureen Kilkenny
Shahla Shapouri
Book Reviews
44 A Taste of the Country: A Collection of Calvin Beale’s Writings
46 Three Faces of Power
This periodical is published
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51 The Political Economy of U.S. Agriculture: Challenges for the
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Rockville, 20849-1608
or call 1-800-999-6779. This .53 Risk Analysis: A Guide to Principles and Methods for Analyzing
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[phone]. 54 Agricultural Risk Management
The Secretary of Agriculture has
determined that the publication of
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Use of funds for printing this
periodical has been approved by
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and Budget.
In This Issue
As a special feature of this issue, the Journal presents policies. She argues that because freer trade reallo-
an article by Michael Weiss on the methodology of cates resources throughout the economy, a computable
dynamical systems. Thirty years ago in Econometrica, general equilibrium model is useful in tracing the
Benoit Mandelbrot wrote the first in a series of arti- effects of a free trade policy. She employs a 10-sector
cles challenging neoclassical economic theory’s imita- CGE model to examine the effects of liberalizing pol-
tion of 19th-century energy physics. Those articles icies under three adjustment scenarios. She allows
upset some ofthe fundamental notions of deterministic government deficit reduction effects ofliberalization to
mechanics inherent in neoclassical theory. According affect outcomes. Her analysis concludes that, if result-
to Mirkowski, Mandelbrot’s contributions to economics ing savings reduce government deficits, additions to
were essentially dormant while he moved on to physics GNP, perhaps $4.5 billion, will ensue from multilateral
and meteorology. However, a significant and growing trade liberalization. Improvements in GNP from trade
group of mathematically inclined economists is now liberalization in the 1990’s are expected to be less.
taking an interest in, and extending, his ideas. Weiss’s
article provides the essentials of chaos for modeling Several ofthe book reviews in this issue vary from our
dynamic behavior. standard economics fare, thus reminding us that, not-
withstanding the title of this journal, we do embrace
The three remaining articles more nearly reflect the all the social sciences. That eclecticism is well repre-
traditional applied analyses of agricultural economics. sented by Calvin Beale’s diverse writings, collected
A
Kim, Lin, and Leath examine the flour milling indus- under the title Taste ofthe Country, insightfully and
try. Huang and Hyberg try their hand at fine-tuning supportingly reviewed by Sonya Salamon.
agricultural price supports. Kilkenny models the off-
Members of the economics club can reach beyond Wal-
farm effects of liberalizing agricultural trade policy. In
rasian optima, witness Kenneth Boulding’s Three
the future, topics included in these papers, such as
firm failures, commodity stocks, and employment fluc- Faces of Power, reviewed by Dwight Gadsby. Bould-
tuations, may be extended with the insights of chaotic ing has p—resented the three dimensi—ons of human
relations threat, exchange, and love in many arti-
dynamics. In the meantime, our near-term analyses
can simply assume that tracks and outcomes are not cles and books. Here, like Lasswell, he shows the rela-
“sensitively dependent on initial conditions.” tionship between political and economic force.
The other books reviewed in this issue are closer to
Kim, Lin, and Leath look at causes and consequences conventional agricultural economics and policy. Daniel
of the reduced number, expanded size, and changed Pick recommends reading the collection of papers on
location of flour mills in the United States. Size effi- imperfect competition and trade theory edited by Car-
ciencies and automation will increase the number of ter, McCalla, and Sharpies. Robert Green is only mod-
large mills, reduce the number of small mills and the erately supportive of the collection of papers on grain
total number of mills, and reduce employment. Mills policies and the world market, prepared by Roberts,
have tended to move from sources of wheat supply Love, Field, and Klijn. Dave Ervin recommends the
toward markets for flour. Despite the estimated collection of papers on the political economy of agricul-
decline of mills from 203 in 1990 to 160 by the year ture, edited by Kramer, but emphasizes that the his-
2000, the authors see no problem in meeting an torical content is stronger than its empirical content.
increased demand for flour.
Michael Wetzstein reviews two recent books on risk.
The Huang and Hyberg alternative to production con- The guide to principles and methods ofrisk analysis by
trols for supply management is a form ofprice discrim- Cohrssen and Covello is directed primarily at health
ination they call variable pricing support schedules. and environmental risks. Never mind, the methods
They propose to manage supplies of commodities with and principles apply to the broad range of problems in
prices rather than with control of land use. The high- which risk is an element of concern. Wetzstein says
est of the variable support prices would go to the first the guide is “an excellent foundation for any student
units of production, with the lowest prices for the last interested in learning how risk analysis is currently
units at below expected market prices. Their program- undertaken.” The book by Fleisher on agricultural risk
ming model yielded price schedules that favor small management is a broad overview of techniques. As
farms over large farms, compared with the traditional such, this short volume may lack some of the detail of
mandatory production control programs. other references, but it is an excellent text with which
to enter the field of risk management.
Kilkenny looks beyond farms and agriculture as she
examines the effects of liberalizing agricultural trade Gene Wunderlich
1
Nonlinear and Chaotic Dynamics: An Economist’s Guide
Michael D. Weiss
Abstract, hi recent ijears, research in both niathenia- state space (where R is the real number line). Sup-
tics and the applied sciences has produced a revolu- pose that the economy evolves deterministically in
tion in the understanding of nonlinear dynamical such a way that its state at any time uniquely deter-
systems. Used widely in economics and other disci- mines its state at all later times. Then, if the initial
plines to model change over time, these systems are position of the economy in R^^ ^t time 0 is Vq, the
now known to be vulnerable to a kind of ''chaotic," evolution of the economy through time will be repre-
unpredictable behavioi-. This article places this revolu- sented by a path in R^-^ starting at Vq and traced out by
tion in historical context, discusses some of its \\ as time, t, moves forward. This path, called the
implications for economic modeling, and explains orbit generated by the initial position Vq, represents a
many ofthe important mathematical ideas on which it “future history” of the system. Questions about the
is based. behavior ofthe economy over time are really questions
about its orbits. We are often interested not so much
Keywords. Limits to predictability, nonlmear and in the near-term behavior of orbits as in their eventual
chaotic dynamical systems, structural stability of behavior, as when we engage in long-range forecasting
economic models, fractals. or study an economy’s response to a new government
policy or an unexpected shock after the initial period of
In the past two decades, the world of science has come adjustment has passed and the economy has settled
to a fundamentally new understanding ofthe dynamics down.
of phenomena that vary over time. Grounded in math-
ematical discovery, yet given empirical substance by Fractals, Sensitive Dependence, and Chaos
evidence from a variety of disciplines, this new per-
spective has led to nothing less than a re-examination Scientists long have known that it is possible for a sys-
of the concept of the predictability of dynamic be- tem’s state space to contain an isolated, unstable point
havior. Our implicit confidence in the orderliness of p such that different initial points near p can generate
dynamical systems, specifically of nonlinear dynamical orbits with widely varying longrun behavior. (For
systems, has not, it turns out, been entirely justified. example, a marble balanced on the tip of a cone is
Such systems are capable of behaving in ways that are unstable in this sense.) What was unexpected, how-
far more erratic and unpredictable than once believed. ever, was the discovery that this type of instability
Fittingly, the new ideas are said to concern chaotic can occur throughout the state space, sometimes actu-
dynamics, or, simply, chaos. ally at every point, but often in strang—ely patterned,
fragmented subsets of the state space subsets typ-
Economics is not immune from the implications of this ically of noninteger dimension, called fractals. Once
new understanding. After all, our subject is replete investigators knew what to look for, they found this
with dynamic phenomena ranging from cattle cycles to phenomenon, termed sensitive dependence on initial
stock market catastrophes to the back-and-forth inter- conditions, to be widespread among nonlinear dynam-
play of advertising and product sales. Ideas related to ical systems, even among the simplest ones. Though
the notion ofchaotic behavior are now part ofthe basic technical definitions vary, systems exhibiting this
mathematical toolkit needed for insightful dynamical unstable behavior have generally come to be called
modeling. Agricultural economists need to gain an “chaotic.”
understanding of these ideas just as they would any
other significant mathematical contribution to their For chaotic systems, any error in specifying an initial
field. This article is intended to assist in this educa- point, even the most minute error (due to, say, com-
tional process. puter rounding in the thousandth decimal place), can
give rise to an orbit whose longrun behavior bears no
What exactly has chaos theory revealed? To address resemblance to that of the orbit of the intended initial
this question, let us consider an economy, subject to point. Since, in the real world, we can never specify a
change over time, whose state at time t can be point with mathematically perfect precision, it follows
described by a vector, v^, of (say) 14 numbers (money that practical longrun prediction of the state of a cha-
supply at time t, inflation rate at time t, and so on). otic system is impossible.
Formally, this vector is a point in the 14-dimensional
Attractors
Weiss is an economist in the Commodity Economics Division,
tEhReS.ERTSheChaauotshoTrhetohra}n^ksSeJmoihnnarMcfCorlemlalnanydsatinmduloatthienrgpdairstciucsispiaonntssoinn For a dynamical system, perhaps the most basic ques-
chaotic dynamics. Carlos Arnade, Richard Heifner, and an anony- tion is “where does the system go, and what does it do
mous referee furnished helpful review comments. when it gets there?” In the earlier view of dynamical
2 THE JOURNAL OF AGRICULTURAL ECONOMICS RESEARCH/VOL. 43, NO. 3, SUMMER 1991
systems, the place where the system went, the point nor the imaging techniques available at the time per-
set in the state space to which orbits converged (called mitted him to explore his intuitions fully.
an attractor), was usually assumed to be a geo-
metrically uncomplicated object such as a closed curve Following Poincare’s work and that of the American
or a single point. Economic modelers, for example, mathematician G.D. Birkhoff in the early part of this
have often implicitly assumed that a dynamic economic century, and despite continuing interest in the Soviet
process will ultimately achieve either an equilibrium, a Union, the subject of dynamical systems fell into rela-
cyclic pattern, or some other orderly behavior. How- tive obscurity. During this period, there was some
ever, another discovery of chaos theory has been that awareness among mathematicians, scientists, and
the attractor of a nonlinear system can be a bizarre, engineers that nonlinear systems were capable oferra-
fractal set within which the system’s state can flit tic behavior. However, examples of such behavior
endlessly in a chaotic, seemingly random manner. were ignored, classified as “noise,” or dismissed as
aberrations. The idea that these phenomena were
Just as an economy can have two or more equilibria, a characteristic of nonlinear dynamical systems and that
dynamical system can have two or more attractors. In it was the well-behaved, textbook examples that were
such a case, the set of all initial points whose orbits the special cases had not yet taken root.
converge to a particular attractor is called a basm of
attraction. A recent finding has been that the bound-
Then, in the 1960’s and 1970’s, there was a flurry of
ary between competing basins of attraction can be a
activity in dynamical systems by both mathematicians
fractal even when the attractors themselves are unex-
A and scientists working entirely independently. Mathe-
ceptional sets. type ofsensitivity to the initial condi-
matician Stephen Smale turned his attention to the
tion can operate here too: the slightest movement
subject and used the techniques of modern differential
away from an initial point lying in one basin of attrac-
topology to create rigorous theoretical models of cha-
tion may move the system to a new basin of attraction
otic dynamics. Meteorologist Edward Lorenz dis-
and thus cause it to evolve toward a new attractor. covered that a simple system of equations he had
devised to simulate the earth’s weather on a primitive
Chaotic behavior within a working model would be
computer displayed a surprising type of sensitivity:
easier to recognize if all orbits initiating near an erra- the slightest change in the initial conditions eventually
tic orbit were also erratic. However, the potentially would lead to weather patterns bearing no resem-
fractal structure of the region of sensitive dependence
blance to those generated in the original run.
can allow initial points whose orbits behave “sensibly”
and initial points whose orbits are erratic to coexist
inseparably in the state space like two intermingled Biologist Robert May used the logistic difference equa-
clouds of dust. Thus, simulation of a model at a few tion x^+i = rx^ (1-Xn) to model population level, x,
trial points cannot rule out the possibility of chaotic over successive time periods. He observed that for
dynamics. Rather, we need a deeper understanding of some choices ofthe growth rate parameter, r, the pop-
the mathematical properties of our models. Nor can ulation level would converge, for other choices it
chaotic dynamics be dismissed as arising only in a few would cycle among a few values, and for still others it
quirky special cases. As we shall see, it arises even would fluctuate seemingly randomly, never achieving
when the system’s law of motion is a simple quadratic. either a steady state or any discernible repeating pat-
tern. When he attempted to graph the population level
against the growth rate parameter, he observed a
The Discovery of Chaos strangely patterned, fragmented set of points.
Recent years have witnessed an explosion of interest
and activity in the area of chaotic dynamics. What Physicist Mitchell Feigenbaum investigated the
accounts for this new visibility, which extends even behavior of dynamical systems whose equations of
beyond the research community into the public media? motion arise from unimodal (hill-shaped) functions. He
To provide an answer, we briefly trace the historical noticed that certain parameter values that sent the
development of the subject. system into repeating cycles always displayed the
same numerically precise pattern: no matter which
The first recognition of chaotic dynamics is attributed dynamical system was examined, the ratios of succes-
to Henri Poincare, a French mathematician whose sive distances between these parameter values always
work on celestial mechanics around the turn of the converged to the same constant, 4.66920- ••. Feigen-
baum had discovered a universal property of a class of
century helped found the study of dynamical systems,
systems in which some structure (perhaps a solar sys- nonlinear dynamical systems. His discovery ultimately
tem, perhaps—as now understood—an economy) clarified how systems can evolve toward chaos.
changes over time according to predetermined rules.
Poincare foresaw the potential for unpredictability in Thus, as these and other examples demonstrate, while
dynamical systems whose equations of motion were mathematicians were developing the theory of non-
nonlinear. However, neither the mathematical theory linear and chaotic dynamics, scientists in diverse disci-
3
plines were witnessing and discovering chaotic Henceforth, for brevity, we denote by:
phenomena for themselves. Ultimately, researchers
learned of one another’s findings and recognized them fL
common origin.
the nth iterate of a function f. Thus, fi(x) = fix), f2(x)
The role of the computer in the emergence of the con- f=i(xf)ifi=x)X)., Ofif(xc)ou=rsefi,fiff"ixi)s))i,tsealnfdasfounocnt.ionB.yIctosnhvoeunltdionno,t
temporary understanding of dynamical systems is dif-
ficult to exaggerate. As we now realize, even the be confused with the nth derivative of f, which is
customarily denoted:
simplest systems can generate bewilderingly compli-
cated behavior. The development of modern computer
f(n)_
power and graphics seems to have been necessary
before researchers could put the full picture of non-
Orbit Diagrams
linear and chaotic dynamics, quite literally, into focus.
Fortunately for expository purposes, many of the
The Mathematics of Chaos important features of dynamical systems are present
in one-dimensional systems. In fact, one of the impor-
We now explain some of the basic mathematical ideas tant findings of chaos research has been that discrete
involved in nonlinear dynamics and chaos. We also dynamical systems generated by iteration of even the
adopt a slightly different perspective. In the above most elementary nonlinear scalar functions are capable
discussion, we have implicitly portrayed dynamical of chaotic behavior. Thus, we shall concentrate on
systems as being in motion in continuous time. How- functions that operate on the number line.
ever, the equations ofmotion of such systems typically
involve differential equations, and a proper treatment For such functions, there is a particularly convenient
often requires advanced mathematical machinery. It is technique for diagramming orbits. Consider a function
generally much easier to work with (and to under- f and an initial point x (fig. 1). Beginning at the point
stand) discrete-time systems, in which time takes only (x, x) on the 45° line, draw a dotted line vertically to
integer values representing successive time periods. the graph off; the point ofintersection will be (x, fix)).
Let us shift our attention to these systems. From that point, draw a dotted line horizontally to the
45° line; the point of intersection will be (fix), fix)).
From there, draw a dotted line vertically to the graph
When the law ofmotion ofa discrete dynamical system
is unchanging over time, the movement of the system toifnfu;etthheispopianttteorfnionftemrosvecitnigonvewritlilcablely(ftiox)t,hef2(gxr))a.phCoonf-f
through time can be understood as a process of iterat-
and then horizontally to the 45° line. The resulting dis-
ing a function. To establish this point, consider a typi-
play, called an orbit diagram, shows the behavior of
cal dynamic economic computer model, M, having k the orbit originating at x. In particular, the orbit may
endogenous variables. To start the model running, one
enters an initial condition vector, Vq, of k numbers.
The model computes an output vector, M(vo), contain-
ing the new values of the k endogenous variables at
the end of the first time period. The model then acts
on M(vo) and computes a new output vector, M(M(vo)),
describing the economy at the end of the second time
period. Successive output vectors are computed in the
same manner. Note that the model itself, the law of
motion, remains unchanged during this process. In
M
effect, acts as a function, mapping k-vectors to new
k-vectors, applying itself iteratively to the last-
computed function value. The state space of the econ-
omy is the k-dimensional space R^, and, for each initial
condition vector Vq, there is a corresponding orbit, Vq,
M(vo), M(M(vo)), M(M(M(vo))), •••, describing the
future evolution of the economy.
More generally, consider any function f. If f maps its
domain (the set of all x for which f(x) is defined) into
itself, then, for each Xq in the domain of f, the
sequence Xq, f(xo), f(f(xo)), f(f(f(xo))), •••is well-
defined and may be considered an orbit of a dynamical
system determined by f through iteration.
4
be visualized from the intersection points marked on Let us first dispense with the case a = 1. In this case,
the 45° line; the dotted lines indicate the direction of if b = 0, then every x is a fixed point of g (that is, g(x)
motion ofthe system. Ofcourse, the points (x,x), (f(x), = x), and (since then, also g"(x) = x) the system
f(x)), (f2(x), f2(x)), ... only look like the orbit. They always remains at any initial point. In contrast, if b
reside in the plane, whereas the actual orbit, consist- 0, then no X is a fixed point ofg; indeed, for any initial
ing of the numbers x, f(x), f2(x), ..., resides in the point X, g"(x) diverges monotonically as n-+^ to either
state space, that is, in the number line. ^ or according to whether b > 0 or b < 0.
Dynamics of Linear Systems In discussing the six remaining cases, that is, the
cases in which a 1, I take b to be an arbitrary num-
Though the basic focus of this paper is nonlinear ber. In these cases, g has exactly one fixed point,
dynamics, examination of linear systems provides b/(l-a), and any orbit originating there remains there.
essential intuition about nonlinear ones. Thus, we I next examine the behavior of orbits originating at
begin with an exhaustive treatment of the linear case. points other than b/(l-a). For this purpose, I assume
that the initial point x is an arbitrary number distinct
Choose any numbers a,b, and consider the function g from b/(I-a).
defined by g(x) = ax + b. To compute a typical orbit of
g, observe that: If a < -1, then g"(x) has no finite or infinite limit.
g-(x) = a(ax-tb) + b Rather, it eventually alternates between positive and
negative numbers as its absolute value diverges
= a^x -t b(l+a), monotonically to
=
If a -I, the fixed point b/(I-a) equals b/2, and:
g^(x) = a[a2x -t b(l+a)J + b
g"(x) = (-l)'i(x-b/2) + b/2
= a^x -t- bd-ta+a^),
_ (b-x if n is odd
g^(x) = a-'x + b(l +a+a-+a3), (x if n is even.
and, in general, g'^(x) = a"x -i- b(l-ta-ta-+a3-h Thus, g'^(x) alternates endlessly between the (distinct)
=
ta'’-i). If a 1, then numbers b-x and x.
g"(x) = X -t bn. If -I < a < 0, g'^(x) converges to b/(l-a) while alter-
nating above and below
it.
mHeotwreivcesre,rieifsagi7v^es1:, the formula for the sum of a geo- If a = 0, then, for all n, g'^(x) = b. Thus, the system
moves from the initial point directly to b and remains
g"(x) = a"x + b there.
Ll-a_
If 0 < a < 1, g"(x) converges monotonically to b/(l-a).
The convergence is from above ifx > b/(l-a) and from
below if X < b/(l-a).
Note that when a is nonnegative, a^ remains nonnega- Finally, if a > 1, then g"(x) diverges monotonically, to
tive, while when a is negative, a'^ alternates between ifX > b/(l-a) and to —^ if x < b/(l-a).
negative and positive. In particular, when a = -1, a"
alternates between -1 and 1. When a ±1, the dis- The possible behaviors of orbits in the one-dimensional
tance between a*^ and 0 either converges monotonically linear system are illustrated in figures 2(a)-2(h). From
to 0 or diverges monotonically to as n-»oc according these figures and the preceding discussion, two lessons
to whether |a| < 1 or |a| > 1. Using these facts, we emerge. First, the fixed point is often at the “center of
now analyze the behavior ofall the orbits generated by the action”: it is to or from this point that orbits typ-
g, according to the various possibilities for the struc- ically converge or diverge. Second, the slope param-
tural parameters a and b and the initial point x. We eter, a, plays a pivotal role in determining orbit
shall find it convenient to organize our analysis around dynamics. These two principles hold as well for non-
We
the possible value of a. distinguish seven cases: (1) linear systems.
a < -1; (2) a = -1; (3) -1< a < 0; (4) a = 0; (5) 0 < a <
I; (6) a = 1; and (7) a > 1. Within each of these cases, Fixed Points and Periodic Points
we consider all possible values of the remaining struc-
tural parameter b and the initial point x, and we deter- It is not a coincidence that, in the linear system, con-
mine the longrun behavior ofthe orbit originating at x vergent orbits always converge to a fixed point of the
when g has structural parameters a and b. underlying function. In fact, this property holds in
5
general. To establish it, suppose a continuous function limit must be a fixed point. Correspondingly, if an
f has a convergent orbit x, f(x), f^Cx), f^(x), ... . economy converges to an equilibrium, the equilibrium
Let L be the limit. Then: state must be a fixed point of the system function.
= Closely related to fixed points are points whose orbits
f(L) may leave but later return (see fig. 2(b)). A point x is
called a periodic point of f with period n if f'’(x) = x.
= lim f"''Tx) The smallest positive n for which the latter equation
n-*oo
holds is called the prime period of x. It can be shown
that any period of x is a multiple of the prime period.
=
L,
Every fixed point of a function f is a periodic point of f
so that L is a fixed point of f. Thus, in partial answer of prime period 1. It is also true that every periodic
to our guiding question, “where does the system go,” point is a fixed point (though not ofthe same function),
we can reply: if it converges to any finite limit, that since the periodicity condition f'^(x) = x is nothing but
6