POST KEYNESIAN PERSPECTIVES AND COMPLEX ECOLOGIC-ECONOMIC
DYNAMICS
J. Barkley Rosser, Jr.
James Madison University
rosserjb@jmu.edu
January, 2010
Abstract:
This paper considers the implications of complex ecologic-economic dynamics
for three broad, Post Keynesian perspectives: the uncertainty perspective, the
macrodynamics perspective, and the Sraffian perspective. Catastrophic, chaotic, and
other complex dynamics will be seen as reinforcing the conceptual foundations of
Keynesian uncertainty. Predatory-prey models will be seen as deeply linked to Post
Keynesian macrodynamic models. Finally, certain cases in ecologic-economic systems
will be seen as generating such Sraffian, capital theoretic conundra as reswitching.
Ecologic-economic models considered besides predator-prey will include fisheries,
forestry, lake dynamics, and global climatic-economic dynamics.
Acknowledgement: I acknowledge valuable input from three anonymous referees.
1
I. Introduction
Post Keynesian economics has been riven by deep splits within its ranks. What
was arguably its first self-conscious school, the Sraffian (Sraffa, 1960), or neo-Ricardian,
school has for all practical purposes been expelled from the group, to the extent that it did
not pick up and leave on its own voluntarily, arguably ironic in that it was a friend of that
group, Joan Robinson, who has long been reported to have coined the term “postKeynesian.”1 A sign of this expulsion is the absence of any chapters relying on the
Sraffian perspective in A New Guide to Post Keynesian Economics (Holt and Pressman,
2001). This group found itself in sharpest conflict with the school that draws on Keynes
(1936) to emphasize the role of fundamental uncertainty in economics, with Davidson
(1982-83, 1994) being the leading advocate of this view, which has criticized the Sraffian
group for its reliance on comparing long-run equilibrium states and downplaying the role
of money in the economy. Between these two groups has been one that has focused more
on specific models of macroeconomic dynamics, or macrodynamics,2 often relying upon
nonlinear relations in the economy that can lead to endogenous fluctuations, including of
various complex varieties. This group often looks to the work of Kalecki (1935, 1971)
for its inspiration, although such figures as Kaldor (1940), Goodwin (1951), and even
Hicks (1950) played important roles in its development. Among those discussing the
development of these divisions have been Harcourt (1976), Hamouda and Harcourt
(1988), and King (2002).
While “post-Keynesian” continues to be used, especially in Great Britain, its use by Paul Samuelson as a
label for the neoclassical synthesis has made it somewhat less popular, becoming supplanted largely by
“Post Keynesian,” which I shall use in this paper.
2
The term “macrodynamics” was first used in print by Kalecki (1935), but it appears that he probably got it
from Ragnar Frisch during a 1933 conference of the Econometric Society, and the original Polish version
of his paper used a term that is derived from the German konjunctur instead (Sawyer, 1985).
1
2
While the deepest of these splits are probably unbridgeable, Rosser (2006) has
proposed that the fact that each of these approaches can be viewed as drawing partly from
or influencing the perspective of complex economic dynamics may provide one way of
seeing some degree of unity in the diversity and conflicts between these schools. This
paper can be seen as a direct extension of this argument, with the focus now being more
specifically on ecologic-economic systems and their various forms of dynamic
complexity. While intellectually there have been influences from ecology onto
economics and vice versa in terms of developing complex nonlinear dynamical models,
the focus here will be on how combined ecological-economic systems such as fisheries or
the global climate-economic system can particularly generate such complex dynamics.
Such dynamics can be viewed as an ontological foundation of Keynesian uncertainty,
given the nature of such complex dynamics.3
This paper will pursue this theme by first reviewing briefly what constitute
complex dynamics. Then it will focus on ecologic-economic dynamics as a foundation of
fundamental uncertainty of the Keynesian sort, including the implications for the broader
debates among Post Keynesians. Then will be a presentation of the links between
complex ecologic and macrodynamic models. Finally, it will be shown that capital
theoretic paradoxes of the Sraffian sort can arise from complex ecologic-economic
systems, with the reminder that these in turn can generate complex dynamics. It will
conclude by a final consideration of the implications for the relations between the schools
of these ideas and arguments.
3
Certainly this discussion can be viewed as part of the broader existence of nonlinear systems in many
parts of nature and society, although the combined ecologic-economic systems are special because of how
often these nonlinear complexities arise out of the human-nature interaction.
3
II. What Are Complex Dynamics?
Asking “what are complex dynamics?” simplifies our discussion somewhat, given
the substantial controversies that swirl about the general concept of “complexity,” even if
one keeps the discussion strictly to “economic complexity.” The MIT physicist, Seth
Lloyd, famously collected at least 45 different definitions of “complexity” (Horgan,
1997, p. 303, footnote 11), with many of these involving some form or variation of
algorithmic or other computationally related definitions of complexity. Some have long
advocated the use of such definitions in economics (Albin with Foley, 1996), with a
recent upsurge of such advocacy (Markose, 2005; Velupillai, 2005). However, while
these approaches may involve more rigorous definitions than other approaches, they are
less useful for the analysis of ecologic-economic systems than more explicitly dynamic
definitions. Indeed, curiously enough, some of the critics of dynamic approaches
criticize them precisely because of their dependence on biological analogies and concepts
(McCauley, 2004, Chap. 9).4
Therefore we shall stick with the definition used by Rosser (1999), which in turn
comes from Day (1994). This definition is that a system is dynamically complex if it
endogenously does not converge on a point, a limit cycle, or an explosion or implosion.5
This definition provides a reasonably clear criterion for distinguishing dynamical systems
in this regard, even if one may have difficulty in determining whether or not a particular
4
Rosser (2009a) provides a detailed discussion of this controversy, noting that while algorithmic
definitions allow for measures of degrees of complexity, they do not generally allow for a clear division
between complex and non-complex systems unless one defines complex systems as those that are not
computable at all. Dynamic definitions allow for such a reasonable criterion for useful such economic
systems.
5
Curiously, Day (2006, p. 63) has since moved toward favoring a more general definition of complexity
taken from the Oxford English Dictionary: “a group of interrelated or entangled relationships.” While we
shall not dispute this, it should be noted that it is not easy to use this definition to distinguish complex from
non-complex systems.
4
real world system meets fulfills it. A characteristic of dynamically complex systems as
we have defined it here is that they will usually involve some degree of nonlinearity,
although the presence of nonlinearity is no guarantee that a system will be dynamically
complex. This is true for a single equation system, although Goodwin (1947) showed
that a system of coupled linear equations with lags might behave in the manner described
here as complex, even though the uncoupled, normalized equivalent is nonlinear. Such
systems were studied by Turing (1952) in his analysis of morphogenesis in complex
systems.
Rosser (1999) characterized this definition as a “broad tent” one, which included
within itself “the four C’s,” cybernetics, catastrophe theory, chaos theory, and “small
tent” complexity, associated with heterogeneous interacting agents models. These four
approaches appeared on the scene publicly in turn decade after decade, one after the
other, even though the mathematical roots of each had been developing over much longer
periods of time going back even from the 19th century (Rosser, 2000, Chap. 2).
Arguably, the first of these has become folded into the last of these currently, while the
other two continue to develop on their own separate paths, with numerous applications in
pure biology and ecology. Broadly speaking, catastrophe theory studies endogenous
discontinuities in certain kinds of dynamical systems that arise as given control variables
change continuously,6 while chaos theory focuses on systems that exhibit sensitive
dependence on initial conditions, also known as “the butterfly effect.” Regarding the
“small tent complexity,” this can be seen as having its origins in certain models from the
1970s (Schelling, 1971; Föllmer, 1974) in which immediate neighbors affect each other
6
Rosser (2007) provides a discussion of how catastrophe theory in particular fell strongly out of favor in
economics, even though it provides useful tools for studying important phenomena, with arguments for
bringing it back more into use. We shall see some of those uses below in this paper.
5
without necessarily directly affecting an entire system, even though these local effects
can lead to broader systemic effects through complex emergence.7 It would be in the
1990s that there would be a fuller development of such approaches.
III. Complex Ecologic-Economic Dynamics and (Post) Keynesian Uncertainty
A. The Debate
Paul Davidson (1994) is the acknowledged leader of what he calls the “Keynes
Post Keynesian” school of economic thought,8 which emphasizes particularly the role of
fundamental uncertainty from the work of Keynes (1921, 1936) and also the importance
of the role of money in the economy. We shall focus here on first of these rather than the
second, which has little relationship with ecological economics.
A long running debate between Davidson (1996) and other Post Keynesians
(Rosser, 2001a, 2006) has involved the relationship between complexity theory and the
concept of Keynesian uncertainty. While Davidson has rejected complexity theory as not
providing an ontological foundation for Keynesian uncertainty. He argues (1982-83) that
the foundation of Keynesian uncertainty is the ubiquity of nonergodicity in economic
dynamics, which must be accepted as axiomatically true. Others have argued that instead
that it is the ubiquity of complex dynamics in economic systems that is the source of this
nonergodicity, providing a theoretically and empirically valid foundation for the concept.
We shall consider some ecologic-economic systems that exhibit forms of dynamic
The concept of “emergence” was developed in the early 20 th century in Britain, ultimately drawing on
arguments of Mill (1843). This concept has been criticized by some of the computability complexity
approach such as McCauley and Markose on grounds that it is not rigorous. However, a recent, rigorous
mathematical presentation that draws on biological examples with economic parallels such as “flocking,”
has been made by Cucker and Smale (2007).
8
Some have labeled this school as being “fundamentalist Keynesian,” (Coddington, 1976), although
Davidson has disliked this label and introduced the “Keynes Post Keynesian” one in his 1994 book.
7
6
complexity that may imply nonergodic Keynesian uncertainty. Indeed, the problem of
non-quantifiable uncertainty has been one of the biggest issues facing both standard
environmental as well as more heterodox ecological economists for some time, with
many of these uncertainties deriving from the limits of our scientific knowledge about the
environment.9
Keynes’s (1937) emphasis in his later direct references to fundamental uncertainty
involve such matters as the price of copper or the nature of the economic system of
Britain twenty years in the future. However, the foundation of this uncertainty in
Keynes’s view is discussed in more detail in his 1921 Treatise on Probability, as
discussed by Rosser (2001a). Thus, whereas it is often argued that the subjectivist
Keynes saw such uncertainty as reflecting situations without any underlying probability
distribution due to free will, in fact he saw a broader array of possible cases. Indeed, he
recognized the possibility of essentially standard classical probability for certain kinds of
situations, such as the throwing of a fair die where there are clear probabilities that can be
analytically determined, allowing for a profit-making insurance industry.
In between these extremes are various intermediate cases. Thus there may be two
series of events with events being able to be ranked ordinally within each series but not
across the series, even when there might be an event common to both series. A source of
such non-cardinality or non-comparability might be if each event has a different type of
probability, such as one being skewed and another not.10 Another case, perhaps more of
Davidson’s critique involves arguing that these complexity approaches and presumably also these
scientific limits of environmental knowledge are “merely epistemological” problems rather than
ontological, and that therefore they are not sufficiently fundamental to found Keynesian uncertainty on. If
somehow our scientific knowledge or our knowledge of the dynamics of complex systems were to become
sufficiently great, then such systems or models would be seen as classical in their essence.
10
Rowley and Hamouda (1987) and O’Donnell (1990) have discussed this in terms of the debate between
Keynes and Tinbergen over econometrics. Koppl and Rosser (2002) argue that infinite regress problems
9
7
an epistemological issue, is where there might be probabilities, but they cannot be
estimated or determined due to data unavailability. Complex dynamics can generate all
of these.
We shall first consider ecologic-economic systems that are susceptible to
catastrophic discontinuities, whose timing and scale are both difficult to predict. Then
consider ecologic-economic systems exhibiting chaotic dynamics, whose tendency to
exhibit the butterfly effect make them unpredictable.
B. Catastrophically Discontinuous Ecologic-Economic Systems
Even without interactions with human beings and their economically driven
conduct in relation to the natural environment, strictly ecological systems are known to
exhibit dynamic discontinuities on their own.11 Some are known to exhibit multiple
equlibria with discontinuities appearing as systems move from one basin of attraction to
another dynamically, even without any human input, including the periodic mass suicides
of lemmings (Elton, 1924), coral reefs (Done, 1992; Hughes, 1994), kelp forests (Estes
and Duggins, 1995), and potentially eutrophic, shallow lakes (Schindler, 1990). The
latter can be exacerbated by human input as well in combined systems, as humans can
flip such a lake from a clear oligotrophic state to a murky eutrophic state by loading
phosphorus from fertilizers or other sources (Carpenter, Ludwig, and Brock, 1999;
associated with the Keynesian beauty contest can also lead to such problems, and Rotheim (1988) argues
that Keynes also at times sympathized with emergentist “organicism” in which “the whole is not equal to
the sum of its parts” and “small changes produce large effects” as in catastrophe or chaos theory.
11
A deep question we shall not pursue here is whether or not evolution itself is fundamentally a continuous
or discontinuous process, with Darwin (1859) arguing the former and Gould (2002) arguing the latter. For
further commentary, see Rosser (1992), Hodgson (1993). See Rosser (2008) for a more complete
discussion of discontinuities in ecologic-economics systems.
8
Wagener, 2003). Figure 1, taken from Brock, Mäler, and Perrrings (2002, p. 277) shows
the basic dynamics of this system as a function of phosphorus loadings.
Figure 1: Hysteresis effects in the management of shallow lakes
Figure 2: Spruce-Budworm Dynamics
A famous example of a cyclical pattern involving two species interacting in which
the explosion of population of one leads to a catastrophic collapse of the other is the
9
spruce-budworm cycle of about 40 years in Canadian forests, wherein budworms eat the
leaves of the spruce trees (Ludwig, Jones, and Holling, 1978). Now, there is a substantial
degree of predictability in this system, given its roughly periodic nature. However,
human intervention can affect it in various ways. In particular, human efforts to avoid or
overcome the cycle can actually lead to greater discontinuities and larger catastrophic
collapses, an observation that underlay Hollings’ (1973) innovation of the concept of a
tradeoff between stability and resilience in ecosystems. Furthermore, Holling (1986) has
argued that this system can be substantially impacted by small changes in quite distant
ecosystems, as for example the draining of wetlands in the mid-US that can lead to fewer
birds arriving in Canada from Mexico that eat the budworms and help keep their
population under control, an example of “local surprise, global change.”
The dynamics of this system are given as follows, from Ludwig, Jones, and
Holling (1978). Let B equal the budworm population, rB their natural population growth
rate, KB the budworm carrying capacity (determined by the amount of leaves on the
spruce trees), α the predator saturation parameter (a proportion of the budworm carrying
capacity), β the maximum rate of predation on the budworms, and u* the equilibrium leaf
volume, then the budworm dynamics in their early stages are given by
dB/Dt = rBB(1 – B/KB) – βB2/(α2 + B2).
(1)
Nonzero equililbria are solutions of
(rBKB/β) = u*/[(α/K2) + u*2)(1 – u*)].
(2)
The set of solutions implied by this system is depicted in Figure 2, with the zone of
multiple equilibria and associated catastrophic hysteresis loops representing an infected
forest. This system is a variation on a predator-prey system, which we shall discuss
10
further below, but note here that the original predatory-prey models studied by Lotka
(1920, 1925) and Volterra (1926, 1931) showed smooth, interconnected cycles rather
than discontinuities, which is somewhat more like what the first empirically studied
predatory-prey cycle, that of the arctic hare and lynx, also tends to show, albeit with some
variations.
In this case as depicted in Figure 2, there are multiple equilibria of the leaf
volume as a function of the actual leaf volume, where the latter determines the budworm
carrying capacity. Thus, as the leaf volume and the budworm carrying capacity reaches
critical levels, there can be discontinuous expansions or collapses of the equilibrium leaf
volume, which are triggered by collapses or explosions of the budworm population
respectively, with such a cycle of approximately 40 years being observed in Canadian
spruce forests.
A classic system subject to multiple equilibria and sudden, catastrophic changes
due to human activity is in desert ecosystems, especially in cases where cattle grazing of
fragile grasslands is involved (Noy-Meir, 1973; Ludwig, Walker, and Holling, 2002,;
Rosser, 2005). In such cases, fragile rangelands can be suddenly overtaken by woody
vegetation quite suddenly after an episode of overgrazing. Of course, this is linked to the
classic problem of open access. Aldo Leopold (1933, pp. 636-637) gives a classic
description of the outcome and its source in the US Southwest:
“A Public Domain, once a velvet carpet of rich buffalo-grass and grama,
now an illimitable waste of rattlesnake-bush and tumbleweed, too impoverished
to be accepted as a gift by the states within which it lies. Why? Because the
ecology of the Southwest is set on a hair trigger.”
11
Yet another system in which catastrophic declines of populations can happen with
this being clearly the result of human activities interacting with the ecosystem, is in
fishery dynamics, especially in the famous case of an open-access fishery subject to a
backward-bending supply curve (Copes, 1970). Collapses of fisheries are a global
problem of enormous consequence and importance, with many such happening, including
among others: Antarctic blue and fin whales, Hokkaido herring, Peruvian anchoveta,
Southwest African pilchard, North Sea herring, California sardines, Georges Bank
herring (and more recently, cod12 also), and Japanese sardine (Clark, 1985, p. 6), with
Jones and Walters (1976) specifically studying the collapse of the Antarctic blue and fin
whales using catastrophe theory. A more general approach is provided by Clark (1990),
Rosser (2001b), and Hommes and Rosser (2001), which is summarized below.
Let x = fish biomass, r = intrinsic fish growth rate, k = ecological carrying
capacity, t = time, h = harvest, and F(x) = dx/dt, the growth rate of the fish without
harvest (but limited by the carrying capacity). Then a sustained yield harvest, drawing on
Schaefer (1957) is given by
h = F(x) = rx(1 – x/k).
(3)
Let E = catch effort in standardized vessel time, q = catchability per vessel per
day, c = constant marginal cost, p = price of fish, and δ = the time discount rate. Then the
basic harvest yield is
h(x) = qEx.
(4)
Hommes and Rosser (2001) show that the supply curve for optimizing fishers is
given by
xδ(p) = k/4{1+(c/pqk)-(δ/r)+[(1+(c/pqk)-(δ/r)2+(8cδ/pqkr)]1/2}.
12
(5)
See Ruitenback (1996) for a discussion of the collapse of the once great cod fishery off Newfoundland.
12
This entire system is depicted in Figure 3, with the backward-bending supply
curve in the upper right and the yield curve in the lower right. The degree of backward
bending is linked to the discount rate, and if it is less than about 2 percent, there is no
backward bend, with the curve simply asymptotically approaching the maximum
sustained yield of the fishery as the price rises. However, the maximum backward bend
occurs when δ is infinite, which gives the case equivalent to the open access case studied
by Gordon (1954).
The basic story of fishery collapse is depicted also in the upper right quadrant,
where it is presumed that there is a gradual increase in demand, which eventually triggers
a sudden increase in price and decrease in the steady-state harvest yield as the system
passes through a single equilibrium zone into a three equilibria zone and finally into the
single equilibrium zone associated with high price and low harvest yield.
13
Figure 3 here: Gordon-Schaefer-Clark Fishery Model
Regarding the implications for Post Keynesian uncertainty theory, while some of
these systems have elements of predictability, such as the approximately 40 year
periodicity of the spruce-budworm cycle, others do not at all, such as the sudden
collapses of overgrazed grasslands or overfished fisheries. The general existence of
ecological thresholds is a ubiquitous phenomenon (Muradian, 2001), with the locations of
these thresholds generally unknown. Rosser (2001b) proposes using the precautionary
principle in such cases, and Gunderson, Holling, Pritchard, and Peterson (2002) see this
as a fundamental problem for maintaining the resilience of threatened ecosystems around
the world. This is not to say that extremal events cannot be modeled or their probability
come to be known (Embrechts, Küppelberg, and Mikosch, 2003). But a critical threshold
of global significance that has not been crossed before with the relevant probability
14
distribution unknown, such as the danger of Greenland or Antarctic ice sheets sliding off
suddenly due to global warming, remains subject to and reinforcing the problem of
Keynesian uncertainty, even if Lloyd’s of London is writing catastrophic insurance
contracts on beachfront housing against massive flooding. This problem becomes more
serious when what the matter involves an irreversibility within the system (Kahn and
O’Neill, 1999). Spath (2002) provides further discussion of the ethics and economics
involved in the complex system of global warming.
B. Chaotic and Other Complex Dynamics in Ecologic-Economic Systems
It was actually from the study of population dynamics in ecology that the term
“chaos” first came to be used for endogenously erratic dynamical systems that exhibit
sensitive dependence on initial conditions (May, 1974). Soon after this, actual chaotic
dynamics were observed in laboratory populations of sheep blowflies (Hassell, Lawton,
and May, 1976). It has since been argued that one is less likely to observe actual chaotic
dynamics in natural populations because of the presence of noise, while at the same time
such noise is likely to increase the amplitude of fluctuations that do occur (Zimmer,
1999).
The Gordon-Schaefer-Clark fishery model of Hommes and Rosser (2001)
described above can also be shown to exhibit chaotic dynamics under certain not
unreasonable conditions. Letting the demand function be linear of the form
D(p) = A – Bpt,
(6)
15
then letting agents follow naïve expectations of the form that next period’s price will be
the same as this period’s price, and Sδ is the supply function that depends on the discount
rate δ, this leads to cobweb13 adjustment dynamics of
pt = D-1Sδ(pt-1) = [A - Sδ(pt-1)]/B.
(7)
For the case where B = 0.25 and A is given a value such that consumer demand will
equal the maximum sustained yield at the minimum possible price, Hommes and Rosser
(2001) show that as δ increases past 2% the supply curve will bend backward and the
system will gradually undergo period-doubling bifurcations. Chaotic dynamics will
occur in a range for δ between about 8% and 10%, with the system simply going to the
high price/low yield equilibrium for discount rate higher than 10%.
Hommes and Rosser then follow earlier work of Hommes and Sorger (1998), who
in turn followed an argument due to Grandmont (1998), which shows that in a chaotic
environment, agents following a relatively simple adjustment rule might be able to “learn
to believe in chaos” and adjust to follow the underlying chaotic dynamic according to a
consistent expectations equilibrium. Such a process of learning with movement from an
initial guess of a steady state to a chaotic dynamic is shown in Figure 4, drawing on
Hommes and Sorger (1998), with time on the horizontal axis and price on the vertical.
What is depicted here involves agents using a simple auto-regressive rule of thumb
decision process in which the parameters of the rule are adjusted according to how well
the rule of thumb performs in predicting the actual price. At first the initial guess
regarding the parameter values performs well, and the price remains fairly constant, only
13
Chiarella (1988) showed for a wide class of cases that chaotic dynamics can arise with cobweb dynamics.
Such dynamics are widespread in agriculture, and various cycles in agriculture, including cattle and pigs,
have been argued to be possibly chaotic. For an overview of possible chaotic dynamics in various subparts of agriculture, see Sakai (2001).
16
to break down as the price begins oscillating in a two-period cycle. As the parameters are
adjusted further, the dynamic converges on the underlying chaotic dynamic.
Figure 4: Learning to believe in chaos
The possibility of chaotic dynamics in fisheries has been studied by others as
well, with Conklin and Kolberg (1994) showing it for reasonable parameters in the case
of a halibut fishery when the supply curve is bending backwards. Furthermore, Doveri,
Scheffer, Rinaldi, Muratori, and Kuznetsov (1993) have shown the possibility of chaotic
dynamics in a multiple-species aquatic ecosystem.
The fishery model laid out above and studied by Hommes and Rosser (2001) can
also be shown to exhibit yet another complex phenomenon that increases the difficulty of
making clear forecasts and of experiencing sudden changes in the dynamic pattern of a
system. This is the phenomenon of the coexistence of multiple basins of attraction in
which the boundaries between these basins may have a fractal shape, leading to a
complex interpenetration of one basin by another or by several others. This phenomenon
has been demonstrated for the Hommes-Rosser model by Foroni, Gardini, and Rosser
(2003).14 An example of how such a system looks is depicted in Figure 5, which shows
14
The possibility of such dynamics in a purely economic model was first demonstrated by Lorenz (1992)
for a variation on the Kaldor (1940) macroeconomic model. Again, with such fractal boundaries, small
changes in parameter values can push the system from one basin of attraction into another with little
predictability.
17
the basins of attraction for a ball held over three magnets, drawn from Peitgen, Jürgens,
and Saupe (1992).
Figure 5: Fractal basin boundaries for three magnets
Finally, we move to a much grander scale perspective to consider the possibility
of chaotic dynamics involving the combined, global economic-climatic system. Chen
(1997) has shown the possibility of such chaotic dynamics at such a level. In his model,
he has two sectors, agriculture and manufacturing. Global temperature is a linear
function of the level of manufacturing, but agriculture is a quadratic function of global
temperature. With some assumptions regarding price setting between the two sectors, he
is able to show the possibility of chaotic dynamics in both global temperature as well as
the sectoral levels of output and prices.
At this point we need to remind ourselves that chaotic dynamics most thoroughly
undermine any form of simple forecasting. Slight changes in initial points or parameter
values can lead to substantial changes in dynamic paths. This is one way in which the
18
possibility of complex dynamics provides a conceptual foundation for the concept of
fundamental Keynesian uncertainty.
IV. Ecological Foundations of Complex Post Keynesian Macrodynamics
We have already noted the fact that the early Post Keynesian economics models
involving nonlinear relations in investment or other macro relations were the foundation
of realization for economics more broadly of the possibility of endogeneity of macro
fluctuations (Rosser, 2006a), a development initiated by Kalecki (1935). These earlier
Post Keynesian models (not labeled that when they first appeared), which contained
nonlinear elements, usually in the relevant investment equation, would later be shown to
imply the possibility of complex dynamics (Rosser, 2000, Chap. 7). These later
manifestations of their inherent complexity came to play an important role in the more
general recognition that nonlinearity can lead to endogenously complex dynamics in
macroeconomic models.
Among the most important of those involved in these efforts was Richard
Goodwin (1947, 1951, 1967), with Strotz, McAnulty, and Naines (1953) showing the
first chaotically dynamic economic model, based on Goodwin’s (1951) with its nonlinear
accelerator, even as they did not understand what they had discovered.15 His 1967
model more explicitly drew on ecological predecessors in the form of the predator-prey
model of Lotka (1920, 1925) and Volterra (1926, 1931).16 Ironically for the old Marxist,
15
This somewhat repeats the experience of van der Pol and van der Mark (1927 when they first observed
chaotic dynamics in radio mechanics without realizing that the “noise” they had found was a manifestation
of something theorized earlier (Poincaré, 1880-1890). See Rosser (2009b) for further discussion.
16
At the same time that Goodwin presented his predatory-prey model of macrodynamics, Samuelson
(1967) did so also, although he focused more on a Malthusian-style model of population and land use
dynamics. Samuelson did not follow up on this development, in contrast to Goodwin (1990), who would
eventually study the chaotic dynamics that could arise from this formulation.
19
Goodwin, in his model it is the workers with their wage demands who play the role of the
“predators,” with their wage demands bringing about the reversal of the investmentdriven capitalist expansion. Goodwin’s (1967) model of business cycles is given by the
following, where W = wages. Y = national GDP, with W/Y = ω, the workers’ share, L =
employed workers, N = population of labor force, hence L/N = the rate of employment,
which is given by λ, with P(λ) being a linear Phillips curve relation between the rate of
employment and the rate of changes of wages,17 K = the capital stock, v = K/Y, the
accelerator relation, α is the rate of technological change, and β is the rate of population
growth. From all of this the Goodwin (1967) model can be given as
dω/dt = ω[ P(λ) – α],
(8)
dλ/dt = λ[(1 – ω)/v – α – β].
(9)
While Goodwin initially only studied the conservative oscillation implied by the classical
form of these equations, Pohjola (1981) would show that with certain parameter values in
a discrete version of this model can generate chaotic dynamics, extending a version of the
model developed by Desai (1973), with Goodwin (1990) himself later fully examining
these implications.
A similar history has occurred with regard to the predator-prey model itself in
ecology, one of the most important and widely used models of population dynamics
among species. The original formulation due to Lotka and Volterra generated
conservative oscillations. Given their regular periodicity, these seemed to fit available
field data, for example from the famous arctic hare and lynx cycle from Canada for which
the Hudson Bay Company had compiled a more then two centuries-long data set, which
17
Curiously, and in contrast to the usual formulation, this Phillips Curve is positively sloped. When it is
linear this corresponds to the classical Lotka-Volterra formulation.
20
showed a cycle of about ten to eleven years in length of the two interrelated populations,
based on pelts sold to the company by Indians who hunted the species. This particular
time series is shown in Figure 6.
Figure 6: Hudson Bay hare-lynx cycles
However, a complication is that whereas in the simple theoretical model
amplitude is constant from period to period, this is not the case in the field data. Such
fluctuations of amplitude with a roughly constant period are consistent with chaotic
variation of the predator-prey model, which can arise with appropriate lagging or further
nonlinearities. Such nonlinearities can enter in through the recognition of the role of
other species, for example in the hare-lynx model recognizing that when the hare
population is sufficiently low, the lynx will switch to pursuing other species. This was
the extension that led Schaffer (1984) to be the first to propose the possibility of chaotic
dynamics within the hare-lynx cycle, with Solé and Bascompte (2006, pp. 38-42)
providing an overview. Shortly, after Schaffer, Brauer and Soudack (1985) showed the
21
possibility of chaotic dynamics in predator-prey fisheries systems with high harvest
thresholds. The dynamics of lemmings and Finnish voles have also been seen to be
possibly chaotic, or at least semi-chaotic, with periods of unstable chaotic dynamics
interspersing with periods of convergence (Ellner and Turchin, 1995; Turchin, 2003).
We note the curiosity here that whereas it was from biology that the study of chaotic
dynamics entered into economics (May, 1974), with regard to variations of classical
predator-prey models, it was economics that discovered the possibility of chaotic
dynamics prior to the ecologists (Pohjola, 1981; Shaffer, 1984), although Gilpin (1979)
showed the possibility of spiral chaos in a special version of a biological predator-prey
model prior to Pohjola.. Work that has followed up on the Goodwin tradition of using
predator-prey dynamics to study complex dynamics in macroeconomic fluctuations has
been done by Chiarella, Flaschel, and Franke (2005).
More recent work has shown the possibility of various other forms of complex
dynamics arising with predator-prey systems. Thus, the phenomenon of fractal basin
boundaries between multiple basins of attraction has been studied by Gu and Huang
(2006) for predator-prey models. Kaneko and Tsuda (2001) an even broader array of
forms of complex dynamics that can arise in coupled dynamical systems.
Finally, the discussion of predator-prey dynamics has moved to a higher level of
macroevolution where chaotically oscillating patterns of phenotype and genotype
variation occur over the long periods of evolutionary change within a predator-prey
context (Solé and Bascompte, 2006; Sardanyés and Solé, 2007). Such longer horizon
coevolutionary processes may occur in Schumpeterian economic dynamics.
22
V. Capital Theoretic Paradoxes in Ecologic-Economic Systems
From almost its beginning as a self-conscious, transdisciplinary discipline,
ecological economics has seen itself as connected to the Sraffian approach to economic
modeling, with this deriving more fundamentally from the ideas of Quesnay and the
physiocrats (Christensen, 1989). While this initial perception had more to do with the
input-output nature of the Sraffian system, it is however not surprising that the capital
theoretic paradoxes such as reswitching should also show up in ecological and
environmental systems when interacting with the economic system run by humans.
Problems relating to time discounting have long been of major concern among
environmental and ecological economists, usually with a strong emphasis on the ethical
issues involved in decisions that affect distant future generations (Howarth and Norgaard,
1990). While most of this discussion has presumed that using a lower discount rate will
lead to more environmentally sound and sustainable outcomes, the possibility of capital
theoretic paradoxes due to complexity of time patterns of effects can complicate this
standards story considerably, showing a curious interconnection between Sraffian Post
Keynesian economics and ecological economics in complex systems.
The problem of time discounting and the environment arose out of the use of costbenefit analysis in the United States government, where it was initially developed for
evaluating possible dam-building for flood control projects by the U.S. Army Corps of
Engineers, dating back to the 1930s. For such projects there was long no accounting for
environmental consequences of building the dams, such as destruction of habitat for
endangered species, whose costs would go on well into the future. Rather, the costs of
the dams were up front in time, associated mostly with the building of the dam and
23
buying any property that would be flooded by the dam. Benefits were seen to accrue in
the future, mostly associated with flood control, although possibly some for recreation or
irrigation as well. Thus, in such simple situations it was unequivocal: using a higher
discount rate would lead to less dam building. Thus, in the period when environmental
impacts were not being counted, environmentalists seeking to block the building of dams
supported the use of higher discount rates in these evaluations, and when a uniform
discount rate of 10% was imposed in 1970 throughout the U.S. government for all
projects by President Nixon’s OMB Director, Roy Ash, many environmentalists made
alliance with cost-cutting conservatives to support this “pro-conservation” move.18
More generally it came to be realized that when it comes to the environment,
particular actions may have upfront costs, but then delayed environmental costs as well,
such as with nuclear power or strip mining of coal, where there is a cost to begin the
activity up front, then a period of positive net benefits, but then delayed environmental
costs associated with waste disposal or cleanup (Herfindahl and Kneese, 1974; Porter,
1982; Asheim, 2008). Prince and Rosser (1985) studied the example of strip mining of
coal in the U.S. Southwest and found that for reasonable figures, reswitching could occur
between strip mining of coal and cattle grazing, with the former more beneficial at rates
between about 2 and 7 percent, with cattle grazing dominating at rates lower than 2
percent and higher than 7 percent, due to this time pattern effect.
An area of ecological economics where such phenomena may well arise is in
forestry, especially when forests can have multiple uses that have varying and
complicated time patterns connected with them. The equation for the optimum rotation
18
Hannesson (1987) has made a similar argument with respect to fisheries with the capital-intensity of
modern, commercial fishing technologies making it possible that a lower interest rate could lead to higher
risks of overfishing fisheries.
24
period, T, of a forest, allowing for functions besides timber, is due to Hartman (1976),
with p being the price of timber, f(t) is the timber growth function, c is the (constant)
marginal cost of harvesting the timber, r is the real discount rate, and g(t) gives the stream
of net amenities that are not due to timber cutting, some of which may be social in nature,
although they may be private and appropriable, such as cattle grazing:
pf’(T) = rpf(T) + r[(pf(T) – c)/(erT – 1)] – g(T).
(10)
The interpretation of this is essentially that one should cut when the trees are
growing at the real rate of interest (an old solution of Irving Fisher, 1907), but corrected
for the benefits of getting more rapidly growing younger trees in the ground and
accounting for the non-timber amenities. The timber growth function tends to decline in
rate with time. However, the non-timber amenities may take a variety of patterns. Thus
on timber lands in Montana, grazing can occur when the forest is young, and the benefits
of grazing tend to rise to about 12.5 years and then sharply decline after that. Combining
this with the timber function gives a non-monotonic function of the present value as T
varies. This is shown for the Montana forests in Figure 7, drawing on Swallow, Parks,
and Wear (1990), with MBD representing the marginal benefit of delaying cutting, and
MOC, the marginal opportunity cost of doing so.
25
Figure 7: Optimal Hartman rotation on Montana forest lands
An even more complicated situation presents itself in the eastern deciduous
forests such as the George Washington National Forest in Virginia. In this forest there is
an effort to maximize net social benefit, taking into account non-marketed benefits such
as to hunters. Given the time pattern of the broader ecosystem after timber harvesting,
one finds three different peaks of benefit for different activities. Thus, about five to six
years after a clearcut, deer population is maximized, with deer liking the edges of recent
clearcuts with new trees just beginning to grow. About 20-25 years in there is a
maximum degree of biodiversity achieved, with much undergrowth. Turkeys and quails
are maximized in this period and environment. However, bear populations maximize in
older growth forests more than 60 years old, as undergrowth disappears and old tree
26
trunks fall that the bear can inhabit. The net benefit function from the standpoint of
various groups of hunters is shown in Figure 8, taken from Rosser (2005, p. 198),
drawing on the FORPLAN analysis of Johnson, Jones, and Kent (1980). While no
comparison with an alternative use was made, the possibility of some sort of paradox and
anomaly of the sort already discussed is clearly much higher in such situations.
time
Figure 8: Virginia Deciduous Forest Hunting Amenity
Thus, while there probably remain such deep differences between the Sraffian
neo-Ricardian branch of Post Keynesian economists and the fundamentalist Keynes Post
Keynesians who focus uncertainty and the role of money that they cannot be bridged, in
the area of complex ecologic-economic analysis, common features present themselves.19
19
See Rosser (1991, Chaps. 8 and 13) for further discussion of these issues. We note here that the link
between Sraffian issues and ecologic-economic analysis exists even when no complex dynamics occur.
27
VI. Policy Implications and Conclusions
The existence of complex dynamics in ecologic-economic systems complicates
policy-making considerably. Rosser (2001) argued that two well-known principles are
more important in the face of the threats of possible sudden discontinuities or more
erratic dynamic patterns: the precautionary principle and the scale-matching principle.
The first is pretty obvious. The existence of thresholds beyond which catastrophic
outcomes can occur should induce considerable caution, especially when irreversibilities
are involved (Kahn and O’Neill, 1999). How we deal with discovering where those
thresholds are and what to do about them remains a problem that veers into that of
Keynesian uncertainty, even though some do make efforts to estimate probabilities in
these situations. In some cases, such as with the various reports on global warming,
probabilities are estimated, but they are done so while ignoring the probabilities of these
more disturbing potentially catastrophic events.
Scale-matching is another matter that involves making sure that any policy action
is directed at the appropriate level of the ecological hierarchy. Global problems should
be dealt with globally; local ones locally. This may seem obvious, but it becomes less
obvious when decisionmaking must interact with the assignment or assessment of
property rights. These two must align themselves relevantly with the environmental or
ecological effects of a policy or an action (Rosser and Rosser, 2006). It should be kept in
mind that mere ownership is not sufficient (Ciriacy-Wantrup and Bishop, 1975), as
implied by such analysts of the “common property resource problem” as Gordon (1954).
Control of access is the key, and the assignment of property rights must align itself with
28
the ability to control access and achieve environmentally sustainable outcomes, a point
emphasized in the work of Ostrom (1990).
Finally, just as argued by Rosser (2006a) that there was an important influence
from Post Keynesian economics on the development of complexity theory, so too here we
see some elements of influence on the more specifically ecological-economic complex
systems theory. Indeed, the influence has been both ways as was seen with the role of
predator-prey models in Post Keynesian macrodynamics. Many of the deepest themes of
the various schools of Post Keynesian economics are most clearly seen in the cases of
complex ecologic-dynamics, from fundamental uncertainty to capital theoretic paradoxes.
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