Learning Center
Plans & pricing Sign in
Sign Out
Your Federal Quarterly Tax Payments are due April 15th Get Help Now >>



  • pg 1

J. Barkley Rosser, Jr. James Madison University rosserjb@jmu.edu

October, 2008

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.


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).


While the deepest of these splits are probably unbridgeable, Rosser (2006a) 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. As in the earlier paper, this argument can be seen as working in both directions. Thus, on the one hand foundational ideas of dynamic complexity have arisen from the study of ecological systems. On the other, the interlinkage of ecological with economic systems can be the source of complex dynamics in the combined systems. An example of an idea from ecology that directly influenced macrodynamic theory is that of the predator-prey model, first applied to macroeconomics by Goodwin (1967). An example of the latter are the variety of complex dynamics that can arise in fisheries through the interaction of nonlinearities in the biodynamics of fish populations with nonlinearities in the behavior of the fishers (Hommes and Rosser, 2001). This paper will pursue this theme by first reviewing briefly what constitute complex dynamics. The major discussion will follow, which will focus on ecologiceconomic dynamics as a foundation of fundamental uncertainty of the Keynesian sort, and the implications for the broader debates among Post Keynesians. Then there 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 the complexities of 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.

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). 3 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 endogenously does not converge on a point, a limit cycle, or an explosion or implosion. 4

Rosser (2006b) 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. 4 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


viewed as non-complex, 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 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 19 th 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

shall not dispute this, it should be noted that it is not easy to use this definition to distinguish complex from non-complex systems.


change continuously, 5 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 without necessarily directly affecting an entire system, even though these local effects can lead to broader systemic effects through complex emergence. 6 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, 7 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, even though Davidson (1967) himself was an early developer and advocate of the environmental economics in the United States while he was at Rutgers University. But he emphasized neither Keynesian uncertainty nor the role of money particularly in his discussion of such issues, and indeed his approach prefigured what are now relatively conventional methods in that

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. 6 The concept of “emergence” was developed in the early 20th 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). 7 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.


area rather than the more heterodox ecological economics approaches developed since that time (Martinez-Alier, 1987). A long running debate between Davidson (1996) and other Post Keynesians (Rosser, 2001a, 2006a) 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, which he insists must be accepted on axiomatic grounds, others have argued that indeed the ubiquity of complex dynamics in economic systems can provide a theoretically and empirically valid foundation for the concept. We shall not regurgitate the details of this debate here further. Rather, we shall consider some ecologic-economic systems that exhibit forms of dynamic complexity that this observer at least believes imply a reasonable form of 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. 8 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

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.


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. 9 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; 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


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.


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 spruce-budworm cycle of about 40 years in Canadian forests (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, r B 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 = r BB(1 – B/KB) – βB2 /(α2 + B2). Nonzero equililbria are solutions of (rBKB/β) = u*/[(α/K2) + u*2)(1 – u*)]. 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 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. 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 (2) (1)


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.” 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, cod 10 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).


See Ruitenback (1996) for a discussion of the collapse of the once great cod fishery off Newfoundland.


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}. (5)

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.


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


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)


then letting agents follow naïve expectations of the form that next period’s price will be the same as this period’s price leads to cobweb 11 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).

Figure 4: Learning to believe in chaos


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).


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).12 An example of how such a system looks is depicted in Figure 5, which shows the basins of attraction for a ball held over three magnets, drawn from Peitgen, Jürgens, and Saupe (1992).


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.


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 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 dynamically economically model, based on Goodwin’s (1951) with its nonlinear accelerator, even as they did not understand what they had discovered.13 His 1967 model more explicitly drew on ecological predecessors in the form of the predatorprey model of Lotka (1920, 1925) and Volterra (1926, 1931).14 Ironically for the old Marxist, 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 investment-driven 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


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). 14 At the same time that Goodwin presented his predatory-prey model of macrodynamics, Samuelson (1967) did so also. 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.


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, 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], dλ/dt = λ[(1 – ω)/v – α – β]. (8) (9)

While Goodwin initially only studied the endogenous limit cycle easily implied by these equations, Pohjola (1981) would show that with certain parameter values 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 only generated simple oscillations, limit cycles. 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 several centuries-long data set, which 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 or semi-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 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, with regard to predator-prey models, it was economics that discovered the possibility of chaotic dynamics prior to the ecologists (Pohjola, 1981; Shaffer, 1984). 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). This pattern in some respects reflects the chaotic long wave evolutionary dynamics studied by Goodwin (1986).

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 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. 15 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 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)


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.


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.

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 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.


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. 16

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

See Rosser (1991, Chaps. 8 and 13) for further discussion of these issues.


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 the ability to control access and achieve environmentally sustainable outcomes. 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.


References Albin, Peter S. with Duncan K. Foley. 1998. Barriers and Bounds to Rationality: Essays on Economic Complexity and Dynamics in Interactive Systems. Princeton: Princeton University Press. Asheim, Geir B. 2008. “The Occurrence of Paradoxical Behavior in a Model where Economic Activity has Environmental Effects.” Journal of Economic Behavior and Organization 65, 529-546. Brauer, Fred and A.E. Soudack. 1985. “Optimal Harvesting in Predator-Prey Systems.” International Journal of Control 41, 111-128. Brock, William A., Karl-Göran Mäler, and Charles Perrrings. 2002. “Resilience and Sustainability: The Analysis of Nonlinear Dynamic Systems.” In Lance H. Gunderson and C.S. Holling, eds. Panarchy: Understanding Transformations in Human and Natural Systems. Washington: Island Press, pp. 261-289. Carpenter, Stephen R., Donald Ludwig, and William A. Brock. 1999. “Management of Eutrophication for Lakes Subject to Potentially Irreversible Change.” Ecological Applications 9, 751-771. Chen, Z. 1997. Can Economic Activity Lead to Climate Chaos? An Economic Analysis on Global Warming. Canadian Journal of Economics 30, 349-366. Chiarella, Carl. 1988. “The Cobweb Model: Its Instability and the Onset of Chaos.” Economic Modelling 5, 377-384. Christensen, Paul P. 1989. “Historical Roots for Ecological Economics – Biophysical versus Allocative Approaches.” Ecological Economics 1, 17-36. Ciricacy-Wantrup, Siegfried V. and Richard C. Bishop. 1975. “’Common Property’ as a Concept in Natural Resources Policy.” Natural Resources Journal 15, 713-727. Clark, Colin W. 1985. Bioeconomic Modelling and Fisheries Management. New York: Wiley-Interscience. Clark, Colin W. 1990. Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd edition. New York: Wiley-Interscience. Coddington, Alan. 1976. “Keynesian Economics: The Search for First Principles.” Journal of Economic Literature 14, 1258-1273. Conklin, James E. and William C. Kolberg. 1994. “Chaos for the Halibut?” Marine Resource Economics 9, 153-182.


Copes, Parzival. 1970. “The Backward-Bending Supply Curve of the Fishing Industry.” Scottish Journal of Political Economy 17, 69-77. Cucker, Felipe and Steve Smale. 2007. “On the Mathematics of Emergence.” Japanese Journal of Mathematics 2, 197-227. Darwin, Charles. 1859. The Origin of the Species by Means of Natural Selection or the Preservation of Favored Races in the Struggle for Life. London: John Murray. Davidson, Paul. 1967. “An Exploratory Study to Investigate and Measure the Benefits of Scenic Federal-Aid Highways.” Highway Research Record, No. 182, 18-21. Davidson, Paul. 1982-83. “Rational Expectations: A Fallacious Foundation for Studying Crucial Decision-making Processes.” Journal of Post Keynesian Economics 5, 182-197. Davidson, Paul. 1994. Post Keynesian Macroeconomic Theory. Aldershot, UK and Brookfield, US: Edward Elgar. Davidson, Paul. 1996. “Reality and Economic Theory.” Journal of Post Keynesian Economics 18, 479-508. Day, Richard H. 1994. Complex Economic Dynamics: An Introduction to Dynamical Systems and Market Mechanism, Volume I. Cambridge, MA: MIT Press. Day, Richard H. 2006. “The Complex Problem of Modelling Economic Complexity.” In Mark Setterfield, ed. Complexity, Endogenous Money and Macroeconomic Theory: Essays in Honour of Basil J. Moore. Cheltenham, UK and Northampton, US: Edward Elgar, pp. 63-73. Desai, Meghnad. 1973. “Growth Cycles and Inflation in a Model of Class Struggle.” Journal of Economic Theory 6, 527-545. Done, T.J. 1992. “Phase Shifts in Coral Reef Communities and their Ecological Significance.” Hydrobiology 247, 121-132. Doveri, F., M. Scheffer, S. Rinaldi, S. Muratori, and Ya. Kuznetsov. 1993. “Seasonality and Chaos in a Plankton-Fish Model.” Theoretical Population Biology 43, 159-183. Ellner, S. and Peter Turchin. 1995. “Chaos in a Noisy World: New Methods and Evidence from Time Series Analysis.” American Naturalist 145, 343-375. Elton, Charles S. 1924. “Periodic Fluctuations in the Number of Animals: Their Causes and Effects.” British Journal of Experimental Biology 2, 119-163. Elton, Charles S. and M. Nicholson. 1942. “The Ten Year Cycle in the Numbers of Lynx in Canada.” Journal of Animal Ecology 11, 215-244.


Embrechts, Paul, Claudia Küppelberg, and Thomas Mikosch. 2003. Modelling Extremal Events for Insurance and Finance. Berlin: Springer. Estes, J.A. and D. Duggins. 1995. “Sea Otters and Kelp Forests in Alaska: Generality and Variation in a Community Ecological Paradigm.” Ecological Monographs 65, 75-100. Fisher, Irving. 1907. The Rate of Interest. New York: Macmillan. Föllmer, Hans. 1974. “Random Economies with Many Interacting Agents.” Journal of Mathematical Economics 1, 51-62. Foroni, Ilaria, Laura Gardini, and J. Barkley Rosser, Jr. 2003. “Adaptive and Statistical Expectations in Renewable Resource Markets.” Mathematics and Computers in Simulation 63, 541-567. Goodwin, Richard M. 1947. “Dynamical Coupling with Especial Reference to Markets Having Production Lags.” Econometrica 15, 181-204. Goodwin, Richard M. 1951. “The Nonlinear Accelerator and the Persistence of Business Cycles.” Econometrica 19, 1-17. Goodwin, Richard. M. 1967. “A Growth Cycle.” In C.H. Feinstein, ed. Socialism, Capitalism and Economic Growth: Essays Presented to Maurice Dobb. Cambridge, UK: Cambridge University Press, pp. 54-58. Goodwin, Richard M. 1986. “The Economy as an Evolutionary Pulsator.” Journal of Economic Behavior and Organization 7, 341-349. Goodwin, Richard M. 1990. Chaotic Economic Dynamics. Oxford, UK: Oxford University Press. Gordon, H. Scott. 1954. “Economic Theory of a Common Property Resource: The Fishery.” Journal of Political Economy 62, 124-142. Gould, Stephen Jay. 2002. The Structure of Evolutionary Theory. Cambridge, MA: The Belknap Press of Harvard University. Grandmont, Jean-Michel. 1998. “Expectations Formation and Stability of Large Socioeconomic Systems.” Econometrica 66, 741-781. Gu, En-Guo and Yibing Huang. 2006. “Global Bifurcations of Domains of Feasible Trajectories: An Analysis of a Discrete Predator-Prey Model.” International Journal of Bifurcations and Chaos 16, 2601-2613.


Gunderson, Lance H., C.S. Holling, Lowell Pritchard, Jr., and Garry Peterson. 2002. “Understanding Resilience: Theory, Metaphors, and Frameworks.” In Lance H. Gunderson and Lowell Pritchard, Jr., eds. Resilience and the Behavior of Large-Scale Systems. Washington: Island Press, pp. 3-20. Hamouda, Omar F. and Geoffrey C. Harcourt. 1988. “Post Keynesianism: From Criticism to Coherence.” Bulletin of Economic Research 40, 1-33. Hannesson, Rognvalder. 1987. “The Effect of the Discount Rate on the Optimal Exploitation of Renewable Resources.” Marine Resource Economics 3, 319-329. Harcourt, Geoffrey C. 1976. “The Cambridge Controversies: Old Ways and New Horizons – or Dead End?” Oxford Economic Papers 28, 25-65. Hartman, R. 1976. “The Harvesting Decision when a Standing Forest has Value.” Economic Inquiry 14, 52-58. Hassell, M.P., J.H. Lawton, and R.M. May. 1976. “Patterns of Dynamical Behavior of Single-Species Populations.” Journal of Animal Ecology 45, 471-486. Herfindahl, Orris C. and Allen V. Kneese. 1974. Economic Theory of Natural Resources. Columbus, OH: Charles Merrill. Hicks, John R. 1950. A Contribution to the Theory of the Trade Cycle. Oxford: Oxford University Press. Hodgson, Geoffrey M. 1993. Economics and Evolution: Bringing Life Back into Economics. Ann Arbor: University of Michigan Press. Holling, C.S. 1973. “Resilience and Stability of Ecological Systems.” Review of Ecology and Systematics 4, 1-24. Holling, C.S. 1986. “Resilience of Ecosystems: Local Surprise and Global Change.” In W.C. Clark and R.E. Munn, eds. Sustainable Development of the Biosphere. Cambridge, UK: Cambridge University Press, pp. 292-317. Holt, Richard P.F. and Steven Pressman, eds. 2001. A New Guide to Post Keynesian Economics. London: Routledge. Hommes, Cars H. and J. Barkley Rosser, Jr. 1981. “Consistent Expectations Equilibria and Complex Dynamics in Renewable Resource Markets.” Macroeconomic Dynamics 5, 180-203. Hommes, Cars H. and Gerhard Sorger. 1998. “Consistent Expectations Equilibria.” Macroeconomic Dynamics 2, 287-321.


Horgan, John. 1997. The End of Science: Facing the Limits of Scientific Knowledge in the Twilight of the Scientific Age, paperback edition. New York: Broadway Books. Howarth, Richard B. and Richard B. Norgaard. 1990. “Intergenerational Resource Rights, Efficiency, and Social Optimality.” Land Economics 66, 1-11. Hughes, T.P. 1994. “Catastrophes, Phase Shifts, and Large-Scale Degradation of a Caribbean Coral Reef.” Science 265, 1547-1551. Jones, Dixon D. and Carl J. Walters. 1976. “Catastrophe Theory and Fisheries Regulation.” Journal of the Fisheries Resource Board of Canada 33, 287-300. Johnson, K.N., D.B. Jones, and B.M. Kent. 1980. A User’s Guide to the Forest Planning Model (FORPLAN). Fort Collins, CO: USDA Forest Service, Land Management Planning. Kahn, James R. and Robert.V. O’Neill. 1999. “Ecological Interaction as a Source of Economic Irreversibility.” Southern Economic Journal 66, 391-402. Kaldor, Nicholas. 1940. “A Model of the Trade Cycle.” Economic Journal 50, 78-92. Kalecki, Michal. 1935. “A Macrodynamic Theory of Business Cycles.” Econometrica 3, 327-344. Kalecki, Michal. 1971. Selected Essays on the Dynamics of the Capitalist Economy. Cambridge, UK: Cambridge University Press. Kaneko, Kunihiko and Ichiro Tsuda. 2001. Complex Systems: Chaos and Beyond, a Constructive Approach with Applications in Life Sciences. Berlin: Springer-Verlag. Keynes, John Maynard. 1921. Treatise on Probability. London: Macmillan. Keynes, John Maynard. 1936. The General Theory of Employment, Interest and Money. London: Macmillan. King, John E. 2002. A History of Post Keynesian Economics since 1936. Cheltenham, UK and Northampton, US: Edward Elgar. Leopold, Aldo. 1933. “The Conservation Ethic.” Journal of Forestry 31, 634-643. Lorenz, Hans-Walter. 1992. “Multiple Attractors, Complex Basin Boundaries, and Transient Motion in Deterministic Economic Systems.” In Gustav Feichtinger, ed. Dynamic Economic Models and Optimal Control. Amsterdam: North-Holland, pp. 411430.


Lotka, Alfred J. 1920. “Analytical Notes on Certain Rhythmic Relations in Organic Systems.” Proceedings of the National Academy of Sciences, United States. 6, 410-415. Lotka, Alfred J. 1925. Elements of Physical Biology. Baltimore: Williams and Williams (reprinted in 1945 as Elements of Mathematical Biology). Ludwig, Donald, Dixon D. Jones, and C.S. Holling. 1978. “Qualitative Analysis of Insect Outbreak Systems: Spruce Budworm and Forest.” Journal of Animal Ecology 47, 315332. Ludwig, Donald, Brian H. Walker, and C.S. Holling. 2002. “Models and Metaphors of Sustainability, Stability, and Resilience.” In Lance H. Gunderson and Lowell Pritchard, Jr., eds. Resilience and Behavior of Large-Scale Systems. Washington: Island Press, pp. 21-48. Markose, Sheri M. 2005. “Computability and Complexity: Markets as Complex Adaptive Systems (CAS).” Economic Journal 115, F159-F192. Martinez-Alier, Juan. 1987. Ecological Economics: Energy, Environment and Society. Oxford, UK and Cambridge, MA: Blackwell Publishers. May, Robert M. 1974. “Biological Populations with Non-Overlapping Generations: Stable Points, Cycles, and Chaos.” Science 186, 645-647. McCauley, Joseph P. 2004. Dynamics of Markets: Econophysics and Finance. Cambridge, UK: Cambridge University Press. Mill, John Stuart. 1843. A System of Logic: Ratiocinative and Inductive. London: Longmans Green. Muradian, R. 2001. “Ecological Thresholds: A Survey.” Ecological Economics 38, 7-24. Noy-Meir, I. 1973. “Desert Ecosystems: Environment and Producers.” Annual Review of Ecology and Systematics 4, 353-364. Peitgen, Hans-Otto, Hartmut Jürgens, and Dietmar Saupe. 1992. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag. Pohjola, Matti T. 1981. “Stable, Cyclic, and Chaotic Growth: The Dynamics of a Discrete Time Version of Goodwin’s Growth Cycle.” Zeitschrhift für Nationalökonomie 41, 7-38. Poincaré, Henri. 1880-1890. Mémoire surles courbes définies par les equations différentialles I-IV, Oeurve I. Paris: Gauthier-Villars. Porter, Richard C. 1982. “The New Approach to Wilderness through Benefit Cost Analysis.” Journal of Environmental Economics and Management 9, 59-80.


Prince, Raymond and J. Barkley Rosser, Jr. 1985. “Some Implications of Delayed Environmental Costs and Reswitching Between Food and Energy Production in the Western United States.” Growth and Change 16, 18-25. Rosser, J. Barkley, Jr. 1991. From Catastrophe to Chaos: A General Theory of Economic Discontinuities, 1st edition. Boston/Dordrecht: Kluwer Academic Publishers. Rosser, J. Barkley, Jr. 1992. “The Dialogue between the Economic and the Ecologic Theories of Evolution.” Journal of Economic Behavior and Organization 17, 195-215. Rosser, J. Barkley, Jr. 1999. “On the Complexities of Complex Economic Dynamics.” Journal of Economic Perspectives 13(4), 169-192. Rosser, J. Barkley, Jr. 2000. From Catastrophe to Chaos: A General Theory of Economic Discontinuities: Mathematics, Microeconomics, Macroeconomics, and Finance, Volume I, 2 nd edition. Boston/Dordrecht: Kluwer Academic Publishers. Rosser, J. Barkley, Jr. 2001a. “Alternative Keynesian and Post Keynesian Perspectives on Uncertainty and Expectations.” Journal of Post Keynesian Economics 23, 545-566. Rosser, J. Barkley, Jr. 2001b. “Complex Ecologic-Economic Dynamics and Environmental Policy.” Ecological Economics 37, 23-37. Rosser, J.Barkley, Jr. 2005. “Complexities of Dynamic Forest Management Policies.” In Shashi Kant and R. Albert Berry, eds. Economics, Sustainability, and Natural Resources. Dordrecht: Springer, pp. 191-206. Rosser, J. Barkley, Jr. 2006a. “Complex Dynamics and Post Keynesian Economics.” In Mark Setterfield, ed. Complexity, Endogenous Money and Macroeconomic Theory: Essays in Honour of Basil J. Moore. Cheltenham, UK and Northampton, US: Edward Elgar, pp. 74-98. Rosser, J. Barkley, Jr. 2006b. “Computational and Dynamic Complexity in Economics.” Mimeo. James Madison University, http://cob.jmu.edu/rosserjb. Rosser, J. Barkley, Jr. 2007. “The Rise and Fall of Catastrophe Theory Applications in Economics: Was the Baby Thrown Out with the Bathwater?” Journal of Economic Dynamics and Control 31, 3255-3280 . Rosser, J. Barkley, Jr. 2008. “Dynamic Disontinuities in Ecologic-Economic Systems.” In Craig Allen and C.S. Holling, eds. Discontinuity in Ecosystems and Other Complex Systems. New York: Columbia University Press, in press.


Rosser, J. Barkley, Jr. and Marina V. Rosser. 2006. “Institutional Evolution of Environmental Management under Global Economic Growth.” Journal of Economic Issues 40, 421-429. Ruitenback, H.J. 1996. “The Great Canadian Fishery Collapse: Some Policy Lessons.” Ecological Economics 19, 103-106. Sakai, K. 2001. Nonlinear Dynamics and Chaos in Agricultural Systems. Amsterdam: Elsevier. Samuelson, Paul A. “A Universal Cycle?” In R. Henn, ed. Methods of Operations Research, III. Muhlgasse: Verlag Anton Hain. Sardanyés, Josep and Ricard V. Solé. 2007. “Chaotic Stability in Spatially-Revolved Host-Parasite Replicators: The Red Queen on a Lattice.” International Journal of Bifurcations and Chaos 17, 589-606. Sawyer, Malcolm C. 1985. The Economics of Michal Kalecki. London: Macmillan. Schaefer, M.B. 1957. “Some Considerations of Population Dynamics and Economics in Relation to the Management of Marine Fisheries.” Journal of the Fisheries Resource Board of Canada 14, 699-681. Schaffer, W.M. 1984. “Stretching and Folding in Lynx Fur Returns: Evidence for a Strange Attractor in Nature?” American Naturalist 124, 796-820. Schelling, Thomas C. 1971. “Dynamic Models of Segregation.” Journal of Mathematical Sociology 1, 143-186. Schindler, D.W. 1990. “Experimental Perturbations of Whole Lakes as Tests of Hypotheses Concerning Ecosystem Structure and Function.” Oikos 57, 25-41. Solé, Ricard V. and Jordi Bascompte. 2006. Self-Organization in Complex Ecosystems. Princeton: Princeton University Press. Spath, Clive L. 2002. Greenhouse Economics: Value and Ethics. London: Routledge. Sraffa, Piero. 1960. Production of Commodities by Commodities: Prelude to a Critique of Economic Theory. Cambridge, UK: Cambridge University Press. Strotz, Robert H., J.C. McAnulty, and Joseph B. Naines, Jr. 1953. “Goodwin’s Nonlinear Theory of the Business Cycle: An Electro-Analog Solution.” Econometrica 21, 390-411. Swallow, S.K., P.J. Parks, and D.N. Wear. 1990. “Policy-Relevant Non-Convexities in the Production of Multiple Forest Benefits.” Journal of Environmental Economics and Management 19, 264-280.


Turchin, Peter. 2003. Complex Population Dynamics: A Theoretical/Empirical Synthesis. Princeton: Princeton University Press. Turing, Alan M. 1952. “The Chemical Basis of Morphogenesis.” Philosophical Transactions of the Royal Society B 237, 37-72. van der Pol, Balthasar and J. van der Mark. 1927. “Frequency Demultiplication.” Nature 120, 363-364. Velupillai, Kumaraswamy Vela. 2005. “Introduction.” In K.V. Velupillai, ed. Computability, Complexity and Constructivity in Economic Analysis. Victoria: Blackwell, pp. 1-14. Volterra, V. 1926. “Fluctuations in the Abundance of a Species Considered Mathematically.” Nature 118, 558-600. Volterrra, V. 1931. Lecons sur la ThéorieMathématique de la Lutte pour la Vie. Paris: Gauthier-Villars. Wagener, Florian O.O. 2003. “Skiba Points and Heteroclinic Bifurcations with Applications to the Shallow Lake System.” Journal of Economic Dynamics and Control 27, 1533-1561. Zimmer, C. 1999. “Life after Chaos.” Science 284, 83-86.


To top