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The impact of the self organisation approach on economic science: why economic theory and history need no longer be mutually exclusive domains
~ i
ELSEVIER
MATHEMATICS AND COMPUTERS N SIMULATION
Mathematics and Computers in Simulation 39 (1995)393-398
The impact of the self organisation approach on economic science: why economic theory and history need no longer be mutually exclusive domains John Foster Department of Economics, The University of Queensland, Queensland 4072, Australia
1. Introduction The self-organisation approach has grown in popularity in the natural sciences in recent years. The purpose of this paper is to suggest how the approach might be applied, empirically, in economics.
2. Economics and self organisation In orthodox economics, the prevailing methodology is to construct a timeless theory from which certain propositions are deduced then translated, through the addition of an auxiliary hypothesis, into a specification which can address historical data. However, it is clear that theories which involve timeless deductive logic have no place in a self-organisation context. In practice, natural scientists investigating selfoorganising structures formulate theories in an inductive way, much in the m a n n e r suggested by Lawson [1] who prefers the term "retroduction". Such abstraction from structural change is of fundamental importance in understanding how self-organising systems develop and survive. A detailed understanding of the microscopic structure of a system cannot reveal macroscopic features which are critically important. Thus, proponents of self-organisation have focussed attention upon macroscopic measures such as, for example, entropy. The operation of the entropy law and the parallel existence of dissipative structures, with evolutionary and irreversible features are taken as axiomatic in nature. However, the axioms in question are not clothed with extensive mathematical deduction, facilitated by the introduction of convenient, but unrealistic, assumptions to address real processes. Instead, the reverse is the case, the axioms concerning processes are the outcome of extensive empirical study, and, thus, mathematical deduction is unnecessary, even though the theory of process favoured may well be specified in mathematical terms. 0378-4754/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 3 7 8 - 4 7 5 4 ( 9 4 ) 0 0 0 8 9 - 3
394
J. Foster~Mathematics and Computers in Simulation 39 (1995) 393-398
The specification of an evolutionary process in mathematical terms enables particular specifications to be estimated in particular cases, permitting a degree of ceteris paribus prediction concerning the developmental possibilities of a system in the near future and, perhaps more importantly, the onset of discontinuities can be given precise parametric representation. It is in the context of discontinuities that mathematics becomes an important tool. In physics, Haken [2] has devised mathematical techniques to deal with discontinuity in "synergetic" systems (see [3]) for an assessment of the economic applicability of these techniques) and in biology similar techniques have been devised.
3. The logistic equation If we examine some spatially-defined dissipative structure, with its external boundaries fixed, with time-related increases in complexity and organisation, what we typically observe in fields as disparate as population ecology and reaction kinetics is a density-dependent logistic path of mass dynamics over time:
d N J d t = rN,[1 - ( N , / K ) ] .
(1)
Where K is the carrying capacity, or boundary, of N development and r is the net birth rate, diffusion rate, etc. This represents a very simple, but useful, theory of process which provides the analytical foundation u p o n which the modelling of particular historical processes can proceed. Despite its simplicity, it can generate very complex nonlinear dynamics in certain circumstances. It conforms to the requirements of a theory as stated by Lawson [1] since it is the product of extensive retroduction in the natural sciences. Whatever the precise form of the logistic, its sigmoid form captures the general character of the density-dependent dynamics which self-organising dissipative structures must exhibit as an axiomatic necessity. The smoothness of the logistic is an assumption which is added to the self-organisation axioms. In particular cases it will not be smooth and the purpose of introducing additional exogenous variables into a logistical equation is to relax this assumption and, in so doing, distil estimates of the underlying logistic parameters. However, smoothness is also dictated by endogenous considerations - it is conditional upon certain parametric configurations which, if exceeded in certain directions, yield discontinuities of the type which we observe in actual historical experience. W h e n a discontinuity occurs, structural change loses its smooth character and there occurs fusion into a larger unit or disintegration, after which the subject/environment relationship must be qualified or redefined. Here, mathematics becomes an important tool for the biologist or chemist to ascertain, from empirically-derived logistic parameters, the stability range of the process in question and any steady states which can exist. However, a limitation of the simple logistic in this regard is that it has to be specified in terms of discrete time intervals between events to generate discontinuities. Many macroscopic mass measures can alter continuously through time yet do not display smooth logistic paths towards steady states. Thus, although the simple logistic equation captures a general endogenous tendency in self-organising systems, it is strictly limited in its empirical application.
J. Foster~Mathematics and Computers in Simulation 39 (1995) 393-398
395
Although Eq. (1) can offer a useful representation of, for example, certain kinds of chemical kinetics and even though it can be regarded as the embodiment of a general theory of a historical process, evolutionary biologists, in particular, view it as special case. A more general specification is as follows
d N J d t = rNt[1 - ( N , / K ) + a( M t / K ) ] .
(2)
Where M is an interacting population. An identical companion equation can be specified for M in terms of N. Eq. 2 is one example of a class of interacting population equations which have been derived from empirical research. It is a two party (Kolmogorov) version of the L o t k a Volterra logistic specified in Eq. (1). The resultant model is extremely simple, but very useful since most biologists agree that it can capture many of the properties of more complicated models. This is particularly the case with respect to discontinuities in otherwise stable-looking developmental trajectories. Murray [4], for example, considers three cases: predator-prey, mutualist and competitive interactions. The latter two can be described in terms of Eq. (2) with mutualism being the case where a > 0 and competition where a < 0. In each case, he examines the parametric ranges within which stable co-evolution can prevail and the critical parametric combinations where stability breaks down into discontinuity (the 'bifurcation point' in the language of chaos). In the case of competition, the "principle of competitive exclusion" tends to operate over wide parameter ranges in population ecology. In other words, competition tends not to be a stable co-evolutionary state. Mutualism, on the other hand, is very stable while predator-prey interaction tends to generate oscillations. There are, in fact, six interactive conditions which, at any point in time, an organism can be in: (1) Prey, (2) Predator, (3) Competitor, (4) Mutualist, (5) Neutral, (6) Combatant. Eq. (1) is essentially condition (5) and, as we have noted, a rather special case. The constant need of a dissipative structure for free energy supplies ensures that neutrality is an ephemeral condition bolstered by the existence of stocks and other homoeostatic mechanisms. Many supposedly neutral conditions are due to unstated mutualist associations (eg. siblings and parents). Thus, as has been noted biologists rarely use Eq. (1) in empirical investigation, except in demonstrable special circumstances (a single predator in an isolated environment with massive, fast replenishing stocks of prey). Thus, population ecologists have focussed upon conditions (1) to (4).
4. The logistic equation in economics
In the history of economic thought, the priority ordering of interactive states which we observe in biology seems to have been reversed. Economists have tended to begin with abstract
396
3". Foster/Mathematics and Computers in Simulation 39 (1995) 393-398
independent individuals in condition (5) and then argued that the economy is a n d / o r should be dominated by mutualism (4), i.e. by trading and the formation of contractual relations. As A d a m Smith emphasised, very much in line with self-organisation theory, people have an inherent desire to "truck and barter" when allowed to do so. Competition (3) is assigned the third rank in the economist's priorities, as an interactive condition which prevents mutualist relations from developing into cross market predator (2)-prey (1). This last type of interaction appears to have the lowest rank in economics and is even considered by many orthodox economists to be outside the discipline of economics, belonging to political science or sociology, along with combat (6). It is viewed as an undesired interactive condition and, over the centuries since A d a m Smith rallied against Mercantilism, a set of legal institutions have evolved to retard the development of predator-prey and combative relations in the economy. If we abandon orthodox economic thinking and regard economic structures as dissipative and self-organisational in character, then the logistic equation immediately offers itself as a possible abstraction for use in understanding economic processes. This is true even though the self-organisation approach is likely to require significant adaptation for use in the economic domain, because the logistic equation is e m b e d d e d in the general notion of a dissipative structure (Kauffman [5] stresses that this connection has existed in all forms of self-organising systems right back to Turing's [6] seminal work on the formation of chemical compounds). It is capable of surviving as a general abstraction with the introduction of, for example, an informational, rather than an energetic, focus for the emergence of economic complexity and organisation (see [7,3,8]). Furthermore, economists interested in technological diffusion have already caught glimpses of the logistic equation in their studies. On examining Eq. (2), it is clear that mutualism and competition operate differently in economic systems. Mutualism involves practices such as trade and contracting where customs and institutional rules make such practices possible. Thus, since mutualism influences, K, directly, there is usually no need to enter a separately defined mutual M. Equally, it is inappropriate to specify competition in terms of a competitive M--relative prices and measures of comparative economic power are more appropriate (however in [9] competition is modelled between techniques using the same mathematics as found in the relevant chapter of [4]). Furthermore, even the initial K cannot be expected to remain constant in economic contexts simply because it is a repository for an array of mutualisms which ebb and flow for institutional, socio-political and environmental reasons. Thus, K, is a variable which has to become the object of empirical inquiry over any selected historical period, using retroduction, rather than a pre-specified magnitude. Furthermore, knowledge of the existence of the logistic equation itself may have to be taken into account (Foster [10] points out, in the context of technical innovation, that a rule of thumb in business is to switch technology at the inflection point o n the relevant logistic curve). Fortunately, there is now growing support for the view that the concept of a dissipative structure and the attendant process of self-organisation have wide applicability in economics [11]. Furthermore, many economists are already aware of the presence of logistical dynamics in certain processes. The logistic has been experimented with most extensively in the area of technological change to understand innovation and diffusion processes. However, studies of this type have tended to look for simple logistic patterns in historical data, rather than to use the profile as a general theoretical characterisation to help explain historical data which need
J. Foster~Mathematics and Computers in Simulation 39 (1995) 393-398
397
not, in fact, look logistical in shape over time. Indeed, even though there has b e e n some success in fitting the simple logistic equation in the case of technological diffusion, there has been little in the way of understanding concerning discontinuities. W h e n we deal with structural change in economics, we are in a position of disadvantage in relation to natural scientists in the sense that our capacity to control external boundaries in laboratory conditions is very limited. However, a great deal of ecological research, built upon the logistic equation, is c o n d u c t e d in field conditions quite similar to those faced by economists. Historical data, collected in non-laboratory conditions for other purposes, are used with attendant statistical problems. The great advantage that economists have is in the availability of a universal m e a s u r e of mass across disparate structures, namely, monetary valuation. This defines the disciplinary boundaries of economics as being concerned with structure which can be traded, lent or borrowed, otherwise no value can be assigned. Economics also deals with the value of time, in the form of money-valued flows. These can e m a n a t e both from valued structure and structure which has no explicit valuation, such as h u m a n capital. Structural change raises the value of the stock of economic structure and will be associated with increases in the use value of time to engage in new processes and to maintain enlarged structure. The use value of time is determined " o n the margin" of irreversible time and economic agents will choose the time use with the highest marginal return. The opportunity cost of time use can, of course, involve activities which have no valuation, such as 'leisure'. All we can do as economists is to infer a value on the margin when switches of time use occur. Such implicit valuations are, in effect, outside the bounds of economic analysis but are important because they offer measurable connections with the greater social and ecological environment in which significant fluctuation occurs. Because structure and process can often be valued in economics, economists are in a particularly favourable position to operationalise the self-organisation approach, utilising a variant of the logistical abstraction, provided suitable time series data exists (for a primitive example see [12]). Such an approach is " o p e n - e n d e d " in several respects: it allows for both nonlinear stability and instability; it can incorporate " o r t h o d o x " factors such as relative prices; socio-political factors have a key role to play; the institutional fabric surrounding an economic structure is vital and must be well u n d e r s t o o d by a researcher. Because the logistic equation is not a formal derivative of a system of deductive logic, but the product of empirical observation it provides an abstraction of endogenous, historical processes which does not preclude any factor of historical relevance, provided some measure of its influence can be discovered. In short, it can be a tool which aids economic explanation and, in many circumstances, a better understanding of future occurrences in economic systems.
References [1] T. Lawson, Abstraction, tendencies and stylised facts: a realist approach to economic analysis. Cambridge J. Econom. 13 (1989) 59-78. [2] H. Haken, Synergetics: An Introduction (Springer, Berlin, 1983). [3] K. Dopfer, Toward a theory of economic institutions: synergy and path dependency, J. Economic Issues 25(?) (1991) 535-550. [4] J.D. Murray, Mathematical Biology (Springer, Berlin, 1989).
398
J. Foster/Mathematics and Computers in Simulation 39 (1995) 393-398
[5] S.A. Kauffman, The evolution of economic webs, in: P.W. Anderson et al. eds., The Economy and an Evolving Complex System, SFI Studies in the Sciences of Complexity (Addison Wesley, Redwood City, 1988). [6] A.M. Turing, The chemical basis of morphogenesis, Royal Society B237 (1952) 37-72. [7] C. Dyke, From entropy to economy: a thorny path, Advances in Human Ecology 1 (1991) 149-176. [8] J. Foster, The thermodynamic approach to economic analysis: an assessment, Der Offentliche Sektor, 18(2/3) (1992) 12-26. [9] B. Amable, Competition among techniques in the presence of increasing returns to scale, J. Evolutionary Econom. 2 (1992) 147-58. [10] R. Foster, Innovation (Summit Books, New York, 1986). [11] J. Foster, Economics and the self-organisation approach: Alfred Marshall revisited? The Econom. J. 103 (1993) 975-991. [12] J. Foster, The evolutionary macroeconomic approach to econometric modelling: a comparison of Sterling and Australian Dollar M3 systems determination, Papers on Economics and Evolution No. 9301, edited by the European Study Group for Evolutionary Economics (1993) 31.
ELSEVIER
MATHEMATICS AND COMPUTERS N SIMULATION
Mathematics and Computers in Simulation 39 (1995)393-398
The impact of the self organisation approach on economic science: why economic theory and history need no longer be mutually exclusive domains John Foster Department of Economics, The University of Queensland, Queensland 4072, Australia
1. Introduction The self-organisation approach has grown in popularity in the natural sciences in recent years. The purpose of this paper is to suggest how the approach might be applied, empirically, in economics.
2. Economics and self organisation In orthodox economics, the prevailing methodology is to construct a timeless theory from which certain propositions are deduced then translated, through the addition of an auxiliary hypothesis, into a specification which can address historical data. However, it is clear that theories which involve timeless deductive logic have no place in a self-organisation context. In practice, natural scientists investigating selfoorganising structures formulate theories in an inductive way, much in the m a n n e r suggested by Lawson [1] who prefers the term "retroduction". Such abstraction from structural change is of fundamental importance in understanding how self-organising systems develop and survive. A detailed understanding of the microscopic structure of a system cannot reveal macroscopic features which are critically important. Thus, proponents of self-organisation have focussed attention upon macroscopic measures such as, for example, entropy. The operation of the entropy law and the parallel existence of dissipative structures, with evolutionary and irreversible features are taken as axiomatic in nature. However, the axioms in question are not clothed with extensive mathematical deduction, facilitated by the introduction of convenient, but unrealistic, assumptions to address real processes. Instead, the reverse is the case, the axioms concerning processes are the outcome of extensive empirical study, and, thus, mathematical deduction is unnecessary, even though the theory of process favoured may well be specified in mathematical terms. 0378-4754/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 3 7 8 - 4 7 5 4 ( 9 4 ) 0 0 0 8 9 - 3
394
J. Foster~Mathematics and Computers in Simulation 39 (1995) 393-398
The specification of an evolutionary process in mathematical terms enables particular specifications to be estimated in particular cases, permitting a degree of ceteris paribus prediction concerning the developmental possibilities of a system in the near future and, perhaps more importantly, the onset of discontinuities can be given precise parametric representation. It is in the context of discontinuities that mathematics becomes an important tool. In physics, Haken [2] has devised mathematical techniques to deal with discontinuity in "synergetic" systems (see [3]) for an assessment of the economic applicability of these techniques) and in biology similar techniques have been devised.
3. The logistic equation If we examine some spatially-defined dissipative structure, with its external boundaries fixed, with time-related increases in complexity and organisation, what we typically observe in fields as disparate as population ecology and reaction kinetics is a density-dependent logistic path of mass dynamics over time:
d N J d t = rN,[1 - ( N , / K ) ] .
(1)
Where K is the carrying capacity, or boundary, of N development and r is the net birth rate, diffusion rate, etc. This represents a very simple, but useful, theory of process which provides the analytical foundation u p o n which the modelling of particular historical processes can proceed. Despite its simplicity, it can generate very complex nonlinear dynamics in certain circumstances. It conforms to the requirements of a theory as stated by Lawson [1] since it is the product of extensive retroduction in the natural sciences. Whatever the precise form of the logistic, its sigmoid form captures the general character of the density-dependent dynamics which self-organising dissipative structures must exhibit as an axiomatic necessity. The smoothness of the logistic is an assumption which is added to the self-organisation axioms. In particular cases it will not be smooth and the purpose of introducing additional exogenous variables into a logistical equation is to relax this assumption and, in so doing, distil estimates of the underlying logistic parameters. However, smoothness is also dictated by endogenous considerations - it is conditional upon certain parametric configurations which, if exceeded in certain directions, yield discontinuities of the type which we observe in actual historical experience. W h e n a discontinuity occurs, structural change loses its smooth character and there occurs fusion into a larger unit or disintegration, after which the subject/environment relationship must be qualified or redefined. Here, mathematics becomes an important tool for the biologist or chemist to ascertain, from empirically-derived logistic parameters, the stability range of the process in question and any steady states which can exist. However, a limitation of the simple logistic in this regard is that it has to be specified in terms of discrete time intervals between events to generate discontinuities. Many macroscopic mass measures can alter continuously through time yet do not display smooth logistic paths towards steady states. Thus, although the simple logistic equation captures a general endogenous tendency in self-organising systems, it is strictly limited in its empirical application.
J. Foster~Mathematics and Computers in Simulation 39 (1995) 393-398
395
Although Eq. (1) can offer a useful representation of, for example, certain kinds of chemical kinetics and even though it can be regarded as the embodiment of a general theory of a historical process, evolutionary biologists, in particular, view it as special case. A more general specification is as follows
d N J d t = rNt[1 - ( N , / K ) + a( M t / K ) ] .
(2)
Where M is an interacting population. An identical companion equation can be specified for M in terms of N. Eq. 2 is one example of a class of interacting population equations which have been derived from empirical research. It is a two party (Kolmogorov) version of the L o t k a Volterra logistic specified in Eq. (1). The resultant model is extremely simple, but very useful since most biologists agree that it can capture many of the properties of more complicated models. This is particularly the case with respect to discontinuities in otherwise stable-looking developmental trajectories. Murray [4], for example, considers three cases: predator-prey, mutualist and competitive interactions. The latter two can be described in terms of Eq. (2) with mutualism being the case where a > 0 and competition where a < 0. In each case, he examines the parametric ranges within which stable co-evolution can prevail and the critical parametric combinations where stability breaks down into discontinuity (the 'bifurcation point' in the language of chaos). In the case of competition, the "principle of competitive exclusion" tends to operate over wide parameter ranges in population ecology. In other words, competition tends not to be a stable co-evolutionary state. Mutualism, on the other hand, is very stable while predator-prey interaction tends to generate oscillations. There are, in fact, six interactive conditions which, at any point in time, an organism can be in: (1) Prey, (2) Predator, (3) Competitor, (4) Mutualist, (5) Neutral, (6) Combatant. Eq. (1) is essentially condition (5) and, as we have noted, a rather special case. The constant need of a dissipative structure for free energy supplies ensures that neutrality is an ephemeral condition bolstered by the existence of stocks and other homoeostatic mechanisms. Many supposedly neutral conditions are due to unstated mutualist associations (eg. siblings and parents). Thus, as has been noted biologists rarely use Eq. (1) in empirical investigation, except in demonstrable special circumstances (a single predator in an isolated environment with massive, fast replenishing stocks of prey). Thus, population ecologists have focussed upon conditions (1) to (4).
4. The logistic equation in economics
In the history of economic thought, the priority ordering of interactive states which we observe in biology seems to have been reversed. Economists have tended to begin with abstract
396
3". Foster/Mathematics and Computers in Simulation 39 (1995) 393-398
independent individuals in condition (5) and then argued that the economy is a n d / o r should be dominated by mutualism (4), i.e. by trading and the formation of contractual relations. As A d a m Smith emphasised, very much in line with self-organisation theory, people have an inherent desire to "truck and barter" when allowed to do so. Competition (3) is assigned the third rank in the economist's priorities, as an interactive condition which prevents mutualist relations from developing into cross market predator (2)-prey (1). This last type of interaction appears to have the lowest rank in economics and is even considered by many orthodox economists to be outside the discipline of economics, belonging to political science or sociology, along with combat (6). It is viewed as an undesired interactive condition and, over the centuries since A d a m Smith rallied against Mercantilism, a set of legal institutions have evolved to retard the development of predator-prey and combative relations in the economy. If we abandon orthodox economic thinking and regard economic structures as dissipative and self-organisational in character, then the logistic equation immediately offers itself as a possible abstraction for use in understanding economic processes. This is true even though the self-organisation approach is likely to require significant adaptation for use in the economic domain, because the logistic equation is e m b e d d e d in the general notion of a dissipative structure (Kauffman [5] stresses that this connection has existed in all forms of self-organising systems right back to Turing's [6] seminal work on the formation of chemical compounds). It is capable of surviving as a general abstraction with the introduction of, for example, an informational, rather than an energetic, focus for the emergence of economic complexity and organisation (see [7,3,8]). Furthermore, economists interested in technological diffusion have already caught glimpses of the logistic equation in their studies. On examining Eq. (2), it is clear that mutualism and competition operate differently in economic systems. Mutualism involves practices such as trade and contracting where customs and institutional rules make such practices possible. Thus, since mutualism influences, K, directly, there is usually no need to enter a separately defined mutual M. Equally, it is inappropriate to specify competition in terms of a competitive M--relative prices and measures of comparative economic power are more appropriate (however in [9] competition is modelled between techniques using the same mathematics as found in the relevant chapter of [4]). Furthermore, even the initial K cannot be expected to remain constant in economic contexts simply because it is a repository for an array of mutualisms which ebb and flow for institutional, socio-political and environmental reasons. Thus, K, is a variable which has to become the object of empirical inquiry over any selected historical period, using retroduction, rather than a pre-specified magnitude. Furthermore, knowledge of the existence of the logistic equation itself may have to be taken into account (Foster [10] points out, in the context of technical innovation, that a rule of thumb in business is to switch technology at the inflection point o n the relevant logistic curve). Fortunately, there is now growing support for the view that the concept of a dissipative structure and the attendant process of self-organisation have wide applicability in economics [11]. Furthermore, many economists are already aware of the presence of logistical dynamics in certain processes. The logistic has been experimented with most extensively in the area of technological change to understand innovation and diffusion processes. However, studies of this type have tended to look for simple logistic patterns in historical data, rather than to use the profile as a general theoretical characterisation to help explain historical data which need
J. Foster~Mathematics and Computers in Simulation 39 (1995) 393-398
397
not, in fact, look logistical in shape over time. Indeed, even though there has b e e n some success in fitting the simple logistic equation in the case of technological diffusion, there has been little in the way of understanding concerning discontinuities. W h e n we deal with structural change in economics, we are in a position of disadvantage in relation to natural scientists in the sense that our capacity to control external boundaries in laboratory conditions is very limited. However, a great deal of ecological research, built upon the logistic equation, is c o n d u c t e d in field conditions quite similar to those faced by economists. Historical data, collected in non-laboratory conditions for other purposes, are used with attendant statistical problems. The great advantage that economists have is in the availability of a universal m e a s u r e of mass across disparate structures, namely, monetary valuation. This defines the disciplinary boundaries of economics as being concerned with structure which can be traded, lent or borrowed, otherwise no value can be assigned. Economics also deals with the value of time, in the form of money-valued flows. These can e m a n a t e both from valued structure and structure which has no explicit valuation, such as h u m a n capital. Structural change raises the value of the stock of economic structure and will be associated with increases in the use value of time to engage in new processes and to maintain enlarged structure. The use value of time is determined " o n the margin" of irreversible time and economic agents will choose the time use with the highest marginal return. The opportunity cost of time use can, of course, involve activities which have no valuation, such as 'leisure'. All we can do as economists is to infer a value on the margin when switches of time use occur. Such implicit valuations are, in effect, outside the bounds of economic analysis but are important because they offer measurable connections with the greater social and ecological environment in which significant fluctuation occurs. Because structure and process can often be valued in economics, economists are in a particularly favourable position to operationalise the self-organisation approach, utilising a variant of the logistical abstraction, provided suitable time series data exists (for a primitive example see [12]). Such an approach is " o p e n - e n d e d " in several respects: it allows for both nonlinear stability and instability; it can incorporate " o r t h o d o x " factors such as relative prices; socio-political factors have a key role to play; the institutional fabric surrounding an economic structure is vital and must be well u n d e r s t o o d by a researcher. Because the logistic equation is not a formal derivative of a system of deductive logic, but the product of empirical observation it provides an abstraction of endogenous, historical processes which does not preclude any factor of historical relevance, provided some measure of its influence can be discovered. In short, it can be a tool which aids economic explanation and, in many circumstances, a better understanding of future occurrences in economic systems.
References [1] T. Lawson, Abstraction, tendencies and stylised facts: a realist approach to economic analysis. Cambridge J. Econom. 13 (1989) 59-78. [2] H. Haken, Synergetics: An Introduction (Springer, Berlin, 1983). [3] K. Dopfer, Toward a theory of economic institutions: synergy and path dependency, J. Economic Issues 25(?) (1991) 535-550. [4] J.D. Murray, Mathematical Biology (Springer, Berlin, 1989).
398
J. Foster/Mathematics and Computers in Simulation 39 (1995) 393-398
[5] S.A. Kauffman, The evolution of economic webs, in: P.W. Anderson et al. eds., The Economy and an Evolving Complex System, SFI Studies in the Sciences of Complexity (Addison Wesley, Redwood City, 1988). [6] A.M. Turing, The chemical basis of morphogenesis, Royal Society B237 (1952) 37-72. [7] C. Dyke, From entropy to economy: a thorny path, Advances in Human Ecology 1 (1991) 149-176. [8] J. Foster, The thermodynamic approach to economic analysis: an assessment, Der Offentliche Sektor, 18(2/3) (1992) 12-26. [9] B. Amable, Competition among techniques in the presence of increasing returns to scale, J. Evolutionary Econom. 2 (1992) 147-58. [10] R. Foster, Innovation (Summit Books, New York, 1986). [11] J. Foster, Economics and the self-organisation approach: Alfred Marshall revisited? The Econom. J. 103 (1993) 975-991. [12] J. Foster, The evolutionary macroeconomic approach to econometric modelling: a comparison of Sterling and Australian Dollar M3 systems determination, Papers on Economics and Evolution No. 9301, edited by the European Study Group for Evolutionary Economics (1993) 31.