John Kemp (1999)

Journal of Artificial Societies and Social Simulation vol. 2, no. 3, <https://www.jasss.org/2/3/1.html>

Abstract

This paper presents a dynamic model of consumer choice incorporating consumption externalities. The model is deliberately minimalist and symmetric, so that there are no endogenous or exogenous factors causing consumers in aggregate to favour one particular commodity rather than another. Yet, the results of simulations show that remarkable switches and reswitches in patterns of demand can arise spontaneously and in ways that are emergent and unpredictable. The model lends force to the view that, changes in economic and social activity can occur even in the absence of any catalyst of change.

Keywords:

Consumption externalities, Dynamic choice

Introduction

1.1

Conventional utility maximising theories of consumer behaviour have not found it easy to handle shifting preferences or fluctuating patterns of demand. Leibenstein (1950, 1976) attempted to introduce externalities in consumption by incorporating what he termed 'bandwagon, snob and Veblen effects', and Lancaster (1966, 1971) has striven to develop an approach capable of incorporating, among other things, the introduction of new products. Less formally, the writings of Galbraith (e.g. 1952, 1967) are replete with discussion of how firms might be able to manipulate preferences through advertising, product differentiation and product innovation.

1.2

More recently, there has emerged a non-mainstream literature on endogenous preferences. Some writers see preferences as influenced by habit formation and/or interdependencies and have sought to demonstrate the conditions for convergent or stable choice. Pollak (1978) provides a brief survey, and also an antidote to the view of Stigler and Becker (1977) that, phenomena which could be construed as inconsistent with the stability of tastes can be reconciled with it. The literature of price dependent preferences is also reviewed by Pollak (1977). From a supply perspective, Pashigian, Bowen and Gould (1995) in studying seasonal variations in prices find that, changes in fashion are related to producers' costs of making style changes. Granovetter and Soong ( 1986) have shown how the interaction of bandwagon and reverse bandwagon effects can lead to the emergence of fluctuating patterns of demand for a commodity. They do this by assuming two thresholds. The lower one represents the percentage of other agents that must make a purchase before a given agent will make a purchase, and the upper threshold represents the percentage above which an agent will no longer make a purchase. Kirman (1993) presents a model of market switching which stems from the empirical observation that foraging ants switch from one food source to another in a way that has often been seen as puzzling, but which is similar to some observed patterns of consumer behaviour.

1.3

Still, other approaches have sought to explain fads and fashions as the result of information cascades (Bikhchandani, Hirshleifer and Welch 1992) or herd behaviour (Scharfstein and Stein 1990). However, in many instances it is not simply the emergence or collapse of a fad which requires to be explained, but its subsequent re-emergence. It is this aspect that is addressed in this paper, which is concerned with fluctuating, cyclical or alternating choices between commodities.

1.4

Despite the foregoing, the dominant paradigm within economics has remained that of orthodox neoclassical consumer theory with its essentially static equilibrium approach, as presented in standard texts (e.g. Gravelle and Rees 1992, Varian 1992 and Nicholson 1998). Even in works dealing with more recent developments in rational choice theory, little attention has been paid to changing tastes (e.g. Hargreaves Heap et al 1992). The predictive power of this approach relies upon the formalistic rigour of its axiomatic basis. A fundamental assumption is that of transitivity of preferences. If changes in taste were admitted then the transitivity assumption could seem as if violated. The theory would then become capable of explaining everything and nothing at the same time. Only by maintaining the constancy of preferences can falsifiable predictions be derived. If changes have to be considered they have to be treated as exogenous, once and for all in their effects and analysed by the comparison of static equilibria.

1.5

The main difficulty in the economic approach is that the central concern of conventional equilibrium consumer theory is to determine the conditions for efficient resource allocation at a given point in time, with a given set of resources, and with a given set of tastes and preferences, rather than to analyse continuously changing patterns of consumption over time. Yet, where there are externalities in consumption the decision to consume is essentially dynamic in nature. Current choices are affected by past choices and then have a continuing effect upon future choices. In such situations an act of choice is better construed as a sequential action. Consequently, equilibrating mechanisms are less likely to be appropriate concepts, and so it is better to attempt to confront the dynamic aspects of consumption head on.^[1]

1.6

There is an important alternative behavioural approach to the understanding of consumer behaviour which stems from social psychology, and which has an extensive literature. A number of contributions from this field are surveyed in Janssen and Jager (1999) who incorporate them to simulate lock-in. In this tradition, behaviour is not seen as resulting from the actions of rational economic man, but as stemming from factors such as needs (Maslow 1954), social learning and imitation (Bandura 1986) and social comparison (Festinger 1954, Moscovici and Faucheux 1972). Similarly, in this tradition Nowak and Latané (1994) incorporate social influence and social impact theory to simulate the emergence of social order from individual behaviour.

1.7

Despite all the work from these alternative traditions, there is still much about the fickleness of human behaviour that is not easily explainable. It is often seen as non-linear and unpredictable. In this paper no attempt is made to model or explain shifts in consumption patterns in terms of identifying distinct causal factors. A very simple and minimalist model is presented which generates behaviour with marked qualitative changes which cannot be attributed to any major underlying cause. The implication of this is that should patterns of consumer behaviour arise which are not capable of being modelled or predicted by existing theories, then it may be that the changes have resulted from the unforeseeable interactions which arise from the micro dynamics. That is it might be that " it jes' growed" as quoted by Liebowitz and Margolis (1990, p.23)

1.8

Shifting patterns of demand are, however, an obvious fact of life. Clearly, orthodox approaches have limitations for the analysis of situations where patterns of demand are in constant flux. This is obviously the case in fashion goods, such as clothing, where one style takes off for a time, but is then caught up and surpassed by some other, only to re-emerge later in an apparent wave like manner. It is also the case in less overtly fashion goods, where the decision to purchase a particular type of commodity is influenced by that commodity's current popularity (e.g. Apple Mac versus the PC), and also in situations where new products emerge to provoke a revision of preferences, gain sales and later settle back (e.g. skateboards).

1.9

This paper provides a framework for the analysis of pure consumption decisions where consumption externalities are important. The approach taken is a minimalist one and shows how a variety of fluctuating, sometimes cyclical, sometimes irregular patterns of demand can arise spontaneously. This is the case, even though there is nothing within the structure of the model, nor any other reason, to favour one commodity over another. The patterns of demand generated are emergent, self-organising, indeterminate and unpredictable, but have a relevance for understanding the formation of real patterns of consumption.

1.10

However, the central purpose of this paper is not to model or explain the demand patterns for any particular commodity. Many of the studies cited earlier have attempted to do this and to identify those factors which have a major influence upon commodity selection. The current purpose is to try to see whether marked changes in demand patterns could arise even in the absence of any of those factors, and when individuals' preference functions are stable. If in some instances it is not possible to attribute changes to any identifiable cause in terms of changes in relative prices, costs, incomes or other variables which are usually seen as important determinants of demand, the implication is that it might then become impossible to identify a conventional commodity demand function, despite knowledge of all the variables which underpin it.

An approach to choice

2.1

Recently there have been new developments in economics and the social sciences which may assist us in our task. There has been an increased interest in the dynamics of economic change. This has partly been a result of progress in the natural sciences with the recognition of the importance of non-linearity, the development of chaos theory and the study of complex systems. In economics, this has meant that greater attention is being given to the view that economic activity should be seen as a developmental or evolutionary process, rather than as the predictable equilibrating behaviour of closed systems. See for example the collections: Anderson, Arrow and Pines (1988), Arthur, Durlauf and Lane (1997), Arthur (1994a) and the non-technical discussions: Waldrop (1992), Ormerod (1995, 1998). A series of essays which embrace a broader social science perspective is to be found in Kiel and Elliott (1996). The importance of all this for economics and social science in general cannot be overstated, for it carries with it an implication of path dependency. That is, a contemporary situation can have arisen, not as the outcome of some equilibrating mechanism, but from the cumulation of a series of chance, independent and perhaps individually insignificant historical events. The present is dependent upon the paths of history in a self-reinforcing way. Thus, what is may well be something that need not, and in other circumstances would not, have inevitably been. Furthermore, recent examples from the economics literature suggest that, such path dependent processes do not always ensure that what transpires in the long run need be in any sense optimal.

2.2

Economics research has been concerned largely with technology selection, that is in the factors which determine why a particular technology should have become dominant over time rather than some other equally viable but non-adopted one. Traditionally, economic analysis has focused upon the ability of markets to ensure the long run adoption of the most economically efficient technology. However, sufficient examples exist where lock-in to an apparently inferior technology has occurred, that it is possible to question whether markets work efficiently to produce optimal outcomes. This has attracted the attentions of both economists and historians of economics. So, there has been a growth of case studies which have examined the historical processes that have led to lock-in to particular technologies (e.g. David 1975, David and Bunn 1988 , Cowan 1990a, 1990b, Cowan and Gunby 1996). A similar methodology is utilised by Kemp and Wilson (1999) to explore lock-in to a socio-economic regime. Alternatively a simulation approach has been adopted by Janssen and Jager (1999) to identify two types of lock-in: spatial and global. Contemporaneously, there has been an attempt to construct a theoretical structure which formalises these processes, and which facilitates greater understanding through the provision of a more rigorous analytical framework (Arthur 1989). That is, there have been attempts to discern why something has come to be, when that something need not inevitably have been. This interest has been given further impetus by recent attempts to recast economics within a new paradigm, which sees economic systems as dynamic, changing and evolutionary (e.g. Hodgson 1993, Nelson 1995) or as self-organising (Krugman 1996) rather than equilibrating.

2.3

In this essay it is contended that the methodology espoused in this literature is equally applicable to the analysis of consumer demand where consumption externalities are important. Consumption patterns often display apparent lock-in. Sometimes, unlike a technology (e.g. QWERTY), this is impermanent and there may be swings back and forth between alternatives.

2.4

Arthur (1988, 1989) proposed a model to explain the selection of competing technologies. However, his approach can readily be adapted to other situations involving dynamic choice. Ostensibly, his model was developed to explain the puzzling phenomenon of why a particular technology could become dominant over time, even though that technology might be technologically inferior to some alternative but non-adopted one. Well known and often cited examples are: (i) the almost universal adoption of the QWERTY keyboard, for it has often been claimed that this layout was initially adopted in order to minimise the possibility of typewriter keys jamming, and that it is notoriously inefficient for speed typing compared to the alternative Dvorak layout (see: David 1975 and Gould 1992 pp.59-75), (ii) the domination of the VHS video recording system over the technologically superior Betamax (Arthur 1990) and (iii) the inability of the acclaimed Apple Mac computer to stave off falling market share in a market dominated by the IBM PC and its clones. Further examples of technology adoption are discussed in Freeman and Soete (1997).

2.5

The reasons for such dominance are usually ascribed to the presence of positive feedback which creates increasing returns to adoption. The many sources of increasing returns to a technology are detailed in Arthur (1988), and additional sources relevant to the adoption of socio-economic regimes are considered in Kemp and Wilson (1999). For example, increasing returns can arise from the presence of learning effects or from the existence of network externalities. Thus in the case of QWERTY, generations of typists have trained and become familiar with a keyboard layout, so that an environment has been created which now militates against change. In the case of video recorders, the more rapid growth of pre-recorded material on VHS tapes for sale and hire led to the ultimate decline and demise of the Betamax system. However, some writers are highly sceptical. In particular Liebowitz and Margolis (1990, 1994) have cast considerable doubt upon the claimed superiority of the Dvorak keyboard and other examples of non-adopted technologies.

2.6

Though Arthur's model was created to explain the selection of competing technologies, it can be thought of as not so much about technology, but more about the general problem of choice by rational agents in a dynamic context. It can readily be adapted to the decision to consume, and to other aspects of choice which have a social rather than an economic dimension. As a prelude to a model of consumption the essential features of Arthur's approach are reviewed. In the subsequent section this is adapted to provide a dynamic model of consumer choice. Some concluding comments are provided in the final section.

2.7

It is assumed that there are n agents of two types J and K, each of whom chooses one unit of either of two technologies A and B. The n agents enter the market sequentially and randomly with equal probabilities for each type. Thus the probability of a given type of agent entering is p_j = p_k = 1/2. Other things equal, J agents have a constant predisposition to favour technology A, and K agents to favour technology B. However, as the number of adoptions of a given technology grows, so the attractiveness of that technology changes. The structure of the model is set out in figure 1, where the linear functions entered in the cells represent the returns to each type of agent from adopting a particular technology as the number of adoptions of that technology increases.

Figure 1: A linear model

The term a_j represents the autonomous return to a J agent from selecting a unit of technology A independent of the numbers of adoptions, i.e. it is the constant predisposition to select A. Likewise b_j is the autonomous return from a unit of B. Similarly, a_k and b_k are the returns to K agents. Since, independent of the number of adoptions, J agents obtain a higher return from adopting technology A, and K agents from B, we have a_j > b_j and a_k < b_k. The total numbers of adoptions of technologies A and B are represented by Na and Nb respectively. As the number of adoptions increase, the returns to a J agent from technologies A and B are enhanced by the amounts j.Na and j.Nb respectively, where j is a constant of proportionality, which may be positive, negative or zero. Similarly k.Na and k.Nb are the enhancements to K agents' returns as the numbers of adoptions of the respective technologies increase. Choice is therefore a continuous dynamic process with the n agents making sequential adoptions of a technology. Arthur considers the adoption process under constant, diminishing and increasing returns.

1. Constant returns j = k = 0:

   With j = k = 0, Na and Nb play no part in decisions. It is obvious that J agents will always adopt A and that K agents will adopt B irrespective of Na and Nb (since a_j > b_j and a_k < b_k ) . The adoption process is such that, the difference between the numbers of adoptions (d = Na - Nb) is a simple random walk, with the shares Na / (Na +Nb) and Nb / (Na + Nb) both tending to 1/2. The adoption profile is entirely dependent upon the random entry of the two types of agents, but the long run shares are predictable. The process is ergodic (i.e. it is not path dependent) since the pattern of previous adoptions has no influence upon future adoptions. Re-running the simulation will always generate a different adoption profile, but it will be one that conforms to these same general characteristics. Thus, different simulations will be qualitatively similar.

2. Diminishing returns j = k < 0:

   With negative j and k, an increase in the number of adoptions of a given technology will cause the returns to diminish, and at some stage fall below the level of returns available from the alternative technology. When this occurs, agents will switch their choice of technology. As the alternative technology becomes increasingly adopted, so its returns will likewise fall and ultimately provoke re-switching. The process is now a simple random walk with reflecting barriers. Again the process is predictable and ergodic. The long run shares of A and B again tend to 1/2. The adoption profile is confined between the reflecting barriers (b_j - a_j)/j and (b_k - a_k)/k, with d the difference in adoptions never able to exceed or fall below these bounds.

3. Increasing returns 0 < j = k:

   With positive j and k there is a distinct and qualitative change in the dynamics of the adoption process. The process is illustrated in the computer simulation of figure 2 where the vertical axis represents the difference between the numbers of adoptions (d = Na - Nb) and the horizontal axis the total number of adoptions (N = Na + Nb). For the purposes of this simulation, the values of the parameters were set at: a_j = 5 , b_j =4, j = 0.1 a_k =4, b_k = 5, k = 0.1. Thus, with positive j and k, as the adoptions of a given technology increase, that technology becomes inexorably more attractive. If one technology, for example A, through chance happens to get sufficiently far in front, then that technology will become more attractive not only to J agents, but also to K agents despite their predisposition to select B. Thus J agents continue to select A, whereas K agents permanently switch from B to A. No further selections of B are made and the process becomes locked-in to technology A.

Figure 2: Increasing Returns 0 < j = k

Under increasing returns the process is a random walk with absorbing barriers given by (b_j - a_j)/j and (b_k - a_k)/k. When a technology receives sufficient adoptions so that the difference in the number of adoptions reaches an absorbing barrier, thereafter that technology is always adopted. The alternative technology becomes extinct. The final outcome is now unpredictable and non-ergodic. It is predictable that one or other of the technologies will ultimately dominate with a share of 100%, but which one that should be is completely unpredictable. Neither can it be known how many adoptions have to be made before lock-in occurs. All is dependent upon the path of previous adoptions, and that is due purely to the random, and unknowable, order in which J and K agents enter. An important implication is that the dominant technology need not necessarily be the one that would have been most efficient in the long run, given equal rates of adoption (e.g. VHS and QWERTY). Thus, in the presence of increasing returns, selection mechanisms are claimed to be inefficient. Adopted technologies are dependent upon the accidents of history, the counterfactual is of relevance, and so it becomes legitimate to ask the question "what if ?" since the situation that exists is one that need not, and in other circumstances might not have been.

An approach to consumption

3.1

It should be clear that there is nothing especially technological about Arthur's model, and that it can equally be applied to the analysis of consumer and or social choice. All that is required is to redefine agents of types J and K as consumers, and technologies A and B as commodities. However, many consumption patterns are not captured by the linear forms of the previous section.

3.2

Real patterns of commodity selection often display greater variety and diversity of form with greater unpredictability. Swarming is a pervasive feature in the sales patterns of many goods. There are swings and roundabouts of fashion, as one fad gives way to another only to re-emerge at a future date. An example can be seen in the case of trouser styles where, over time, there have been swings back and forth between tight and flared varieties. In other instances the demand for a good can collapse only to reappear later. Thus, most men now go hatless, though this was once uncommon, and similarly successive generations discover, tire of and then rediscover the yo-yo. Individuals differ in their psychological propensities, for some may buck a trend whilst others swim with it. In markets where they do so, externalities in consumption form an argument within individuals' utility functions. In such situations it is helpful to conceptualise consumption as a dynamic process. Consumers enter a market sequentially to purchase a good influenced in part by the prior purchases of others. Indeed for many goods, and not just high fashion goods, this may be a more appropriate characterisation of the consumption process than that of the conventional equilibrium model of economists in which all consumption decisions are made simultaneously.

3.3

As more consumers purchase a commodity, for some, it can become fashionable and increasingly desirable. After further purchases it may become commonplace and lose attractiveness with consumers switching to some alternative. However, individuals are influenced in different ways. Society comprises both extroverts and introverts with some consumers gaining utility from being 'in with the crowd', whilst others wish to stand apart, for there are both leaders and followers of fashion. The dynamics of opinion diffusion are discussed in Shiller (1989 Chs. 1 and 2). Whatever the precise nature of utility functions, it is clear that for many goods, the utilities derived are not independent of the number of previous sales, and that the utility functions are not linear in numbers of selections. A more general formulation is appropriate, as shown in figure 3, where the cell entries now represent ordinal utility functions.

Figure 3: A general model

3.4

The assumptions that underlie this adapted model are: There are n utility maximising consumers of types J and K, each of which selects a unit either of commodity A or of B. The consumers enter the market randomly and sequentially where the probability of a given type entering is p_j = p_k =1/2. For simplicity, it is assumed that commodities have constant and identical prices, and so relative prices play no part in a consumer's decision ^[2]. Consumers have ordinal utility schedules defined over a single unit of A and a single unit of B. This is because it is assumed that individuals only select one unit of A or one unit of B at the moment of choice. This is not a heroic assumption, for there are many commodities where only one unit can be consumed at a point in time. One can have either a Chinese or an Indian meal at a restaurant, visit either 'United' or 'City' on a Saturday afternoon, but never both or more than one at the same time. In such situations the concept of marginal utility is less directly relevant for the analysis of an individual choice. Thus, the dependent variable U_j^A is the utility derived from one unit of A, as a function of the numbers (N_A) of adoptions of A.

3.5

Numbers of prior selections apart, J consumers have a preference for A, and K consumers for B (i.e. in terms of autonomous utilities: a_j>b_j and a_k<b_k ). In deciding between A and B, a J consumer will compare U_j^A with U_j^B and select whichever is the greater. At the moment of choice the autonomous utilities a_j and b_j are enhanced by amounts which are functionally dependent upon the numbers of prior selections of A and B (i.e. by j(Na) and j(Nb) respectively). In a similar manner K consumers compare U_k^A with U_k^B and select a unit of A or B accordingly. Thus the situation is formally equivalent to Arthur's model of figure 1, with the exception that it is not restricted to linear forms. An illustrative algorithm of this selection process is provided in the appendix.

3.6

Many natural phenomena from electromagnetics to hydrodynamics are cyclical or wavelike and are described by one, or a combination of, the trigonometric functions. Assume that individuals' attitudes to certain commodities are, also, wave like in form, being dependent upon the numbers of prior purchases. For our immediate purpose, it neither matters which particular functions we adopt, nor whether they are realistic approximations of attitude formation. This is because the intention is to abstract from all those specific factors from economics and social science which might be thought of as influencing changes in behaviour. If having so done we observe marked change then it might just be that there are other influences at work which arise out of the micro dynamics of the system. So, employing Occam's razor, we adopt a simple sine function for consumers' and commodities. It is not contended that a sine function is a realistic proxy for individuals' utility functions or that it could apply to all commodities. The main requirement is that the utility function be non-linear in N and have a wave form, so to reflect the varying strength of preferences with numbers of adoptions. In any case, such a function appears to be no less philosophically objectionable, than to posit, for example, the principle of diminishing marginal utility. It is clearly possible to think of commodities where a single unit becomes more or less attractive dependent upon the numbers of prior adoptions. It is rather like Arthur's (1994b) El Farol problem where the attractiveness of visiting a particular bar increases with the numbers of other visitors but declines should it become overcrowded, only to regain its attraction as clients leave. The sine function captures this property.

3.7

Thus in figure 3, let j(Na)=j.sin(Na), j(Nb)=j.sin(Nb), k(Na)=k.sin(Na) and k(Nb)=k.sin(Nb), so the utility functions become:

U_j^A = a_j + j.sin(Na)
U_j^B = b_j + j.sin(Nb)

U_k^A = a_k + k.sin(Na)
U_k^B = b_k + k.sin(Nb)

Consider the selection process. If, |j| < (a_j-b_j)/2 and |k| < (b_k-a_k)/2, then it will always be the case that U_j^A > U_j^B and U_k^A < U_k^B . Thus, J agents will always choose A, and K agents will always choose B. This gives a selection profile similar to the case of constant returns discussed in the previous section. It is the familiar simple random walk, dependent solely upon the entry order of J and K agents. As we saw in the previous section, the process is ergodic with predictable shares tending to 1/2.

3.8

However, for greater values of |j| and |k|, quite remarkable and unpredictable patterns develop with agents switching and re-switching commodities. To see how this occurs it is helpful to look, firstly, at the position which would face J consumers in isolation.

Figure 4: Utility functions of type J consumers

Consider figure 4. Since a_j>b_j and sin(0) = 0, the initial J consumer would select A and thus cause j.sin(Na) to increase. Successive J consumers would then select A, and there would be a movement along the upper curve until point c was reached. At that point any further selections of A would yield U_j^A < b_j. Consequently subsequent J consumers would then switch to B and there would be a movement along the lower curve until point d was reached. At that point there would be alternate selections of A and B until points e and f were arrived at. Once point f was arrived at there could be no further selections of B, since any movement to the right around the lower curve would yield, for J, a level of utility, U_j^B , which was below the minimum attainable from A. This characterises what would be the case if there were an absence of K consumers. J consumers would become permanently locked-in to A. However, K consumers do exist, and since for them a_k < b_k, they will initially make selections of B. Observe that Nb represents the total selections of B by both J and K consumers. Thus, irrespective of what J consumers are or are not doing, Nb and j.sin(Nb) will change. Therefore successive J consumers will experience a movement to the right beyond f, but this will be as a consequence of the actions, not of J, but of K consumers. Once the actions of K consumers cause Nb to grow until point g is reached, switching and re-switching by J consumers will re-commence and then continue indefinitely. For K consumers the precise analogue applies. Thus U_j^B and U_k^B are interdependent, as are U_j^A and U_k^A. There will be continuous switching and re-switching between A and B by both types of consumers, and no lock-in by any consumer type to any commodity type.

3.9

It might be helpful to stress that the functions in figure 4 should not be interpreted as depicting how a given individual's utility continuously varies with their possession of greater or lesser amounts of commodities. An individual at a point in time only purchases and possesses one unit of a commodity. For an individual, at the moment of choice, there will be finite and constant single values of Na and Nb. Thus that individual will select either one unit of A or one unit of B by comparing the two values of U_j^A and U_j^B. Having made that selection the next individual will make a selection in a like manner but on the basis of the now changed values of Na and Nb.

3.10

The selection path of this process is analytically intractable, but can be explored through computer simulation. Arbitrarily we set a_j = b_k = 5, a_k = b_j = 4, and j = k = 2, and adopt the sine functions sin(p/20.Na) and sin(p/20.Nb) which have wavelength 40. These values have been chosen to provide deliberate symmetry in the utility functions. This emphasises that there is nothing , within either the structure or the parameter values of the model to favour the selection of one commodity over the other. Figures 5 and 6 show typical plots for N=Na+Nb=500. The thick and thin monotonically rising lines represent the cumulative sales of Na and Nb respectively. The third line plots the difference (d = Na-Nb).

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The simulation can be reproduced using an Excel spreadsheet that can be downloaded to run on the reader's computer.

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3.11

Consider figure 5(i). There is clear symmetry here. Both commodities are growing continuously over time with regular cycling, though in terms of cumulative sales one commodity is always in the lead. In figure 5(ii), after an initial lead taken by one commodity the path appears to settle down to a similar pattern. Again, in figure 5(iii), there is a pattern of cycling, but this time the lead switches from one to the other and then re-switches back. There is neither the random walk nor the lock-in of the linear model of section II.

Figure 5 (i) to (iii): Simulations Displaying Cycling

3.12

The regularity in these three figures might, at first glance, be thought unsurprising. It might also be thought, given the underlying sine functions and their deliberate symmetry, that this regular pattern of behaviour has been imposed upon the system, and that it would form a distinguishing characteristic. That this is not so, is clearly seen from a perusal of the additional simulations in figures 6(i) to (xvii). If the wave like patterns are dependent upon the underlying nature of the sine function, then the absence of such a pattern must seem surprising, as in figure 6(xv) where cyclicity appears and thereafter disappears.

(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
(xiii)
(xiv)
(xv)
(xvi)
(xvii)
Figure 6: Simulations (i) to (xvii)

3.13

The most striking feature that emerges from figures 6(i) to (xvii) is the rich diversity of behaviour. Sometimes there is regularity and sometimes there is not. In some cases there is clear cycling, in others there is not. Sometimes abrupt changes are apparent, sometimes not. Sometimes, one commodity maintains and keeps the lead, sometimes the lead switches and re-switches. Yet, the structure of the model is deliberately of the simplest and symmetric. The sequential pattern of choices is unpredictable, non-ergodic and determined by the path of prior choices. None of the patterns displayed could be said to be inconsistent or incompatible with what could occur in a real market ^[3]. If a pattern were to represent observed time series data, we might be forgiven if we were to look for some exogenous shock to explain any abrupt change such as that in figure 6(iii). Here, there is a lengthy period of apparent regular behaviour (for approximately 300 selections) where both commodities grow together. There is then a distinct change in activity with the appearance of regular cycling. Similarly in figure 6 (xiv), pari passu growth turns to cycling, reverts to equal growth and then back to cycling. If we were living in either of those worlds we might try to attribute such changes to some distinct cause. What might have caused such a change we might well ask? However, the truth is that, in these instances, there has been no unitary cause, and the effects simply reflect the underlying micro dynamics. The changes have occurred through spontaneous self organisation resulting from the cumulation of individually small random historical events. It should be noted that, although the profile of selections is unpredictable it is not chaotic. Chaotic systems are deterministic and re-running a simulation with the same parameter values will re-create, identically, the same series (see Kemp 1997) ^[4]. There is no such determinism here, for the re-running of a simulation will always produce a qualitatively different series where the frequency, order and pattern of cycles and switches will differ. Unlike chaotic simulations, these simulated series can never be reproduced or replicated.

3.14

The remarkable feature is that nothing is exogenous. There are no policy shocks, no manipulative behaviour on the part of market participants, and neither is there any asymmetry in the power wielded by agents. The utility functions, and the proportions of consumer types which enter have been kept deliberately symmetric in order to highlight the possibility of spontaneously generated changes. Of course, in real systems the true, but unknowable, utility functions may not be as simple as the arbitrary ones adopted here, and the underlying parameters are unlikely to remain constant throughout time. Nevertheless, it would appear that running simulations with varied functional forms and/or asymmetry or extending the number of commodities under consideration does not appear to alter the general observation that, fluctuations and changes in those fluctuations can arise without the existence of any clear ascertainable external or internal stimulus. They can arise simply through the interaction of individuals' utilities which is conditioned by the random entry of agents of different types. For example, figures 7(i) and (ii) depict situations where after 250 selections the proportion of consumers with an autonomous preference for the commodity that is, at that stage, in the lead has been increased to 2/3. That is, there is a general change of tastes in favour of the most successful commodity. Such a change need not necessarily wipe out the other commodity. A dispersion in the rates of growth arises, but even so, there are still clear swings of popularity. Casual empiricism might suggest that such a profile could reflect something similar to the struggle between the Apple Mac and the PC. Over time the market has moved predominantly in favour of the PC, but there still remain small surges in the popularity of the lagging Apple Mac. Clearly, such shifts in probabilities can result from marketing policies of competing firms. Similarly, to the extent that firms' policies widen or narrow the difference between the autonomous utilities (a_j & b_j and a_k & b_k), they will reduce or heighten the influence of the externalities in consumption, as also will policies which reduce or increase the amplitude of the adoption functions through reducing or raising the values of j and k and so affecting the responsiveness to changes in N ^[5]. Likewise shifts in the probabilities could arise from the sorts of factors which social psychologists see as important in influencing human behaviour.

Figure 7(i) and (ii): Asymmetry

Conclusion

4.1

It has been shown that a simple model can generate a variety of profiles that are unpredictable, but which may have a relevance for the understanding of real time series. This suggests that observed regularity, irregularity and changes therein may sometimes have no special cause, other than the fact that consumer preferences are influenced by prior purchases. This suggests that abrupt change need not necessarily occur as a consequence of a special shock or policy change. That is, there may be no identifiable cause. Changes can arise spontaneously through self-organisation, as small chance events in a system's past can, over time, have a large collective influence. It is known that emergence and self organisation are features of natural phenomena, and it has been suggested that they may also characterise economic systems (Waldrop 1992, Krugman 1996). The model provides support for this view. Thus, continuously shifting and fluctuating patterns of demand, the swings and roundabouts of fashion, may well be endemic, arising without apparent cause. These are features of a model that is symmetric and of the simplest. Setting different values for the parameters, introducing asymmetry or varying the utility function ^[6], appears to have little qualitative effect upon the conclusion that, change may sometimes arise spontaneously with 'neither rhyme nor reason'. In such cases, searching for catalysts of change may be fruitless.

4.2

The implication of the foregoing is that, in those markets where consumers select commodities sequentially, there may be no clear determinate long run equilibrium profile, or clear static equilibrium optimal allocation of resources. Thus, patterns of demand may change unpredictably. This is particularly so where current purchases are influenced by past choices. In many goods there is no continuous trade off through substitution. That is, at the moment of sale we purchase one unit of one commodity or nothing: for example, one pair of trousers, one film at the cinema, one restaurant meal, one football match, one wedding dress, one graduation gown, and so on. Selection processes can lead to markets which evolve, with the past influencing the present in an indeterminate and changing way.

4.3

Of course, given that it is possible to generate patterns similar to observable patterns without any extraneous cause does not mean that there are no extraneous causes. Indeed, in any real system it is likely that discernible events could be expected to produce effects that could overshadow or obscure the underlying dynamics of the changes suggested in this paper. Some of those changes may well be attributable to the sorts factors which both economists and social psychologists see as important in wreaking change, and which form the substance of the works referred to in section I. However, the current approach is deliberately shorn of all those possibilities. Having then abstracted from all those possible causal factors, change can still occur. If this is so, then there might not be any discernible cause of change.

4.4

The approach in this paper does not suggest the existence of some universal law of human behaviour in the sense that Newton's law of universal gravitation can be said to be a law. What it does suggest is that it may not be possible to develop a model capable of prediction in the way that is possible from Newton's law. What it also does is to offer insights into how unanticipated swarming can take place without any conscious design. Why, for no apparent reason, has there been a decline in the general demand for headwear within the U.K., but a contemporaneous surge in the demand for baseball caps? What the approach suggests is that there need not be anyone sitting there with a master plan, and that fashions and the collapse of fads and their re-emergence can arise out of individual decisions which then contribute towards a collective influence on those decisions.

4.5

Of course, nothing in the foregoing is to deny the influence of policies of firms and governments, or of prices, or of other exogenous and identifiable shocks in wreaking major change, but only to deny that such change has to be explainable solely in terms of such identifiable extraneous influences. Thus, cause entails change, but change does not necessarily entail a distinct cause.

4.6

Finally the results may also have tentative implications for explanation of changes in social as well as economic activity. The decision to buy a baseball cap is an economic one the decision to wear it back to front is social. Both are influenced at the moment of decision by the prior decisions of others. There are many decisions in life that are one-off and made by individuals sequentially: choice of A level subjects, career, marriage etc., all of which display patterns of varying popularity.

Acknowledgements

A version of this paper was first presented at the 23rd Annual Convention of the Eastern Economic Association, Washington D.C. April 1997, and I am grateful to the participants for comments. I am also particularly grateful to John Avery, Enrico Bellino, Edmund Chattoe, Derek Leslie, Scott Moss, Ian Steedman and George Zis for comments upon an earlier draft. I also give thanks to two anonymous referees. Any remaining deficiencies are mine alone.

Notes

¹ This is not to deny the multiperiod nature of general equilibrium analysis (Debreu 1959 pp.28-29), or considerations of dynamic stability. However, these cannot be thought of as involving a dynamic process in the sense that is to be discussed in this paper. Likewise, the analysis of intertemporal choice, despite the explicit introduction of time, is conventionally treated within a comparative static rather than a dynamic framework.

² Economists in particular might object to the absence of relative prices in this model. However, in many substitute and/or fashion goods there is no reason to suppose different costs of production. For example, the costs of producing different styles of trouser are unlikely to differ. Under perfect competition with constant returns to scale, this implies constant and identical prices. Clearly relative prices and incomes must form important determinants of demand, but we abstract from them here in order to isolate other issues.

³ It should be noted that the variety displayed within figures 5 and 6 has NOT been contrived by making a small selection from a large number of simulations. The variety is inherent, for only three or four simulations have been discarded from the total performed.

⁴ However, see Benhabib and Day (1981) for an approach that produces deterministic endogenous fluctuations which are chaotic and shown to arise because of the presence of some cyclicity in preferences.

⁵ For example, Pashigian, Bowen and Gould (1995) claim that reductions in style changes by automobile manufactures have led to a decrease in the amplitude of the seasonal fluctuation in automobile prices.

⁶ As already pointed out in Paragraph 3.6, the particular utility function adopted is not meant to be realistic nor to encapsulate consumers' attitudes to any particular commodity, and it is not suggested that it is, even theoretically, applicable to all commodities. To discover the true utility function for a commodity is clearly impossible. All that is being said is that, if preferences are not linear and do not monotonically vary with consumption, then unpredictable and apparently spontaneous change is possible. Any other of, or a composite of, the circular functions could equally serve to illustrate this proposition.

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