Catastrophe Theory As a Model For Describing Consumer Behavior

Terence A. Oliva, Louisiana State University
Alvin C. Burns, Louisiana State University
ABSTRACT - Consumer behavior models do not have mechanisms to handle divergent and discontinuous behavior. Rene Thom's Catastrophe Theory seems to be a promising means for modeling such phenomena. This paper describes the cusp catastrophe and illustrates how it can describe consumer adoption and consumer complaining behavior.
[ to cite ]:
Terence A. Oliva and Alvin C. Burns (1978) ,"Catastrophe Theory As a Model For Describing Consumer Behavior", in NA - Advances in Consumer Research Volume 05, eds. Kent Hunt, Ann Abor, MI : Association for Consumer Research, Pages: 273-276.

Advances in Consumer Research Volume 5, 1978      Pages 273-276

CATASTROPHE THEORY AS A MODEL FOR DESCRIBING CONSUMER BEHAVIOR

Terence A. Oliva, Louisiana State University

Alvin C. Burns, Louisiana State University

ABSTRACT -

Consumer behavior models do not have mechanisms to handle divergent and discontinuous behavior. Rene Thom's Catastrophe Theory seems to be a promising means for modeling such phenomena. This paper describes the cusp catastrophe and illustrates how it can describe consumer adoption and consumer complaining behavior.

INTRODUCTION

It is an understatement to contend that the behavioral scientist encounters frustration when he attempts to model the phenomena under his study. Unlike his counterparts in the physical sciences, the behavioral scientist finds his subject matter highly resistant to straightforward conceptualization. Simple models cannot possibly cope with the myriads of multiple determinants, complex interactions, non-linear relationships, discontinuities and seemingly unsystematic variations.

Consumer behaviorists have confronted these problems for some years now. While we believe we have transcended simplistic "black box" models and can portray the behavior of a consumer in the context of ladder or stairs concepts, decision trees, and elaborate flow diagrams, the behavior inevitably escapes the boundaries of the model when we attempt to describe it in appreciable detail. One possible reason for the disappointing experiences thus far may be in the implicit assumption in most consumer behavior models that behavior follows a smooth transition through and between stages in the process. As a more analytical posture is assumed, however, it soon becomes evident that such smooth transitions are rare. More commonly observed are sudden shifts, stalls, and divergent behavior. A mechanism for describing these phenomena must be included if our models are to faithfully depict the process.

CATASTROPHE THEORY

As indicated above, a major shortcoming of consumer behavior models has been the inability to handle abrupt (catastrophic) changes in behavior. Modeling of what might be called "misbehaving phenomena" has defied mathematical analysis for a long time. The reason for this is that such analysis usually requires some form of continuum. Relatively recently, however, French mathematician Rene Thom (1972) using a branch of mathematics known as Topology developed some ideas which generate models that can handle messy phenomena.

Topology is a kind of rubber sheet geometry that allows elastic movements (Arnold, 1962). It is the branch of mathematics that deals with the fundamental properties of literally anything; and, as such, focuses on invariance through transformation (Arnold, 1962; Barr, 1964). Thom uses that part of topology that deals with multidimensional surfaces because of its ability to represent underlying forces in nature under different states of equilibrium (Zeeman, 1976).

Structurally, catastrophe models are topological surfaces generated by setting the first derivative of a function which relates various combinations of control variables (independent variables) equal to zero (Starobin, 1976). Thom has shown that only seven models are needed to describe all possible abrupt transitions in situations that are controlled by up to four independent variables (Zeeman, 1976). The mathematical formulations of these models are contained in Table 1.

TABLE 1

RENE THOM'S SEVEN ELEMENTARY CATASTROPHES

The way in which catastrophe models are able to describe phenomena that diverge, are subject to sudden shifts, and are discontinuous is through folds and pleats in the surfaces that relate different equilibrium states. Love and hate, for example, represent two different states.

Transition from one to the other does not necessarily progress along a unidimensional continuum. At the same time, it would be incorrect to say that they occupy two disconnected continua such that an individual cannot go from one to the other. A catastrophe model would relate the two by means of a surface fold. (See Figure 1)

FIGURE 1

A CATASTROPHE BEHAVIOR SURFACE

The movement from hate to love or vice versa can occur a number of ways on the surface. A smooth transition develops if the process is such that the individual misses the fold. On the other hand, an abrupt transition occurs when the individual "falls off" one surface to the other by crossing the fold. Thus the theory describes equally well both the persistent suitor who slowly wins his heartthrob's affection and the wife who finds her cheating husband in bed with another woman.

Some of the situations where catastrophe models have been used are: the development of hostilities between nations, cathartic release from self-pity, aggression in dogs, behavior of the stock market, buckling of an elastic beam, anorexia nervosa, territoriality among reef fish (Zeeman, 1976), population dynamics (Starobin, 1976), and collective bargaining (Oliva and Capdevielle, 1977). All the proceeding situations have conditions where abrupt transitions occur, making the use of other modeling techniques virtually impossible on anything but a trivial level.

Perhaps the best way to understand what catastrophe theory is all about is by close examination of one of its models. The model we have chosen is the cusp catastrophe since it is easily understood and has been the most widely used of the models. A Cusp Model is generated by setting the first derivative of the following function equal to zero for all combinations of the variables a, b, and x, where a and b are two control dimensions, and x represents the system's behavior dimension:

f (a, b, x) = 1/4x4 - ax - 1/2 bx2   (1)

df/dx = 0 = x3 - a bx   (2)

The response surface generated is peculiar (as indicated earlier) in that it has a fold in it. (See Figure 2) By projecting the fold onto the control surface, an area of overlap known as the bifurcation set is defined. It is domain where two modes of behavior are possible depending on the initial conditions.

For a situation to be suitable for description by the cusp catastrophe, five conditions are requisite: bimodality, sudden transitions, hystersis, unaccessibility, and divergence (Zeeman, 1976).

FIGURE 2

A CUSP CATASTROPHE

Referring to Figure 3, we can see how the five conditions stated above are related by the model. It should be apparent from the diagram that the fold is the focal point of the model. Taking each condition in the abstract we can consider how the model operates in general.

FIGURE 3

A CUSP CATASTROPHE'S 5 REQUIREMENTS

Divergent behavior is accounted for since as one moves out from the edge toward the singularity, which is the starting point of separation between the two surfaces, the system is forced to adopt one of the two opposing behaviors (e.g., love or hate; attack or flee). The trajectory that will be followed will depend on the values of the control variables (state of the system) just prior to reaching the singularity.

Bimodality is created in the model by the top and bottom surfaces of the fold. It represents the domain where the system tends to hold equal amounts of both control variables at the same time. In a love/hate situation, the bififuraction set would represent the area of ambivalence.

The middle sheet of the fold represents the inaccessible area which is the least likely behavior. In the love/ hate situation, this would be the condition of neutrality or non-emotion. To see why it is inaccessible, consider the following. If we hold control variable b constant at a value of 3 and increase the value of a, the system trajectory will proceed in a smooth fashion along the bottom surface until it reaches (2,3,-1) where it will Jump to (2,3,2) on the upper surface and continue smoothly. Consequently, there is no means for the behavior to reach the middle sheet.

Sudden transitions (catastrophes) are accounted for as suggested in the preceding paragraph at the edges of the fold. That is, as the system moves along the surface toward the pleat at some point a small increase in the control variable will cause a sudden shift in behavior. To use Zeeman's (1976) example, if an angry dog is made more fearful, at some point his behavior will abruptly change from attack to retreat; conversely, if a fearful dog is progressively enraged, at some point he will stop retreating, turn and attack. It is the ability of the model to represent sudden shifts that gives it great advantage over other conceptualizations.

The fact that the catastrophes do not occur at the same place but depend on the prior state of the system incorporates lags into the model which match real-world phenomena. For example, if a person starts out "in love" and is made to increasingly dislike the other, the complete changeover to hate will not occur at the same point as a person who starts out hating another. That is, a person in love endures more grief from a second individual than does one who hates the second individual; at the same time, a person who hates a second will require more kindness to changeover to love than does one who already cares for the second. Simply then, movement coming from one direction precipitates a catastrophe in a different place than does movement from a different direction, and this lag is called hysteresis.

The above is not intended to be a complete examination of catastrophe theory. What has been presented is a review of one catastrophe model to help explain the application of the theory to two consumer behavior examples.

APPLICATION OF THE CUSP CATASTROPHE TO TWO CONSUMER BEHAVIOR SITUATIONS

Brief reflection reveals that the cusp catastrophe is an alternative model for several areas of study in consumer behavior. The remainder of the paper is devoted to our interpretations of two such examples. Our intent is purely illustrative, consequently, although a wealth of research and thought may exist for both of our examples, we will not endeavor to review or critique. Also, our comments will be restricted to the cusp catastrophe model. Hopefully, those who are working in the areas of our two examples will envision the value of the conceptualization and be inspired to apply it in more detailed fashion.

Example 1: Consumer Adoption

As a model of individual behavior, the cusp catastrophe offers an alternative view of the adoption decision. Its five conditions are satisfied in the following manner. Divergence exists in the form of the adopt/ no adopt decision facing the consumer. The phase of evaluation prior to the adopt/no adopt decision satisfies the bimodality condition while inaccessibility is evident in that a consumer will resist ambivalence or half-adoption. Sudden transitions in behavior occur at the point of adoption, especially for later adopters. Finally, the hysteresis condition is evidenced by the fact that the decisions to adopt or to unadopt are unlikely to occur under the same conditions for all consumers.

To further illustrate adoption as a cusp catastrophe, Figure 4 represents the process as adoption/no adoption behavior overlaid on two control surfaces. In this interpretation, the control dimensions are: (1) conformity pressure as perhaps measured by the urgings of others or market penetration and (2) individual predispositions such as desire for novelty, recognition, acceptance, or some other relatively stable personal attribute. These are subsumed into a general attitude position ranging from liberal (acceptance of change) through moderate (deliberative and skeptical of change) to traditional (resistant to change), reminiscent of characteristics of various adopter groups. Seen this way, the cusp catastrophe model can simultaneously describe all classes of adopters (and non-adopters) across the time dimension, and it affords insight to the differences in their adoption decisions.

FIGURE 4

CUSP CATASTROPHE INTERPRETATION OF CONSUMER ADOPTION

Innovators progress through the adoption process by traversing the smooth surface of the cusp catastrophe defined by liberal attitudes and introduction of the innovation to the market. Their behavior is non-catastrophic in that they do not cross the bifurcation set. Adoption of new behavior is natural and uninhibited; moreover, the decision to stop adoption is a smooth transition back during this early stage. The majority of the population, however, faces a very different circumstance, for their moderate attitudes hold them at a state of no adoption until conformity pressure increases to the point at which they must evaluate the adoption decision. In the terminology of catastrophe theory, they are in the domain of the bifurcation set. Those with somewhat less moderate attitudes (e.g., the early majority) reach the edge of the no adoption behavior surface pleat and drop (up) to the adoption behavior surface. A snowball effect is seen as large segments of the population adopt, increase conformity pressure, and cause (more skeptical) others to adopt. Ultimately, the innovation is adopted by all but a very few members of the population. At this point, the laggards or traditionalists occupy the behavior surface where the pleat is widest. Divergence for this group is greater than for any other adopter group; consequently, the change in behavior for them is literally greater than for anyone else. For this reason, they will resist change until they are forced to adopt.

Example 2: Consumer Complaining Behavior

Our illustration of the adoption cusp catastrophe utilizes a dynamic control dimension (conformity pressure) matched against a stable psychological state. The model easily handles two dynamic control dimensions, as will be demonstrated in this example. Consumer complaining behavior has received an increasing amount of attention in recent years although no widely accepted models exist in our knowledge. In our present interpretation, consumer complaining behavior can be seen as a function of consumer dissatisfaction and company promotional efforts. Dissatisfaction for the individual consumer grows and declines with product use, changes in expectations, and/or information acquisition. Company promotional efforts exist in the magnitude and quality of persuasion, education, demonstration or other such efforts. The behavior surface is comprised of complaining or not complaining (divergence). Under the same circumstances some individuals complain while others do not (bimodality); sudden changes in consumers from noncomplainers to complainers and vice versa have been observed, and complaining behavior is dependent on prior conditions (hysteresis).

Viewed as the cusp catastrophe in Figure 5, complaining behavior and even the intensity of complaints becomes clear. When a consumer's experiences engender dissatisfaction with a brand or company, he will traverse the behavior surface to the complaint mode. However, it is likely that company promotional efforts will counteract dissatisfaction to some degree early in the process. Perhaps as a consequence of personal style, some consumers will move out on the behavior surface and enter the bifurcation set on the no complain surface while others will choose the complaint surface (A). These two groups of consumers in the region outlined by the bifurcation set are undoubtedly the firm's marginal customers. One group persists in mild complaints to dealers, salesmen, or even company executives. The other group does not complain, but undoubtedly will welcome a competitor who will respond to their needs more effectively. Given that there are no satisfactory alternatives available, the situation can easily become escalated. That is, greater dissatisfaction as a result of greater product usage will draw the noncomplainers closer to the edge of complaining behavior (B). It will simultaneously push the mild complainers into a more militant posture. The reaction of the company to these symptoms is quite natural: it increases its promotional activities in an attempt to counteract the situation. Seen as a cusp catastrophe, however, the net result is to draw the consumers further out on the pleat and to thereby increase the potential of intensive complaining behavior.

At this point, the company has a tiger by the tail, even though it may not realize the situation. It cannot reduce its promotional activities; nor can it back off from its previous claims, for the model predicts an immediate and catastrophic consequence in the form of a flood of complaints to the company and probably consumer protection and other regulatory agencies as well. Moreover, when consumers change from noncomplainers to complainers under these circumstances, hysteresis dictates that massive amounts of company promotion will be required to restore them to a noncomplaining attitude (C).

FIGURE 5

CUSP CATASTROPHE INTERPRETATION OF CONSUMER COMPLAINING BEHAVIOR

Fortunately, however, the model affords a prescription for the company. It must reduce dissatisfaction through product modification or restoration of damages. Actions, not words, will return consumers to the noncomplaining equilibrium state by reducing their dissatisfaction.

CONCLUSION

These two examples are illustrative of the ability of one catastrophe model to describe consumer behavior. It should be clear that while mathematically complex, the conceptual aspects of catastrophe theory are easy to comprehend and are sufficiently flexible to serve our needs. Given that consumer behavior is multidimensional and subject to abrupt changes, it will be to our advantage to scrutinize the applicability of the cusp and other catastrophe models in the near future.

REFERENCES

B. H. Arnold, Intuitive Concepts in Elementary Topology (Englewood Cliffs, New Jersey: Prentice-Hall, 1962).

Stephen Barr, Experiments in Topology (New York: Thomas Y. Crowell Co., 1964).

T. Oliva and C. Capdevielle, "Catastrophe Theory: A Collective Bargaining Application," Forthcoming in the Proceedings of the National Academy of Management Annual Meetings, (August, 1977).

L. Starobin, "Our Changing Evolution: Strategies for 1980," in General Systems XXI (Washington, D.C.: Society for General System Research, 1976) 3-46.

R. Thom, Stabilite Structurelle et Morphongenese (New York: Buyamin, 1972).

E. C. Zeeman, "Catastrophe Theory," Scientific American (April 1976), 65-83.

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