KINKS AND CATASTROPHES: A NOTE ON THE RELEVANCE OF CATASTROPHE THEORY FOR ECONOMICS

J.S. DODGSON

Universit-v of Liverpool and University of British Columbia

I. CATASTROPHE THEORY

In a recent Paper Harris [ l ] considered catastrophe theory and its relation to economics. In that Paper economic theory and catastrophe theory were characterised as representing alternative approaches to the analysis of particular problems. The purpose of this Note is t o argue that catastrophe theory is not an alternative to existing economic theory, but is instead a n addition to the (many) mathematical tools which are available to analyse economic models. As a mathematical technique, the economic content of catastrophe theory is of course zero, and use of the theory cannot provide a n alternative to economic theorising. Its use may, however, permit improvements to be made in the modelling of economic phenomena.

Catastrophe theory was developed by the mathematician Renk Thom [6].1 Its best- known application is in dealing with situations in which continuous changes in “independent” (or “control”, or “slow”) variables lead to abrupt, or discontinuous, changes in “dependent” (or “state”, or “fast”) variables. Although the proofs of Thom’s theorems are based on the methods of differential topology, the approach itself is best seen as an extension of the calculus rather than as a distinct and separate mathematical approach. One of the main features of elementary catastrophe theory is its demonstration of the ways in which discontinuous change can occur in continuously differentiable, that is smooth, systems.

The theory deals with smooth systems in which there is a vector, a, of control variables and a vector, x, of state variables. These are related to each other by a function F(x, a ) which yields the equilibrium values of the system. For economic systems the most obvious application would be one where F was a function to be maximised (profit, utility) or minimised (cost, expenditure). However, the approach could equally well be considered where the maximand or minimand was not explicitly stated, as for example in the case of models which determined stable equilibrium levels of national income in macro-economic systems2 or in models where equilibrium values were defined in terms of the most probable behaviour of the system. Catastrophe theory and the associated theory of bifurcations consider the nature of the points of the function F(x, a), termed stationary points, at which the first derivative of the

‘For an extensive survey of catastrophe theory and its applications see Poston and Stewart [2]. Clear short introductions to the subject are to be found in Stewart [5] and Zeeman [lo].

*For an example see Varian [7].

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function is zero, and those stationary points, termed bifurcations, which yield singularities at which the nature of the equilibrium solution to the system changes.

In his classification theorem of the elementary catastrophes Thom showed the types of discontinuous change which can occur in structurally stable systems3 with different numbers of control and state variables. Thus in the case of a smooth system with two control variables and one state variable, Thom’s theorem proves that the type of behaviour described by the cusp catastrophe analysed in detail in Section I of Harris’ paper is locally the most complicated thing that can happen. This means that if discontinuous change can occur in a smooth system with only two control variables then the stationarity surface of the behaviour graph in such a system must exhibit local behaviour which is qualitatively equivalent to the cusp catastrophe.4 More generally, if there are four or fewer control variables, then there are only seven types of discontinuous change, termed the seven elementary catastrophes, which can occur. These elementary catastrophes are usually described in terms of standard algebraic forms, known as canonical forms, which have been chosen on the basis that they yield the simplest forms of graph which are qualitatively equivalent to the stationarity surfaces which can arise in each case.

While this result is often claimed to reduce considerably the types of behaviour which might be observed, one purpose of this Note is to demonstrate that even rather simple economic models can yield quite complex types of discontinuous change. We illustrate this by showing the implications of the convex kink demand curve analysed by Joan Robinson [3] and recently reconsidered by Walters [8]. The model is shown in Figure 1. A demand curve of this type can occur when demand becomes relatively more elastic as price falls. This might occur if the price fall brings the commodity within the reach of new groups of consumers, or if (as in the case of electric power) the price fall enables the commodity to be used for a wider range of purposes. In such circumstances the marginal revenue curve will decline, then rise and finally decline again. In developing the implications of this model it will be clear that we need to know the underlying economic model before we can use catastrophe theory to analyse it. In addition, it will become clear that the necessary condition for the existence of the type of discontinuous change posited by catastrophe theory is the existence of multiple equilibria. Hence when we know or suspect that multiple equilibria do exist we can be alerted to the fact that discontinuous change is possible. Further, occurrence of discontinuous change can alert us to the possible existence of multiple equilibria and to the underlying structure of the system we are trying to model.

11. THE CONVEX KINK DEMAND CURVE

We now illustrate the way in which smooth changes in control variables can lead to discontinuous changes in state variables (that is to changes in the nature of the solution

3A system (x, a ) is structurally stable if, with a small perturbation 6n on n, the perturbed system (x , 6 a ) is topologically equivalent to the unperturbed system; that is, a small change in the control variables does not alter the underlying topology.

4Note however that the global behaviour could be more complicated, since the possibility exists of there being multiple cusps in such systems.

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FIGURE I

Price Determination in Monopoly Market with Convex Kink Demand Curve

Price

P4

P3

P? PI

;; b:

to the system) using the kinked demand curve model. Suppose that the monopolist’s marginal cost curve takes the linear form b + cQ, where, purely for expositional convenience, we assume that both b and c 2 0 . By regarding the position of the demand curve as fixed, we can then interpret the model as a model of a system with two control variables, b and c, and one state variable, price, p , linked by a profit function, T@. b, c). We then derive values of profit-maximising price for all non-negative combinations of b and c. These yield what is termed the stationarity surface.5 In the three-dimensional Figure 2 the resulting price surface will have a fold for those combinations of b and c which yield multiple equilibria. In the horizontal plane of Figure 2, which is drawn separately as Figure 3, these values lie in the cusp shaped area. For the values of b and c within the cusp there are three levels of price for which MC = MR, the two local maxima which lie on the two upper surfaces of the fold, and the local minimum which lies on the undersurface between them. Points on the boundary of the cusp are bifurcation points, for which there are two levels of price for which MC = MR. One of these is a local maximum; the other a point of inflexion where 8 r / a Q 2 =O. The point z is known as the cusp-point.

Now the path taken by price, output, and profits as the parameters of the cost function change can be seen to depend on the quality of the firm’s information.

SAlternatively referred to as the behaviour graph, or catastrophe manifold.

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FIGURE 2

Price Stationarity Surface: Values of Profit-Maximising Price for Values of b and c

Profit-maximising price

FIGURE 3

Horizontal Plane of Figure 2: The Cusp

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Consider the situation shown in Figure 1, where c is fixed at CI and only b changes. Figure 4 then shows the vertical cross-section of Figure 2’s profit-maxircising price surface for c equal to cI. If firms are fully aware of their demand and cost functions, then as b declines there will be a sudden discontinuous fall in price when b reaches bo, where the two local maxima yield the same level of profit.6 If b were to rise from below bo, there would be a discontinuous increase in price, and fall in quantity, at bo. The path taken by price in Figure 4 would be kjifed in the former case and the reverse of this in the latter.

FIGURE 4

Vertical Cross-section of Price Stat ionari ty Surface for c = c,

However, if the monopolist does not have full knowledge of his demand and cost curves, but is instead only able to ascertain whether profits will rise or fall if he makes small adjustments in price away from the existing position, then we find a different type of adjustment, one which exhibits hysteresis.’ As b declines the firm will gradually

hThis particular adjustment rule is sometimes referred to in the catastrophe theory literature as the Maxwell convention.

’This adjustment rule is referred to as the perfect delay rule. and is the one most commonly found in applications of catastrophe theory. In the case of the kinked demand curve model it depends on the assumption that the economic decision-maker is myopic. This is the situation envisaged by Joan Robinson: “The net monopoly revenue at each point would be different, but it is unlikely that any monopolist would have sufficient knowledge of the situation to enable him to choose the greatest one from among them. If the monopolist had reached one equilibrium point. there would be no influence luring him towards another at which his gains might be greater” (131). p. 58).

Another example of such an assumption of myopia. but on the part of consumers rather than firms is to be found in Harris [I] . Note that discontinuous change (of one sort or another) occurs because of the existence of multiple equilibria; the assumption of myopia merely determines the particular adjustment rule that the catastrophic change follows.

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adjust price down and output upwards until b falls below bl, when price will jump frompl topl . The path taken in Figure 4 will be kjihed. On the other hand if b were to rise from below bl, then the discontinuous jump would occur when b reached bz; price would rise from p2 to p 4 and the path taken in Figure 4 would be defgik. Different paths would occur if both b and c were to change, and thus the surface in Figure 2 shows the behaviour of price for any conceivable path of the two parameters of the marginal cost function.

Instead of considering the behaviour of price we could instead consider the behaviour of output. Since price and output are related inversely by the monotonically declining demand curve of Figure 1 , the profit-maximising stationarity surface of output will also exhibit the cusp pattern, but with output inversely rather than directly related to the values of b and c.

Finally we may note that the behaviour surface of the maximand profit (or what Harris terms “the energy function”) takes a different form. Consider again the situation where c is fixed at CI and b changes. If firms are myopic, then as b declines profits will rise continuously, but then as b falls below bl there will be a sudden discontinuous increase in profits (equal to the area in Figure lbetween the marginal cost curve bl+ CI Q and the marginal revenue curve betwen outputs Q3 and QI). If instead b were to rise from below bl, there would be a continuous fall in profits, interrupted by a discontinuous increase, as b rose above b2. However, if the firm had perfect information and was fully aware of the position of its demand and cost functions, then as b fell from above b2, profits would rise and when b reached the value of bo (at which two levels of output and price yield the same level of profits) there would be a discontinuous increase in output and fall in price, but no discontinuous change in profits. If b were to rise from below bo, the reverse would occur. The pattern of profits is shown in Figure 5(ii). With myopia profits follow the path defgik if b rises, and kjfied if b falls. With perfect information profits would follow the path desjk, or its reverse kjyed.

If c were to take the value of c?, then profits would take the values shown in Figure 5(i). There will be a kink in the profit curve at the point z’ which corresponds to the cusp-point z in Figures 2 and 3. For higher values of c the profit curve will be smooth and monotonically decreasing as b increases. If c were equal to zero, then the pattern of profits would be as shown in Figure 5(iii). The three-dimensional profit surface is then found by stacking the vertical cross-sections for different values of c, of which the three diagrams of Figure 5 are three examples, over the horizontal plane of Figure 3. The resulting profit surface consists of two smooth surfaces which intersect each other vertically above zbs. (This line zbs in the horizontal plane links all combinations of b and c for which there are two local profit maxima yielding equal levels of profit.)

With perfect information profits will lie on the upper surface and there will be no discontinuous changes in profits as b and c change, although the profit surface will be kinked vertically above zbs. With myopia discontinuous changes in profits can occur, and there may be discontinuous increases (but never decreases) as the values of b and c change.

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FIGURE 5

Profit Curves for Different Values of c (vertical cross-sections of three-dimensional profit surface)

Profit I (i) c = c 2

(ii) c = c ,

0 L Profit

0 1

(iii) c = 0

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111. THE RELEVANCE OF CATASTROPHE THEORY FOR ECONOMICS

Economics has been regarded as a likely area for the application of catastrophe theory but, despite interest among economists, few applications have emerged, and these have tended to be speculative in nature.8 One reason for the lack of applications is that while sudden (and sometimes catastrophic in the popular sense of the word) changes may occur in economic systems, this may be the result of changes within systems which are not themselves continuously differentiable, and hence not amenable to analysis by catastrophe theory. The claims of generality for Thom’s classification theorem do sometimes appear to have diverted attention from the fact that it does only apply to continuously differentiable systems, and so cannot describe all types of sudden change which could occur. Further, as the example discussed in this Note demonstrates, catastrophic change in continuously differentiable systems requires the existence of multiple equilibria. But many economic models are deliberately set up in such a way that multiple equilibria cannot occur. For example, consumer theory and welfare theory are generally (and for good reasons) based on the assumption of strict convexity of indifference curves, and this yields unique equilibria given consumers’ incomes.9 Similarly, the shapes of functions in many macroeconomic models ensure that the models yield unique equilibrium levels of national income and of other dependent variables. Where the shapes of functions are more complex, and do yield multiple equilibria, empirical testing is more difficult.

However, I believe that catastrophe theory does have relevance for economists because it can alert them to the ways in which discontinuous change can occur in economic systems as a result of continuous changes in independent variables, once multiple equilibria are possible, and, consequently, can aid in the modelling of such changes. Thus, an understanding of the workings of the three-dimensional cusp catastrophe in models with two independent variables can pave the way to an intuition for the higher order catastrophes which are possible in systems with more than two independent variables. Furthermore, once we know (or suspect) the structure of the economic model, computer programmes are available to determine the types of singularities, and their associated types of discontinuous change, which are (or could be) present in these more complex models.10

In conclusion, catastrophe theory is not, and cannot be, a substitute for economic modelling since, as the simple kinked demand curve example shows, we do need to know the underlying structure of the economic system if we are successfully to model its behaviour. As with other mathematical techniques, catastrophe theory might help

*See Harris [ I ] on the collapse of expectations at the peak of the trade cycle: Varian [7] on Kaldor’s trade cycle model; and Zeeman [9] on the behaviour of stock exchanges. While the example in the present Note deals only with short-term adjustment of the dependent (fast) variable to changes in the independent (slow) variables, the models in the latter two papers also incorporate feedback from fast to slow variables and consider the nature of the longer-run adjustment paths of the slow variables.

9The implications for consumer behaviour of relaxing the assumption of convexity, and permitting indifference curves to have intermediate concave sections, are discussed in Harris’s paper on catastrophe theory [I]. For the implications for Pareto equilibria of such relaxations of convexity of indifference curves see Smale [4].

“See Poston and Stewart ([2], pp. 431-441).

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us t o d o this by suggesting the types of hypothesis which might be formulated and by indicating the types of discontinuous behaviour which systems might exhibit. But of course catastrophe theory is not an alternative to existing economic models, merely an addition to the mathematical techniques which economists can use in their attempts to model economic phenomena.

First version received April 1981 Final version accepted June 1981

(Editors)

REFERENCES I . L. Harris, “Catastrophe Theory. Utility Theory and Animal Spirit Expectations”. Ausrralian

Economic Papers. vol. 18. 1979. 2. T. Poston and I. Stewart, Catastrophe Theory and Its Application.\ (Imndon: Pitman. 1978). 3. .I. Robinson, Tl7e Economics of Imperfect Comperitioii (London: Macmillan, 1933). 4. S. Smale. “Global Analysis and Economics 1: Pareto Optimum and a GCneJahItion of Morse

Theory” in M.M. Peixeto (ed.) DI~nan7ical S):c.tenu (New York: Academic Press. 1973). 5. 1. Stewart. “The Seven Elementary Catastrophes”. NeH, Scienrisr, vol. 68. 1975. 6. R. Thom. Structural Stabilitj~ and Morpho~ene.si.s (Reading. Mass.: W.A. Benjamin. 1975). 7. H.A. Varian, “Catastrophe Theory and the Business Cycle”. Economic Inyuir1,. vol. 17, 1979. 8. A.A. Walters, “A Note on Monopoly Equilibrium”, Economr~~ Journal. vol. 90. 1980. 9. E.C. Zeeman. “On the Unstable Behaviour of Stock Exchanges”, Journal of Marhematical

Econoniics. vol. I , 1974. 10. E.C. Zeeman. “Catastrophe Theory”. Scient~fic American. vol. 234. 1976.