Cynthia Kinnan Chaos and the Logistic Map February 2001

Tasks 1-7 (MS Word document)

Applications of Chaotic Models in Economics

Historically, economists have, whenever possible, used linear equations to model economic phenomena, because they are easy to manipulate and usually yield unique solutions. However, as the mathematical and statistical tools available to economists have become more sophisticated, it has become impossible to ignore the fact that many important and interesting phenomena are not amenable to such treatment. Unfortunately, linear models fail to account for the fact that, while average behavior of components of a system has an important predictive role, so do ``structural qualitative changes brought about by the presence of non-average components and conditions within the system'' (Allen 9).

Important phenomena for which linear models are not appropriate include ``depressions and recessionary periods, stock market price bubbles and corresponding crashes, persistent exchange rate movements . . . and the occurrence of regular and irregular business cycles'' (Creedy and Martin 1). Therefore, economic theorists are turning to the study of non-linear dynamics and chaos theory as possible tools to model these and other phenomena. In this brief report I will discuss the application of logistic models to advertising and the introduction of new technology, and the use of another non-linear model to describe business cycles. In the process I will discuss some of the advantages and disadvantages to using logistic and other non-linear models to describe economic phenomena, and evaluate the extent to which such models are supported by empirical data.

The logistic model applied to advertising

It is intuitive to assume that the amount a firm can spend on advertising next month is proportional to its current profits, and that next month's profits will be affected by the amount of advertising. A simple dynamic model of advertising can be constructed to reflect these facts as follows (Creedy and Martin 8):

$X_t$ is profit, $Y_t$ is advertising expenditure, $\lambda$ is measure of the return generated by advertising expenditure, and $\gamma$ is the fraction of profits devoted to advertising.

These equations combine to yield:

which is the familiar logistic equation. Given what we have concluded about the behavior of the logistic model, it is clear that deciding how much should be spent on advertising is no easy matter. For instance, in light of the sensitivity of the equation's behaviour to the parameter $a$, or in this case $\gamma \cdot \lambda$, even a small error in estimating the return generated by advertising account expenditure could lead to an outcome very different than what the model predicted. Furthermore, assume a firm decides, based on use of the model, to spend, say, $1455.62391 on advertising in its first month of business. If they rounded this to $1456 or even $1455.62 and $\gamma \cdot \lambda$ happened to be between $a_{\mathrm{\infty }}$ and $4$, the result after a year might be totally different from what they hoped.

These results seem reasonable, although obviously simplistic (in that advertising dollars may be spent in many ways, which do not all have equal yields, and in that profit is certainly dependent on variables other than advertising). After all, the amount of uncertainty and the difficulty in making long-term predictions indicated by the model are in agreement with the reality that two similar firms might spend almost identical amounts on advertising, but after a year one is wildly successful while the other is on the edge of bankruptcy. When the complex relationship between profit and other variables is considered as well, it seems plausible that chaos theory might provide a more accurate representation of this situation than, for instance, a linear supply and demand model.

The logistic model applied to the diffusion of CT scanners

Logistic-type models can also be constructed that provide a more realistic description of a process than did the simple advertising model. For instance, it was found that the adoption by US hospitals of a new type of costly medical technology could be modelled in this manner by the following equation (Creedy and Vance 65):

where $X_t-X_{t-1}$ is the change in the number of community hospitals with more than 100 beds having a CT scanner over one month, $D$ is a dummy variable to reflect a goverment policy passed midway through the interval studied which slowed CT scanner diffusion, $\alpha$ reflects the extent to which diffusion slowed in the second period, and $K$ is the ceiling value.

This example is presented to demonstrate that logistic-type equations can model economic/institional behavior reasonably well. With coefficients determined using a non-iterative least squares method, the model was found to have an $R^2$ value of $0.670$ (Creedy and Vance 65), meaning that 81.2% of variation in the empirical data could be explained by this model.

The Kaldor business cycle model

Finally, we will examine a macroeconomic model that attempts to model the complex behavior of the cycles of growth and recession that economies experience. The Kaldor business cycle model consists of the following 4 equations (Creedy and Vance 20):

Where $Y$ is output, $K$ is capital, $I$ is gross investment, $S$ is savings, and the other variables are parameters. According to Creedy and Vance, the behavior is determined by the size of $\alpha$: ``The model displays a unique stationary point for small values of $\alpha$, whereas for larger values there is no closed orbit. For very large values of $\alpha$, there appears to be no realtionship between $Y$ and $K$ ... [a] supposed chaotic pattern'' (20). The evidence of chaotic behavior in this model is strengthened by the fact that it is highly sensitive to initial conditions: changes in $Y_1$ and $K_1$ from $65$ to $65.1$ and $265$ to $265.1$, respectively, when $\alpha =20$ are shown to cause significant divergence in within 15 cycles.

A chaotic model of business cycles seems highly plausible. Long-term predictions of economic performance are notoriously difficult and unreliable. For instance, Daniel Kadlec reports that a sample of ten Wall Street analysts, when asked how long they expected it to take before the NASDAQ recovered to its record high close of 5,049, gave answers ranging from 12 months to seven years (122).

Conclusion

There is certainly much promise in the use of non-linear dynamic models to model economic phenomena. The fact that so many such phenomena, ranging from stock values and foreign exchange rates to bank runs and agricultural prices, have failed, despite years of scrutiny, to exhibit behavior amenable to linear analysis, indicates that more powerful tools are needed, and there is a growing body of evidence suggesting that chaos theory has a role to play. Unfortunately, this role is proving to be a somewhat nihilistic one, for chaos theory seems to indicate that long-term economic forecasting is, in many cases, simply impossible, because even the tiniest measurement error will cause predictions to differ hugely from actual behavior. However, this is not to say that the contribution of chaos theory cannot be fruitful. It can indicate when we can expect chaotic behavior to occur, and often can indicate the range of values a variable can be expected to assume, which allows a limited sort of long-range planning. So, while chaos theory is not the definitive tool of economic analysis, it does have applicability to many economic situations.

Works Cited

Allen, Peter M. In Evolutionary Economics and Chaos Theory: New Directions in Technology Studies, Leydesdorff and van den Besselaar Eds. St. Martin's Press: New York, 1994, pp 1-17.

Creedy, John and Vance L. Martin. Chaos and Non-Linear Models in Economics. Edward Elgar Publishing Limited: Aldershot, 1994.

Kadlec, Daniel. ``Bubble Trouble,'' Time, 11 Dec 2000, p 122.