Evolutionary competition between prediction rules and the emergence of business cycles within Metzler’s inventory model

Abstract

We revise Metzler’s well-known inventory model by developing a framework in which producers can either apply a simple but cheap extrapolative or a more sophisticated but expensive regressive forecasting rule to predict future sales. Producers are boundedly rational in the sense that they tend to select forecasting rules with a high evolutionary fitness. Using a mixture of analytical and numerical tools, we attempt to describe the characteristics of our model’s dynamical system. Our results suggest that the evolutionary competition between heterogeneous predictors may lead to irregular fluctuations in economic activity.

Appendix: Aggregation of sales expectations

In this appendix, we illustrate the conditions under which expected sales of consumption goods, U _t , may be expressed as a weighted average of both types of expectations. For the sake of simplicity, we restrict our analysis to the simplest case in which producers have no memory at all and have no preference for simple linear heuristics, i.e. m = 0 and μ = 0. Given this, Eqs. 10 and 11 will reduce to

$$\begin{array}{rll} A_t^E&=&-\left(C_{t-1}-U_{t-1}^E\right)^2,\\ A_t^R&=&-\left(C_{t-1}-U_{t-1}^R\right)^2. \end{array}$$

Moreover, we want to omit all cases of asynchronous updating. Hence, the fractions of agents using either the extrapolative or the regressive forecasting rule are respectively denoted by

$$ w_t^E=\frac{\exp[gA_t^E]}{\exp[gA_t^E]+\exp[gA_t^R]} $$

(41)

and

$$ w_t^R=\frac{\exp[gA_t^R]}{\exp[gA_t^E]+\exp[gA_t^R]}. $$

(42)

Let us now suppose that there are N firms in the market. All firms shall be equal in size in the sense that each firm is able to sell the same amount of consumption goods. Nevertheless, each firm still forms its individual expectations about its future sales, which are now specified by

$$U_{i,t}^E=\frac{C_{t-1}}{N}+c\biggl(\frac{C_{t-1}}{N}-\frac{C_{t-2}}{N}\biggr)$$

(43)

$$=\frac{1}{N}\Bigl(C_{t-1}+c\bigl(C_{t-1}-C_{t-2}\bigr)\Bigr)$$

(44)

$$=\frac{1}{N}U_t^E $$

(45)

and

$$ U_{i,t}^R=\frac{C_{t-1}}{N}+f\biggl(\frac{\overline{C}}{N}-\frac{C_{t-1}}{N}\biggr)$$

(46)

$$=\frac{1}{N}\Bigl(C_{t-1}+f\bigl(\overline{C}-C_{t-1}\bigr)\Bigr)$$

(47)

$$=\frac{1}{N}U_t^R. $$

(48)

Accordingly, the individual performance of each predictor may be written as

$$ A_{i,t}^E=-\biggl(\frac{C_{t-1}}{N}-U_{i,t-1}^E\biggr)^2$$

(49)

$$=-\biggl(\frac{C_{t-1}}{N}-\frac{U_{t-1}^E}{N}\biggr)^2$$

(50)

$$=\frac{1}{N^2}A_t^E $$

(51)

and

$$ A_{i,t}^R=-\biggl(\frac{C_{t-1}}{N}-U_{i,t-1}^R\biggr)^2$$

(52)

$$=-\biggl(\frac{C_{t-1}}{N}-\frac{U_{t-1}^R}{N}\biggr)^2$$

(53)

$$=\frac{1}{N^2}A_t^R. $$

(54)

The individual weighting function

$$ w_{i,t}^E=\frac{\exp[g_iA_{i,t}^E]}{\exp[g_iA_{i,t}^E]+\exp[g_iA_{i,t}^R]} $$

(55)

may now be interpreted as the (individual) probability with which a particular firm chooses the extrapolative forecasting rule. Hence, expected future sales of a particular firm are determined by the following probability function

$$ U_{i,t}=\begin{cases} U_{i,t}^E & \text{with probability $w_{i,t}^E$}\\ U_{i,t}^R & \text{with probability $1-w_{i,t}^E$}\\ \end{cases} $$

(56)

while overall expected sales are simply given by

$$ U_t=\sum\limits_{i=1}^NU_{i,t}. $$

(57)

If we furthermore suppose that all producers react equally sensitive to differences in the performance of both predictors, we are able to scale parameter g so that the intensity of choice of each producer is given by

$$ g=\frac{g_i}{N^2}. $$

(58)

With this simple transformation, it is now straightforward to see that our individual probability function given by Eq. 55 corresponds to Eq. 41. Thus, for a sufficiently large number of firms, approximately $w_t^E N$ firms will choose the extrapolative predictor, while approximately $(1-w_t^E)N$ firms will select the regressive forecasting rule.

Accordingly, the aggregate of expected sales of consumption goods may properly be approximated by

$$ U_t=w_t^EU_t^E+w_t^RU_t^R. $$

(59)