## 1/4/11

### Empirically correct model of real economic growth (France)

We continue our GDP modelling with France. This is one of the biggest developed countries providing information on the population age distribution. As in the previous cases, the model for the GDP growth for France was obtained by the trial-and-error method using a discrete form of equation (1.1). The empirical constant A and the defining age have been varied in order to fit the amplitude and timing of observed peaks and troughs. The best fit values are: \$320 and eighteen years of age. In the upper panel of Figure 1, the observed and predicted curves for the period between 1970 and 2009 are presented. Superficial visual inspection allows us to suggest that the agreement between the curves does not contradict our concept. The only principal difference between the US and France is that the defining age for France is eighteen years, as was the case for Japan.

Figure 1. Observed and predicted growth rate of real GDP in France. The predicted curve is obtained from relationship (1.4) with A=\$320 (1990 US dollars). Upper panel: Original curves. Lower panel: The original curves smoothed with MA(3).

There are original estimates of the number of 18-year-olds in France, which can be used for the prediction of the past GDP figures. The future GDP can be predicted only by extrapolation of younger age populations. For example, the number of 10-year-olds in 2000 can be used as a proxy to the number of 18-year-olds in 2008. Moreover, it is possible to transform the age pyramid for a given year into the distribution of 18-year-olds, with the accuracy of extrapolation decaying with age. In this study, the number of 5-year-olds in 2001 is the reference distribution. So, using this age we are able to estimate the evolution of GDP up until 2014.

A better prediction could be obtained after censuses, which usually provide a well balanced single-year-of-age distribution. In France, the last general population census with published results was in 1999. By itself, the accuracy of the population estimates is difficult to evaluate, but many features unveil artificial character of the population age pyramid. In any case, one cannot help observing very good correspondence between the slowdowns in both curves in the beginning of 1990s and 2000s in Figure 1.

A high-amplitude fluctuation in the first derivative is a common feature for most measured macroeconomic variables. This is a direct manifestation of measurement errors associated with numerous limitations in relevant measuring procedures and inappropriately small time step. In the USA, the average annual growth in real GDP per capita during the latter 20 years is around 2% with the average uncertainty of 1 percentage point, i.e. the annual estimates are of the same order of magnitude as the corresponding uncertainty. In the absence of adequate improvements in the measurement methodology per se, better accuracy could be achieved via stretching the time horizon of corresponding GDP readings, i.e. the time step should be larger than one year.

As an intermediate measure one can smooth all time series in order to cancel out measurement noise. There is a variety of smoothing techniques, some of them very complicated, but even a moving average is enough for the original data in Figure 1. In the lower panel, both original curves are smoothed with a three-year moving average, MA(3). After 1985, the curves are very close. That observation supports the assumption that the fluctuations were chiefly induced by high uncertainty of the measurements, and thus, are effectively suppressed by destructive interference. Before 1985, the curves suffer a slight divergence, which can be an indication of the problems with the extrapolation over 20 years back in the past as well as with the reliability of the GDP measurements. According to the predicted curves, France will not suffer a protracted recession in the next four to six years. It is, however, likely that an insignificant GDP decline period will hit France in the near future.

Figure 2. Observed and predicted number of 18-year-olds in France. The former variable is extrapolated from the number of 5-year-olds in 2001 with a 13-year shift, and the latter from the observed real GDP per capita.

With the knowledge of annual GDP estimates it is possible to further double-check our model using the reversed equation (1.6) thus calculating the number of 18-year-olds in France. Figure 2 illustrates the inversion results between 1963 and 2009. In general, the observed and predicted curves are very close after 1985. Before 1985, the curves diverge in minor details, but both show a sharp increase in the 18-year-old population after 1960. This is a major feature which has higher importance for the model than smaller deviations. In the past, annual population estimates in developed countries were highly unreliable.