A Frequency Analysis of Economic Complexity (II)

The altered peaks and troughs of the sinusoida...

A non sinusoidal waveform displayed on a oscilloscope. (Photo credit: Wikipedia)

In a previous article, I discussed the possibility of using mathematical models in order to predict the evolution of the economy of a company or a country. In this article, I want to continue that discussion from a more real case. In the former case, the benefit function was the integration of three different frequencies, trying to simulate the effect of periodical monthly, quarterly, and yearly events, however, the work was done with single-frequency signals but in a real case payments are done in single packets instead of being distributed through a sinusoidal wave.

In order to work with a real case, I will simulate the evolution of a little business with the following behavior:

          A single employee that receive his salary the last day of the month.

          A single client that pays every three months the first day of the quarter.

          A margin of thirty percent for the incomes.

          A single payment of taxes on benefits of twenty five percent, the last day of the year.

And a second case with the same quantity of incomes but perceived daily.

The objective of these two cases is analyzing the evolution of the effect of the uncertainty on a possible model.

The benefit function of the former case would have the following aspect:

Generated Periodic Benefit Function in the domains of time and frequency. Trimestral case

Graph 1 Generated Benefit Function in the domains of time and frequency. Trimestral Case

As we can see the signal is not totally band limited but most of the energy is concentrated in the low frequencies as it was expected, the reason is that incomes and expenses are done instantly and the frequency response of the kind of single (impulse) is flat all over the spectrum.

The second example is similar but in that case is incomes are constant, the frequency response of them produces only energy at zero frequency. The signal has more energy at the low frequencies and less energy at the higher ones, if we remember the previous article, this fact implies that the business should be much more robust and predictable.

Generated Periodic Benefit Function in the domains of time and frequency. Daily case

Graph 2 Generated Benefit Function in the domains of time and frequency. Daily case

In order to show you that fact, I am going to introduce the effect of the uncertainty. I am going to consider that the costs are fixed (salaries usually are) but the incomes proceed from a market with high uncertainty, although the mean value of our incomes is preserved, they can change up or down a fifty percent (a very high uncertainty). We can see the effect of that kind of uncertainty related to incomes at the following graphs.

 Original and noisy signals for trimestral and daily incomes

Graph 3 Original (blue) and noisy (green) signals for trimestral and daily incomes

This graph shows that the daily incomes reproduce a more similar behavior between the theoretical situation and the uncertain one, as we expected from the difference between low and high frequencies.

On the other hand, it is interesting to analyze our capability to make a model of both businesses. In order to do this, we are going to try to look at the differences between the real behavior and a model extracted from only low frequencies. In order to do this, we will filter the signal at the frequency domain with a low pass filter, and we will compare the result at the time domain with the initial signal. Thinking about Nyquist’s frequency, and our most frequent event is every thirty days, Nyquist’s frequency should be events every fifteen days however, we know that the signal will not be recovered exactly due to the harmonics of impulse inputs.

Noisy signals and filtered noisy signals at a frequency of events with a period of fifteen days

Graph 4 Original noisy signals (blue) and filtered noisy signals (green) at a cutting frequency of events with a period of fifteen days

Additionally, I am going to include some harmonics increasing the cutting frequency for events produced every six days (near a week).

 Noisy signals and filtered noisy signals at a frequency of events with a period of six days

Graph 5 Original noisy signals (blue) and filtered noisy signals (green) at a cutting frequency of events with a period of six days


As we can see it is possible to recover the behavior of the business, and it seems possible to make a mathematical model of this system if we obtain data of the business every week, and visually it is better the result of the second case.

I know that this discussion is not done in the language of economists, and that is the reason why now I am going to translate it talking about the conclusions.

Most businesses are much more complex that these ones, but we can make some analogies, for instance, the initial model can be similar to the way as a national state works. It pays salaries every month and gets incomes from VAT every three months and finally yearly from tax on rents and societies. Of course there are many other expenses of different periods that would not be considered here. The second example is more similar to great private companies of consumption goods, it gets incomes any time and it pays periodically. From this analogy we can find that private management is more robust (in the complexity sense) than public one in most cases, namely, the way that they get their incomes make them less exposed to unexpected events (as showed Graph 3).

Another conclusions that we can see, is that increasing VAT is better than increasing other taxes as taxes on rent or societies with higher periodicity in order to improve the manageability of public accountancy. Then, this is not a happy idea of officials at Brussels, this can be demonstrated mathematically.

Looking at Graph 5, when we cannot discriminate well between signal and noise (between our capability to get incomes and the uncertainty of the markets) we cannot get a model to predict the future. We have modeled well the past behavior of the first case, but it does not imply that we can extrapolate it to forecast the future (see Graph 3). With the second case, the effect of the uncertainty is integrated and we could get a more reasonable model.

As many people know investments in highly volatile markets as stock exchange only following the expected future results of the business has mathematical sense if it is done thinking on the long term, because the information of the business usually is sampled quarterly instead of daily, however, the stock exchange has the advantage of letting us to adjust our investments daily to every event that we can receive from the economic press.

In the same way as daily incomes produce more robust businesses even under an uncertain scenario, having the capability to act on the costs (with a more flexible labor market) can produce more robust business, again, this is not a happy idea of the officials at Brussels, this is perfectly shown through mathematics, but on the other hand, a problem of public debt should be approached making more flexible the public costs and the staff of the governments, instead of acting only increasing taxes. This has the same mathematical sense but it is not usually considered by any public official at Brussels or at the national states.



 Chaqueta Azul Azul Lagrimas 0003


Mr. Luis Díaz Saco

Executive President


advanced consultancy services


   Nowadays, he is Executive President of Saconsulting Advanced Consultancy Services. He has been Head of Corporate Technology Innovation at Soluziona Quality and Environment and R & D & I Project Auditor of AENOR. He has acted as Innovation Advisor of Endesa Red representing that company in Workgroups of the Smartgrids Technology Platform at Brussels collaborating to prepare the research strategy of the UE



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