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Мир в экономических индексах, не зависящих от размера страны.
IX. Одномерные распределения стран по Индексу Перфектности и Индикатору Равновесия в 2016-2021 гг.
Сейдаметова Зарема Сейдалиевна, доктор педагогических наук, профессор
Темненко Валерий Анатольевич, кандидат физико-математических наук, доцент
Крымский инженерно-педагогический университет имени Февзи Якубова, Симферополь, Республика Крым
Цель исследования – изучение структурной устойчивости одномерных распределений стран по Индексу Перфектности pf и Индикатору Социально-Экономического Равновесия b в течение шестилетнего временного промежутка с 2016 по 2021 гг. Метод исследования – построение ежегодных одномерных распределений в виде Парето-диаграмм и гистограмм, и визуальное сопоставление этих ежегодных распределений. Выявлено, что эти одномерные распределения сохраняют свою структуру в течение рассматриваемого временного промежутка, хотя отдельные страны могут иметь значительные смещения вдоль этих распределений. Научная новизна заключается в предварительном, качественном выявлении самого факта структурной устойчивости такой неуправляемой и нерегулируемой сложной системы, как мировая экономика. Представленные данные позволяют в результате поставить задачу количественного описания этой структурной устойчивости и задачу эмпирического исследования устойчивости этой структурной устойчивости на более длительном временном промежутке.
Ключевые слова: мировая экономика; экономические индексы; Индекс Перфектности; Индекс Равновесия; одномерные распределения по экономическим индексам.
Цитировать: Seidametova Z.S., Temnenko V.A. The World in economic indices that do not depend on a country's size. IX. One-dimensional distributions of countries by Perfectness Index and by Equilibrium Indicator in 2016-2021 // KANT. – 2023. – №4(49). – С. 127-135. EDN: JDZVCI. DOI: 10.24923/2222-243X.2023-49.24
Seidametova Zarema Seidalievna, DSc of Pedagogical sciences, Professor
Temnenko Valerii Anatolievich, Ph.D. of Physics and Mathematical sciences, Associate Professor
Fevzi Yakubov Crimean Engineering-Pedagogical University, Simferopol, RC
The purpose of the research is to study of the structural stability of one-dimensional distributions of countries according to the Perfectness Index pf and the Socio-Economic Equilibrium Indicator b over a six-year time period from 2016 to 2021. The research method is the construction of annual one-dimensional distributions in the form of Pareto-diagrams and histograms, and visual comparison of these annual distributions. These one-dimensional distributions are found to maintain their structure over the time period under consideration, although individual countries may have significant shifts along these distributions. The scientific novelty lies in the preliminary, qualitative identification of the very fact of structural stability of such an uncontrollable and unregulated complex system as the world economy. The presented data make it possible, as a result, to pose the task of a quantitative description of this structural stability and the task of empirically studying the stability of this structural stability over a longer time period.
Keywords: world economy; economic indices; Perfectness Index; Equilibrium Index; one-dimensional distributions by economic indices.
УДК 339.97:330.43
5.2.5
Seidametova Z.S., Temnenko V.A.
The World in economic indices that do not depend on a country’s size.
IX. One-dimensional distributions of countries by Perfectness Index and by Equilibrium Indicator in 2016-2021
Introduction
In the paper [1] a new pair of economic indices was introduced, replacing the basic economic indices EPI and CPI when describing the world economy. Let us remind the reader that CPI is the Corruption Perception Index, determined annually using a special methodology based on data provided by several independent organizations and published on the website [2]. This is a hybrid expert assessment of the level of corruption in each country on a 100-point scale: the higher the CPI, the presumably lower the level of corruption in the country. It is obvious that corruption as a set of shadow, illegal processes cannot be measured directly. But the CPI has proven to be a very useful measure of the health of a society. There is a strong statistical relationship between the state of society, as measured by the CPI, and the capabilities of the economy, as measured by the EPI (see, for example, [1]). The Economic Productivity Index that we introduced in [3] has the following form:
EPI=(GDP/PC)/(max{GDP/PC})∙100 (%), (1)
where GDP/PC is the Gross Domestic Product per capita for each country, and max{GDP/PC} is the maximum value of GDP/PC achieved in some country in the world in the same year. The numerator and denominator in formula (1) must be expressed in current US dollars. The index EPI is expressed as a percentage, but we will usually omit the percentage symbol (%) when writing EPI.
The new pair of economic indices we introduced, the Perfectness Index pfand the Socio-Economic Equilibrium Indicator b are defined as follows:
pf=100∙√((r∙s)/max{r∙s} ), b=√(λ r/s). (2)
In formulas (2) r and s are “intermediate” variables introduced for convenience by transforming the indices:
r=10√EPI, s=(CPI/10)^2, (3)
and λ is a numerical constant, the choice of which is motivated in [1]: λ=√(2⁄5). The expression max{r∙s} in formulas (2) means the maximum value of the product r∙s, achieved in a certain country in the same year for which the indices r, s, pf and b are calculated.
The variables r and s, called “rectified” variables in [4], are normalized, like the basic variables CPI and EPI: the maximum possible value r is 100, the maximum value sis also 100. The paper [4] describes the statistical characteristics of the world economy in “rectified” indices {r, s} in 2016-2021.
If we substitute formulas (3) into formulas (2), we can obtain an explicit connection between the pair {b, pf} and the pair {CPI, EPI}, which does not contain “intermediate” indices {r, s}:
pf=100∙(CPI∙∜EPI)/max{CPI∙∜EPI} ,
b=〖10〗^(3/2)∙√λ∙∜EPI/CPI. (4)
It is significant that the pair of economic variables {b, pf} contains, instead of the index EPI, the fourth root of EPI: the Perfectness Index pf is proportional to the product ∜EPI and the index CPI, the Equilibrium Indicator b is proportional to the ratio ∜EPI to the index CPI. The linear statistical correlation between EPI and (CPI)^4 (or, which is the same, between ∜EPI and CPI) was first noted by us in [5] when studying the state of the world economy in 2017. In the paper [6], the correlations of the Perfectness Index pf with “rectified” economic indices {r, s} according to statistical data for 2016-2021 were studied in detail.
The purpose of this research is to study the structural stability of one-dimensional distributions of countries according to the Perfectness Index pf and the Equilibrium Indicator b. This goal structures the following tasks: 1) build annual Pareto diagrams of the distribution of countries according to the Perfectness Index pf according to statistical data for 2016-2021; 2) identify the characteristic features of each annual Pareto diagram in order to divide the world economy into Perfectness Groups, which we previously introduced based on statistical data from only one year (2021); 3) provide an approximate mathematical description of the Pareto diagram of the Perfectness Index using the Beta distribution; 4) construct annual histograms of the distribution of countries by the Perfectness Index pf in 2016-2021 and indicate the location of the Perfectness Groups on these histograms; 5) construct annual histograms of the distribution of countries according to the Equilibrium Indicator b according to statistical data for 2016-2021 and indicate on these histograms the intervals corresponding to some models of socio-economic equilibrium, first introduced according to statistical data of only one year (2021) [7]; 6) formulate a conclusion about the existence/non-existence of the structural stability of the world economy in 2016-2021.
Main part
Pareto diagrams of perfectness. The fig. 1 shows Pareto diagrams of the distribution of countries by the Perfectness Index pf for 2016-2021. On each Pareto diagram, countries are arranged in descending order of the Perfectness Index pf. The country’s number n in this list in descending order pf is called the perfectness ranking. The diagrams are compiled according to the data table given in the electronic supplement (https://t.ly/teD2a) to the paper [6]. The diagrams highlight countries that separate perfectness levels from each other. The idea of perfectness levels was introduced in [1] based on statistical data for 2021. The six perfectness levels introduced in [1] have the following names and abbreviations (in descending order of the Perfectness Index values): 1) VeryHigh (VH); 2) High (H); 3) UpperMiddle (UM); 4) LowMiddle (LM); 5) Low (L); 6) VeryLow (VL).
The names of the countries separating the levels of perfectness are shown in fig. 1 in abbreviated form, in accordance with the three-letter ISO standard for abbreviated country names [8].
For example, on the Pareto diagrams of perfectness for 2021, ARE (23) means United Arab United that having n=23 by perfectness in 2021. The abbreviation EST (24) means Estonia, which has a perfectness ranking of n=24 in 2021. Arab Emirates completes the list of 23 countries that make up the VeryHigh perfectness level in 2021. Estonia starts the list of 29 countries that make up the High perfectness level in 2021, which is completed by Croatia (HRV), which has a perfectness ranking n of 52. Bahrain (BHR), with a ranking of n=53, starts the list of 43 countries belonging to the UpperMiddle perfectness level. This list ends with Mexico (MEX, n=95).
Indonesia (IDN, n=96) starts the list of 25 LowMiddle countries, ending with the Philippines (n=120). Burkina Faso (BFA, n=121) begins the list of 31 countries belonging to the Low perfectness level in 2021. This list ends with Mali (MLI, n=151). The latest level of Perfectness, the VeryLow level, starts in 2021 with Niger (NER, n=152). In total, in 2021, the list of countries with known EPI and CPI values (and, therefore, with known Perfectness Index values pf) contains 176 countries.
The same markings displaying the boundaries of perfectness groups are contained in fig. 1 Pareto diagrams of perfectness for other years, starting from 2016.
Fig. 1 - Pareto diagram of the distribution of countries according to the Perfectness Index pf in 2016-2021
The concept of “levels of perfectness” is a rather formal, accounting concept. We introduced it for the convenience of describing a fairly large array of countries (about 170 countries annually) by structuring this array. But we tried to combine the boundaries between the levels of perfectness with some characteristic features of the Pareto diagram. For example, the boundary between the VeryHigh and High levels of perfectness is quite clearly expressed. It is tied to a visually noticeable jump between the values of the Perfectness Index pf of two countries – a country completing the highest level of perfectness VH, and a country starting the next level of perfectness, level H. It is thanks to this visually noticeable jump in perfectness in fig. 1 you can see that in 2016, the VeryHigh level of perfectness was contained in all 24 countries belonging to the EPI-group Hot (paper [9] describes the dynamics of the economic productivity ranking of the countries of the EPI-group Hot in 2016-2021). From 2017 to 2021 Israel, whose Corruption Perceptions Index CPI decreased annually (and, therefore, the Perfectness Index pf decreased) “fell out” from the list of countries belonging to the VeryHigh level of Perfectness and dropped to the High level of perfectness. After the “fallout” of Israel, 23 countries remained at the VH level. All of them belong to the EPI-group Hot. Estonia’s CPI index systematically increased in 2016-2021 and this country confidently took the top spot in the list of countries at the High level of Perfectness, displacing Israel lower on the list of countries at the High level.
The boundaries between some levels of perfectness in fig. 1 were set by us so as to formally correspond to certain values of the Perfectness Index pf. For example, pf10 separates the Low Perfectness level from the VeryLow level; pf15 separates the LowMiddle level of perfectness from the Low level, but in 2021 and, to a lesser extent in 2017, this boundary also corresponds to a visually detectable jump in the Perfectness Index during the transition between these levels.
A Perfectness Index pf20 separates the UpperMiddle level of perfectness from the LowMiddle level. At approximately the same value pf there would be an inflection point of some continuous smooth graph approximating the Pareto diagram.
The boundary between the High and UpperMiddle levels of perfectness formally corresponds to pf35. But in some years (2019, 2018) we shifted this boundary to a nearby and visually noticeable jump in the Perfectness index pf.
It can be seen that the perfectness rankings of those countries that mark the boundaries between perfectness levels do not change very significantly over the years, although the countries corresponding to these rankings may change from year to year.
This gives us some evidence of the structural stability of the Pareto diagram.
Anchor points of the Pareto diagram of perfectness. Table 1 presents the anchor points of the Pareto diagram of perfectness for 2016-2021. As such anchor points, perfectness rankings are selected corresponding to the Perfectness Index values from pf=90 to pf=10 with step pf=10. If the corresponding decimal value pf was between two countries according to the perfectness ranking, then the table shows both rankings – above and below the decimal value of the Perfectness Index separating them pf. If one of these two countries was much closer than the second to the decimal value separating them pf, then the table shows only the ranking of this closer country. This table 1 is compiled based on data from the electronic supplement (https://t.ly/teD2a) to the paper [6].
Table 1 - Anchor points of the Pareto diagram of perfectness (rankings of perfectness of countries closest to the decimal value of the perfectness index pf)
year
pf 2021 2020 2019 2018 2017 2016
90 5 5 5 – 6 5 4 – 5 5 – 6
80 10 – 11 10 12 12 13 14
70 19 – 20 20 20 19 – 20 20 20
60 24 – 25 24 – 25 24 – 25 24 24 24
50 32 – 33 34 – 35 36 37 34 32 – 33
40 45 – 46 45 – 46 46 44 – 45 45 45 – 46
30 62 – 63 63 – 64 64 – 65 63 – 64 63 – 64 64
20 95 – 96 96 – 97 94 – 95 96 – 97 94 88 – 89
10 151 – 152 151 153 152 148 – 149 144
Total number of countries N 176 170 167 170 169 168
Comparing the columns of Table 1 for different years is made difficult by the fact that the total number of countries N displayed on the Pareto diagram changed from year to year (see the last line of Table 1). But nevertheless, table. 1 quite convincingly demonstrates the structural stability of Pareto diagrams: the anchor points of the Pareto diagram hardly changed from year to year. This structural stability is especially clear at the top of the Pareto diagram of perfectness, at pf>20. The “tail” of the Pareto diagram at pf20 is less stable. This may be due to both significant real fluctuations in perfectness in the zone of low perfectness, and large relative errors of the indices EPI and CPI (and, therefore, perfectness) for countries with low productivity and high levels of corruption (see, for example, [4] and [6]). This almost fixedness of the anchor points means the structural stability of the general form of the Pareto diagram: no parts of this diagram “sag” or “tighten”. This, in turn, can be understood as the structural stability of the world economy - or at least a significant part of the world economy, for countries with Perfectness Index values pf>20 (this is about a hundred countries).
A clear idea of the structural stability of the global economy is given in fig. 2. This figure combines the six Pareto diagrams of perfectness shown in fig. 1 separately for each year from 2016 to 2021. Pareto diagrams for each year are shown with different symbols. This “summary” Pareto diagram has some “frayed” areas, corresponding to areas of noticeable annual changes in the distribution of countries by perfectness. But in general, according to fig. 2 we can draw a conclusion about the structural stability of the Pareto perfectness diagram.
Fig. 2 - Six combined annual Pareto Charts of Perfectness for 2016-2021
On the approximation of Pareto diagrams of perfectness using the Beta distribution. The functional form of the Pareto diagram of perfectness can be described as follows. Let’s move from an integer variable n (1nN) to a continuous variable x linearly related to n (x∈(0,1)). A continuous curve that approximates Pareto diagrams is a curve with one inflection point x=x_ip, concave at 0<x<x_ip and convex at x_ip<x<1. When x→0 we can assume that the curve (let’s denote it P(x)) grows without limit, and when x→1 P(x)→0, but the derivative dP/dx at x→1 increases without limit. There is one well-known function in mathematics that matches this description; it is used in probability theory and mathematical statistics. It is called Beta distribution [10], [11]:
P(x)=Cx^α (1-x)^(β-1), (5)
where 0<α<1; 1<β<2, and C is the normalization constant. In mathematical statistics, the constant C is determined from the condition of normalizing the integral of function (5) to unity. This integral is expressed in terms of Euler's Beta function, hence the name of this distribution. When applying a function P(x)to an approximation of the Pareto Diagram of Perfectness, the constant C can be determined using the Pareto Diagram itself. You can connect a continuous function of type (5) or some other type with the empirical Pareto diagram by assuming that with ideal approximation the equalities must be satisfied for all n:
P(x_n )=pf_n, (6)
where pf_n is the Perfectness Index for countries with a perfectness ranking equal to n.
In formula (6) we can put that
x_n=(n-b)/(N+a-b), (0<a<1; 0<b<1) (7)
where a and b are unknown in advance adjustment constants that describe the conditional “expansion” of the change interval n to the right (by the amount a) and to the left (by the amount b).
The problem of finding a continuous function that best approximates a discrete Pareto diagram of perfectness can be solved as the problem of minimizing the sum of squared differences P(x_n )-pf_n over all countries:
S=∑_(n=1)^N▒(P(x_n )-pf_n )^2 =min (8)
in a five-dimensional parameter space {C,α,β,a,b}.
This is a rather labor-intensive computational task, to which the authors intend to devote a separate publication. Such a computational determination of descriptive parameters {C,α,β,a,b} would be extremely useful if we had any independent mathematical-economic model that allows us to a priori calculate such structural functions of the world economy as Pareto Perfectness Diagrams, or Pareto Diagrams of the Economic Productivity Index EPI, or Pareto Diagrams for Corruption Perceptions Index CPI. Currently, such a model is unknown.
Histograms of the distribution of countries by the Perfectness Index in 2016-2021. Fig. 3 shows histograms of the distribution of countries according to the Perfectness Index in 2016-2021. These histograms contain the same information as their corresponding Pareto diagrams in fig. 1. But this information is presented in fig. 2 in a different form, more clearly demonstrating the discrete nature of the corresponding distribution. On each histogram, the entire interval of change in the Perfectness Index from 〖pf〗_max=100 to a certain value 〖pf〗_min (changing its value from year to year) is divided into 40 identical intervals. All intervals, except the first one, are semi-closed (the left edge of the interval is open, the right edge is closed). The leftmost first interval is closed. The height of the bar above each interval is determined by the number of countries with a Perfectness Index within that interval. The areas corresponding to the six perfectness groups are shown in each figure using an abbreviated name.
Fig. 3 - Histogram of the distribution of countries by the Perfectness Index pf in 2016-2021.
Abbreviated names of Perfectness groups: VH – VeryHigh, H – High, UM – UpperMiddle, LM – LowMiddle, L – Low, VL – VeryLow
We did not make quantitative comparisons of perfectness histograms across different years. But according to the visual impression, the histogram of perfectness is structurally stable, changing little from year to year, despite the obvious and time-varying discrete “ruggedness” of the histogram.
Histograms of the distribution of countries according to the Equilibrium Indicator b. Fig. 4 shows histograms of the distribution of countries according to the Socio-Economic Equilibrium Indicator b in 2016-2021. Each histogram has an interval on the axis b corresponding to countries with significantly disturbed socio-economic equilibrium b. The number of countries in this interval varies from year to year. This is partly due to the fact that some countries with low CPI and EPI values are not included in global databases every year. For example, South Sudan and Venezuela were on our list of countries with significant disequilibrium in 2021 and 2016. But in 2017-2020, the CPI value was not known for these countries (and, therefore, the values of the Perfectness Index pf and the Equilibrium Indicator b could not be calculated).
Fig. 4 - Histogram of the distribution of countries by Equilibrium Indicator b in 2016-2021.
D – interval on the axis b, corresponding to disequilibrium countries
In the histograms in fig. 4 in addition to the area D corresponding to countries with significantly disturbed equilibrium, fig. 4 shows the area Model III, corresponding to countries with a slight violation of socio-economic equilibrium. “Models of socio-economic equilibrium”, like some of the zones we have identified on the plane {Equilibrium Indicator b, Perfectness Index pf} are described in paper [7]. Countries with other equilibrium models (basic model or Model I, Model II, as well as countries with “underutilized economies”) in the histograms of fig. 3 are mixed in the area to the left of the interval corresponding to Model III.
In the narrow range of changes in the Equilibrium Indicator b on these histograms from b0.65 to b1.16 there is an almost constant number of countries (126-128 countries in 2017-2020, 120 countries in 2016, 135 countries in 2021). This interval corresponds predominantly to the Model I zone with a small admixture of countries with “underutilized” economies and a small group of countries with Model II socio-economic equilibrium.
From these descriptions, it can be concluded that the global economy was structurally stable in 2016-2021. A sign of such stability is the low variability in the distribution structure of countries according to the Equilibrium Indicator b.
Conclusions
The paper presents the authors’ arguments indicating the structural stability of the world economy in 2016-2021. These arguments are based on the analysis of one-dimensional distributions of countries on the Perfectness Index pf and the Socio-Economic Equilibrium Indicator b. In any case, we can confidently speak about the structural stability of a significant part of the world economy, including countries with a Perfectness Index pf>20 (this is about a hundred countries). This structural stability is probably not an eternal law of the world economy. It would be more correct to say that there are quite long periods of structural stability of the world economy. Empirically, we found such a period of six years.
The paper presents the idea of continuous approximation of the Pareto perfectness diagram using the Beta distribution. At present, it is difficult to judge whether the very possibility of using the Beta distribution in this context is simply a mathematical curiosity, or whether this possibility lifts the veil of secrecy over some fundamental law about the nature of competition in the world economy.
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