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Мир в экономических индексах, не зависящих от размера страны. III. Индекс Перфектности и Индикатор Социально-Экономического Равновесия
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 – введение новой пары экономических индексов: Индекс Перфектности pf и Индикатор Социально-Экономического Равновесия b. Это новая пара экономических индексов вводится с помощью двух последовательно выполняемых нелинейных преобразований координат на базовой плоскости экономических индексов {Индекс Экономической Продуктивности EPI; Индекс Восприятия Коррупции CPI}. С помощью этих преобразований выявлено, что большинство стран мира имеет значения индикатора равновесия b, принадлежащие очень узкому интервалу, что свидетельствует, по нашему мнению, о присущей многим странам тенденции к формированию некоторого вида социально-экономического равновесия. Научная новизна заключается в выявлении этой тенденции. Представленные данные позволяют в результате выявить фундаментальные черты распределения стран на плоскости {Индекс Перфектности; Индикатор Равновесия}.
Ключевые слова: мировая экономика; экономические индексы; «выпрямляющее» преобразование; преобразование «сжать и встряхнуть».
Цитировать: Seidametova Z.S., Temnenko V.A. The World in economic indices that do not depend on a country's size. III. The Perfectness Index and the Socio-Economic Equilibrium Indicator // KANT. – 2023. – №2(47). – С. 84-90. EDN: EAAPNP. DOI: 10.24923/2222-243X.2023-47.16
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 study is to introduce a new pair of economic indices: the Perfectness Index pf and the Indicator of Socio-Economic Equilibrium b. This new pair of economic indices is introduced using two successively performed non-linear coordinate transformations on the base plane of economic indices {Economic Productivity Index EPI; Corruption Perception Index CPI}. With the help of these transformations, it was revealed that most countries of the world have the values of the equilibrium indicator b, which belong to a very narrow interval. It indicates, in our opinion, a tendency inherent in many countries to form some kind of socio-economic equilibrium. The scientific novelty of the research lies in the identification of this tendency. The presented data make it possible, as a result, to identify the fundamental features of the distribution of countries on the plane {Perfectness Index; Equilibrium Indicator}.
Keywords: world economy; economic indices; “rectifying” transformation; “squeeze and shake up”-transformation.
УДК 339.97:330.43
ВАК РФ 5.2.5
Сейдаметова З.С., Темненко В.А.
Seidametova Z.S., Temnenko V.A.
The World in economic indices that do not depend on a country’s size. III. The Perfectness Index and the Socio-Economic Equilibrium Indicator
Introduction
In the papers [1] and [2], the state of the world economy in 2021 was presented and studied on the plane of economic indices {CPI, EPI} and in the three-dimensional space of economic indices {CPI, EPI, BLI}. The economic indices mentioned here have the following meaning. CPI is the Corruption Perception Index, borrowed by us from the database of the international organization Transparency International [3]. This is a hybrid expert index that indirectly assesses the level of corruption in the country on an integer hundred-point scale: the higher the index CPI, the lower the level of corruption in the country.
The Economic Productivity Index introduced in [4] is defined as follows:
EPI=(GDP/PC)/(max{GDP/PC})∙100 (%). (1)
The numerator of formula (1) contains the annual value of the Gross Domestic Product Per Capita of a given country, expressed in current US dollars. The denominator contains the maximum value of GDP/PC in the same year, achieved in some country of the world and also measured in current US dollars. The GDP/PC champion-country is determined among those countries for which the CPI index is known for the same year. Luxembourg has been such a champion-country in recent years (since 2016).
According to formula (1) the EPI index is expressed as a percentage, but we usually omit the percent symbol (%) when writing EPI.
The Budget Loading Index introduced in [4] (or simply “Budget Index”) BLI is defined as follows:
BLI=BR/GDP∙100 (%). (2)
The numerator of formula (2) contains the budget revenues of the government of the given country, and the denominator is the annual value of the Gross Domestic Product of the given country. The numerator and denominator in formula (2) must be measured in the same currency. The budget index is measured as a percentage, but we usually omit the percent symbol (%) when writing BLI. In the paper [1], we studied the distribution of countries on the basic plane of economic indices {CPI, EPI} and identified six discrete levels of economic productivity and, accordingly, six groups of countries belonging to these levels. These EPI-groups are named in descending EPI order: Hot, Warm, UpperCold (UC), MiddleCold (MC), LowCold (LC) and the Least Developed Countries (LDC). The paper [1] describes the distribution of countries by these EPI-groups.
The purpose of this study is to move from the CPI and EPI indices to a new pair of economic variables characterizing each country: the Perfectness Index pf and the Indicator of Socio-Economic Equilibrium b. This transition is based on the property of the world economy that we noted earlier [5], [6]: the EPI⁄(CPI)^4 ratio for most countries of the world changes in a very narrow range; it is a global “almost-constant”. There is a small group of countries that do not obey this rule and, apparently, have problems maintaining socio-economic equilibrium. This transformation of variables is performed in two steps. After performing the transformation, we consider the one-dimensional distribution of countries by pf and b. This goal: the transformation of variables and the description of the distribution of countries in new variables, structures the following tasks: 1) to perform a “rectifying” transformation of variables by passing from the pair {CPI, EPI} to the pair {s, r} in the form of two independent monotonic transformations for each of the indices CPI and EPI; 2) describe the distribution of countries on the plane “rectified” variables {s, r}; 3) describe the rule of statistical interconditionness of “rectified” economic indices based on the corresponding rule for the original indices {CPI, EPI}; 4) perform the second transformation of economic indices, passing from the pair {s, r} to the pair {pf, b}; 5) construct and describe a one-dimensional distribution of countries according to the Perfectness Index pf in the form of a Pareto diagram and identify six discrete levels of perfectness; 6) describe the “ranking shake-up” when moving from the ranking of countries according to the Economic Productivity Index EPI to the ranking of countries according to the Perfectness Index pf; 7) build and describe a histogram of the distribution of countries according to the Indicator of Socio-Economic Equilibrium b.
Main part
Rectifying transformation. Let us carry out the following transformation on the basic plane of economic variables {CPI, EPI}:
■(r=10√EPI,@s=(CPI/10)^2.) (3)
each of the two transformations (3) affects only one index; each transformation is monotonic (it does not change the order of countries according to the corresponding index); each transformation does not change the normalization of the index (the maximum possible value of r is 100, the maximum possible value of s is also 100).
We used the transition from EPI to √EPI earlier [7]. This transition slightly stretches the EPI-axis for EPI-groups with low EPI, allowing us to consider the details of the distribution of countries by economic productivity at low EPI.
The transition from CPI to (СPI)^2 in formulas (3) with the simultaneous transition from EPI to √EPI is dictated by the above-mentioned feature of the world economy. This feature in the language of the pair of indices {s, r} can be described as follows: for most countries of the world, the ratio r/s lies within a very narrow interval.
We have called transformation (3) a “rectifying” transformation. Fig. 1 shows the distribution of countries in the world in 2021 on the plane of “rectified” economic indices {s, r}. In this figure, the dotted line shows the straight line of the best fit. It can be seen that the EPI-groups are concentrated along this straight line. Fig. 1 shows the “normal ovals” for the three “senior in rank” EPI-groups (Hot, Warm, UpperCold). For other EPI-groups, “normal ovals” are not shown in fig. 1 so as not to clutter up the drawing. Nearly 80% of the countries of each EPI-group are in the “normal ovals” [2]. The same figure shows the lines along which the outsider countries of each EPI-group are located (countries with insufficient or excess values of the s-index for this EPI-group). Table 1 shows the Rule of statistical interconditionness of economic indices s and r for all EPI-groups: if a country belongs to this EPI-group, then with a probability of 80% its index s belongs to the normal range for s for this EPI-group. This table is built on the basis of the corresponding table 1 of the paper [3] with the transition from EPI and CPI to s and r according to formula (3).
Fig. 1. Distribution of countries on the plane of “rectified” economic indices {s,r} (2021).
straight line of the best approximation, ––– “normal ovals” for three EPI-groups, --- lines of outsiders of three EPI-groups with low/high values of s-index.
Table 1. Statistical conditionness rule for “rectified”
economic indices s and r for all EPI-groups (2021)
EPI-group name Range r Normal range for s
Hot [29.43; 100] [50.4; 77.4]
Warm [12.00; 26.69] [21.2; 46.2]
UpperCold [4.0; 11.12] [9.0; 34.8]
MiddleCold [2.0; 4.0) [9.0; 24.0]
LowCold [0.9; 2.0) [5.3; 18.5]
LDC [0.166; 0.9) [3.6; 17.6]
“Shake-up and squeeze” transformation. Let's now call this pair of variables {s, r} the “old” pair. Let's move from the “old” pair of economic variables {s, r} to another, new, pair of variables, pf and b, according to the following scheme. One of the new variables (pf) must represent the normalized geometric mean of the old variables s and r (i.e., must be proportional to √(r∙s)). The second “new” variable (b) must be proportional to the square root of the ratio of the “old” variables, √(r⁄s). We used a similar transformation scheme earlier when we described the picture of the world in economic indices depending on the size of the country [7]. Taking into account the normalization condition for the new variable pf, the transformation with the necessary properties can be written as follows:
■(pf=100√((r∙s)/max{r∙s} ) (%),@b=√(λ ( r )/s), ) (4)
where λ is a numerical factor that we will choose later, and max{r∙s} is the maximum value of the product of indices r and s achieved in some country in the world in a given year. We will name the variable pf as the “Perfectness Index”. This index is expressed as a percentage, but the percent symbol (%) will usually be omitted when writing pf. The maximum value of the Perfectness Index, in accordance with formulas (4), is 100%. The Perfectness Index equally weighted takes into account the state of the economy (described by the index r) and the state of society (described by the index s). For this reason, the term “Perfectness Index” seems to us appropriate, convenient for pf.
Earlier in [8] and [9] we introduced the so-called “parameter of socio-economic equilibrium a ̃” as follows:
a ̃=(4∙EPI)/(CPI)^4 ∙〖10〗^5. (5)
In the “Introduction” to this paper it is noted that the value thus defined is “almost constant”. The numerical factors in formula (5) are chosen so that for most countries a ̃ differs little from 1.0.
Using the definition of b in formula (4), raising this formula for b to the fourth power, we get that
b^4=λ^2 ( r^2 )/s^2 . (6)
Using formulas (3), which determine the indices r and s, from formula (6) we can obtain that
b^4=〖10〗^6 λ^2 ( EPI )/(CPI)^4 =5/2 λ^2 a ̃. (7)
If we choose λ=√(2⁄5) (0.6324…), then formula (7) implies a simple relationship connecting the quantities b and a ̃:
b=∜(a ̃ ). (8)
Since, as noted above, for most countries of the world a ̃ differs little from 1.0, then, according to (8), the value of b for most countries of the world also differs little from 1.0. We called the new economic variable b the “Indicator of Socio-Economic Equilibrium”. This is an unnormalized variable. This variable does not contain a comparison of this country with some other “reference” country. For these reasons, we prefer not to use the term “index” when naming this new economic variable. It is convenient to keep the term “Index” for normalized economic values. For variable b, we use the term “indicator”. There are only a few countries in the world for which the value of this indicator significantly exceeds one. This deviation of b from one indicates a significant violation of the socio-economic equilibrium in these countries.
Fig. 2 shows the distribution of countries on the plane of economic indices {r, pf} and {s, pf}. These distributions show a measure of the statistical correlation between the “new” Perfectness Index pf and the “old” indices r and s. In these figures, a diagonal is drawn with a dotted line (i.e., straight lines pf=r and pf=s). The solid line in these figures shows the straight line of the best fit. These figures show that the “new” variable pf correlates almost equally well with the “old” variables r and s. However, this correlation is characterized by a significant scatter. Transformation (4) quite noticeably “shakes up” the ranking of countries in terms of economic productivity r. Within each EPI-group, a country’s ranking in terms of economic productivity, r, and the country’s ranking in Perfectness, pf, can be very different if the country is not within the range of the norm in terms of the s-index (see table 1). For this reason, we call transformation (4) “shaking”. The second part of the transformation name (4) is the “squeezing” transformation. Transformation (4) dramatically changes the appearance of the distribution of countries on the plane of economic variables. On the {s, r} plane (see fig. 1), this distribution of countries is extended almost along the diagonal (r=s), but country-points “freely” fill some large part of the 100100 index square. Transformation (4), shifting the distribution of countries to the plane {b,pf}, drastically compresses this distribution along the b-axis and significantly shakes up the ranking of countries by the Perfectness Index pf compared to the ranking of countries by economic productivity r.
a
b
Fig. 2. Statistical correlation of the Perfectness Index pf with “rectified” indices r and s, 2021.
---- diagonal pf=r and pf=s, ––– straight line of least deviations. a. Distribution of countries on the plane of economic indices {r,pf}, b. Distribution of countries on the plane of economic indices {s,pf}.
The one-dimensional distribution of countries by Perfectness Index pf and Perfectness groups. Fig. 3 shows the Pareto-diagram of the distribution of countries according to the Perfectness Index pf in 2021. Fig. 3a is shown on a linear scale along the pf axis. Fig. 3b is presented on a logarithmic scale along the pf axis. The logarithmic scale allows us to see some details in the tail of the distribution at small pf. The vertical axis shows the Perfectness index, the horizontal axis shows the number of the country in descending order pf (a “country ranking by Perfectness”). When analyzing any one-dimensional distribution, it is convenient to divide it into a small number of intervals. In this case, we will call these intervals “levels of Perfectness”. The boundaries of these intervals are conveniently combined with any noticeable details of the Pareto-diagram. On the axis of perfectness, we have identified six discrete levels, naming them in order of shift from top to bottom along the axis of perfectness:
the level VeryHigh (VH),
the level High (H),
the level UpperMiddle (UM),
the level LowMiddle (LM),
the level Low (L),
the level VeryLow (VL).
All countries that are at a certain level of Perfectness form a Perfectness group with the same name.
a
b
Fig. 3. Pareto diagram of the distribution of countries according to the Perfectness Index pf, 2021. a. Linear scale along the perfectness axis pf, b. Logarithmic scale along the perfectness axis pf.
Fig. 3a and fig. 3b show the three-letter ISO [10] coded names of the countries that close the corresponding Perfectness groups.
The VeryHigh Perfectness group is closed by the United Arab Emirates (ARE, n=23, pf=64.65, b=0.875). There are 23 countries in this group. By composition, this is described in [1] and [2] the EPI-group Hot minus Israel. Israel has the lowest CPI in the EPI-group Hot (CPI=59), not in line with its very high productivity (EPI=39.05). With this discrepancy, the “shaking and squeezing” transformation (4) “pushed” Israel down the rankings (n(EPI)=15, n(pf)=26).
Croatia (HRV, n=52, pf=35, b=1.021) closes the High Perfectness group. There are 29 countries in this group. This is a fairly homogeneous group in terms of productivity: 24 of these 29 countries belong to the EPI-group Warm. One country, Israel, as mentioned above, “fell down” into this Perfectness group from above, from the EPI-group Hot, due to excessive levels of corruption (low CPI). Four countries, after the “shaking” transformation (4), “moved” into this Perfectness group from the EPI-group UpperCold, in which they had “unbalanced high” CPI. These are Seychelles (SYC, CPI=70), Costa Rica (CRI, CPI=58), St. Vincent and the Grenadines (VCT, CPI=59) and St. Lucia (LCA, CPI=56). The CPI-norm interval for the EPI-group UpperCold is [30; 59] (see [2]). Seychelles falls outside the normal range. The other three countries named here are near the upper limit of the norm for the EPI-group UC.
Mexico (MEX, n=95, pf=20.04, b=1.343) closes the list of countries at the UpperMiddle Perfectness level. There are 43 countries in this Perfectness group. This group of Perfectness is less homogeneous in productivity. Twenty-seven countries of them belong to the EPI-group UpperCold (this is about 2/3 of this EPI-group). Twelve countries in this Perfectness group belong to the EPI-group MiddleCold. Some of them (but not all) have excess CPI values for the EPI-group MiddleCold. One country jumped up to the UpperMiddle Perfectness level from the EPI-group LowCold. Three countries moved down in the rating to the UpperMiddle Perfectness level from the EPI-group Warm due to the excessive level of corruption for the EPI-group Warm (low CPI values). These are Bahrain (CPI=42), Hungary (CPI=43), Trinidad and Tobago (CPI=41), these CPI values are below the lower limit of the CPI-norm of the EPI-group Warm in 2021 (CPImin=46, see table 1 in paper [2]).
Philippines (PHL, n =120, pf=16.34, b=0.867) indicated in fig. 3 close the list of countries belonging to the LowMiddle Perfectness level. There are 25 countries in this level. Thirteen of them belong to the EPI-group MiddleCold. Five countries fell to this level of Perfectness from the EPI-group UpperCold due to the results of the “ranking shake-up”. All of them are countries with a very high level of corruption for the EPI-group UpperCold. The index CPI for these five countries is 29-31. The norm CPI for UpperCold is the interval [30; 59] (see table 1 in [2]). Six countries fell into this Perfectness level as a result of a “ranking shake-up” from the EPI-group LowCold. For these countries, the index CPI is close to the upper limit of the norm CPI for the EPI-group LowCold. One country moved up to this Perfectness level from the EPI-group LDC due to an exceptionally high CPI score for LDC. This is Rwanda (RWA, n(pf)=108, n(EPI)=159, CPI=53, pf=18.33, b=0.420). The Rwanda index CPI is much higher than the upper limit of the CPI-norm of the EPI-group LDC (CPImax=42).
The list of countries belonging to the Low Perfectness level closes Mali (MLI, n=151, pf=10.18, b=0.78). There are 32 countries at the Low Perfectness level. This level of Perfectness is a mixture of two types of countries: 1) countries with a very low CPI (high level of corruption) and more or less high productivity (EPI-groups UC, MC and LC); 2) countries with very low economic productivity (EPI-group LDC) and moderate or high CPI.
To the lowest level of Perfectness, the VeryLow level (pf<10) belong 26 countries. Eighteen of them are countries with the lowest productivity, i.e. countries of the EPI-group LDC. In addition to them, at this level of Perfectness there are countries with a low CPI value from the EPI-group LowCold and even one country from the EPI-group UpperCold with CPI=17 (Lybia).
Distribution of the “ranking shake-up”. Let n(EPI) is the number of the country in the descending list of the economic index EPI, and n(pf) is the number of the same country in the descending list of the Perfectness index pf. Denote by the difference of these numbers:
∆=n(EPI) – n(pf). (9)
Let's call the value of “ranking shake-up”.
Above, this phenomenon of “ranking shake-up” has been described for some specific countries. Fig. 4 shows a histogram of the distribution of the “ranking shake-up” in 2021. The entire interval of the ranking shake-up from the smallest value =-76 (Turkmenistan) to the largest value =+61 (Bhutan) when constructing the histogram in fig. 4 is divided into 40 identical small half-open intervals (the right end belongs to the interval, the left end does not, except for the leftmost closed interval). The height of the bar above each small interval is equal to the number of countries that have a “ranking shake-up” belonging to this small interval.
Fig. 4. Histogram of the distribution of the “ranking shake-up” during the transition from the country ranking by EPI to the country ranking by the perfectness index pf.
In the distribution of countries according to the “ranking shake-up”, 9 countries have large negative values of (<-30). These are countries that have an excessively low CPI compared to the norm of their EPI-group. Ten countries have large positive values (>30).
Each significant change in the ranking of one country leads to a change by one in the ranking of some sufficiently large group of countries: by +1 if this significant change in the ranking of one of the countries was negative and by -1 if it was positive. These, in a sense, random shakes, lead to the fact that the distribution of “ranking shake-up” in the fig. 4 in the central part, for a very large number of countries (about a hundred countries) is almost random (“almost Gaussian”).
Distribution of countries according to the Equilibrium Indicator b. The histogram of the distribution of countries according to the Indicator of Socio-Economic Equilibrium b is shown in fig. 5(a, b). Fig. 5a shows all countries of the world with known values of indicators b. This figure shows that there are a small number of countries with large values of b (b>1.45). These eight countries we will call non-equilibrium. It may be necessary to note some excess of countries in the narrow range from b1.2 to b1.4. There are several countries with very low b values, from b=0.42 to b0.6-0.7. For most countries of the world (130-140 countries), the Equilibrium Indicator b is focused on a fairly narrow range, from b0.7 to b1.2. Fig. 5b presents the same histogram of the distribution of countries according to the Equilibrium Indicator b on the interval [0.420, 1.215]. It can be seen that despite the discrete “jaggedness” of the histogram, the distribution of countries by b is quite evenly between b0.7 and b1.10. About 120 countries are located on this interval.
a
b
Fig. 5. Histogram of the distribution of countries according to the indicator of socio-economic equilibrium b (2021). a. The entire interval of change of the equilibrium indicator b is presented from b_min=0.420 (Rwanda) to b_max=2.278 (Equatorial Guinea) (178 countries). b. The interval of change of the equilibrium indicator is presented b from b_min=0.420 to b=1.215 (Paraguay) (154 countries).
This fact that we have established – the narrowness of the range of variation of the indicator b for most countries of the world – indicates as we think the existence of a fundamental law of the world economy: the law of the existence of a tendency towards a certain socio-economic equilibrium between the capabilities of society (they are measured by the index CPI) and the opportunities of the economy (they are measured by the index EPI). Recall that the ratio of the largest EPI to the smallest exceeds 500; the ratio of the largest CPI value in the world to the smallest is also quite a lot (eight-fold superiority). But built from them according to formulas (3) and (4) Indicator b is a global “almost constant” for most countries in the world. Perhaps the correct term to describe the “near-constancy” of b is the term “diffuse” constant.
Conclusions
The paper proposes the transformation of the base pair of economic indices CPI and EPI into a new pair of economic variables {Perfectness Index pf, Indicator of Socio-Economic Equilibrium b}. The one-dimensional distribution of countries according to the Perfectness index pf is studied. Six groups of Perfectness were singled out, the composition of the Perfectness groups was analyzed. A one-dimensional distribution of countries according to the equilibrium indicator b is described. The “narrowness” of this distribution is noted: about 120 countries have b values that belong to a narrow b interval between b0.7 and b =1.10. Only eight countries have b >1.45. We interpret this “narrowness” of the distribution of b as an indication of the existence of a global economic law – the tendency of many countries to maintain a certain kind of equilibrium between the state of society and the state of the economy. This tendency is spontaneous, natural and does not rely on any conscious decisions of governments, businesses or opinion leaders.
Natural science laws are often expressed in the existence of strictly defined eigenstates of physical systems and the existence of some well-defined physical constants that can be measured with high accuracy, sometimes in 8-10 decimal places. The laws of the global economy, apparently, are expressed in the existence of certain tendencies and the existence of some diffuse constants.
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