Revisiting the Kuznets Curve: A New Approach to Describe a Bilateral Nonlinear Kuznets Curve

doi:10.21203/rs.3.rs-2491535/v1

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Revisiting the Kuznets Curve: A New Approach to Describe a Bilateral Nonlinear Kuznets Curve

https://doi.org/10.21203/rs.3.rs-2491535/v1

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The prior studies regarding the Kuznets curve have obtained inconsistent results regarding the existence of the Kuznets curve. One of the possible reasons for this inconsistency might be a specification error in modeling the Kuznets curve. In other words, the one-sided causality from per capita income to inequality might not make sense. In this paper, we assume a two-sided causality in the per capita income-inequality nexus. Also, we assumed a nonlinear relationship between per-capita income and income inequality as a quadratic form of these two variables. In other words, changes in per capita income may have a nonlinear impact on the inequality level, and inequality levels can determine per-capita income in a nonlinear fashion. For justifying this two-sided nonlinear relationship which means a bilateral nonlinear Kuznets curve, firstly, we have reviewed the literature to explain this type of per capita income-inequality nexus. Second, we have justified this nonlinear relationship by the wavelet coherence approach and finally, we have applied our hypothesis in a dynamic simultaneous panel data model with a quadratic form of both per capita income and inequality. We have used two samples of countries, including G7 + BRICS during 1970–2019, and 129 countries consist of both developing and developed countries during the period of 1980–2019. The empirical results of the wavelet coherence approach indicate that there are two different phases including the same-phase and opposite phase in different time horizons which might reflect the possibility of a nonlinear relationship in per capita income-inequality nexus as well as a two sided causality between per capita income and inequality. Finally, we have estimated different simultaneous equations with different inequality indexes during the different periods using the GMM method. The empirical results indicate that there is a bilateral nonlinear relationship in per capita income-inequality nexus in both samples and different periods. In other words, the results support a bilateral nonlinear Kuznets curve.

JEL Classification: D63, D31, C49, C33

Kuznets Curve

Inequality

Per-Capita Income

Wavelet Coherence Approach

Simultaneous Panel Data Model

The inequality-per capita income nexus has attracted a lot of attention from many researchers and policy makers in the last decades. It is very important for policy makers to have a precise knowledge about the nexus between the inequality and per capita income to make economic decisions. For instance, if per capita income has a beneficial impact on income inequality, policy makers can easily promote high economic growth policies. However, in the case of a detrimental impact on income inequality, policy makers should opt between the high economic growth and low inequality policies. Furthermore, if the nexus is nonlinear, as suggested by the famous Kuznets curve, policy makers can promote growth-enhancing policies with the objective to achieve low income inequality in the long-run but at the cost of a non-permanent increase in inequality in the short-run. Therefore, it is vital to know about the per capita income-inequality nexus to design successful policies from a social welfare perspective. To meet this objective, finding the suitable econometric specification plays a critical role. For modeling the Kuznets curve, Kuznets (1955) has assumed a one-sided causality from per-capita income to income inequality, so the possibility of two-sided causality between per-capita income and income inequality has been ignored. In this study, we assume a two-sided causality between per-capita income and income inequality to achieve a more realistic model of Kuznets curve. Also, the literature, that will be reviewed in next section, suggests that the per-capita income-inequality nexus can be nonlinear in both sides. In other words, changes in mean income may have a nonlinear impact on the inequality level (Kuznets, 1955), and inequality levels can determine per-capita income in a nonlinear fashion. Therefore, it is suitable to consider the per-capita income-inequality nexus as a nonlinear bilateral relationship that both may have a nonlinear inverse U-shaped impact on each other.

Therefore, in this paper, our novelty is modeling Kuznets curve by adding the assumption of two-sided causality between per-capita income and income inequality as a nonlinear bilateral relationship. For justifying our hypothesis, firstly, we use a wavelet coherence approach and second, we use a simultaneous panel data with a quadratic form of both inequality and per-capita income to capture nonlinearities between these two important variables.

This paper is organized by five sectors. The next section is devoted to review of literature. In section 3, we present the methodology and in section 4, we show the empirical results. Final section is devoted to conclusion.

The study of Kuznets (1955) about the inequality-per-capita income nexus, which is one of the most important relationships, has opened a new chapter in economics. Kuznets (1955) speculated an inverse U-shaped relationship between the inequality level and per capita income that was later confirmed by Gupta & Singh (1984), Campano & Salvatore (1988), Jha (1996), Thornton, (2001) and Bahmani-Oskooee & Gelan (2008). The intuition is the following: In the first stage of the development process, the per capita income increases as a result of the emergence of the industrial sector and movements of workers from the agriculture sector to the industrial sector. However, this process results in the increasing income inequality because of the different wage levels between these two sectors. After this first stage and the development of the industrial sector in economy, most workers will move to the industrial sector and benefit from high wages: increasing per capita income is then associated with decreasing income inequality.

To explain the effect of mean income changes on inequality, many hypotheses including international trade, technical change and organizational change have been proposed. They mostly explain that these processes alter the wage differential between skilled and unskilled workers, leading to the inequality of wages between workers (Aghion et al., 1999).

In the other hands, there are some studies which have not confirmed the Kuznets curve hypothesis like Saith (1983), Papanek & Kyn (1986), Anand & Kanbur (1993), Toniolo et.al (1999) and Desbordes & Verardi (2012). So, the contrary results in the prior studies may be arisen by a specification error in modeling Kuznets curve which assumes a one-sided causality from per-capita-income to inequality and the inequality effects on the per capita income have been ignored. In the other hands, inequality also affects per capita income (Galor, & Zeira, 1993, Brueckner, & Lederman, 2015). So, when we consider per-capita income-inequality nexus, it is reasonable to assume a bilateral model rather than a one-sided causality model like the Kuznets. There are different mechanism regarding the impact of inequality on per capita income. First, there are some reasons regarding possible positive effects of inequality on per capita income that probably these effects occur in low inequality regimes:

a-According to conventional economics, inequality foster investment by increasing incentives for producers and entrepreneurs (Aghion et al, 1999). So, inequality may increase per-capita income as well as economic growth.

b- Also, according to conventional economics again, inequality increase aggregate saving (Koo, & Song,2016) which results in the increase in investment, and therefore, accelerates the production, per-capita income and growth. The positive inequality-saving nexus is a key factor in neoclassical growth model of Solow (1956) as well as in neo-Keynesian growth models including Lewis (1954), Kaldor (1957) and Pasinetti (1962) growth models (Schmidt-Hebbel, & Serven, 2000).

Second, there are some reasons regarding possible negative effects of inequality on per capita income that probably these effects occur in high inequality regimes:

a- Political and economic instabilities due to rising inequalities deter investment and obstructs the social consensus (Alesina and Perotti, 1996; Knack and Keefer, 2000). So, inequality might have a negative impact on per-capita income especially in a country with a high inequality.

b- According to the human capital accumulation theory, inequalities reduce accumulation of human capital by the poor as a result of investment delays in human capital (Galor and Zeira ,1993, 1998).

c- According to the development’s book of Todaro (1997), high inequality increases unproductive investment by the rich. Also, in developing countries, the rich decades consume more luxury or imported goods and they invest their savings on some developed counties that this phenomenon is known as capital flight (Todaro and Smith, 2011).

d- Inequality may decrease the incentives for investment as a result of the endogenous fiscal policy theory (Bertola 1993; Bénabou, 1996; Perotti 1996; Cingano, 2014). In other words, higher inequality does not accept by voters and they pressure to policy makers to do redistribution policies including increasing taxation on rich people that this policy may decrease incentive for investment and it has a diminishing impact on per-capita income and growth.

In some developing countries, such as Zimbabwe and Zambia, we see high inequality and low per capita income but in developed countries like France, Germany and Canada, we see low inequality and high per capita income. Therefore, it is possible to exist a nonlinear relationship between income inequality and per capita income at a cross country level including both developing and developed countries. In other words, in countries with low inequality, we can see a positive relationship between inequality and per capita income, but after inequality exceeds its threshold, we see the opposite effect of inequality on per capita income in countries with high inequality. Thus, income inequality can be divided into low and high inequality regimes. In a low inequality regime, increasing inequality can be seen by increasing per capita income by increasing the savings of high-income groups and turning it into investment as a result of a developed financial system. This will also increase the motivation of investors and entrepreneurs because more entrepreneurial effort is met with more income. So, income inequality acts as a positive sign to encourage entrepreneurship and more effort. In other words, due to the inherent difference in human talents, some of the inequality in income distribution due to the inherent inequality in the talent or effort of individuals is considered a natural and necessary for the economic growth and development process. But crossing the threshold of inequality and being in a regime of high inequality reduces the motivation of the lower and middle classes to work harder, because they see a great distance to reach the upper classes, and this great distance motivates them to reduce further effort. On the other hand, the existence of severe inequalities in income distribution can reflect the unequal distribution of opportunities and facilities, including access to quality education, health and well-being in society. Also, in the regime of high inequality, we see social and political instabilities and protests of the people which increases the risk of investing in these countries. Also, the pressure of voters to decrease high inequality may cause that the politicians peruse the policies in favor of the endogenous fiscal policy theory in democratic countries or even populist policies in some developing countries with weak democracy that harm production, per-capita income and growth. Also, according to the theory of limited access order that introduced by North (2007), in some developing countries, the politicians using unequal distribution of rents- which increase income inequality- can control violence and establish order and security, but the cost of this policy is underdevelopment and low level of per-capita income.

Therefore, we believe that there is a nonlinear relationship between inequality and per-capita income as Fig. 1. In other words, there is a threshold level of inequality which maximizes per-capita income. Before this threshold level, inequality can increase per-capita income but after this threshold level, inequality decreases per-capita income. Our description regarding per-capita income-inequality can solve the ambiguous results of prior studies regarding the Kuznets curve.

First of all, we use a wavelet coherence approach¹ to consider coherency between per capita income and inequality. To do it, we have used the package of Biwavelet in R. The wavelet coherence approach is an Econophysics approach to consider the correlation between two variables at different time scales. This method can identify the leading or lagging variables which is a useful metrics to specify the causality between variables. Also, the wavelet coherence approach can capture co-movement between per capita income and inequality in both time and frequency domains. The wavelet coherency can be applied in both conditions of nonstationary and structural breaks in the time series (Chang et al, 2018). The continuous wavelet transform is defined as:

${\text{W}}_{\text{X}}\left(\text{u},\text{s}\right)={\int }_{-{\infty }}^{+{\infty }}\frac{1}{\sqrt{\text{s}}}\stackrel{-}{{\Psi }\left(\frac{\text{t}-\text{u}}{\text{s}}\right)} \text{d}\text{t}$ , $(u, s)\mathcal{ }\in \mathcal{ }\mathcal{R}$ and s > 0 (1)

where s is the scale of the wavelet and u is the shift of the wavelet in time (Khochiani and Nademi, 2020). Using the continuous wavelet transform, we can obtain a figure like Fig. 2 for relationship between per capita income and inequality.

In Fig. 2, there are some instruments that help us to analyze the results. The direction of arrows in the figures of wavelet results show the situation of phases of the two variables. For instance, if the direction of arrows is on the right side, we can say that the variables are in the same phase or they have a positive correlation. On the other hand, if the direction of arrows is on the left side, we can say that the variables are in the opposite phases or they have a negative correlation. Regarding the lagging or leading variables, if the direction of arrows is up, this means that the second variable causes the first variable or the second variable is a leading variable and if the direction of arrows is down, this means that the first variable causes the first variables or the first variable is a leading variable. Also, the area with warm or dark colors indicates a strong coherency between the variables and the area with cold or light colors indicates a weak coherency between the variables.

Then, we use the econometric methodology to model the per-capita income-inequality nexus. In econometric modeling, the model is obtained from theories and literature reviews. According to the literature review, we have applied simultaneous equations models by adding the quadratic form of both per-capita income and inequality to test the existence of a bilateral nonlinear relationship between per-capita income and inequality.

We suggest dynamic simultaneous panel data models as follows:

$${Log(Y}_{it})=\alpha +\theta Log({Y}_{it-1})+{\beta }_{0}{Inequality}_{it}+{\beta }_{1}{{Inequality}_{it}}^{2}+{\epsilon }_{it}$$

$${Inequality}_{it}=\alpha +\phi {Inequality}_{it-1}+{\beta }_{0}Log\left({Y}_{it}\right)+{\beta }_{1}(Log{{Y}_{it})}^{2}+{ϵ}_{it}$$

Where ${Log(Y}_{it})$ is the logarithm of per-capita income, ${Inequality}_{it}$ is the income inequality index, ${{Inequality}_{it}}^{2}$is the square of income inequality, $(Log{{Y}_{it})}^{2}$ is the square of Logarithm of per-capita income. $Log\left({Y}_{it-1}\right)$ is the first lag of logarithm per-capita income and ${Inequality}_{it-1}$ is the first lag of income inequality index. The letters $i and t$ denote country and time, respectively. For the inequality index, we have used three different indexes, including the Gini coefficient, top 10% share, and top 1% share.² For per capita income, we have used the per-adult national income³.

For estimation of the model, we have used the system GMM method. Also, for system estimation of GMM method, we have used the first to the third lags of both dependent and explanatory variables as the instrumental variables. GMM method has several benefits including dealing with endegeneity problem and capturing the dynamic behavior of dependent variables.

Also, the data have been collected from the World Inequality Database (WID)⁴. We have applied two different samples, including G7⁵ + BRICS⁶ countries and 129 countries⁷ including both developing and developed countries during the 1980–2019 period. The descriptive statistics of all variables in both samples have been presented in Tables 1 and 2.

Table 1

Descriptive Statistics of the Variables in the First Sample or G7 + BRICS Countries
Variable	Mean	Median	Min	Max	St. Dev.
${{L}{o}{g}({Y}}_{{i}{t}})$	9.75	10.04	7.42	10.87	0.843
Gini Coefficient	0.51	0.49	0.31	0.74	0.090
Top 10% Share	0.38	0.36	0.21	0.65	0.095
Top 1% Share	0.12	0.10	0.03	0.30	0.056
Source: Own Calculation

Table 2

Descriptive Statistics of the Variables in the Second Sample or 129 countries
Variable	Mean	Median	Min	Max	St. Dev.
${{L}{o}{g}({Y}}_{{i}{t}})$	12.02	11.65	4.34	20.30	2.47
Gini Coefficient	0.52	0.53	0.21	0.75	0.086
Top 10% Share	0.41	0.42	0.07	0.69	0.098
Top 1% Share	0.13	0.13	0.02	0.43	0.04
Source: Own Calculation

[1] . For more information about the wavelet coherence approach, please see Torrence & Compo (1998).

[2] . Methodology of calculation of inequality indexes can be found in https://wid.world/methodology/.

[3] . Calculated by Euro/Constant 2020/PPP.

[4] . https://wid.world/

[5] . G7 countries include the USA, UK, France, Germany, Italy, Canada, and Japan.

[6] . BRICS countries include Brazil, Russia, India, China, and South Africa.

[7] . Afghanistan, Albania, Algeria, Angola, Argentina, Armenia, Australia, Austria, Azerbaijan, Bahrain, Bangladesh, Belarus, Belgium, Benin, Bhutan, Bosnia and Herzegovina, Brunei Darussalam, Bulgaria, Burundi, Cambodia, Canada, Chad, China, Comoros, Costa Rica, Croatia, Cuba, Cyprus, Czech Republic, Denmark, Djibouti, DR Congo, Ecuador, Egypt, Equatorial Guinea, Estonia, Ethiopia, Finland, France, Gabon, Georgia, Germany, Ghana, Greece, Guinea, Hungary, Iceland, India, Indonesia, Iran, Iraq, Ireland, Israel, Italy, Japan, Jordan, Kazakhstan, Korea, Kosovo, Kuwait, Kyrgyzstan, Latvia, Lebanon, Liberia, Libya, Lithuania, Luxembourg, Macao, Malaysia, Malta, Mauritania, Mauritius, Moldova, Mongolia, Montenegro, Morocco, Myanmar, Nepal, Netherlands, New Zealand, Niger, Nigeria, North Korea, North Macedonia, Norway, Pakistan, Palestine, Papua New Guinea, Paraguay, Philippines, Poland, Portugal, Qatar, Romania, Russian Federation, Rwanda, Sao Tome and Principe, Saudi Arabia, Serbia, Seychelles, Singapore, Slovakia, Slovenia, South Africa, South Sudan, Spain, Sri Lanka, Sudan, Sweden, Switzerland, Syrian Arab Republic, Taiwan, Tajikistan, Tanzania, Timor-Leste, Togo, Trinidad and Tobago, Tunisia, Turkey, Turkmenistan, Uganda, Ukraine, United Kingdom, Uruguay, USA, Uzbekistan, Viet Nam, Yemen, and Zanzibar.

First of all, we have presented the results of wavelet coherency between per capita income and inequality indexes. Figures 2, 3, and 4 indicate wavelet coherency between the logarithm of per capita income and different inequality indexes including Gini coefficient, top 10% share, and top 1% share in the both G7 and BRICS countries during the 1980–2019 period. Also, the same figures for 129 countries have been shown in Figs. 5, 6, and 7. These figures show that there are two different phases, including the same-phase or a positive correlation and the opposite-phase or a negative correlation in different time horizons, which might reflect the possibility of a nonlinear relationship in per capita income-inequality nexus because, in a nonlinear relationship like inverse U-shape, we confront with two different positive and negative relationships. Also, the direction of the arrows cannot show the leading or lagging variables that this may be because of a two-way relationship between per capita income and inequality. So, the results of the wavelet coherency approach between per capita income and inequality indexes might support our hypothesis that there is a two-sided causality between per-capita income and income inequality as a nonlinear bilateral relationship. Therefore, the results of the wavelet coherence approach might be a sensible reason for modeling a simultaneous equation model with a quadratic form of both per capita income and inequality. In other words, we have used the econophysics approach to justify econometric modeling, which can be one of our novelties in this paper.

In the next step, we have presented the estimation results of econometric models. Before presenting the results, we have tested unit root for all variables. Table 3 shows the Levin, Lin & Chu Test for all variables of the samples.

Table 3

Unit Root Test (Levin, Lin & Chu Test)
Variable	Sample = G7 + BRICS			Sample = 129 Countries
Variable	Levin, Lin & Chu t*	Test Type	P-Value	Levin, Lin & Chu t*	Test Type	P-Value
${Log(Y}_{it})$	-2.25	Intercept and Trend	0.01	-3.07	Intercept and Trend	0.00
$Log\left({Y}_{it-1}\right)$	-1.77	Intercept and Trend	0.03	-2.61	Intercept and Trend	0.00
$(Log{{Y}_{it})}^{2}$	-1.73	Intercept and Trend	0.04	-3.26	Intercept and Trend	0.00
Gini Coefficient	-1.82	Intercept	0.03	-7.68	Intercept	0.00
The First Lag of Gini Coefficient	-1.78	Intercept	0.03	-6.28	Intercept	0.00
Square of Gini Coefficient	-1.66	Intercept	0.04	-8.25	Intercept	0.00
Top 10% Share	-2.18	Intercept and Trend	0.01	-196.49	Intercept	0.00
The First Lag of Top 10% Share	-2.24	Intercept and Trend	0.01	-196.52	Intercept	0.00
Square of Top 10% Share	-2.47	Intercept and Trend	0.00	-160.96	Intercept	0.00
Top 1% Share	-2.82	Intercept and Trend	0.00	-3.49	Intercept	0.00
The First Lag of Top 1% Share	-2.72	Intercept and Trend	0.00	-1.77	Intercept	0.03
Square of Top 1% Share	-2.43	Intercept and Trend	0.00	-5.37	Intercept and Trend	0.00
Source: Own Calculation

The results of Table 3 indicate stationary in all variables in both samples. So, in the next step, we have estimated different econometric models for the two samples.

Table 4

Estimation Results in G7 + BRICS countries during 1970–2019
Equation	Variables	Model 1 (Inequality Index = Gini Coefficient)			Model 2 (Inequality Index = Top 10% Share)			Model 3 (Inequality Index = Top 1% Share)
Equation	Variables	Coefficient	St. Error	Prob	Coefficient	St. Error	Prob	Coefficient	St. Error	Prob
Equation 1	Intercept	-0.0049	0.03	0.89	0.06	0.02	0.00	0.10	0.017	0.00
	$Log\left({Y}_{it-1}\right)$	0.98	0.001	0.00	0.98	0.001	0.00	0.98	0.001	0.00
	${Inequality}_{it}$	0.53	0.12	0.00	0.30	0.07	0.00	0.21	0.10	0.03
	${{Inequality}_{it}}^{2}$	-0.51	0.11	0.00	-0.38	0.08	0.00	-0.72	0.33	0.02
Equation 2	Intercept	-0.089	0.05	0.08	-0.07	0.04	0.13	-0.07	0.03	0.03
	${Inequality}_{it-1}$	0.99	0.005	0.00	1.003	0.003	0.00	1.002	0.006	0.00
	$Log\left({Y}_{it}\right)$	0.02	0.01	0.04	0.017	0.01	0.09	0.01	0.008	0.02
	$(Log{{Y}_{it})}^{2}$	-0.001	0.0006	0.03	-0.0009	0.0005	0.08	-0.0009	0.0004	0.02
J-Statistic		0.13			0.22			0.14
R-Square		0.99			0.99			0.97
Source: Own Calculation

Table 5

Estimation Results in G7 + BRICS countries during 1980–2019
Equation	Variables	Model 1 (Inequality Index = Gini Coefficient)			Model 2 (Inequality Index = Top 10% Share)			Model 3 (Inequality Index = Top 1% Share)
Equation	Variables	Coefficient	St. Error	Prob	Coefficient	St. Error	Prob	Coefficient	St. Error	Prob
Equation 1	Intercept	-0.007	0.04	0.86	0.05	0.02	0.01	0.10	0.01	0.00
	$Log\left({Y}_{it-1}\right)$	0.98	0.001	0.00	0.98	0.001	0.00	0.98	0.001	0.00
	${Inequality}_{it}$	0.57	0.14	0.00	0.32	0.08	0.00	0.377	0.10	0.00
	${{Inequality}_{it}}^{2}$	-0.54	0.12	0.00	-0.405	0.09	0.00	-1.19	0.33	0.00
Equation 2	Intercept	-0.089	0.05	0.12	-0.06	0.04	0.12	-0.06	0.04	0.13
	${Inequality}_{it-1}$	0.98	0.005	0.00	0.99	0.003	0.00	0.99	0.006	0.00
	$Log\left({Y}_{it}\right)$	0.02	0.01	0.07	0.017	0.009	0.06	0.015	0.009	0.09
	$(Log{{Y}_{it})}^{2}$	-0.001	0.0006	0.05	-0.001	0.0005	0.04	-0.0008	0.0004	0.07
J-Statistic		0.14			0.09			0.14
R-Square		0.99			0.99			0.99
Source: Own Calculation

Table 6

Estimation Results in G7 + BRICS countries during 1990–2019
Equation	Variables	Model 1 (Inequality Index = Gini Coefficient)			Model 2 (Inequality Index = Top 10% Share)			Model 3 (Inequality Index = Top 1% Share)
Equation	Variables	Coefficient	St. Error	Prob	Coefficient	St. Error	Prob	Coefficient	St. Error	Prob
Equation 1	Intercept	-0.15	0.04	0.00	-0.0005	0.03	0.98	0.095	0.02	0.00
	$Log\left({Y}_{it-1}\right)$	0.98	0.001	0.00	0.98	0.001	0.00	0.98	0.002	0.00
	${Inequality}_{it}$	1.10	0.16	0.00	0.62	0.10	0.00	0.56	0.14	0.00
	${{Inequality}_{it}}^{2}$	-1.004	0.14	0.00	-0.72	0.11	0.00	-1.68	0.44	0.00
Equation 2	Intercept	-0.09	0.06	0.16	-0.08	0.06	0.18	-0.084	0.05	0.10
	${Inequality}_{it-1}$	0.98	0.006	0.00	0.98	0.004	0.00	0.99	0.008	0.00
	$Log\left({Y}_{it}\right)$	0.026	0.01	0.08	0.022	0.013	0.09	0.02	0.01	0.07
	$(Log{{Y}_{it})}^{2}$	-0.001	0.0008	0.06	-0.001	0.0007	0.05	-0.001	0.0005	0.05
J-Statistic		0.08			0.119			0.16
R-Square		0.99			0.99			0.99
Source: Own Calculation

Table 7

Estimation Results in 129 countries during 1980–2019
Equation	Variables	Model 1 (Inequality Index = Gini Coefficient)			Model 2 (Inequality Index = Top 10% Share)			Model 3 (Inequality Index = Top 1% Share)
Equation	Variables	Coefficient	St. Error	Prob	Coefficient	St. Error	Prob	Coefficient	St. Error	Prob
Equation 1	Intercept	-0.109	0.02	0.00	-0.02	0.01	0.14	0.01	0.007	0.13
	$Log\left({Y}_{it-1}\right)$	1.0001	0.0004	0.00	0.99	0.0004	0.00	0.99	0.0004	0.00
	${Inequality}_{it}$	0.52	0.09	0.00	0.23	0.08	0.00	0.12	0.06	0.06
	${{Inequality}_{it}}^{2}$	-0.54	0.09	0.00	-0.30	0.09	0.00	-0.34	0.19	0.08
Equation 2	Intercept	0.001	0.002	0.40	-0.001	0.002	0.50	-0.003	0.001	0.03
	${Inequality}_{it-1}$	0.99	0.001	0.00	0.98	0.001	0.00	0.98	0.002	0.00
	$Log\left({Y}_{it}\right)$	0.0006	0.0002	0.03	0.001	0.0003	0.00	0.0009	0.0002	0.00
	$(Log{{Y}_{it})}^{2}$	-0.00002	0.00001	0.04	-0.00004	0.00001	0.00	-0.00003	0.00001	0.00
J-Statistic		0.069			0.02			0.05
R-Square		0.99			0.99			0.99
Source: Own Calculation

Table 8

Estimation Results in 129 countries during 1990–2019
Equation	Variables	Model 1 (Inequality Index = Gini Coefficient)			Model 2 (Inequality Index = Top 10% Share)			Model 3 (Inequality Index = Top 1% Share)
Equation	Variables	Coefficient	St. Error	Prob	Coefficient	St. Error	Prob	Coefficient	St. Error	Prob
Equation 1	Intercept	-0.14	0.04	0.00	-0.02	0.02	0.29	0.008	0.009	0.36
	$Log\left({Y}_{it-1}\right)$	1.0001	0.0005	0.00	0.99	0.0005	0.00	0.99	0.0005	0.00
	${Inequality}_{it}$	0.64	0.15	0.00	0.25	0.13	0.05	0.10	0.09	0.22
	${{Inequality}_{it}}^{2}$	-0.63	0.14	0.00	-0.31	0.15	0.03	-0.19	0.24	0.42
Equation 2	Intercept	0.003	0.002	0.19	-0.001	0.002	0.61	-0.003	0.002	0.07
	${Inequality}_{it-1}$	0.98	0.001	0.00	0.98	0.001	0.00	0.98	0.003	0.00
	$Log\left({Y}_{it}\right)$	0.0007	0.0003	0.03	0.001	0.0003	0.00	0.001	0.0003	0.00
	$(Log{{Y}_{it})}^{2}$	-0.00002	0.000013	0.05	-0.00004	0.00001	0.00	-0.00003	0.00001	0.00
J-Statistic		0.07			0.03			0.05
R-Square		0.99			0.99			0.99
Source: Own Calculation

Tables 4, 5, and 6 indicate the estimation results of the simultaneous equations in the sample of G7 and BRICS countries during three different periods, including 1970–2019, 1980–2019, and 1990–2019. These empirical results show that there is a bilateral nonlinear relationship between per capita income and inequality. In other words, in the first equation, inequality indexes have a nonlinear impact as an inverse U-shape on the logarithm of per capita income. In the second equation, the logarithm of per capita income has a nonlinear as an inverse U-shape on different inequality indexes. For checking the robustness of the results, we have used three different inequality indexes and three different periods. The results indicate a robust bilateral nonlinear relationship between per capita income and inequality. Also, for more consideration and for ensuring robustness, we have estimated the models for another sample of 129 countries, including both developing and developed countries. Tables 7 and 8 indicate the empirical results of the models in these countries for two different time periods including the periods of 1980–2019 and 1990–2019. The empirical results show a robust bilateral nonlinear relationship between per capita income and inequality in the second sample in both time periods. Therefore, the Kuznets curve existence has been confirmed in all samples and all periods. Our results suggest that the Kuznets curve should be considered by simultaneous equation models with a nonlinear form of both inequality and per capita income. Regarding the Kuznets curve in 129 countries, we can use the initial interpretation of Kuznets because in this sample, we have two groups of countries including developed and developing countries and many of these developing countries has not a modern section like a developed industrial sector and these economies are rely on agricultural, mineral or an industrial sector with low level of technology, therefore we can say that in the first stage of the development process or in developing countries, the per capita income increases as a result of the emergence of some few modern industrial sectors and movements of workers from the agriculture or traditional sector to these new industrial sectors which means increasing in income inequality because of the different wage levels between these two sectors. After this first stage and the development of the industrial sector in economy and moving to a country with large modern industrial sector, most workers will move to the industrial sector and benefit from high wages: increasing per capita income is then associated with decreasing income inequality. Also, we can interpret the Kuznets curve in G7 + BRICS countries in a similar way. We can say that in a country in BRICS with lower level of per capita income comparing with G7 countries, most of workers work in an industrial sector with a low technology level comparing with high tech industrial sector, but also these BRICS countries also have a small high tech industrial and increasing in per capita income and moving some workers from the traditional industrial sector to high tech industrial sector increases the inequality because of high difference between wages of workers in these two type of industrial sector. After this stage and the development of the high tech industrial sector in economy and moving to a country with large high tech industrial sector like G7 countries, most workers will move to the high tech industrial sector and benefit from high wages and increasing per capita income is then associated with decreasing income inequality.

Regarding the nonlinear impact of inequality on per capita income, we can say that when inequality is relatively low (in low regime of inequality), increases in inequality can increase the saving and investment and per capita income as well. The increase in inequality, if it is caused by the expansion of high technology industries and intense innovations, can not only lead to more accumulation of capital and more investment in the economy and thus increase per capita income, but even this inequality is considered as a positive phenomenon by the society because it can lead to the improvement of the level of well-being and quality of life of people as a result of the use of new high tech goods or services. In other words, inequality foster investment by increasing incentives for producers and entrepreneurs of high tech sectors. But in contrast, when inequality is relatively high, political and economic instabilities due to rising inequalities deter investment and obstructs the social consensus (Alesina and Perotti, 1996; Knack and Keefer, 2000). So, inequality might have a negative impact on per-capita income especially in a country with a high inequality. Also, According to the human capital accumulation theory, inequalities reduce accumulation of human capital by the poor as a result of investment delays in human capital (Galor and Zeira ,1993, 1998). Furthermore, According to Todaro (1997), high inequality increases unproductive investment by the rich. Also, in developing countries, the rich decades consume more luxury or imported goods and they invest their savings on some developed counties that this phenomenon is known as capital flight (Todaro and Smith, 2011). Finally, high Inequality may decrease the incentives for investment as a result of the endogenous fiscal policy theory (Bertola 1993; Bénabou, 1996; Perotti 1996; Cingano, 2014).

Finally, Figs. 8, 9, and 10 show the impact curves of inequality indexes for both samples in all periods. These figures indicate a nonlinear impact of inequality indexes on the logarithm of per capita income.

Previous studies on the existence of the Kuznets curve have provided conflicting results. Our paper claims that this may be due to a specification error in econometric modeling. In other words, previous studies have considered one-way causality from per capita income to inequality, while inequality also affects per capita income. We also believe that inequality affects per capita income non-linearly. This paper attempts to provide sufficient empirical evidence to support this hypothesis. For this purpose, first, the wavelet coherence approach has been used to examine the correlation between per capita income and different inequality indicators. The results of wavelet coherence represent two important facts. First, there are both same-phase and opposite-phase in the diagrams, which can indicate a non-linear relationship between per capita income and inequality indicators. The second fact is that based on the wavelet correlation diagrams, it is not possible to identify the leading and lagging variables, which can be a reflection of the two-way causality between per capita income and inequality. These two facts and the theoretical literature presented in the literature review section, have been used to model the relationship between per capita income and inequality using the simultaneous panel method. Also, to investigate the nonlinear effects of per capita income and inequality, the quadratic form of these two variables has been used. Finally, two samples, including G7 and BRICS countries, and 129 countries, including developing and developed countries have been used to evaluate the results. Also, different periods have been considered for estimating the model. The results of the estimated models show that there is a two-way nonlinear relationship between inequality and per capita income in the samples and these results also are robust. Also, the existence of the Kuznets curve in the samples has been confirmed. Therefore, we suggest that for testing the existence of Kuznets curve, it is necessary to consider a two-way causality between per capita income and inequality. Also, in addition to the two-way causality, it is necessary to include both equations in the model nonlinearly.

For future studies, we suggest using a simultaneous Markov switching model to consider relationship between per capita income and inequality to capture endogenous nonlinearity between the low and high regimes of inequality and per capita income.

Ethical Approval

This declaration is “not applicable”.

Competing interests

This paper is extracted from a research project that financed by the French government with the file number 102715U in Campus France. The CERDI Université Clermont Auvergne CNRS in the Clermont-Ferrand was our host for two months to do some parts of this research project. We would like to appreciate Government of France, and Campus France because of their financial supports. Also, we would like to highly appreciate Professor Florent Bresson for hosting us in the CERDI. Finally, we would like to thank the scientific section of French Embassy in Tehran.

The findings of this article do not necessarily represent the views of the French government.

Authors' contributions

Haniyeh Sedaghat Kalmarzi had written “Introduction and Literature Review” and Younes Nademi had written other parts including “Methodology, Empirical Results, and Conclusion”. All authors reviewed the manuscript.

Funding

This paper is extracted from a research project that financed by the French government with the file number 102715U in Campus France.

Availability of data and materials

Upon request, we can provide the data to the readers of the article via email.

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Competing interest reported. This paper is extracted from a research project that financed by the French government with the file number 102715U in Campus France. The CERDI Université Clermont Auvergne CNRS in Clermont-Ferrand was our host for two months to do some parts of this research project. We would like to appreciate Government of France, and Campus France because of their financial supports. Also we would like to highly appreciate Professor Florent Bresson for hosting us in the CERDI. Finally, we would like to thank the scientific section of French Embassy in Tehran. The findings of this article do not necessarily represent the views of the French government.

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Revisiting the Kuznets Curve: A New Approach to Describe a Bilateral Nonlinear Kuznets Curve

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Abstract

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1. Introduction

2. Literature Review

3. Methodology

4. Empirical Results

5. Conclusion

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Variable	Mean	Median	Min	Max	St. Dev.
\({{L}{o}{g}({Y}}_{{i}{t}})\)	9.75	10.04	7.42	10.87	0.843
Gini Coefficient	0.51	0.49	0.31	0.74	0.090
Top 10% Share	0.38	0.36	0.21	0.65	0.095
Top 1% Share	0.12	0.10	0.03	0.30	0.056
Source: Own Calculation