Income inequality, carbon emissions, and green development efficiency

Income inequality affects not only social well-being, health, and carbon emissions but also the strategy for green development in China. Based on the panel data of 205 cities in China from 2010 to 2020, a panel model with partial linear functional coefficient is used to analyze and test the relationship between income inequality, carbon emissions, and green development efficiency under different regional economic development levels. The empirical results show that the impact of income inequality on carbon emissions and green development efficiency is significant. The worsening of income inequality could aggravate carbon emissions, but the effect of income inequality on carbon emissions shows an increase–decrease-flattening with the continuous improvement of regional economic development. In terms of affecting the green development efficiency, the effect of income inequality on the efficiency of green development presents an inverted U shape. However, the number of cities where income inequality has an inhibitory effect on carbon emissions and green development efficiency has increased over time, and the impact of income inequality in a few cities on green development efficiency is not significant. These findings provide new insights into the understanding of shared prosperity and the strategy of carbon peaking and carbon neutralization in China.


Introduction
With the continuous occurrence of a large number of environmental pollution incidents, the stability of China's economic development and the unsustainability of economic growth have been seriously challenged (Dong et al. 2021;Liao et al. 2021). Therefore, exploring the internal mechanism of income inequality and carbon emissions is of great significance for exploring the reform of the social and ecological civilization system, forming a rational and orderly income distribution pattern, and promoting the realization of the dual-carbon strategic target (Yang et al. 2022a, b). In 2017, China made it clear that the economy has shifted from high-speed growth to high-quality development (Jahanger 2021). China moves towards a new stage of development with relatively abundant products. If we blindly pursue GDP growth, it could bring about a negative chain reaction, such as the widening gap between the rich and poor, environmental problems, and low development quality (Taghizadeh-Hesary et al. 2022). Therefore, promoting the high-quality development of the urban economy has become a new idea and new requirement for the economic development in China (Du et al. 2022a, b). To this end, in the first year of the "14th Five-Year Plan," China took a series of measures to achieve high-quality urban economic development such as narrowing the income gap, improving the system and mechanism of common prosperity, and realizing the beauty of harmony. This also urgently needs to speed up the peaking of carbon emissions, realize clear waters and lush mountains as soon as possible, and restore the beauty of ecology. Therefore, to build a harmonious society and realize the high-quality development of urban green economy, it is necessary to improve the income distribution system and mechanism, implement energy-saving and emission-reduction measures, Communicated by Eyup Dogan. and take into account fairness issues while improving the efficiency of green development (Du et al. 2022a, b;Khan et al. 2022). Green development efficiency is an important indicator to measure the quality of green economic development, including green technical efficiency and green total factor productivity (GTFP) (Zhang et al. 2015;Yang and Ni 2022). Thus, this paper uses green development efficiency to measure economic development more comprehensively and also provides a new perspective for the impact of income inequality.
Income inequality, environmental issues, and economic development have long been a common concern (World Bank 2010; Arslan et al. 2022). Many previous studies have done a lot of research on the pairwise relationship among the three (Khan and Hou 2021). For example, how income inequality affects economic growth (Aghion et al. 1999;Acemoglu 2002;Acemoglu et al. 2005;Dabla-Norris et al. 2015;Mdingi and Ho 2021). Most studies believe that income inequality could promote economic growth in the period of low economic development, while income inequality could hinder economic growth as economy grows. On the other hand, controversial studies exist on the relationship between income inequality and carbon emissions. Income inequality and climate change have been the two unsolved challenges facing humanity.
In addition, it is worth noting that the differences in factor endowments such as natural resources and location advantages between regions. As a result, the impact of income inequality on carbon emissions and green development efficiency could be restricted by the level of regional economic development and show heterogeneity. Jorgenson et al. (2017) argued that income inequality can lead to competition for consumption, which shifts the consumption preferences of groups with below-average incomes toward less protection of the environment, leading to increased demand for highly polluting goods and services, including larger cars and frequent travel. Figure 1 exhibits the trend of the distribution of real GDP per capita (LNPRGDP) for 205 cities in China. As shown in Fig. 1, although the economic gap between cities is significant, the economic development level of Chinese cities has been continuously improving, and the impact mechanism of income inequality on carbon emissions and green development efficiency may also change. Therefore, this paper aims to analyze the specific impact of income inequality on carbon emissions and green development efficiency considering different levels of the economic development in Chinese cities.
The contributions of this paper relative to the existing literature are threefold. First, the indicator of economic growth is simply measured by GDP when the existing literature studies the link between income inequality and economic growth, while this paper uses the indicator of green development efficiency, which can reflect high-quality economic development. The calculation of green development efficiency indicator is also quite distinctive. The calculation model adopts the super slack-based measure (DEA-SBM) model with data envelopment analysis and the global Malmquist-Luenberger (GML) index based on the directional distance function (DDF). The desirable output index is in addition to GDP at constant price, as well as the green coverage. The undesirable output index is except for waste water and smoke, sulfur dioxide, and included PM2.5 for measurement, making the measurement results of the indicator of high-quality economic development more reasonable. Second, in terms of research methods, the previous literature studies the linear or non-linear relationship between variables, but the relationship between variables is not so absolute as the economy moves forward. Therefore, this paper employs a partial linear function coefficient (PLFC) panel model to study the relationship between these variables. In addition, most of the literature investigate the link between income inequality and the environment or economic growth. This paper integrates the three into a framework to study the relationship. Third, we test the hypothesis that the impact of income inequality on carbon emissions and green development efficiency depends on the economic development level. Besides, existing studies have all taken provinces in China as the object, but this paper takes prefecture-level cities as the object of study, which further refines the heterogeneity among regions.

Income inequality and carbon emissions
Income inequality and climate change are the two unsolved challenges facing humanity today (Uddin et al. 2020). Khan and Hou (2021) examined the impact of socioeconomic and environmental sustainability on CO 2 emissions in the presence of combustible renewable energy sources and found that socioeconomic sustainability can contribute to increased CO 2 emissions. Zhang et al. (2022) applied the DH causality test to show a bidirectional link between income and CO 2 emissions, whether income inequality could exacerbate carbon dioxide emissions and thus contribute to climate change remains to be explored.
Existing studies generally summarize the link between income inequality and ecological environment quality into three types (Safar 2022), including the political economy approach (PEA) (Boyce 1994), and Veblen's emulation theory (Veblen and Galbraith 1973). Among them, they believed that income inequality is positively associated with declining environmental quality, that is, the increase of income inequality could lead to the increase of environmental pollution. Some studies also confirmed this conclusion (Fitzgerald et al. 2015;Knight et al. 2017;Uzar and Eyuboglu 2019;Baloch et al. 2020;Yang et al. 2022a, b). The third is the marginal propensity to emit (MPE) theory, which held that income inequality helps reduce environmental degradation (Ravallion et al. 2000;Berthe and Elie 2015;Hailemariam et al. 2020;Wan et al. 2022;Safar 2022).
Although many literatures examined the relationship between income inequality and carbon emissions (Uddin et al. 2020), they did not have an agreed point of view. Some literature suggests that higher income inequality leads to lower CO 2 emissions (Wan et al. 2022). On the contrary, income inequality boosts carbon emissions (Knight et al. 2017;Zhu et al. 2018;Sager 2019). Of course, income inequality and carbon emissions may be not related (Wolde-Rufael and Idowu 2017). However, some studies also have inconsistent results due to the sample selection, measurement methods, and income inequality measures (Safar 2022). Grunewald et al. (2017) found that income inequality and carbon emissions are negatively correlated in lowand middle-income economies, while the two are positively correlated for middle-and high-income economies. Analogously, Jorgenson et al. (2017) used the Gini coefficient to measure income inequality and found that there was no correlation between income inequality and carbon emissions. In addition, financial development and income can reduce natural resource price volatility (Liu et al. 2022), which in turn affects carbon emissions.

Income inequality and economic growth
Distribute income and resources equally among the population is also one of the challenges for countries (Zahoor et al. 2022). The income inequality gap (measured by the Gini coefficient) between the rich and the poor is high whether in developed or developing economies (Dabla-Norris et al. 2015). To this end, China has attached great importance to the issue of income inequality in recent years and has included common prosperity in its 14th Five-Year Plan. At the same time, targeted poverty alleviation is also listed as one of the three major battles. There is very little literature on the relationship between income inequality and green development efficiency, and most of the research on income inequality is mainly conducted in terms of causes and effects on economic development. The findings vary widely due to the various transmission mechanisms linking income inequality and economic development (Mdingi and Ho, 2021). That is, some are positive (Balcilar et al. 2021), some are negative (De La Croix and Doepke 2003), and some are uncertain (Aiyar and Ebeke 2020). Therefore, the results of the relationship between income inequality and economic development remain controversial.
Specifically, Berg et al. (2018) found that lower income inequality is associated with faster economic growth, and inequality appears to influence growth through human capital accumulation and reproductive channels. Based on the U.S. Tax Reform Act of 1986 (TRA-86), Chen (2020) indicated that per capita GDP growth deteriorates less when a more equal distribution of income is pursued after 1986. Conversely, inequality without poverty does not seem to have a significant impact on economic growth (Breunig and Majeed 2020). Aiyar and Ebeke (2020) confirmed that ignoring intergenerational mobility leads to uncertainty about the relationship between income inequality and growth. But beyond that, Balcilar et al. (2021) believed that income inequality has a positive impact on economic growth before the average Gini coefficient threshold and has a negative impact on economic growth after this threshold is exceeded. Besides, Benos and Karagiannis (2018) suggested that changes in income inequality could not affect US growth in the short or long term.
To the best of our knowledge, the existing literature examines the relationship between income inequality, environmental issues, and economic development extensively from multiple perspectives and methods. But most of them are based on the relationship between one of the two variables. The relationship between income inequality, carbon emissions, and green development efficiency is rarely included in a framework to explore, so there is a lack of an accurate grasp of the relationship between the three and the exploration of the impact mechanism. Furthermore, this paper incorporates green development efficiency into the research on income inequality and carbon emissions and considers the differences in regional economic development levels to study the impact of income inequality on carbon emissions and green development efficiency. As China faces the development needs of the new era, it is crucial to clarify the relationship between income gaps, carbon emissions, and the high-quality development of green economy in China in the future.

Methodology
In order to study the impact of income inequality on carbon emissions (CO 2 ) and green development efficiency (GTFP), this paper firstly constructs the following linear model: where X it is the degree of income inequality of the city i in the period t. Z it represents a series of control variables. i represents the unobserved individual effect. it is a random error.
Considering that the impact of income inequality on carbon emissions and green development efficiency may be affected by the levels of regional economic development, some studies may construct an interaction term between the level of regional economic development and income inequality, but this strategy may lead to estimation bias (Du et al. 2021a, b). Therefore, the variable coefficient W it refers to the regional economic development level expressed by per capita real GDP. Part of the variable coefficient panel model is as follows: where the coefficient (W it ) represents the heterogeneous impact of income inequality on carbon emissions and green development efficiency under different regional economic development levels, which is a non-parametric function of functional coefficients and does not contain linear parameters. Based on the panel data, An et al. (2016) extended a partially linear variable coefficient model with fixed effects and used a series method to estimate the model. The specific estimation process is as follows (Du et al. 2021a, b). First, the difference method is used to eliminate the fixed effect i . Second, the substitution function coefficients γ(W it ) are approximated by linear combinations of the k basis functions: where p(W it ) is a k × 1 basis function, and θ is an unknown parameter of k × 1. If k is large enough, there is a linear combination of p(W it ) that can approximately replace any smoothing coefficient (W it ) , and the mean square error is the smallest. Then the Eq. (4) can be rewritten as: where the error term Δε it = Δμ it + Δ(γ(W it )X it )-Δ(Z it p(W it ))′θ. (1) Finally, a least squares estimation is performed as: Furthermore, the coefficient function γ(‧) is estimated as: Therefore, the final settings of the estimated PLFC model in this paper are as follows: where the two explained variables in this paper are green development efficiency (GTFP) and carbon emissions (CO 2 ). The explanatory variable LNTHEIL it is the degree of income inequality (Theil index). (LNPRGDP it ) is the function coefficient of income inequality. The control variables include internet penetration rate (LNINTER), economic openness (LNTRA), green innovation capability (LNGPAT), digital financial inclusion (LNFINC), information technology (LNICT), and industrial structure advanced (ISH). i represents the unobserved individual effect. it is a random error.

Data
The data used in this paper are collected from the provincial statistical yearbooks, statistical yearbooks of prefectures and cities during the period of 2010-2020, and the digital financial inclusion adopts the "Peking University Digital Financial Inclusion Index." The PM2.5 concentration data comes from satellite remote sensing data released by NASA. Excluding some cities with serious missing variables, this paper selects 205 prefecture-level cities and above. The three key variables in this paper, including green development efficiency, carbon emissions, and income inequality, cannot be obtained directly, and need to be obtained through measurement.
Regarding green development efficiency, China's urban economic development has entered a stage of high-quality development. Based on the DEA-SBM model, this paper employs the GML index to measure the green total factor productivity (GTFP) of 205 cities in China. Based on the global production possibility set, we combine the non-radial SBM model (Tone 2001;Färe and Grosskopf 2010) and the (Oh 2010). Then the GTFP index is expressed as: where x represents the input indicators. y is the desirable output indicators. b represents the undesirable output indicator. g represents the direction vector. � ⃗ S G (x, y, b;g y , g b ) represents the global directional distance function. The input indicators used in this paper to measure urban GTFP of the high-quality development include labor, capital stock, and energy inputs. Employment numbers are used to measure labor input in 205 Chinese cities (Du et al. 2021a, b). The capital stock is measured by the perpetual inventory method (Du et al. 2022a, b), and the price is deflated with 2010 as the base period to eliminate the price factor. The energy input is measured by the city's annual electricity consumption. In the case of the carbon peaking and carbon neutrality goals, the economic growth of cities could be constrained by energy use and carbon emissions. Under the pressure of dual control of energy consumption to meet the standard, and the shortage of power supply, the most vulnerable industries are the high energy consumption industries. Thus, electricity consumption is one of the best indicators of energy consumption. Desirable output indicators include economic output and green output. Economic output is GDP at constant price with 2010 as the base period and the influence of price factors removed, while green output is measured by the green coverage rate of each city, which represents the quality of life of residents. Undesirable output indicators are selected from industrial soot emissions, industrial wastewater emissions, industrial sulfur dioxide emissions, and PM2.5 concentrations.
With respect to carbon emissions (CO 2 ), the calculation is based on the carbon emission inventory compiled according to the latest revision of energy data from the National Bureau of Statistics in China. Various energy data sources for calculating carbon emission indicators are the Urban Statistical Yearbook and the Urban Construction Statistical Yearbook.
For the core explanatory variables, the index used to measure income inequality is the Theil index. The indicators for calculating Theil index include urban and rural household disposable income, urbanization rate, total population, total urban population, total rural population, total urban income, total rural income, and total income. The urbanization rate is measured using the resident population rather than the registered population. And the per capita disposable income in rural areas is the per capita net income before 2013. In addition, the regional economic development degree is measured by using per capita GDP (LNPRGDP). (10) As for the control variable, internet penetration (LNIN-TER) is represented by the number of Internet broadband access users, and the impact of the internet on our economic life is self-evident. Economic openness (LNTRA) is expressed by the total import and export volume, which not only affects the high-quality development of the urban economy but also affects our trade due to the frequency of trade. Green innovation capability (LNGAPT) is used by the sum of green invention patent applications and green utility model patent applications. Due to the time lag in the granting of patents, this paper applies the number of green patent applications instead of grants. Digital financial inclusion (LNFINC) is represented by the Peking University Digital Financial Inclusion Index. This indicator can indirectly affect carbon emissions and the high-quality development of urban green economy by optimizing the industrial structure and promoting the level of technological innovation (Juuti 2022). Since this indicator has been standardized in the process of processing, the logarithm is no longer taken here. The level of information technology (LNICT) is expressed by information transmission, computer service, and software practitioners. This indicator is an important indicator to measure the regional economic growth ability, competitiveness, and modernization level, and should be highly valued. Advanced industrial structure (ISH) is expressed by the value-added ratio of the tertiary industry and the secondary industry and standardized processing. This indicator not only affects the environmental conditions such as clean and green production but also affects the high-quality development of urban green economy. index. Among them, since the industrial upgrading (ISH) has been standardized, and the urban green development efficiency (GTFP) is measured, there are no need to take the logarithm. The other variables are all taken the logarithm. The descriptive statistics of the variables involved in this paper are shown in Table 1. Table 2 displays estimation results of the impact of income inequality on the development of green economy in cities without considering different regional economic development by the linear panel model. The estimation results suggest that income inequality has a significant effect on the green development efficiency of Chinese cities. The estimated coefficient of LNTHEIL is statistically significantly − 0.183, implying that income inequality is negatively related the development efficiency of green economy, that is, the increase of income inequality could reduce the efficiency of green development. Specifically, for 0.01% increase in income inequality, the efficiency of green development may decrease by 0.183. The estimated coefficient of income inequality on carbon emissions is positive and significant, indicating that for 0.01% increase in income inequality, carbon emissions may increase by 0.068%. These estimates provide preliminary evidence that rising income inequality over time is bad for both the environment and the economy. In terms of control variables, their impacts on carbon emissions appear to be more significant than green development efficiency. Next, we discuss the heterogeneous effects of income inequality on the economic environment of Chinese cities at different levels of economic development.

Estimated results of PLFC model
In this section, the impact of income inequality on carbon emissions and green development efficiency is analyzed based on a PLFC panel model. The estimated results of PLFC model are shown in Table 3. Figures 2 and 3 exhibit the heterogeneous influences of income inequality on the economic environment of Chinese cities for the situation of economic development in different regions. In addition, we also examine that the effect of income inequality on CO 2 emissions and GTFP could vary with the level of economic development in different regions.
In terms of carbon emissions, it can be seen intuitively in Fig. 2 that with the continuous improvement of regional economic development, the impact of income inequality on carbon emissions begins to decrease after increasing to a certain peak, and finally stabilizes. Concretely, as the economy continues to develop, the increase of economic and financial activities, and the frequent foreign trade in the early stage, China's economic policy allowed some people to get rich first, which aggravated income inequality and eventually led to an increase in carbon emissions. However, as the level of regional economic development reaches a certain level, people's material life is more abundant, and their attention to the environment is constantly increasing. Especially in recent years, the Chinese government has taken a number of measures to manage the ecological environment. The proposal of the dual-carbon goal has forced enterprises to transform, and the practice of industrial structure upgrading and green innovation has gradually reshaped the competitiveness of enterprises. So, these may be the reasons why the influence of income inequality on carbon emissions has gone from increasing to decreasing to leveling off. As for the development efficiency of city's green economy in Fig. 3, we can find that income inequality helps to improve the green development efficiency in the early stage of relatively low economic development. This just confirms the correctness of the national policy that some people get rich first. However, with the continuous increase of economic output value, the impact of income inequality on the efficiency of urban green development presents an inverted U shape. This result suggests that the previous income inequality is conducive to the development of urban green economy. While with the improvement of regional economic development level, the aggravation of income inequality is not conducive to the development of urban green economy. Therefore, in order to eliminate the adverse effects of income inequality, China has adopted a series of policies that are beneficial to the development of the country and the prosperity of the people, such as targeted poverty alleviation, rural revitalization, and other shared prosperity policies. With the emphasis on the ecological environment and the development of new energy, green, and low-carbon technologies, urban green development is also constantly improving. As shown in Fig. 3, the curve first rises and then falls, indicating that the above-mentioned influencing factors could gradually weaken the impact of income inequality on the efficiency of green economic development.

Heterogeneous results across time periods
Furthermore, we provide a more nuanced analysis of the heterogeneous effects of different types of urban income inequality in China on carbon emissions and the efficiency of green economic development. To this end, this paper uses the per capita real gross domestic product from 2010 to 2020 and divides 205 cities into three groups: regions with the low, the medium, the moderate and high level of economic development. Figure 4a-c show the impact of income inequality in different cities on carbon emissions and green development efficiency in 2010, 2015, and 2020, respectively.
It can be seen intuitively in Fig. 4a that regardless of the degree of regional economic development in 2010, income inequality has significant implications for carbon emissions. The effect on the green development efficiency of most cities is significant, and the impact on the green development efficiency of a few cities is minimal. To be specific, for cities with lower economic development, income inequality reduces carbon emissions, but only slightly. However, income inequality affects green development efficiency in three stages: reduced, insignificant, and increased. For cities with a moderate level of regional economic development and The function coefficient of income inequality on green development efficiency most cities with the high economic development, the increase in income inequality could increase carbon emissions and green development efficiency. For a few cities with a high level of regional economic development, income inequality could inhibit carbon emissions and the efficiency of green economic development. In some cities with the high economic development, the effect of income inequality on the efficiency of green development is not significant.
We can be found in Fig. 4b that in 2015, for cities with the low regional economic development, the increase in income inequality could inhibit carbon emissions and green development efficiency. For cities with the moderate regional economic development, the effect of income inequality on carbon emissions and green development efficiency is gradually increasing. However, for cities with the moderate economic development in some regions, the effect of income inequality on the efficiency of green development is minimal. For most cities with the high regional economic development, income inequality has a promoting effect on carbon emissions and green development, but a few cities have an inhibitory effect.
In addition, as shown in Fig. 4c in 2020, in cities with the low economic development and a few medium-sized cities, income inequality dampens carbon emissions and green development efficiency. However, in cities with the moderate economic development and a few highlevel cities, the effect of income inequality on carbon emissions is relatively moderate and has no influence on the efficiency of green development. In most cities with the high economic development, income inequality positively impacts carbon emissions and green development efficiency.
Overall, the results of Fig. 4 suggest that as economic development continues to improve over time, the number of cities where income inequality inhibits carbon emissions and green development efficiency continues to increase. For cities with the high regional economic development, the number of cities where income inequality promotes carbon emissions and green development efficiency is decreasing, which is inseparable from China's environmental policies in recent years. At present, China's economy is no longer only "GDP" performance in the new era, but also has higher requirements for the environment, and the government's emphasis on the environment continues to increase. According to data publicly reported by the government, Chinese environmental pollution control investment accounts for more than 3.5% of GDP, far exceeding the international standard. Moreover, China has also launched the trading of the carbon market to help lowcarbon transformation and green development.

Conclusions and policy implications
Continuously improving the income distribution mechanism, promoting shared prosperity, and improving the quality of the ecological environment are the inherent requirements for high-quality urban economic development in the new era. Taking into account the heterogeneity of regional economic development levels, this paper adopts a PLFC panel model to study the impact of income inequality on carbon emissions and green development efficiency. Our results show that first, income inequality has a significant impact on CO 2 emissions and green development efficiency. Specifically, the worsening of income inequality could increase carbon emissions, but as the continuous improvement of the regional economic development, the effect of income inequality on carbon emissions could increase and gradually decrease, and finally the region could level off. With regard to affecting the green development, the impact of income inequality on the efficiency of green development presents an inverted U shape. Second, we take the periods of 2010, 2015, and 2020 as examples to prove the heterogeneous effects of income inequality on CO 2 emissions and GTFP over time. Overall, income inequality has a significant impact on CO 2 emissions and green development efficiency. Specifically, the number of cities where income inequality inhibits CO 2 emissions and green development efficiency is increasing. While cities with higher economic development, the number of cities where income inequality promotes carbon emissions and green development efficiency is decreasing. Besides, the impact of income inequality in some cities on green development efficiency is not significant.
The empirical results suggest that in the case of different economic development levels, income inequality has different effects on CO 2 emissions and urban green development efficiency. Therefore, it is necessary to promulgate relevant supporting policies to adapt to regional economic development, continuously improve the income distribution mechanism, promote common prosperity, and improve the ecological environment, which are the requirements for high-quality urban economic development. Meanwhile, in view of the differences in the regional economic development, we should actively guide the industry to carry out technological innovation and energy conservation and emission reduction. Therefore, our findings illuminate the following policy implications. First, the government needs to be committed to maintaining a steady rise in residents' income, actively promoting targeted poverty alleviation, and meanwhile attaching importance to the quality of economic development. Of course, the government must completely change the GDP-only official evaluation system, establish the concept of green development, and actively promote the innovation of the environmental tax system, so as to move towards a path of high-quality development that is win-win for the economy and the environment (Du et al. 2022a, b). Second, income distribution is the source of people's livelihood. If the income gap is widened, it may negatively affect the quality of the environment such as CO 2 emissions and the quality of green economic development. The government must continuously improve people's livelihood, improve the income distribution mechanism, strengthen social security, and promote a fair social and economic environment. Meantime, it is necessary to improve the heterogeneous impact caused by the inequality of regional development and coordinate the inequality of income distribution caused by regional development. Third, as the continuous development of regional economy, income inequality may affect carbon emissions, and may also harm social well-being and health. Therefore, the government must continue to promote targeted poverty alleviation, promote common prosperity, balance income, and protect the environment. In addition, there is a regional imbalance in carbon emissions in China. Therefore, the formulation of energy conservation and emission reduction policies should take into account the emission reduction responsibilities and emission reduction capabilities of different groups of people, ensure equal rights, responsibilities and interests, and avoid the phenomenon of the poor subsidizing the rich. Only in this way can we better promote the realization of the dual-carbon strategic goal.
It is worth noting that the limitation of this paper may be that it ignores the spatial and geographical differences between cities when describing the differences in different levels of economic development. The use of GDP per capita in this paper may not fully reflect the differences in economic development such as resource endowments among different regions. Studying the spatial spillover effects of income inequality on carbon emissions and green development efficiency may be a topic worth exploring in the future. In addition, there are many channels through which income inequality affects carbon emissions and green development efficiency. This paper cannot identify them all, so it is difficult to explore their true mechanisms.
Funding This paper was supported by Guangzhou Office of Philosophy and Social Science (2021GZGJ29), Guangdong Provincial Key Scientific Research Platform Project (2020ZDZX1064), and Guangdong Educational Science Planning Project(2022GXJK673).

Data availability
The data of this study come from the official published data. For example, National Bureau of Statistics, Provincial Statistical Yearbooks, Statistical Bulletins and Statistical Yearbooks of some prefectures and cities, Institute of Digital Finance Peking University, etc.