Global warming leads to a rapid increase in the probability of extreme climate events, and sustainable economic and social development faces various risks. The main reason for global warming is the greenhouse gas emissions generated by human economic activities [1]. In 2020, the World Meteorological Organization (WMO) released the ‘Greenhouse Gas Bulletin’, which shows that the radiation intensity caused by greenhouse gases in 2019 increased by approximately 45% compared with 1990, and carbon dioxide accounted for approximately 80% of the increase. Global politicians made the proposal to control carbon dioxide emissions to adapt to global climate change [2]. As the largest developing country, and the most significant carbon emission country, China actively participates in global governance and responds to climate change and promises to reduce carbon dioxide emissions per unit of Gross Domestic Product (GDP) by between 60 and 65% by 2030 compared with 2005.

However, there is still a significant gap between the living standards of Chinese residents and developed countries, and there is a strong demand for development [3]. Coordinating the relationship between carbon emission reduction and economic development is an urgent problem for governments to address.

It is generally believed that the motivation of economic growth is to increase input factors and improve productivity. However, with the increase of input factors, the marginal benefit decreases. Thus, the source of long-term economic growth are productivity gains [4]. Carbon productivity refers to a region that provides products and services to meet human needs with fewer carbon emissions, i.e., the ratio of production to carbon emissions is an important indicator of sustainable regional economic development [5]. Carbon productivity is based on economic growth and carbon emission reduction. With the increase in carbon productivity, the economy may increase energy consumption and, subsequently, carbon emissions will increase, which is known as the ‘rebound effect’ [6,7]. However, carbon productivity can significantly inhibit the excessive growth of this emission increment, so carbon productivity is still essential for a low-carbon economy in developing countries [8,9]. It should be noted that there are gaps in resource endowments, the industrial structure, and the economic base in different regions, and there are differences in the spatial distribution of carbon productivity. Is this difference likely to converge over time? At the same time, inter-regional linkages continue to strengthen and the emergence of economic agglomeration and carbon productivity agglomeration have also emerged. Different regions will affect the convergence of carbon productivity through spatial effects such as capital, labor, and technology spillovers. ‘Efficient reproductive effects’ of carbon productivity allow regions to learn from each other and improve carbon productivity, i.e., when carbon productivity in adjacent regions increases, the region will improve its carbon productivity by imitating its technology and management practices [10]. The convergence of carbon productivity has important practical significance for the government to formulate relevant environmental policies, develop a low-carbon economy, and promote regional low-carbon cooperation. Given convergence, the existing literature includes studies about energy intensity convergence [11], carbon emission convergence [12], and total factor productivity convergence [13]. However, they are all based on the traditional Solow model, lacking theoretical support in the field of spatial convergence, and the empirical results cannot be reasonably explained.

The neoclassical growth model believes that the economy has a ‘steady state’ and ‘conditional convergence’; that is, under the assumption of diminishing returns to the scale of input factors, economic growth will eventually reach equilibrium [14]. Economists have frequently discussed whether there is economic convergence in a region’s economic growth. Many scholars use different regional sample data to prove that the region with lower per capita income has faster economic growth than the region with higher per capita income [15,16,17]. The convergence of regional economic growth can be divided into absolute convergence, relative convergence, and club convergence. However, some scholars believe that there is no ‘steady state’ in the economy, changes in initial conditions will have a long-term impact on the economy, and there is no mechanism to ensure that economies tend to ‘converge’ [18,19,20]. Romer improves the endogenous growth model and thinks that knowledge spillover produces economies of scale, making developed countries have higher per capita output [21]. Lucas uses the optimal technological progress model, which assumes that the renewable capital’s returns remain unchanged, and concludes that the per capita output growth rate is independent of the initial per capita output level [22]. Therefore, after relaxing the hypothesis of diminishing marginal returns to capital, it is impossible to obtain the economic growth convergence. However, Bloom et al. (2002) improved the technology diffusion model, considering that the technology diffusion utility shortens the regional technology gap, which makes the economy that of conditional convergence [23].

With the deepening of research, the convergence hypothesis extends to other fields, such as energy efficiency, environmental quality, financial development, and so on [24,25,26]. The sustainable development of the economy has always been the focus of global attention. According to the Environmental Kuznets Curve (EKC), there is an inverted ‘U’ relationship between income and environmental quality [27]. Will economic convergence make the environment converge? Scholars have conducted in-depth studies on this issue [28]. Some scholars have incorporated environmental pollution into the Solow model and found that countries with low environmental efficiency/regions catch up with high environmental efficiency, which verifies the environmental convergence hypothesis [29,30,31,32]. At the same time, scholars’ studies on the externality of environmental pollution have found that environmental efficiency has a spatial spillover effect [33]. Some scholars use Chinese data to find that the convergence rate of carbon productivity in the spatial panel model is higher than that in the non-spatial panel model, which verifies that resource concentration externalities play an important role in improving carbon productivity, narrowing regional disparities, and achieving sustainable growth convergence [5,12]. Economic development brings various resources and environmental problems, especially global warming. Governments are committed to energy conservation and emission reduction, and sustainable growth convergence under carbon emission constraints, that is, carbon productivity convergence, has attracted widespread attention [10,34].

Scholars mainly focus on the convergence analysis of carbon intensity and carbon emissions and seldom study carbon productivity [35,36]. Scholars use empirical data to analyze the convergence of construction [37], manufacturing, industry, and energy industries [38]. Regional carbon productivity convergence has also received attention. Dong et al. (2013) studied the convergence of regional carbon productivity in China. They found a convergence trend in Chinese carbon productivity, but there is a gap in the convergence rate of regional carbon productivity [39]. Shen et al. (2021) analyzed the convergence of Chinese carbon productivity from urban agglomeration and found that Chinese carbon productivity showed noticeable stickiness and spatial dependence in adjacent areas [40]. Scholars also discussed the influencing factors of carbon productivity. They found that carbon productivity is affected by socio-economic, policy, and energy factors. Technological progress can increase carbon productivity and reduce regional disparities in carbon productivity. With the wide application of space measurement technology, more and more literature strengthens the research on the spillover effect when analyzing the factors affecting carbon productivity. Wu et al. (2021) analyzed 17 cities in Central and Western China. They found that the industrial development and urbanization patterns affecting carbon productivity are homogeneous and mutually imitated and have apparent spatial spillover effects [41]. Zheng et al. (2020) also proved the multiple effects of the economic development level, the industrial structure, and urbanization on carbon productivity [42].