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<journal-id journal-id-type="publisher-id">Front. Environ. Sci.</journal-id>
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<journal-title>Frontiers in Environmental Science</journal-title>
<abbrev-journal-title abbrev-type="pubmed">Front. Environ. Sci.</abbrev-journal-title>
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<issn pub-type="epub">2296-665X</issn>
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<article-id pub-id-type="publisher-id">1741575</article-id>
<article-id pub-id-type="doi">10.3389/fenvs.2025.1741575</article-id>
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<subj-group subj-group-type="heading">
<subject>Original Research</subject>
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<title-group>
<article-title>Spatio-temporal patterns of energy efficiency in Chinese prefecture cities under green low-carbon transition</article-title>
<alt-title alt-title-type="left-running-head">Yang et al.</alt-title>
<alt-title alt-title-type="right-running-head">
<ext-link ext-link-type="uri" xlink:href="https://doi.org/10.3389/fenvs.2025.1741575">10.3389/fenvs.2025.1741575</ext-link>
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<contrib contrib-type="author" equal-contrib="yes">
<name>
<surname>Yang</surname>
<given-names>Changpeng</given-names>
</name>
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<sup>&#x2020;</sup>
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<contrib contrib-type="author" equal-contrib="yes">
<name>
<surname>Chen</surname>
<given-names>Liming</given-names>
</name>
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<sup>&#x2020;</sup>
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<uri xlink:href="https://loop.frontiersin.org/people/3218088"/>
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<contrib contrib-type="author" corresp="yes">
<name>
<surname>Cao</surname>
<given-names>Qinglou</given-names>
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<aff id="aff1">
<institution>College of Tourism and Resource Environment, Qiannan Normal University for Nationalities</institution>, <city>Duyun</city>, <country country="CN">China</country>
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<author-notes>
<corresp id="c001">
<label>&#x2a;</label>Correspondence: Qinglou Cao, <email xlink:href="mailto:caoqinglou@sgmtu.edu.cn">caoqinglou@sgmtu.edu.cn</email>
</corresp>
<fn fn-type="equal" id="fn001">
<label>&#x2020;</label>
<p>These authors share first authorship</p>
</fn>
</author-notes>
<pub-date publication-format="electronic" date-type="pub" iso-8601-date="2026-02-13">
<day>13</day>
<month>02</month>
<year>2026</year>
</pub-date>
<pub-date publication-format="electronic" date-type="collection">
<year>2025</year>
</pub-date>
<volume>13</volume>
<elocation-id>1741575</elocation-id>
<history>
<date date-type="received">
<day>07</day>
<month>11</month>
<year>2025</year>
</date>
<date date-type="rev-recd">
<day>16</day>
<month>12</month>
<year>2025</year>
</date>
<date date-type="accepted">
<day>22</day>
<month>12</month>
<year>2025</year>
</date>
</history>
<permissions>
<copyright-statement>Copyright &#xa9; 2026 Yang, Chen and Cao.</copyright-statement>
<copyright-year>2026</copyright-year>
<copyright-holder>Yang, Chen and Cao</copyright-holder>
<license>
<ali:license_ref start_date="2026-02-13">https://creativecommons.org/licenses/by/4.0/</ali:license_ref>
<license-p>This is an open-access article distributed under the terms of the <ext-link ext-link-type="uri" xlink:href="https://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution License (CC BY)</ext-link>. The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.</license-p>
</license>
</permissions>
<abstract>
<p>The green and low-carbon economy has become a pivotal direction for global development, representing both an imperative response to climate change and an intrinsic requirement for economic structural transformation. China&#x2019;s economic development currently occupies a critical phase of structural transition, yet research on the impact of energy efficiency measurement across prefecture-level cities in low-carbon economic development remains scarce. Addressing this gap, this paper conducts an empirical examination of China&#x2019;s energy efficiency and green low-carbon economic development levels based on data from 285 prefecture-level cities spanning 2012 to 2022. Specifically, the Systemic Behavioural Modelling (SBM) approach is employed to systematically measure energy efficiency and low-carbon economic development levels (measurement values are presented in Appendices 1 and 2). Furthermore, the Dagum Gini coefficient and kernel density estimation are utilised to analyse and interpret the characteristics of energy efficiency disparities and their spatio-temporal distribution. Finally, the empirical investigation examines the impact of low-carbon economic development on energy efficiency. The findings reveal: Firstly, national energy efficiency exhibits an overall upward trend, manifested by a sustained growth rate exceeding 1 since 2013, with the fastest increase occurring in 2021 at 1.184. Concurrently, the overall Gini coefficient declined from 0.164 in 2013 to 0.108 in 2022, indicating significant inter-regional disparities. Both intra-regional and inter-regional variations, alongside hyper-dispersion differences, exert substantial influence on energy efficiency inequality. Secondly, a positive correlation exists in the spatial distribution of energy efficiency. Specifically, the highest regional density of energy efficiency is observed around 1.2, indicating that energy efficiency is concentrated within this range in most years, exhibiting a tendency towards spatial clustering. Finally, the research conclusions and policy implications proposed herein offer implementation pathways for enhancing China&#x2019;s energy efficiency and developing a green, low-carbon economy, providing directional guidance and theoretical reference for advancing China&#x2019;s green productive forces in the new era.</p>
</abstract>
<kwd-group>
<kwd>Dagum Gini coefficient</kwd>
<kwd>energy efficiency</kwd>
<kwd>green low-carbon economy</kwd>
<kwd>kernel density</kwd>
<kwd>non-expected output SBM model</kwd>
</kwd-group>
<funding-group>
<funding-statement>The author(s) declared that financial support was not received for this work and/or its publication.</funding-statement>
</funding-group>
<counts>
<fig-count count="15"/>
<table-count count="6"/>
<equation-count count="27"/>
<ref-count count="42"/>
<page-count count="18"/>
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<custom-meta>
<meta-name>section-at-acceptance</meta-name>
<meta-value>Environmental Economics and Management</meta-value>
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</front>
<body>
<sec sec-type="intro" id="s1">
<label>1</label>
<title>Introduction</title>
<p>The development of a green, low-carbon economy represents not only a vital pathway to enhancing energy efficiency but also a pivotal driver for comprehensive green transformation across socio-economic sectors. Through technological innovation and the application of green, low-carbon technologies, it optimises energy production and consumption, elevates energy utilisation efficiency, and facilitates the establishment of an efficient, clean, and low-carbon energy system, thereby propelling high-quality economic development. As a major global energy consumer, China&#x2019;s distinctive energy structure and commitments to carbon reduction render its green economic transition particularly urgent. According to data from the World Energy Statistical Yearbook 2024, China ranks first globally in energy consumption, reaching 170.74 exajoules (EJ) &#x2013; accounting for 27.6% of the world&#x2019;s total consumption. This figure underscores China&#x2019;s immense scale of energy use while highlighting the significant responsibility it bears in advancing energy efficiency.</p>
<p>The Chinese government places high priority on carbon reduction, viewing it as a vital task in advancing ecological civilisation and building a Beautiful China. At the 75th session of the United Nations General Assembly in 2020, China formally announced its &#x2018;dual carbon&#x2019; goals, pledging to peak carbon emissions by 2030 and achieve carbon neutrality by 2060 (<xref ref-type="bibr" rid="B8">Chu, 2022</xref>). This commitment was institutionalised and reinforced through dedicated provisions in the report to the 20th National Congress of the Communist Party of China, establishing it as central to China&#x2019;s future development blueprint. The Congress explicitly stated that China must accelerate the transition to a green economy and advance carbon peaking and carbon neutrality in a proactive yet prudent manner. To achieve this objective, the Chinese government has implemented a series of measures. In May 2024, the State Council issued the Action Plan for Energy Conservation and Carbon Reduction (2024&#x2013;2025), proposing initiatives to enhance energy efficiency and reduce carbon emissions in energy-consuming products and equipment. This includes accelerating the renewal and upgrading of energy-consuming products, facilities, and key equipment, expediting energy-saving and carbon-reduction retrofits for data centres (<xref ref-type="bibr" rid="B29">State Council, 2024</xref>). Furthermore, emphasis was placed on enhancing the recycling of waste products and equipment, accelerating the development of a waste materials recycling system, and strengthening the alignment of supply and demand in the recovery and disposal of waste products and equipment (<xref ref-type="bibr" rid="B29">State Council, 2024</xref>).</p>
<p>This paper examines data from 285 prefecture-level cities in China (excluding Tibet, Xinjiang, Hong Kong, Macao, and Taiwan) to investigate the impact of the green low-carbon economy on energy efficiency. It analyses the promotional role of the green low-carbon economy in energy efficiency development and its spatio-temporal evolution trends. The paper&#x2019;s marginal contribution lies in constructing a multi-model analytical framework to systematically dissect energy efficiency, revealing its evolutionary characteristics from temporal and spatial dimensions, thereby offering new perspectives and methodologies to existing research. Furthermore, addressing existing research gaps, studies on energy efficiency within China&#x2019;s low-carbon economy context have predominantly focused on developed regions such as the Yangtze River Delta economic belt and eastern urban clusters, with limited research on prefecture-level cities nationwide. Consequently, this paper undertakes supplementary research covering all 285 prefecture-level cities. Finally, the conclusions drawn herein offer valuable insights for the Chinese government in formulating targeted energy strategies, advancing green and low-carbon economic development, optimising energy consumption structures, and enhancing pathways to achieve greater energy utilisation efficiency.</p>
</sec>
<sec id="s2">
<label>2</label>
<title>Literature review</title>
<sec id="s2-1">
<label>2.1</label>
<title>Energy efficiency and the green low-carbon economy</title>
<p>Research concerning the green low-carbon economy and energy efficiency In 1995, the World Energy Council defined energy efficiency as the reduction of energy inputs required to provide equivalent energy services (<xref ref-type="bibr" rid="B17">Hui and Jie, 2022</xref>). The concept of a &#x2018;low-carbon economy&#x2019; was first introduced by the UK government in its 2003 Energy White Paper, whose core tenets involve reducing energy consumption and carbon dioxide emissions through technological innovation, policy adjustments, and industrial restructuring under the guidance of sustainable development principles, thereby achieving a win-win outcome for economic growth and environmental protection. China&#x2019;s development of &#x2018;new quality productive forces&#x2019; was first proposed by General Secretary Xiinping in 2023. Its core lies in productivity generated by science and technology innovation, achieving breakthroughs in key disruptive technologies, thereby transcending traditional productive forces (<xref ref-type="bibr" rid="B10">Du, 2024</xref>). This aligns seamlessly with the development concepts of energy efficiency and the green low-carbon economy. On one hand, China&#x2019;s green and low-carbon economic development must adhere to the fundamental principle and overarching orientation of green development, embedding green and low-carbon requirements throughout the entire process and all aspects of high-quality development (<xref ref-type="bibr" rid="B17">Hui and Jie, 2022</xref>). On the other hand, enhancing energy efficiency entails not merely reducing energy consumption but achieving sustainable economic, social, and environmental development through optimised energy utilisation. Within energy production, the green low-carbon economy propels the transformation and upgrading (<xref ref-type="bibr" rid="B30">Sun and Min, 2024</xref>) of conventional energy sources, bolstering the resilience of fossil fuels while elevating standards for their clean extraction and utilisation. In energy consumption, it drives energy-saving and carbon-reduction retrofits, advancing breakthroughs in emerging frontier technologies for decarbonisation and emissions reduction (<xref ref-type="bibr" rid="B18">Yiang and Wu, 2025</xref>). The seamless integration and dual-engine propulsion of new energy and digital technologies constitute the defining characteristic of this new wave of technological revolution and industrial transformation (<xref ref-type="bibr" rid="B2">Lin and Yuqiang, 2024</xref>). Concurrently, empirical research on green productivity within the green and low-carbon economy demonstrates that its evolution can be decomposed into several key drivers: changes in technical efficiency (reflecting catch-up effects from management optimisation and institutional improvements), technological progress (reflecting outward shifts in the frontier of green technologies), and changes in scale efficiency (reflecting adjustments towards optimal production scales). This decomposition aids in identifying the primary sources and constraints of green productivity growth in Chinese cities (<xref ref-type="bibr" rid="B2">Lin and Yuqiang, 2024</xref>).</p>
<p>Energy efficiency constitutes a pivotal factor in green and low-carbon economic development, with enhancing energy efficiency being crucial for reducing carbon emissions and achieving low-carbon economic growth. <xref ref-type="bibr" rid="B22">Liu and Wang, (2022)</xref> employed the LMDI full decomposition model to examine the decarbonisation factors within China&#x2019;s manufacturing sector from 2000 to 2018, demonstrating that the energy intensity effect made a particularly significant contribution to carbon emissions within this industry. Concurrently, this paper examines energy efficiency from the perspectives of power generation and coal consumption. Research by <xref ref-type="bibr" rid="B9">Dongie and Qiaoianqiang (2022)</xref> indicates that under the dual-carbon framework, coal-fired power is transitioning from a primary power source to a foundational safeguard. Next-generation coal-fired technologies focus on three key directions: clean utilisation, efficient power generation, and low-carbon emissions. Among these, high-parameter, large-capacity units leverage system integration advantages to significantly enhance thermal efficiency and fuel adaptability, thereby robustly supporting the construction of a green energy efficiency system (<xref ref-type="bibr" rid="B9">Dong and Qiao, 2022</xref>). Tang Zhenyin (2023), (<xref ref-type="bibr" rid="B32">Tang, 2021</xref>), usingilin Province as a case study, employed a CGE model to analyse the impact of low-carbon development policies on the energy economy. The findings indicate that low-carbon policies have promoted the optimisation of the energy structure and the advancement of new energy technologies.</p>
</sec>
<sec id="s2-2">
<label>2.2</label>
<title>Spatiotemporal evolution of energy efficiency</title>
<p>From 2005 to 2014, the spatiotemporal pattern of urbanisation efficiency within the Yangtze River Economic Belt (YREB) exhibited significant growth and regional disparities. Research by <xref ref-type="bibr" rid="B19">Jin et al. (2018)</xref> indicates that overall urbanisation efficiency rose from 0.34 to 0.53 during this period, representing a cumulative growth rate of approximately 54.07%. This assessment, based on input-output data from 110 cities analysed using Stochastic Frontier Analysis (SFA), revealed the trend of improving urbanisation efficiency and its spatial correlation characteristics within the region. Conversely, (<xref ref-type="bibr" rid="B13">Guo and Han, 2024</xref>) proposed a Generalised Luenberger Productivity Index (GLPI) employing distance elasticity shares as input weights to measure China&#x2019;s urban green total factor productivity (GTFP) from 2000 to 2019, subsequently applying SFA for parametric decomposition. Findings indicate that China&#x2019;s urban GTFP grew cumulatively by 140.59% during this period, primarily driven by technological progress (TC), while technological efficiency change (TEC) and scale efficiency change (SEC) exhibited significant spatio-temporal heterogeneity. Theoretical validation and empirical data matching confirmed the scientific rigour and applicability of this indicator. Additionally, (<xref ref-type="bibr" rid="B42">Zhu et al., 2020</xref>) employed principal component analysis to examine the spatiotemporal patterns of China&#x2019;s energy efficiency from 2000 to 2018. They noted that energy efficiency levels across Chinese regions remain generally low, with the nation&#x2019;s overall energy efficiency development characterised as moderately efficient. <xref ref-type="bibr" rid="B12">Guan et al. (2020)</xref> employed the SBM model to measure energy efficiency in the Yellow River Basin from 1979 to 2017, revealing significant spatial disparities in energy efficiency that transitioned from non-equilibrium to equilibrium. <xref ref-type="bibr" rid="B5">Chen and Zhang (2023)</xref> employed methods such as standard deviation ellipse analysis to study China&#x2019;s agricultural energy efficiency over the past 2&#xa0;decades. They concluded that spatially, China&#x2019;s agricultural energy efficiency exhibits a stepped distribution pattern with higher efficiency in the east and lower in the west. Temporally, no clear convergence or divergence trend in efficiency distribution was observed.</p>
</sec>
<sec id="s2-3">
<label>2.3</label>
<title>Research on factors influencing energy efficiency</title>
<p>Regarding factors affecting energy efficiency, (<xref ref-type="bibr" rid="B20">Liu and Lu, 2024</xref>) employed a DEA (Data Envelopment Analysis) game-based cross-efficiency model to evaluate China&#x2019;s energy efficiency from 2012 to 2021. Building upon this, they constructed a Spatial Durbin Model (SDM) to analyse influencing factors. Findings indicate that technological progress and industrial structure exert positive impacts on China&#x2019;s energy efficiency, whereas urbanisation levels and energy consumption structure exert negative impacts, with all effects being statistically significant. Concurrently, (<xref ref-type="bibr" rid="B11">F&#xe4;re et al., 2026</xref>) extended Prieto and Zof&#xed;o&#x2019;s (2007) activity analysis model by incorporating non-freely disposable adverse outputs into the analytical framework. Using Danish input-output table data from 1990 to 2007, the study compared the differences between the aggregate joint production model in Data Envelopment Analysis (DEA) and the input-output model in measuring productivity changes, providing a methodological comparison for efficiency assessment where both desired and undesired outputs coexist. Furthermore, (<xref ref-type="bibr" rid="B14">Hu and Liu, 2011</xref>) demonstrated that investment levels, industrial structure, and energy consumption patterns are the primary factors significantly influencing regional disparities in energy efficiency. <xref ref-type="bibr" rid="B31">Suo and Yang (2025)</xref> analysed energy development in 11 western provinces from 2010 to 2021 using a three-stage DEA model, finding that levels of openness and shared development significantly promote energy efficiency improvements. Furthermore, scholars have indicated that relative energy prices (<xref ref-type="bibr" rid="B37">Yang et al., 2011</xref>), government policies, and digital financial development all exert certain influences on energy efficiency progress (<xref ref-type="bibr" rid="B21">Liu and Tian, 2019</xref>; <xref ref-type="bibr" rid="B39">Zhang and Shaofang, 2022</xref>).</p>
<p>In summary, existing research has yielded substantial findings in the respective domains of green low-carbon economies and energy efficiency. However, a review reveals that most studies focus on national, provincial, or regional scales, with limited analysis at the city level. Analytical paradigms and methodologies tend to be rather uniform, with few innovations in multi-model approaches. Furthermore, research linking these two areas remains scarce, and no unified perspective or consensus has yet emerged. Consequently, there remains scope for improvement and potential for marginal discussion in exploring these two concepts. Building upon existing research, this paper undertakes a systematic analysis of the spatiotemporal evolution of energy efficiency and its influencing factors, aiming to provide a valuable supplement and refinement to the current body of research.</p>
</sec>
</sec>
<sec id="s3">
<label>3</label>
<title>Theoretical analysis of indicators and framework construction</title>
<sec id="s3-1">
<label>3.1</label>
<title>Analysis of theoretical basis for indicators</title>
<p>This study, grounded in the context of green and low-carbon economic development, employs scientifically selected indicators to construct factors. Specifically, energy efficiency indicators are systematically evaluated from both input and output dimensions (<xref ref-type="bibr" rid="B40">Zhang and Shi, 2021</xref>). Simultaneously, building upon validated prior research, the most representative secondary indicators were purposefully selected for corresponding empirical data analysis. Regarding input indicators, the metric of urban employment per unit reflects human resources&#x2019; contribution to energy production. Drawing from Solow&#x2019;s economic growth model, labour constitutes a core factor in the production function, exhibiting a non-linear relationship with energy efficiency. An appropriate labour scale can stimulate technological spillovers, thereby optimising energy efficiency (<xref ref-type="bibr" rid="B28">Solow, 1956</xref>). Furthermore, the aforementioned research indicates (Zhang and Shi) that labour scale influences energy utilisation efficiency improvements to a certain extent. Regarding national fiscal education expenditure, this indicator measures human capital investment. According to human capital theory, educational investment can indirectly optimise energy utilisation efficiency by enhancing workers&#x2019; skill levels (<xref ref-type="bibr" rid="B27">Schultz, 1961</xref>). Secondly, the year-on-year growth rate of fixed asset investment represents a key factor in green infrastructure investment for low-carbon transition (<xref ref-type="bibr" rid="B41">Zhong and Wang, 2022</xref>). Particularly within a low-carbon economic framework, green infrastructure investment (such as smart grid development) holds significant importance for low-carbon transition, effectively enhancing energy utilisation efficiency. The total energy consumption indicator directly reflects the scale of energy usage, while energy intensity per unit of GDP serves as a core monitoring metric for low-carbon policies, providing an intuitive representation of energy efficiency (<xref ref-type="bibr" rid="B7">Chu, 2009</xref>). The relationship between total energy consumption and economic growth constitutes a focal point in energy economics research (<xref ref-type="bibr" rid="B23">Liu et al., 2019a</xref>), with its trends offering crucial reference value for evaluating energy efficiency. Electricity consumption reflects energy structure issues (<xref ref-type="bibr" rid="B35">Xia and Wang, 2018</xref>). As a relatively clean energy source, shifts in electricity consumption reveal the extent of energy structure adjustment and optimisation, thereby influencing energy efficiency. Finally, research indicates that excessively high coal consumption inhibits low-carbon productivity (<xref ref-type="bibr" rid="B24">Liu et al., 2019b</xref>). Consequently, within a low-carbon economy, reducing coal consumption and increasing the proportion of clean energy represent key pathways to enhancing energy efficiency and achieving low-carbon development.</p>
<p>In terms of output dimensions, the regional gross domestic product (GDP) indicator serves as the core metric for measuring economic output. Energy productivity (GDP/energy consumption) directly quantifies the efficiency with which energy is converted into economic value (<xref ref-type="bibr" rid="B40">Zhang and Shi, 2021</xref>). By analysing the relationship between GDP and energy consumption, the economic efficiency of energy utilisation can be assessed, thereby providing a basis for formulating energy policies. Secondly, electricity generation capacity reflects the efficiency of energy supply output within a low-carbon economy context (<xref ref-type="bibr" rid="B35">Xia and Wang, 2018</xref>). Meanwhile, industrial SO<sub>2</sub> emissions serve as an indicator of the pollution costs associated with energy use. According to decoupling theory (<xref ref-type="bibr" rid="B25">OECD, 2002</xref>), promoting the decoupling of economic growth from pollutant emissions is essential. Reducing industrial SO<sub>2</sub> emissions is therefore crucial for lowering the environmental costs of energy utilisation and enhancing energy efficiency. Finally, industrial wastewater discharge indirectly reflects the environmental externalities of the energy industry (<xref ref-type="bibr" rid="B3">Camilo, 1997</xref>). By internalising negative environmental outputs as efficiency evaluation costs, the true benefits of energy utilisation can be assessed more comprehensively, driving energy consumption towards green and low-carbon development.</p>
</sec>
<sec id="s3-2">
<label>3.2</label>
<title>Indicator composition</title>
<p>Based on the aforementioned theoretical analysis and adhering to fundamental principles of scientific rigour, comprehensiveness, effectiveness, and operational feasibility in indicator establishment, the energy efficiency system level (see <xref ref-type="table" rid="T1">Table 1</xref>) employs a guideline-level indicator system measured across input and output dimensions. Input indicators include: the guideline-level indicator system measures performance across input and output dimensions. Input indicators comprise: urban employment per unit area, year-on-year growth in fixed-asset investment, state-funded education expenditure, total energy consumption, electricity consumption, and coal consumption. Output indicators consist of regional GDP, electricity generation, industrial SO<sub>2</sub> emissions, and industrial wastewater discharge. Collectively, these form a comprehensive energy efficiency assessment framework.</p>
<table-wrap id="T1" position="float">
<label>TABLE 1</label>
<caption>
<p>Energy efficiency measurement indicator system.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th align="center">Principle layer</th>
<th align="center">Indicator Layer</th>
<th align="center">Unit</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td rowspan="6" align="center">Input</td>
<td align="center">Urban employment</td>
<td align="center">10,000 persons</td>
</tr>
<tr>
<td align="center">Year-on-year growth of fixed asset investment</td>
<td align="center">%</td>
</tr>
<tr>
<td align="center">Government expenditure on education</td>
<td align="center">10,000 yuan</td>
</tr>
<tr>
<td align="center">Total energy consumption</td>
<td align="center">10,000 tons of SCE (standard coal equivalent)</td>
</tr>
<tr>
<td align="center">Electricity consumption</td>
<td align="center">100 million kWh</td>
</tr>
<tr>
<td align="center">Coal consumption</td>
<td align="center">10,000 tons</td>
</tr>
<tr>
<td rowspan="4" align="center">Output</td>
<td align="center">Regional GDP</td>
<td align="center">10,000 yuan</td>
</tr>
<tr>
<td align="center">Electricity generation</td>
<td align="center">100 million kWh</td>
</tr>
<tr>
<td align="center">Industrial SO<sub>2</sub> emissions</td>
<td align="center">tons</td>
</tr>
<tr>
<td align="center">Industrial wastewater discharge</td>
<td align="center">10,000 tons</td>
</tr>
</tbody>
</table>
</table-wrap>
</sec>
</sec>
<sec id="s4">
<label>4</label>
<title>Introduction to the data processing model</title>
<sec id="s4-1">
<label>4.1</label>
<title>Data sources and preprocessing</title>
<p>This study focuses on 285 prefecture-level cities in China from 2012 to 2022 (excluding Tibet, Xinjiang, and Hong Kong, Macao, and Taiwan). The data primarily comes from the China Statistical Yearbook, the China Urban Statistical Yearbook, the China Energy Statistical Yearbook, the China Electricity Yearbook, and the statistical yearbooks of various prefecture-level cities. In terms of outlier handling, for observations that significantly deviate from the normal range, this study primarily employs outlier removal methods. Regarding missing value handling, given the absence of data for certain indicators in 2022 and minor missing data in historical years (with a missing rate below 5%), this study uses linear interpolation to supplement the missing data.</p>
</sec>
<sec id="s4-2">
<label>4.2</label>
<title>Non-expected output SBM model (Un_SBM_CRS)</title>
<sec id="s4-2-1">
<label>4.2.1</label>
<title>Model introduction</title>
<p>DEA generally considers producing more output with fewer resources to be an efficient production method (<xref ref-type="bibr" rid="B15">Hu et al., 2011</xref>). When considering non-desired outputs, regardless of the amount of inputs, we do not want to produce more industrial waste. Therefore, the most efficient production method in today&#x2019;s society must be a green production method, i.e., producing more desired outputs and fewer non-desired outputs with fewer inputs.</p>
<p>Assume there are n decision units, each of which includes three elements: inputs, desired outputs, and undesired outputs (such as wastewater, carbon dioxide, and particulate matter), represented by three vectors (<inline-formula id="inf1">
<mml:math id="m1">
<mml:mrow>
<mml:mi>X</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf2">
<mml:math id="m2">
<mml:mrow>
<mml:mi>Y</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf3">
<mml:math id="m3">
<mml:mrow>
<mml:mi>Z</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>). Use the SBM model with non-expected outputs to evaluate DMU (<inline-formula id="inf4">
<mml:math id="m4">
<mml:mrow>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf5">
<mml:math id="m5">
<mml:mrow>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf6">
<mml:math id="m6">
<mml:mrow>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>) as shown in <xref ref-type="disp-formula" rid="e13">Equation 13</xref>:<disp-formula id="e1">
<mml:math id="m7">
<mml:mrow>
<mml:mi>&#x3c1;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>min</mml:mi>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>m</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mstyle displaystyle="true">
<mml:msubsup>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>m</mml:mi>
</mml:msubsup>
</mml:mstyle>
<mml:mfrac>
<mml:mrow>
<mml:msubsup>
<mml:mi>s</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>x</mml:mi>
</mml:msubsup>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo>&#x2b;</mml:mo>
<mml:mfrac>
<mml:mn>1</mml:mn>
<mml:mrow>
<mml:msub>
<mml:mi>s</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi>s</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mstyle displaystyle="true">
<mml:msubsup>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>k</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:msub>
<mml:mi>s</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:msubsup>
</mml:mstyle>
<mml:mfrac>
<mml:mrow>
<mml:msubsup>
<mml:mi>s</mml:mi>
<mml:mi>k</mml:mi>
<mml:mi>y</mml:mi>
</mml:msubsup>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mrow>
<mml:mi>k</mml:mi>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#x2b;</mml:mo>
<mml:mstyle displaystyle="true">
<mml:msubsup>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>l</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:msub>
<mml:mi>s</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:msubsup>
</mml:mstyle>
<mml:mfrac>
<mml:mrow>
<mml:msubsup>
<mml:mi>s</mml:mi>
<mml:mi>l</mml:mi>
<mml:mi>z</mml:mi>
</mml:msubsup>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mrow>
<mml:mi>l</mml:mi>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
<label>(1)</label>
</disp-formula>
<disp-formula id="e2">
<mml:math id="m8">
<mml:mrow>
<mml:mtable columnalign="left">
<mml:mtr>
<mml:mtd>
<mml:mrow>
<mml:mi>s</mml:mi>
<mml:mo>.</mml:mo>
<mml:mi>t</mml:mi>
<mml:mo>.</mml:mo>
</mml:mrow>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mrow>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mstyle displaystyle="true">
<mml:munderover>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:munderover>
</mml:mstyle>
<mml:msub>
<mml:mi>&#x3bb;</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:msubsup>
<mml:mi>s</mml:mi>
<mml:msup>
<mml:mi>i</mml:mi>
<mml:mo>&#x2032;</mml:mo>
</mml:msup>
<mml:mi>x</mml:mi>
</mml:msubsup>
<mml:mo>&#x2200;</mml:mo>
<mml:mi>i</mml:mi>
<mml:mo>;</mml:mo>
</mml:mrow>
</mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:mrow>
</mml:math>
<label>(2)</label>
</disp-formula>
<disp-formula id="e3">
<mml:math id="m9">
<mml:mrow>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mrow>
<mml:mi>k</mml:mi>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mstyle displaystyle="true">
<mml:munderover>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:munderover>
</mml:mstyle>
<mml:msub>
<mml:mi>&#x3bb;</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:mo>&#x2212;</mml:mo>
<mml:msubsup>
<mml:mi>s</mml:mi>
<mml:msup>
<mml:mi>k</mml:mi>
<mml:mo>&#x2032;</mml:mo>
</mml:msup>
<mml:mi>y</mml:mi>
</mml:msubsup>
<mml:mo>&#x2200;</mml:mo>
<mml:mi>k</mml:mi>
<mml:mo>;</mml:mo>
</mml:mrow>
</mml:math>
<label>(3)</label>
</disp-formula>
<disp-formula id="e4">
<mml:math id="m10">
<mml:mrow>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mrow>
<mml:mi>l</mml:mi>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mstyle displaystyle="true">
<mml:munderover>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:munderover>
</mml:mstyle>
<mml:msub>
<mml:mi>&#x3bb;</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:mo>&#x2212;</mml:mo>
<mml:msubsup>
<mml:mi>s</mml:mi>
<mml:msup>
<mml:mi>l</mml:mi>
<mml:mo>&#x2032;</mml:mo>
</mml:msup>
<mml:mi>z</mml:mi>
</mml:msubsup>
<mml:mo>&#x2200;</mml:mo>
<mml:mi>l</mml:mi>
<mml:mo>;</mml:mo>
</mml:mrow>
</mml:math>
<label>(4)</label>
</disp-formula>
<disp-formula id="equ1">
<mml:math id="m11">
<mml:mrow>
<mml:msubsup>
<mml:mi>s</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>x</mml:mi>
</mml:msubsup>
<mml:mo>&#x2265;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
<mml:msubsup>
<mml:mi>s</mml:mi>
<mml:mi>k</mml:mi>
<mml:mi>y</mml:mi>
</mml:msubsup>
<mml:mo>&#x2265;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
<mml:msubsup>
<mml:mi>s</mml:mi>
<mml:mi>l</mml:mi>
<mml:mi>z</mml:mi>
</mml:msubsup>
<mml:mo>&#x2265;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>&#x3bb;</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:mo>&#x2265;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
<mml:mo>&#x2200;</mml:mo>
<mml:mi>i</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>j</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>k</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>l</mml:mi>
<mml:mo>&#x2265;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>;</mml:mo>
</mml:mrow>
</mml:math>
</disp-formula>
<inline-formula id="inf7">
<mml:math id="m12">
<mml:mrow>
<mml:msup>
<mml:mi>s</mml:mi>
<mml:mi>x</mml:mi>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> &#x2208;<inline-formula id="inf8">
<mml:math id="m13">
<mml:mrow>
<mml:msup>
<mml:mi>R</mml:mi>
<mml:mi>m</mml:mi>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>,s<sup>z</sup>&#x2208; <inline-formula id="inf9">
<mml:math id="m14">
<mml:mrow>
<mml:msup>
<mml:mi>R</mml:mi>
<mml:mrow>
<mml:mi>s</mml:mi>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> represent the excess amounts of inputs and non-expected outputs, respectively, while <inline-formula id="inf10">
<mml:math id="m15">
<mml:mrow>
<mml:msup>
<mml:mi>s</mml:mi>
<mml:mi>y</mml:mi>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> &#x2208;<inline-formula id="inf11">
<mml:math id="m16">
<mml:mrow>
<mml:msup>
<mml:mi>R</mml:mi>
<mml:mrow>
<mml:mi>s</mml:mi>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> represents the shortage of expected outputs. <inline-formula id="inf12">
<mml:math id="m17">
<mml:mrow>
<mml:mi>&#x3c1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> represents the efficiency value of the decision-making unit, and m, <inline-formula id="inf13">
<mml:math id="m18">
<mml:mrow>
<mml:msub>
<mml:mi>s</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf14">
<mml:math id="m19">
<mml:mrow>
<mml:msub>
<mml:mi>s</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> represent the number of variables for inputs, expected outputs, and non-expected outputs.</p>
<p>When <inline-formula id="inf15">
<mml:math id="m20">
<mml:mrow>
<mml:mi>&#x3c1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; 1, which is <inline-formula id="inf16">
<mml:math id="m21">
<mml:mrow>
<mml:msup>
<mml:mi>s</mml:mi>
<mml:mi>x</mml:mi>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; 0, <inline-formula id="inf17">
<mml:math id="m22">
<mml:mrow>
<mml:msup>
<mml:mi>s</mml:mi>
<mml:mi>y</mml:mi>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; 0, <inline-formula id="inf18">
<mml:math id="m23">
<mml:mrow>
<mml:msup>
<mml:mi>s</mml:mi>
<mml:mi>z</mml:mi>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; 0, represents DMU, it is valid. However, when <inline-formula id="inf19">
<mml:math id="m24">
<mml:mrow>
<mml:mi>&#x3c1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> &#x3c;1occurs, it indicates that the DMU is not effective and there is room for improvement.</p>
</sec>
<sec id="s4-2-2">
<label>4.2.2</label>
<title>Malmquist index and decomposition</title>
<p>The Malmquist index is essentially the ratio of two distance function values and is a theoretical index (CCD, 1982). Based on the M index formula, the M index with variable returns to scale can also be directly derived and decomposed into PEC pure technical efficiency change and PTC pure technical change. The decomposition formula is shown below:</p>
<p>Case of constant returns to scale (CRS):<disp-formula id="e5">
<mml:math id="m25">
<mml:mrow>
<mml:msup>
<mml:mi>M</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>E</mml:mi>
<mml:mi>C</mml:mi>
<mml:mo>&#x2a;</mml:mo>
<mml:mi>T</mml:mi>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:math>
<label>(5)</label>
</disp-formula>
<disp-formula id="e6">
<mml:math id="m26">
<mml:mrow>
<mml:msup>
<mml:mi>M</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msup>
<mml:mi>x</mml:mi>
<mml:mi>t</mml:mi>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mi>t</mml:mi>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mi>x</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="|">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:msup>
<mml:mi>E</mml:mi>
<mml:mi>t</mml:mi>
</mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msup>
<mml:mi>x</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
<mml:mrow>
<mml:msup>
<mml:mi>E</mml:mi>
<mml:mi>t</mml:mi>
</mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msup>
<mml:mi>x</mml:mi>
<mml:mi>t</mml:mi>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mi>t</mml:mi>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#x2a;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msup>
<mml:mi>E</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msup>
<mml:mi>x</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
<mml:mrow>
<mml:msup>
<mml:mi>E</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msup>
<mml:mi>x</mml:mi>
<mml:mi>t</mml:mi>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mi>t</mml:mi>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:mfrac>
</mml:msup>
</mml:mrow>
</mml:math>
<label>(6)</label>
</disp-formula>
<disp-formula id="e7">
<mml:math id="m27">
<mml:mrow>
<mml:mi>E</mml:mi>
<mml:mi>C</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msup>
<mml:mi>E</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msup>
<mml:mi>x</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
<mml:mrow>
<mml:msup>
<mml:mi>E</mml:mi>
<mml:mi>t</mml:mi>
</mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msup>
<mml:mi>x</mml:mi>
<mml:mi>t</mml:mi>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mi>t</mml:mi>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
<label>(7)</label>
</disp-formula>
<disp-formula id="e8">
<mml:math id="m28">
<mml:mrow>
<mml:mi mathvariant="normal">T</mml:mi>
<mml:mi>C</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="|">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:msubsup>
<mml:mi>E</mml:mi>
<mml:mi>c</mml:mi>
<mml:mi>t</mml:mi>
</mml:msubsup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msup>
<mml:mi>x</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
<mml:mrow>
<mml:msubsup>
<mml:mi>E</mml:mi>
<mml:mi>c</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msubsup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msup>
<mml:mi>x</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#x2a;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msubsup>
<mml:mi>E</mml:mi>
<mml:mi>c</mml:mi>
<mml:mi>t</mml:mi>
</mml:msubsup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msup>
<mml:mi>x</mml:mi>
<mml:mi>t</mml:mi>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mi>t</mml:mi>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
<mml:mrow>
<mml:msubsup>
<mml:mi>E</mml:mi>
<mml:mi>c</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msubsup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msup>
<mml:mi>x</mml:mi>
<mml:mi>t</mml:mi>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mi>t</mml:mi>
</mml:msup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:mfrac>
</mml:msup>
</mml:mrow>
</mml:math>
<label>(8)</label>
</disp-formula>
</p>
<p>Where <inline-formula id="inf20">
<mml:math id="m29">
<mml:mrow>
<mml:msup>
<mml:mi mathvariant="normal">E</mml:mi>
<mml:mi>t</mml:mi>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> represents the efficiency value of the SBM model in period t, <inline-formula id="inf21">
<mml:math id="m30">
<mml:mrow>
<mml:msup>
<mml:mi>E</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> represents the efficiency value of the SBM model in period t&#x2b;1, the input vector, and <inline-formula id="inf22">
<mml:math id="m31">
<mml:mrow>
<mml:mi mathvariant="normal">y</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> represents the output vector.</p>
</sec>
<sec id="s4-2-3">
<label>4.2.3</label>
<title>Dagum Gini coefficient</title>
<p>The Gini coefficient method proposed by Dagum (1997) (<xref ref-type="bibr" rid="B4">Chen, 2023</xref>) was used to conduct a decomposition analysis of the differences in energy efficiency between China and its eight major regions. According to the Gini coefficient proposed by Dagum and his subgroup decomposition method, the Gini coefficient of energy efficiency can be defined as:<disp-formula id="e9">
<mml:math id="m32">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>i</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mi>w</mml:mi>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
<label>(9)</label>
</disp-formula>
</p>
<p>In the above equation, <inline-formula id="inf23">
<mml:math id="m33">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> represents the overall Gini coefficient; <inline-formula id="inf24">
<mml:math id="m34">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mi>w</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> represents the contribution of horizontal disparities within regions; <inline-formula id="inf25">
<mml:math id="m35">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> represents the contribution of net value disparities between regions; and <inline-formula id="inf26">
<mml:math id="m36">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> represents the contribution of super-density.<disp-formula id="e10">
<mml:math id="m37">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>i</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mstyle displaystyle="true">
<mml:msubsup>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>k</mml:mi>
</mml:msubsup>
</mml:mstyle>
<mml:mrow>
<mml:mstyle displaystyle="true">
<mml:msubsup>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>h</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>k</mml:mi>
</mml:msubsup>
</mml:mstyle>
<mml:mrow>
<mml:mstyle displaystyle="true">
<mml:msubsup>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:mstyle>
<mml:mrow>
<mml:mstyle displaystyle="true">
<mml:msubsup>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>r</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:mstyle>
<mml:mrow>
<mml:mfenced open="|" close="|" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mrow>
<mml:mi>h</mml:mi>
<mml:mi>r</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mrow>
</mml:mrow>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
<mml:msup>
<mml:mi>n</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mi>&#x3bc;</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mo>,</mml:mo>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>h</mml:mi>
</mml:msub>
<mml:mo>&#x2264;</mml:mo>
<mml:msub>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:mo>&#x2264;</mml:mo>
<mml:mo>&#x22ef;</mml:mo>
<mml:mo>&#x2264;</mml:mo>
<mml:msub>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>k</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
<label>(10)</label>
</disp-formula>
<disp-formula id="equ2">
<mml:math id="m38">
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>,</mml:mo>
<mml:mn>2</mml:mn>
<mml:mo>,</mml:mo>
<mml:mn>3</mml:mn>
<mml:mo>,</mml:mo>
<mml:mo>&#x22ef;</mml:mo>
<mml:mo>,</mml:mo>
<mml:mi>k</mml:mi>
<mml:mo>;</mml:mo>
<mml:mi>n</mml:mi>
<mml:mi>j</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>,</mml:mo>
<mml:mn>2</mml:mn>
<mml:mo>,</mml:mo>
<mml:mn>3</mml:mn>
<mml:mo>,</mml:mo>
<mml:mo>&#x22ef;</mml:mo>
<mml:mo>,</mml:mo>
<mml:mi>n</mml:mi>
<mml:mi>j</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</disp-formula>
</p>
<p>Through this decomposition, it can be divided into k regions and n cities, where <inline-formula id="inf27">
<mml:math id="m39">
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mi>j</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> is the number of cities in region <inline-formula id="inf28">
<mml:math id="m40">
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf29">
<mml:math id="m41">
<mml:mrow>
<mml:mtext>yji</mml:mtext>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mtext>yhr</mml:mtext>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> represents the energy efficiency level of any city in region <inline-formula id="inf30">
<mml:math id="m42">
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, and <inline-formula id="inf31">
<mml:math id="m43">
<mml:mrow>
<mml:mi>&#x3bc;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> represents the average energy efficiency level of all cities.<disp-formula id="e11">
<mml:math id="m44">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mfrac>
<mml:mn>1</mml:mn>
<mml:mrow>
<mml:mn>2</mml:mn>
<mml:msub>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:mrow>
<mml:mstyle displaystyle="true">
<mml:msubsup>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:mstyle>
<mml:mrow>
<mml:mfenced open="|" close="|" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mrow>
<mml:mi>h</mml:mi>
<mml:mi>r</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mrow>
<mml:mrow>
<mml:msup>
<mml:mi>n</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
<label>(11)</label>
</disp-formula>
<inline-formula id="inf32">
<mml:math id="m45">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> represents the Gini coefficient of energy efficiency in region <inline-formula id="inf33">
<mml:math id="m46">
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>.<disp-formula id="e12">
<mml:math id="m47">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mi>w</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mstyle displaystyle="true">
<mml:munderover>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>k</mml:mi>
</mml:munderover>
</mml:mstyle>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:msub>
<mml:mi>P</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
<label>(12)</label>
</disp-formula>
<inline-formula id="inf34">
<mml:math id="m48">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mi>w</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> represents the contribution rate of regional differences within the region to the overall difference.<disp-formula id="e13">
<mml:math id="m49">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mstyle displaystyle="true">
<mml:msubsup>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:mstyle>
<mml:mrow>
<mml:mstyle displaystyle="true">
<mml:msubsup>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>r</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msubsup>
</mml:mstyle>
<mml:mrow>
<mml:mfenced open="|" close="|" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mrow>
<mml:mi>h</mml:mi>
<mml:mi>r</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mrow>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>h</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>h</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
<label>(13)</label>
</disp-formula>
<inline-formula id="inf35">
<mml:math id="m50">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> represents the interregional Gini coefficient for regions <inline-formula id="inf36">
<mml:math id="m51">
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf37">
<mml:math id="m52">
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>.<disp-formula id="e14">
<mml:math id="m53">
<mml:mrow>
<mml:msub>
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mo>&#x3d;</mml:mo>
</mml:mrow>
</mml:msub>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
<label>(14)</label>
</disp-formula>
<disp-formula id="e15">
<mml:math id="m54">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mo>&#x3d;</mml:mo>
</mml:mrow>
</mml:msub>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mi>&#x3bc;</mml:mi>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
<label>(15)</label>
</disp-formula>
<disp-formula id="e16">
<mml:math id="m55">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mstyle displaystyle="true">
<mml:munderover>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
<mml:mi>k</mml:mi>
</mml:munderover>
</mml:mstyle>
<mml:mstyle displaystyle="true">
<mml:munderover>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>h</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:munderover>
</mml:mstyle>
<mml:msub>
<mml:mi>G</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>P</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>h</mml:mi>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi>P</mml:mi>
<mml:mi>h</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:msub>
<mml:mi>D</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
<label>(16)</label>
</disp-formula>
<inline-formula id="inf38">
<mml:math id="m56">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> represents the contribution rate of the net value difference in that region to the overall difference.<disp-formula id="e17">
<mml:math id="m57">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mrow>
<mml:mstyle displaystyle="true">
<mml:munderover>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
<mml:mi>k</mml:mi>
</mml:munderover>
</mml:mstyle>
<mml:mrow>
<mml:mstyle displaystyle="true">
<mml:munderover>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>h</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:munderover>
</mml:mstyle>
<mml:mrow>
<mml:msub>
<mml:mi>G</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>P</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>h</mml:mi>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi>P</mml:mi>
<mml:mi>h</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>D</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mrow>
</mml:mrow>
</mml:mrow>
</mml:math>
<label>(17)</label>
</disp-formula>
<inline-formula id="inf39">
<mml:math id="m58">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> represents the contribution of super-density.<disp-formula id="e18">
<mml:math id="m59">
<mml:mrow>
<mml:msub>
<mml:mi>D</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mi>d</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mi>d</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
<label>(18)</label>
</disp-formula>
<inline-formula id="inf40">
<mml:math id="m60">
<mml:mrow>
<mml:msub>
<mml:mi>D</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> represents the relative impact of energy efficiency levels between regions <inline-formula id="inf41">
<mml:math id="m61">
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf42">
<mml:math id="m62">
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>.<disp-formula id="e19">
<mml:math id="m63">
<mml:mrow>
<mml:msub>
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msubsup>
<mml:mo>&#x222b;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mi>&#x221e;</mml:mi>
</mml:msubsup>
<mml:mi>d</mml:mi>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>y</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:msubsup>
<mml:mo>&#x222b;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mi>y</mml:mi>
</mml:msubsup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>y</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>x</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mi>d</mml:mi>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>h</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
<label>(19)</label>
</disp-formula>
<disp-formula id="e20">
<mml:math id="m64">
<mml:mrow>
<mml:msub>
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msubsup>
<mml:mo>&#x222b;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mi>&#x221e;</mml:mi>
</mml:msubsup>
<mml:mi>d</mml:mi>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>h</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>y</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:msubsup>
<mml:mo>&#x222b;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mi>y</mml:mi>
</mml:msubsup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>y</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>x</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mi>d</mml:mi>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>y</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
<label>(20)</label>
</disp-formula>
<inline-formula id="inf43">
<mml:math id="m65">
<mml:mrow>
<mml:msub>
<mml:mi>d</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> represents the difference in the comprehensive energy efficiency scores between regions; <inline-formula id="inf44">
<mml:math id="m66">
<mml:mrow>
<mml:msub>
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the super-variable first-order distance; <inline-formula id="inf45">
<mml:math id="m67">
<mml:mrow>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>h</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> is the cumulative rate distribution function for region <inline-formula id="inf46">
<mml:math id="m68">
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>.</p>
</sec>
<sec id="s4-2-4">
<label>4.2.4</label>
<title>Kernel density estimation</title>
<p>Kernel density estimation is a nonparametric estimation method that uses a smooth peak function to fit sample data (<xref ref-type="bibr" rid="B38">Zhang and Hao, 2020</xref>) and describes the distribution pattern of random variables using a continuous density curve. Assuming that <inline-formula id="inf47">
<mml:math id="m69">
<mml:mrow>
<mml:mi>f</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> is the density function of random variable <inline-formula id="inf48">
<mml:math id="m70">
<mml:mrow>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and the probability density at <inline-formula id="inf49">
<mml:math id="m71">
<mml:mrow>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, this paper selects the commonly used Gaussian kernel as the expression form of the kernel function, such as:<disp-formula id="e21">
<mml:math id="m72">
<mml:mrow>
<mml:mi>f</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mn>1</mml:mn>
<mml:mrow>
<mml:mi>N</mml:mi>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mrow>
<mml:mstyle displaystyle="true">
<mml:munderover>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>N</mml:mi>
</mml:munderover>
</mml:mstyle>
<mml:mrow>
<mml:mi>K</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mi>h</mml:mi>
</mml:mfrac>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:mrow>
</mml:mrow>
</mml:math>
<label>(21)</label>
</disp-formula>
<disp-formula id="e22">
<mml:math id="m73">
<mml:mrow>
<mml:mi>K</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:msqrt>
<mml:mrow>
<mml:mn>2</mml:mn>
<mml:mi>&#x3c0;</mml:mi>
</mml:mrow>
</mml:msqrt>
</mml:mrow>
</mml:mfrac>
<mml:mi>exp</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msup>
<mml:mi>x</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
<label>(22)</label>
</disp-formula>
<inline-formula id="inf50">
<mml:math id="m74">
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is the sample size, <inline-formula id="inf51">
<mml:math id="m75">
<mml:mrow>
<mml:mi>h</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is the bandwidth, <inline-formula id="inf52">
<mml:math id="m76">
<mml:mrow>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the independently and identically distributed observations, and <inline-formula id="inf53">
<mml:math id="m77">
<mml:mrow>
<mml:mi>K</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="|">
<mml:mrow>
<mml:mo>&#xb7;</mml:mo>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> is the kernel function.</p>
</sec>
<sec id="s4-2-5">
<label>4.2.5</label>
<title>Markov chain</title>
<p>A Markov chain is a stochastic process used to describe the change in system state over time (<xref ref-type="bibr" rid="B36">Xiaohong and Qi, 2013</xref>), in which the probability of each state transition depends only on the current state and not on previous states. This paper uses a Markov chain to analyze the dynamic evolution trend of China&#x2019;s energy efficiency level. Following the principle of discrete equal division, the discrete values are divided into four levels: low, medium-low, medium-high, and high. The model is constructed as follows:<disp-formula id="e23">
<mml:math id="m78">
<mml:mrow>
<mml:mi>P</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mfenced open="{" close="}" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>j</mml:mi>
<mml:mo>&#x7c;</mml:mo>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>i</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>&#xb7;</mml:mo>
<mml:mo>&#xb7;</mml:mo>
<mml:mo>&#xb7;</mml:mo>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>i</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mfenced open="{" close="}" separators="|">
<mml:mrow>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>j</mml:mi>
<mml:mo>&#x7c;</mml:mo>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
<label>(23)</label>
</disp-formula>
<disp-formula id="e24">
<mml:math id="m79">
<mml:mrow>
<mml:msub>
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
<label>(24)</label>
</disp-formula>
</p>
<p>Among them: <inline-formula id="inf54">
<mml:math id="m80">
<mml:mrow>
<mml:msub>
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the probability that the energy efficiency level of a city at time t will transfer from type i to type j at time t&#x2b;1; <inline-formula id="inf55">
<mml:math id="m81">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the number of cities in type i at time t; <inline-formula id="inf56">
<mml:math id="m82">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the number of cities in type i at time t that will transfer to type j at time t&#x2b;1. By calculating the conversion, a traditional Markov probability transition matrix is obtained to reflect the dynamic evolution of national energy efficiency.</p>
<p>Spatial Markov chains incorporate spatial lag factors into traditional Markov transition matrices, where spatial lag factors reflect the collaborative development status at the neighborhood level, specifically measured by the level of spatially weighted neighboring regions. To avoid the island effect when constructing the spatial matrix, this paper adopts a spatial inverse distance matrix. Based on this, this paper decomposes the formula into n independent transition matrices to explore the changing trends in energy efficiency under different spatial lag scenarios.<disp-formula id="e25">
<mml:math id="m83">
<mml:mrow>
<mml:mi>P</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="|">
<mml:mrow>
<mml:mtable columnalign="left">
<mml:mtr>
<mml:mtd>
<mml:msub>
<mml:mi>p</mml:mi>
<mml:mn>11</mml:mn>
</mml:msub>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:msub>
<mml:mi>p</mml:mi>
<mml:mn>21</mml:mn>
</mml:msub>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mo>&#x22ee;</mml:mo>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:msub>
<mml:mi>p</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mtd>
</mml:mtr>
</mml:mtable>
<mml:mtable columnalign="left">
<mml:mtr>
<mml:mtd>
<mml:msub>
<mml:mi>p</mml:mi>
<mml:mn>12</mml:mn>
</mml:msub>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:msub>
<mml:mi>p</mml:mi>
<mml:mn>22</mml:mn>
</mml:msub>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mo>&#x22ee;</mml:mo>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:msub>
<mml:mi>p</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mtd>
</mml:mtr>
</mml:mtable>
<mml:mtable columnalign="left">
<mml:mtr>
<mml:mtd>
<mml:mrow>
<mml:mo>&#xb7;</mml:mo>
<mml:mo>&#xb7;</mml:mo>
<mml:mo>&#xb7;</mml:mo>
</mml:mrow>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mrow>
<mml:mo>&#xb7;</mml:mo>
<mml:mo>&#xb7;</mml:mo>
<mml:mo>&#xb7;</mml:mo>
</mml:mrow>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mrow>
<mml:mo>&#xb7;</mml:mo>
<mml:mo>&#xb7;</mml:mo>
<mml:mo>&#xb7;</mml:mo>
</mml:mrow>
</mml:mtd>
</mml:mtr>
</mml:mtable>
<mml:mtable columnalign="left">
<mml:mtr>
<mml:mtd>
<mml:msub>
<mml:mi>p</mml:mi>
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:msub>
<mml:mi>p</mml:mi>
<mml:mrow>
<mml:mn>2</mml:mn>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mo>&#x22ee;</mml:mo>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:msub>
<mml:mi>p</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
<label>(25)</label>
</disp-formula>
</p>
</sec>
</sec>
</sec>
<sec id="s5">
<label>5</label>
<title>Energy efficiency measurement analysis</title>
<sec id="s5-1">
<label>5.1</label>
<title>Overall analysis</title>
<p>Using the non-expected output SBM model to measure energy efficiency across 285 prefecture-level cities yielded Malquist indices presented at provincial level (due to space constraints, SBM Malquist results are shown in <xref ref-type="sec" rid="s16">Supplementary Appendix Table 2</xref>). These results illustrate the evolution of energy efficiency from 2013 to 2022&#xa0;at both the national level and across eight major regions (see <xref ref-type="fig" rid="F1">Figure 1</xref>). Overall, energy efficiency across all regions remained at a relatively high level throughout the study period, exhibiting a trend of &#x2018;steady growth&#x2019;. The energy efficiency growth rate in 2013 compared to the previous year was 1.137, and it has maintained a growth rate greater than 1 since 2013, gradually increasing. The fastest growth in energy efficiency measurement occurred in 2021, with a growth rate of 1.184. Since 2013, China&#x2019;s energy efficiency has grown rapidly, benefiting from multiple positive factors. Firstly, the implementation of national energy-saving and carbon-reduction policies alongside dual-control measures for energy consumption has effectively driven industrial restructuring and optimisation, fostering the development of energy-saving and carbon-reduction technologies and industries. Secondly, technological progress has played a pivotal role in enhancing energy utilisation efficiency, particularly through the advancement of new energy sectors such as solar and wind power, which have significantly boosted energy efficiency. Furthermore, the optimisation of the energy consumption structure, notably the decline in coal&#x2019;s share of consumption, has also contributed positively to energy efficiency gains. The elevation of economic development levels and household consumption standards has guided shifts in energy utilisation patterns, thereby reducing energy intensity. Policy guidance has also played a vital role in adjusting the energy mix and enhancing energy usage efficiency. Concurrently, improvements in China&#x2019;s energy supply-demand balance and the vigorous development of clean energy, particularly the expansion of non-fossil fuel power generation capacity, have further contributed to energy efficiency gains. The combined effect of these factors has enabled China&#x2019;s energy efficiency to achieve steady growth.</p>
<fig id="F1" position="float">
<label>FIGURE 1</label>
<caption>
<p>Energy efficiency growth rates nationwide and by region.</p>
</caption>
<graphic xlink:href="fenvs-13-1741575-g001.tif">
<alt-text content-type="machine-generated">Line graph showing regional trends from 2012 to 2022 with 10 distinct lines. Each line represents a region: Eastern, Northern, Southern coastal areas, Northeast, Great Northwest, Yangtze River, Southwest, and Yellow River regions. Values range from 0.9 to 1.4.</alt-text>
</graphic>
</fig>
</sec>
<sec id="s5-2">
<label>5.2</label>
<title>Regional analysis</title>
<p>Analysed by region, China&#x2019;s eight major areas exhibit an overall upward trend in energy efficiency (see <xref ref-type="fig" rid="F4">Figure 4</xref>), though significant disparities exist between regions. Generally, coastal regions demonstrate higher energy efficiency growth rates than riverine regions, which in turn exceed those of inland areas. The eastern coastal and northern coastal regions recorded the most rapid growth in energy efficiency. The energy efficiency growth rate in the Northwest fluctuated around 1% between 2013 and 2020, before accelerating rapidly post-2021 in line with other regions. This indicates that the eastern and northern coastal regions benefit from higher levels of regional economic development, technological sophistication, and prioritisation of energy efficiency. Their geographical advantages have enabled them to prosper earlier than inland areas, leading to significantly higher economic development. with greater emphasis on ecological conservation and energy efficiency utilisation, alongside more mature technologies. Following the 18th CPC National Congress in 2012, which positioned ecological civilisation as a vital component of the &#x2018;Five-in-One&#x2019; overall layout, coastal regions have continuously driven inland development. The implementation of China&#x2019;s Western Development Strategy and Central Region Rise Strategy has also spurred rapid energy efficiency growth in riverine and inland areas, with the most pronounced effects and significant growth rate shifts evident after 2020. Concurrently, the disparity in energy efficiency growth rates among the eight major regions has progressively narrowed since 2020, indicating a gradual reduction in regional variations across China.</p>
<p>Examining regional growth trajectories reveals the Northeast exhibits the most pronounced rate of change, followed by the Middle Yellow River region. Other areas also demonstrate corresponding fluctuations, suggesting these two regions are pursuing divergent approaches to enhancing energy efficiency and undertaking corresponding experiments. The most notable variations in energy efficiency rates are observed in Heilongjiang and Henan provinces.</p>
<p>As shown in <xref ref-type="fig" rid="F2">Figure 2</xref>, both the low-carbon economy and energy efficiency exhibit an overall upward trend, albeit at differing rates. Compared to energy efficiency, the growth of the low-carbon economy has been more moderate. Energy efficiency reached its peak in 2016, experienced a dip in 2017, saw a significant improvement in 2018, and subsequently stabilised. This pattern likely reflects the positive impact of policy initiatives and technological advancements on energy efficiency during this period. Observing the Sankey diagram (see <xref ref-type="fig" rid="F3">Figure 3</xref>), significant regional disparities exist in national energy efficiency and low-carbon economic development. Coastal regions in the east and south demonstrate particularly strong performance in both low-carbon economy and energy efficiency, likely benefiting from higher levels of economic development, advanced technology application, and more comprehensive policy support systems. In contrast, the Northwest and Southwest regions exhibit relatively weaker performance in low-carbon economic development and energy efficiency, potentially linked to their industrial structures, energy mixes, and levels of economic development. Combining these insights reveals two key findings: firstly, regional development imbalances constitute a significant constraint on advancing national low-carbon economic development and energy efficiency; secondly, policy support and technological innovation serve as pivotal drivers for enhancing both low-carbon economic development and energy efficiency.</p>
<fig id="F2" position="float">
<label>FIGURE 2</label>
<caption>
<p>Low-carbon economy and energy efficiency change chart.</p>
</caption>
<graphic xlink:href="fenvs-13-1741575-g002.tif">
<alt-text content-type="machine-generated">Line graph depicting trends from 2012 to 2022, with two vertical axes. Left axis represents low-carbon economy values, indicated by light green bars; right axis represents energy efficiency, shown by a green line with dots. Data shows fluctuation over the years with a general upward trend.</alt-text>
</graphic>
</fig>
<fig id="F3" position="float">
<label>FIGURE 3</label>
<caption>
<p>Energy efficiency Sankey diagram.</p>
</caption>
<graphic xlink:href="fenvs-13-1741575-g003.tif">
<alt-text content-type="machine-generated">Sankey diagram illustrating the distribution of a low-carbon economy and energy efficiency across China&#x27;s regions. The regions include Northern coastal, Northeast, Eastern coastal, Middle Yellow River, Southern coastal, Southwest, and Middle Yangtze River. Streams indicate the flow from the whole of China towards these areas, highlighting regional contributions.</alt-text>
</graphic>
</fig>
</sec>
</sec>
<sec id="s6">
<label>6</label>
<title>Analysis of inter-regional differences in energy efficiency development and convergence</title>
<sec id="s6-1">
<label>6.1</label>
<title>Overall differences</title>
<p>To characterise regional disparities in energy efficiency across China and its eight major regions, along with their sources, the Dagum Gini coefficient and subgroup decomposition method were employed to calculate the overall variance, intra-group variance, inter-group variance, and sources of variance in China&#x2019;s energy efficiency from 2013 to 2022.</p>
<p>The dynamic trend of the national overall Gini coefficient for energy efficiency is analysed (see <xref ref-type="table" rid="T2">Table 2</xref>; <xref ref-type="fig" rid="F4">Figure 4</xref>). Regarding the evolution of disparities, the national overall Gini coefficient decreased from 0.164 in 2013 to 0.108 in 2022, indicating a substantial reduction in the overall disparity of energy efficiency nationwide. With the development of the internet, connections between China&#x2019;s major cities have become increasingly close, and the overall disparity in energy efficiency between cities has narrowed year by year.</p>
<table-wrap id="T2" position="float">
<label>TABLE 2</label>
<caption>
<p>Trends in the dynamics of the overall national Gini coefficient of energy efficiency.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th rowspan="2" align="center">Vintages</th>
<th colspan="4" align="center">Gini coefficients</th>
<th colspan="3" align="center">Contribution (%)</th>
</tr>
<tr>
<th align="center">Totally</th>
<th align="center">Within-group Gini coefficient Gw</th>
<th align="center">Inter-group Gini coefficient Gb</th>
<th align="center">Gini coefficient of hypervariant density Gt</th>
<th align="center">Within-group contribution Gw</th>
<th align="center">Intergroup contribution Gb</th>
<th align="center">Hypervariable density contribution Gt</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="center">2013</td>
<td align="center">0.164</td>
<td align="center">0.022</td>
<td align="center">0.036</td>
<td align="center">0.107</td>
<td align="center">13.191%</td>
<td align="center">21.718%</td>
<td align="center">65.091%</td>
</tr>
<tr>
<td align="center">2014</td>
<td align="center">0.152</td>
<td align="center">0.020</td>
<td align="center">0.055</td>
<td align="center">0.077</td>
<td align="center">12.918%</td>
<td align="center">36.301%</td>
<td align="center">50.781%</td>
</tr>
<tr>
<td align="center">2015</td>
<td align="center">0.150</td>
<td align="center">0.020</td>
<td align="center">0.025</td>
<td align="center">0.105</td>
<td align="center">13.338%</td>
<td align="center">16.772%</td>
<td align="center">69.889%</td>
</tr>
<tr>
<td align="center">2016</td>
<td align="center">0.155</td>
<td align="center">0.021</td>
<td align="center">0.034</td>
<td align="center">0.101</td>
<td align="center">13.365%</td>
<td align="center">21.640%</td>
<td align="center">64.994%</td>
</tr>
<tr>
<td align="center">2017</td>
<td align="center">0.205</td>
<td align="center">0.027</td>
<td align="center">0.039</td>
<td align="center">0.139</td>
<td align="center">13.040%</td>
<td align="center">18.952%</td>
<td align="center">68.008%</td>
</tr>
<tr>
<td align="center">2018</td>
<td align="center">0.123</td>
<td align="center">0.016</td>
<td align="center">0.037</td>
<td align="center">0.070</td>
<td align="center">12.775%</td>
<td align="center">30.411%</td>
<td align="center">56.815%</td>
</tr>
<tr>
<td align="center">2019</td>
<td align="center">0.150</td>
<td align="center">0.020</td>
<td align="center">0.019</td>
<td align="center">0.111</td>
<td align="center">13.448%</td>
<td align="center">12.576%</td>
<td align="center">73.976%</td>
</tr>
<tr>
<td align="center">2020</td>
<td align="center">0.131</td>
<td align="center">0.017</td>
<td align="center">0.043</td>
<td align="center">0.071</td>
<td align="center">13.118%</td>
<td align="center">32.510%</td>
<td align="center">54.372%</td>
</tr>
<tr>
<td align="center">2021</td>
<td align="center">0.122</td>
<td align="center">0.017</td>
<td align="center">0.021</td>
<td align="center">0.084</td>
<td align="center">13.577%</td>
<td align="center">17.305%</td>
<td align="center">69.119%</td>
</tr>
<tr>
<td align="center">2022</td>
<td align="center">0.108</td>
<td align="center">0.015</td>
<td align="center">0.017</td>
<td align="center">0.076</td>
<td align="center">13.499%</td>
<td align="center">16.104%</td>
<td align="center">70.397%</td>
</tr>
</tbody>
</table>
</table-wrap>
<fig id="F4" position="float">
<label>FIGURE 4</label>
<caption>
<p>Overall variance map.</p>
</caption>
<graphic xlink:href="fenvs-13-1741575-g004.tif">
<alt-text content-type="machine-generated">Line graph showing data from 2013 to 2022. Values start at 0.164 in 2013, dip to 0.150 in 2015, peak at 0.205 in 2017, drop to 0.123 in 2018, rise to 0.150 in 2019, then decline to 0.108 in 2022.</alt-text>
</graphic>
</fig>
</sec>
<sec id="s6-2">
<label>6.2</label>
<title>Intra-regional variation</title>
<p>Regarding intra-group variation (see <xref ref-type="table" rid="T3">Table 3</xref>; <xref ref-type="fig" rid="F5">Figure 5</xref>), the middle reaches of the Yellow River exhibited the highest intra-group energy efficiency variation (0.250), while the southern coastal region showed the lowest (0.066). The northwest and eastern coastal regions followed with Gini coefficients of 0.245 and 0.224 respectively, with the remaining four major regions displaying fluctuating patterns. Regarding trends, the Gini coefficients across the eight major regions exhibited similar patterns, showing a wavelike, modest downward trajectory. The northern coastal region experienced minimal fluctuation, with an average annual coefficient of 0.1283. In 2017, all eight regions demonstrated pronounced inequality, marked by a sharp rise followed by a steep decline. Since 2020, the coefficients have gradually decreased in a wavelike pattern.</p>
<table-wrap id="T3" position="float">
<label>TABLE 3</label>
<caption>
<p>Table of coefficients of variation within regions.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th rowspan="2" align="center">Vintages</th>
<th colspan="8" align="center">Intraregional coefficient of variation</th>
</tr>
<tr>
<th align="center">Northeastern region</th>
<th align="center">Eastern seaboard</th>
<th align="center">North coastal area</th>
<th align="center">southern coastal region</th>
<th align="center">Great northwest region</th>
<th align="center">Southwest region</th>
<th align="center">Middle reaches of Yangtze river</th>
<th align="center">Middle reaches of the Yellow river</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="center">2013</td>
<td align="center">0.174</td>
<td align="center">0.166</td>
<td align="center">0.128</td>
<td align="center">0.191</td>
<td align="center">0.153</td>
<td align="center">0.152</td>
<td align="center">0.138</td>
<td align="center">0.169</td>
</tr>
<tr>
<td align="center">2014</td>
<td align="center">0.097</td>
<td align="center">0.080</td>
<td align="center">0.129</td>
<td align="center">0.123</td>
<td align="center">0.174</td>
<td align="center">0.155</td>
<td align="center">0.120</td>
<td align="center">0.214</td>
</tr>
<tr>
<td align="center">2015</td>
<td align="center">0.169</td>
<td align="center">0.101</td>
<td align="center">0.140</td>
<td align="center">0.122</td>
<td align="center">0.137</td>
<td align="center">0.122</td>
<td align="center">0.121</td>
<td align="center">0.215</td>
</tr>
<tr>
<td align="center">2016</td>
<td align="center">0.154</td>
<td align="center">0.099</td>
<td align="center">0.112</td>
<td align="center">0.163</td>
<td align="center">0.153</td>
<td align="center">0.176</td>
<td align="center">0.142</td>
<td align="center">0.165</td>
</tr>
<tr>
<td align="center">2017</td>
<td align="center">0.146</td>
<td align="center">0.245</td>
<td align="center">0.180</td>
<td align="center">0.186</td>
<td align="center">0.224</td>
<td align="center">0.207</td>
<td align="center">0.162</td>
<td align="center">0.250</td>
</tr>
<tr>
<td align="center">2018</td>
<td align="center">0.125</td>
<td align="center">0.090</td>
<td align="center">0.120</td>
<td align="center">0.087</td>
<td align="center">0.149</td>
<td align="center">0.126</td>
<td align="center">0.114</td>
<td align="center">0.113</td>
</tr>
<tr>
<td align="center">2019</td>
<td align="center">0.161</td>
<td align="center">0.135</td>
<td align="center">0.160</td>
<td align="center">0.125</td>
<td align="center">0.087</td>
<td align="center">0.172</td>
<td align="center">0.18</td>
<td align="center">0.096</td>
</tr>
<tr>
<td align="center">2020</td>
<td align="center">0.111</td>
<td align="center">0.112</td>
<td align="center">0.116</td>
<td align="center">0.105</td>
<td align="center">0.093</td>
<td align="center">0.160</td>
<td align="center">0.133</td>
<td align="center">0.117</td>
</tr>
<tr>
<td align="center">2021</td>
<td align="center">0.134</td>
<td align="center">0.140</td>
<td align="center">0.104</td>
<td align="center">0.093</td>
<td align="center">0.092</td>
<td align="center">0.116</td>
<td align="center">0.125</td>
<td align="center">0.131</td>
</tr>
<tr>
<td align="center">2022</td>
<td align="center">0.124</td>
<td align="center">0.140</td>
<td align="center">0.094</td>
<td align="center">0.066</td>
<td align="center">0.074</td>
<td align="center">0.104</td>
<td align="center">0.109</td>
<td align="center">0.114</td>
</tr>
<tr>
<td align="center">Average value</td>
<td align="center">0.140</td>
<td align="center">0.131</td>
<td align="center">0.128</td>
<td align="center">0.126</td>
<td align="center">0.134</td>
<td align="center">0.149</td>
<td align="center">0.134</td>
<td align="center">0.158</td>
</tr>
</tbody>
</table>
</table-wrap>
<fig id="F5" position="float">
<label>FIGURE 5</label>
<caption>
<p>Decomposition diagram of regional difference coefficient.</p>
</caption>
<graphic xlink:href="fenvs-13-1741575-g005.tif">
<alt-text content-type="machine-generated">Line chart titled &#x22;Decomposition of the internal bikini coefficient within the group&#x22; from 2013 to 2022. It compares coefficients across eight regions: Northeast, Eastern coastal, Northern coastal, Southern coastal, Great northwest, Southwest, Middle reaches of the Yangtze River, and Middle reaches of the Yellow River. Each region has a distinct line style and color. Values fluctuate, peaking around 2017 and generally declining towards 2022.</alt-text>
</graphic>
</fig>
</sec>
<sec id="s6-3">
<label>6.3</label>
<title>Inter-regional disparities</title>
<p>Regarding inter-regional disparities (see <xref ref-type="table" rid="T4">Table 4</xref>), the southern coastal regions and the Great Northwest region, as well as the Great Northwest region and the southern coastal regions, exhibit significant differences in the inter-group Gini coefficient (0.099), frequently presenting a &#x2018;single-peak-double-peak&#x2019; phenomenon. This was observed in 2013 (0.178) and 2017 (0.178). Southern Coastal Regions exhibited the most pronounced inter-group Gini coefficient disparity (0.099), frequently displaying a &#x2018;unimodal-bimodal&#x2019; pattern. This reached peaks in 2013 (0.178) and 2017 (0.209), before experiencing sharp declines in 2014 and 2018 respectively. The smallest inter-group differences were observed between Northern Coastal Regions and Middle Yangtze River Regions and Eastern Coastal Regions and Southwestern Regions.</p>
<table-wrap id="T4" position="float">
<label>TABLE 4</label>
<caption>
<p>Coefficient of variation between regions.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th align="center">Region pair</th>
<th align="left">2013</th>
<th align="left">2014</th>
<th align="left">2015</th>
<th align="left">2016</th>
<th align="left">2017</th>
<th align="left">2018</th>
<th align="left">2019</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="center">Northeast &#x26; eastern coastal</td>
<td align="left">0.196</td>
<td align="left">0.112</td>
<td align="left">0.145</td>
<td align="left">0.130</td>
<td align="left">0.210</td>
<td align="left">0.127</td>
<td align="left">0.151</td>
</tr>
<tr>
<td align="center">Northeast &#x26; northern coastal</td>
<td align="left">0.165</td>
<td align="left">0.163</td>
<td align="left">0.160</td>
<td align="left">0.137</td>
<td align="left">0.173</td>
<td align="left">0.130</td>
<td align="left">0.163</td>
</tr>
<tr>
<td align="center">Northeast &#x26; southern coastal</td>
<td align="left">0.194</td>
<td align="left">0.120</td>
<td align="left">0.150</td>
<td align="left">0.163</td>
<td align="left">0.173</td>
<td align="left">0.113</td>
<td align="left">0.152</td>
</tr>
<tr>
<td align="center">Northeast &#x26; northwest</td>
<td align="left">0.188</td>
<td align="left">0.143</td>
<td align="left">0.160</td>
<td align="left">0.162</td>
<td align="left">0.199</td>
<td align="left">0.140</td>
<td align="left">0.135</td>
</tr>
<tr>
<td align="center">Northeast &#x26; southwest</td>
<td align="left">0.178</td>
<td align="left">0.143</td>
<td align="left">0.149</td>
<td align="left">0.175</td>
<td align="left">0.185</td>
<td align="left">0.127</td>
<td align="left">0.176</td>
</tr>
<tr>
<td align="center">Northeast &#x26; Yangtze river middle reaches</td>
<td align="left">0.160</td>
<td align="left">0.141</td>
<td align="left">0.152</td>
<td align="left">0.149</td>
<td align="left">0.159</td>
<td align="left">0.121</td>
<td align="left">0.176</td>
</tr>
<tr>
<td align="center">Northeast &#x26; Yellow river middle reaches</td>
<td align="left">0.181</td>
<td align="left">0.169</td>
<td align="left">0.196</td>
<td align="left">0.164</td>
<td align="left">0.212</td>
<td align="left">0.121</td>
<td align="left">0.139</td>
</tr>
<tr>
<td align="center">Eastern coastal &#x26; northern coastal</td>
<td align="left">0.153</td>
<td align="left">0.116</td>
<td align="left">0.123</td>
<td align="left">0.112</td>
<td align="left">0.229</td>
<td align="left">0.112</td>
<td align="left">0.152</td>
</tr>
<tr>
<td align="center">Eastern coastal &#x26; southern coastal</td>
<td align="left">0.184</td>
<td align="left">0.109</td>
<td align="left">0.125</td>
<td align="left">0.143</td>
<td align="left">0.222</td>
<td align="left">0.135</td>
<td align="left">0.145</td>
</tr>
<tr>
<td align="center">Eastern coastal &#x26; northwest</td>
<td align="left">0.164</td>
<td align="left">0.165</td>
<td align="left">0.142</td>
<td align="left">0.146</td>
<td align="left">0.245</td>
<td align="left">0.147</td>
<td align="left">0.124</td>
</tr>
<tr>
<td align="center">Eastern coastal &#x26; southwest</td>
<td align="left">0.162</td>
<td align="left">0.123</td>
<td align="left">0.114</td>
<td align="left">0.161</td>
<td align="left">0.235</td>
<td align="left">0.139</td>
<td align="left">0.170</td>
</tr>
<tr>
<td align="center">Eastern coastal &#x26; Yangtze river middle reaches</td>
<td align="left">0.165</td>
<td align="left">0.104</td>
<td align="left">0.112</td>
<td align="left">0.124</td>
<td align="left">0.213</td>
<td align="left">0.121</td>
<td align="left">0.169</td>
</tr>
<tr>
<td align="center">Eastern coastal &#x26; Yellow river middle reaches</td>
<td align="left">0.176</td>
<td align="left">0.162</td>
<td align="left">0.171</td>
<td align="left">0.140</td>
<td align="left">0.250</td>
<td align="left">0.113</td>
<td align="left">0.128</td>
</tr>
<tr>
<td align="center">Northern coastal &#x26; southern coastal</td>
<td align="left">0.163</td>
<td align="left">0.145</td>
<td align="left">0.143</td>
<td align="left">0.144</td>
<td align="left">0.202</td>
<td align="left">0.127</td>
<td align="left">0.149</td>
</tr>
<tr>
<td align="center">Northern coastal &#x26; northwest</td>
<td align="left">0.146</td>
<td align="left">0.217</td>
<td align="left">0.158</td>
<td align="left">0.140</td>
<td align="left">0.230</td>
<td align="left">0.145</td>
<td align="left">0.129</td>
</tr>
<tr>
<td align="center">Northern coastal &#x26; southwest</td>
<td align="left">0.141</td>
<td align="left">0.152</td>
<td align="left">0.133</td>
<td align="left">0.156</td>
<td align="left">0.213</td>
<td align="left">0.135</td>
<td align="left">0.174</td>
</tr>
<tr>
<td align="center">Northern coastal &#x26; Yangtze river middle reaches</td>
<td align="left">0.137</td>
<td align="left">0.128</td>
<td align="left">0.132</td>
<td align="left">0.130</td>
<td align="left">0.187</td>
<td align="left">0.125</td>
<td align="left">0.173</td>
</tr>
<tr>
<td align="center">Northern coastal &#x26; Yellow river middle reaches</td>
<td align="left">0.151</td>
<td align="left">0.192</td>
<td align="left">0.183</td>
<td align="left">0.150</td>
<td align="left">0.235</td>
<td align="left">0.120</td>
<td align="left">0.133</td>
</tr>
<tr>
<td align="center">Southern coastal &#x26; northwest</td>
<td align="left">0.178</td>
<td align="left">0.168</td>
<td align="left">0.134</td>
<td align="left">0.162</td>
<td align="left">0.209</td>
<td align="left">0.123</td>
<td align="left">0.109</td>
</tr>
<tr>
<td align="center">Southern coastal &#x26; southwest</td>
<td align="left">0.174</td>
<td align="left">0.142</td>
<td align="left">0.128</td>
<td align="left">0.173</td>
<td align="left">0.198</td>
<td align="left">0.110</td>
<td align="left">0.151</td>
</tr>
<tr>
<td align="center">Southern coastal &#x26; Yangtze river middle reaches</td>
<td align="left">0.170</td>
<td align="left">0.131</td>
<td align="left">0.132</td>
<td align="left">0.157</td>
<td align="left">0.175</td>
<td align="left">0.108</td>
<td align="left">0.156</td>
</tr>
<tr>
<td align="center">Southern coastal &#x26; Yellow river middle reaches</td>
<td align="left">0.183</td>
<td align="left">0.174</td>
<td align="left">0.181</td>
<td align="left">0.169</td>
<td align="left">0.222</td>
<td align="left">0.113</td>
<td align="left">0.113</td>
</tr>
<tr>
<td align="center">Northwest &#x26; southwest</td>
<td align="left">0.157</td>
<td align="left">0.192</td>
<td align="left">0.142</td>
<td align="left">0.166</td>
<td align="left">0.219</td>
<td align="left">0.139</td>
<td align="left">0.136</td>
</tr>
<tr>
<td align="center">Northwest &#x26; Yangtze river middle reaches</td>
<td align="left">0.158</td>
<td align="left">0.194</td>
<td align="left">0.148</td>
<td align="left">0.156</td>
<td align="left">0.198</td>
<td align="left">0.135</td>
<td align="left">0.139</td>
</tr>
<tr>
<td align="center">Northwest &#x26; Yellow river middle reaches</td>
<td align="left">0.165</td>
<td align="left">0.210</td>
<td align="left">0.189</td>
<td align="left">0.173</td>
<td align="left">0.243</td>
<td align="left">0.137</td>
<td align="left">0.092</td>
</tr>
<tr>
<td align="center">Southwest &#x26; Yangtze river middle reaches</td>
<td align="left">0.151</td>
<td align="left">0.141</td>
<td align="left">0.123</td>
<td align="left">0.169</td>
<td align="left">0.186</td>
<td align="left">0.124</td>
<td align="left">0.178</td>
</tr>
<tr>
<td align="center">Southwest &#x26; Yellow river middle reaches</td>
<td align="left">0.164</td>
<td align="left">0.189</td>
<td align="left">0.176</td>
<td align="left">0.185</td>
<td align="left">0.233</td>
<td align="left">0.125</td>
<td align="left">0.139</td>
</tr>
<tr>
<td align="center">Yangtze river middle reaches &#x26; Yellow river middle reaches</td>
<td align="left">0.158</td>
<td align="left">0.180</td>
<td align="left">0.178</td>
<td align="left">0.159</td>
<td align="left">0.213</td>
<td align="left">0.116</td>
<td align="left">0.143</td>
</tr>
</tbody>
</table>
</table-wrap>
<p>The Gini coefficients for the Eastern Coastal and Northern Coastal Regions and Eastern Coastal and Middle Yellow River Regions exhibited the most pronounced fluctuations in 2017, while the Northeast and Southern Coastal Regions and Northeast and Southwest Regions showed the smallest fluctuations during that year.</p>
</sec>
<sec id="s6-4">
<label>6.4</label>
<title>Super-density variation</title>
<p>Considering <xref ref-type="table" rid="T4">Table 4</xref>; <xref ref-type="fig" rid="F6">Figure 6</xref> together, the super-efficiency contribution rate emerges as the primary determinant of energy efficiency, with an average contribution rate of 64.344%. Moreover, the variation between groups exceeds that within groups. From 2013 to 2022, the line depicting super-density exhibits significant fluctuations, yet the overall trend remains upward. This may indicate that this factor&#x2019;s influence on energy efficiency has intensified during this period, potentially linked to global energy price fluctuations, the introduction of new technologies, or shifts in energy policy. For inter-group variations, this may relate to factors such as technological advancement, policy support, and resource allocation differences between groups. Intra-group analysis reveals a relatively stable trend with a slight downward trajectory, indicating that energy efficiency disparities among group members have not significantly widened. This suggests a convergence in energy efficiency performance, potentially stemming from improved mechanisms for sharing technical knowledge and the application of systematic management measures. Such approaches foster coordinated enhancement of group-level energy efficiency by reducing practical heterogeneity.</p>
<fig id="F6" position="float">
<label>FIGURE 6</label>
<caption>
<p>Comparison of hyperdensity differences.</p>
</caption>
<graphic xlink:href="fenvs-13-1741575-g006.tif">
<alt-text content-type="machine-generated">Line graph showing Gini coefficient trends from 2013 to 2022. Three lines represent within-group Gini (solid circles), between-group Gini (inverted triangles), and super-variable density Gini (stars). Within-group remains steady around 0.02. Between-group fluctuates between 0.02 to 0.05. Super-variable density varies significantly between 0.08 and 0.13.</alt-text>
</graphic>
</fig>
</sec>
</sec>
<sec id="s7">
<label>7</label>
<title>Preliminary analysis of spatio-temporal evolution</title>
<sec id="s7-1">
<label>7.1</label>
<title>National dynamic distribution</title>
<p>Based on the national energy efficiency kernel density map (see <xref ref-type="fig" rid="F7">Figure 7</xref>), the national energy efficiency exhibits the following four characteristics.<list list-type="alpha-lower">
<list-item>
<p>Two distinct peaks are discernible in <xref ref-type="fig" rid="F13">Figure 13</xref>. These peaks exhibit an increasing trend over time, with their positions shifting progressively to the right. This indicates an overall upward trajectory in national energy efficiency. The emergence of a peak at the far right suggests heightened national emphasis on energy efficiency development, signifying continuous improvement. The presence of two peaks may imply the existence of two distinct economic entities or sectors exhibiting divergent trends in energy efficiency. One may be maintaining its existing energy efficiency level while the other is improving;</p>
</list-item>
<list-item>
<p>The height of the primary peak has markedly increased with a narrowing trend, indicating that regional disparities in national energy efficiency are widening, yet absolute differences are diminishing. This may suggest that while energy efficiency in some regions may be rising rapidly, overall regional disparities are not further widening and may even be narrowing;</p>
</list-item>
<list-item>
<p>The right tail is progressively shortening year by year, with distribution extension exhibiting a certain degree of contraction. This indicates that regional disparities in national energy efficiency are gradually diminishing;</p>
</list-item>
<list-item>
<p>The highest regional density of energy efficiency is observed around 1.2, suggesting that energy efficiency is concentrated within this range in most years.</p>
</list-item>
</list>
</p>
<fig id="F7" position="float">
<label>FIGURE 7</label>
<caption>
<p>Map of nuclear density in national regions.</p>
</caption>
<graphic xlink:href="fenvs-13-1741575-g007.tif">
<alt-text content-type="machine-generated">Three-dimensional surface plot showing variations in energy efficiency from 2008 to 2022. The plot features years on the x-axis, energy efficiency on the y-axis, and a height dimension representing values ranging from zero to six, with peaks around 2014.</alt-text>
</graphic>
</fig>
</sec>
<sec id="s7-2">
<label>7.2</label>
<title>Analysis of distribution dynamics evolution</title>
<p>Regional variations are primarily manifested in the following aspects:</p>
<p>All eight major regions nationwide exhibit multi-peak phenomena, with peak heights showing an overall upward trend. Notably, the main peaks of the kernel density curves narrow in the middle Yangtze River region (see <xref ref-type="fig" rid="F8">Figure 8</xref>), the Eastern Coastal Region (see <xref ref-type="fig" rid="F9">Figure 9</xref>), the Northwest Region (see <xref ref-type="fig" rid="F10">Figure 10</xref>), and the Middle Yellow River Region (see <xref ref-type="fig" rid="F11">Figure 11</xref>) exhibit narrowing widths of their principal peaks in the kernel density curves. Conversely, the Northern Coastal Region (see <xref ref-type="fig" rid="F12">Figure 12</xref>) and Southern Coastal Region (see <xref ref-type="fig" rid="F13">Figure 13</xref>) show widening principal peaks, while the Northeast Region (see <xref ref-type="fig" rid="F14">Figure 14</xref>) and Southwest Region (see <xref ref-type="fig" rid="F15">Figure 15</xref>) maintain largely unchanged principal peak widths.</p>
<fig id="F8" position="float">
<label>FIGURE 8</label>
<caption>
<p>Map of nuclear density in the middle reaches of the Yangtze River.</p>
</caption>
<graphic xlink:href="fenvs-13-1741575-g008.tif">
<alt-text content-type="machine-generated">Three-dimensional surface plot showing energy efficiency over time. The x-axis represents years from 2012 to 2022, the y-axis is labeled from zero to ten, and the z-axis indicates energy efficiency values. Peaks are visible, indicating changes in efficiency.</alt-text>
</graphic>
</fig>
<fig id="F9" position="float">
<label>FIGURE 9</label>
<caption>
<p>Map of nuclear density along the eastern seaboard.</p>
</caption>
<graphic xlink:href="fenvs-13-1741575-g009.tif">
<alt-text content-type="machine-generated">Three-dimensional surface plot showing variations in energy efficiency over time, from 2012 to 2022. The graph features colored peaks and valleys, with energy efficiency on the X-axis and years on the Y-axis.</alt-text>
</graphic>
</fig>
<fig id="F10" position="float">
<label>FIGURE 10</label>
<caption>
<p>Map of nuclear density in the greater north-west.</p>
</caption>
<graphic xlink:href="fenvs-13-1741575-g010.tif">
<alt-text content-type="machine-generated">A 3D surface plot illustrates changes in energy efficiency over time, ranging from 2012 to 2022. Peaks and valleys represent varying levels of efficiency, with the y-axis indicating magnitude up to 15.</alt-text>
</graphic>
</fig>
<fig id="F11" position="float">
<label>FIGURE 11</label>
<caption>
<p>Map of nuclear density in the middle reaches of the Yellow River.</p>
</caption>
<graphic xlink:href="fenvs-13-1741575-g011.tif">
<alt-text content-type="machine-generated">3D line graph depicting energy efficiency over time. The x-axis represents years from 2014 to 2022, the y-axis shows values up to 25, and the z-axis indicates energy efficiency ranging from 0.8 to 1.4. Peaks are visible around 2016, 2018, and 2020.</alt-text>
</graphic>
</fig>
<fig id="F12" position="float">
<label>FIGURE 12</label>
<caption>
<p>Map of nuclear density on the northern coast.</p>
</caption>
<graphic xlink:href="fenvs-13-1741575-g012.tif">
<alt-text content-type="machine-generated">Three-dimensional line graph depicting energy efficiency over time. The x-axis represents years from 2012 to 2022, the y-axis indicates values up to 10, and the z-axis shows energy efficiency from 0.8 to 1.6. Peaks occur between 2014 and 2018, and 2020 and 2022.</alt-text>
</graphic>
</fig>
<fig id="F13" position="float">
<label>FIGURE 13</label>
<caption>
<p>Map of nuclear density along the southern coast.</p>
</caption>
<graphic xlink:href="fenvs-13-1741575-g013.tif">
<alt-text content-type="machine-generated">Three-dimensional surface plot showing energy efficiency trends over time. The plot displays fluctuations with peaks, indicating changes in efficiency from 2012 to 2022. The horizontal axes represent the year and energy efficiency levels.</alt-text>
</graphic>
</fig>
<fig id="F14" position="float">
<label>FIGURE 14</label>
<caption>
<p>Map of nuclear density in the north-east.</p>
</caption>
<graphic xlink:href="fenvs-13-1741575-g014.tif">
<alt-text content-type="machine-generated">Three-dimensional graph showing energy efficiency trends over time from 2015 to 2020. The x-axis represents energy efficiency, and the y-axis shows the year. Peaks are visible at specific efficiency levels, indicating variations in energy performance over the years.</alt-text>
</graphic>
</fig>
<fig id="F15" position="float">
<label>FIGURE 15</label>
<caption>
<p>Map of nuclear density in the south-west.</p>
</caption>
<graphic xlink:href="fenvs-13-1741575-g015.tif">
<alt-text content-type="machine-generated">Three-dimensional surface plot depicting changes in energy efficiency over time from 2014 to 2022. The x-axis represents years, the y-axis shows energy efficiency values, and the z-axis represents a numeric scale from zero to eight. Peaks and valleys indicate variations in energy efficiency during this period.</alt-text>
</graphic>
</fig>
<p>This indicates a trend towards narrowing regional disparities in energy efficiency within the middle Yangtze River region, eastern coastal region, Northwest China region, and middle Yellow River region. whilst regional disparities in energy efficiency within the northern coastal and southern coastal regions are expanding. Regional differences in the northeast and southwest regions remain largely unchanged. The energy efficiency kernel density curves for the Northwest, Middle Yellow River, and eastern coastal regions exhibit a wave-like pattern transitioning from &#x2018;uniform distribution&#x2013;bimodal&#x2013;unimodal&#x2019;, with peaks distinctly shifting to the right. This indicates a marked improvement in energy efficiency across these areas. The energy efficiency kernel density curve for the southern coastal region exhibits a fluctuating pattern of &#x2018;uniform distribution &#x2013; narrow bimodal &#x2013; broad bimodal&#x2019;, accompanied by polarisation. This suggests that energy efficiency in the southern coastal region may be linked to urban development, with coastal cities potentially demonstrating higher energy efficiency.</p>
<p>Regarding extensibility, the energy efficiency and density curves for the Northeast, Eastern Coastal, Southwest, and Middle Yangtze regions exhibit pronounced right-tailing phenomena alongside broadening of the primary peaks. This indicates gradually widening regional disparities in energy efficiency across these areas. The northern coastal region&#x2019;s energy efficiency kernel density curve exhibits an overall quadratic shape, indicating diminishing regional disparities with efficiency values converging around 1.3. The middle Yellow River region and the Greater Northwest region show kernel density curves that initially peak before gradually broadening and declining, suggesting that regional differences in energy efficiency within these areas may be narrowing due to policy interventions or other factors.</p>
<p>The southern coastal region exhibits a curve with a low central peak flanked by high values, indicating pronounced polarisation and significant intra-regional disparities. Polarisation is also present in the middle Yellow River region, the Northwest, the middle Yangtze region, and Southwest China, though these areas are stabilising over time with values highly clustered around 1.2, 1.1, 1.0, and 0.9 respectively. Notably, the middle Yellow River region showed high concentration in 2014 and 2019, 2016 and 2019, while exhibiting polarisation in 2015 and 2017, with this trend markedly diminishing after 2019.</p>
<p>Regarding volatility and stability, the energy efficiency kernel density curve for the Northeast region remained relatively stable overall, with the peak consistently at 1.2 and the width largely unchanged, indicating steady development in energy efficiency. In contrast, the energy efficiency kernel density curves for the other seven major regions exhibited significant variations, reflecting differing patterns of energy efficiency development across these areas.</p>
</sec>
</sec>
<sec id="s8">
<label>8</label>
<title>In-depth analysis of spatio-temporal evolution</title>
<p>To further reflect the internal flow direction and locational shift characteristics of high-quality economic development, this study employs a Markov transition probability matrix for analysis. The results are presented in <xref ref-type="table" rid="T5">Table 5</xref>. The probabilities of provinces maintaining their original tier after 1&#xa0;year are 32.09%, 31.93%, 35.46%, and 36.53%, respectively. This indicates relative stability between different tiers of the energy efficiency development index, with diagonal elements consistently exceeding non-diagonal values. This suggests that prefecture-level cities maintaining their original tier exhibit a relatively high probability of sustaining their energy efficiency level. Club convergence phenomena are observed across both low-level and high-level tiers. The probabilities of upward transition for low-level, lower-middle-level, and upper-middle-level regions are 29.06%, 28.31%, and 29.41% respectively, exceeding the probabilities of downward transition for lower-middle-level, upper-middle-level, and high-level regions (23.34%, 19.12%, and 20.03%). Prefectural-level cities should strive to achieve upward transitions while maintaining their existing energy efficiency levels.</p>
<table-wrap id="T5" position="float">
<label>TABLE 5</label>
<caption>
<p>Markov chain transfer concept matrix results.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th align="center"/>
<th align="center">&#x2160;</th>
<th align="center">&#x2161;</th>
<th align="center">&#x2162;</th>
<th align="center">&#x2163;</th>
<th align="center">observed value</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="center">&#x2160;</td>
<td align="center">0.321</td>
<td align="center">0.291</td>
<td align="center">0.174</td>
<td align="center">0.214</td>
<td align="center">695.000</td>
</tr>
<tr>
<td align="center">&#x2161;</td>
<td align="center">0.233</td>
<td align="center">0.319</td>
<td align="center">0.283</td>
<td align="center">0.164</td>
<td align="center">664.000</td>
</tr>
<tr>
<td align="center">&#x2162;</td>
<td align="center">0.160</td>
<td align="center">0.191</td>
<td align="center">0.355</td>
<td align="center">0.294</td>
<td align="center">612.000</td>
</tr>
<tr>
<td align="center">&#x2163;</td>
<td align="center">0.242</td>
<td align="center">0.192</td>
<td align="center">0.200</td>
<td align="center">0.365</td>
<td align="center">594.000</td>
</tr>
</tbody>
</table>
</table-wrap>
<p>The aforementioned findings indicate the necessity of incorporating spatial factors to establish a spatial Markov transition probability matrix (see <xref ref-type="table" rid="T6">Table 6</xref>). Firstly, the four transition probability matrices differ under various spatial lag types. This demonstrates that when neighbouring prefecture-level cities exhibit disparities in energy efficiency development, the probability of a city&#x2019;s energy efficiency trajectory shifting due to such influences varies accordingly. Secondly, the diagonal elements of the transition probability matrices under different spatial lag types are not entirely greater than the off-diagonal elements. This indicates that the probability of energy efficiency &#x2018;level locking&#x2019; under spatial spillover effects is reduced, a phenomenon particularly pronounced under Type IV lag conditions. Furthermore, non-zero elements exist on both sides of the diagonal. This indicates instability in energy efficiency development: while upward transitions to ideal states are achievable, downward transition risks persist. Moreover, transitions occur only between adjacent tiers, with inter-tier leaps being rare. Furthermore, different lag types exert varying influences on the same tier. Under high-level lag conditions, the probability of low-level cities transitioning to medium-low levels is 37.94%, significantly higher than the 31.03% probability under low-level lag conditions. Finally, the same lagging category exerts varying effects across different tiers. Under high-level lagging conditions, the probabilities of upward transition by one tier for low-level, medium-low level, and medium-high level are 24.05%, 26.67%, and 21.74% respectively, exhibiting an increasing and decreasing relationship. This indicates that transition probabilities are influenced not only by lagging categories but also by the initial energy efficiency tier and the prefecture-level city&#x2019;s developmental context.</p>
<table-wrap id="T6" position="float">
<label>TABLE 6</label>
<caption>
<p>Results of the spatial markov concept transfer matrix.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th align="center"/>
<th align="center">Type of spatial lag</th>
<th align="center">t/(t&#x2b;1)</th>
<th align="center">&#x2160;</th>
<th align="center">&#x2161;</th>
<th align="center">&#x2162;</th>
<th align="center">&#x2163;</th>
<th align="center">observed value</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td rowspan="16" align="center">spatial</td>
<td rowspan="4" align="center">&#x2160;</td>
<td align="center">&#x2160;</td>
<td align="center">0.310</td>
<td align="center">0.296</td>
<td align="center">0.168</td>
<td align="center">0.227</td>
<td align="center">203.000</td>
</tr>
<tr>
<td align="center">&#x2161;</td>
<td align="center">0.234</td>
<td align="center">0.369</td>
<td align="center">0.234</td>
<td align="center">0.162</td>
<td align="center">111.000</td>
</tr>
<tr>
<td align="center">&#x2162;</td>
<td align="center">0.148</td>
<td align="center">0.198</td>
<td align="center">0.358</td>
<td align="center">0.296</td>
<td align="center">81.000</td>
</tr>
<tr>
<td align="center">&#x2163;</td>
<td align="center">0.231</td>
<td align="center">0.209</td>
<td align="center">0.165</td>
<td align="center">0.396</td>
<td align="center">91.000</td>
</tr>
<tr>
<td rowspan="4" align="center">&#x2161;<break/>
</td>
<td align="center">&#x2160;</td>
<td align="center">0.314</td>
<td align="center">0.268</td>
<td align="center">0.165</td>
<td align="center">0.253</td>
<td align="center">194.000</td>
</tr>
<tr>
<td align="center">&#x2161;</td>
<td align="center">0.212</td>
<td align="center">0.359</td>
<td align="center">0.283</td>
<td align="center">0.146</td>
<td align="center">212.000</td>
</tr>
<tr>
<td align="center">&#x2162;</td>
<td align="center">0.183</td>
<td align="center">0.156</td>
<td align="center">0.361</td>
<td align="center">0.300</td>
<td align="center">180.000</td>
</tr>
<tr>
<td align="center">&#x2163;</td>
<td align="center">0.258</td>
<td align="center">0.153</td>
<td align="center">0.177</td>
<td align="center">0.411</td>
<td align="center">124.000</td>
</tr>
<tr>
<td rowspan="4" align="center">&#x2162;</td>
<td align="center">&#x2160;</td>
<td align="center">0.315</td>
<td align="center">0.324</td>
<td align="center">0.187</td>
<td align="center">0.174</td>
<td align="center">219.000</td>
</tr>
<tr>
<td align="center">&#x2161;</td>
<td align="center">0.215</td>
<td align="center">0.299</td>
<td align="center">0.311</td>
<td align="center">0.175</td>
<td align="center">251.000</td>
</tr>
<tr>
<td align="center">&#x2162;</td>
<td align="center">0.151</td>
<td align="center">0.205</td>
<td align="center">0.328</td>
<td align="center">0.317</td>
<td align="center">259.000</td>
</tr>
<tr>
<td align="center">&#x2163;</td>
<td align="center">0.242</td>
<td align="center">0.200</td>
<td align="center">0.219</td>
<td align="center">0.340</td>
<td align="center">265.000</td>
</tr>
<tr>
<td rowspan="4" align="center">&#x2163;</td>
<td align="center">&#x2160;</td>
<td align="center">0.380</td>
<td align="center">0.241</td>
<td align="center">0.177</td>
<td align="center">0.203</td>
<td align="center">79.000</td>
</tr>
<tr>
<td align="center">&#x2161;</td>
<td align="center">0.333</td>
<td align="center">0.222</td>
<td align="center">0.267</td>
<td align="center">0.178</td>
<td align="center">90.000</td>
</tr>
<tr>
<td align="center">&#x2162;</td>
<td align="center">0.152</td>
<td align="center">0.217</td>
<td align="center">0.413</td>
<td align="center">0.217</td>
<td align="center">92.000</td>
</tr>
<tr>
<td align="center">&#x2163;</td>
<td align="center">0.237</td>
<td align="center">0.202</td>
<td align="center">0.211</td>
<td align="center">0.351</td>
<td align="center">114.000</td>
</tr>
</tbody>
</table>
</table-wrap>
</sec>
<sec id="s9">
<label>9</label>
<title>Conclusion and recommendations</title>
<sec id="s9-1">
<label>9.1</label>
<title>Conclusion</title>
<p>Against the backdrop of low-carbon economic development, this study employs provincial and prefecture-level panel data from mainland China (excluding Tibet, Xinjiang, Hong Kong, Macao, and Taiwan) spanning 2012&#x2013;2022. By comprehensively evaluating energy efficiency across multiple dimensions&#x2014;including socioeconomic development, green ecological progress, and technological innovation capacity&#x2014;the following conclusions are drawn:</p>
<p>First, employing the non-expected output SBM model to measure and analyse the green low-carbon economy and energy efficiency of China&#x2019;s 285 prefecture-level cities (measurement results presented in Appendices 1 and 2), the findings indicate a positive trajectory in the growth rate of energy efficiency across Chinese regions. Fluctuations in this growth rate reveal the challenges and opportunities faced by different regions in their energy economic development processes.</p>
<p>Second, Dagum Gini coefficient analysis identified the sources of energy efficiency disparities. The national aggregate Gini coefficient decreased from 0.164 in 2013 to 0.108 in 2022, indicating a substantial reduction in overall regional energy efficiency differences. Intra-regional disparities, inter-regional disparities, and hyper-variability density disparities were identified, with hyper-variability density contributing most significantly to overall disparities (average contribution rate: 64.344%). Kernel density estimation revealed multi-modal distributions across all eight major regions, with overall peak height increases, indicating a national upward trend in energy efficiency and gradually diminishing inter-regional disparities. Markov chain analysis indicates that under the high-level lag type, the probability of transition from low to medium-low levels (37.94%) exceeds that under the low-level lag type (31.03%). Concurrently, under high-level lag conditions, the probabilities of upward transitions from low, medium-low, and medium-high levels to the next tier are 24.05%, 26.67%, and 21.74% respectively, reflecting a relationship of growth and decline. This indicates that the dynamic evolution of energy efficiency levels exhibits relative stability, albeit with inherent instability.</p>
<p>Thirdly, empirical analysis of aggregate data reveals a substantial narrowing of overall national energy efficiency disparities. Both inter-regional and intra-regional variations in energy efficiency are diminishing, suggesting gradual improvement in regional disparities within China&#x2019;s energy efficiency landscape.</p>
</sec>
<sec id="s9-2">
<label>9.2</label>
<title>Recommendations</title>
<p>Firstly, grounded in the theory of coordinated regional development and the intrinsic requirements of green transition, empirical research by Pan et al. (2019) demonstrates that China&#x2019;s green productivity index reflects balanced progress in green economic growth over recent decades. This indicates that systematic efforts can yield policy improvements conducive to low-carbon economic transformation (<xref ref-type="bibr" rid="B26">Pan et al., 2019</xref>). It is recommended to establish a differentiated policy support system, prioritising targeted assistance for China&#x2019;s central-western regions and the old industrial bases in the northeast: on the one hand, by establishing specialised green and low-carbon development funds to focus on supporting regional infrastructure modernisation, clean technology innovation, and industrial transformation and upgrading, thereby fostering high-tech industrial clusters; Secondly, establishing an &#x2018;East-Central/Western&#x2019; industrial synergy platform to refine industrial transfer coordination mechanisms and factor market allocation systems, thereby promoting cross-regional resource optimisation and complementary advantages to systematically enhance regional green and low-carbon development capabilities.</p>
<p>Secondly, it is recommended that the government establish an incentive policy framework for green technological innovation. Research by Wang et al. (2022) demonstrates that low-carbon pilot city policies can motivate enterprises to pursue green technological innovation, thereby achieving more efficient energy utilisation and emissions reduction (<xref ref-type="bibr" rid="B33">Wang et al., 2022</xref>). This further corroborates the synergistic effects of technological innovation and policy intervention in driving the transition towards a green, low-carbon economy. This should be achieved by increasing fiscal investment in science and technology, implementing tax incentives such as additional deductions for R&#x26;D expenditure, and establishing diversified science and technology financial support mechanisms. The focus should be on cultivating enterprises&#x2019; innovation capabilities in key areas including energy-saving and emission-reduction technologies, new energy development and utilisation, and resource recycling. Concurrently, the green transformation and upgrading of traditional industries should be accelerated. This involves enhancing industrial energy efficiency through intelligent technological upgrades, phasing out high-energy-consuming outdated production capacity in an orderly manner, and prioritising the cultivation of strategic emerging industries such as new energy, new materials, and high-end equipment manufacturing. Regarding energy structure adjustment, it is recommended to implement a diversified energy development strategy, systematically reducing the proportion of fossil fuel consumption. Technological innovation and infrastructure improvements should enhance the absorption capacity and supply stability of renewable energy sources such as wind, solar, and hydro power, thereby constructing a modern energy system that is clean, low-carbon, secure, and efficient.</p>
<p>Thirdly, establish a multi-tiered energy efficiency governance system.<xref ref-type="bibr" rid="B6"> Chen et al. (2021)</xref>; <xref ref-type="bibr" rid="B6">Z. Liu et al. (2021)</xref> examined China&#x2019;s regional green governance policies, indicating that technological and environmental factors are key determinants of regional energy efficiency performance, with effective governance models narrowing inter-regional efficiency gaps (<xref ref-type="bibr" rid="B6">Chen et al., 2021</xref>). Similarly, Gabriela Araujo-Vizuete et al. (2025) proposed a hybrid governance model for Ecuador&#x2019;s energy transition, balancing top-down and bottom-up approaches to foster social participation and enhance energy governance effectiveness (<xref ref-type="bibr" rid="B1">Araujo-Vizuete and Robalino-L&#xf3;pez, 2025</xref>). Consequently, the following recommendations are proposed for this paper: Firstly, establish a lifecycle-based energy management system. Implement mandatory energy efficiency standards and tiered energy conservation targets to guide comprehensive development towards energy-efficient production and consumption patterns. Secondly, adopt a low-carbon development paradigm in urban planning. Optimise urban form and structure using spatial econometric methods, prioritising resource-efficient cities, enhancing green transport networks, and promoting ultra-low energy building technologies. Finally, establish cross-regional energy coordination mechanisms, leveraging big data platforms to optimise energy infrastructure allocation. Through interregional energy complementarity and resource sharing, systematically enhance overall energy utilisation efficiency, ultimately constructing a multi-stakeholder governance framework for energy conservation and carbon reduction involving government, market, and society.</p>
<p>Fourthly, research by <xref ref-type="bibr" rid="B34">Wei and Ma (2024)</xref> demonstrates that green technological progress significantly enhances carbon reduction capabilities, with particularly pronounced effects when supported by active carbon markets, highlighting its pivotal role in achieving the dual carbon goals (<xref ref-type="bibr" rid="B34">Wei and Ma, 2024</xref>). <xref ref-type="bibr" rid="B16">Hu et al. (2023)</xref> regional research further reveals the multidimensional characteristics of green development: taking China&#x2019;s Yangtze River Delta economic belt as an example, multidimensional economic development can positively enhance green technology development efficiency, with regional disparities gradually diminishing over time (<xref ref-type="bibr" rid="B16">Hu et al., 2023</xref>). Consequently, efforts should focus on establishing a comprehensive innovation system for green technology development: by increasing R&#x26;D intensity, deepening strategic collaborations with universities and research institutions, and establishing collaborative innovation mechanisms integrating industry, academia, research, and application to achieve breakthroughs in key energy-saving and emission-reduction technologies. Concurrently, systematic advancement of energy-efficient retrofits for production equipment should be pursued, incorporating intelligent energy management systems and advanced energy-saving process technologies. This will sustainably enhance green total factor productivity and micro-level energy utilisation efficiency, thereby providing a robust microfoundation for constructing a clean, low-carbon, secure and efficient modern energy system, and propelling the realisation of China&#x2019;s sustainable energy development objectives.</p>
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<title>Author contributions</title>
<p>CY: Writing &#x2013; review and editing, Writing &#x2013; original draft, Methodology, Supervision, Data curation. LC: Data curation, Methodology, Conceptualization, Writing &#x2013; review and editing, Supervision. QC: Project administration, Writing &#x2013; review and editing, Data curation, Software, Writing &#x2013; original draft, Supervision.</p>
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<bold>Edited by:</bold> <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/1107854/overview">Tsun Se Cheong</ext-link>, Hang Seng University of Hong Kong, China</p>
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<ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/2032733/overview">Xiaoyun Du</ext-link>, Zhengzhou University, China</p>
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