Abstract The inequitable redistribution of electricity price cross-subsidies constitutes a critical issue, as it compromises the implementation efficiency of tiered electricity pricing (TEP) policies and impedes the equalization of basic public services in the power sector. Drawing on residential TEP data from Hebei Province spanning 2016 to 2020, this paper employs the Gini coefficient method and reveals that high-income residential users receive substantially larger electricity price cross-subsidies than their low-income counterparts. Overall, the degree of such inequality has been rising annually. Furthermore, both high-income and low-income groups exhibit greater inequity in subsidy allocation relative to the middle-income group. Against this backdrop, this paper proposes a more rational tiering framework for TEP by adopting the rank-sum ratio (RSR) method, thereby identifying a viable pathway for residential users across all income brackets to share electricity costs equitably. This research contributes to the sound management of electricity price cross-subsidies, mitigates the inequity in subsidy distribution, and guides residents toward rational electricity consumption behaviors. 2. Research Methods This study takes residential electricity consumption data from Hebei Province (2016–2020) as a case study, applies the Gini coefficient method to calculate and analyze electricity price cross-subsidization and the income distribution effects among residential users in different income groups. It reveals the trends and causes of unfair income redistribution via cross-subsidization during the implementation of the TEP policy, and proposes a rational TEP categorization approach using the rank-sum ratio (RSR) method. The aim is to identify effective approaches to ensure that various user groups bear electricity costs fairly; finally, the inter-group and intra-group Gini coefficient methods are used to verify whether this categorization approach guarantees the relative fairness of subsidies among different user groups. 2.1. Cross-Subsidy Mechanism Power market cross-subsidy is an income redistribution mechanism driven by price regulation. Regulators set differentiated electricity tariffs for various user groups. Industrial and commercial users pay tariffs that exceed marginal supply costs, thereby generating surplus revenue. This revenue offsets the structural losses incurred by residential and agricultural users, whose tariffs fall below marginal costs, in order to meet policy objectives such as providing affordable electricity to households. This study employs the Price-Gap Approach proposed by the International Energy Agency to quantify the income transfer associated with cross-subsidies. Specifically, the method measures the deviations between actual regulated tariffs and cost-reflective benchmark tariffs (i.e., tariffs based on marginal supply costs) to calculate both the absolute amount and the relative intensity of the cross-subsidy. S ub i = ( M i − P i ) ୍ଠ Q i D S ub i = ( M i − P i ) / M i (1) In the above equation, S u b i denotes the amount of cross-subsidization for residential customers, D S u b i denotes the degree of cross-subsidization. A positive D S u b i indicates residential electricity underpricing, meaning that residents receive implicit subsidies. A negative value signifies overpricing, implying that residential users provide subsidies to the power system. Larger absolute values of D S u b i indicate more severe price distortion and a more pronounced redistributive effect of the cross-subsidy mechanism.; M i denotes the cost of electricity, which is the base price of electricity for residential customers, The benchmark electricity cost (CNY/kWh) comprises the provincial weighted average power purchase price, the regulated transmission and distribution tariff (including integrated line losses) for the corresponding voltage level, and the legally mandated governmental funds and surcharges; P i denotes the actual price of electricity for residential customers; and Q i denotes the actual electricity consumption of residential customers. 2.2. Gini Coefficient Method Power tariff cross-subsidies essentially act as an implicit form of income redistribution. Evaluating their fairness requires quantitative tools that can measure unequal resource allocation. The Dagum Gini coefficient serves as a classic welfare economics method for depicting relative gaps in the distribution of residential subsidies. Researchers have widely applied this indicator to measure disparities in income, property, consumption, and living standards [ 22]. This indicator possesses distinct theoretical merits that align well with our research scope. First, it maintains solid consistency with fundamental economic theories. Built on the Lorenz curve, the Gini coefficient quantifies inequality by calculating the area ratio between the absolute equality line and the actual distribution curve. Larger distribution gaps correspond to higher Gini values [ 22, 23]. The coefficient ranges from 0 to 1. Values closer to 0 reflect fairer income redistribution from power tariff cross-subsidy policies, while values approaching 1 indicate poorer policy fairness. Second, the Gini coefficient supports international comparison and standardized evaluation. Major global institutions, including the World Bank and the International Energy Agency, adopt this index to assess the distributional effects of energy subsidies. The United Nations Development Programme (UNDP) sets 0.4 as the warning threshold for income disparity, thereby providing an objective criterion for judging the redistribution fairness of power tariff cross-subsidies. Third, this method offers strong scalability. Unlike the Theil index, which captures only overall inequality, the Dagum Gini coefficient decomposes total inequality into within-group inequality, between-group inequality, and transvariation [ 24]. This decomposition allows us to identify the sources and contribution degrees of unfair subsidy distribution across and within power consumption groups. Accordingly, this study applies the Gini coefficient approach to examine inequality in the income redistribution driven by power tariff cross-subsidies. G T = ∑ j = 1 k ∑ h = 1 k ∑ i = 1 n j ∑ r = 1 n h S ub j i − S ub h r 2 n 2 S u b ପ୍ତ (2) The calculation method of the Gini coefficient is referred to the literature of Dagum [ 25], as shown in Equation (2). Where G T is the overall Gini coefficient, n is the number of samples, u denotes the average value of electricity price cross-subsidies enjoyed by all sample users, k denotes the number of groups clustered according to electricity consumption after processing the sample overall, S u b j i and S u b h r denote the electricity price cross-subsidies enjoyed by any user in groups of j and h ( j = 1, 2, …, k, h = 1, 2, …, k), n j and n h denote the number of sample users in subgroups j and h. To further explore the disparities in income redistribution between and within groups, the Gini coefficient can be decomposed as follows: G j j = 1 2 u j ∑ i = 1 n j ∑ r = 1 n j S u b j i − S u b j r n j 2 (3) G j h = ∑ i = 1 n j ∑ r = 1 n h S u b j i − S u b h r n j n h ( S u b j ପ୍ତ + S u b h ପ୍ତ ) (4) G j j is the intra-subgroup Gini coefficient, denotes the intra-group Gini coefficient when residential electricity consumption is in group j; G j h is the inter-group Gini coefficient, denotes the inter-subgroup Gini coefficient of residential electricity consumption in different groups. G w = ∑ j = 1 k G j j p j s j G n b = ∑ j = 2 k ∑ h = 1 j − 1 G j h ( p j s h + p h s j ) D j h G t = ∑ j = 2 k ∑ h = 1 j − 1 G j h ( p j s h + p h s j ) ( 1 − D j h ) (5) G w is the contribution of intra-subgroup differences, which represents the distributional gap of electricity price cross-subsidies enjoyed by users in j( h) electricity consumption group. G n b is the difference contribution between groups, p j = n j / n is the ratio of the number of sample users in subgroup j to the number of all sample users, and s j = n j u j / n u is the ratio of the electricity price cross-subsidies enjoyed by sample users in subgroup j to the electricity price cross-subsidies enjoyed by all sample users. G t is the ultra-high density contribution rate, reflecting the phenomenon of cross-overlap and reflecting the relative gap situation. Among them: D j h = d j h − p j h d j h + p j h (6) d j h = ∫ 0 ∞ d F j ( y ) ∫ 0 y ( y − x ) d F h ( x ) p j h = ∫ 0 ∞ d F h ( y ) ∫ 0 y ( y − x ) d F j ( x ) (7) D j h is the relative influence between the sample users in groups j and h. d j h represents the weighted average of the absolute value of the positive difference in the degree of cross-subsidy between groups j and h. y( x) in Equation (7) represents the amount of cross-subsidy in group j( h), and F( x) is the cumulative distribution function. p j h represents the weighted average of the absolute value of the negative difference in the degree of cross-subsidy between groups j and h. 2.3. Rank-Sum Ratio Method The optimal design of power tariff cross-subsidies is a multi-criteria decision-making problem. Policymakers must balance multiple objectives, including fairness, efficiency, and policy feasibility. The rank-sum ratio (RSR) method is a comprehensive non-parametric statistical evaluation tool. This study selects this method based on three theoretical considerations. First, this approach is well suited to the inherent characteristics of the data distribution. Residential electricity consumption data generally follow a strongly right-skewed distribution. Heterogeneous factors, including income level, climatic conditions, housing area, and consumption habits, lead to substantial differences across individual samples. Conventional parametric statistical methods, such as analysis of variance (ANOVA) and regression models, require data to satisfy normal distribution and homogeneity of variance. In contrast, the RSR method conducts statistical analysis based on rank values and imposes no constraints on the population distribution. It is highly robust to outliers and suits research scenarios characterized by highly heterogeneous residential electricity consumption behaviors [ 16, 17]. Second, this method offers strong capabilities for multi-dimensional comprehensive evaluation. Optimizing electricity pricing tiers requires considering multiple indicators, including power consumption distribution, electricity expense burden ratio, and the Gini coefficient. The rank-sum ratio method converts multi-dimensional indicators into dimensionless rank-sum ratios. It standardizes and integrates indicators with diverse dimensions and attributes, and eliminates biases arising from subjective weight assignment [ 26]. Third, this method ensures accurate and feasible tier classification. It determines grouping schemes based on the internal structural features of sample data and generates repeatable outcomes. This feature distinguishes it from the analytic hierarchy process and other approaches that rely on subjective expert judgment. This method not only performs sample classification but also ranks tier performance and estimates probabilities [ 27]. Considering the research object and available data in this study, factors such as income, environmental conditions, electricity expenditure, and consumption habits lead to significant heterogeneity in residential electricity consumption. Therefore, the data conditions in this paper satisfy the requirements for applying the rank-sum ratio method to investigate the optimal number of pricing tiers. The calculation steps of the rank-sum ratio method are as follows: Step 1 involves selecting relevant indicators. Select indicators that effectively capture variations in residential electricity consumption, as this choice critically affects the accuracy of the final classification. Step 2 entails calculating the rank of each indicator. In statistics, the rank is an ordinal number, represented by R i j , where i is the year ( i = 1, 2, …, 5), j is the index ( j = 1, 2, …, 5), m represents the total number of years, and n is the total number of indicators. The calculation method is as follows. If an indicator is of the beneficial type, apply Formula (8): R i j = 1 + ( n − 1 ) X i j − min X j max X j − min X j (8) If the indicator is a cost type, apply Formula (9): R i j = 1 + ( n − 1 ) max X j − X i j max X j − min X j (9) Step 3 consists of calculating the rank-sum ratio R S R i based on the ranks obtained, then sorting these ratios in ascending order. Step 4 involves calculating the frequency of each group, along with the cumulative frequency ∑ f , the average rank R ପ୍ତ , and the cumulative frequency distribution P for each group. The cumulative frequency distribution is then converted to derive the probit value: the corresponding probit Y is obtained by consulting a probit table, after which the regression equation is fitted: R S R ˜ = a + b P robit (10) Step 5 is to calculate the estimated RSR value R S R ˜ using Formula (10), followed by implementing the grading procedure. Subsequently, Bartlett’s test for homogeneity of variance is conducted sequentially; a significant result indicates that the final grading method is optimal. 2.4. Complementary Mechanisms of the Gini Coefficient and Rank-Sum Ratio Methods This study couples the Dagum Gini coefficient and the rank-sum ratio (RSR) method to construct a dual-layer empirical analysis framework. The two methods do not simply overlap; instead, they form endogenous complementarity across four dimensions: mathematical logic, evaluation hierarchy, statistical properties, and policy orientation. This coupling compensates for the inherent limitations of each individual model at the methodological level. Specifically, the Dagum Gini coefficient traces structural inequality in subsidy distribution and identifies the sources of distributional unfairness [ 28]. The RSR method conducts multi-objective constrained optimization to determine rational tier boundaries. This method calculates optimal tier thresholds under multiple constraints including equity, efficiency and cost. The two methods jointly establish a closed-loop paradigm spanning diagnosis and optimization. The complementary mechanisms are explained from four perspectives below. First, deep coupling in mathematical logic: static tracing and dynamic optimization form a closed loop. From the perspective of mathematical logic, the Dagum Gini coefficient, based on the Lorenz curve, structurally decomposes the continuously distributed subsidy amounts. It quantifies the contributions of within-group disparity, between-group disparity, and transvariation, thereby precisely locating the institutional causes of imbalance in subsidy distribution. For example, it can clearly answer: under the current tiered electricity pricing, to what extent do high-consumption users “capture” subsidies, and how severe is the insufficient coverage for low-income groups? However, the Gini coefficient is essentially a static inequality measurement tool. It can only “diagnose” the current situation and lacks the ability to “prescribe” solutions. The RSR method differs. It transforms multidimensional indicators such as fairness, efficiency, and cost into dimensionless rank-sum ratios based on non-parametric rank transformation, and solves for the optimal electricity pricing tier thresholds through sorting and classification. It belongs to the category of dynamic constrained optimization models and can answer: “Given the constraints, what should the tier cutoffs be?” Therefore, the structural pain points identified by the Gini coefficient (e.g., excessively high subsidies for third-tier users) can directly serve as constraints or weighting criteria in the optimization process of the RSR method. The two thus form a mathematical closed loop of “problem identification to quantitative solution,” rather than operating as independent analyses. Second, nested complementarity in evaluation hierarchy: macro-level structural diagnosis integrates with micro-level tier implementation. The two methods form a nested structure of “macro, meso, and micro” levels in their evaluation hierarchy. The Dagum Gini coefficient primarily operates at the macro and meso levels. On the one hand, it measures the overall inequality of subsidy distribution across all residential users (macro level). On the other hand, it decomposes the contributions of disparities between different electricity consumption groups and identifies institutional allocation barriers between groups (meso level). This allows researchers to judge the policy defects of the current tiered electricity pricing from an overall institutional perspective, rather than relying on isolated or local observations. The RSR method descends to the micro-level regulatory layer. It focuses on heterogeneous characteristics of individual household electricity consumption (such as consumption distribution and electricity expenditure burden ratio), and determines tier divisions and calibrates tier cutoffs based on the intrinsic distribution of the sample rather than subjective settings. This solves the practical problem that the “one-size-fits-all” tier division of traditional tiered pricing deviates from actual consumption distributions and misclassifies user groups. The complementarity between the two lies in the following: the macro-level structural judgments provided by the Gini coefficient (e.g., “between-group disparity accounts for more than 90% of total inequality”) define constraint boundaries for the micro-level tier optimization of the RSR method. That is, the optimization cannot undermine the basic between-group fairness lower bound. Conversely, the tier classification results obtained from the RSR method can be re-evaluated by the Gini coefficient to verify whether they have improved overall and between-group inequality. This creates a hierarchical verification and feedback mechanism, rather than having the two methods work in isolation. Third, statistical complementarity: parametric precision integrates with non-parametric robustness. From a statistical perspective, the two methods complement each other in addressing the complex characteristics of residential electricity consumption data, which are skewed, heterogeneous, and contain extreme values. The Dagum Gini coefficient is a parametric measurement method. It is highly sensitive to continuous subsidy amounts and can precisely capture small distributional differences, making it suitable for quantifying detailed gaps in welfare distribution. However, the cost of this sensitivity is that the Gini coefficient is susceptible to extreme outliers and has limited adaptability to skewed distributions. In residential electricity consumption data, high-consumption users may receive subsidy amounts far above the mean, and such extreme values can significantly inflate the Gini coefficient, potentially masking the true distribution among low- and middle-income groups. The RSR method is the opposite. It is a typical non-parametric method that does not require the data to follow a normal distribution or satisfy homogeneity of variance. It performs analysis based on rank ordering rather than raw values, thereby naturally mitigating the influence of extreme high or low consumption samples and demonstrating strong data robustness. However, this robustness also has a cost: rank transformation discards some fine-grained information contained in the raw values. The practical benefit of combining the two is as follows. The Gini coefficient provides precise measures of subsidy disparities, but its results undergo cross-validation through the robustness of the RSR method. Meanwhile, the tier classification results from the RSR method can be subjected to sensitivity tests using the Gini coefficient, for example, by examining whether the Gini coefficient changes dramatically after excluding extreme samples. This two-way verification of precision and robustness provides greater confidence in the empirical findings than any single method alone. Fourth, two-way balancing in policy orientation: fairness as a rigid constraint cooperates with comprehensive benefit optimization. At the policy application level, the two methods represent respectively “fairness-first” and “efficiency-first” policy logics, and their coupling achieves two-way balancing rather than a simple compromise. The core function of the Dagum Gini coefficient is to provide a rigid fairness constraint. It offers a set of quantifiable bottom-line indicators. For example, an overall Gini coefficient exceeding 0.4 enters the internationally recognized “inequality warning zone,” and an excessively high contribution rate of between-group disparity indicates serious institutional allocation defects. In the policy optimization process, any tier scheme must not significantly worsen these fairness indicators. This prevents excessive sacrifice of low-income groups’ welfare in pursuit of economic efficiency or administrative convenience. The RSR method aims to maximize comprehensive benefits. Within the fairness constraint boundaries set by the Gini coefficient, it solves for the optimal tier scheme by balancing multiple dimensions, including economic cost, operational efficiency, and policy feasibility. For instance, it can select the tier threshold that minimizes the subsidy leakage rate without increasing the overall Gini coefficient. The two-way balancing mechanism can be summarized as follows: the Gini coefficient defines the lower bound of fairness (i.e., “no worse than this”), and the RSR method searches within this bound for the best possible comprehensive solution. This approach is more operational and verifiable than merely emphasizing the slogan of “balancing fairness and efficiency,” and provides a quantifiable technical pathway for the long-term improvement of the cross-subsidy system in tiered electricity pricing. 4.2. Recalculation and Analysis of the Overall Gini Coefficient The overall Gini coefficient following grading via the rank-sum ratio (RSR) method was recalculated and analyzed. A recalculated value lower than the pre-regrouping overall Gini coefficient of electricity price cross-subsidies indicated that cross-subsidy distribution was relatively more equitable under this grouping approach.
