📈 economics

Inequality at a Location: A Local Inequality Index Based on Mean Log Deviation

This paper introduces the Local Inequality Index (LII), a novel measure based on mean log deviation that satisfies both Locality and Spatiality properties by weighting income deviations with a distance-decay kernel, thereby revealing localized inequality patterns in Chinese cities that conventional global measures fail to detect.

Original authors: Dongyuan Mu

Published 2026-10-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Dongyuan Mu

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Inequality is usually thought of as a statistic for a whole country or a large group, a single number that summarizes how far apart the rich and the poor are on average. But people do not experience inequality as a national average; they experience it from where they stand. A person in a wealthy neighborhood might feel secure even if the country is poor, while someone in a struggling town might feel left behind even if the nation is getting richer. This feeling depends on the immediate surroundings: the incomes of the neighbors, the local opportunities, and the visible gaps between the person's own situation and those nearby. Economists call this the reference group, the circle of people we compare ourselves to, which is often drawn from our physical location rather than the entire population. Understanding how income is distributed in these specific local environments is crucial because it shapes everything from how safe a street feels to how people vote on national issues.

A new study by Dongyuan Mu at Kindai University tackles the difficulty of measuring this local experience. For decades, researchers have tried to capture local inequality by taking a standard national formula and applying it to a small, fixed area, like a county or a circle drawn around a city. The problem with this approach is that it treats the boundary as a hard wall. If a wealthy city sits just outside the line, it is ignored, even though it might be the most important comparison for someone living right on the edge. This method creates a "modifiable areal unit problem," where the result changes simply because the researcher drew the line in a slightly different place. The paper argues that a true measure of local inequality should not just select a sample of people to look at; it should read the position of every single unit in the country, weighing them based on how far away they are.

To solve this, the author proposes a new tool called the Local Inequality Index. Instead of cutting off the world at a specific distance, this index looks at the entire country from the perspective of a single city. It acknowledges that a disparity with a neighbor who lives ten miles away feels much more significant than a disparity with someone living a thousand miles away, even if the difference in income is exactly the same. The index uses a mathematical rule where the influence of a distant city fades as the distance grows, ensuring that the immediate surroundings shape the measurement more than the distant regions. This allows the index to satisfy two key requirements: it is specific to the location being measured, and it responds to the exact arrangement of cities, meaning that if a city moves slightly, the measure of inequality for its neighbors changes.

The researchers tested this new index using data from nearly 300 prefecture-level cities in China, covering the years from 2000 to 2020. They compared the results of their new Local Inequality Index against the traditional national measure of inequality. The traditional measure showed a clear trend: the gap between rich and poor across the entire country was shrinking over these two decades. The new index confirmed this general trend, but it also revealed a much more complex and hidden story that the national average completely missed. While the country as a whole was becoming more equal, the local experience was not uniform. The index uncovered isolated pockets of high inequality that were invisible in the national data, as well as regions where local disparities were quietly worsening even as the national numbers improved.

One of the most striking findings involved the Northeast region of China. According to the traditional national statistics, inequality in this region was falling, and the cities there were becoming more similar to one another. However, the Local Inequality Index told a different story. It showed that while the cities within the Northeast were indeed becoming more equal among themselves, they were falling further behind the rest of the country. Because the new index weighs distant cities based on their income difference, the growing gap between the Northeast and the wealthy coastal regions began to dominate the local experience. As a result, the index showed that local inequality in the Northeast was actually rising, driven by the distance to the richer parts of the nation, a reversal that the standard national measure failed to detect.

The study also looked at the famous economic hubs along the coast, such as the Yangtze River Delta and the Pearl River Delta. In these areas, the index revealed that high local inequality was not always caused by a single super-rich city surrounded by poor ones. In the Yangtze River Delta, for example, the cities surrounding Shanghai were themselves very wealthy, creating a relatively smooth gradient of income. In contrast, the Pearl River Delta showed a steeper local gradient, where the gap between the richest city and its immediate neighbors was much sharper. This distinction helped explain why some wealthy regions felt more unequal than others, based on the specific shape of their local income landscape rather than just their overall wealth.

By applying this method, the researchers demonstrated that inequality is not a single number that applies everywhere. It is a landscape that changes depending on where you stand. The Local Inequality Index provides a way to map this landscape, showing exactly where the income gaps feel the sharpest to the people living there. It turns the abstract concept of inequality into a concrete map of local environments, revealing patterns of isolation and disparity that remain hidden when we only look at the national average. This approach suggests that to truly understand economic well-being, we must look not just at how much money a country has, but at how that money is distributed in the immediate world of every individual.

Technical Summary: Inequality at a Location: A Local Inequality Index Based on Mean Log Deviation

Problem Statement
Conventional inequality measures treat inequality as a property of a population aggregate, ignoring the spatial context in which individuals experience economic disparity. While theories of relative deprivation suggest that individuals assess their status against local reference groups rather than the national population, existing empirical methods fail to capture "inequality at a location" effectively. Current approaches typically compute standard indices (e.g., Gini, MLD) within fixed administrative units or fixed-radius buffers. These methods suffer from two primary limitations:

Modifiable Areal Unit Problem (MAUP): Results depend heavily on arbitrary boundary definitions and scale.
Lack of Spatial Sensitivity: Position is used only to select a sample; once selected, the spatial arrangement of units within that sample is ignored. Moving a unit within a buffer does not change the index value.

The paper argues that a measure of local inequality must treat position as an intrinsic variable that shapes the measurement itself, rather than merely a filter for data selection.

Methodology: The Local Inequality Index (LII)
The paper proposes the Local Inequality Index (LII), constructed by adapting the Mean Log Deviation (MLD) to a spatial context using a distance-decay kernel.

Core Construction: For a focal location $j$ , the LII ( $LII_j$ ) is calculated as a weighted MLD where the weights are determined by a kernel function $w(d_{ij})$ based on the distance between unit $i$ and focal unit $j$ .
$LII_j := \ln \bar{X}^w_j - \sum_i S_j(p_i) \ln X_i$
Where:
- $\bar{X}^w_j$ is the spatially weighted mean income centered on $j$ .
- $S_j(p_i)$ and $S_j(s_i)$ are the spatially weighted population and income shares, respectively.
- The kernel $w(d_{ij})$ ensures that closer units contribute more heavily to both the reference mean and the dispersion calculation.
Kernel Specification: The paper recommends a homogeneous kernel $w(d) = d^{-\alpha}$ with a decay parameter $\alpha \ge 0$ .
- Self-Distance ( $d_{jj}$ ): To include the focal unit in its own environment, a strictly positive self-distance is defined as $d_{jj} = k \cdot \min_{i \neq j} d_{ij}$ (where $0 < k \le 1$ ). This prevents the focal unit from being excluded (as in traditional spatial weights) and ensures the global MLD is nested when $\alpha = 0$ .
- Decay Parameter ( $\alpha$ ): Controls the spatial scale of the reference environment. As $\alpha$ increases, the influence of distant units diminishes.
Theoretical Properties:
The paper defines two necessary properties for a valid measure of inequality at a location:
1. Locality: The measure must be defined at each location and yield different values for different locations.
2. Spatiality: The measure must respond to the spatial configuration of units; moving a unit must change the measure at some location.
  The LII satisfies both properties for any $\alpha > 0$ . It inherits standard inequality properties (scale invariance, continuity) from the GE family and satisfies a "Spatial Transfer Principle" (inequality decreases if a progressive transfer occurs between units equally weighted from the focal point).

Key Contributions

Conceptual Framework: The paper formalizes "inequality at a location" by establishing Locality and Spatiality as verification criteria, distinguishing true local measures from population measures applied to subsamples.
Index Construction: It constructs the LII, a measure that reads the position of every unit via a kernel function rather than using position to select a sample. This allows for a granular, location-specific value for every observation in the dataset.
Analytical Characterization: The paper derives the behavior of the LII under the homogeneous kernel. It shows that:
- At $\alpha = 0$ , LII collapses to the global MLD.
- As $\alpha \to \infty$ , LII converges to the inequality of the immediate neighborhood (or zero if the self-weight dominates).
- The response of LII to changes in $\alpha$ is governed by the covariance between local income deviations and log distance.

Empirical Results
The author applies the LII to Chinese prefecture-level cities from 2000 to 2020 ( $N \approx 290$ ).

National Trends: The LII tracks the decline in the conventional national MLD over the period, confirming that the index captures broad distributional shifts.
Spatial Heterogeneity: The LII reveals patterns masked by aggregate data:
- Central/Western Regions: Consistently show low LII values, reflecting spatially homogeneous but low-income environments rather than equitable distributions.
- Yangtze River Delta (YRD): The region maintains high LII values, but the paper notes it is not organized around a single dominant center; the cities surrounding Shanghai are themselves highly developed, resulting in a less dispersed local income environment compared to other high-inequality clusters.
- Pearl River Delta (PRD): This region presents a configuration that sustains high LIIs due to a steep local income gradient. Shenzhen's income is approximately twice that of the cities within 100 km, creating a sharp contrast that drives local disparity. Until 2015, Guangzhou recorded the highest LII in the PRD, as its income lagged significantly behind Shenzhen while remaining close enough to elevate the spatially weighted neighborhood mean.
- Northeast China: A critical finding is that while the Northeast's internal inequality (regional MLD) declined, its LII increased relative to the national median. This is driven by the widening income gap between the Northeast and the rest of China. As the kernel extends beyond regional borders, the "poverty" of the Northeast relative to the national average becomes the dominant factor in its local inequality measure.
Parameter Sensitivity: Increasing $\alpha$ shifts the focus to immediate neighbors. For major hubs like Beijing and Shanghai, high $\alpha$ values cause their LII to drop (as they become their own reference), while their poorer immediate neighbors (e.g., Baoding) show rising LII values due to the sharp contrast with the hub.

Significance and Claims
The paper claims that the LII provides a necessary tool for understanding inequality as a spatially experienced phenomenon. Its primary significance lies in its ability to:

Unmask Local Disparities: It detects isolated high-inequality agglomerations and regions where local disparities are worsening despite national or regional improvements (as seen in the Northeast case).
Avoid MAUP: By weighting all units continuously based on distance, it avoids the arbitrary boundary issues inherent in fixed-radius or administrative unit approaches.
Granularity: It generates a map of inequality at the resolution of the data itself (e.g., city-level), without requiring aggregation into provinces.

The author maintains a modest tone, noting that the choice of $\alpha$ and $k$ depends on researcher assumptions regarding the spatial scale of inequality perception. They suggest future work could focus on principled selection rules for $\alpha$ and the application of the LII's additive decomposability to sectoral or demographic groups. The framework is also noted to be applicable to non-geographic "distances" (e.g., age or social network proximity).

Technical Summary: Inequality at a Location: A Local Inequality Index Based on Mean Log Deviation

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