Estimating Income Inequality Using Single-Parameter Lorenz Curves: A New Proposal

Parametric models are valuable tools for the empirical analysis of income and wealth distributions. Previous research suggests that parametric functional forms provide more accurate estimates of inequality measures than nonparametric methods, especially when data are limited to a few income shares (Dhongde and Minoiu 2013; Jorda et al. 2021). This superiority of parametric models appears to hold even for simple functional forms with only two parameters. As a result, parametric models become essential when individual records are unavailable, and only aggregated income distribution data are accessible.

The unavailability of individual data was particularly common in the 1980s and 1990s, and it remains a significant challenge today. For instance, when analyzing the distribution of income in Bhutan, one must rely on grouped data, as individual records are not accessible to the general public. In fact, grouped data are often more readily available than individual records. Several comprehensive secondary databases have been developed to provide access to aggregated income distribution data, including the World Income Inequality Database (WIID) hosted by the World Institute for Development Economics Research, the Poverty and Inequality Platform (PIP) managed by the World Bank, and the World Inequality Database (WID) operated by the Paris School of Economics. While these resources are invaluable, they typically rely on grouped data, making parametric models essential for an accurate analysis of income distributions.

The challenge of identifying an adequate functional form for income distributions remains unresolved, with several alternative specifications proposed.¹ Among these, the log-normal distribution is likely the most widely used model. However, despite its popularity, the log-normal model has notable limitations and is often outperformed by alternative functional forms, including those with just two parameters (Bandourian et al. 2002; Jorda et al. 2021). Consequently, scholars have devoted significant attention to finding a functional form that better fits observed distributions. Among the range of alternatives, the generalized beta family of distributions stands out as the most promising option (Kleiber and Kotz 2003).²

Another branch of the literature on statistical distributions has focused on developing ad hoc Lorenz curves. Since the pioneering work of Kakwani and Podder (1973), directly modeling the Lorenz curve has become a common practice, as it provides an excellent fit to observed income shares while maintaining a straightforward estimation process. The World Bank has a long-standing tradition of using such models to estimate income distributions from grouped data. Notably, Kakwani’s (1980) beta Lorenz curve and Villaseñor and Arnold’s (1989) quadratic Lorenz curve are both integrated into the PIP methodology to derive poverty estimates. Alternative models with fewer parameters, such as those proposed by Aggarwal (1984), Chotikapanich (1993), Kakwani and Podder (1973), Gómez-Déniz (2016), and the Lamé Lorenz curves (Henle et al. 2008; Sarabia et al. 2017), also provide good approximations of the Lorenz curve. Developing simpler models remains crucial for ensuring ease of implementation while retaining accuracy, particularly when working with grouped data.

Paul and Shankar (2020) introduced a one-parameter model that appears to provide an excellent fit for Australian data (hereafter referred to as the PS Lorenz curve). In this paper, we explore new properties of their model, offering a new representation that enables us to define the Lorenz ordering and derive closed-form expressions for various inequality measures. Additionally, we demonstrate that, despite the model’s strong apparent performance, it is only suitable for representing income distributions with high levels of inequality, as the Gini index is bounded below by 0.418. To address this limitation, we propose a new model that, while still bounded, has a Gini index range starting at 0.164, allowing for a more flexible representation of income distributions across a broader range of inequality levels.³ We also compare the performance of our model with the PS Lorenz curve across various countries exhibiting different levels of inequality. Our proposed model demonstrates superior flexibility and accuracy, even for countries with high inequality, where the constraint imposed by the Gini index of the PS model is not binding.

The contents of this paper are structured as follows. In Section 2, we explore a new representation of the PS Lorenz curve and introduce closed-form expressions for several inequality measures. Section 3 introduces an alternative Lorenz curve model, which offers a much wider range for the Gini index. Additionally, this new model is simpler and provides closed-form expressions for both the quantile function and the cumulative distribution function (CDF), making it easier to compute several economic quantities and improving the accuracy of inequality estimates, particularly in cases where the PS Lorenz curve is limited by its bounded Gini index. In Section 4, we analyze the performance of our new model compared to the PS Lorenz curve across various countries. The paper concludes by discussing the practical and theoretical implications of our findings.

In this section, we introduce a new representation of the PS Lorenz curve, which simplifies the definition of Lorenz ordering. We also present closed-form expressions for several inequality indices, including the Gini index and the Donaldson, Weymark, and Kakwani (DWK) index.

The PS proposal has limitations which deem it unsuitable for distributions with low inequality levels. In this section, we propose a simpler functional form than (1) and with more satisfactory properties. The new Lorenz curve is defined as follows:with \(\gamma >0\) (hereafter referred to as the SJT Lorenz curve). Relevant properties of this new family are obtained in the following sections.

In this section, we compare the performance of the PS Lorenz curve and our proposed model in estimating income inequality across various countries using goodness-of-fit (GOF) measures. This analysis is based on grouped data in the form of income shares from the WIID, which includes 24,366 datasets from 201 countries, spanning from the beginning of the nineteenth century. To ensure a globally representative sample of income distributions, we selected 10 countries from different world regions. The income shares and inequality measures for these countries are presented in Appendix (Table 3). The selected countries exhibit a wide range of inequality levels, with Gini indices between 0.2846 and 0.4880. Consequently, we believe our results can be generalized to most countries worldwide.

We turn now our attention to the estimation of the curves given in (1) and (10). Let x be a random sample of size N of a continuous random variable X. Assume that the sample \(({\textbf {x}})\) is divided into J mutually exclusive intervals \(H_j = [h_{j-1},\) \( h_j) , j= 1, \dots , J\). Let \(\varvec{1}_{[h_{j-1}, h_j)}(x_i)\) be an indicator function, taking the value 1 if \(x_i \in [h_{j-1}, h_j)\) and zero otherwise. Denote \(c_j= \sum _{i=1}^N\varvec{1}_{[h_{j-1}, h_j)}(x_i)x_i/\sum _{i=1}^Nx_i, j=1,\dots , J\) as the proportion of total income held by individuals in the \(j^{th}\) interval and the cumulative proportion by \(s_j= \sum _{k=1}^j c_k\). Let \(u_j = \sum _{i=1}^N\varvec{1}_{[h_{j-1}, h_j)}(x_i)/N , j=1,\dots , J\) denote the frequency of the sample x in the \(j^{th}\) interval and \(p_j= \sum _{k=1}^j u_k\) the cumulative frequency.

Therefore, the proportion of observations in each group is determined before sampling, so the population proportions (\(p_j\)) are fixed, while the income shares (\(s_j\)) are treated as random variables. Estimating parametric distributions by maximum likelihood would be misspecified in this case, due to the non-stochastic nature of the group frequencies (Hajargasht and Griffiths 2020). To avoid this potential issue, nonlinear least squares are conventionally used to estimate the vector of parameters of interest by minimizing the distance between the observed income shares and the functional form of the Lorenz curve. In this context, nonlinear least squares can be referred to as the equally weighted minimum distance (EWMD) estimator, which takes the following form:where \(\varvec{M}(\varvec{\theta })'= [m_1(\gamma ), \dots , m_{J-1}(\gamma )]\) is the vector of moment conditions, given bywhere \(\varvec{s}' = (s_1, \dots , s_{J-1})\) is a vector of cumulative income shares associated with the population proportions \(\varvec{p}' =(p_1, \dots , p_{J-1})\). Closed expressions for \( L(\varvec{p}; \gamma )\) are given in (1) and (10).

The EWMD estimator given in (14) overlooks that, by definition, the sum of the income shares is equal to one, introducing dependence between the income shares. As a result, EWMD provides consistent but inefficient estimates of parameters, including those that depend on the \(\gamma \) parameter, such as inequality measures. To obtain more efficient estimates, optimally weighted minimum distance (OMD) estimators could be used. However, Jorda et al. (2021) find that, in most cases, EWMD yields more accurate estimates of inequality measures than OMD. Therefore, since our primary goal is to obtain consistent estimates and examine the practical implications of using these two Lorenz curves to estimate inequality measures, we opt to use the EWMD estimator.

Table 2 presents the estimates of the gamma parameter for the PS Lorenz curve and the new proposal (SJT curve), along with the bootstrap standard errors of these estimates, the mean squared errors (MSE) for each model, and the estimated Gini index. As expected, the gamma parameter for the PS curve approaches zero in countries with Gini indices below 0.418, reflecting the model’s limitation in capturing lower inequality levels. This limitation is further evidenced by the higher MSE for these countries, indicating a poor fit for the PS curve. Our model also shows better performance in countries with high inequality levels, where the Gini index constraint of the PS Lorenz curve is not binding. Notably, the MSE for the PS Lorenz curve is three to five times higher in Honduras, the USA, and Brazil, where it reaches 0.0009, compared to the SJT model’s MSE, which ranges between 0.0002 and 0.0003.

In terms of the Gini coefficient, the SJT curve provides a much closer approximation to empirical inequality estimates in countries with low inequality levels. However, in Brazil, Honduras, and the USA, which exhibit empirical Gini indices between 0.47 and 0.49, the PS model seems to provide more accurate estimates of this inequality measure, although the differences with the SJT curve are not important.

To investigate this surprising result, Figure 4 presents the observed income shares in Brazil in 2021 (black points) alongside the estimated PS and SJT models. The graph shows that the SJT model provides a highly accurate fit and clearly outperforms the PS model, consistent with the MSE values. However, the gap between the observed and estimated Gini index is three times smaller for the PS Lorenz curve (see Table 2). This occurs because the Gini index is proportional to the area between the Lorenz curve and the egalitarian line. As a result, underestimations in some income shares can be offset by overestimations in others. Thus, the seemingly better performance of the PS model is a statistical artifact arising from the way the Gini coefficient is calculated.

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In this paper, we have derived new and important economic properties of the PS Lorenz curve, focusing on its enhanced representation through convex linear combinations of Lorenz curves. We have provided explicit formulas for the Gini index, as well as the DWK index, and established the Lorenz ordering of the curve. Additionally, we demonstrated that the Gini index for this model is constrained by a lower bound of 0.418, which significantly limits its applicability to countries with low inequality.

To address this limitation, we proposed a new Lorenz curve with a broader range for the Gini index, offering greater flexibility in measuring inequality in income distributions. Our proposal also presents several additional advantages. First, it offers a simpler functional form compared to the PS curve, making it more practical to use. Second, the Gini index derived from our model has a more straightforward expression than that of the PS curve. Finally, both the quantile function and the CDF are available in closed form (see equations 12 and 13).

Our results suggest that the SJT curve consistently outperforms the PS curve, yielding lower MSE statistics in all cases. For countries with low inequality, the Gini index of the PS model tends to its lower bound, limiting its accuracy. In contrast, for countries with higher inequality, both models provide reasonably accurate estimates of inequality measures. However, the apparent good performance of the PS model seems to be influenced by the way the Gini coefficient is defined, as underestimations in some income shares can be offset by overestimations elsewhere. Overall, the Lorenz curve proposed in this paper surpasses the PS model by offering a wider range of Gini estimates and consistently delivering a better fit, making it a more robust tool for measuring inequality across diverse income distributions.