مناسب بودن اقدامات نابرابری درآمد برای تجزیه و تحلیل منطقه ای : برخی از شواهد به دست آمده از تجزیه و تحلیل شبیه سازی و آزمایش راه اندازی

The paper looks at the sensitivity of commonly used income inequality measures to changes in the ranking, size and number of regions into which a country is divided. During the analysis, several test distributions of populations and incomes are compared with a ‘reference’ distribution, characterized by an even distribution of population across regional subdivisions. Random permutation tests are also run to determine whether inequality measures commonly used in regional analysis produce meaningful estimates when applied to regions of different population size. The results show that only the population weighted coefficient of variation (Williamson’s index) and population-weighted Gini coefficient may be considered sufficiently reliable inequality measures, when applied to countries with a small number of regions and with varying population sizes.

The study of inequality across regions is rather different to the study of inequality between individuals. This derives from the fact that regions are groups formed by individuals. This is not as obvious as it may sound. For example, a tradition exists in the regional income convergence literature that treats regions as individual observations regardless of the size of the former (cf. e.g., [1]). As such, large and small regions are assumed to carry equal weight, just as fat and thin people are treated equally when looking at inequality between them. The computational issues associated with multi-group comparisons of income inequality were noticed (apparently for the first time) by the American economist Max Lorenz. In his seminal paper published in 1905, Lorenz highlighted several drawbacks associated with the comparison of wealth concentrations between fixed groups of individuals. In particular, he found that while an increase in the percentage of the middle class is supposed to show the diffusion of wealth, a simple comparison of percent shares of persons in each income group may often lead to the opposite conclusion. For instance, while the upper income group in a particular period may constitute a smaller proportion of the total population, the overall wealth of this group may be far larger compared to another time period under study ([2]: 210–211). The remedy he suggested was to represent the actual inter-group income distribution as a line, plotting ‘along one axis cumulated percents of the population from poorest to richest, and along the other the percent of the total wealth held by these percents of the populations’ (ibid. p. 217). In an essay published in 1912, the Italian statistician Corrado Gini moved Lorenz’s ideas a step further, suggesting a simple and easy comprehendible measure of inequality known as the Gini coefficient. Graphically, the calculation of this coefficient can be interpreted as follows: Ginicoefficient=AreabetweenLorenzcurveandthediagonalTotalareaunderthediagonal Turn MathJax off Mathematically, the Gini coefficient is calculated as the arithmetic average of the absolute value of differences between all pairs of incomes, divided by the average income (see Table 1).1 The coefficient takes on values between 0 and 1, with zero interpreted as perfect equality [3]. A few years later, Dalton [4] carried out the first systematic attempt to compare the performance of different inequality measures against ‘real world’ data. As he noted, many inequality measures, though having intuitive or mathematical appeal, react to changes in income distribution in an unexpected fashion. For instance, if all the incomes are simply doubled, the variance quadruples the estimates of income inequality. Dalton’s second observation was that some inequality measures do not comply with a basic principle of population welfare set forward by Arthur Pigou and formulated as follows: ‘if there are only two income-receivers, and a transfer of income takes place from the richer to the poorer, inequality is diminished’ (ibid. p. 351). After applying the ‘principle of transfers’ to various inequality measures, Dalton found that most measures of deviation (e.g., the mean standard deviation from the arithmetic mean, and the coefficient of variation) are perfectly sensitive to transfers and pass the ‘test with distinction’ (ibid. p. 352). The Gini index, commonly used in empirical studies, was also found by Dalton sufficiently sensitive to income transfers. He also found that the standard deviation is sensitive to transfers among the rich, while the standard deviation of logarithms is less sensitive to transfers among the rich than to transfers among the poor but still changes when a transfer among the rich takes place. Two other fundamental requirements for a ‘robust measure’ of inequality, set forward by Dalton, are the principle of proportional addition to incomes, and the principle of proportional increase in population. According to the former, a proportional rise in all incomes diminishes inequality, while the proportional drop in all incomes increases it. According to the latter principle, termed by Dalton the ‘principle of proportional additions to persons,’ a robust inequality measure should be invariant to proportional increase in the population sizes of individual income groups. Dalton’s calculations showed that most commonly used measures of inequality comply with these basic principles. Only the most ‘simple’ measures, such as absolute mean deviation, absolute standard deviations and absolute mean difference, fail to indicate any change, when proportional additions to the numbers of persons in individual income groups are applied (ibid. pp. 355–357, see also [5], pp. 87–112).2 Yitzhaki and Lerman [6]noted another deficiency inherent to most inequality measures, viz., insensitivity to the position which a specific population subgroup occupies within an overall distribution. Their Gini decomposition technique (see below) takes group-specific positions into account. In particular, they suggested weighting subgroups by the average rank of their members in the distribution. This is in contrast to the weighting system used more conventionally according to which between-group inequality is weighted by the rank of the average [7] and [8]. This latter system results in a large residual when inequality is decomposed into within and between groups. In contrast, the Yitzhaki approach results in a more concise decomposition with no residual [9]. Other empirical studies proposed and used a variety of additional inequality measurements, such as the population weighted coefficient of variation (Williamson’s index), Theil index, Atkinson index, Hoover and Coulter coefficients [3], [6], [10], [11], [12], [13], [14] and [15]. However as the Gini measure ranges between 0 and 1 and is unaffected by change of scale (the population principle), it has become probably the most attractive measure for inequality in regional analysis. While there have been numerous attempts to test the conformity of commonly used inequality measures with basic inequality criteria – e.g., principles of transfer, proportional addition to incomes, and proportional addition to population – (see inter alia [4], [5] and [11]), there appears to be no systematic attempt to verify whether all of these measures are equally suitable for regional analysis, in which individual countries may be represented both by a different numbers regions and by regions of different population sizes. The lack of interest to this aspect of inequality measurement may have a simple explanation. Since commonly used inequality indices (some of which appear in Table 1) are abstract mathematical formulas, one can assume that they can be applied to both large and small countries alike and provide fully comparable results. However, it is well known that the use of different measurement indices in regional analysis gives rise to highly variable results. For example, the notion of optimal regional convergence (i.e., that point where regional convergence also reduces overall nation-level inequality) has been shown to be highly dependent on the type of inequality index used [16] as is the measurement of regional price convergence [17]. The present paper attempts to determine whether commonly used inequality measures are sufficiently sensitive to changes in the ranking, size and number of regions into which a country is divided. The paper is organized as follows. First, we look at the specificity of measuring regional inequality. Given the fact that the unit of observation (i.e., a region) is a group measure, it presumably needs some weighting as regions of a country come in different sizes. We then proceed to discuss the general principles that should govern in our view, the selection of robust inequality measures. Then we move to testing the compliance of different commonly used inequality measures against the set of criteria that should characterize a robust inequality measure. The tests are run in two phases. First, we use a number of pre-designed distributions, to verify whether a particular inequality measure meets our intuitive expectations concerning inequality estimates. Then, in the second stage of the analysis, we run more formal permutation tests to verify whether different inequality measurements respond sensibly to changes in the population distribution across the space.

Though individual studies of regional disparity may deal with separate development measures – population growth, wages, welfare, regional productivity, etc. – the use of an integrated indicator is often essential, particularly if a comparative (cross-country) analysis is required. In order to measure the extent of disparities, various indices of inequality are commonly used. These indices may be classified into two separate groups [14]: • Measures of deprivation (Atkinson index, Theil redundancy index, Demand and Reserve coefficient, Kullback-Leibler redundancy index, Hoover and Coulter coefficients, and the Gini index); • Measures of variation, such as the coefficient of variation and Williamson’s index. In this paper, we did not attempt to assess whether these measurements reflect either the ‘true meaning’ or ‘underlying causes’ of regional inequality. Neither did we try to establish whether geographic inequality is a positive socio-economic phenomenon or a negative one. We shall leave these philosophical questions for future studies. Our task was simple: we attempted to determine whether commonly used inequality measures produce meaningful estimates when applied to countries of different size and with different number of regional subdivisions, thus making it possible to directly compare the results of analysis obtained for one particular country with those obtained elsewhere. This task is not as abstract as it may sound. There has been a tradition in the regional income convergence literature to treat regions of a country as individual observations, regardless of the numbers of regions into which a country is subdivided and of their population sizes (cf. e.g., [1]). As such, all regions, big and small, are assumed to carry equal weight, an assumption which is hardly justified empirically, considering a large variation in the number of regions and their population sizes that are found in most countries across the globe. In order to formalize these distinctions, we designed a number of simple empirical tests, in which several hypothetical income and population distributions were compared with the ‘reference’ distribution in which the population was distributed evenly across regional divisions and assumed to be static. In the first test, we checked whether the overall number of regions matters. In the second, we checked whether different inequality indices respond to differences in the regional distribution of population, viz., evenly spread population in the reference distribution vs. unevenly spread population in the test distribution. Finally, in the third test, we verified whether different inequality indices were sensitive to the sequence in which regions are introduced into the calculation. Somewhat surprisingly, none of the indices we tested appeared to pass all the tests, meaning that they may produce (at least theoretically) misleading estimates if used for small countries. However, several indices – CC, WI and Gini (W) – appeared to exhibit only minor flaws and may thus be considered as more or less reliable regional inequality measures, when used empirical studies for comparing the extent of regional inequalities across the time and space, that is, over different time periods and between different countries. Mitigating regional development disparities between rapidly developing and economically successful core regions and lagging peripheral areas, has been a prime objective of regional development policy in most developed countries worldwide. In this respect, efficient regional inequality measures may become an indispensible tool for gauging the success of failure of such a policy, enabling policy-makers and regional scientists to compare the extent of development disparities before and after the policy intervention, or in relationship to other countries, characterized by similar development levels. However, this important objective can be achieved only if inequality measures used in the analysis faithfully reflect the extent of interregional disparities but no less importantly, the number of people facing them. For instance, if in a country with e.g., 20 million residents, the population size of the least developed region may be 50,000 or, alternatively, 5,000,000 residents. This difference may have different implications for “regional equalization” policies, both in sense of policy tools to be used and their magnitude. However, if regions are assumed to be indivisible units, equal in size, as with many widely used inequality measures, this important distinction may go unnoticed thereby distorting development policy overall. In this respect, the present analysis serves to caution regional analysts against using different inequality measures indiscriminately or comparing their values directly, between countries and time periods. Although an inequality measure may be sufficiently sensitive to differences in the population size across regions this sensitivity may not always be sufficient, unless intra-regional disparities are also taken into account. Even when controlling for regional size, treating a region as the prime unit of observation in the analysis of regional inequality may hide considerable internal income disparities among its residents. Therefore future research efforts will need to develop regional inequality indices that account for both disparities between individual regions (in terms of their sizes and development levels), as well as their internal income heterogeneities. Further research may also be needed to develop and a set of mathematical tools that can detect potential performance problems of various regional inequality measures, without a need to carry out extensive empirical simulations, such as those undertaken in the present study. Although future work on the performance of different inequality indices may thus be needed to verify the generality of our observations, the present analysis clearly cautions against indiscriminate use of inequality indices for regional analysis and comparison.