نابرابری درآمد، شبه تقعر، تغییرات تدریجی جمعیت

An income distribution is a mixture of two given income distributions if the relative frequency it associates with each income level is a convex combination of the relative frequencies associated with it by the given two income distributions—e.g., the income distribution of a country is obtained as a mixture of the income distributions of its regions. In this article, it is established that all inequality measures commonly considered in the literature—the class of decomposable inequality measures and the class of normative inequality measures based on a social welfare function of the rank-dependent expected utility form—satisfy quasi-concavity properties, which imply, loosely speaking, that mixing income distributions increases inequality. These quasi-concavity properties are then shown to greatly reduce the possible patterns describing the evolution of inequality in the overall income distribution (a mixture) during a process in which population gradually shifts from one of its constituent income distributions to another over time.

In the analysis of income inequality, it is often useful to view the income distribution of interest as being composed of several constituent income distributions, e.g., the income distributions corresponding to different regions, sectors, or genders. The question of how inequality in the overall income distribution is affected if the constituent income distributions change, has received considerable attention in the form of decomposability analysis.1 By contrast, the complementary question of how overall inequality changes if the population shares corresponding to the constituent income distributions change, has not been studied much. Nevertheless, the latter question is interesting both from the empirical and the theoretical perspective. There are several empirical phenomena that involve a shift of the population from one constituent income distribution to another. Take as an example the phenomenon of demographic ageing. In this case, the overall income distribution changes over time because population gradually shifts from the income distribution of working consumers to the income distribution of retired consumers. Another (particularly natural) example is that of a country with two regions that have different population growth rates: here population shifts from the income distribution corresponding to the region with the lower growth rate to that corresponding to the region with the higher growth rate. As a final example, consider the development process studied by Kuznets (1955) which involves a gradual population shift from the income distribution of the agricultural sector to that of the industrial sector. Besides being of empirical interest, the gradual population shift process is relevant theoretically. In order to see this, assume that the overall income distribution is constituted of two perfectly equal income distributions: one in which everyone has income 10 and another in which everyone has income 50. Now, suppose that we start off with the entire population in the former income distribution, and that population gradually shifts to the latter over time. The income distribution will take, among others, the following three forms at various stages of this simple process: View the MathML source Turn MathJax off Thinking about how inequality evolves as the income distribution changes from A to B and from B to C obviously means thinking about how inequality judgements are influenced by the relative population sizes of the “rich” and “poor.” For this reason, this simple case of the gradual population shift process has been considered of importance for the theoretical question of how inequality comparisons ought to be made in the first place. It has been studied in this way by Fields, 1987 and Fields, 1993, among others. The key to tackling the question of how inequality evolves during a gradual population shift lies in the behaviour of inequality measures with respect to mixing income distributions. Let us first explain what we mean by mixing income distributions. Assuming that income distributions are defined in terms of relative frequencies, each income distribution can be defined as a mixture, i.e., a convex combination, of its constituent income distributions. As an illustration, consider a country with two regions: “region P” and “region Q,” representing population shares of α and 1 − α, respectively. Indeed, if px and qx are the proportions of the population with income x in regions P and Q, respectively, then the proportion of the population with income x in the country is equal to αpx + (1 − α)qx. Now, during a gradual population shift process, the income distribution at each stage is a mixture of the income distribution at any earlier stage and the income distribution at any later stage—as an illustration, note that income distribution B in the example above of the simple case of the process, is a fifty-fifty mixture of income distributions A and C. In order to describe the evolution of inequality during a gradual population shift process, the important question is whether income inequality in a mixture is greater than, smaller than, or equal to, income inequality in each of its constituent income distributions. Moreover, can a general answer even be given to this question, or does the answer depend on the specifics of the constituent income distributions and on the particular inequality measure that is used? In this article, we show that a general answer can indeed be given to the question of how inequality measures behave with respect to mixing income distributions. It is demonstrated that virtually all inequality measures that are studied in the literature on inequality measurement—viz., the class of decomposable inequality measures and the class of normative inequality measures based on the general social welfare function of the rank-dependent expected utility form—satisfy quasi-concavity properties, which say, loosely speaking, that mixing income distributions tends to increase inequality. For instance, the properties imply the following for the case where inequality is equal in the two constituent income distributions that are mixed: inequality in the mixture is at least as great as that in each of its constituent income distributions, and if the mean incomes of the constituent income distributions are not equal, then inequality in the mixture is strictly greater than that in each of its constituent income distributions. We emphasise that while all well known inequality measures satisfy these quasi-concavity properties, the properties are not implied by the fundamental Lorenz type axioms on their own. With respect to the problem of how inequality evolves during a gradual population shift process, the quasi-concavity properties are shown to reduce the possible patterns describing the evolution of inequality to only three: (i) an increasing pattern in which inequality increases during the entire process, (ii) a decreasing pattern in which inequality decreases during the entire process, and (iii) an inverted-U pattern in which inequality increases in the first stages of the process and decreases afterwards. This result generalises some results of Kakwani (1988) and Anand and Kanbur (1993) in the same context. The article is structured as follows. Section 2 deals with notation and basic concepts. In Section 3, we show axiomatically that the quasi-concavity properties are satisfied by all inequality quasi-orderings satisfying the transfer principle, a weak invariance axiom, and decomposability. Instead of focusing exclusively on relative inequality concepts, as is common in the literature, we consider the weak invariance axiom of Bossert and Pfingsten (1990) that allows for relative and absolute inequality concepts as well as intermediate ones. While the result of Section 3 applies to, among others, the inequality measures based on a social welfare function of the expected utility form, it does not apply to its rank-based alternatives, the generalised Gini indices, as these are not decomposable. Therefore, we consider in Section 4 the class of inequality measures (absolute, relative as well as intermediate cases) based on a social welfare function of the rank-dependent expected utility form, which generalises both the class of expected utility inequality measures and the class of generalised Gini indices. Benefiting from functional representability of the given inequality orderings, it is shown that the quasi-concavity properties are also satisfied by all members of this general class of normative inequality measures. In Section 5 we spell out the implications of the results of 3 and 4 for the question of how inequality evolves during a gradual population shift process. Section 6 concludes. All the proofs are contained in an Appendix A.

The literature on inequality measurement has focused exclusively on the specific strategy of supplementing the fundamental axioms, TP and βINV, with decomposability ideas, i.e., ideas concerning how changes in the inequality of constituent income distributions have to relate to changes in overall inequality—directly, in the form of the DEC axiom, or, indirectly, by basing inequality measures on an RDEU social welfare function, which incorporates a weak decomposability condition. It was demonstrated in this article that all inequality measures considered in the literature satisfy the quasi-concavity properties QC and CSQC. Moreover, it was shown that the latter property allows only three patterns describing how inequality evolves during a process in which population gradually shifts from one constituent income distribution to another. On the one hand, the latter result reveals an attractive feature of CSQC: the property facilitates the study of empirical phenomena in which gradual population shifts occur. On the other hand, it may be argued that the three patterns allowed by CSQC are not the only plausible ones. If it is concluded that the other—non CSQC consistent—inequality views should also be expressible within a theory of inequality measurement, then our results show that one should focus on supplementing the fundamental axioms, TP and βINV, in alternative ways, rather than with decomposability ideas.