Introduction

Lorenz curves and the corresponding Gini coefficients, widely used measures of statistical variability, inform a large body of literature that focuses typically on inequalities in household or individual income. These measurement techniques, however, are highly generalizable: For instance, they have been used to assess variability in the intensity of migratory streams (Plane and Mulligan, 1997), variability in educational attainments (Hicks, 1997), and variability in life expectancies (Hicks, 1997). In the present study we emphasize ways of conceptualizing and measuring trends and differentials in income distributions, with a view toward strengthening research on determinants and consequences of such income inequalities as they occur within spatial or ``ecological" aggregates of households.

Regarding trends and differentials in income distributions, Bishop et al. (1997) and Braun (1995) conclude that the U.S. economy as a whole has experienced increased inequality over the past few decades. In a study of trends in income inequality by state, Langer (1999:63-64) finds that while several large states experienced steadily increasing inequality from 1949 to 1989, others changed direction (e.g., Virginia) or even followed ``cyclical" patterns (e.g., Texas). Addressing the relationship between income inequality and economic development (Kuznets, 1955) and presenting data by county, Nielsen and Alderson (1997) discern a ``Kuznetsian pattern" of declining inequality associated with economic development, while concluding that this pattern has been followed more recently by an upswing in inequality.

Regarding determinants of income inequality, Bishop et al. (1997) and Levernier (1999:164) suggest that, ceteris paribus, racial composition has little impact on Gini income inequality and that age effects have declined over time. However, large numbers of female-headed households and variations in educational composition seem to have a large impact in increasing inequality. Similar results are reported by Nielsen and Alderson (1997), who conclude that the recent increase in Gini inequality has been due, in large part, to increasing proportions of female-headed households. Along with Levernier (1999), Nielsen and Alderson (1997) do not find strong evidence of an impact of unemployment on income inequality.

We know of relatively few studies that attempt to assess the consequences of Gini inequality for social behavior. However, many studies have used comparable measures of inequality, to be discussed in a moment. These studies often focus on ``pathological" behaviors, e.g., crime or serious family disorganization. Braun (1995) and Hsieh and Pugh (1993) carried out meta-analyses of a large number of such studies, concluding that poverty and income inequality are both associated with violent crime. Hsieh and Pugh report that among many positive coefficients examined, 80 per cent were at least of moderate strength. Wilson's findings suggest that poverty concentration, net of poverty per se and other factors, may have an impact on family disorganization. However, studies of this relationship usually employ limited measures of inequality---e.g., differences in family income by sex of household heads---and they typically assess the effects of divorce on the post-divorce household income of men and women studied comparatively (Peterson, 1996; Weitzman, 1996). We know of no studies that follow up systematically on Wilson's suggestion that this causal presumption be reversed, and that Gini concentration of poverty be assessed as to its impact on family disorganization.

In examining the literature on inequality and its correlates, we often find that a given investigator eschews highly generalizable indices such as the Gini coefficient in favor of much simpler measures based on, say, income differences among a few nominal categories. Usually, as in several studies cited above, income differences by sex/gender or by race/ethnicity are taken to be either a major independent or dependent variable. This practice places researchers in a situation in which, with limited justification, they are taking the relationship between, say, income and race, to be a key variable. It is arguable that a more appropriate and logical procedure would involve a more general univariate measure, such as the Gini coefficient, that embodies all forms of variability regardless of their determinants or consequences and also has utility in cross-cultural and historical studies due to its ``scale invariability" and similar properties, to be discussed below. Having found substantial variability in some sort of inequality, we would then seek specific mechanisms (Bunge, 1997; Bennett and Lynch, 1997), such as race/ethnicity or sex/gender, that would account for such variability. Finally, one should also be attuned to the possibility that if, for instance, family disorganization is both cause and consequence of Gini inequality---a hypothesis strongly implied by the literature---then we may be in the presence of strong positive feedback between these two variables as they move through time. Positive feedback appears to be an inescapable dimension of the ``growing-tangle-of-pathology" metaphor. Tangles sustain themselves and get worse through positive feedback.

Paradox

Wilson (1987:111) employs various statistical techniques, including Gini coefficients, to describe and explain the recent historical development of socioeconomic (SES) inequality within the black American population. He describes contemporary patterns of poverty by saying that ``... low-income families and individuals are, in several important respects, more socially and economically isolated than before the great civil rights victories, particularly in terms of high joblessness and the related problems of poverty, family instability, and welfare dependency." He attributes these changes to the growing income differences, within the black population, between poor families and families of higher socioeconomic status. In so doing, he cites data of the sort used in constructing Lorenz curves (Wilson, 1987: Table 5.1), and he points out that the corresponding Gini coefficients reveal ``... that income inequality is greater and has increased at a faster rate among black families than among white families from 1966 to 1981." One must be aware, however, that Wilson's major point pertains not only to a growing disparity between poor and wealthier minority families. Rather, he develops a definitive hypothesis that it is the social and economic isolation brought about by poverty concentration that is crucial---and concentration, within the neighborhoods where it occurs, implies low Gini variability. In statistical terms, Wilson implies that income variance between poor and relatively wealthy black families has grown much larger, while income variance within poor black neighborhoods has grown much smaller. As we move forward a few pages (Wilson, 1987:137-38) it becomes clear that, in Wilson's view, it is the uniformity of poverty within minority ghettoes that is a major cause of social disorganization and increasing ``aberrant behavior."

Straight Lines and Curves

The following procedures, employing a computer mathematical system, plot several Lorenz curves, obtain several Gini coefficients, show how these techniques may be misleading under the circumstances described by Wilson, and suggest a modified technique for assessing distributions of SES likely to generate what Wilson calls ``concentration effects."

In carrying out the Lorenz-Gini procedures, in effect, we place all income recipients in rank order from poorest to richest---left to right on a horizontal axis. Then, moving left to right, we plot on a vertical axis the income received by each unit, summed along with the income of all ``poorer" units to the left. We thus obtain, for any given place in the rank order, the cumulative wealth received by the first n income recipients, i.e., cumulative wealth from the lowest position to the current position on the horizontal axis. If there is inequality, then each income recipient, as we move from left to right, will have a higher income than anybody (or any household, organization, etc.) to his/her/its left in the lowest-to-highest rank order.

Not so for the hypothetical case of New Harmony, Indiana, circa 1850, with its presumably complete equality: Here, income recipients may appear in any random order, and the proportion of total wealth allocated to any fixed-length segment of a given ordering would always be uniform. The rate of change for the vertical axis, then, would remain fixed. Therefore, complete income equality appears as a straight line in a Lorenz diagram, and the corresponding Gini coefficient would be zero. When inequality does exist, the Lorenz curve bends away from the complete-equality line: The greater the bend, the higher the inequality and the higher the Gini coefficient.

We measure inequality, again, by moving along the ranking of unequals from left to right (i.e., poorest to richest). We find that for any given proportion of those ranked, as we move toward the rich end of the distribution we capture an increasingly large proportion of available wealth. If the size of these income segments accelerates rapidly as we move rightward along the horizontal axis, there is a relatively large disparity between the straight line that represents equality and the high-acceleration curve representing a given (unequal) distribution. Again, the greater the bend in this curve, the greater its departure from the equality line, and the higher will be its Gini coefficient. The Gini coefficient, then, measures the discrepancy between the equality line and the Lorenz curve, and then divides this discrepancy by its maximum---the total area through which the Lorenz curve would move in a situation of total inequality, where one member of the population possessed all wealth. The Gini coefficient varies between zero and a value very close to unity (Allison, 1978:869-70).

Let us, then, set up a complete equality function,

Specious Equality

The above procedure is very parsimonious; however, it does not entirely capture Wilson's argument. A major thesis of the Wilson book holds that poverty within the lower SES levels of the black population has become highly homogeneous, with residents of poor neighborhoods experiencing highly concentrated, uniform poverty that creates concentration effects, i.e., a range of attitudes, beliefs, and behaviors that express a sense of hopelessness and help to sustain a tangle of pathology. If such a pattern exists, what this means in Lorenz/Gini terms is that within the lower strata of the black population there is, ironically, substantial equality. In other words---and here is the real paradox---a contemporary Lorenz curve for poor central-city black households should look a lot like a historical Lorenz curve for New Harmony.

Let's draw a straight-line representing income homogeneity over a portion of the income distribution, place it on our plot, and see what we can do with it (Figure 2). It will apply only to the poorest segment of the population, and it will therefore allocate among the poor a relatively small segment of total economic wealth. Assuming, as a starting point [note 1], that the poorest 50 per cent of the central-city black population is likely to exemplify Wilson's concentration effects, we extend a straight line across our plot from the origin to the point ( , ) representing the proportion of family income received by the poorest 50 per cent of the population. This line will graph an equation of the form . We know that the -intercept is zero and that , so that the slope of the line, b, is obtained by solving

Controlling Central Tendency

Although Gini coefficients tell us about the degree of income variability in neighborhoods surrounding a given household, they do not tell us anything about central tendency. The assumption of this study, almost certainly shared by Wilson, is that ``concentration effects" occur in neighborhood contexts in which there is both a high poverty rate and high uniformity of poverty. In research designed to assess Lorenz-Gini variability of poverty and its causes and effects, low-income households should be selected in a way that ensures that the level of living of surrounding neighborhoods is also low. Many social surveys provide such selection criteria, e.g., economic characteristics of the census tract in which a given household is located (Cohen, 1998). A poor household located in a poor neighborhood that is characterized by high uniformity of poverty is precisely the sort of situation that, in Wilson's view, gives rise to ``pathological" behavior. It is local-level Gini coefficients, compared against similar distributions for larger surrounding communities, that capture the essential features of Wilson's argument.

Resolution

We draw a main diagonal, representing complete equality, as before:

> y[4] := x;

An Index of Concentration

We propose a measure, called the index of concentrated poverty ( ), in which the coefficient will be divided by a standard Gini coefficient, called , pertaining to the truly disadvantaged segment of the population, i.e.,

Applications

If the ---or a measure comparable to it---were obtained for a large number of American central cities, it is possible that it would have predictive and explanatory power in relation to Wilson's hypothesized concentration effects. When we examine Wilson's argument we realize that it invokes a range of social and socio-psychological mechanisms, such as extreme social isolation, thought to have a role in sustaining the terrible tangle of pathology that, in Wilson's view, is growing ever more severe (Wilson, 1987:21-29,46-62) within the ghettoes. A measure such as the asks us to contemplate not only what it is like to be cast haplessly and hopelessly upon an undifferentiated ocean of misery, what it is like to live in a place uniformly poor; it also asks us to contemplate the ``reference group" impact (Merton, 1957:225-386) of the victims' ability to discern large and lovely islands of glittering prosperity, barely visible at distances that would truly test one's ability ever to traverse them, while the nearby spatial environment provides almost nothing of comparable allure. The several studies cited in the introduction to this article lend considerable support to the thesis that, in general, concentrated poverty creates invidious comparisons, especially across class, race/ethnic, and sex/gender boundaries, with resultant conflict and pathological behaviors. As noted, however, these studies often have limited concepts of poverty concentration and, as in the case of research on family disorganization, they often fail to ascertain causal direction.

The tangle of pathology is a long-lived, widely-cited, fascinating and compelling metaphor that might well induce us to ask whether the same conditions that, in Wilson's analysis, influence it might also influence the vital processes, including fertility and, especially, the ultimate pathologies of mortality and associated morbidity (Hicks, 1997). The high and perhaps increasing excess mortality occurring within the lower strata of American society (Kitagawa and Hauser, 1973:25; Pappas et al., 1993; Duleep, 1995a, 1995b) seems to arise largely from ``exogenous" causes (Stockwell et al., 1995)---causes that reflect environmental conditions---and it is likely that these causal factors would include the same sorts of social and psychological mechanisms that sustain the infamous tangle of social pathology. In brief, strong evidence suggests that low levels of SES per se produce excessive levels of mortality, and perhaps one should ask whether the extremely low dispersion---the low Gini coefficients---around such low averages might enhance conditions that make for excess mortality and morbidity.[note 5] Concentrated poverty, for instance, is likely to suppress access to information about health maintenance, it is likely to suppress access to health-care facilities not solely because of their high costs but also because of their virtual absence, and it is likely to suppress the development of community self-help organizations that may have an impact on health and mortality. Further hypotheses abound, and a search for specific mechanisms (Bunge, 1997) should be undertaken.

Conclusions

The ``methodological" proposal of this paper was inspired largely by the theoretical writings of Wilson (1987). The proposed index seems to capture important dimensions of the socio-economic dynamics of poverty, while also meeting several ``basic criteria for measures of inequality" (Allison, 1978:866)---criteria that tend to eliminate measures such as those based on the variance. The highly generalizable Lorenz/Gini apparatus encourages researchers to think in terms of the most widely applicable sorts of theories: Reference-group comparisons, for instance, may occur across class, sex/gender, or race/ethnic lines, and may involve other criteria not clearly discerned by past studies, e.g., employment in core vs. peripheral industries. The Lorenz/Gini procedures, because of scale invariability and other such advantages, also facilitate comparative and historical research, and it is highly likely that studies investigating, say, time-series aspects of the interaction of family disorganization and Gini inequality, would arrive at sophisticated feedback hypotheses in this realm, hypotheses comparable to those suggested by Kuznets (1955) in a slightly different context. Finally, research dealing especially with mortality and morbidity suggests that concentrated poverty, as a causal agent, may involve causal clusters or configurations, i.e., social situations with effects reaching well beyond the sum of their parts. Findings supportive of such a pattern would be amenable to interpretation based on chaos theory (Boyce and DiPrima, 1997:533-39).

Again, the assumption of this study, almost certainly shared by Wilson, is that ``concentration effects" occur in neighborhood contexts in which there is both a high poverty rate and high uniformity of poverty. Poor households located in a poor neighborhood characterized by high uniformity of poverty---this is clearly Wilson's image of conditions that give rise to pathological behavior. In this paper, we propose to measure these conditions with precision. In brief, we attempt to refine ways of conceptualizing and measuring trends and differentials in income distributions with a view toward strengthening research on determinants and consequences of income inequalities as they occur within ecological aggregates of households. In statistical terms, Wilson implies that increasing proportions of income variance between poor and relatively wealthy minority families, combined with reduced income variance within poor minority neighborhoods, create the powerful engine of increasing social disorganization and aberrant behavior.

References

Allison, Paul D. 1978. ``Measures of inequality." American Sociological Review 43:865-80.

Bishop, John A., John P. Formby, and W. James Smith. 1997. ``Demographic change and income inequality in the United States, 1976-1989." Southern Economic Journal 64: 34-44.

Blau, Judith R. and Peter M. 1982. ``The Cost of Inequality: Metropolitan Structure and Violent Crime." American Sociological Review 47: 114-129.

Boyce, William E. and Richard C. DiPrima. 1997. Elementary differential equations and boundary value problems. New York: Wiley.

Braun, Denny. 1995. ``Negative consequences to the rise of income inequality." Research in Politics and Society 5: 3-31.

Bunge, Mario. 1997. ``Mechanism and explanation." Philosophy of the social sciences 27:410-65.

Bureau of the Census. 1993. ``Money income of households, families, and persons in the United States: 1992." Current Population Reports, Series P60-184. Washington, D.C.: U.S. Government Printing Office.

Cohen, Philip N. 1998. ``Black concentration effects on black-white and gender inequality: Multilevel analysis for U.S. metropolitan areas." Social Forces 77:207-29.

Creedy, John. 1997. ``Evaluating alternative tax and transfer schemes with endogenous earnings." Oxford Economic Papers 49:43-56.

Duleep, Harriet O. January, 1995. ``Occupational experience and socioeconomic variations in mortality." Population Studies Center Discussion Paper Series, No. UI-PSC-20. Urban Institute, Population Studies Center: Washington, D.C.

Duleep, Harriet O. Summer, 1995. ``Mortality and income inequality among economically developed countries." Social Security Bulletin, Vol. 58, No. 2. Washington, D.C.

Hanneman, Robert A. 1980. ``Income Equality and Economic Development in Great Britain, Germany, and France: 1850 to 1970." Comparative Social Research 3: 175-184.

Hicks, Douglas A. 1997. ``The inequality-adjusted human development index: A constructive proposal." World Development 25:1283-98.

Hsieh, Ching-Chi and M. D. Pugh. 1993.``Poverty, income inequality, and violent crime: A meta-analysis of recent aggregate data studies." Criminal Justice Review: 18,182-202.

Kennedy, Bruce P. et al. 1998. ``Social capital, income inequality, and firearm violent crime." Social Science and Medicine 47: 7-17.

Kitagawa, Evelyn and Philip M. Hauser. 1973. Differential mortality in the United States: A study in socioeconomic epidemiology. Cambridge: Harvard University Press.

Kuznets, Simon. 1955. ``Economic growth and income inequality." American Economic Review 45:1-28.

Langer, Laura. 1999. ``Measuring income distribution across space and time in the American states." Social Science Quarterly 80:55-67.

Levernier, William. 1999. ``An analysis of family income inequality in metropolitan counties." Social Science Quarterly 80:154-65.

Merton, Robert K. 1957. Social theory and social structure. Glencoe: The Free Press.

Nielsen, Francois and Arthur S. Alderson. 1997. ``The Kuznets curve and the great u-turn: Income inequality in U.S. counties, 1970 to 1990." American Sociological Review 62:12-33.

Pappas, Gregory, Susan Queen, Wilbur Hadden, and Gail Fisher. July 8, 1993. ``The increasing disparity in mortality between socioeconomic groups in the United States, 1960 and 1986." New England Journal of Medicine, Vol. 329, No. 2.

Peterson, Richard R. 1996. ``A re-evaluation of the economic consequences of divorce." American Sociological Review 61:528-36.

Plane, David A. and Gordon F. Mulligan. 1997. ``Measuring spatial focusing in a migration system." Demography: 34:251-62.

SELFREF, 1998.

SELFREF, 1999a; 1999b.

Shihadeh, Edward S. and Darrell J. Steffensmeier. 1994. ``Economic inequality, family disruption, and urban black violence: Cities as units of stratification and social control." Social Forces 73: 729-751.

Smeeding, Timothy M. and Peter Gottschalk. 1998. ``Cross-national income inequality: How great is it and what can we learn from it?" Focus 19, Number 3:15-19. University of Wisconsin---Madison: Institute for Research on Poverty.

Stockwell, Edward G., Franklin W. Goza, and Jack L. Roach. 1995. ``The relationship between socioeconomic status and infant mortality in a metropolitan aggregate, 1989-91." Sociological Forum 10:297-308.

Weitzman, Lenore J. 1996. ``The economic consequences of divorce are still unequal: Comment on Peterson." American Sociological Review 61:537-38.

Wilson, William J. The truly disadvantaged. 1987. Chicago: University of Chicago Press.

Notes

(1) In this article we do not experiment with different ways of arriving at this cutting point. The cutting point should probably be determined empirically, in a manner that would involve extensive experimentation with the ICP measure to be introduced below.

(2) We resist the temptation to call this a mini-Gini.

(3) Recall the discussion above, in which we explain why multi-stage samples for this investigation would require initial selection of low-SES census tracts, or other comparable area units.

(4) In the software package used for this paper, all derivatives---not merely partial derivatives---contain Greek characters.

(5) At <http://csa.berkeley.edu>, it is possible to analyze data from the General Social Survey. In cross-tabulating respondent's subjective sense of quality of health against, say, educational achievement (degrees attained), one observes a j-shaped curve in which substantial excess morbidity appears primarily within the lowest educational strata.