Population and Sample Selection Effects in Measuring International Income Inequality*

A long tradition of theoretical work in the economics literature on economic growth predicts the convergence of poorer with richer economies, as a result of capital movements from areas of relatively high capital utilization to those where capital intensity is relatively lower. Th e work of neoclassical economists such as Samuelson (1948) and Solow (1956, 1957) was complemented by the work of heterodox theorists like Gerschenkron (1962), who emphasized the importance of technology transfer in contrast and complement to that of capital movement. Whatever the mechanisms, the new countries of the south were expected to grow rapidly in the decades following formal political independence. Empirically, however, this growth did not materialize, and a sociological critique emerged that capital transfers, far from promoting convergence in income between rich and poor countries, might actual promote and perpetuate divergence. Building on the theoretical work of Frank (1969), Emmanuel (1972), and Amin (1974), Chase-Dunn (1975) demonstrated empirically that international economic dependence—the fl ipside of capital movement—actually retarded


INTRODUCTION
A long tradition of theoretical work in the economics literature on economic growth predicts the convergence of poorer with richer economies, as a result of capital movements from areas of relatively high capital utilization to those where capital intensity is relatively lower.Th e work of neoclassical economists such as Samuelson (1948) and Solow (1956Solow ( , 1957) ) was complemented by the work of heterodox theorists like Gerschenkron (1962), who emphasized the importance of technology transfer in contrast and complement to that of capital movement.Whatever the mechanisms, the new countries of the south were expected to grow rapidly in the decades following formal political independence.
Empirically, however, this growth did not materialize, and a sociological critique emerged that capital transfers, far from promoting convergence in income between rich and poor countries, might actual promote and perpetuate divergence.Building on the theoretical work of Frank (1969), Emmanuel (1972), and Amin (1974), Chase-Dunn (1975) demonstrated empirically that international economic dependence-the fl ipside of capital movement-actually retarded 10 economic growth.Th e ensuing empirical debate on foreign capital dependence continues to this day, with recent contributions from Kentor (1998), Dixon andBoswell (1996a, 1996b), and Firebaugh (1992Firebaugh ( , 1996)).Only a hiatus in the production of new data seems to have slowed down the controversy.
Although rarely citing the sociology literature on dependence, economists have now largely abandoned the concept of convergence driven by exogenous capital fl ows.Th e "New Growth Th eory" of Romer (1994) and Lucas (1988), essentially an elaboration on Gerschenkron, postulates that convergence between low and high income countries will only occur where suffi cient human capital exists in the poorer countries to take advantage of the technological advances made in the richer countries.Empirical studies of "conditional convergence"convergence conditional on levels of educational attainment-have generally supported this view.Barro (1991), Barro and Lee (1993) and Barro and Sala-I-Martin (1995) fi nd that educational levels are signifi cantly related to growth rates, and that once education is controlled statistically, low income countries do grow faster than high income countries.When no controls are made, however, Barro and his colleagues fi nd no evidence of convergence, but instead fi nd a measurable (but statistically non-signifi cant) divergence between high and low income countries over the past few decades.
Meanwhile, within sociology a new debate has opened up on the measurement of international income inequality, sparked by Firebaugh' s (1999) critique and elaboration on Korzeniewicz and Moran (1997).In a wide-ranging article on trends in world income inequality and their sources, Korzeniewicz and Moran (1997) found that increasing between-country inequality was the main source of an overall increase in inequality over the period .Th eir approach is based on the use of World Bank GNP per capita fi gures evaluated in U.S. Dollars at offi cial exchange rates.In reaction, Firebaugh (1999) claimed that between-country inequality had actually been stable over the period 1965-1992, arguing that the earlier observed increase in inequality was the result of an inappropriate choice of data series.Using Penn World Table GDP data evaluated at purchasing power parity, Firebaugh described a "plateau" of inequality, and attempted to demonstrate that this plateau was not sensitive to the results for any particular country.Korzeniewicz and Moran (2000) presented a comment on Firebaugh' s (1999) article, which was published along with a reply by Firebaugh (2000).As they emphasized in their respective comments, the key scientifi c issue separating Firebaugh from Korzeniewicz and Moran is one of operationalization, although they also diff er normatively.Both camps fi nd that reported levels of inequality and changes in levels of inequality over time are highly sensitive to the specifi c measures of inequality chosen to be studied.Both agree with the other camp' s empirical results, once the choice of operationalization has been determined.Both camps argue that the measures chosen should represent theoretically grounded operationalizations of the authors' idiosyncratic understandings of the (ambiguous) term "inequality."What both camps fail to see, however, is that the data does not in fact diff erentiate between their views.Instead, I suggest in this paper that much of the controversy surrounding international income inequality stems from a failure to understand the theoretical implications of chosen operationalizations of the concept of inequality.
Firebaugh' s own avowed aim was "to provide the foundation for a general sociological literature on intercountry income inequality" (p.1598).Accepting this ambition at face value, I will organize my own contributions in this article around Firebaugh' s agenda.Firebaugh broke new ground on several fronts: through methodological advances in the handling of exchange rates, through his ANOVA metaphor for studying cross-country inequality, and, most importantly, through his specifi cation of a demographic theory of world income convergence.In formulating my ideas, I will examine each of these advances in detail.While the fi rst two areas are mainly of interest to specialists in the fi eld, the third is of general importance for all social scientists and policy-makers working in the global arena, and will receive special attention.

NATIONAL CONTRIBUTIONS TO INTERNATIONAL INEQUALITY
Several measures of cross-national income inequality coexist in the literature.Firebaugh incorporates four of them in his article: "V 2 ," "Th eil," "VarLog," and "Gini."A "key fi nding" (p.1608) of Firebaugh' s article is that all four of these indices can be represented in general terms by a single computational formula: each index, Firebaugh argues, can be expressed as the sum over all countries of each individual country' s world population share times a function of its income ratio (the ratio of its income per capita to the average world income per capita): where I = inequality index j = country label p j = country j's proportion of world population r j = country j's proportion of world income f = some functional form used to measure inequality (p.1608, eq. 2) Population and Sample Selection Effects 12 try in the study (through their eff ect on the mean), Firebaugh' s decomposition would be correct, and a major result, as Firebaugh claims.As it stands, world income inequality cannot be decomposed as Firebaugh suggests.
Th is in itself would not create a major problem for the paper-although it would invalidate the analyses of country contributions to inequality presented on pp.1617-1619 (which I have, in any case, been unable to replicate).Th e real problem comes when population weights are introduced.Firebaugh' s equation 2 implies that population weights can be applied to the already-computed income ratios.While this is possible computationally, it is a highly idiosyncratic way to conduct weighted analyses.Typically, in weighted analyses, weights are applied to each case according to some weighting variable (in this case, population).Each case is then essentially entered into the analyses as many times as its weighting indicates-i.e., if country A has ten times the population of country B, country A is entered ten times for every entry of country B. Th is would mean that the population weights would be used fi rst to compute a weighted mean world income level, which would be used in calculating the national income ratios.Only then would the income ratios, based on the weighted mean income, be entered into the analysis.If Firebaugh' s equations 2 and 3, as they stand, were used to calculate "weighted" inequality measures, the results would be truly odd-in the case of VarLog, weighted squared deviances from the unweighted mean of the logged income ratios.
Luckily, Firebaugh does not actually apply equations 2 and 3 in computing his own weighted results in table 2 of his article (p. 1606).Using Firebaugh' s data, I have confi rmed computationally that in fact he uses conventional weighted analysis to arrive at the fi gures in his Table 2, not the methods detailed in his text.His (seeming) application of equations 2 and 3, however, in investigating the contributions to world inequality of individual countries have led him to ignore some important aspects of the data.I have not been able to replicate his results on the contributions of individual countries.

DATA PROBLEMS AND THE EFFECT OF CHINA ON WORLD INEQUALITY
In the study that kicked off the current debate, Korzeniewicz and Moran (1997) used GNP per capita fi gures reported by the World Bank.Th ese fi gures are based on U.N. System of National Accounts (SNA) data generated and published by individual countries and compiled by the World Bank, which converts the local currency fi gures reported by all countries into U.S. Dollars.Th e SNA fi gures are on aggregate value-added estimates on the production side, computed bottom-up from surveyed prices and reported quantities of goods produced.Th e resulting aggregate fi gures are converted into U.S. Dollars at offi cial exchange rates.
The function f is particular to the inequality measure used: V 2 :f(r j ) = (r j -1) 2 Theil:f(r j ) = r j log(r j ) VarLog:f(r j ) = Var[log(r j )] Gini:f(r j ) = r j (q j -Q j ), where q j is the proportion of units poorer than j and Q j is the proportion of units richer than j (p. 1608, eq. 3) Formulating inequality in this way, Firebaugh traces each country' s "contribution" to world income inequality simply by multiplying its own population share by a function of its income ratio.If this were correct, it surely would represent a "key finding," as Firebaugh claims.
Th e reality, however, is more complicated.Leave aside the fact that the Gini coeffi cient obviously cannot be computed from r j alone as Firebaugh implicitly claims.Consider the simplest and mathematically most elegant measure of income inequality: the variance of the logs of national income per capita (VarLog).Th is measure has the advantage of being invariant under conditions of balanced economic growth: when all countries' income per capita grow by the same percentage, the variance of the logs of income per capita does not change.
(Th e simple variance of income per capita would rise with economic growth.)Computing the unweighted variance of the logs is straightforward-one simply logs the national income per capita fi gures, and then takes the variance of the logged fi gures.Firebaugh has done this for 120 countries for the years 1960-1989 in his table 2. (p. 1606) Is each country' s "contribution" to unweighted income inequality in this case just its squared deviation from the mean, as Firebaugh would imply?One could make this argument, but it is not unambiguously correct.Each country contributes not just to the fi nal variance score (which in this case is just the total squared deviation from the mean logged income per capita), but to the mean itself as well.When a country is taken out of the analysis, the new resulting VarLog measure does not perfectly correspond to the old measure minus the "contribution" of the excluded country, since by taking the country out of the analysis, the mean (around which the variance is calculated) changes as well.
Why didn't Firebaugh notice this?In his notation for equations 2 and 3, he creates the notation "r j " to stand for the ratio of a country' s income to the mean world income.Th e subscript "j," referring to the country, is here misleading, since r j is not just a descriptor of a particular country, but also of the world as a whole, through the denominator of the expression.However, using r j to stand for the ratio in subsequent analyses tends to lead the reader to think that each country' s income ratio depends only on that country' s data.If this were true, and a country' s income ratio did not depend on the national incomes of every other coun-Population and Sample Selection Effects 14 Firebaugh' s work is based on data from the Penn World Table, Mark 5.6a (Summers et al 1994), henceforth PWT.Th e Penn World Table (PWT) is a set of purchasing power parity (PPP) based estimates of income per capita fi gures for over 140 nations, built around the benchmarks established by the United Nations' International Comparisons Program.(See Summers and Heston 1991 for a summary of the PWT methodology; see Ahmad 1994 for a description of the UN ICP.)In contrast to the SNA method, PWT real gross domestic product is estimated in a top-down fashion, beginning from fi nal consumption and working towards implied output.Th ey take as their base the fi nal SNA consumption fi gures, but do not depend on a host of other SNA intermediate assumptions.Output fi gures in the PWT are ultimately computed at international, rather than domestic, prices.
Th e PWT methodology is only well-suited to environments in which prices are determined in an open market through the interaction of consumer' s individual demand functions.Given its consumption-side approach, at the heart of the PWT methodology is the assumption that fi nal consumer prices refl ect in some meaningful way the producer prices of inputs to production processes.In the absence of free consumer markets, or worse in the presence of government manipulated producer markets, this methodology breaks down.In general, the PWT methodology is not well-suited to estimating GDP per capita in planned economies, and Summers et al recognize this by the inclusion in the table of a precautionary planned economy dummy variable.For example, Good and Ma (1999) show that the Penn World Table signifi cantly overstates gross domestic product (GDP) per capita growth rates for eastern Europe and the Soviet Union in the post-war period.
In his country-by-country sensitivity analyses (p.1617-1619), Firebaugh seems to fi nd relatively small contributions by specifi c countries to changes in world income inequality (the lack of units in Table 5 makes the results diffi cult to interpret more specifi cally).Th is would lead one to conclude that his major results do not depend heavily on the contribution of any one observation.However, as I argued above, his methodology for isolating individual country eff ects is fatally fl awed.On account of its huge population, any change in the position of China in the scale of world incomes does in fact have a huge impact on weighted between-country inequality.Th is point is especially relevant given China' s high rate of economic growth in the 1970s and 1980s, as reported in the PWT.
Taking a much more straightforward approach than Firebaugh' s, I fi nd that when China (the world population leader) is simply excluded from the study of weighted income inequality, the percent change from 1960-1989 in the VarLog measure jumps from +5 to +24.In other words, with China in the analysis (n=120), world income inequality is found to have increased 5 in the period, while without China in the analysis (n=119), inequality is found to have increased 24.Is the China eff ect real, or a methodological artifact arising from the application of the PWT methodology in an inappropriate environment?Th e online appendix to the PWT documentation includes the warning that "estimates for China and the large size of their diff erence from the exchange rate suggest the great uncertainty attached to these numbers."(Summers et al 1994, Appendix) In a section that runs over 3400 words (in an unpaginated computer fi le), Summers et al detail the problems with and uncertainties surrounding the Chinese fi gures.Given the problems of applying the PWT methodology in planned economies, this China eff ect is suspicious.
China is only one of fi ve east-bloc non-market economies included in the 120 country subset of the Penn World Table for which income and population data are available for 1960-1989 (the others are the USSR, Romania, Yugoslavia, and Czechoslovakia).Until better data is available for all of these countries, it would seem premature to come to defi nitive conclusions about world income inequality.At the present time, prudence suggests that our analyses be restricted to changes in international income inequality among market economies only.

INEQUALITY AMONG MARKET ECONOMIES
In addition to data concerns, there are strong theoretical arguments for excluding the fi ve east-bloc countries based on the nature of their economic systems.Th e non-market economies made concerted attempts to exclude themselves from the western world-economy, insulating themselves from the very world-systemic forces that increased world income inequality in the period under study.Although Wallerstein (1979:100) explicitly considers the east bloc to have been integrated into the modern world-economy, other scholars (notably Szymanski 1983) have argued that the socialist countries formed a distinct, competing world-system.If this is the case, it makes theoretical sense to exclude them from measures of inequality that are designed to measure the inequalityinducing eff ects of the modern world-system.
If, as Emmanuel (1972) and others have argued, the exploitation of the periphery of the world-economy is accomplished through unequal exchange, it stands to reason that higher levels of international trade could be associated with higher levels of exploitation.Chase-Dunn, Kawano, and Brewer (2000) have observed that global trade has increased dramatically since 1945, accelerating especially after 1960.It would be interesting to observe whether or not this increase in global trade has been paralleled by an increase in inequality among When I replicate Firebaugh' s analyses with the fi ve east-bloc counties in the sample excluded, I arrive at very diff erent results from those reported in Firebaugh' s Table 2 (p. 1606).My weighted and unweighted results for world income inequality, using the VarLog measure, appear in Table 1.In the fi rst two columns, I replicate Firebaugh' s (n=120) results, while in the last two columns, I report the results for market economies only, with the fi ve east-bloc economies are excluded (n=115).
Th e diff erences between my and Firebaugh' s results are not so striking as when China alone is excluded, but the reported increase in world income inequality is still three times that reported by Firebaugh (+15 versus +5).Th e pattern is similar to that reported by Firebaugh, however, with inequality increasing sharply between 1960 and 1965, then leveling out or dropping after 1965.Th e discontinuity in both series between 1960 and 1965 is somewhat disturbing.Are the results for 1960 aberrantly low?Or are those for 1965 aberrantly high?If the former, we should exclude 1960, begin our analyses in 1965, and report relatively stable levels of inequality, whether or not planned economies are included in the analyses.If the latter, we should begin our analyses in 1960, and report dramatically increasing inequality over the period.If we had data reaching back to, say, 1950, it would be easy to tell whether 1960 or 1965, or neither, stuck out of trend as an aberrant point.
Unfortunately for this solution, many of the countries in these analyses did not exist in 1950, and we may never have an internationally-comparable set of GDP per capita estimates for the colonial administrative areas corresponding to the independent countries studied here.Th e Penn World Table does, however, contain data for the entire period 1950-1989 for n=59 countries, and results for these are reported in Table 2.Note that none of these 59 countries had planned economies.Th e results seem to confi rm the admissibility of the fi gures for 1960; that year appears to have been the end point of a period of stable inequality that extended back at least as far as 1950.Between 1960 and 1965 there does appear to have been a jump in inequality, but a real jump, not a spurious jump arising from random error.
Another way to approach this problem would to examine a corresponding income per capita time series derived from foreign currency exchange rates rather than purchasing power parity.Th e World Bank' s (1999) World Development Indicators include such a series for gross national product (GNP) per capita beginning in 1960.If analyses based on the foreign exchange (FX) series also contain a discontinuity between 1960 and 1965, it would cast further doubt on the reliability of the 1960 estimate.If, on the other hand, the results derived from the FX series show a smooth trend from 1960 forward, it would suggest that perhaps it is the 1965 fi gure that is out of line.Results based on World Bank data  Population and Sample Selection Effects 18 for all available countries (n=91) and for market economies only (n=89; China and Hungary excluded) are presented in Table 3.
As with the PPP results, our interpretation of the trend in international inequality as reported in Table 3 depends on whether we choose to include China in the analysis (Hungary has only a minor impact on the weighted fi gures).If, as I argue above, we concentrate on the results for market economies only, there appears to be a smooth increasing trend in inequality.Th is would suggest that it is the 1965 fi gures that are spurious in Table 1.On the other hand, those including the planned economies would reach the opposite conclusion.It should be noted, however, that FX-derived income fi gures are even more severely fl awed for planned economies than are PPP fi gures.Th e exchange rates used by the World Bank are the offi cial rates reported by the respective governments; in the case of planned economies, these rates are set by the government, with no regard to market conditions, since currency exchanges occur only with government permission.
Since both the restricted-n, 1950-1989 PPP results for market economies and the 1960-1989 FX results for market economies lend support to the thesis that the (relatively) low inequality score for 1960 is a valid observation, it seems safe to conclude that international inequality did, in fact, increase over the period.Moreover, the high consistency of the estimated growth in weighted inequality among market economies in each of the three tables above (15, 17, 21; see Table 4) shows that these results are somewhat robust with respect to period, sampling, and method of analysis.Th is is in marked contrast to the wild swings found both in the unweighted analyses and in the analyses conducted with planned economies included in the sample.In sum, it seems safe to use population weights, as Firebaugh advocates, so long as planned economies, China in particular, are excluded.Whether it will be possible to include China in weighted analyses in the future, as its economy becomes more open, is unknown.Given the sheer size of China' s population, it is imperative that we look for ways to properly handle the Chinese case.However, until such time as we can reliably include China in our analyses, a conservative estimate for the increase in inequality in the three decades after 1960 seems to be around 15-20.

ASSESING FIREBAUGH'S CONTRIBUTIONS
As the foregoing arguments demonstrate, Firebaugh' s article does in fact mark a critical juncture in the study of international inequality.First, Firebaugh makes a strong case for the use of purchasing power parity exchange rates in computing inequality.Although in the end the choice of exchange rate seems to have little eff ect on measured changes in inequality, the practical arguments in favor of purchasing power parity are strong.Scholars in the world-system tradition have tended to use market exchange rates in making international comparisons, since their objective has been to measure international purchasing power, not domestic purchasing power.Th e root problem with this approach is not its emphasis on international versus domestic prices, but the fact that true, freemarket exchange rates exist for only a handful of countries for the periods under study.In any case, using purchasing power parity measures, unweighted world inequality has increased dramatically since 1960, an increase almost twice that found using "market" exchange rates.Firebaugh' s advocacy of purchasing power parity represents an advance in the fi eld that should be endorsed by world-system scholars and their critics alike.N/A = Not Applicable Source: Tables 1-3 above.

Note:
Population and Sample Selection Effects 20 Firebaugh' s second major contribution, the promulgation of an "ANOVA" model for studying inequality, is prefi gured in Korzeniewicz and Moran (1997) and mirrored in recent work by Alderson and Nielsen (1999).Firebaugh uses this model to make a strong argument for the use of population-weighted measures.Although changes in unweighted inequality may be easier to interpret, Firebaugh' s model suggests that weighted inequality corresponds more closely to what sociologists mean by the concept.However, taking Firebaugh' s principles one step further, I have argued that inequality be studied using only market economies.Th ere are strong theoretical reasons to do so.Sociological theories of the creation and maintenance of international inequality are largely grounded in concepts like investment dependence (Alderson and Nielsen 1999) and economic diff erentiation (Gustafsson and Johansson 1999).By largely removing themselves from the world-economy, the planned economies insulated themselves from these eff ects.It is perhaps telling that the income levels of the formerly planned economies of the east bloc dropped precipitously upon integration with the world-economy.Th e strongest arguments for restricting analyses to market economies may, however, be simply pragmatic.Good data just isn't available for the planned economies, and, until it is, their income trends relative to the rest of the world must remain a mystery.
Th e major contribution of Firebaugh' s article, however, is the graphic demonstration that demographics matter.Even if Firebaugh may be premature in claiming to have discovered a "great plateau" (p.1622) in world income inequality in the period 1960-1989, he is right to stress the importance of demographic eff ects on inequality.It is in this respect that his work is seminal.

DEMOGRAPHY AND INEQUALITY
Since national incomes are generally compared on a per capita basis, population is an important component of all international inequality measures.Th e impact of population issues is compounded when income data is weighted by population.I will discuss the population-GDP per capita nexus briefl y below; it is too complicated to be treated fully here.I will, however, attempt to isolate some of the eff ects on measured inequality of changes in population acting through population weights in inequality measures.
Ceteris paribus, an increase (decrease) in population will yield a decrease (increase) in GDP per capita.Th ere are at least two mechanisms through which a change in population would not be expected to yield a corresponding change in GDP.First, the change in total population may be associated with a change in the age structure of the population.For example, an increase in the birth rate will lead to a higher proportion of (non-working) children in the population.
In this case, population may rise with no corresponding rise in GDP, yielding a decline in GDP per capita.Second, population growth may lower the marginal productivity of labor.Labor is only one of many inputs in economic production.If population grows more quickly than the supply of other inputs, the marginal value of an additional unit of labor will decline over time.As a result, GDP growth will not keep pace with population growth.Similarly, slowly growing or declining populations may be associated with GDP per capita growth due to the higher marginal productivity of labor under these conditions.Such eff ects are not easy to estimate; the estimation of the eff ects of population growth and the demand elasticity of labor on a wide cross-national basis would require a dedicated monograph.
Th ere is however another, more tractable issue with how population issues aff ect inequality measures.Firebaugh was right to emphasize it, although his methodological errors (discussed in section 1 above) make nonsense of his results.At issue is the extent to which changes in weighted inequality over time have been the result of changes in the relative population weights of the various countries, as opposed to resulting from changes in GDP per capita.
A crude way to study the eff ects of changing population weights on measured international income inequality is simply to hold the weights constant, hundred million people, is a good thing…yet it has reduced "inequality," by bringing these countries closer to the world mean.Diffi culties of interpretation notwithstanding, demographic changes are driving much of the vertical mobility of countries in the world-economy.As Firebaugh argues, GDP per capita declines when populations grow quickly, not because the output per worker declines, but because fast-growing populations are young populations, with relatively fewer workers and relatively more dependents.Extending Firebaugh' s reasoning, we should not expect economic models to do very well at predicting changes in GDP per capita.Models of per capita income should, where appropriate, incorporate demographic analyses, since population growth rates may completely obliterate the impact of GDP growth, as suggested by Table 5 (above).Although Firebaugh does not make the conclusion, his reasoning suggests that we should model demographic and economic change separately, only weaving the two threads at the fi nal stage, when we analyze changes in GDP per capita as the synthesis of the two.It may turn out that the forces driving economic growth (change in GDP) are not the primary forces driving improvements in welfare (change in GDP per capita).It is likely that the latter have a strong demographic coloring (population age structure, labor force participation, population growth, etc.).

TOWARDS AN IMPROVED INEQUALITY MEASURE
An alternative operationalization of international income inequality would be to consider the gaps not between individual countries, but between levels in while allowing the levels of GDP per capita to change.By my calculations, if population growth had been identical in all countries from 1960 through 1989, and countries' population shares thus remained constant at their 1960 levels over the entire period, weighted inequality would have increased 30 in the non-Communist world, instead of the 15 observed.(Table 5) In eff ect, diff erential population growth seems to have slowed the increase in inequality by half.Th is sounds like good news-population processes are working to reduce income inequality.
Another way to look at the same issue is to hold GDP per capita constant at 1960 levels, while allowing the population weights to change as observed.In this analysis, we would have seen a 12 drop in measured inequality among market economies simply through the action of changing weights over time: population growth in countries farther in the tails of the income distribution is not keeping up with population growth in the middle of the distribution.(Table 5) On both counts, then, population eff ects seem to be reducing inequality.
But is this a good thing?Unfortunately, these seemingly optimistic results are merely a chimera arising from the diffi culty of interpreting weighted analyses.Using population-weighted measures, "inequality" is reduced whenever a child is born in a country that is near the weighted mean world income level.Since income distributions are skewed with a long right tail, even rather poor countries are closer to the mean than are moderately rich countries.As a result, increasing populations in poor countries reduce measured "inequality."Th is eff ect can lead to perverse policy prescriptions.For example, a policy promoting rapid population growth in relatively poor countries would tend to reduce "inequality" as measured by any of the conventional metrics.Th e use of weighted inequality measures may be scientifi cally appropriate in most cases, but in every case they should be clearly labeled "HANDLE WITH CARE." On another front, however, it may be a mistake to concentrate on reducing or eliminating inequality as a policy goal.Leaving demography aside, decreasing inequality, however measured, may not actually be a good thing.For example, the rise of east Asia is now increasing world income inequality dramatically; as east Asian economies advance beyond the world (weighted or unweighted) mean income, inequality increases in lock-step.An equality-minded global public policy would be obliged to discourage such economic adventurism.Th is is absurd.In a similar vein, consider changes in the world since 1989.With the collapse of the Soviet Union, national income levels in Russia, Ukraine, Belarus, and the other former Soviet republics have declined precipitously.Since living standards in the Soviet Union in 1989 were well above the world mean, these precipitous declines have had the eff ect of reducing world income inequality.Few would argue that the plunging into poverty of a vast region, home to some three 24 the world-system hierarchy.Th is approach shifts the unit of analysis from the country to the hypothesized components of the system.Th is has the benefi t of comparing like with like, largely separating within-level mobility from inter-level convergence.
Using Arrighi and Drangel' s (1986) categorization of the world-economy into "organic" core, semiperipheral, and peripheral zones, I have calculated the mean logged GDP per capita for each zone.Th e results are reported in Table 6.All three levels have grown at roughly the same rate over the 39 study years.
As a fi rst attempt at constructing a new inequality measure, I have followed  in calculating the diff erences in means between each of the zones.(Table 7) Each of the gaps has increased over the four decades, albeit modestly.Th e gap that has widened the most dramatically has been the gap between the semiperipheral and peripheral means.Th e core has been leaving the semiperiphery behind in terms of average level of development only slowly, but the gap between the semiperiphery and the periphery has widened more dramatically.In addition to the secular trend, there is also a great wave in the coresemiperiphery gap.Th e semiperiphery rapidly advanced on the core in the 1960s and 1970s, but fell back substantially over the course of the 1980s.
Another interesting perspective to be gained from looking at the data by level is on the behavior of international income inequality within zones of the world-economy.Separating the data out by zone eliminates many of the skew eff ects that plague similar analyses conducted on all countries at once, while adding an important new dimension.Th e results of this analysis are striking.(Table 8) Convergence within the core was quite strong over the 39 year period, with the VarLog measure for core countries declining by over 50.Income inequality in the semiperiphery, by contrast, was relatively stable.Income inequality in the periphery, however, ballooned by over 200.Interpreting and understanding these trends will require both better data and additional research.

CONCLUSIONS
Hopefully, the results reported above may help to put the methodological arguments of the convergence/divergence debate in perspective.Contra Firebaugh and Korzeniewicz and Moran, the choice of data series makes little difference, once the planned economies are excluded.Whether weighted by population, or unweighted, whether using purchasing power parity or "market" exchange rates, inequality among market economies was on the rise in the three decades after 1960.Political independence has not led to economic independence for most of the countries of the world, nor for most of the people of the world.Quite the opposite: the post-colonial world has been marked by even greater inequality in incomes than was the world in 1960.Rich countries have gotten richer, while poor countries have not.Moreover, rich countries have become more similar in their incomes, while poor countries have become more diverse.Keeping in mind the fact that there is a floor effect in operation at the poorer end of the world income distribution (incomes cannot go below zero), these results are all the more discouraging.
i.e., those economies exposed to the extraction of surplus value through the channel of unequal exchange.