FRANCISCO H. G. FERREIRA^{**}

JULIE A. LITCHFIELD^{***}

^{*} We are grateful to the Leverhulme Trust, the CNPq in Brasília and STICERD, LSE for financial support; to Frank Cowell, Stephen Howes, Rodolfo Hoffman, one anonymous referee and seminar participants at IPEA (Rio de Janeiro) for helpful comments; and to Phillippe G. Leite and Kaspar Richter for excellent research assistance. ^{ **} Department of Economics, Pontifícia Universidade Católica do Rio de Janeiro

^{***} Poverty Research Unit, University of Sussex

*JEL Classification:* D3, I3, N3.

**ABSTRACT **

This paper investigates the increases in inequality observed in Brazil during the 1980s, as well as the declines in the first half of the 1990s. It also documents the more cyclical trends in poverty during the same period. Using static decompositions of inequality by household characteristics, it quantifies the importance of education, race, geographic location and demographic structure of the household as determinants of inequality levels. Decomposing inequality by factor components reveals that almost half of overall inequality is due to the distribution of self-employment incomes. The causes of changes in inequality differ across the two decades. The rise in inequality in the 1980s appears to have been driven by increases in the educational attainment of the population, in a context of highly convex returns, and by high and accelerating inflation. In the 1990s, the fall in inequality was associated with increasing equality between urban and rural areas, declining returns to education, and falling inflation. Poverty dynamics were closely associated with real wage levels.

**RESUMEN **

Este trabajo investiga la desigualdad observada en Brasil durante los años 80, como también las declinaciones observadas en la primera mitad de los años 90. Además documenta las tendencias más cíclicas en pobreza durante el mismo período. Usando descomposiciones estáticas de desigualdad por características de los hogares, cuantifica la importancia de la educación, raza, locación geográfica y estructura demográfica de los hogares como determinantes de niveles de inequidad. La descomposición de inequidad por los componentes de factores revela que casi la mitad de la desigualdad total se debe a la distribución de ingresos de autoempleos. Las causas de los cambios en inequidad difieren a lo largo de dos décadas. El aumento en la inequidad en los años 80 parece haber sido provocado por aumentos en logros educacionales de la población, en un contexto de retornos extremadamente convexos, y por una inflación alta y acelerada. En los años 90, la caída en la inequidad estuvo asociada con una creciente equidad entre las áreas urbanas y rurales, retornos en declinación en educación y una inflación descendente. Las dinámicas de la pobreza estaban estrechamente asociadas con los niveles de salario real.

**Keywords**: Brazil; Income Distribution; Inequality; Poverty.

**1. INTRODUCTION **

Ever since moderately reliable household-level data became available, with the 1960 Census, economists have struggled to understand the determinants of the dynamics of income distribution in Brazil. From the outset, different investigators agreed about the trends, but were puzzled about their causes. The 1960s were a decade of unambiguous and pronounced increases in inequality: the Gini coefficient rose from around 0.500 in 1960 to 0.565 in 1970 (see Bonelli and Sedlacek, 1989). But while they agreed on the diagnosis, analysts differed markedly as to the causes of this increase. One group, led by Carlos Langoni, argued that the primary cause was a convexification of the returns to education. Another faction, led by Albert Fishlow, felt that repressive labour market policies were the crucial factor in determining the high level of Brazilian inequality.^{1}

The 1970s displayed a more complex evolution. Income inequality rose between 1970 and 1976, reached a peak on that year, and then fell _ both for the distribution of total individual incomes in the economically active population (PEA) and for the complete distribution of household per capita incomes _ from 1977 to 1981. See Bonelli and Sedlacek (1989), Hoffman (1989) and Ramos (1993). But those were the heady days of the "Brazilian Economic Miracle", with annual per capita GNP growth rates often in excess of 6% and declining poverty throughout the period: there was little patience for squabbling about the dynamics of inequality.

This changed again in the 1980s, when the economic stagnation of the Debt Crisis meant that changes in inequality translated directly on to changes in welfare. Always one of the highest in the world, Brazilian income inequality resumed its upward trend in 1981, and increased significantly during that decade. Apart from being important in their own right, these increases in inequality more than offset whatever limited growth there was in the period, causing poverty to rise as well, albeit with sharp cyclical fluctuations. From 1990 to 1995 - during the early structural reforms of the Brazilian economy, including some openness to trade and a successful stabilisation of the price level - inequality fell again.

The purpose of this paper is to return to the debate about the determinants of these dynamics, during the last two decades. We briefly summarise the evolution of inequality and poverty in Brazil from 1981 to 1995, and discuss the main factors behind the high levels of inequality, before moving on to a discussion of the elusive determinants of the changes in inequality and poverty. We find that inequality increased unambiguously (but not monotonically) during the 1980s, before falling steadily in the first half of the 1990s. Microeconomic determinants - such as education, demographic composition and spatial variables - perform well in explaining inequality levels, and provide some guidance as to what might be behind the trends in inequality. Their explanatory power for inequality dynamics is higher in the 1990s, however, and it appears to be impossible to account for the steady rise in inequality in the 1980s without recourse to its time-series correlation with the rate of inflation.

The paper is structured as follows. Section 2 contains a brief description of the data sets used in this analysis and of the main trends in poverty and inequality over the period. Section 3 reports on the static inequality decompositions carried out with three inequality measures, for the years 1981, 1990 and 1995.^{2} These decompositions follow the method employed by Cowell and Jenkins (1995), and aim to separate total inequality *levels* into its components within and between groups, where the groups are defined by specific household attributes, such as regional location, urban-rural status, or age, gender, race or education of the head.

However, the personal distribution of income does not only reflect differences in these household characteristics, but also differences in the extent to which households have access to formal employment, vis-à-vis a reliance on self-employment, or indeed variation in their access to capital or transfer incomes. Therefore, this section also examines the income sources of each household and their relationship with inequality of total household income per capita. These sources are earnings, incomes received under the state social insurance system, and other receipts including rental income, interest on savings, dividends, gifts and any other sources.

The next two sections turn to inequality dynamics, and search for both micro and macroeconomic explanations for them. Section 4 discusses a dynamic decomposition methodology due to Mookherjee and Shorrocks (1982), which separates *changes* in inequality into components due to changes in the mean incomes of different groups, changes in the composition of these groups, and unexplained changes. Section 5 then investigates the potential role of changes in macroeconomic aggregates, focusing on the rate of inflation. It suggests that there may be an important link between high and accelerating inflation, and the growth of inequality. Unlike previous studies which focused on labour earnings in metropolitan areas, we work with a broader income concept and a nationally representative sample. Section 6 concludes.

**2. THE DATA AND WHAT IT SAYS **

The data sets are the *Pesquisa Nacional por Amostra de Domicílios *(PNAD) for 1981-1995, produced by the *Instituto Brasileiro de Geografia e Estatística *(IBGE)^{3}. Data were collected each year from a representative national sample of households, with a sample size ranging from 286,000 to 517,000 individuals. The survey reports each year on a range of variables that form the basic data set. Questions are asked on subjects pertaining to the household and to individuals within the household. Information is recorded on the geographic location of the household; characteristics of the dwelling; household size; relationships between individuals in the household; activities of individuals; income from labour, transfers and other sources (such as land rents and capital); occupation and other labour characteristics; age; gender; education; ethnicity and literacy. The definition of income throughout the main analysis is gross monthly household income per capita and the population is all individuals in the population.^{4 } Monetary amounts are all measured in 1995 Brazilian Reais, with a dollar exchange rate of US$ 0.953. The Brazilian *INPC* official consumer price index is used to convert nominal incomes into real incomes. For a more detailed description of the data set and methodology see Litchfield (2001).

This section presents summary statistics of the income distributions. Mean and median incomes are presented for each comparable year of the series along with four summary measures of inequality. These are the Gini coefficient

Variation (CV). The Generalised Entropy class of measures is chosen because its members satisfy all of the desired axioms of inequality measures^{5}. Whilst the Gini will only satisfy these principles under certain conditions it is included in the analysis to allow some degree of comparability with other studies^{6}. The values for these indices for the period 1981-1995, along with the corresponding mean and median incomes, are presented in Table 1 below.

Two main features of the data jump out from Table 1. The first is the difference between mean and median income. In each year, median income is only approximately half of mean income. This indicates that the distribution was extremely skewed to the right, with 50% of the population receiving incomes less than half of the arithmetic mean.

The second key feature of Table 1 is the growth in inequality over the period, as demonstrated by the four summary measures. Between 1981 and 1995, the Gini coefficient rose by 3%, GE (0) rose by 7%, GE (1) by 9% and the CV by just over 10%. However, this rise in inequality was not monotonic over the period. During the 1980s, the Gini coefficient increased by more than five percent, GE (0) and GE (1) both rose by 15%, and the CV increased by nearly twenty-three percent, while during the 1990s inequality fell, with all measures falling: the Gini by 3%, GE (0) by 5%, GE (1) by 6% and the CV by 7%. The larger proportionate changes in the CV during the 1980s suggest that the increase in inequality was driven by a relatively large increase in incomes in the upper tail. Changes in inequality during the 1990s are smaller and fairly similar across the four measures, but the slightly larger decline in the CV may be due to smaller proportionate gains in incomes at the top of the distribution.

The summary statistics also shed some light on the relationship between the macro-economic cycle and the distribution of income. All four measures increased substantially during the recession of 1981-83, fell with the resumption of growth in 1984, and then resumed an upward trend, peaking in 1989, before declining until 1995. 1986 was an atypical year, in that both the Theil indexes and the Gini fell, indicating falling inequality with respect to the bottom and the middle of the distribution. The sharp rise in the CV suggests a greater dispersion amongst higher incomes. These changes go against the general trend and are almost surely due to the redistributional effects of lower inflation brought about by the 1986 Cruzado Plan. This plan lowered inflation substantially, with a positive impact upon those least able to protect their incomes against imperfect indexation. In addition to lower inflation, the lower inequality amongst the relatively poor in 1986 may also reflect the accumulated effect of three years of growth. The fall in all four inequality measures in 1990, albeit to levels much higher than the decade average - and than any year up to 1987 - also coincides with a strong, if short-lived, reduction in inflation in the second and third quarters. Similarly the fall between 1993 and 1995 may also reflect the distributional benefits of lower inflation after the *Plano Real* of 1994.

How about the dynamics of poverty over this period? In order to identify the poor, we use a set of regionally specific poverty lines calculated by Rocha (1993) for use with PNAD 1990 data. Rocha begins by computing the minimum cost of food baskets required to attain the FAO recommended caloric requirements. Because of substantial differences across the country's regions - and within these regions, from metropolitan to other urban areas and then to rural areas - in both consumption patterns and prices, a food basket was calculated for each area specifically.^{7 }The food costs for each area therefore respect not only price differences, but also differences in tastes and local food availability.^{8 } Rather than using the inverse of an Engel coefficient to obtain the poverty line, Rocha estimated non-food expenditure amongst the poor directly for each separate metropolitan area^{9}. The sum of the non-food expenditure amongst the poor and the cost of the food basket gives the set of regional poverty lines. The values of the regionally specific poverty lines, in 1995 Reais, for the relevant PNAD regions are reported in Table A1 in the Appendix, which is converted from table XIII in Rocha (1993).

APPENDIX

Three measures were chosen to summarise poverty in each year, and changes in poverty during the decade. These indices - all of which are members of the parametric FGT(a) class - are the headcount index (for a = 0), the normalised poverty deficit (for a = 1) and the FGT2 measure (for a *=* 2). Combined, the three measures take into account the "three *I*s" of poverty - incidence, intensity and inequality amongst the poor.^{10}

Poverty estimates using each measure are presented below in Table 2. Over the period as a whole, the proportion of people in poverty fell, the poor were on average less poor and inequality amongst the poor also fell. The rise in the headcount index indicates that a slightly larger proportion of the population were poor by the end of the decade than in the beginning. In addition, the fact that the poverty gap grew by proportionately more than the headcount index (6% versus 1%) is evidence that the poor were, on average, further away from the poverty line. Finally, the 10% rise in FGT(2) suggests that incomes among the poor were also distributed more unequally.

Like that of inequality, the time-series of poverty in this period was not monotonic. In fact, poverty appears to have behaved more (anti-) cyclically than inequality, with sharp increases during recession periods and substantial declines when growth resumed. All three measures indicate a sharp increase in poverty from 1981 to 1983, due to the recession. Indeed, all of the measures have 1983 as their peak year for the whole period. All measures then decline monotonically until 1986. All three measures are at their minimum in 1986 and then rise until 1993, except for a temporary decline in 1989.^{11 } The poverty reduction between 1993 and 1995 was sufficiently large to bring all poverty measures below their 1981 levels. In all cases, 1995 saw the second lowest values in the series, after 1986. It is unlikely that this is unrelated to the expansionary nature of Brazil's stabilisation plans of 1986 (*Cruzado Plan*) and 1994 (*Real Plan*).^{12}

**3. STATIC DECOMPOSITIONS OF BRAZILIAN INEQUALITY**

We now turn to an investigation of the structure of inequality in Brazil, both as relates to the nature of the households that receive income, and to the composition of the income flows they receive. Decompositions are carried out for three years: 1981, 1990 and 1995. In the first instance, we examine the role played by certain individual and family characteristics, through a set of static inequality decompositions by population subgroups.^{13} The analysis in this paper concentrates on seven attributes of the household: its regional location; its urban/rural status; its demographic composition; as well as the age, gender, race and educational attainment of the household head^{14},^{15}. Choosing the partitions themselves, for example the break points between age groups, can be somewhat arbitrary. Our choices are based on those used in other studies, or on standard classifications such as the five official geographic regions of Brazil and the IBGE classification of urban and rural areas. The partitions are as follows:

§ *Age of household head*. Households are grouped into six categories by the age of the household head: i) under 25, ii) 25-34, iii) 35-44, iv) 45-54, v) 55-64 and vi) 65+ years. This follows Bonelli and Ramos (1993).

§ *Educational attainment of household head.* This is measured as years of schooling, categorised into five groups: i) illiterates or those with less than one year schooling, ii) elementary school - 1-4 years, iii) intermediate school - 5 to 8 years, iv) high school - 9 to 11 years, and v) college education, with

12 or more years of schooling. Again this follows Bonelli and Ramos(1993).

§ *Gender of household head. *Simply male or female.

§ *Race of household head.* This is split into three categories: i) white, ii) Asian and iii) black and mixed race, including indigenous. Unfortunately very little data is available for the entire period. In 1981 the question did not appear in the core questionnaire and in 1985 less than 5% of the sample responded to the question. Only for the last two or three years of the 1980s was there a significant response rate to the question. Hence race will only be used for the analysis of 1990 and 1995. Following the standard practice in studies of Brazil, mixed race heads of households are grouped together with black and indigenous heads.

§ *Household type. *Five types of households are identified: i) "single adult" households comprised of only 1 adult; ii) "couple, no kids" households comprised of only adults, i.e. all aged over 14 or over; iii) "couples with kids" households with more than 1 adult plus children; iv) "single parent" households with a single adult plus children and v) elderly households whose head is aged 65 or over, with or without children. This is a simplification of the categories used by Tanner (1987) for Northeast Brazil.

§ *Region. * There are five official, standard geographical regions in Brazil: North, Northeast, Southeast, South and Centre-West. See Figure 4.

§ *Urban/Rural location of household. *Urban and rural areas are those defined by IBGE and used in the PNAD.

The point of the static decompositions is to separate total inequality in the distribution into a component of inequality between the above groups in each partition (I_{B}) - the explained component and the remaining within-group inequality (I_{W}) the unexplained component. Unfortunately, many widely used inequality measures are not decomposable, in the sense that overall inequality can not be related consistently to the constituent parts of the distribution. In particular, we are interested in measures where I_{B} + I_{W} = I. This is not generally true, for instance, of the Gini coefficient, but it is true of all members of the Generalised Entropy class of measures (see Cowell, 1995).

We conduct the decompositions for the three members of this class which we introduced in Section 2: GE(a), a = 0, 1, 2. Before presenting the results of the actual decomposition, considerable insight may be gained by looking at the within-group means and inequality levels for each partition. Table 3 presents mean incomes, population shares and values for each of the inequality measures defined above, for each of the subgroups in the Region and Urban/Rural partitions. Table 4 contains the same information for the partition by Age of Head, whereas Table 5 does it for the Household Type, or demographic partition. Table 6 comprises the same information for the partitions by Education, Gender and Race of the Head.

These four tables contain a wealth of information. Consider Table 3, which summarises the pattern of inequality across spatial partitions. Both the urban/rural split and the partition by region suggest that geographic location was an important explanatory factor of the level of overall inequality in each year. Mean incomes varied considerably across regions and across the urban/rural divide. On average the urban population was about three times richer than the rural population, and this ratio appears to have been stable over the whole period.^{16 } Within-group inequality levels, particularly when measured by GE(0) and GE(1), whilst not low, were lower than overall inequality and this, together with the large between group differences in average incomes, suggests that between group inequality may be important in determining the overall level of inequality in each year.

Turning now to the regional pattern of incomes and inequality, a similar story emerges. The richest regions in Brazil throughout the period were the Southeast and South: incomes in the Southeast were on average about two and a half times those of the Northeast and this ratio is consistent over time. Regional inequality levels varied, with some regions - for example the Northeast - showing more inequality than the country as a whole, and than other regions such as the richer Southeast. Poorer areas _ those with lower average income _ had higher levels of inequality than richer areas. Over time the measures moved in line with overall inequality, with the largest changes occurring in the Southeast, where inequality was generally lower than in the rest of the country. Hence the summary data suggest that while regional differences may be important in explaining the level of inequality in any one year, the change in overall inequality is likely to be due to changes in within-region inequality.

Table 4, which contains summary statistics for each age group (of household heads) suggests that the age of the household head is not a promising candidate for explaining much of total inequality, either in any one year or over time. The mean incomes per age group were fairly close to each other, varying only slightly around the overall mean, although they did follow a rough "life-cycle" path in each year, rising from youth through to middle age, with a slight drop around the age when families are probably at their largest, rising again until retirement age. Differences between groups were not great, and were fairly constant over the period. Inequalities within each age group also appear to have been close to the overall level. Over time the measures of within-group inequality behaved in a similar way to overall inequality, rising in the 1980s and falling in the 1990s.

Table 5 reveals that the age of the head was not the crucial demographic variable to look at. When we classify households by their composition, there is considerable variation across types: mean per capita incomes were highest for single adults, with their income being between 4 and 5 times the average per capita income of single parents, and about 3 times the overall average. Elderly heads were on average no worse off than the overall average, and better off than any other families with children.^{17}

Most households are comprised of both adults and children, although the structure has changed slightly over time. Between 1981 and 1995 the population share of households classified as "couple, with children", i.e. adults aged less than 65 and children living together, fell from 74% to 67% to be replaced mainly by adult only households, but also by households with elderly heads and single parent households. The pattern of inequality over time within each group does not always follow the pattern of overall inequality. Couples, with and without children, and those households headed by the elderly, saw a rise in inequality during the 1980s followed by a small fall in the 1990s, whereas inequality amongst the remainder, i.e. single adults and single parents, rose through the whole period. Given that couples, with and without children, form the bulk of the population, it is likely that this drives the changes in overall inequality, and hence that changes in inequality are largely due to changes in within-group inequality.

The partition by years of schooling of the household head appears more promising as a candidate for explaining the level of total inequality, as Table 6 illustrates. Sub-group means rise monotonically with education level and display substantial variation around the overall mean. In 1981, the mean income of households with a functionally illiterate head was half of those with elementary education and only around 10% of those with a college education. By 1995 the gap between those with an illiterate head and those with a college-educated head had widened somewhat. These large differences between households with heads with different education suggests that educational attainment is likely to be a powerful determinant of overall inequality in each year and that widening differences may explain some of the change in inequality.

The same can not be said about the partition by gender of the household head. Mean incomes were similar across male- and female-headed households, and inequality levels were close to the overall mean across all three years. It should be noted that this result - which will be confirmed by the actual decompositions in Table 7 - is not about earnings inequality between men and women in the labour market. It is based on per capita household incomes, and on a definition of household head which is open to widely different interpretations (see footnote 12). Neither does it contain any information on the intra-household allocation of income or resources, so that the fact that gender of household head is unimportant in accounting for inequality should not be interpreted as a statement about either labour market or intra-household discrimination.^{18}

The final partition, also described in Table 6 but only for 1990 and 1995, is by race of household head. This partition seems to suggest that race is an important determinant of overall inequality. Mean incomes by racial group vary considerably, with households with black or mixed race heads earning on average substantially less than either white or Asian heads, and have a mean income below the upper most poverty line. In 1990 households headed by a black (or mixed race or indigenous) person received incomes just over half of the national average, around a third of the mean income of white headed households and around a quarter of households headed by an ethnic Asian. Similar differences existed in 1995. These large racial differences in incomes suggest that the level of inequality can be partly explained by race, although is unlikely to explain changes over time.^{19}

While observing subgroup means and inequality measures can be informative, there is a more formal way to appraise the contributions of each of these household attributes to overall inequality. This is through the static decomposition analysis suggested by Cowell and Jenkins (1995), which is described below.^{20 }When total inequality *I*, as measured by any of the three indices reported in the foregoing tables, is decomposed by population subgroups, the Generalised Entropy class of measures can be expressed as the sum of within-group inequality, *I _{W}*, and between-group inequality,

*I*. Within-group inequality,

_{B}*I*, is calculated and weighted as follows:

_{W}

where f_{j} is the population share and v_{j} the income share of each subgroup j, j=1,2,....k. Between-group inequality, I_{B}, is measured by assigning the mean income of group j, m(y_{j}) to each member of the group and calculating:

Cowell and Jenkins (1995) show that the within- and between-group components of inequality, defined as above, can be related to overall inequality in the simplest possible way: I_{B }+ I_{W} = I. They then suggest an intuitive summary measure, R_{B} , of the amount of inequality explained by a particular characteristic or

.

This statistic can be interpreted as the share of total inequality which can be accounted for or explained by the attributes defining partition P. Table 7 below presents values of R_{B} for partitions by each characteristic discussed earlier. This is done for each of the three inequality indices used in this paper, and for 1981, 1990 and 1995. Clearly, the share of inequality explained by any or all of the household attributes varies according to the measure being decomposed. Our discussion focuses on GE(0) and GE(1). The explanatory power of the decompositions is smaller for G(2), which is more sensitive to higher incomes.

As expected age and gender of the household head have negligible explanatory power. The most important determinant of overall inequality is the educational attainment of the household head. Differences between groups, arising because of substantial differences in mean incomes, account for between 37% and 42% of overall inequality, and this share remains constant over time. Family type, race, region and the urban or rural location of the household are also important determinants of overall inequality. Differences between households of different family type account for between 8% and 12% of total inequality. Racial differences explain between 11% and 13% of total inequality. Regional differences account for between 8% and 12% of total inequality and differences between urban and rural areas explain as much as 17% of total inequality. Whereas the importance of race and family type, like that of education, is roughly constant over time, the explanatory power of the spatial partitions declines over time. The rural/urban decompositions, in particular, suffers a rather pronounced loss in importance, indicating a likely convergence of the income distributions across the rural and urban areas of the country. We return to this finding in Section 4.

The importance of education as the chief factor accounting for income inequality levels in Brazil bears remarking upon. This variable - however coarsely based only on years of schooling, and taking no account of quality differences - is three to four times as important as any of the other structural and demographic factors considered in this analysis. Subject to the proviso made above that for variable factors these results can not be used to infer the direction of causation - which is particularly relevant in the case of years of schooling this is an informative exercise.

This section concludes with a brief examination of how the structure of income inequality relates to income sources, following a methodology of inequality decomposition by factor components developed by Shorrocks (1982). Table 8 presents the results of this decomposition. For each income type f, mean income, GE(2), plus the correlation with total household income are shown. Sf is the absolute share of a particular income source f and so summing across this row gives the value of GE(2) overall. A large value indicates a large contribution to overall inequality. *sf *is the proportionate share of total inequality, and so this row sums to one. Again a large value indicates a large contribution.

The value of GE(2) varies a lot by income source. This is because it shows the level of inequality across all households, regardless of whether they actually receive a particular type of income. This means that for some types of income many households will have a zero entry: for example employee earnings are received by most households (73%) whereas private transfers are received by only 5% of households. The value of GE(2) drops considerably for all income sources once only those households with a particular income source are included. The last two rows of Table 8 present the population share of households receiving positive amounts of each income source, and GE(2) for positive incomes only. We can see that the very high value of GE(2) for capital income, 38.43 is largely driven by the large proportion of households (90%) with zero income from that source, but drops to under 3 when only those households with capital income are considered. However, for the decomposition of total household income inequality, all households must considered in each calculation.

Earnings for employees show the lowest inequality (1.49) probably because most households have earnings and on average earnings make up over 50% of total household income. Self-employment earnings inequality is higher, because of its higher contribution to total household income at the extremes of the distribution. Social insurance transfers show more inequality than either of the two earnings sources, partly because only 32% of households receive income from public transfers, so there are a very large number of zeroes in the vector of household receipts from social insurance. In addition most transfers are related to past earnings so their inequality are likely to be replicated in the distribution of transfers.

The main insight from this decomposition is that the informal sector - where most self-employment income is generated - appears to be the key not only to Brazilian poverty, which we have known for some time, but also to inequality. The largest contribution to overall inequality comes from self-employment incomes. This income type is responsible for 48% of total household inequality. Earnings are the next most important source of household inequality, contributing 36% towards total inequality.^{21}

**4. THE DYNAMIC DECOMPOSITION OF BRAZILIAN INEQUALITY**

Now that we know something about the relative importance of the factors behind the high *levels* of inequality in Brazil - such as educational attainment, geographic location, family composition and race - we ask whether these household characteristics can also help explain the *changes* in inequality which took place between 1981 and 1995, as reported in Section 2. To do so, we use a dynamic decomposition of GE(0), due to Mookherjee and Shorrocks (1982).

Accounting for changes in an overall measure of inequality such as GE(0) by means of a partition of the distribution into subgroups defined by some household attribute must entail at least two components to the change: one caused by a change in inequality between the groups, and one by a change in inequality within the groups. The first one is naturally the part of the total change *explained* by the partition, whereas the second is a "pure inequality" or unexplained effect. But the explained component can be further disaggregated into an effect due to changes in relative mean incomes between the subgroups an "income effect" - and one due to changes in the size or membership of the subgroups - an "allocation effect". The Mookherjee and Shorrocks (1982) procedure captures these three effects in an intuitive way. It allows the change in overall inequality to be decomposed into four terms as follows^{22}:

where D is the difference operator, f_{j} is the population share of group j, l_{j} is the mean income of group j relative to the overall mean, ie m(**y**_{j})/m(**y**), and the overbar indicates a simple average. The first term (a) in the equation above captures the unexplained, or pure inequality effect. The second and third terms (b and c) capture the allocation effect, holding within-group inequality and relative mean incomes constant in turns. The final term (d) corresponds to the income effect.

By dividing both sides through by G(0)_{t}, proportional changes in overall inequality can be compared to proportional changes in the individual effects (Jenkins, 1995). It is then straightforward to draw conclusions about the importance of each effect in explaining changes in the total. Changes in terms b, c or d indicate the extent to which changes in mean incomes for the different groups, or in their composition, explain the observed changes in total GE(0). Changes in the first component - the pure inequality effect are the unexplained changes, due to greater inequality within the groups. Table 9 shows the dynamic decomposition results for the three time periods, 1981 to 1990 when inequality rose substantially, 1990 to 1995 when inequality fell, and 1981 to 1995 when overall inequality rose.

The picture that emerges from Table 9 is much less clear than the one we painted for the level decompositions in the previous section. The overall message is that neither changes in the compositions of the various groupings over these fifteen years, nor changes in their relative means explain the observed trends very well. Indeed, for all time periods and for all of the structural factors considered, the pure inequality effect (a) is largest, i.e. changes in inequality within each sub-group were the chief determinants of the overall change, whether inequality fell or rose.

This is particularly marked for the decompositions by age, gender and race of the household head. In these three cases, the terms measuring the allocation (b and c) and income effects (d) are close to zero. Consequently, the component of the changes in each sub-period which are unexplained by the exercise is very close to the actual change in the mean log deviation.

The decompositions for region and household type shed a little more light. In the case of regions, although the bulk of observed changes are not accounted

for, it seems worth noting that the *explained *terms are all negative, throughout the period. During the 1980s, when migratory flows were higher, their overall effect (net of differential population growth) was mildly equalising. During the 1990s, when migration became less widespread, the allocation effect vanishes. During this period, though, one thirteenth of the decline in inequality is attributable to a convergence in regional incomes.

The decomposition by household types is interesting in that its unexplained term always overshoots the actual change. This suggests that in both sub-periods considered, the allocation and income effects went against the dominant trend. In the 1980s, a convergence in mean incomes across all demographic categories meant that demographic factors contributed to a decline in inequality, ceteris paribus. During 1990-1995, this tendency was reversed, with mean incomes pulling apart across family types, while inequality overall was falling.

But the two partitions that really account for a substantial part of the observed change are those by educational attainment and by urban/rural area. Recalling the data in Tables 6, note that the percentage of households headed by illiterate individuals or those with only elementary schooling fell throughout the period, with a rise in the percentage of heads with intermediate or higher levels of education. During the 1980s, this gradual upgrading of schooling levels led to an increase in inequality, as part of the mass of the distribution of education started climbing along the steep and convex returns curve. During the 1990s, as average returns to education fell across the range (see Ferreira and Paes de Barros, 1999), we observe a negative income effect for the education partition, indicating that some of the decline in inequality in this period can be ascribed to the combined quantity and price effects of the educational expansion.

But the largest contribution to an understanding of inequality dynamics in the first half of the 1990s comes from the urban/rural partition, where a combination of allocation and income effects generate 3.7 percentage points of the actual 6.4 percentage points change in GE(0). Only 2.7 percentage points of the decline are left unexplained by this partition. This can be ascribed both to continued urbanization and to the relative increase in prosperity in the rural areas in the South, Southeast and Centre-West of the country, partly linked to the growth in non-agricultural employment in the period. See Ferreira and Lanjouw (2001).

**5. THE IMPACT OF MACROECONOMIC PERFORMANCE **

The dynamic decompositions in the previous section contributed to our understanding of the determinants of changes in Brazilian inequality. In particular, a convergence in mean incomes between urban and rural areas, continued urbanization, and a reduction in average returns to education seem to account for part of the decline in inequality in the early 1990s. The picture was less clear for the 1980s, however, when changes in the distribution of education appear to account only for a small part of the substantial overall increase in inequality, and the other decompositions fail to explain much at all.

Bearing in mind that the outstanding economic fact of the 1980s in Brazil was hyperinflation, this section changes the line of approach and seeks to investigate whether there are any suggestive relationships between macroeconomic variables and inequality (and poverty). This is motivated by the frequent suggestions to the effect that high and accelerating inflation has distributional consequences. The inflation tax tends to be a regressive wealth tax, since the ability to protect capital from it through portfolio adjustments is generally held to be increasing in income, at least over an initial range. In addition, some have suggested that the ability to index one's wages is also increasing in education.

In particular, Neri (1995) discusses five separate channels through which higher inflation can lead to increases in inequality, by imposing greater costs on poorer households than on richer ones. In each case, he presents substantial supportive empirical evidence from Brazil. The five channels are: (i) economies of scale in financial transactions: while shoe-leather costs may not vary with the amount involved in a financial transaction aimed at protecting assets from inflation, the benefits do. This would remain the case even if there were no barriers to entry into certain asset markets. (ii) But these barriers to entry are widespread, and mean that access to some assets particularly effective in avoiding the inflation tax are only open to depositors disposing of more substantial sums. Neri presents revealing evidence about the incidence of ownership of overnight deposits and credit cards across the distribution of income. (iii) Tighter labour markets, usually associated with higher skill levels, are better at preserving real salary values. Indexation is less perfect for unskilled, poorer workers. (iv) In addition to financial assets, one can protect the value of one's wealth against inflation by reallocating portfolio from cash to consumption goods. The effectiveness of this strategy declines with the share of goods in one's expenditure which is perishable, and this is higher for poorer households, due to Engel's law and the fact that a higher share of foodstuffs is perishable than for most other categories of goods. (v) Finally, it also depends on the storage technology available to households. Neri presents evidence on the positive correlation between freezer ownership and household income, which adds another reason why the ability to defend one's wealth against inflation increases with income.^{23}

The unequalising effect of high inflation is felt exclusively within the partition groupings in Table 9, since its impact on household welfare varies only with wealth, and not education, location, or other attributes. It may thus provide a candidate explanation for the large unexplained component in changes in inequality during the 1980s. After all, it would be almost surprising if the increase in Brazil's inflation rate from 80% p.a. in 1980 to 1509% in 1990 had no distributional effects.

In the absence of a more detailed theoretical framework, and given the limitations of the time-series data, our analysis is based only on simple bivariate (Rank-Spearman) correlation coefficients between the Theil index (for inequality) and the FGT(2) index (for poverty), on the one hand, and the four macro variables (inflation, unemployment, the real wage rate and GDP growth)^{24} on the other. These are meant merely as descriptive tools, and should not be interpreted in any way as establishing causation. The correlation coefficients and their p-values are reported in Table 10.^{25}

While there are no significant relationships between inequality and the real wage rate or the rate of growth of GDP, the correlation coefficients between inequality and unemployment, and between inequality and inflation are significant. Since there is little reason to expect unemployment to be negatively correlated with inequality, we regard this as likely to be spurious, and due to the negative correlation (-0.3795) between the inflation and unemployment variables themselves in the period.^{26 }The time series for inequality, log inflation and unemployment are plotted in Figure 1. The only macroeconomic variable in the set we considered which was significantly related to poverty was the real wage rate. It was significant at the 0.1% level, and the two time-series are plotted in Figure 2.

Table 10 and Figures 1 and 2 suggest that macroeconomic fluctuations may indeed have played a role in the dynamics of poverty and inequality in Brazil over the period of study. Interestingly, the main factors affecting poverty (real wages and, to a lesser extent, unemployment) are different from those impacting on inequality (inflation). High and unstable inflation was perhaps the single most notable feature of the Brazilian macroeconomic scenario throughout the 1980s. Given the various reasons (discussed above) that would lead us to expect it to increase inequality, it is perhaps not surprising that the two are so closely related in that decade, but that the correlation weakens into the 1990s.

These tentative results are at odds with the view prevalent in Anglo-Saxon economies that unemployment has an inequality-augmenting effect, while inflation has an (insignificant) equalizing effect, as reported for the cases of the US by Blinder and Esaki (1978) and of the UK by Nolan (1987). They do confirm previous findings for Brazil as regards inflation, although not for unemployment.^{27 } It may be the case that whereas in low-inflation economies, an increase in inflation merely proxies for an increase in aggregate demand, leading to higher wages for the bottom of the distribution, in high-inflation economies such as Brazil, the regressive effect of the inflation tax dominates.

**6. CONCLUSIONS**

This paper has described and analysed both the structure and the evolution of inequality and poverty in Brazil during 1981-1995. We found that inequality rose steadily during the 1980s and fell a little in the 1990s. From the beginning to the end of the period, inequality rose according to all four measures investigated. Unlike inequality, poverty fell for the whole period, and its behaviour was more pronouncedly cyclical. It rose markedly during the recessions of 1983 and 1993, and fell rapidly during periods of stabilization and economic growth (notably in 1986 and 1994).

Using standard Theil decompositions, we found that the high level of inequality in the country could be reasonably well accounted for by a number of structural factors, such as the distribution of education, spatial differences, and racial heterogeneity. Educational attainment was by far the most important explanatory factor, accounting for 37-42% of overall dispersion on its own. Causality can not be inferred, but the finding is descriptively significant. Race, the demographic make-up of households, regional location and urban/rural status also accounted for some 10% of total inequality each, but age and gender of head were unimportant as sources of inequality. A Shorrocks decomposition by factor components, although hampered by the relatively aggregated nature of the PNAD questionnaire, revealed that income from self-employment, although smaller in magnitude than earnings, contributed most to overall inequality, accounting for almost half of all dispersion in the GE(2).

Changes in inequality were harder to explain than levels, particularly in the 1980s. Changes in the relative mean incomes across partition sub-groups, or in their composition, accounted for very little in that decade. The only exception was that some of the overall increase in inequality can be attributed to an allocation effect in education, due to an increase in the number of those with middle- or high-school attainments, at the expense of those with four or less years of schooling. Relative proportions and the convex structure of returns meant that this poverty-reducing development was inequality-augmenting. A cursory look at the correlation between inequality and macroeconomic variables suggests that high and accelerating inflation might also bear some of the responsibility for increases in inequality during the 1980s.

In the 1990s, four factors seem to account for most of the overall decline in inequality: a decline in the differences between the mean incomes accruing to households with different levels of educational attainment; a decline in the relative means across urban and rural areas; continued urbanisation; and the decline in inflation from 1994 onwards.

The overall lesson, to the extent that there is one, is that the main cause of Brazil's unenviable record inequality levels remains its combination of inequality in educational achievements and high returns to education in the labour market. Regional and urban/rural differences are still important, but their magnitude has been declining over time. Racial differences are substantial, and discrimination must be fought, both in the labour market and in access to education. And none of this should serve as an excuse for a return to inflationary deficit financing. Brazil has a very unequal distribution of income, and it must address the structural factors which underpin it, beginning with educational opportunities. But it must do so within the macroeconomic constraints which ensure low inflation, macroeconomic stability and sustainable growth.

^{1} See, e.g., Fishlow (1972), Langoni (1973) and Bacha and Taylor (1978).

^{2} These three years are chosen so as to enable us to make broad assessments about the decade of the 1980s, as well as of the period up to the middle of the following decade. Additionally, they are neither periods of unusual macroeconomic volatility nor of shock treatment against it.

^{3} Three years are missing from the time series presented below: 1982 is excluded on advice from the IBGE, since income questions that year had a different reference period, and the answers are therefore not comparable with those from other surveys. 1991 was a Census year, during which PNADs are not fielded. The survey was not fielded in 1994 either, for cost-related reasons.

^{4} In this paper, we do not deflate the raw PNAD incomes by a regional price index, nor do we impute rents, since the assumptions required about the stability over fifteen years of certain estimated relationships were deemed too strong. In other words: the consumption surveys which could be used to generate nation-wide regional price indices are so far apart (1975 and 1996), as to make sensible comparisons of regionally deflated data over the period we are concerned with in this paper hazardous. See, however, Ferreira *et al. *(2000) for results when these adjustments are carried out, for a single point in time.

^{5 }These axioms are as follows: anonymity, the transfer principle, scale and translation invariance, population replication invariance and decomposability (see Cowell, 1995).

^{6 }The Gini coefficient is not perfectly decomposable unless sub-groups of the population do not overlap in the space of incomes.

^{7} In fact, this was done for the nine metropolitan areas (Belém, Fortaleza, Recife, Salvador, Belo Horizonte, Rio de Janeiro, São Paulo, Curitiba and Porto Alegre), as well as Brasília and Goiânia, using the 1987 expenditure survey - Pesquisa de Orçamentos Familiares (POF). For the other urban and rural areas, conversion factors were borrowed from an earlier work by Fava (1984), which was based on the most recent available data for these areas, the 1975 Estudo Nacional da Despesa Familiar (ENDEF). These were updated to 1990 prices using the INPC price index.

^{8} For an alternative approach to dealing with regional differences in the cost of living, using a regional price index defined for a fixed basket, see Ferreira *et al.* (2000).

^{9} `The poor' amongst whom she computes non-food expenditures are those who, according to information recorded in the POF, were unable to meet** ***minimum* caloric requirements as specified by FAO.

^{10} See Foster, Greer and Thorbecke (1984).

^{11} Note that the minimum in 1986 is a particularly pronounced one. Poverty incidence was a full ten percentage points below that of any other year in the series, other than 1995. This reflects the expansionary and redistributive nature of the 1986 Cruzado stabilization plan, and is in line both with the rise in mean income and the decline in inequality, described in Table 1 above. The role of these macroeconomic factors is further discussed in Section 5 below.

^{12 }For a more detailed description of these poverty and inequality trends, including a treatment of stochastic dominance and an assessment of sensitivity to equivalence scales, see Ferreira and Litchfield (2000).

^{13 }These techniques were pioneered by Bourguignon (1979), Cowell (1980) and Shorrocks (1980).

^{14 }Whilst it is possible to draw some inferences about the direction of causality between *fixed* attributes, such as gender or race, and incomes, it is difficult to do so between *variable* attributes, such as education or demographic choices, and incomes.

^{15 }PNAD interviewers were instructed to register as household head the person "responsible for the household or so perceived by the remaining members" (IBGE, 1993, p.16).

^{16} Although see Ferreira *et al.* (2000) for a discussion of the measurement errors likely to introduce a downward bias into PNAD rural income data.

^{17} These magnitudes are naturally sensitive to the equivalence scale used. See Ferreira and Litchfield (2000) for a discussion of the robustness of poverty and inequality trends in this period with respect to the choice of equivalence scale.

^{18} Apparently, however, Brazil is not exceptional as regards the unimportance of gender of household head as a variable to explain income differences. Quisumbing *et al.* (1995) use stochastic dominance to investigate whether poor male-headed households fared significantly better than those headed by females in ten developing countries, and were able to statistically reject that hypothesis in most cases. The notable exceptions were rural Ghana and Bangladesh.

^{19} Note that each of these decompositions is univariate, and does not control for the other attributes. Regression analysis suggests that the importance of race is quite closely correlated with education (See Paes de Barros, Henriques and Mendonça, 2000).

^{20} Their approach draws on Bourguignon (1979), Cowell (1980) and Shorrocks (1980 and 1984).

^{21} This was to be expected, since earnings are such an important share of total household income at all income levels.

^{22} This is actually an approximation to the true decomposition, but both Mookherjee and Shorrocks (1982) and, later, Jenkins (1995) argue that for computational purposes this approximation is sufficient.

^{23} While the effects of channels (iv) and (v) are not captured by PNAD income data, the first three channels affect capital or labour incomes, and their effects should therefore be registered.

^{24 }The inflation rates were obtained from IDB (1991, 1996); unemployment rates are for September of each year, from the PME (IBGE); real wage rates are for São Paulo only, from the FIESP series; growth rates are derived from the official IBGE GDP series. The time-series data are not presented due to space constraints, but are available on request.

^{25 }We also ran simple OLS time-series regressions of the inequality and poverty measures on the macroeconomic indicators, and of decile shares on the same RHS variables. However, the small sample size meant that it was impossible to obtain statistical significance when all variables were included. Statistical significance did obtain when we restricted the estimation to pairs of explanatory variables, but these regressions were likely to suffer from omitted variable bias. Since the exercise is purely descriptive in any case, we preferred to omit the regression results, which are available from the authors on request.

^{26} The absence of a significant *positive* relationship is evidence that unemployment in Brazil - as in other developing countries with large informal sectors and undeveloped social safety nets - is not a labour status likely to be reported by the very poorest. They may respond to negative labour demand shocks by retreating to an informal sector characterised by self-employment with low productivity rates, or by employment at flexible wages.

^{27 }Urani (1993) and Cardoso, Paes de Barros and Urani (1995) are some of the authors to have also found that inflation increased inequality in the 1980s. The differences with respect to the role of unemployment are due to the fact that they focused on the distribution of labour earnings, and relied on data from the Pesquisa Mensal de Emprego (PME) surveys, which cover only the six largest metropolitan areas in the country (Porto Alegre, São Paulo, Rio de Janeiro, Belo Horizonte, Salvador and Recife), whereas we use the PNAD sample, which covers smaller urban and rural areas as well.

**REFERENCES**

Bacha, E.L. and Taylor, L. (1978), "Brazilian Income Distribution in the 60s: facts, model results and the controversy", *Journal of Development Studies*, 14 (3): 271-97. [ Links ]

Barros, R. P., R. Henriques and R. Mendonça (2000), "Education and Equitable Economic Development", *Economía*, 1 (1), pp.111-144. [ Links ]

Blinder, A. and H. Esaki (1978), "Macroeconomic Activity and Income Distribution in the Postwar United States", *Review of Economics and Statistics*, LX(4), 604-9. [ Links ]

Bonelli, R. and L. Ramos (1993), "Income Distribution in Brazil: an evaluation of long-term trends and changes in inequality since the mid-1970s", paper presented to the 12th Latin American Meeting of the Econometric Society, Tucumán, Argentina, August 17-20th, 1993. [ Links ]

Bonelli, R. and G. L. Sedlacek (1989), "Distribuição de Renda: evolução no último quarto de século", in Sedlacek, G. L. and R. Paes de Barros, Mercado de Trabalho e Distribuição de Renda: Uma Coletânea, (Rio de Janeiro: IPEA; Série Monográfica #35). [ Links ]

Bourguignon, F. (1979), "Decomposable Income Inequality Measures", *Econometrica*, 47, 901-20. [ Links ]

Cardoso, E., R. Paes de Barros and A. Urani (1995), "Inflation and Unemployment as Determinants of Inequality in Brazil: the 1980s", Chapter 5 in Dornbusch, R. and Edwards, S. (eds): Reform, Recovery and Growth: Latin America and the Middle-East; (Chicago: University of Chicago Press for the NBER). [ Links ]

Cowell, F.A. (1980), "On the Structure of Additive Inequality Measures", *Review of Economic Studies*, 47, 521-31. [ Links ]

Cowell, F.A. (1995), Measuring Inequality, 2nd edition, (Hemel Hempstead: Harvester Wheatsheaf). [ Links ]

Cowell, F.A. and S.P. Jenkins (1995), "How much inequality can we explain? A methodology and an application to the USA", *Economic Journal*, 105, 421-430. [ Links ]

Fava, V.L. (1984), "Urbanização, Custo de Vida e Pobreza no Brasil", Instituto de Pesquisa Economica Aplicada (Rio de Janeiro)/ University of São Paulo, mimeo. [ Links ]

Ferreira, F.H.G. and P. Lanjouw (2001), "Rural Nonfarm Activities and Poverty in the Brazilian Northeast",* World Development*, 29, 3, 509-528. [ Links ]

Ferreira, F.H.G., P. Lanjouw and M.C. Neri (2000), "A New Poverty Profile for Brazil using PPV, PNAD and Census Data", Discussion Paper Nº 418, Economics Department, PUC-Rio. [ Links ]

Ferreira, F.H.G. and J.A. Litchfield (2000), "Desigualdade, Pobreza e Bem-Estar Social no Brasil: 1981/95", Ch. 2 in R. Henriques (ed.): Desigualdade e Pobreza no Brasil (Rio de Janeiro, IPEA). [ Links ]

Ferreira, F.H.G. and R. Paes de Barros (1999), "The Slippery Slope: Explaining the Increase in Extreme Poverty in Urban Brazil, 1976-1996", *Brazilian Review of Econometrics*, 19 (2), pp.211-296. [ Links ]

Fishlow, A. (1972), "Brazilian Size Distribution of Income", *American Economic Review* Proceedings, 62, pp.391-402. [ Links ]

Foster, J., J. Greer, and E. Thorbecke (1984), "A class of decomposable poverty measures". *Econometrica*, 52, pp.761-65. [ Links ]

Hoffman, R. (1989), "Evolução da Distribuição da Renda no Brasil, Entre Pessoas e Entre Famílias, 1979/86", in Sedlacek, G.L. and R. P. de Barros (eds.) Mercado de Trabalho e Distribuição de Renda: Uma Coletânea, Rio de Janeiro: IPEA, Série Monográfica No. 35. [ Links ]

Instituto Brasileiro de Geografia e Estatística (IBGE) (1991), Para Compreender a PNAD, (Rio de Janeiro: IBGE). [ Links ]

Instituto Brasileiro de Geografia e Estatística (IBGE) (1993), Pesquisa Nacional por Amostra de Domicílios: Síntese de Indicadores da Pesquisa Básica - 1990, (Rio de Janeiro: IBGE). [ Links ]

Inter-American Development Bank (IDB) (1991, 1996), Economic and Social Progress Report, (Washington: IDB). [ Links ]

Jenkins, S.P. (1995), "Accounting for Inequality Trends: Decomposition Analyses for the UK, 1971-86", *Economica*, 62, 29-63. [ Links ]

Langoni, C.G. (1973), Distribuição de Renda e Crescimento Econômico do Brasil, (Rio de Janeiro: Expressão e Cultura). [ Links ]

Litchfield, J.A. (2001), Welfare and Income Distribution in Brazil, 1981-1995, Ph.D. Dissertation, London School of Economics. [ Links ]

Mookherjee, D. and A. Shorrocks (1982), "A Decomposition Analysis of the Trend in UK Income Inequality", *Economic Journal*, 92, 886-902. [ Links ]

Neri, M. (1995), "Sobre a Mensuração dos Salários Reais em Alta Inflação", *Pesquisa e Planejamento Econômico*, 25, Nº 3, 497-525. [ Links ]

Nolan, B. (1987), Income Distribution and the Macroeconomy, (Cambridge: Cambridge University Press). [ Links ]

Quisumbing, A.R., L. Haddad and C. Peña (1995), "Gender and Poverty: New Evidence from Ten Developing Countries", FCND Discussion Paper No. 9, IFPRI, Washington. [ Links ]

Ramos, L. (1993), A Distribuição de Rendimentos no Brasil: 1976/85, (IPEA: Rio de Janeiro). [ Links ]

Rocha, S. (1993), "Poverty Lines for Brazil: New Estimates from Recent Empirical Evidence", IPEA Rio de Janeiro, mimeo. [ Links ]

Shorrocks, A.F. (1980), "The Class of Additively Decomposable Inequality Measures", *Econometrica*, 48, pp.613-25. [ Links ]

Shorrocks, A.F. (1982), "Inequality decomposition by Factor Components", *Econometrica*, 50(1), 193-211. [ Links ]

Shorrocks, A.F. (1984), "Inequality Decomposition by Population Subgroup", *Econometrica*, 52, 1369-85. [ Links ]

Tanner, C. (1987), "Malnutrition and the Development of Rural Households in the Agreste of Paraíba State, North-East Brazil",* Journal of Development Studies*, 23, 2. [ Links ]

## No comments:

## Post a Comment