Gomes, Fábio A. R.

English

Research Papers in Economics

Convergence in Income 
Inequality: the Case of Brazilian 
Municipalities
Fábio Augusto Reis Gomes
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Convergence in Income Inequality: the Case of Brazilian Municipalities
 ∗ 
 
 
 
Fábio Augusto Reis Gomes
# 
Ibmec São Paulo 
 
 
 
Abstract 
This paper investigates income inequality convergence among 5507 
Brazilian municipalities. Two periods are used, 1991 and 2000, and 
inequality is measured by the Gini index. For the country as a whole, the 
results suggest that the Brazilian municipalities are converging to an 
inequality level greater than the current (year 2000) level. However, when 
regional differences are controlled for, the South region converges to a 
lower inequality level while the other four regions remain converging to a 
higher level. 
 
 
Key-words: Inequality, Convergence, Brazil. 
 
Jel Codes: C31, O15, O18, O54 
 
                                                 
∗ The author is grateful to Cleomar Gomes da Silva and Márcio Laurini for their comments. The usual disclaims 
apply. 
# Mailing address: Ibmec São Paulo. Rua Quatá, 300, Sala 422, Vila Olímpia. São Paulo, SP, Brasil - CEP: 04546-
042. Phone: (55) 11-4504-2776. E-mail: FabioARG@isp.edu.br.   2
1. Introduction 
 
According to the neoclassical growth model, economies converge to their own steady 
state and the speed of convergence is inversely related to the gap between effective income and 
its steady state value (Solow, 1956). This proposition, known as conditional convergence, gave 
rise to an enormous literature on the subject.
1 Another proposition of most versions of the 
neoclassical growth models is the convergence in distribution (Bénabou, 1996). Instead of 
focusing on the average income (the first moment), it is possible to examine if countries (or 
regions) with similar fundamentals tend towards the same invariant distribution of wealth or pre-
tax income. Bénabou (1996) and Ravallion (2001, 2003) test if there is evidence in favor of 
income inequality convergence among various countries. By using international data sets and the 
Gini index as a measure of inequality, their findings support the convergence hypothesis. 
Bleaney and Nishiyama (2003) find that inequality convergence amongst (advanced) OECD 
countries is significantly faster than amongst developing countries. Indeed, in one of their test 
specifications, the authors do not find evidence in favor of inequality convergence for developing 
countries. 
A common drawback in using international data is the lack of homogeneity. In fact, 
comparability becomes a greater problem when one wants to study inequality convergence than 
when the need is to study income convergence, because the surveys that estimate the Gini index 
are less standardized than National Accounts (Ravallion, 2001). Such lack of homogeneity 
inevitably casts some doubt on the robustness of results reported by previous papers, despite the 
effort to use reliable data. In an attempt to overcome this problem, Ezcurra and Pascual (2005) 
used data of European regions supplied by the European Community Household Panel, and their 
results support the inequality convergence hypothesis.  
Surveys within a country can be used to obtain reliable data sets once they tend to employ 
the same methodology throughout the entire time series. Another reason to use intra-country data 
is that the neoclassical model predicts conditional convergence and, restricting the analysis to a 
single country, the fundamentals are expected to be identical or, at least, similar. Despite these 
advantages, Panniza (2001) seems to be the only one who explores this possibility in his 
investigation of the US states. And his results do not reject the inequality convergence 
hypothesis.  
The aim of this paper is to test income inequality convergence among the 5507 Brazilian 
municipalities, using a reliable data set built by the Brazilian Human Development Report 
(2003), hereafter BHDR. Therefore, this paper overcomes the problem of data homogeneity 
across surveys and eliminates the need to control for specific parameters related to different 
countries. Indeed, in order to interpret the convergence test as a conditional convergence test, we 
go one step further and control for regional differences of each of the five Brazilian macro 
regions (South, Southeast, Center-West, North and Northeast). Initially, the results suggest that 
the Brazilian municipalities are converging to a higher level of inequality. However, when 
regional differences are controlled for, the South region converges to a lower inequality level 
while the other four regions remain converging to a higher inequality level. 
                                                 
1 For an extensive literature review, see Temple (1999) and de la Fuente (2000).   3
2. Econometric Methodology 
 
As mentioned before, the data set includes all the 5507 Brazilian municipalities and it 
was extracted from BHDR, which is based on the Brazilian census from 1991 and 2000. It is 
worth mentioning that in both years the Gini index is based on the household income per capita 
and the establishment of several towns that occurred in this period is taken into account by the 
BHDR.  
Following Bénabou (1996), Ravallion (2003) and Bleaney and Nishiyama (2003),
2 the 
test equation is based on Gini level: 
 
,2000 ,1991
0 1 ,1991 9
ii
ii
GG
Ge δδ
−
= ++  (1) 
where  t i G ,  is the Gini index for municipality i in period t,  i e  is the residual term in the 
municipality i,  0 δ  and  1 δ  are the parameters. Therefore, the dependent variable is the annual 
average change in the Gini index, the convergence hypothesis is not rejected if  0 1 < δ  and the 
long-run equilibrium value of the Gini index is estimated by  1 0 /δ δ − .   
Ravallion (2003) also used the logarithm of the Gini by means of the following 
estimation: 
  ()
,2000
0 1 ,1991
,1991
1
ln ln
9
i
ii
i
G
Ge
G
γγ

= ++  

 (2) 
 
The dependent variable becomes the annual average growth rate of the Gini index. Besides, 
0 1 < γ  means convergence as before, but the long-run value of the Gini index 
becomes () 1 0 / exp γ γ − . Equations (1) and (2) are estimated by means of ordinary least squares 
(OLS) and least absolute deviations (LAD).
3 The latter has the advantage to be robust to outliers. 
  Equations (1) and (2) are also re-estimated, including dummies for the 5 macro Brazilian 
regions:  South, Southeast, Center-West, Northeast and North. The dummies are: 1) 1 1 = D  for 
the Center-West and zero otherwise; 2)  1 2 = D  for the Northeast and zero otherwise; 3)  1 3 = D  
for the North and zero otherwise; 4)  1 4 = D  for the Southeast and zero otherwise. These 
intercept dummies are included in equations (1) and (2) in order to take into consideration some 
effects which are specific to each region.  Besides, we multiply the initial inequality measure by 
each dummy in order to allow a specific  1 γ  for each region. These are important steps in order to 
interpret the convergence test as conditional convergence test. 
Table 1 reports the descriptive statistics. During the period under analysis, there was an 
increase, from 52.595 to 56.073, of the Gini index, which also had both its maximum and the 
minimum values increased. While the standard deviation implies a rise in the dispersion, the 
variation coefficient indicates a minor decrease. Both the mean annual average change and 
growth rate of the Gini index are positive, indicating that inequality within each municipality 
increased. In fact, inequality increased in 3654 municipalities (66.35%). The correlation between 
                                                 
2 Indeed, Bleaney and Nishiyama (2003) used an equality index given by 100 minus Gini. 
3 Bénabou (1996) and Bleaney and Nishiyama (2003) applied only the OLS estimator. Ravallion (2003) applied 
OLS and IV estimators, but the results were very close. To apply the IV estimator, one needs at least a three-period 
information of the Gini index in order to use the first period as an instrument.     4
the average change in the Gini index and the Gini initial value is -0.5533. The average growth of 
Gini and its respective initial inequality measure show a correlation of about -0.5879. These 
results are a first indication of inequality convergence. If confirmed, Brazilian municipalities are 
likely to be converging to a higher level of inequality.  
 
3. Results 
 
Table 2 presents the results for equations (1) and (2). Equation (1) shows that the 
parameters of both estimators (OLS and LAD) are significant, at 1% significance level, and the 
sign of the initial inequality variable is negative. As a consequence, the convergence hypothesis 
is not rejected. The implied long-run value of Gini is approximately 58 for the OLS regression 
and 57 for the LAD regression. Estimations of Equation (2) yield similar results and, as a 
consequence, the inequality convergence hypothesis is not rejected.  
To visualize the empirical findings, Figure 1 displays the Kernel Density related to the 
Gini index in both periods (1991 and 2000) and it also displays a vertical line in the mean value 
of the implied long-run Gini index (57.5978).
4 The distribution is moved to the right, towards the 
long-run value of the Gini index. Indeed, in 1991, 4489 municipalities (81.51%) had a Gini index 
lower than 57.5978 whereas in 2000 the total decreased to 3414 (61.99%).   
Equations (1) and (2) are also estimated with controls for regional differences. Both 
estimators (OLS and LAD) and both equations (1 and 2) have coefficients significant at the 1% 
level. There is strong evidence of convergence: the initial inequality measure and its interactions 
with the dummies have a negative coefficient, in all cases. For each macro region the long-run 
implied Gini index is similar across estimators and equations. The average long-run value of the 
Gini index is displayed on Table 4, together with basic information for each region. It is clear 
that such value is greater than the average observed Gini in 2000 for all regions, with an 
exception for the South region. The number of municipalities with an observed Gini index below 
its long-run value is greater in 1991 than in 2000 for all regions, except for the South region. 
These results mean that, controlling for regional differences, the South region converges to a 
lower inequality level while the other four regions converge to a higher inequality level. 
Figures 2 to 6 display the Kernel Density of the Gini index in 1991 and 2000, together 
with a vertical line in the mean value of the implied long-run Gini index, for each region. For the 
Center-West, Northeast, North and Southeast the distribution is moved to the right, towards the 
long-run value of the Gini index. For the South, the densities related to both periods (1991 and 
2000) are very close to each other, and the distribution does not seem to become worse. Lastly, it 
is worth mentioning that the regions with the higher mean income (South, Southeast and Center-
West) have also a relatively lower long-run Gini. 
 
 
4. Conclusions 
 
This paper investigates, and it does not reject, the income inequality convergence 
hypothesis for Brazilian municipalities. As we are restricted to a single country, and we 
controlled for regional differences, it is possible to interpret the empirical tests as conditional 
convergence tests, differently from other papers. The results indicate that the Brazilian 
                                                 
4 The density is estimated using the Epanechnikov kernel. The mean long-run Gini is the average of the LR Gini 
from Table 2.   5
municipalities are converging to a higher level of inequality, with an exception for the 
municipalities located in the South region. 
 
 
References 
 
Benabou, R. (1996) “Inequality and growth”, in NBER Macroeconomics Annual (Eds) B. 
Bernanke and J. Rotemberg, MIT Press, Cambridge, 11-73. 
 
Bleaney, M. and Nishiyama, A. (2003) “Convergence in inequality: differences between 
advanced and developing countries”, Economics Bulletin 4, 1-10. 
 
Brazilian Human Development Report (2003). Fundação João Pinheiro (FJP-MG). 
 
de la Fuente, A. (2000) “Convergence across countries and regions: theory and empirics”, CEPR 
Discussion Paper no. 2465.  
 
Ezcurra, R. and Pascual, P. (2005) “Is there a convergence in income inequality levels among the 
European regions?” Applied Economics Letters 12, 763-767. 
 
Panizza, U. (2001) “Convergence in income inequality”, Journal of Income Distribution 10, 5-
12. 
 
Ravallion, M. (2001) “Inequality convergence”, World Bank, Policy research working paper 
2645. 
 
Ravallion, M. (2003) “Inequality convergence”, Economics Letters 80, 351-361. 
 
Solow, R. (1956) “A Contribution to the Theory of Economic Growth”, Quarterly Journal of 
Economics 70, 65-94. 
 
Temple, J. (1999) “The New Growth Evidence”, Journal of Economic Literature 37, 112-156. 
   6
 
Table 1 – Descriptive Statistics 
Variable Mean  Standard 
Deviation
Variation 
Coefficient Maximum Minimum 
1991 G   52.595 5.659  0.108 79.000  35.000 
2000 G   56.073 5.866  0.105 82.000  36.000 
() 9 / 1991 2000 G G −   0.386 0.736 1.904 3.333  -2.222 
() 9 / / ln 1991 2000 G G   0.007 0.013 1.877 0.060  -0.039 
 
 
 
Table 2 – OLS and LAD applied to equations (1) and (2) 
Equation (1)  Equation (2)  Independent 
Variables  OLS LAD OLS LAD 
Constant  4.1703* 3.9861* 0.2977* 0.2883* 
 (0.0843)  (0.0984)  (0.0059)  (0.0067) 
1991 G   -0.0719* -0.0694*     
 (0.0016)  (0.0019)     
() 1991 ln G      -0.0734*  -0.0712* 
         (0.0015)  (0.0017) 
R
2 0.3061  0.1582  0.3456  0.1831 
N 5507  5507  5507  5507 
LR  Gini  57.9670 57.4000 57.6422 57.3818 
Note: Parenthesis reports the standard error and * indicate significant at 1%. N is the 
number of observations. LR Gini is the implied long-run value for the Gini index. 
   7
 
Table 3 – OLS and LAD applied to equations (1) and (2) plus regional dummies 
Equation 1  Equation 2  Independent 
Variables  OLS LAD OLS LAD 
Constant 2.9047*  2.5694*  0.2046*  0.1876* 
1 D    1.1555* 1.0265* 0.0816* 0.0669* 
2 D   1.8319* 2.0037* 0.1302* 0.1388* 
3 D   2.7094* 3.1898* 0.1778* 0.2087* 
4 D   0.6674* 0.4861* 0.0566* 0.0438* 
1991 G   -0.0545* -0.0486*     
1991 1G D   -0.0137* -0.0120*     
1991 2G D   -0.0237* -0.0274*     
1991 3G D   -0.0344* -0.0440*     
1991 4G D   -0.0096* -0.0069*     
() 1991 ln G      -0.0515*  -0.0472* 
() 1991 1 ln G D      -0.0186*  -0.0151* 
() 1991 2 ln G D      -0.0302*  -0.0325* 
() 1991 3 ln G D      -0.0409*  -0.0488* 
() 1991 4 ln G D         -0.0135*  -0.0105* 
R
2  0.4553 0.2664 0.4941 0.2918 
N  5507 5507 5507 5507 
LR Gini         
South 53.2550  52.8571  52.9562  53.0002 
Center-West 59.5334  59.3333  59.2143  59.3446 
Northeast   60.5080  60.1538  60.0972  60.0002 
North   63.1300  62.2000  62.6833  62.0002 
Southeast 55.6736  55.0000  55.4715  55.0002 
Note: * indicate significant at 1%. N is the number of observations. LR Gini is the implied 
long-run value for the Gini index. 
 
 
 
 
 
 
 
 
 
 
   8
Table 4 – Regional Analysis 
Mean Gini  Municipalities below         
the LR Gini 
1991 2000  Region  Number of 
Municipalities 
Mean 
Income 1991 2000 
Mean 
LR 
Gini 
N % N % 
South  1159  233.69 53.33 53.29 53.02  591 50.99 598 51.60
Southeast  1666  222.25 52.51 54.34 55.29 1205  72.33  1039  62.36
Center-West  446  209.16 54.42 57.56 59.36  376 84.30 295 66.14
North  449  120.47 54.14 61.34 62.50  407 90.65 291 64.81
Northeast  1787  85.16 51.35 57.80 60.19 1667  93.28  1271  71.12
Note: LR Gini means implied Long-Run Gini. 
 
 
 
 
 
 
.0 0
.0 1
.0 2
.0 3
.0 4
.0 5
.0 6
.0 7
.0 8
30 40 50 60 70 80 90
Gi ni
Gini, 2000 Gini, 1991
F igure 1 -  K ernel D ensity: B razil
.0 0
.0 1
.0 2
.0 3
.0 4
.0 5
.0 6
.0 7
.0 8
30 40 50 60 70 80 90
Gi ni
Gini, 2000 Gini, 1991
Figure 2 - K ernel D ensity: C enter-W est
 
   9
.0 0
.0 1
.0 2
.0 3
.0 4
.0 5
.0 6
.0 7
.0 8
30 40 50 60 70 80 90
Gi ni
Gini, 2000 Gini, 1991
Figure 3 - Kernel Density: Northeast
.0 0
.0 1
.0 2
.0 3
.0 4
.0 5
.0 6
.0 7
.0 8
.0 9
30 40 50 60 70 80 90
Gi ni
Gini, 2000 Gini, 1991
Figure 4 -  K ernel Density: N orth
 
 
.0 0
.0 1
.0 2
.0 3
.0 4
.0 5
.0 6
.0 7
.0 8
30 40 50 60 70 80 90
Gi ni
Gini, 2000 Gini, 1991
Figure 5 - Kernel Density: Southeast
.0 0
.0 1
.0 2
.0 3
.0 4
.0 5
.0 6
.0 7
.0 8
30 40 50 60 70 80 90
Gi ni
Gini, 2000 Gini, 1991
Figure 6 - Kernel Density: South
 
 
 
 
 
 
 
 
 

A Contribution to the Theory of Economic Growth”,

Convergence across countries and regions: theory and empirics”,

Convergence in income inequality”,

Convergence in inequality: differences between advanced and developing countries”,

Development Report

Inequality and growth”,

Inequality convergence”,

Inequality convergence”, World Bank, Policy research working paper 2645.

Is there a convergence in income inequality levels among the European regions?”

The New Growth Evidence”,

Convergence in Income Inequality: the Case of Brazilian Municipalities

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Convergence in Income Inequality: the Case of Brazilian Municipalities

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