Slide 1

The World Distribution of Income (from Log-Normal Country Distributions)Xavier Sala-i-MartinColumbia UniversityJune 2008 1GoalEstimate WDIEstimate Poverty Rates and CountsEstimate Income Inequality across the worlds citizens2DataGDP Per capita (PPP-Adjusted). We usually use these data as the mean of each country/year distribution of income (for example, when we estimate growth regressions)Note: I decompose China and India into Rural and UrbanUse local surveys to get relative incomes of rural and urbanApply the ratio to PWT GDP and estimate per capita income in Rural and Urban and treat them as separate data points (as if they were different countries)Using GDP Per Capita we know

3GDP Per Capita Since 1970

4Annual Growth Rate of World Per Capita GDP

5-Non-Convergence 1970-2006

6-Divergence (191 countries)

7Histogram Income Per Capita (countries)

8

9

10

11

12Adding Population Weights13

14

15

16

17

Back18Population-Weighted -convergence (1970-2006)

19We can use Survey DataProblemNot available for every yearNot available for every countrySurvey means do not coincide with NA means

But NA Numbers do not show Personal Situation: Need Individual Income Distribution20Surveys not available every yearCan Interpolate Income Shares (they are slow moving animals)RegressionNear-ObservationCubic InterpolationOthers21

22

23

24Missing CountriesCan approximate using neighboring countries25Method: Step 1: InterpolateBreak up our sample of countries into regions(World Bank region definitions).Interpolate the quintile shares for country-years with no data, according to the following scheme, and in the following order:Group I countries with several years of distribution dataWe calculate quintile shares of years with no income distribution data that are WITHIN the range of the set of years with data by cubic spline interpolation of the quintile share time series for the country. We calculate quintile shares of years with no data that are OUTSIDE this range by assuming that the share of each quintile rises each year after the data time series ends by beta/2^i, where i is the number of years after the series ends, and beta is the coefficient of the slope of the OLS regression of the data time series on a constant and on the year variable. This extrapolation adjustment ensures that 1) the trend in the evolution of each quintile share is maintained for the first few years after data ends, and 2) the shares eventually attain their all-time average values, which is the best extrapolation that we could make of them for years far outside the range of our sample.Group II countries with only one year of distribution data.We keep the single year of data, and impute the quintile shares for other years to have the same deviations from this year as does the average quintile share time series taken over all Group I countries in the given region, relative to the year for which we have data for the given country. Thus, we assume that the countrys inequality dynamics are the same as those of its region, but we use the single data point to determine the level of the countrys income distribution.Group III countries with no distribution data.We impute the average quintile share time series taken over all Group I countries in the given region.26Step 2: Find the of the lognormal distribution using least squares

27Step 3: Compute the resulting normal distributions, and the poverty and inequality statistics

28Step 4 (to generate confidence intervals): Generate a new data set of quintilesHaving obtained our point estimates, we obtain our standard errors by reproducing our original set of income distribution data by drawing samples of the sample size given in the country information sheets for the WIDER database from each estimated lognormal distribution corresponding to a country-year with data, calculating the sample quintile shares for each of these samples, discarding the sample

29Step 5: Repeat steps 1 through 4 using the original values of and to generate samples in step 4Repeat the steps 1 through 4 to generate a new set of poverty and inequality measures for each country-year and the world as a whole over the 34 years. We repeat the procedure N (300) times. Note that we do not use our estimates to generate income shares for country-years with no data, but we obtain the data by the procedure described above in order to keep the data-generating process identical to the one we used to obtain point estimates. Note also that in all iterations, we generate our samples from the lognormals with parameters given by the point estimates we obtain from the true, rather than synthetic data.

30Step 6: Find the mean and the standard deviation of poverty and inequality measuresNote that we have as many observations of the poverty and inequality measures as we have iterations of step 5. For this paper we used N=300. We can now estimate the mean and standard deviation of these observations. If our assumption about the nature of the sampling in the surveys as roughly i.i.d., our assumption that the country-year distributions are lognormal, and our assumption that the interpolation provides reasonable estimates of quintile shares for country-years with no data are all correct, the standard deviation of the estimates for the N iterations should converge to the population standard deviation of the (complicated) estimator that we use to obtain our point estimates.

31Results32

33

34

35

36

Back37

38

39

40

41

42

43

44

45

Back46

47

48

49

50

51

52

53

54

55

Back56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

Back78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

Back98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

Back115

116

117

118

119

120

121Poverty Rates

122Rates or Headcounts?Veil of Ignorance: Would you Prefer your children to live in country A or B?(A) 1.000.000 people and 500.000 poor (poverty rate = 50%)(B) 2.000.000 people and 666.666 poor (poverty rate =33%)If you prefer (A), try country (C)(C) 500.000 people and 499.999 poor.123Poverty Counts

124

125

126Gini and Atkinson Index (coef=1)

127Sen Index (=Income*(1-gini))

128Atkinson Welfare Level

129MLD and Theil

130

131

132

133MLD Decomposition (t=total, w=within, and b=between country inequality)

134Theil Decomposition (t=total, w=within, and b=between country inequality)

135Regional Analysis136Sub Saharan Africa

137East Asia

138South Asia

139Latin America

140Middle East and North Africa

141Eastern Europe

142Former Soviet Union

143Counts (all regions, $1/day)

144

145

146

147

148

149Sensitivity of Functional form: Poverty Rates ($1/day) with Kernel, Normal, Gamma, Adjusted Normal, Weibull distributions

150Sensitivity of Functional form: Gini ($1/day) with Kernel, Normal, Gamma, Weibull distributions

151Sensitivity of GDP Source: Poverty Rates ($1/day) with PWT, WB, and Maddison

152Sensitivity of Source of GDP: Gini with PWT, WB, and Maddison

153Sensitivity of Interpolation Method: Poverty Rates 1$/day with Nearest, Linear, Cubic and Baseline

154Sensitivity of Interpolation Method: Gini with Nearest, Linear, Cubic and Baseline

155Preliminary Results on Confidence Intervals with Lognormal: Gini

156Preliminary Results on Confidence Intervals with Lognormal: MLD

157