1. Xavier Sala-i-Martin Columbia University June 2009
The World Distribution of Income (from Log-Normal Country Distributions)
Xavier Sala-i-Martin
Columbia University
June 2009

2. Goal
Estimate WDI consistent with the empirical growth evidence (which uses GDP per capita as the mean of each country/year distribution).
Estimate Poverty Rates and Counts resulting from this distribution
Estimate Income Inequality across the world’s citizens
Estimate welfare across the world’s citizens
Analyze the relation between poverty and growth, poverty and inequality

3. Data GDP 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)

4. Note: I decompose China and India into Rural and Urban
Use local surveys to get relative incomes of rural and urban
Apply 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…

5. GDP Per Capita Since 1970

6. Annual Growth Rate of World Per Capita GDP

7. β-Non-Convergence

8. σ-Divergence (191 countries)

9. Histogram Income Per Capita (countries)

14. Adding Population Weights

19. Back

20. β-Non-Convergence

21. Population-Weighted β-convergence (1970-2006)

22. But NA Numbers do not show Personal Situation: Need Individual Income Distribution
We can use Survey Data
Problem
Not available for every year
Not available for every country
Survey means do not coincide with NA means

23. Surveys not available every year
Can Interpolate Income Shares (they are slow moving animals)
Regression
Near-Observation
Cubic Interpolation
Others

24. Strategy 1: (Sala-i-Martin 2006)

28. Missing Countries
Can approximate using neighboring countries

29. Strategy 2: Pinkovskiy and Sala-i-Martin (2009)

30. Method: Interpolate Income Shares
Break 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 data
We 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 country’s inequality dynamics are the same as those of its region, but we use the single data point to determine the level of the country’s 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.

31. Method 2: Step 1: Find the σ of the lognormal distribution using least squares for the country/years with survey data

32. Step 2: Compute the resulting normal distributions for each country-year

33. Step 3: Estimate implied Gini coefficients for country/years with available surveys

34. Step 4: Three Types of countries
Countries with multiple surveys
Intrapolate ginis
Estimate location parameter as a function of sigma(Gini) for intrapolated years and then estimate the mean with sigma and GDP per capita
Countries with ONE survey
We keep the single year of data, and impute the Ginis for other years to have the same deviations from this year as does the average Gini 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 (ie, we assume that the country’s inequality dynamics are the same as those of its region, but we use the single data point to determine the level of the country’s income distribution.)
Countries with NO distribution data
We impute the average Gini time series taken over all Group I countries in the given region.

35. Step 5: Integrate across countries and get the WDI

36. Summary of Baseline Assumptions
We use GDP data from PWT 6.2
Sensitivity: WB, Madison
We break up China and India into urban and rural components, and use POVCAL surveys for within country inequality.
Sensitivity: China and India are treated as unitary countries
We use piecewise cubic splines to interpolate between available survey data, and extrapolate by horizontal projection.
Sensitivity Interpolation: 1) nearest-neighbor interpolation, 2) linear interpolation.
Sensitivity Extrapolation: 1) assuming that the trends closest to the extrapolation period in the survey data continue unabated and extrapolating linearly using the slope of the Gini coefficient between the last two data points, and 2) a mixture of the two methods in which we assume the Gini coefficient to remain constant into the extrapolation period, except if the last two years before the extrapolation period both have true survey data.
Lognormal distributions
Sensitivity: 1) Gamma, 2) Weibull, 3) Optimal (Minimum Squares of residuals), 4) Kernels

37. Results

42. Back

51. Back

61. Back

83. Back

103. Back

120. Back

127. Poverty Rates: $1/day

128. Poverty Rates

130. Rates or Headcounts?
Veil of Ignorance: Would you Prefer your children to live in country A or B?
(A) people and poor (poverty rate = 50%)
(B) people and poor (poverty rate =33%)
If you prefer (A), try country (C)
(C) people and poor.

131. Poverty Counts

132. Poverty Counts

134. Regional Analysis

135. Poverty Rates

136. Counts $1/day

137. Poverty Rates $2/day

138. Counts $2/day

151. Inequality

157. MLD and Theil

158. Decomposable Measures (Generalized Enthropy, GE)

166. Welfare

167. Sen Index (=Income*(1-gini))

168. Atkinson Welfare Index
certainty equivalent for a person with a CRRA utility with risk aversion parameter gamma of a lottery over payoffs, in which the density is equal to the distribution of income.
Hence, the Atkinson welfare index is the sure income a CRRA individual would find equivalent to the prospect of being randomly assigned to be a person within the community with the given distribution of income

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

179. Sensitivity of Functional form: Gini with Kernel, Normal, Gamma, Weibull distributions

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

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

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

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

184. Missreporting Rich don’t answer Poor don’t have houses
Eliminate quintiles 1 and 3 and repeat the procedure

187. The WB revised China and India GDP (PPP)
Following the conclusion of the International Comparisons Project (ICP) in November 2007, the World Bank has changed its methodology with respect to calculating country GDPs at PPP.
This change lowered Chinese and Indian GDPs by 40% and 35% respectively
Several criticisms have been made of this finding;
It considers prices in urban China only (so prices are too high and real income too low).
Chinese GDP in 1980 is implied to be $465, and by applying the old WB growth rates, it is $308 in 1970, which may be below the lower limit of survival
In comparing the original and revised World Bank series, we see that the effect of the revision was largely to multiply each country’s GDP series by a time-invariant constant, which is the expected effect of applying the PPP adjustments derived from the ICP to all years from 1980 to 2006.
We nevertheless compare the poverty and inequality estimates arising from the new WB series to our baseline estimates.

191. END

192. All is not money! Easterlin Paradox:
1) Within a society, rich people tend to be much happier than poor people. 2) But, rich societies tend not to be happier than poor societies (or not by much). 3) As countries get richer, they do not get happier (after a given threshold)
Implications
Economic Sociologists: Relative Income
UN: Human Development Index (as opposed to GDP)
Environmental Movement: No growth

193. Problem with Easterlin Paradox:
Old data (1974)
No poor countries in the data set
Gallup conducted a poll in 2006.
Analyzed by Stevenson and Wolfers (2008)

194. Source: Stevenson and Wolfers (2008)
Source: Stevenson and Wolfers (2008). Economic Growth and Subjective Well-Being: Reassessing the Easterlin Paradox*