The World Distribution of Income (from Log-Normal Country Distributions)

1. The World Distribution of Income (from Log-Normal Country Distributions) Xavier Sala-i-Martin Columbia University June 2008

2. Goal • Estimate WDI • Estimate Poverty Rates and Counts • Estimate Income Inequality across the world’s citizens

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) • 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…

4. GDP Per Capita Since 1970

5. Annual Growth Rate of World Per Capita GDP

6. β-Non-Convergence 1970-2006

7. σ-Divergence (191 countries)

8. Histogram Income Per Capita (countries)

13. Adding Population Weights

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

20. 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

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

25. Missing Countries • Can approximate using neighboring countries

26. Method: Step 1: Interpolate • 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.

27. Step 2: Find the σof the lognormal distribution using least squares

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

29. Step 4 (to generate confidence intervals): Generate a new data set of quintiles • Having 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

30. Step 5: Repeat steps 1 through 4 using the original values of σ and μ to generate samples in step 4 • Repeat 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.

31. Step 6: Find the mean and the standard deviation of poverty and inequality measures • Note 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.

32. Results

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