The world distribution of income from log normal country distributions
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The World Distribution of Income (from Log-Normal Country Distributions). Xavier Sala-i-Martin Columbia University April 2009. Goal. Estimate WDI consistent with the empirical growth evidence (which uses GDP per capita as the mean of each country/year distribution).

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The world distribution of income from log normal country distributions

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

Xavier Sala-i-Martin

Columbia University

April 2009


Goal Distributions)

  • 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


Data Distributions)

  • GDP Per capita (PPP-Adjusted –See Note next page).

    • 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 Distributions)

    • 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…




Non convergence 1970 2006
β Distributions)-Non-Convergence 1970-2006


Divergence 191 countries
σ Distributions)-Divergence (191 countries)




Back Distributions)


Non convergence 1970 20061
β Distributions)-Non-Convergence 1970-2006


Population weighted convergence 1970 2006
Population-Weighted Distributions)β-convergence (1970-2006)


But na numbers do not show personal situation need individual income distribution
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


Surveys not available every year
Surveys not available every year Individual Income Distribution

  • Can Interpolate Income Shares (they are slow moving animals)

    • Regression

    • Near-Observation

    • Cubic Interpolation

    • Others


Strategy 1 sala i martin 2006
Strategy 1: (Sala-i-Martin 2006) Individual Income Distribution


Missing countries
Missing Countries Individual Income Distribution

  • Can approximate using neighboring countries


Strategy 2 pinkovkiy and sala i martin 2009
Strategy 2: Individual Income DistributionPinkovkiy and Sala-i-Martin (2009)


Method interpolate income shares
Method: Interpolate Income Shares Individual Income Distribution

  • 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.


Method 2: Step 1: Find the Individual Income Distributionσof the lognormal distribution using least squares for the country/years with survey data



Step 3 estimate implied gini coefficients for country years with available surveys
Step 3: Estimate implied country-yearGini coefficients for country/years with available surveys


Step 4 three types of countries
Step 4: Three Types of countries country-year

  • Countries with multiple surveys

    • Intrapolateginis

    • 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.



Summary of baseline assumptions
Summary of Baseline Assumptions country-year

  • 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


Results
Results country-year


Back country-year


Back country-year


Back country-year


Back country-year


Back country-year


Back country-year


Poverty rates 1 day
Poverty Rates: $1/day country-year


Poverty rates
Poverty Rates country-year


Rates or headcounts
Rates or Headcounts? country-year

  • 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.


Poverty counts
Poverty Counts country-year


Poverty counts1
Poverty Counts country-year


Regional analysis
Regional Analysis country-year


Poverty rates1
Poverty Rates country-year


Counts 1 day
Counts $1/day country-year


Poverty rates 2 day
Poverty Rates $2/day country-year


Counts 2 day
Counts $2/day country-year


Inequality
Inequality country-year


Mld and theil
MLD and country-yearTheil


Decomposable measures generalized enthropy ge
Decomposable Measures (Generalized country-yearEnthropy, GE)


Welfare
Welfare country-year


Sen index income 1 gini
Sen country-year Index (=Income*(1-gini))


Atkinson welfare index
Atkinson Welfare Index country-year

  • 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


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


Sensitivity of functional form gini with kernel normal gamma weibull distributions
Sensitivity of Functional form: country-yearGini with Kernel, Normal, Gamma, Weibull distributions


Sensitivity of gdp source poverty rates 1 day with pwt wb and maddison
Sensitivity of GDP Source: country-yearPoverty Rates ($1/day) with PWT, WB, and Maddison


Sensitivity of source of gdp gini with pwt wb and maddison
Sensitivity of Source of GDP: country-yearGini with PWT, WB, and Maddison


Sensitivity of interpolation method poverty rates 1 day with nearest linear cubic and baseline
Sensitivity of Interpolation Method: country-yearPoverty Rates 1$/day with Nearest, Linear, Cubic and Baseline


Sensitivity of interpolation method gini with nearest linear cubic and baseline
Sensitivity of Interpolation Method: country-yearGini with Nearest, Linear, Cubic and Baseline


Missreporting
Missreporting country-year

  • Rich don’t answer

  • Poor don’t have houses

  • Eliminate quintiles 1 and 3 and repeat the procedure


The wb revised china and india gdp ppp
The WB revised China and India GDP (PPP) country-year

  • 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.


  • END country-year


All is not money
All is not money! country-year

  • 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


  • Problem with Easterlin Paradox: country-year

    • Old data (1974)

    • No poor countries in the data set

  • Gallup conducted a poll in 2006.

    • Analyzed by Stevenson and Wolfers (2008)


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


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