Is There a Kuznets Curve? John Luke Gallup Portland State University 1721 SW Broadway Cramer Hall, Suite 241 Portland OR 97201 [email protected]tel: 503-725-3929 fax: 503-725-3945 September 25, 2012 Abstract No. There has never been good evidence for a pattern of rising inequality in low-income countries and falling inequality in higher income countries. The only evidence that appears to support the Kuznets hypothesis is the cross-sectional pattern of inequality levels across countries, although the Kuznets hypothesis is an assertion about the path of inequality within countries. Numerous cross-sectional studies established the Kuznets curve as a stylized fact, dominating empirical and theoretical research on the effect of economic growth on income inequality since then. New international panel data with the first internally consistent time series for a large number of countries shows no evidence of a Kuznets curve. The data show an anti- Kuznets curve: inequality decline in low-income countries, and inequality increase in high- income countries. The U-shaped pattern shows up strongly in a non-parametric trend, in stochastic kernel estimation, but weakly in a quadratic fixed effect trend. JEL Codes: D31, O15, O47 Keywords: Income inequality, Kuznets curve, Inequality panel data
Transcript
Is There a Kuznets Curve?
John Luke Gallup Portland State University
1721 SW Broadway Cramer Hall, Suite 241
Portland OR 97201 [email protected] tel: 503-725-3929 fax: 503-725-3945
September 25, 2012
Abstract
No. There has never been good evidence for a pattern of rising inequality in low-income countries and falling inequality in higher income countries. The only evidence that appears to support the Kuznets hypothesis is the cross-sectional pattern of inequality levels across countries, although the Kuznets hypothesis is an assertion about the path of inequality within countries. Numerous cross-sectional studies established the Kuznets curve as a stylized fact, dominating empirical and theoretical research on the effect of economic growth on income inequality since then. New international panel data with the first internally consistent time series for a large number of countries shows no evidence of a Kuznets curve. The data show an anti-Kuznets curve: inequality decline in low-income countries, and inequality increase in high-income countries. The U-shaped pattern shows up strongly in a non-parametric trend, in stochastic kernel estimation, but weakly in a quadratic fixed effect trend. JEL Codes: D31, O15, O47 Keywords: Income inequality, Kuznets curve, Inequality panel data
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1 Introduction In late 1954, Simon Kuznets gave a bold Presidential Address to the American Economic
Association (Kuznets, 1955). He chose a topic which had been largely unstudied due to
lack of data, and used the scant data he had created himself or found elsewhere to
propose a law of motion for the distribution of income.
Kuznets had data for the change in income distribution in the United States, United
Kingdom, and two states in Germany. The change he was confident of was a sharp
decline in inequality in the U.S. and the U.K. after World War I. He also noted
substantial income growth in the two countries during the same period.
Kuznets combined his observations about the U.S. and the U.K. with the historical shift
from agriculture to industry in the course of economic development to propose a typical
pattern of change for income distribution. Kuznets assumed that rural agricultural
incomes are lower and more equally distributed than urban industrial incomes. In that
case, a shift into nascent industry will raise income inequality, as a rising fraction of
workers earn higher industrial wages. Beyond a tipping point, the predominance of
industrial employment will improve income distribution, as most workers earn similar
industrial wages. This theory predicts an inverted U-shaped relationship between
income levels and inequality.
The international income distribution data necessary to evaluate Kuznets’s hypothesis
remained severely limited until the mid 1970s. By that time, at least one national
income distribution observation was available for enough countries to look at the cross-
country relationship between income levels and inequality. The cross-sectional data were
consistent with the Kuznets hypothesis: middle-income countries, especially in Latin
2
America, tended to have higher levels of inequality than low- or high-income countries.
This lead to scores of empirical papers over the next forty years looking for an inverted
U-shaped curve in cross-country data with gradual improvements in country coverage
and econometric technique. A partial list of studies includes Kravis (1960), Paukert
Kuznets showed that inequality fell in two high-income countries as they grew richer
after World War I, but he had no evidence of rising inequality at low income levels.
Ironically, Kuznets’ prediction that inequality will rise during the early stages of
development, for which he had no evidence, is better remembered than his prediction
that inequality will fall at higher incomes.
One reason that Kuznets's paper had a big impact was his model explaining the path of
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inequality. In its simplest form, if agricultural workers all earn a low wage and
industrial workers earn an identical higher wage, then the transition from agriculture to
industry will create an inverted-U curve in inequality. The movement of the first
workers out of agriculture into higher wage industry will increase inequality, but beyond
a certain point, inequality will fall as the majority of workers receive the constant
industrial wage.
A vulnerability of Kuznets’s theory is that minor changes in the story change the
prediction. For instance, if agricultural incomes are more unequal than industrial wages,
perhaps due to unequal land ownership, movement out of agriculture into industry could
reduce inequality right away. Furthermore, all kinds of other dynamics of economic
development are likely to have implications for inequality besides the movement of labor
out of agriculture into industry. International trade, the spread of education, and
transportation linking previously isolated regions, to name just a few dynamics, are all
likely to have major impacts on income distribution in ways that do not naturally
suggest an inverted-U shaped curve.
Besides Kuznets’ scanty data and model of inequality change, what caused his
hypothesis to so thoroughly capture the imagination of economists? The country cross-
sectional data.
Figure 3 shows recent estimates of inequality for 156 countries. The Gini coefficients are
the most recent observation for each country from the panel data set used in this paper
for 87 countries augmented with the most recent Gini coefficients for 69 other countries
from the World Development Indicators (World Bank, 2011).2 GDP per capita
2 The region categories in Figure 3 and other figures are OECD90, Latin America, Eastern Europe and the
Former Soviet Union, Asia, and Africa. OECD90 indicates members of the Organization of Economic
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estimates are from the Penn World Tables (Heston et al., 2010).
This is the evidence, such as it is, that the Kuznets curve exists. A quadratic fit is
significantly concave, although most of the curvature comes from the low inequality
levels at high incomes rather than low inequality at low incomes. The curvature is also
entirely dependent on using a logarithmic scale for income. Figure 4 shows that the
Kuznets curve disappears when the quadratic fit is made using a unit GDP per capita
scale rather than a logarithmic scale.
The level of inequality in different countries relative to income does not necessarily tell
us anything about the typical path of inequality within countries, which was the object
of Kuznets hypothesis. Figure 5 shows that if countries differ in their level of inequality,
inequality could follow a U-shaped trend within each country, but the quadratic fit
across countries could be an inverted-U curve.
The level of inequality could differ across countries in a simple Kuznets model of
inequality change if some countries have regional variation in incomes and others don’t.
If each region industrializes separately, the countries with regional income differences
will have higher inequality (due to the interregional income differential) even if every
region follows the trajectory of rising then falling inequality in the course of
industrialization.
Looking at within-country inequality trends and allowing for different levels of inequality
across countries requires panel data. The panel data assembled for this study is
Cooperation and Development as of 1990, which includes the highest income countries of the world except for oil exporting countries. More recent OECD members Mexico, South Korea, Chile, Israel, Czech Republic, Slovakia, Estonia, Hungary, Poland, and Slovenia are excluded from the OECD90 group, since most of these countries still have income levels substantially lower than OECD90 members. The excluded OECD members countries are included in the other regional groups.
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described next.
3 Data and quadratic trends
The lack of comparable data across countries has always plagued research on income
inequality. Unlike most other basic national statistics, there is no international
organization that collects standardized measures of income distribution worldwide.
Good time series of inequality require data from a uniform household survey design over
time because the scope of the questions about household income or household
expenditure can have a big effect on the calculated dispersion of income or expenditure.
Surveys should cover the whole countries' population and all sources of income and/or
expenditure. Inconsistencies over time in the method used to estimate inequality levels
also cause inaccuracies in the time series.
Only very recently have consistently constructed time-series of income distribution
become available for a large number of countries. Four organizations have created series
of inequality statistics from raw survey data spanning more than a decade for a large
number of countries in certain regions of the world. The four organizations are Eurostat
(2011) for European Union members, the TransMONEE database created by UNICEF
for Eastern Europe and former Soviet Union countries (TransMONEE, 2011), SEDLAC
(2011) at the Universidad Nacional de la Plata for Latin America and the Caribbean,
and the Luxembourg Income Study database (LIS, 2011) for selected high-income
countries. These four organizations provide statistics for all of Europe (East and West),
Central Asia, Latin American, and several additional high-income countries.
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Major parts of the world still do not have organizations which collect standardized
income distribution statistics: East and South Asia, the Middle East, and Africa. Of
these, only the Asian regions have a large number of countries with the raw material for
the statistics: household income and/or expenditure surveys spanning a substantial
number of years.
This study supplements the data from the four regional organizations above with
statistics come from the UNU-WIDER World Income Inequality Database (WIID 2011)
and a few other sources. WIID, unlike the other organizations, compiles secondary data
rather than generating statistics directly from household survey microdata. WIID is a
continuation of the work of Deininger and Squire (1996, 1998) to collect inequality series
from national statistical agencies or academic studies. Unlike the original Deininger and
Squire data, however, this study does not patch together data from disparate studies
into a time series. Crucially, the data in this study only includes country series which
are internally consistent over time in terms of the source household survey and the
method used to calculate the inequality statistics.
The country times series used in this study hew to the following criteria:
1) They are calculated from surveys of household income (covering all income
sources) or household consumption expenditure drawn from a national sample of
all households.
2) The time series are calculated from surveys with the same survey design each
year.3
3) The time series of inequality statistics are calculated using the same method
and definitions within each country.
3 Minor changes of survey questions and survey design still occur over time in many of the standard national surveys, but statisticians usually address these inconsistencies when they calculate a time series of inequality estimates.
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Criteria 2 and 3 are particularly important for ensuring an accurate measurement of the
change of inequality over time. Failure to meet these criteria is what caused
inaccuracies in Deininger and Squire's inequality series.
These data cover a later period than the data in Deininger and Squire. Most of the data
from Eurostat, TransMONEE, and SEDLAC are derived from national surveys
established in the 1990s which were not available when Deininger and Squire compiled
their data. Even most Western European countries did not have annual household
income surveys before the establishment of a European-wide survey in 1995.
For the most part, low-income countries measure inequality using household
consumption, and high-income (and Latin American) countries measure inequality using
household income. Different countries use different weighting methods for household
members. Some use income per adult equivalent, some use income per capita, and a few
use total household income. These differences can affect the level of inequality across
countries, but that won’t be an issue in this study due to controls for country-specific
inequality levels.4
The data include time series from 87 countries, split regionally so that about a quarter of
the countries come each from the OECD902, Latin America, Eastern Europe and the
Former Soviet Union, and Asia and Africa combined. Asia and especially Africa are
underrepresented. The data for Eastern Europe and the former Soviet Union before
1994 are excluded to avoid picking up the sudden inequality changes after the collapse of
central planning.
4 See more details about the definitions of inequality in each country in Gallup (2012). It was this kind of inconsistent definitions within countries which caused the Deininger and Squire data not to have reliable time series.
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Gini coefficients are paired with income levels for each country and year. Income levels
are measured by gross domestic product (GDP) per capita from the Penn World Tables
version 7.0 (Heston and others, 2011). The GDP per capita figures are adjusted for
purchasing power parity and reported in 2005 constant international dollars.
The inequality time series for 87 countries are graphed in Figure 6. One can see the
inverted-U shape of inequality levels similar to the cross section graph in Figure 3.
However, if we control for inequality level differences in each country with a quadratic
fixed effects estimator, there is a slight U shape to the fit, although not statistically
significant, as shown in Table 1. There is no sign of the inverted U of the Kuznets
hypothesis in the typical within-country trend.
Table 1: Fixed Effect Inequality Trend
Gini coefficient
ln(GDP per capita) -2.173
(0.62)
ln(GDP per capita)2 0.135
(0.68)
Constant 44.981
(2.93)**
R2 0.00
N 852
* p<0.05; ** p<0.01
In the cross-section, the appearance of an apparent Kuznets curve depends on whether
or not GDP per capita is transformed in logarithms. This suggests that testing for a
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Kuznets curve may depend sensitively on the functional form used. The next section
looks at the relationship of inequality to income level without parametric assumptions
about the trend.
4 Non-parametric trend
Kuznets’ hypothesis is about the change in inequality as income grows, not about the
level of inequality. Inequality increases at a decreasing rate up to a middle income level
and then decreases at an increasing rate. This relationship can be expressed as
dgdy
= f (y) where f (y) ≥ 0 for y ≤ ym and f (y) < 0 for y > ym .
The slope of the trend of inequality, dgdy
, increases up to a middle income, ym , and then
decreases. The commonly used quadratic trend line takes f (y) = β1 + β2y which implies
that g = β0 + β1y + β2y2 for some β0 , which could be country specific. The Kuznets
hypothesis is that β1 > 0 and β2 < 0 .
With non-parametric methods, it is not necessary to specify the functional form of f (y) ;
the shape of f can be inferred from the data. Using m ≡ dgdy
for the slope of the
inequality trend, we can use non-parametric smoothing to fit the equation
mit = f (yit )+ ε it (1)
to data where ε it is a random error, i is the country indicator, and t is the time
indicator. mit =git − gi,t−1yit − yi,t−1
. Equation 1 is independent of the initial level of inequality in
each country, so it can be used to estimate the typical trend in inequality across
countries.
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f (⋅) is estimated by kernel-weighted local polynomial smoothing (Stata, 2011, p. 1001-
1010). This method takes a weighted polynomial regression of neighboring values to
predict the level f (yit ) for each level of income, where yit is the natural log of GDP per
capita.
This method fails due to the large influence of a small number of outliers. In some
countries, GDP is virtually unchanged over the period of two inequality surveys. Due to
sampling variation, the inequality estimates are different in the two periods, causing the
estimate of the slope to explode: for git − gi,t−1 > 0 as yit − yi,t−1→ 0 , mit →∞ .
A solution is to smooth each country’s inequality trend separately beforehand, and
calculate mit from the smoothed values. Since the ultimate purpose is to find the
average smoothed trend in inequality, smoothing each country’s trend first doesn’t
introduce any bias, but it does eliminate the large outliers due to sampling error.
Figure 7 shows the smoothed trend in the inequality slope for different income levels.
The dotted lines are the 95% confidence bounds. The results are the opposite of the
Kuznets hypothesis. At low income levels, inequality falls with income, and at higher
income levels, inequality rises with income.
The preferred specification shown in the figure is a linear smooth (polynomial of degree
1) using an Epanechnikov kernel with a bandwidth determined by the ROT algorithm of
0.46. Under certain assumptions, the ROT bandwidth is optimal (Fan and Gijbels,
1996). The smooth in Figure 7 turned out not to be sensitive to variations in
bandwidth, kernel choice, or degree of polynomial. The bandwidth was varied from 0.1
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to 10, with virtually no effect on the smooth. Gaussian, triangle, and rectangle kernels
all produced qualitatively similar trends, although the rectangular kernel causes the
point of inflection where the slope crosses the axis to occur at a lower income level. A
polynomial of degree 0 (local mean smoothing) flattens the trend somewhat, and
polynomials of degree 2 and 3 increase the confidence interval somewhat, but the upward
trend remains the same. In each case, the parameters are used for pre-smoothing each of
the individual country series are varied along with the final smooth.
To help visualize the results, we can construct a curve in inequality levels from the
estimated slope function. Given an estimated slope mit = f (yit ) , the inequality level can
be calculated recursively: git = gi,t−1 + mit (yit − yi,t−1) . The level of g i0 is fixed so that the
average level of the curve is equal to the average inequality in the sample. Figure 8
shows the trend line of inequality superimposed on the smoothed country trends. The
confidence bounds were calculated by bootstrapping. Like the estimated slope curve, the
shape and precision of the level curve is not very sensitive to the choice of bandwidth,
kernel, and degree of polynomial.
The non-parametric trend in inequality in Figure 8 shows a strong downward trend in
inequality up to relatively high income levels, where the trend becomes upward sloping.
The non-parametric trend of inequality is clearly U-shaped, not inverted-U shaped.
The trend in Figure 8 is consistent with Kuznets’ data, such as it was, but it is also
consistent with very long historical series that have recently been compiled for top
income shares for almost two dozen countries over the course of the twentieth century.
Almost all of 22 countries discussed in the survey by Atkinson, Picketty, and Saez
(2011) have graphs of the share of income going to the richest 1% over the last hundred
years which have remarkably similar shape to the graph in Figure 8, although some
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middle European countries and Japan show no sign of the upturn at the end.
5 Stochastic Kernel Estimation The previous sections used quadratic and non-parametric trends in inequality
(controlling for country fixed effects) to evaluate the change in average cross-country
inequality as income grows. A fuller assessment would consider the evolution of the
whole distribution of country inequalities as income rises, rather than just the trend. A
single trend, for instance, could mask a dynamic where some kinds of countries tend
towards one level of inequality and other countries tend towards a different level of
inequality. The data show a lot of diversity within countries in the path of inequality,
with some countries’ distribution becoming more equal and others less equal whether
they start out with high or low inequality.
Stochastic kernel estimation is flexible enough to model the net outcome of complex
dynamics of inequality change. It can incorporate a dynamic path where all countries
tend towards a single level of inequality or where they tend towards two or more
different levels of inequality. It can also incorporate a dynamic path where countries
switch places between different levels of inequality, while the overall distribution of
cross-national inequalities remains the same.
Stochastic process models are typically applied to processes that evolve over time.
However, most hypotheses about income distribution concern the path of inequality as
income grows, not as time passes. For this reason, we model the cross-country
distribution of inequalities in the income domain rather than in the time domain.
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The stochastic kernel model represents the evolution of continuous distributions from
period to period. It is the continuous analogue of a Markov chain, which represents the
and applies them to the distribution of income levels across countries. Since stochastic
kernel models use continuous distribution functions, Quah defines them using measure
theory instead of the more accessible matrix algebra of Markov chains.
Continuous income distributions are attractive conceptually, but in practice, stochastic
kernel models are estimated by discrete approximation. Digital computers must
approximate continuous transition surfaces with discrete grids, so a stochastic kernel
model is actually estimated as a fine-grained Markov chain. Since the model is
ultimately estimated using a discretized distribution, we will present stochastic kernel
estimation using the simpler Markov chain notation. There is no loss of generality since
a continuous distribution can be arbitrarily well approximated by a high dimension
discrete distribution. Refer to Quah (1997, 2007) for the continuous formulation.
The usual difference in practice between the stochastic kernel and Markov chain
estimation is the method of calculating the transition matrix. Markov transition
matrices are typically estimated from crude frequency counts of transitions. Stochastic
kernel estimation, in contrast, typically uses bivariate kernel density estimation. 5
5 Although stochastic kernel estimation and kernel density estimation both include the term “kernel”,
meaning distribution function, they are referring to different uses of a distribution function. Stochastic kernel estimation refers to the estimation of the transition from one period's distribution to the next period's distribution of the variable of interest (here, inequality). Kernel density estimation refers to the weighting scheme for averaging neighboring frequency observations. The weights decline moving away from the cell of interest according to the value of the kernel, or distribution function, chosen (e.g. Gaussian, Epanechnikov, triangle, etc.)
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The stochastic kernel estimation of the transition matrix can be finer-grained with more
rows and columns because the kernel density estimation uses information from
neighboring cell frequencies to smooth the density estimates. Whereas a typical Markov
transition matrix of frequencies from a sample of several hundred observations might be
a 5x5 matrix to avoid small sample sizes in each transition cell, a bivariate kernel
density estimator would commonly produce a 50x50 matrix smoothly approximating the
surface. So in practice a stochastic kernel estimation is a Markov chain estimation with
a smoothed transition matrix.
The Markov chain in this application models the evolution of the distribution of country
inequalities with transitions from one income level to another, rather than the more
conventional transition from one time period to another. First divide inequality into N
possible levels, with values g1,…,gN . The set G = {g1,…,gN } is the state space, and
countries move from one inequality level, gi , to another, gj , at each step of income.
A fundamental assumption of the Markov model is the Markov property. Let Xs ∈G be
the inequality level of a country at income level s. The Markov property is that
assumption that E Xs+1 X0,…,Xs( )= E Xs+1 Xs( ) . That is, the inequality level at the next
higher income level depends only on the inequality level at the current income level, and
not on the earlier history of inequality levels at lower income levels. In addition, we
assume homogeneity of transitions across income levels: E Xs+1 Xs( )= E Xs Xs−1( ) for all s.
Then we can define the transition probability as pij = E Xs+1 = gi Xs = gj( ) . The N by N
matrix of all the transition probabilities is P= pij⎡⎣ ⎤⎦ .
Let us be an N-dimensional probability row vector which represents the state of the
Markov chain at income level s. The ith component of us represents the probability
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that the chain is at inequality level gi . Then us+1 = usP . If we assume that for every
inequality level gi except for g1 there is a positive probability of inequality falling to
gi−1 at the next income level, and at every inequality level gi except for gN there is a
positive probability of inequality rising to gi+1 at the next income level, these are
sufficient conditions for the Markov chain to be ergodic, which means there is a
possibility of going from every inequality state to every other inequality state, although
not necessarily in one step. By Doeblin's Theorem (Stroock, 2000, p. 28), ergodic
Markov chains tend towards a unique stationary probability vector as income levels
increase without bound. The stationary probability vector w is defined by
w=s→∞lim u0P
s . w shows us where the distribution of inequalities will end up if the
current distributional dynamic continues indefinitely. The stationary distribution w is
equal to the first left eigenvector of the transition matrix P, and is independent of the
initial distribution u0 (Theorem 8.6, p. 106, Billingsley, 1979).
This Markov model allows for a broad range of inequality dynamics including churning
between inequality levels. However, it does rule out a Kuznets curve, the tendency for
inequality to rise at low income levels and then switches to moving towards lower
inequality beyond a certain income threshold income. The assumption of homogeneity,
E Xs+1 Xs( )= E Xs Xs−1( ) , means that if inequality tends to rise with income, it does so at
all income levels.
To allow for a Kuznets curve dynamic where inequality increases at low income levels
but falls at high income levels, separate Markov processes are estimated for observations
below middle income and for those above middle income. The Kuznets hypothesis
predicts convergence towards higher inequality levels in the low-income sample and
convergence towards lower inequality in the high-income sample.
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Estimating the transition matrix requires some conditioning of the data. We need to
observe the change in inequality across regular income level intervals. Country
inequality levels are measured at regular time intervals (every year in countries with an
annual income survey), not at regular income intervals. To construct a progression of
inequality over equally spaced log income intervals, income is quantized into 200 equal
log income levels. If more than one sequential inequality observation falls within a given
income interval (when income grows too little to progress to the next income level), the
inequality observations are averaged.
The categorization of the data into regular income levels produces an inequality “income
series” (as opposed to a time series) with a lot of gaps, especially in countries with rapid
economic growth, because income may jump several levels between inequality
observations. These gaps are bridged by linear interpolation between income levels in a
given country to maintain a connected series (just as observations are typically
connected by straight lines in graphs, like Figure 1). To prevent rapidly growing
countries with more interpolated observations having disproportionate influence, the
observations are weighted by the number of actual, non-interpolated, data points when
estimating the transition matrix. The regridding of inequality takes the original 861
observations and interpolates them up to 2,035 inequality transition observations.
To allow for a different dynamic below and above middle income, the sample is split at
the mean income level of $15,000 GDP per capita (rounded to the nearest thousand),
and separate transitions are estimated for each subsample. The transition matrices P
are estimated using an Epanechnikov bivariate kernel density. The estimation generates
50 by 50 transition matrices which smooth the raw transition frequencies of the Gini
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coefficient from one income level to the next. The top panel in Figure 9 displays the
stationary kernel for the low-income sample and the bottom panel shows low-income
sample, both using an Epanechnikov bivariate kernel with a bandwidth of 3. In each
case, the stationary kernel is superimposed on the smoothed distribution of inequalities
in the sample.6
The stationary kernel w represents the ultimate distribution of inequality levels if the
transitions observed in the data continue indefinitely. The stationary kernel in low-
income countries (the top panel) shows a strong decline of inequality to quite low
inequality levels. The stationary kernel for high-income countries (the bottom panel)
shows a modest increase in inequality levels compared to the observed sample.
The dotted lines indicate the mean inequality for each distribution. For the low-income
sample, the mean inequality has a large statistically significant decline from the observed
sample at 41.3 to the stationary distribution at 27.1 (t = 30.98; p = 0.0000), even lower
than the mean observed Gini in the high-income sample, at 29.5. In the low-income
sample, the mean inequality has a statistically significant rise from the observed sample
at 29.5 to the stationary distribution at 32.1 (t = -6.42; p = 0.0000).7
These results are not very sensitive to the choice of kernel or bandwidth for the bivariate
kernel density smoothing, but they are quite sensitive to the cut-off point between the
low and low-income sample. The same qualitative pattern appears with a Gaussian or a
6 The distributions of observed Gini coefficients in Figure 9 are smoothed with a univariate kernel smoother
using an Epanechnikov kernel with the “rule of thumb” bandwidth of 2.39 for the low-income sample and 1.25 for the low-income sample. The shape of the univariate kernel smooth was not sensitive to the choice of kernel (Epanechnikov, Gaussian, or rectangular) or the bandwidth within a substantial neighborhood of the bandwidth used.
7 The stationary distribution w is independent of the initial distribution u0 , so the difference in means can be tested with an ordinary t test.
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rectangle kernel, which are among the common kernels most dissimilar to the
Epanechnikov. A similar degree of smoothing requires a lower bandwidth for the
Gaussian kernel and a higher bandwidth for the rectangle kernel. Lower bandwidths
produce somewhat more compact stationary distributions, with a slightly lower mean for
the low-income stationary distribution and a slightly higher mean for the high-income
stationary distribution. Higher bandwidths have the opposite effect. The chosen
bandwidth of 3 for the Epanechnikov kernel in Figure 9 is the lowest bandwidth which
makes the stationary distribution reasonably smooth.
The results are strongly affected by the cut-off point between low income and high
income. The lower income sample shows similarly strong convergence to very low
inequality levels when the cut-off point between low and high income is lowered, but the
convergence becomes weaker as soon as the cut-off point is raised above $15,000. The
high-income sample shows convergence to higher inequality with sample cut-offs between
$12,000 and $18,000, but outside that range shows almost no distributional change as
income grows. This suggests that a cutoff point of $15,000 is about right to distinguish
between the falling inequality in low-income countries and the modestly rising inequality
in high-income countries.
As with the non-parametric trend in the previous section, the stochastic kernel
estimation shows a clear U-shaped relationship between inequality and income levels, the
opposite of Kuznets’ hypothesis. The stochastic kernel estimation allows for more much
more complex dynamics than an average trend across countries, but still shows a strong
decline in inequality in low-income countries and a modest increase in inequality in high-
income countries. The convergence to very low inequality levels is particularly clear in
the low-income sample.
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6 Conclusion The empirical analysis using new, higher quality inequality data shows no sign of a
Kuznets curve. This confirms the lack of any compelling evidence in the literature that
the Kuznets hypothesis describes the typical change in inequality as income grows.
There has never been general evidence for the Kuznets hypothesis except for the huge
number of cross-sectional studies, which we have no reason to believe capture the typical
path of inequality within countries.
Kuznets himself would probably not have been too surprised by the failure to find
evidence for the validity of his hypothesis. He was quite clear about the speculative
nature of his hypothesis, saying that his “excuse for building an elaborate structure on a
such a shaky foundation [of data] is a deep interest in the subject.” (Kuznets, 1955, p.
26)
This study is the first to test the Kuznets hypothesis using internally consistent time
series of inequality for a large number of countries. The non-parametric trend and the
stochastic kernel estimation both show an anti-Kuznets curve: a strong tendency for
inequality to fall with economic growth at low to middle income levels and a weaker
tendency for inequality to rise at middle to high income levels. The quadratic fixed
effects trend, though convex, is not as clear – it essentially shows no relationship
between inequality and income levels.
The U-shaped curve shown in the non-parametric trend and the stochastic kernel
estimation is consistent with the twentieth century history of top income shares for a
number of currently high-income countries. Most of the 22 countries surveyed in
Atkinson, Picketty, and Saez (2011) show the share of income going to the richest 1%
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falls during most of the century and then rises somewhat towards the end.
Although there is no sign of a Kuznets curve, a pattern of convergence of inequality as
income levels rise is quite clear in the data. This is explored in a separate paper
(Gallup, 2012).
It is intriguing to think about what may cause a pattern of inequality change the
opposite of that proposed by Kuznets, but I don’t recommend the same devotion to this
new pattern that has been given to the original Kuznets curve for fifty years.
25
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Figure 1: Inequality Data from Kuznets (1955)
India
Ceylon
PuertoRico
Saxony
Prussia
UnitedKingdom
(richest 15%) UnitedStates45
5055
60In
com
e sh
are
of th
e ric
hest
20%
0 2000 4000 6000 8000 10000GDP per capita
n.b. GDP per capita estimates are from Maddison, 2010. The Puerto Rico GDP is for 1950 while the income share is for 1948. Prussia and Saxony use GDP estimates for Germany.
Figure 2: Estimates of Pre-Industrial Income Inequality
Bihar (India) 1807
Brazil 1872
British India 1947
Byzantium 1000
China 1880
England/Wales 1688
England/Wales 1801-3
Holland 1561
Holland 1732
Kingdom of Naples 1811
Moghul India 1750
Nueva Espana 1790
Old Castille 1752
Roman Empire 14
Modern median gini
Modern 75th percentile gini
34
43.7
20
30
40
50
60
Gin
i coe
ffici
ent
0 1000 1500 1700 1900Year
Source: Milanovic and others (2007).
Figure 3: Inequality in a Cross Section of Countries with a Quadratic Fit
