Download The Gini Index, Pietra Ratio and Mean Division

The Gini Index, Pietra Ratio and Mean Division Point of Income
Distribution
By
Mingliang Frank Shao
School of Economics, Henan University, Kaifeng, Henan, China, 475004;
Phone: 86-15039030907. Email: [email protected]
March 2014
Abstract
The mean division point (MDP) consists of all in a population whose individual income is
less than or equal to the mean income in an economy. It involves the mean population share
(MPS) and mean income share (MIS) that occur at the point of unit slope on a differentiable
Lorenz curve. We find that the Gini index and MPS show an inverted U-shape curve with
TFP but not with GDP.
An interesting result is that an increase of human capital may not reduce income
inequality if TFP is not high enough. The aggregate effects of a change in TFP or human
capital on income inequality are negative only when TFP is higher than a particular level,
which is a convex function of human capital.
We find that MIS has been slowly decreasing with GDP, capital stock and TFP, that MPS
increases with TFP and capital stock, and that MPS tends to increase too after the early stage
of development. Thus, we conclude that the population of the low income segments has
become relatively worse off (with lower MIS and larger Pietra ratio and Gini index) as the
economies have become richer (with higher capital stock, TFP, and per capita GDP) in the
panel of countries.
Key Words: Gini index, Pietra ratio, Mean division point
JEL Classification: D31, E25, O15.
1
I Introduction
Economists have analyzed income inequality and its economic effects for more than a
century. However, there is no consensus within academia regarding how to define a group of
people as relatively low income, nor is it clear how a relatively low income segment of the
population fares as an economy grows.
Income inequality can be described by different dimensions and measurements, and it
changes depending on how income is defined. To discuss the economic effects of income
inequality, we need to define a proper measurement. There are two sets of measurement for
income inequality. One is a multidimensional metric that may consist of all inequalities of
interested variables; for example, inequalities of income, wage, consumption, longevity,
education, and gender, etc. The other is a summary index, such as the Gini index, Theil index,
Pietra ratio, 20/20 ratio, top income share, etc. A proper measurement matters in estimating
and ranking Lorenz curves (Atkinson, 1970; Shorrocks, 1983; Aaberge, 2009; etc.) and
explaining the economic effects of income inequality (Galor and Zeira, 1993; Forbes, 2000;
Easterly, 2006; Banerjee and Duflo, 2003; etc.).
A summary index of income inequality does not tell us how many people are categorized
as part of a relatively low-income group, or how much income share they represent. In fact,
no summary index describes the relative size of either income share or population share for
the relatively low income population. We also, at present, do not have any way to define a
micro-foundation for a summary index of income inequality. This paper introduces the mean
division point to explore these problems.
Barro (2000), using a short panel data, finds that “the Kuznets curve shows up as a clear
empirical regularity across countries and over time and ….. (the curve is) not weakened over
time.” Our evidence does not support this finding with a much larger panel data.
2
Acemoglu (2002) summarizes how technical change affects the distribution of wages and
income. “When skill-biased (or skill-replacing) techniques are more profitable, firms will
have greater incentives to develop and adopt such techniques.” Income inequality changes
due to technical progress, international trade, fiscal policy and institutions (Aghion, at el.,
1999). Skill-biased techniques, trade, fiscal policy and institutions all are typical ingredients
of TFP, so that TFP can work as a comprehensive causal factor of income inequality as well.
We use TFP and human capital as covariates in our model to explain income inequality.
Voitchovsky (2005) finds that the profile of income inequality matters for economic
growth, and shows potential limitations to using a single inequality statistic to explore the
impact of income distribution on growth. This finding implies that a summary index may lose
the power to explain different effects of a profile of income inequality, and that innovations
in measurement of income inequality are required to describe more information of the profile
of income distribution. Our introduction of mean division point gives a simple way of using
the profile of income distribution to explain economic effects.
Atkinson et al. (2011) investigates top incomes (gross incomes before tax) for more than
twenty countries and finds that the gross income share of the population, excluding the top
segment, had decreased, or at least not increased, as economies developed during the last
thirty years. This finding is further strengthened by the evidence of
the Pietra ratio and
mean income share with a much larger panel data in this study.
This paper studies income inequality using the mean division point on the Lorenz curve.
It is the point of unit slope on a differentiable Lorenz curve, and its coordinates are called
mean income share and mean population share, respectively. They define a division of
income distribution at the mean income point of the income probability density function. This
point gives a clear expression of how many people earn less than the mean income and how
these people are relatively poorer than those earning more than the mean income. The mean
3
income share is a relative description of income inequality, and the mean population share
gives the size of a relatively poor population group.
This measurement of income inequality allows us to study how the relatively poor
population and their income would be affected by policies. We use the definition of
disposable income in the calculation of income inequality since disposable income, rather
than gross income, plays the dominant role in households’ decisions.
The paper is organized as follows: part II introduces the mean division point and
discusses the framework of a causal model of income inequality; part III analyzes the data,
while variable definition and statistic issues are also discussed; part IV gives the empirical
results; and Part V concludes.
II The Model
2.1 The Mean Division Point of Income Distribution
Let 𝑓(𝑤) be the probability density function of income distribution, with w denoting
income level. Accordingly, 𝐹(𝑤) is the cumulative probability function of population share
with individual income no more than w, 𝜇 is the mean of per capita income in the economy,
and (𝑥, 𝑦) is a point on the Lorenz curve. Then, we have the following Lorenz curve,
𝑦(𝑥) = (1/𝜇) ∫
𝐹 −1 (𝑥)
0
𝑥
𝑤𝑑𝐹(𝑤) = (1/𝜇) ∫ 𝐹 −1 (𝑡)𝑑𝑡
(2.1)
0
Taking the first derivative of the Lorenz curve (2.1) and letting the slope be unit, then we
have the following equation for the population share, in which each person has an income
level less than or equal to 𝜇,
𝑑𝑦⁄𝑑𝑥 |𝑥 ∗ = 𝐹 −1 (𝑥 ∗ )/𝜇 = 1 → 𝑥 ∗ = 𝐹(𝜇)
Then, the mean income division point (𝑥 ∗ , 𝑦 ∗ ) on Lorenz curve is defined by the
following equations:
4
𝑥 ∗ = 𝐹(𝜇)
{
∗
𝑦 = (1/𝜇) ∫
𝐹(𝜇)
𝐹 −1 (𝑡)𝑑𝑡 ,
(2.2)
0
Definition 2.11 The mean (income) division point (𝑥 ∗ , 𝑦 ∗ ) of income distribution on a
differentiable Lorenz curve is defined by the condition
𝑑𝑥/𝑑𝑦|(𝑥 ∗,𝑦 ∗) = 1.
(2.3)
The mean division point (MDP) defines its mean division shares (MDS). The mean
population share (MPS) consists of all people whose individual income is no more than the
mean income in an economy. The mean income share (MIS) is the relative size of total
income holding by all those whose personal income is no more than the mean income in the
economy.
2.2 Modelling of Income Inequality
We use a dynamic model to explain income inequality. Lagged income inequalities,
endogenous variables and strict exogenous variables are included as explanatory variables in
our regression models. This type of dynamic model, with low-order moving-average
correlation in the idiosyncratic errors or predetermined variables, has been well discussed by
Arellano-Bover (1995) and Blundell-Bond (1998) for panel data with small panel numbers
and large time periods.
The package Stata uses GMM on the first difference equation of dynamic models. All
lagged endogenous variables uncorrelated with explanatory covariates are used as GMM-type
instruments for the first differenced equation, and standard instruments are the differenced
covariates of strictly exogenous variables. We use the Sargan test to test the validity of
instrument over-identification.
1
This definition was called the fair division point of income distribution in Shao (2011), where it was
defined by the point of unit slope on a Lorenz curve, not by probability density function of income distribution.
But the two definitions are actually identical.
5
Regarding the endogenous variable issue, we treat lagged income inequality and GDP as
endogenous, and physical capital stock and human capital as predetermined. Empirical
studies show that schooling is significantly affected by income inequality, as is human capital.
Capital stock is treated as predetermined since income inequality affects aggregate savings.
There are two tricky issues we must address for model specification: one is the best
orders of ARMA and the other is the functional form of ARMA. There is a tradeoff between
the orders of ARMA, which affects the over-identification test, and the number of
instruments, which affects estimates and their significance. We need rules to choose the best
orders of ARMA. We choose the minimum orders for ARMA so that the over-identification
test does not reject the null that over-identifying restrictions are valid. We use the ArellanoBond test -- the command is estat abond -- for an autocorrelation test in the first-differenced
errors. Of course, functional form and missing variables also affect the over-specification test,
so we must consider these issues as well before we choose the orders of ARMA.
For functional forms, we include level form for all covariates, and consider quadratic and
cross effect forms only when they give significant estimates after considering multicolinearity. Human capital strongly affects TFP development (Erosa, et al., 2010), thus, we
include the product of the two variables as a covariate. It turns out that the product of human
capital and TFP works significantly in our regressions. We use the RESET2, F-statistic, to test
its nonlinearity specification; when there are multiple valid nonlinear functions and we
choose the one with larger adjusted R2, which justifies the squared log form for capital stock.
We can use robust regression to deal with heteroskedasticity, the option of robust is
available for dynamic regressions on Stata.
It has been common to discuss how income inequality evolves with development ever
since the Kuznets hypothesis was proposed, and GDP is a typical indicator of development.
2
See Wooldridge “Introductory Econometrics” 5 th edition, Chapter 8 for the RESET test and Chapter 9 for
the following Davidson-Mackinnon test.
6
People often talk about changes in income inequality as GDP grows. This might not be a
proper way to describe the causal evolution of income inequality, since income is primarily
distributed by input factors and institutions. GDP is neither an input factor nor an institutional
cause of income distribution; it is an outcome of a set of interacted factors including all input
factors, economic policies and institutions, and income inequality itself (representing
resource allocation) as well. But GDP also may work as a proxy of some missing factors, in
which economic institutions are the typical ones; thus GDP may also have some explanatory
power in explaining income inequality. Multi-collinearity is another concern with GDP as a
covariate. GDP is highly correlated with other factors, for instance with capital stock, which
may greatly affect other factors’ estimation and their significance in a regression. Capital
stock may become insignificant if GDP is dropped in our regression, but it is definitely a
principal factor to affect income inequality.
As to whether GDP should be included as a covariate to explain income inequality, a test
of joint significance of GDP terms would not be a good solution to the issue if there were
missing variables that are correlated with GDP (for example, fiscal policy and economic
institutions). We will therefore compare models with GDP and without GDP by using the
Davidson-Mackinnon test. If the test does not specify which model is preferred, we will
choose the one with the higher adjusted R2. We use the following procedures to adress the
problem.
Step one, we build three models, one includes current period GDP and its quadratic item,
and the other one includes one-period lagged GDP and its quadratic terms. The third does not
include GDP at all; We compute the adjusted R2 on the level equation of each model;
Step two, we use the RESET test to verify the nonlinearity specification of the three
models; F-statistic and R2 are calculated on the first difference equation;
7
Step three, we use the Davidson-Mackinnon test to compare the above three models that
passed the RESET. If no model is solely preferred in this procedure, we compare their
adjusted R2 calculated after step two for the models that passed the RESET and choose the
one with higher adjusted R2.
It turns out that GDP does not give significant advantage in explaining MIS; and we
cannot drop GDP in explaining the Gini index and MPS (see the Table 4.2.1 in the following
subsection 4.2).
Now, we write the first difference of our model as follows3
𝑦 =∑
Where,
,
1
1,
𝑦
−
𝑥
1
𝑤
,
= {1 2
}, 𝑡 = {1 2
}
are parameter vectors to be estimated, and p is the order of lagged 𝑦 ;
𝑦 : Dependent variable measuring income inequality;
𝑥 : A vector of strictly exogenous variables; here it consists of employment, government
consumption share in GDP, and labor share;
𝑤 : A vector of endogenous and predetermined variables; here it consists of the lags of
income inequality, TFP, human capital and capital stock (and GDP for the model of testing
the Kuznets hypothesis);
: i.i.d or a low-order moving-average process.
III Data and Statistic Issues
3.1 Data Sources and Statistic Unit
The data of GDP and population are retrieved from the PWT8.0 (Penn World Table).
The PWT8.0 provides national income accounts with purchasing power parity converted to
For technic details of the ARMA model, please refer to http://www.stata.com/manuals13/xtxtdpd.pdf
on page 14.
3
8
international prices for 167 countries/territories for some or all of the years from 1950 to
2011.
The data of income distribution are borrowed from “WIID2” (World Income Inequality
Database) of WIDER at the United Nations University. “WIID2” is a panel data built from
different earlier works of income distribution of countries all over the world; these data were
collected by different agents, and thus vary in the definition of income, coverage of sample
area, household, age, and population. The team claims that the data-set can be comparable
without correction and adjustment in definition, statistical unit and survey method; but we
find some data using different statistic units and income definitions, thus we collect and
adjust data with the following criteria to accommodate our regression model.
For the data of income distribution, income is defined as disposable income4 measured by
household per capita. Sample coverage is nationwide and over all ages of the population.
However, if the data available were not sampled by these standards, we will either transfer it
by the method specified in the next subsection 3.2, or take an approximation for a few
observations; for instance, disposable income is approximated by the square root of economic
family equivalence for Canada.
“WIID2C” ranks quality of an observation according to its income definition, survey
quality, and sample methods. Quality 1 is assigned to those in which “the underlying
concepts are known”, and “the quality of the income concept and the survey can be judged as
sufficient” according to some strict criteria. Quality 2 is assigned to observations that “the
quality of either the income concept or the survey is problematic or unknown or we have not
been able to verify the estimates (the sources were not available to us)”5, and all other criteria
are satisfied. We choose only the observations rated as quality 1 or quality 2 in this study.
4
Its definition refers to the “World Income Inequality Database User Guide”, on the table 1 of page 6, and
the definition on page 10.
5
World Income Inequality Database User Guide and Data Sources, page 13.
http://website1.wider.unu.edu/wiid/wiid-documentation1.php
9
In the data-set there are some observations whose surveyed income was gross (monetary)
income or disposable monetary income, or whose income is measured by household or
household equivalence (OECD method, HBAI, square root, see the WIID2 user guide);
which can be used to impute the Gini index that agrees with our standards. This enables us to
enlarge our data set as large as possible.
An observation of income distribution will be kept in our data-set even if it does not
satisfy our standards, but the Gini index satisfying our standards is available for the year of
the observation. In this case the Gini index is used to estimate the MDS. In the next
subsection 3.2, we discuss how to estimate the MDS to satisfy our statistical standards.
One country’s data may come from different sources to enlarge the data set size, while
one observation for the same year from one source may significantly differ from those of
other sources even if they used the same statistical method. When this occurs, we choose the
data with more observations and/or the latest version. However, we have to accept that there
exist statistical errors in the pool of different data sources, which cannot be completely
overcome.
From the two data resources, PWT8.0 and “WIID2C”, we choose 571 observations of 59
countries from 1962 to 2006, which are all capable of being cleaned using the above statistic
standards and available to all variables we use.
3.2 Estimation of MDS with Given Gini Index
There are many cases in which a data set was not made with the methodology we are
using. In these cases, we should find a way to estimate MDS so that it will satisfy our needs.
The Lorenz curve changes with measuring units in income and population, or with the
definition of income. In these cases, the changed Lorenz curve is unknown but if the Gini
10
indices are known before and after the change, then the MDS of the changed Lorenz curve
can be approximated by the following method.
To simplify the question, we consider only such changes that make the Lorenz curve shift
along the orthogonal direction of the tangent line at the MDP. This simplification assumes
that the change in measuring unit or income definition proportionally affects each point on
the Lorenz curve. Figure 3.2.1 below shows the shifting of a triangle Lorenz curve. Let
the Gini index of the Lorenz curve ̂ , and (𝑥1 , 𝑦1 ) be its MDP;
1
be
is the Gini index and
(𝑥 , 𝑦 ) is the MDP of the Lorenz curve ̂ after a change in measuring unit or income
definition.
Figure 3.2.1 Shifting of Lorenz Curve
D
w
C
E
A
B
450
O
𝑥1 𝑥
p
The assumption of a shifting change is not enough to describe point B by point A, but the
problem can be resolved if we further approximate the ratio,
, of Gini indices by the
squared ratio of heights on the common bottom line ̅̅̅̅ of the two triangles,
and
. That is, if the two triangles OAD and BDD are assumed to be similar when the
shift is very small, then,
/
1
(
/
) =
(3.2.1)
Employing the assumption (3.2.1), we have the following results
11
√
1/
/
= (𝑥1
𝑥
{
𝑦
Employing the Gini ratio
𝑥1
𝑦1
2
)⁄(𝑥
𝑥1
𝑦1
2
)=(
𝑥1
.5𝑥1 (1
√ )
.5𝑦1 (1
√ )
.5𝑥1 (1
√ )
.5𝑦1 (1
√ )
𝑦1
2
𝑥1
𝑦1 )⁄(
𝑦1
2
𝑦 )
(3.2.2)
and MDP before the shift of Lorenz curve, we can estimate
the new MDP after the shift with equations (32.2). Fortunately, the Gini index is widely
available with different definitions and statistical methods in the data set “WIID2C”, and so
that the Gini ratio
can be easily computed, which enables us to estimate point B of the shift.
We can directly get the Gini ratio when the Gini indices are available before and after a
change of the statistical method; otherwise we choose the average of the Gini ratio of
previous two or three periods when they satisfy our statistic standards.
The transference from household equivalence by the OECD method to household per
capita may suffer some errors when we use the equations (3.2.2), but we can only take (3.2.2)
as an approximation since we do not have the data of household size for the two population
groups.
Finally, we have enlarged our data set using all current income resources and kept
uniform statistic unit within each country, but we are still unable to apply the uniform statistic
unit to all countries since no enough information were available to do the estimation
according to our standards and methods. There are 20 countries with a total number of 223
observations that do not use disposable income and household per capita as the statistic unit,
which is about 39% of the observations in the data set. For instance, income was defined with
disposable monetary income for Republic of Korea, Belgium, Switzerland, and Australia, it
was gross monetary income for New Zealand and Argentina, gross income for Honduras,
Ukrain, and Uruguay, respectively. The statistic unit of income was square rooted household
equivalence for Republic of Korea and Norway, square rooted economic family equivalence
for Canada, OECD method household equivalence for Australia, Austria, France, Greece,
12
Ireland, Netherland, and Portugal, household equivalence (HBAI) for United Britain, tax unit
per capita for Switzerland, respectively.
3.3 Data Summary
The following variables are used in our data: the Gini index, the Pietra ratio, mean
division shares, average human capital, TFP, gdpe (an index of per capita GDP), physical
capital stock, labor share, employment, and government consumption as a share of GDP are
also used. Their definitions and notations are as follows.
rgdpe: Expenditure-side GDP at constant 2005 reference prices (in mil. 2005 US$);
gdpe: rgdpe observation divided by the highest value in the panel, $12,482,813, which is
the rgdpe of USA in 2004;
empr: Ratio of employed population in the total population;
rkna: Capital stock at constant 2005 national prices (in mil. 2005 US$);
capital: Index of physical capital stock, which is rkna divided by the highest value of all
capital stock observations, $35,589,712, which is the capital stock of USA in 2004;
hc: Index of human capital per person, based on years of schooling at age 25+ (Barro/Lee,
2010) and returns to education (Psacharopoulos, 1994); it is standardized by dividing an
observation with the highest value of all human capital observations in the panel data, 3.565,
which is the human capital of USA in 2004;
tfp: TFP at constant national prices (2005=1);
gov: Share of government consumption in GDP at current reference price;
labsh: Share of labor compensation in GDP at current national prices;
x : MPS at the MDP;
y: MIS at the MDP;
Pietra: 𝑥
𝑦, the maximum Pietra ratio.
13
The data statistics are listed in the following Table 3.3.1.
Table 3.3.1 Data Summary
Variable
Obs
Mean
x
y
Pietra
Gini
gdpe
589
589
589
589
589
0.632
0.383
0.249
0.347
0.275
empr
capital
hc
tfp
labsh
589
589
579
571
580
gov
589
Std. Dev.
Min
Max
0.065
0.036
0.078
0.104
0.164
0.502
0.281
0.101
0.163
0.012
0.832
0.501
0.508
0.685
1.000
0.423
0.316
0.768
0.910
0.580
0.060
0.205
0.106
0.123
0.097
0.298
0.009
0.378
0.403
0.241
0.626
1.000
1.000
1.619
0.776
0.195
0.081
0.044
0.587
The average MPS is 0.632 with a standard deviation of 0.065. The average MIS is 0.383
with a standard deviation of 0.036. The average Pietra ratio is 0.249 with a standard error of
0.078, and the average Gini index is 0.347 with a standard error of 0.104. Among the three
statistics, MIS has the least standard error, and the Gini index has the largest standard error.
We now take a look at the observational relationship between the Gini index and MDS
regarding GDP. Figure 3.3.1 below shows that the Gini index falls during the early stages of
GDP growth; it stops falling and appears to rise a little bit later as an economy crosses near
the mean GDP (which is 0.275) in the panel.
0.700
0.300
0.200
0.400
MIS
0.300
0.500
0.600
Gini
0.500
0.400
MPS
0.600
0.800
0.700
Figure 3.3.1 Observations of Gini Index and MDS with GDP
0.000
0.200
0.400
0.600
GDP
0.800
1.000
0.000
0.200
0.400
0.600
0.800
1.000
GDP
14
Figure 3.3.1 shows that MPS shows in a similar pattern to the Gini index, but MIS
appears to be stable around its mean, which is 0.383 in the early stage of development and
declines a little bit later. The observations of MDS show that the Pietra ratio falls in the early
stages of GDP, and then it seems to be stable or rises a little bit for a much longer period than
the early stages, which is similar to the trend of the Gini index as well.
Figure 3.3.2 below shows that there is a clear pattern between the Gini index and MDS. A
larger Gini index reflects a higher MPS and a lower MIS, and thus larger Pietra ratio. That is,
the Gini index is positively correlated with MPS and the Pietra ratio and negatively correlated
with MIS. Thus the Gini index shows a linear correlation with MDS and the Pietra ratio.
0.700
0.400
0.300
MIS
0.500
0.600
MPS
0.800
Figure 3.3.2 Observations of MDS with Gini Index
0.200
0.300
0.400
0.500
Gini Index
0.600
0.700
We also see the following observational relationships between income inequality and
human capital, employment, labor share and TFP, respectively. Related figures of the
observations can be found as Figure A3.3.3 ~ A3.3.7 in the Appendix I.

The Gini index and MPS increase with TFP, but fall with labor share and
employment. The two statistics fall with capital stock and human capital when capital
stock is less than 0.3 and human capital is less than 0.8, respectively. Beyond these
points, the two statistics tend to reverse the trend.
15

MIS falls slightly with TFP and does not change much with labor share, human
capital, employment, and capital stock.
These observational results will be econometrically analyzed in subsection IV of the
following context.
IV Empirical Results
4.1 Correlation among the Gini Index, Pietra Ratio and MDS
When the Lorenz curve is a triangle, there is a simple relationship between the Gini index
and MDS. Let (𝑥, 𝑦) be the vertex of a triangle Lorenz curve and g be the Gini index, then
the Gini index equals the maximum Pietra ratio6 (Robin Hood index), which is the difference
between the MPS and MIS,
𝑥
(4.1.1)
𝑦
We run a dynamic model to test the correlation between the Gini index and MDS. Table
4.1.1 in Appendix II records the regression result. We write the estimated model as follows,
̂ = .2 3
= 4 5,
(3.24)
. 45𝑥
−1
( .14)
(
. 52𝑦
. 21 ,
. 2)
(. 1),
(4.1.2)
= .
3
The number in parenthesis is the z-value. At the steady state, equation (4.1.2) becomes
̂ = 1.3𝑥
1.31𝑦
. 2
(4.1.3)
Thus, we have the following empirical proposition,
Proposition 4.1.1 The Gini index has a significantly positive correlation with MPS, and a
significantly negative correlation with MIS. The Gini index is about 1.3 times the Pietra ratio
in the panel of countries.
6
The Pietra ratio at a point on Lorenz curve is the difference between the population share and income
share. The equation (4.1.1) is clearly discussed in (Shao, 2011).
16
4.2 GDP-Type Kuznets Curve
We define the Kuznets hypothesis as a GDP-type Kuznets curve when we use GDP to
describe development. TFP will be used to denote development in the next subsection.
We run income inequality on the quadratic functional form of GDP and other covariates
with a dynamic model to test the Kuznets hypothesis. By the definition of MDS, we know
that MPS and MIS are positively correlated; hence we include them and their lags as
covariates in the regression of MDS. One period lagged Gini index and labor share are also
included as covariates. The logarithm form of a variable is denoted by the prefix lg, the
squared form of a variable is denoted by the suffix sqr. For instance,
𝑡𝑓
=[
𝑡𝑓 =
(𝑡𝑓 ),
(𝑡𝑓 )] .
Table 4.2.1 below summarizes the regression results. The regression codes and results can
be found as Table 4.2.2 ~ Table 4.2.5 in the Appendix II. x_kuz, y_kuz and Gini_kuz are
dependent variable of x, y and Gini index, respectively, in each regression that GDP has been
included as an explanatory variable. y_caus denotes MIS as dependent variable that is
explained with covariates excluding GDP. One period lagged GDP is used on the model
Gini_kuz because this is preferred to current GDP by the Davidson-Mackinnon test.
The Davidson-Mackinnon test does not tell us which model is preferred between y_kuz
and y_caus, but y_caus gives much higher adjusted R2 than y_kuz. Thus, y_kuz will only be
used to discuss the effects of GDP on MIS, other where we use the model y_caus to explain
MIS in the contexture.
We include a cross effect regressor, th, which is the product of TFP and human capital.
This means that the marginal effects of human capital on income inequality rely on TFP, and
the marginal effects of TFP on income inequality are affected by human capital as well. The
cross-effect regressor describes the correlation between human capital and TFP.
17
Table 4.2.1 Summary of Regression Results
Variable
y_kuz
y
L1.
--.
y_caus
0.205*
x_kuz
0.295**
x
--.
L1.
0.633***
-0.178*
0.611***
-0.247**
labsh
--.
L1.
0.089***
-0.057*
0.082***
-0.060*
lggdpe
L1.
--.
Gini_kuz
-0.219
0.841***
0.184
-0.049
-0.036
-0.091***
lggdpesqr
L1.
--.
lgth
lgtfp
lgtfpsqr
hc
hcsqr
lgcapitalsqr
lgempr
gov
lggov
0.088***
0.153***
0.038***
-0.440***
-0.137*
-0.318***
-1.515***
0.892***
-0.008*
-0.053**
0.042***
-0.598***
-0.237***
-0.294***
-2.076***
1.116***
-0.005
-0.090***
-0.003
-0.006
-0.024***
0.417***
0.164***
0.191***
1.386***
-0.715***
0.007*
0.053**
0.063
0.346**
0.094**
0.107*
1.120**
-0.656**
0.004*
0.011
0.017
Gini
L1.
0.464***
_cons
-0.687***
N
-0.442*
390
0.936***
390
1.168***
390
390
legend: * p<0.05; ** p<0.01; *** p<0.001
The adjusted R2 is 0.178 for the model y_kuz, 0.283 for y_caus, 0.611 for x_kuz, and
0.743 for Gini_kuz, respectively. The fitted MDS and Gini index with GDP are shown in
Figure A4.2.1 in Appendix III. The fitted MIS is for the model y_kuz in Figure A4.2.1.
Let
denote the growth rate of GDP in period t. The marginal effects of inequalities
regarding GDP are as follows,
𝑦
⁄
𝑑
−1
𝑥
⁄
𝑑
=
⁄
𝑑
−1 (.
=
(.
=
1
.
(. 153
( 𝑑
. 4
( 𝑑
.
4
( 𝑑
)),
−1 )),
𝑑
)),
∗
𝑥
𝑑
∗
𝑦
𝑑
= .15
= .314
∗
= .1 2
18
The star point is the turning point for each inequality measure. We find a negative sign on
the coefficient of the quadratic term of gdpe in the regression of MIS 𝑦
; the turning
point of GDP is .15 and there are a few observations less than .15. We conclude, therefore,
that the GDP-type Kuznets curve is weakly supported with MIS since the turning point is
very small.
We find positive signs on the coefficient of the quadratic term of gdpe in the regressions
of the Gini index and MPS. The turning points of GDP for the Gini index and MPS are .162
and .314, respectively; after which the Gini index and MPS increase with GDP. This is just
the opposite of the Kuznets hypothesis.
In summary, the Kuznets hypothesis is weakly supported with MIS, but absolutely not
with the Gini index and MPS regarding GDP (GDP-type Kuznets curve); actually the Gini
index and MPS show a flattened U-shape with GDP in the panel data. All factors of income
inequality have opposite effects on the two dimensions of MDP.
4.3 TFP-type Kuznets Curve
We call the correlation between income inequality and TFP as a TFP-type Kuznets curve.
This definition differentiates itself from the case of GDP.
Figure A4.3.1 in Appendix III show the fitted Gini index and MDS with TFP. The fitted
MIS is for the model y_caus. We can see from the figure that the estimated Gini index and
MPS show an inverted U-shape with TFP (TFP-type Kuznets curve). The turning point is
about 1, but there are only a few observations after the turning point. The estimated MIS falls
slowly with TFP. Therefore, the TFP-type Kuznets hypothesis might be demonstrated with
the Gini index and MPS, but not with MIS. This means that the Pietra ratio increases in
almost all stages of the development of TFP, and tends to narrow after TFP crosses a value
near one.
19
The marginal effects of TFP on income inequality are as follows,
= [. 34
𝑦
𝑥
=
[. 44
=
)
.
(
)
.13
[. 5
=
𝑡
(
[.
(
(
)
)
.214
(𝑡𝑓 )]/𝑡𝑓
(4.3.1)
. 3
(𝑡𝑓 )]/𝑡𝑓
(4.3.2)
4
.23
.231
.5
. 5
(𝑡𝑓 )]/𝑡𝑓
(𝑡𝑓 )]/𝑡𝑓
(4.3.3)
(4.3.4)
Figure 4.3.1 below draws the above curves, the subscript denotes the regarded variable of
partial derivative. The intervals of TFP and human capital are borrowed from the panel data.
We find that the marginal effects of TFP on income inequality can be positive, neutral, or
negative depending on the combination of TFP and human capital.
Figure 4.3.1 Marginal Effects of TFP on Income Inequality
There are three special points on Figure 4.3.1. Point A gives zero marginal effects of TFP
on MDS and the Pietra ratio; Point B shows zero marginal effects of TFP on the Gini index
and MPS; and point C has zero marginal effects of TFP on the Gini index and MIS.
The triangle ABC forms a set of combinations of TFP and human capital, where an
increase of TFP would partially increase the Pietra ratio and MPS, but decrease the Gini
20
index and MIS. So within the region of the triangle ABC, the marginal effect of TFP on the
Gini index varies from that on the Pietra ratio.
On the upper right corner of the figure, the area is above the curves 𝑦
𝑥
𝐹
=
and
= , an increase of TFP would reduce income inequality (the Gini index, Pietra
𝐹
ratio and MDS) at any combinations of TFP and human capital. Similarly we have the
opposite conclusion on the left and lower corner of the figure, where is below the curves
𝑦
𝐹
=
and 𝑥
𝐹
= .
On the upper left corner of the figure, the area is between the curves of 𝑦
and 𝑥
𝐹
𝐹
=
= , an increase of TFP will decrease MIS and increase MPS at any
combinations of TFP and human capital; and there are also points in this area, where an
increase of TFP does not affect either the Gini index or Pietra ratio.
On the lower right corner of the figure, the area is between the curves of 𝑦
and 𝑥
𝐹
𝐹
=
= , an increase of TFP will partially increase MDS at any combinations of
TFP and human capital, and while the Gini index falls in most of the points.
Proposition 4.3.1 TFP must be higher than a minimum level to partially reduce income
inequality (the Gini index, Pietra ratio or MDS) through improving TFP, and the minimum
level of TFP is a convex function of human capital across countries or over time.
Corollary 4.3.1 A necessary condition to partially reduce the Gini index, Pietra ratio and
MDS through improving TFP is that TFP must be above a minimum level; which is 0.8 in the
panel data.
Corollary 4.3.2 The marginal effects of TFP on the Gini index, Pietra ratio or MDS are
zero when the two variables satisfy the corresponding quadratic functions (4.3.1) ~ (4.3.4).
Proposition 4.3.2 The TFP-type Kuznets hypothesis is supported with the Gini index, the
Pietra ratio and MPS; and MIS shows a flattened U-shape curve with TFP in the panel data.
21
Proposition 4.3.3 At any point between the two curves
−
𝑡
=
and
= , the marginal effects of TFP is negative on the Gini index, but positive on
the Pietra ratio.
4.4 Marginal Effects of Human Capital on Income Inequality
Figure A4.4.1 in the Appendix III show the fitted Gini index and MDS with human
capital. We find that the Gini index and MPS present a slanted-to-the-right U-shape curve
and MIS seems stable with human capital. The Pietra ratio and the Gini index fall in most
stages of human capital accumulation but it reverses the trend somewhat due to the increasing
of MPS when the human capital index goes over the value 0.8, which is a little bit larger than
the mean of human capital index.
The marginal effects of human capital on income inequality are as follows,
= [. 34
𝑦
𝑥
=
[. 44
[. 5
=
𝑡
=
[.
(𝑡𝑓 )⁄
(𝑡𝑓 )⁄
(𝑡𝑓 )⁄
(𝑡𝑓 )⁄
1.12
1.515
1.312
]
(4.4.1)
1.
]
(4.4.2)
4
2.
2. 35
]
2.232
3.
]
(4.4.3)
(4.4.4)
Figure 4.4.1 below shows the equations (4.4.1) ~ (4.4.4), on which we find that the
marginal effects of human capital on income inequality depend on the combination of human
capital and TFP as well.
There are three areas in the figure that deserve special attention. The upper left part is a
set of combinations of TFP and human capital such that the marginal effects of human capital
relieve income inequalities; the lower right part is a set such that the marginal effects of
human capital worsen income inequalities; and between the two is a set in which the marginal
effects of human capital reduce the Gini index and MIS, and increase MPS and the Pietra
ratio.
22
Figure 4.4.1 Marginal Effects of Human Capital on Income Inequality
It is easy to verify that the four curves on Figure 4.4.1 are convex to human capital, and
we have the following empirical propositions and corollary.
Proposition 4.4.1 TFP must be higher than a minimum level to reduce income inequality
through improving human capital, and the minimum level of TFP is a convex function of
human capital across countries and over time.
Corollary 4.4.1 A necessary condition to reduce the Gini index, Pietra ratio or MDS
through improving human capital, is that TFP must be above a minimum level; which is .501
in the panel data.
Corollary 4.4.2 The marginal effects of human capital on the Gini index, Pietra ratio or
MDS are zero when the two variables satisfy the corresponding quadratic functions (4.4.1) ~
(4.4.4).
Proposition 4.4.2 The Gini index, Pietra ratio, and MPS show a slanted-to-the-right Ushape relationship with human capital; MIS shows an inverted U-shape curve with human
capital.
23
𝑡
Proposition 4.4.3 At any point between the two curves
=
and
=
−
, the marginal effects of human capital is negative on the Gini index, but positive on the
Pietra ratio.
4.5 Aggregate Effects of Human Capital and TFP on Income Inequality
Subsections 4.3 and 4.4 show that TFP and human capital can have either positive or
negative effects on income inequality. Since human capital and TFP are closely related, now
we discuss the aggregate marginal-effects (aggregate effects for short) of human capital and
TFP on income inequality. We define the aggregate effects as the total of marginal effects of
human capital and TFP on income inequality.
Using the equations of the subsections 4.3 and 4.4, the aggregate effects of MDS, Gini
index and Pietra ratio are as follows, respectively.
𝑦−
= 𝑡𝑓 ∗ 𝑦−
(𝑡𝑓 ) . 1
(
)
𝑥−
= 𝑡𝑓 ∗ 𝑥−
(𝑡𝑓 ) .4
(
−
(𝑡𝑓 )
2
∗ 𝑥−
)
= 𝑡𝑓 ∗
1. 5
1.
𝑡 = 𝑡𝑓 ∗
(𝑡𝑓 ) 1. 11
∗ 𝑦−
2.343
1.4
2
𝑡
∗
1.
=
1. 5
∗
.5 4
−
2
𝑡
.4
= →
.1 =
−
(
)
=
(
(4.5.1)
→
.12 =
(4.5.2)
=
.2 =
(4.5.3)
→
) .141 =
(4.5.4)
Figure 4.5.1 below graphs the following equations (4.5.1) ~ (4.5.4). There are three sets
deserving special attention on the Figure 4.5.1. The upper part is a set of TFP and human
capital, where an increase of TFP or human capital leads to a reduction of income inequality.
The bottom part on the figure is a set of TFP and human capital, where income inequality
worsens as TFP or human capital is improved. The third part on the figure is the one between
the above two sets, where the Gini index and MIS are reduced, and the Pietra ratio and MPS
are increasing as TFP or human capital increases.
24
Figure 4.5.1 Aggregate Effects of Human Capital and TFP on Income Inequality
We can easily come up with the following empirical propositions; and the corollary 4.5.1
can be reached by computing the minimum point for each curve on the figure.
Proposition 4.5.1 An increase of TFP or human capital can either increase or decrease
the Gini index, Pietra ratio and MDS depending on the combination of the two variables. TFP
must be higher than a particular level for the two factors to have negative aggregate effects
on income inequality (the Gini index, Pietra ratio or MDS).
Corollary 4.5.1 The global minimum TFP is

.846 to reduce the Gini index, Pietra ratio and MDS through increasing TFP or human
capital;

.833 to reduce the Gini index, Pietra ratio, and MIS through increasing TFP or human
capital;

.809 to reduce the Gini index and MIS through increasing TFP or human capital;

.722 to reduce the Gini index increasing TFP or human capital.
Corollary 4.5.2 The aggregate effects of human capital and TFP on income inequality are
zero when the two variables satisfy the corresponding quadratic functions (4.5.1) ~ (4.5.4).
25
Proposition 4.5.2 At any point between the two curves
𝑡
=
and
−
=
, the aggregate effects of TFP and human capital is negative on the Gini index, but positive
on the Pietra ratio.
4.6 Effects of Capital Stock, Employment and Labor Share
Figure A4.6.1 in Appendix III shows the fitted Gini Index and MDS drawn with capital.
The fitted Gini index falls fast at the early stage of capital accumulation before the mean
value, 0.31, and then turns to increase slowly during the long period of capital accumulation,
and so does the fitted MPS; but the fitted MIS falls slowly before the mean value of capital
stock and then tends to be stable around its mean value 0.38 of MIS.
Table 4.2.1 shows that capital accumulation significantly increases MPS, and reduces MIS
with a larger size, and insignificantly increases the Gini index; so that an increase of capital
stock significantly increases the Pietra ratio.
Figure A4.6.2 in Appendix III shows the fitted Gini Index and MDS drawn with labor
share. The fitted Gini index and MPS has been falling with labor share but the estimations are
insignificant. The fitted MIS falls slowly with labor share and the estimation is significant.
All the three statistics tend later to stay near their mean values. Table 4.2.1 in the subsection
4.2 shows that current labor share significantly increases the MIS, thus the Pietra ratio would
be falling with an increasing of labor share.
Figure A4.6.3 in Appendix III shows the fitted Gini Index and MDS drawn with
employment. The fitted Gini index and MPS falls with employment before their mean values,
0.347 and 0.632, respectively, and tend to stay around their mean value later. The fitted MIS
looks stable around its mean value .383. Employment significantly reduces the Gini index
and MPS, and it has an insignificant and positive effect on MIS.
26
The share of government consumption in GDP does not have significant effect on the
three statistics of income inequality.
4.7 Policy Implication
The MDP views overall income inequality by population share and income share defined
at the mean income point. It presents a simple way to evaluate how a policy would affect the
relatively low (mean) income segment of a population.
Since all factors affect the MPS and MIS in opposite ways, MDS and the Pietra ratio have
different economic implications for policymakers. An economic policy may vary with the
two statistics of income inequality.
The most striking implication to policy makers is that a change in TFP or human capital
could worsen income inequality if investment in human capital does not match the TFP level
in an economy. TFP must be higher than a particular level to reduce income inequality
through the improving of human capital. Special attention should be paid to the case that the
Gini index and Pietra ratio move oppositely with a change in TFP or human capital in a
particular set of the two variables.
In reality, some skilled workers may not be able to find proper jobs that occupy their
complete skills. Therefore a worker’s investment in education would not be well
compensated, and that the marginal effects of human capital could be negative for these
workers.
If we want to decrease the Gini index and MPS, then raising employment is, universally,
an effective channel. Investment in human capital is also effective for an economy with
relatively low human capital or high TFP, but this may not be effective for an economy with
a human capital level larger than 0.8 (see Figure A4.4.1 in Appendix III) or TFP index lower
than .846 (see Figure 4.5.1 or Corollary 4.5.1).
27
If we focus on increasing MIS, policymakers should work on increasing labor share, TFP,
and employment. Labor share has a more significant effect on MIS than employment does.
For an economy with TFP index lower than 0.501, an effective tool to reduce income
inequality is firstly to focus on policies that improve TFP (see Figure 4.4.1).
Investment in capital stock increases MPS, has no much impact on MPS, and does not
significantly affect the Gini index in developed economies. Capital tax may strengthen
income inequality due to the potential of decreasing employment, but if the tax revenue on
capital accumulation is invested in education for an economy with a relatively high level of
TFP relative to human capital, then the positive effects on income inequality would be
decreased or even removed; and this policy may also reduce the Pietra ratio and Gini index.
The Gini index can be reduced through better education with a lower level TFP than the MDS
and the Pietra ratio require.
V Conclusion
The mean division point (MDP) consists of all the population whose individual income is
less than or equal to the mean income in an economy. It involves mean population share
(MPS) and mean income share (MIS) that occur at the point of unit slope on a differentiable
Lorenz curve. The difference of MPS from MIS is the maximum Pietra ratio. The Gini index
is about 1.3 times the Pietra ratio in the panel data.
We find that the Gini index and MPS show an inverted U-shape curve with TFP but not
with GDP. An interesting result is that investment in human capital may not be helpful with
the reduction of income inequality if TFP is not high enough. The aggregate effects of a
change in TFP or human capital can be negative on income inequality only when TFP is
higher than a particular level, which is a convex function of human capital. The marginal
effects of a change in TFP or human capital can be different between the Gini index and
Pietra ratio in a particular set of TFP and human capital.
28
We see that capital stock significantly worsens income inequality by increasing MPS and
decreasing MIS as capital stock crosses the mean value in the panel of countries. Labor share
significantly reduces income inequality through increasing MIS. And employment
significantly reduces income inequality through decreasing MPS and the Gini index.
We find that MIS has been slowly decreasing with GDP and TFP, stable with capital
stock; and MPS increases with GDP, TFP and capital stock after the early stage of
development. Income inequality goes up with human capital after the early stage of human
capital development. Thus, we conclude that the population of the low income segments has
become relatively worse off (with lower MIS and larger Pietra ratio and Gini index) as
economies become richer (with higher capital stock, TFP, and per capita GDP) in the panel of
countries. This result requires further discussion when the division of income distribution is
defined at another point on the Lorenz curve by a different slope.
Acknowledgement
I am grateful to the participants in the seminars at Henan University School of Economics
and University of Miami Department of Economics. Special thanks go to Professor Carlos
Flores, Kaixiang Peng and Dr. Manuel Santos for their helpful comments. I thank my good
friend Mr. Brandon Fitzpatrick for his proofreading.
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