Decomposing Household Income by Source and Subgroup - Alex Eble

Decomposing Household Income by Source and Subgroup:
A Methodological Investigation Using Survey Data from Rural China
Alex Eble 1
Indiana University, Bloomington
1 Alex Eble, correspondence Address: 9093 Sweet Bay Court, Indianapolis, IN 46260.
Thanks are due to Ethan Michelson for impeccable guidance and access to his data.
The Department of East Asian Languages and Cultures and the Hutton Honors
College at Indiana University, Bloomington, provided much needed resources to help
with the production of this paper.

Decomposing Household Income by Source and Subgroup
I. Introduction:
Gross inequality is widely considered an undesirable social condition to be mitigated
by policy, as evidenced by diverse social programs instituted by government to
encourage more equitable distribution of resources. This social conviction, in turn,
encourages policy makers and social scientists to better understand the current
condition and sources of inequality. Massive ascent from poverty and rapid uneven
development experienced in countries such as China and India only amplify this need.
With the cooperation and guidance of Ethan Michelson I have been working to draw
on economics to make a contribution to sociological research on income inequality.
Sociologists focusing on China have given extreme amounts of attention to inequality
at individual and regional (and even international) levels while overlooking
investigation into sources of income. This paper looks into various methods, designed
by economists, of describing inequality in terms of its contributing factors. It then
applies these methods to a data set of rural household survey responses from China.
The inequality indices used can “decompose,” meaning that the overall measure can
be further explained or broken down into the sum of individual contributions to
inequality. In one method of break-down, the paper looks at the contribution to
inequality from each of five sources of income. In another method, the paper shows
what proportion of inequality comes from income differences within villages, within
counties but between villages, and between counties. The first method can identify
income sources which are equalizing or, conversely, disproportionate contributors to
overall inequality. The second measure can help identify where to further investigate
geographical inequality.
The second section of this paper briefly explains the relevant research on inequality
indices and how it can be used to choose a few indices from a broad field of
possibilities. The third and fourth sections explain two of these indices’ strengths and
weaknesses. The fifth gives a brief explanation of how these indices are applied to the
data. The sixth section displays the results and the last section interprets issues raised
by the data while suggesting further investigation.
II. Narrowing the Field of Income Inequality Indices
There are numerous mathematical measures to describe inequality and several
methods with which to choose these measures. Among the most successful attempts at
the latter are those performed by Francois Bourguignon and Anthony Shorrocks. In
specifying the conditions for selection of an inequality index for income distribution,
Bourguignon and Shorrocks agree upon a set of simple characteristics which define a
sound inequality index. Namely, they suggest that the index, a function of the set of
incomes:
a) Be continuous and differentiable over all individuals’ incomes
1

Decomposing Household Income by Source and Subgroup
b) Be symmetric (the personality of income earners should not affect the value of the
index)
c) Does not vary when all incomes are multiplied by a scalar
d) Satisfy the symmetry axiom for population (the index for a given distribution must
be the same as that for a distribution obtained by replicating any number of times
each individual income in the initial distribution)
e) Satisfy the Pigou-Dalton condition (the inequality measure must decrease with a
transfer from rich to less rich people that does not reverse their relative position in
the distribution)
f) Be decomposable (the index must be able to be expressed as the sum of a within
group inequality term and a between group inequality term) (Bourguignon 1979,
and Shorrocks 1980, 1982)
These conditions, agreed upon by Bourguignon and Shorrocks, limit the possible
indices to the class of generalized entropy measures given by the following equation:
n 1 1 y
p
GE ( )
1
(1)
2
n p1
y
y p represents income of an individual p, y is the group’s mean income and n is
population. The most widely used indices in the generalized entropy family are the
two Theil indices and half the square of the coefficient of variation (CV), 2
corresponding to equal to zero, one, and two, respectively in the above class.
Shorrocks looked more closely into the nature of decomposing income inequality and
the traits of these indices. Going one step beyond those assumptions agreed upon by
Bourguignon, Shorrocks suggests the following two assumptions to further narrow the
field of acceptable, decomposable inequality indices:
The contribution from a source to inequality must be zero if income from that
source is equal
Two income components who are identical and together comprise total income
must have the same contribution to income inequality
The importance of these extra constraints is that they ensure that the decomposition
rule for any inequality index is unique and that the relative value of income
components’ contribution to inequality is independent of the choice of index. This is a
major asset to social scientists wishing to decompose inequality as it ensures that
comparison can be made across inequality indices that satisfy these assumptions. Both
Theil’s T index and the CV satisfy these constraints and are the two indices which will
be used in the following analyses. (Shorrocks 1982)
2 Half of the square of the coefficient of variation will be referred to as “the CV” in this paper.
2

Decomposing Household Income by Source and Subgroup
III. Theil’s T
Theil’s T index, the member of the general entropy family of inequality indices
corresponding to = 1, is the sum of each individual’s contribution to total inequality.
Theil’s T index weights a data point’s (individual’s) population share and distance
from the mean through the following equation:
T
1
n
p
p
* * ln
p1 n y y
y
In this index, a data point gives a contribution to the overall index based on a
decreasing function of the probability of its occurrence. In other words, given a
normal distribution, the further from the mean an individual’s income is the greater is
that individual’s contribution to the inequality index. (Theil 1967)
Another interesting characteristic is Theil’s T’s non-linearity. As the richest half of the
population’s share of income increases linearly Theil’s T index increases at a more
than linear rate. This phenomenon is due to the decreasing nature of the negative
contribution to overall inequality. When there is total equality in the group, Theil’s T
reaches its minimum, zero. (Conceição and Ferreira, 2000) 3
According to scholars at the University of Texas Inequality Project, (UTIP) a think
tank focusing primarily on the measure and analysis of inequality, the main advantage
of Theil’s T index is the facility with which it decomposes inequality into between and
within group components. Another strength of Theil’s T is its capacity to analyze
inequality from aggregated data is this manner. Several other indices, including the
Gini Coefficient and the CV, require comprehensive individual-level data which is
often unavailable to social scientists. (UTIP 2005) Sicular and Morduch (2002)
show that Theil’s T index can also readily be decomposed among factor incomes. A
prior complaint about Theil’s T was that due to its logarithmic nature, it would be
undefined under negative and zero incomes. Sicular and Morduch show that Theil’s T
can be decomposed for income components and furthermore that Theil’s T is in fact
defined for zero and negative factor income values in the following equation:
s
k
TT
1
n
1
n
n
p1
n
p1
y
y p
ln
y
y p
ln
y
They go on to laud the benefits of Theil’s T as compared to other more frequently
used inequality measures: “The Gini coefficient falls if an income source is increased
by a constant amount for all members of a population”, a desirable characteristic, “but
none of the components of the standard decomposition of the Gini are affected,”
ignoring what we hope to measure as a decrease in income inequality for the given
3 For graphical representation of this phenomenon, please also refer to Conceição and Ferreira, 2000.
k
p
p
(2)
(3)
3

Decomposing Household Income by Source and Subgroup
component. Decomposed, Theil’s T shows such a reduction in inequality. Sicular and
Morduch add that “the Theil-T decomposition provides a better indicator of why the
overall index takes its given value in the first place…the Theil-T index is thus
potentially of greater use to researchers.” (Morduch and Sicular, 2002)
The downsides of Theil’s T, according to UTIP, are that it has no intuitive motivating
picture, cannot directly compare populations with different sizes or group structures,
and that it is comparatively mathematically complex. (University of Texas Inequality
Project 2005) In the words of Amartya Sen, Theil’s T “is an arbitrary formula, and
the average of the logarithms of the reciprocals of income shares weighted by income
is not a measure that is exactly overflowing with intuitive sense.” (Sen 1997)
Theil’s T index also yields a simple hierarchical decomposition among nested regions.
Decomposing Theil’s T index by regions renders the following equations. Total
inequality as the sum of between and within group equality is given by:
With between group inequality given by:
T
B
k
gi1
Y g is income in group g. Y is total income, g
T T T
(4)
B
W
Yg
Yg
ng
ln
(5)
Y Y N
n is population of group g and N is total
population. The between group component is obtained by replacing the income of an
individual with the mean income of the individual’s respective subgroup. (Shorrocks
and Wan 2004)
Within Group Inequality is given by:
where
T
k Yg
T W '
* TW
gi1
Y
(6)
y
ng
gp
w
p1 Yg
g
y
gp
1
ln /
(7)
Yg
n
Akita (2000), recognizing that even this decomposition method resulted in undesirable
use of regional mean incomes as opposed to household level incomes, devised a
method of hierarchical inequality decomposition by which a three-tiered country
structure (region, province, and district) could be decomposed into within province,
between province, and between region contributions to overall inequality. 4 The
equations for further decomposition are then:
4 Note that the sum of between province and within province contributions is identically equivalent to the sum of
the within region contributions.
4

given
and
Decomposing Household Income by Source and Subgroup
T
WP
T T T T
(8)
WP
BP
BR
Yij
yijk
yijk
/ Yij
ln (9)
j k Y
Yij
nijk
/ N
i ij
T
BP
T
BR
Yij
Yij
/ Yi
ln
(10)
Y
N ij / N
i j i
Yi
Yi
/ Y
ln
(11)
i Y Ni
/ N
Where i, j, and k represent regions, provinces, and districts, respectively. (Akita 2000)
IV. The CV - Half of the Square of the Coefficient of Variation
Another widely used index to measure inequality, the coefficient of variation is the
quantity squared of the standard deviation divided by the mean value of the set of
responses:
var( y)
I CV ( y)
(12)
2
One half of the square of this function is the equation derived from the general
entropy class of inequality indices for α = 2. This index, herein referred to as the CV
as mentioned before, also satisfies the requirements set out by Bourguignon and
Shorrocks.
Initial arguments for use of the CV included the fact that it is much more intuitive
than Theil’s T. Also, in using group data weighted by population size, the CV is not
easily skewed by small outliers. Furthermore, it is defined under any form of zero and
negative incomes, a characteristic that many thought Theil’s T lacked. The main
drawback of the CV is that it is particularly sensitive to income transfers in the upper
tail of the income distribution, a less than ideal condition. (University of Texas
Inequality Project 2005) Bourguignon explains that the CV “offers the inconvenience
of referring implicitly to a utilitarian welfare function with convex individual
utilities,” pointing to the upper tail sensitivity. (Bourguignon 1979)
Decomposing the CV, proportional contributions of income factor components are
given by:
k
k cov( y , y)
SCV
(13)
var( y)
5

Decomposing Household Income by Source and Subgroup
k
y is income received from income component k and y is total income.
Sicular and Morduch (2002) suggest that satisfying the property of uniform additions
is another desirable characteristic for an inequality index. The property of uniform
additions “holds that measured inequality should fall if everyone in the population
receives a positive transfer of equal size (or, conversely, that inequality should
increase if everyone receives an equal, negative transfer).” They go on to show that
the coefficient of variation, as well as the CV, do not satisfy the property of uniform
additions. Theil’s T index, on the other hand, does satisfy this property. (Morduch and
Sicular, 2002)
V. Application to the Data
Data from the 2002 Rural Household Survey are used for the following analysis. This
survey was conducted in 5 provinces – Shaanxi, Jiangsu, Henan, Hunan, and
Shandong – and in rural areas contained in the jurisdiction of the autonomous city of
Chongqing. These locations were chosen so as to maximize regional and economic
variation within the sample and thus are not necessarily representative of greater rural
China. Differences within the sample, both geographic and economic, are great.
The respondents comprise almost 3,000 households clustered in 37 villages.
Incidentally, these responses are surprisingly representative of rural China as a whole
when compared to those official estimates for Chinese income listed in the China
County (City) Social and Economic Statistical Yearbook 2002. Per-capita household
income within the survey responses, for example, differs from official values by a
narrow margin of one to seven percent depending on income measure. The survey
respondents are not representative of the actual distribution in the counties surveyed
(the mean age was 64 and 55% of respondents are male), however the household
information seems to resemble the general regional distribution much more closely.
(Michelson 2005)
The main data used in this analysis are household income data. The data are responses
to 6 questions that inquire about income values. Overall income was asked for as well
as revenues from five possible component parts of income. These parts are agriculture,
sidelines 5 , family business, remittances, and craftsmanship. The overall income will
be called “Single Income Response” for the purposes in this paper, and the sum of the
five component parts will be called “Income Composite.”
In this exercise I calculated Theil’s T and the CV for the data set. 6 In terms of
5 The sidelines entry includes income from animal husbandry, fish farming and other sources.
6 Within the Stata® statistical software package, I used Stephen Jenkins “ineqdeco” program to calculate income
inequality and its various decompositions. This program calculates a variety of inequality indices, including the
three of the family of Atkinson’s Indices and four forms of the generalized entropy family. In light of the previous
discussion, Theil’s T index and the CV are the inequality indices used in this analysis. Due to the logarithm in
6

Decomposing Household Income by Source and Subgroup
regional contributions to inequality, I calculated “within group” and “between group”
values for decomposing inequality at both the village and county level. The next
calculation made is to compute the “within-county, between village” component of
income inequality. This is the difference of between-village inequality and between
county inequality. The sum of within-village inequality, within-county, between
village inequality, and between county inequality is also overall inequality. 7
A discussion of sample size
Decomposing inequality across income components raised concern about issues of
sample size. There were very few survey respondents who recorded income from all
five sources. It seemed that decomposing income across the sources meant looking at
income inequality within groups of only those who earned income from a given
source. This would lead to a difference in sample size, as the number of farmers was
not equal to that of entrepreneurs, for example.
After discussion and consultation, I decided to count those not reporting income from
a given source as having zero income from that source. With time-series data, it is
reasonable to assume that most farmers would have a mean positive income from
farming whereas non-farmers would still have no farming income. The fact that our
data is cross-sectional leaves the question of how to differentiate between those
farmers who earned no income that year from farming and non-farmers. This is a
vagary enforced by the nature of the data. To determine an income source’s
contribution to total income inequality, the index by definition must include all
individuals’ income from the given source. Inherently, then, the sample size for all
income components will be identical. 8
VI. Results
Through Stata’s ‘ineqdeco’ inequality function, I investigated the regional
decomposition and income component decomposition of two inequality indices,
Theil’s T index and the CV. The results are shown numerically (proportionally) in
table one:
Theil’s T and the nature of Jenkins’ algorithm, “ineqdeco” cannot include negative and zero values for income or
any of its components. To adjust, zero and negative values were assigned an arbitrarily low positive value, 1% of
the mean This method was suggested by Glenn Firebaugh of Pennsylvania State University in personal
correspondence with Ethan Michelson on Thursday, April 28 th .
7 This method was verified by calculation and initially suggested by Ethan Michelson.
8 This method was suggested by several social scientists in personal correspondence with Alex Eble and Ethan
Michelson. These individuals include Jonathon Morduch (June 19 th , 2005), Stephen Jenkins (June 20 th , 2005),
Dwayne Benjamin (June 21 st , 2005), and Terry Sicular (June 24 th 2005).
7

Decomposing Household Income by Source and Subgroup
Table 1 –Income Components Decomposed Across Regions
Table 2 – Income Component Contributions to Total Inequality
Table 3 – Regional Decomposition of Inequality
The analysis begins with regional contributions to inequality, the results of which are
graphically displayed in figures three and four. For both Theil’s T’s index and the CV,
inequality within villages contributed the most to total inequality. This finding is in
harmony (indeed, almost perfect correspondence) with the large-scale income
8

Decomposing Household Income by Source and Subgroup
inequality investigation recently performed by Benjamin, Brandt and Giles. (2005)
Inequality between counties also contributed significantly to overall inequality.
Contribution to inequality from the differences within a county between villages,
however, was low. The high within-village contribution shows large inequality
between individuals within a village, perhaps the difference between government
officials/businessmen and farmers. The contribution from inter-county inequality
could also suggest preferential political treatment for certain counties, although this is
much more likely the product of differences in physical and human capital
endowments and the physical characteristics of the area that condition the amount of
government investment received. (Jalan and Ravallion 2002: 343) Inequality between
villages within a county does not exceed ten percent of total inequality for either
index, a marginal contribution when compared to the other two regional contributors.
This suggests that the average incomes of villages are fairly evenly distributed within
each county.
Contriubution to Income Inequality
(Theil's T)
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Single Income
Response
Income
Composite
Inter-county inequality
Within-county, intervillage
inequality
Intra-village inequality
Figure 1 – Regional Contributions to Overall Income for the CV
9

Contribution to Inequality (CV)
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
Decomposing Household Income by Source and Subgroup
0%
Single Income
Response
Income Composite
Inter-county inequality
Within-county, intervillage
inequality
Intra-village inequality
Figure 2 – Regional Contributions to Overall Income for Theil’s T
To further test theories about natural resources I examine the regional contributions to
inequality for different income factors. As shown in figures five and six, inequality
stemming from within-village income differences still dominates inequality
contributions for each income factor. The character of its dominance, however, varies
among factors. In agriculture, differences between villages within a county and
between counties account for 30 percent of overall inequality according to Theil’s T
index. 9 Again, this seems likely to be the result of geographical and political factors
as in Jalan and Ravallion, i.e. geographically-conditioned access to public resources.
Inequality in earnings from family business, on the other hand, stems almost entirely
from differences within a village. This is also to say that revenues from family
business are fairly equally distributed between villages and counties.
Income from remittances, however, has a different structure of contributions to
inequality. Over 20 percent of inequality in income from remittances is accounted for
by differences across counties. To understand this, one has to better understand the
nature of migration in China. Regional economic disparity, a local population’s level
of human capital, a location’s proximity to urban centers, and a given location’s
population density are all positive correlates of the likelihood to migrate and thus of
the amount of remittances as they are calculated in this analysis. 10 (Fan 2005) Over
10 percent of inequality springing from income from handicrafts can be accounted for
by revenue differences in handicrafts between villages. This could be indicative of
cultural differences between villages and as well as the presence of village-level
cottage industries. According to field studies of Thai ethnic craft making, the success
9
Notice that this is only 10 percent according to the CV. The significance of this difference will be discussed
shortly.
10
Remittances are the response to question A6 of the survey: “In Question A1, you mention that some household
members are leaving the village to work. What’s their total income this year?”
10

Decomposing Household Income by Source and Subgroup
of these crafts depends on commercial ties outside the village, which in turn is also
dependent upon infrastructure and, to an extent, proximity to population centers.
(Cohen 2000)
Of income inequality in revenue from sidelines, a mere 5 percent can be explained by
differences between villages within counties and between counties. This is also to say
that almost all of the inequality in revenues from sidelines, as in family business, rises
from differences between individuals within a village. Said differently, revenue from
sidelines is relatively equally distributed across villages and counties.
Contribution to Inequality (CV)
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Agriculture
Family Business
Remittances
Handicrafts
Source of Income
Sidelines
Inter-county inequality
Within-county, intervillage
inequality
Intra-village inequality
Figure 5 – Income Factor Inequality Decomposed by Regional Contribution Given by
the CV
Contribution to Inequality
(Theil's T)
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Agriculture
Family Business
Remittances
Handicrafts
Source of Income
Sidelines
Inter-county inequality
Within-county, intervillage
inequality
Intra-village inequality
11

Decomposing Household Income by Source and Subgroup
Figure 6 – Income Factor Inequality Decomposed by Regional Contribution Given by
Theil’s T
The next task is to analyze how each income source contributes to overall inequality. 11
Figures seven through twelve illustrate these contributions graphically. Looking at the
following six graphs, there are several obvious conclusions:
Family business contributes the most to inequality within villages by a
fantastically large margin.
Agriculture contributes very little to inequality at any level.
Within counties but between villages, sidelines, craftsmanship and family
business all play large roles in contributing to inequality, although family
business still plays the most prevalent role.
Between counties, remittances are the largest contributor by far to inequality.
This also shows that between counties the incidence of and/or revenue from
family business and sidelines is more equal than that coming from individuals
leaving home to work.
In the overall income inequality decomposition of Theil’s T as in that of the
CV, family business is by far the main contributor to inequality. This is an
unsurprising finding. Jenkins (1995) and Papatheodorou (1998) both find
similar contributions to inequality in their analysis of income factors’
contribution to inequality in the United Kingdom and Greece, respectively.
One of the more interesting results found in this investigation is the difference
between the two indices that the algorithm yields for proportional contributions to
inequality. As stated before, Shorrocks (1982) asserts that “the relative [sic]
importance of different income components is independent of the choice of inequality
measure.” If “relative proportion” is taken to mean proportional contribution, the
values that ineqdeco yields for the contribution of income components for Theil’s T
and the CV violates Shorrocks’ assertion. Numerous tests of Jenkins’ ineqdeco
algorithm all yielded the same result: the proportional contributions of income
components are not independent of the choice of inequality measure. This warrants
further investigation into the nature of Jenkins’ algorithm and its relation to
Shorrocks’ assertions, both which lie beyond the scope of this paper.
11 Shorrocks cautions against two ways of interpreting this data. In the past, this contribution has been interpreted
as either a) the inequality that would be observed if income component k was the only source of income
differences; or b) the amount by which inequality would fall if differences in a factor’s income receipts were
eliminated. Unfortunately, neither provides a consistent decomposition rule. This inconsistency leads to the
conclusion that they are both to be avoided when making policy recommendations.
12

Contribution to Total Inequality
(CV)
Decomposing Household Income by Source and Subgroup
40
35
30
25
20
15
10
5
0
Agriculture
Family Business
Remittances
Handicrafts
Source of Income
Sidelines
Inter-county inequality
Within-county, intervillage
inequality
Intra-village inequality
Figure 7 – Absolute Income Factor Inequality Decomposed by Regional Contribution
Given by the CV
Contribution to Inequality
(Theil's T)
3.5
3
2.5
2
1.5
1
0.5
0
Agriculture
Family Business
Remittances
Handicrafts
Source of Income
Sidelines
Inter-county inequality
Within-county, intervillage
inequality
Intra-village inequality
Figure 8 – Absolute Income Factor Inequality Decomposed by Regional Contribution
Given by Theil’s T
13

Income Source Contribution to Inequality (CV)
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Decomposing Household Income by Source and Subgroup
Intra-village
inequality
Withincounty,
inter-village
inequality
Inter-county
inequality
Total
Inequality
Sidelines
Handicrafts
Remittances
Family Business
Agriculture
Figure 9 – Regional Contributions to Income Inequality Decomposed Proportionally
by Income Component Given by the CV
Income Source Contribution to Inequality
(Theil's T)
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Intra-village
inequality
Withincounty,
inter-village
inequality
Inter-county
inequality
Total
Inequality
Sidelines
Handicrafts
Remittances
Family Business
Agriculture
Figure 10 – Regional Contributions to Income Inequality Decomposed Proportionally
by Income Component Given by Theil’s T
14

Income Source Contribution to
Inequality (CV)
50
45
40
35
30
25
20
15
10
5
0
Decomposing Household Income by Source and Subgroup
Intra-village
inequality
Within-county,
inter-village
inequality
Regional Contribution
Inter-county
inequality
Sidelines
Handicrafts
Remittances
Family Business
Agriculture
Figure 11 – Absolute Regional Contributions to Income Inequality Decomposed by
Income Component Given by the CV
Income Source Contribution to
Inequality (Theil's T)
6
5
4
3
2
1
0
Intra-village
inequality
Within-county,
inter-village
inequality
Regional Contribution
Inter-county
inequality
Sidelines
Handicrafts
Remittances
Family Business
Agriculture
Figure 12 – Absolute Regional Contributions to Income Inequality Decomposed by
Income Component Given by Theil’s T
VII. Conclusions
This paper offers a foundation from which to choose possible measures of income
inequality. From this basis, it goes on to explain the capabilities of those measures
which are seen most attractive under the lens of Shorrocks and Bourguignon’s rules
15

Decomposing Household Income by Source and Subgroup
for a measure of inequality. Two of these measures, Theil’s T index and the half the
square of the coefficient of variation, herein referred to as the CV, are then applied to
the responses to the 2002 Rural Household Survey. Inequality across geographical
regions is measured through decomposition of the two indices. The same process is
applied to identify and analyze contributions to inequality from five mutually
exclusive income sources.
Overall, inequality stemming from income differences within villages contributes
much, much more to overall inequality than inequality from differences in income
between villages within a county and between counties. Of this inequality, much was
accounted for by the income inequality in earnings from family business. Differences
between counties, however, still contribute almost 35 percent to overall inequality
according to Theil’s T Index. This is seen as exposing inequality in resource allocation,
likely both political and natural.
According to Theil’s T for inequality between counties, income inequality from
remittances and handicrafts contributes 70 percent of all inequality. The prominence
of these two factors suggests differences in natural resource distribution and proximity
to infrastructure as a likely cause of between county income inequality. Pro-poverty
migration policies, targeted investment programs, and expansion of infrastructure are
all possible means to alleviate this inequality. Income inequality rising from family
business, sidelines and handicrafts was responsible for all but 15 percent of inequality
between villages within counties. This again points to differences in natural resources
as a potential point of departure from income equality (some areas lend themselves
more easily to fish farming or souvenir sales than others, for example) and also points
to the aforementioned means to alleviate inequality stemming from this source.
Income inequality within counties but between villages, however, contributed only a
very small portion of overall income inequality and should form a policy focus point.
The obvious culprit as the main income factor contributor to overall inequality was
family business. Income inequality stemming from family business revenues accounts
for nearly 80 percent of overall inequality according to the CV and nearly 40 percent
according to Theil’s T index. The policy conclusion to be drawn from this mirrors that
of Papatheodorou (1998):
“The reduction of the inequality of entrepreneurial income appears to be
the most effective way to reduce total inequality in Greece. It is, therefore,
of great importance to redesign the current tax system in Greece in order
to eliminate the tax evasion among the recipients of entrepreneurial
income. This policy could prove the most efficient, if not only way, to
significantly reduce income inequality.”
Similarly, in the areas investigated in this paper family business is a significant
contributor to every regional slice of inequality and also contributes the lion’s share to
16

Decomposing Household Income by Source and Subgroup
overall inequality. Through instituting new tax policy and/or revitalizing existing
policy to curb the gross inequality stemming from this source of income, China could
make enormous strides in reducing income inequality. Conversely, without attention
paid to reducing income inequality stemming from family business, government
policy aimed at reducing income inequality could at best finish only a little more than
half of the job.
Looking at the other end of the spectrum, agriculture contributes almost nothing to
overall inequality (never more than five percent in any index). This finding has
equally strong policy implications. Policies geared at increasing income have
potentially inequality-inducing effects. Whereas a government-sponsored business
incubator would likely increase inequality in light of the results of this paper,
government investment in raising income from agriculture would almost certainly
reduce inequality.
Also interesting, remittances are the main contributor to inequality between counties.
Further investigation into literature on migration could reveal what factors contribute
to an individual’s decision to leave a village and remit money. Existing networks of
villagers, limited information, prosperity, and proximity to travel infrastructure are all
possible inputs. Policy towards alleviating inequality could improve access these for
the poor and thus affect the amount of resources flowing into a region from
remittances to accordingly even distribution across counties.
The main surprise that came from this investigation is a methodological one. The
relative contributions of income components and regional income inequality as
measured by the ineqdeco function in Stata® may violate Shorrocks’ assertion that the
relative contribution to income inequality of various components is independent of the
choice of inequality index. Certainly, the question remains: is family business
responsible for 80 percent of all income inequality or 40 percent? As can be seen
numerically in figure one and graphically in several of the preceding graphs, there are
numerous other instances in which the relative contribution of an income component
or regional contribution is different according to the CV than that given by Theil’s T.
In most cases the general structure of inequality contributions remains the same.
Family business and intra-village inequality, for example, contribute the lion’s share
to overall inequality for both indices. The proportional contributions of the several
income components, however, differ frequently and noticeably between the two
indices, as does regional decomposition. This result suggests one of four possibilities.
The most obvious possibility is that I committed a mathematical error in calculations.
It is also quite possible that I am misinterpreting Shorrocks’ use of the phrase “relative
contribution.” The other two less likely although more interesting possibilities are that
either Shorrocks made an error in his assertion or that Jenkins did so in writing his
algorithm.
For further investigation, time series data would help distinguish non-farmers from
17

Decomposing Household Income by Source and Subgroup
farmers and yield a more reliable data set from which to draw analyses. The analyses
themselves could be improved greatly by resolving the apparent contradiction
between Shorrocks’ claim and Jenkins’ algorithm. As it stands, the differences in the
results between the two indices are large enough to weaken any policy advice derived
from this analysis. Also, adaptation of Sicular and Morduch’s Theil’s T decomposition
into the ineqdeco or ineqdec0 algorithm would allow for greater precision –
preventing the substitution of one percent of the mean for the negative values and
zeroes. Akita’s nested decomposition equation would also add to the precision and
scope of the decomposition algorithm.
18

References:
Decomposing Household Income by Source and Subgroup
1. Akita, Takahira. 2000 “Decomposing Regional Income Inequality using a
Two-Stage Nested Theil Decomposition Method.” International Development
Working Paper Series 2, IUJ, Research Institute, International University of
Japan
2. Benjamin, Dwayne, Loren Brandt and John Giles. 2005. “The Evolution of
Income Inequality in Rural China.” Economic Development and Cultural
Change. 53(4)
3. Bourguignon, Francois. 1979. “Decomposable Income Distribution
Measures.” Econometrica. 47
4. Conceição, Pedro and Pedro Ferreira. “The Young Person’s Guide to the Theil
Index: Suggesting Intuitive Interpretations and Exploring Analytical
Applications.” UTIP Working Paper Number [14 February 29, 2000]
Available: http://utip.gov.utexas.edu/abstract.htm#UTIP14
5. Cohen, Erik. 2000. The Commercialized Crafts of Thailand: Hill Tribes and
Lowland Villages; Collected Articles. Honolulu: University of Hawaii Press,
6. Fan, C. Cindy. 2005. “Modeling Interprovincial Migration in China,
1985-2000.” Eurasian Geography and Economics. Palm Beach: Apr/May
2005. 46(3)
7. Jenkins, Stephen. 1995. “Accounting for inequality trends: decomposition
analyses for the UK 1971-86.” Economica, 62
8. Michelson, Ethan. 2005 “Peasants’ Burdens’ and State Response: A
Preliminary Explanation of State Concession to Popular Tax Resistance in
Rural China,” Unpublished manuscript
9. Morduch, Jonathan and Terry Sicular. 2002. “Rethinking Inequality
Decomposition, with Evidence from Rural China.” The Economic Journal
112(93)
10. Papatheodorou, Christos. 1998 “Inequality in Greece: An Analysis by Income
Source.” Discussion Paper No. DARP 39
11. Sen, Amartya. 1997. On Economic Inequality. Oxford: Clarendon Press.
12. Shorrocks, Anthony. 1982 “Inequality Decomoposition by Factor
Components” Econometrica, 50(1)
19

Decomposing Household Income by Source and Subgroup
13. Shorrocks, Anthony. 1980. “The Class of Additively Decomposable Inequality
Measures.” Econometrica 48
14. Shorrocks, Anthony and Guaghua Wan. 2004. “Spatial decomposition of
Inequality.” WIDER Discussion Paper No. 2004/01.Available at
http://www.wider.unu.edu/publications/publications.htm
15. Theil, Henri. 1967. Economics and Information Theory. Amsterdam:
North-Holland
16. University of Texas Inequality Project. 2005. “Measuring Inequality.” Austin,
Texas: University of Texas Inequality Project. Retrieved June 12 th , 2005.
Available:
http://utip.gov.utexas.edu/web/Tutorials_Techniques/Introduction%20to%20In
equality%20Studies.ppt
20