WPS6645
Policy Research Working Paper 6645
A Note on the Simple Algebra of the Shared
Prosperity Indicator
David Rosenblatt
Tamara J. McGavock
The World Bank
Development Economics Vice Presidency
Operations and Strategy
October 2013
Policy Research Working Paper 6645
Abstract
One of the two goals of the World Bank Group’s new 40 percent and the growth of the average income of
strategy is to promote shared prosperity, defined as the total population. This paper presents: (i) a brief
the income growth of the bottom 40 percent of the discussion of the properties of the indicator; (ii) the
population. The simple monitoring indicator then simple decomposition in algebraic form; (iii) a graphical
is the income per capita of the bottom 40 percent method for displaying the combinations of the two
of the population. The growth of this indicator can components of the decomposition; (iv) simulations of the
be decomposed into two components: the change decomposition for hypothetical countries; and (v) some
in the share of total income accruing to the bottom illustrative data.
This paper is a product of the Operations and Strategy Unit, Development Economics Vice Presidency. It is part of a larger
effort by the World Bank to provide open access to its research and make a contribution to development policy discussions
around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author
may be contacted at drosenblatt@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
A Note on the Simple Algebra of the Shared Prosperity Indicator 1
David Rosenblatt and Tamara J. McGavock 2
Keywords: Economic growth, income distribution, inequality, development
JEL Codes: D6, D63, E01, I3, O1, O4
Economic Policy Sector Board
1
The authors would like to thank Kaushik Basu for helpful discussions.
2
World Bank and Cornell University, respectively. Questions and comments can be sent to:
drosenblatt@worldbank.org
I. Introduction
The World Bank Group has established two goals to guide its strategy in the coming
years (World Bank, 2013a). The first goal is to end extreme poverty, as defined by the global
extreme poverty measure of $1.25 a day, adjusted for Purchasing Power Parity (PPP). More
specifically, the World Bank Group seeks to orient its programs so that the global extreme
poverty rate reaches 3 percent by 2030. The second goal is to promote “shared prosperity” by
helping every country to foster income growth of the bottom 40 percent of the population. These
two goals recognize that the process must be sustained over time and across generations: in other
words, this requires promoting environmental, social and economic sustainability.
The second goal has its intellectual origins in the concept of “quintile income” (Basu,
2000). Quintile income was proposed as a simple, easy-to-calculate, and easily understood
welfare measure, defined as the income per capita of the bottom quintile (20 percent) of the
population. It draws on Rawlsian notions of promoting the welfare of the least fortunate
members of society, while also possessing the pragmatic feature of comparability with traditional
macroeconomic welfare measures like income per capita. In a developing country context, one
may consider the bottom two quintiles – or 40 percent – as more appropriate for a variety of
practical considerations. Most notably there is the fact that – even in rapidly growing emerging
market economies – the people in the bottom 40 percent have incomes that are well below the
poverty line in rich countries. 3 On the other end of the spectrum, in very poor countries, all of
the people in the bottom 40 percent are below the extreme poverty line. One would hope that –
with economic growth, job creation, and equity-enhancing social programs—these poor people
would not only escape this extreme degree of poverty but also progress to more prosperous
living standards.
II. Shared Prosperity: Definition and General Properties
Following Basu (2000), the income profile of a country with n persons can be written as
the vector x = (x1, x2, … xn) of nonnegative numbers, where each element of the vector represents
the income of person i. Assume that the elements of the vector x are organized from lowest to
highest income, or more formally: x1 ≤ x2 ≤ … ≤ xn. The number n is the total population size of
the country, and one can define r as the largest integer such that r/n ≤ 0.4. The shared prosperity
indicator can then be written as:
() = [1 + 2 + ⋯ + ]/
3
See Milanovic (2012) for a discussion of these cross-country differences.
2
This indicator is based upon the concept of “quintile income” (Basu, 2000); however,
SP(x) here is based on a larger population share – the bottom two quintiles, or bottom 40 percent
of the population – rather than just the bottom quintile. The shared prosperity indicator,
however, has the same broad properties as quintile income. A thorough and in-depth discussion
of the axiomatic properties of this welfare measure is beyond the scope of this paper. In general
terms, following the standard definitions (Campbell and Kelly, 2002), SP(x) is a complete,
transitive ordering over the set X of all possible vectors x. This derives directly from the
definition of SP(x) as an average: the relation takes any vector and converts it into a number that
can be compared in the same way that all real numbers can be compared. It is not a one-to-one
relation: infinitely many vectors have average values for its elements that are equal to the same
number.
As noted in Basu (2000), quintile income and the SP(x) satisfy the criterion of
anonymity: if two countries have the same income vector, but with a random reassignment of
incomes across people, then the two societies would still have the same value of SP(x). In other
words, it does not matter who is the first or rth member of that society. The shared prosperity
indicator also satisfies the weak Pareto principle: if every individual’s income rises – that is,
every xi rises – then society is considered “better off.”
Following Basu (2000), it is important to note that the shared prosperity indicator does
not satisfy the “weak transfer axiom,” since lump-sum transfers from a richer individual within
the bottom 40 percent of the distribution to a poorer individual within the bottom 40 percent
would not lead to any improvement in the value of the indicator (Basu, 2000). More generally,
any change in the distribution of income that maintains the same total share accruing to the
bottom 40 percent would not affect the value of SP(x). In other words, two different income
profiles that add up to the same share of total income for the bottom 40 percent, but which have
different degrees of inequality, would still have the same value of SP(x). In Figure 1, the Lorenz
curves that cross at the 40-20 point represent two distributions of income where the bottom 40
percent receive 20 percent of income. A transformation that leads to more inequality within the
bottom 40 percent – a shift from the solid red (B) to the dashed blue line (A) – would have no
impact on measured shared prosperity. In addition, one can construct Lorenz curves, A and B,
where A weakly dominates B but where the curves touch at the 40th percentile of population line.
In that case, the value of the shared prosperity indicator would be equal in both cases, despite the
weak dominance from an inequality perspective (Basu, 2013).
3
Figure 1: Illustrative Lorenz Curves
100
90
80
70
Share of Income (%)
60
A
50
40
B
30
20
C
10
0
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Share of Population (%)
Differences in SP(x) across countries will be driven by differences in per capita
household income and differences in the share of income. One can imagine two countries with
income profiles x and xꞌ whereby xi > xiꞌ, for every i. Clearly SP(x) > SP(xꞌ), and this is simply
an illustration of the Pareto Principle. It should be noted, however, that it could conceivably be
the case that the Lorenz curve associated with xꞌ is similar to the blue-dashed Curve A in Figure
1, while the Lorenz curve for income profile x is associated with the double green line, Curve C.
In other words, the income profile that dominates from a relative inequality perspective might be
Pareto inferior from an absolute income level perspective. This is all fairly obvious; however, it
is important to keep this nature of the measure in mind when comparing the SP(x) indicator of
shared prosperity across countries or over time. The SP(x) is certainly not an inequality
measure. 4 It is, however, a general welfare measure that is focused on the bottom 40 percent of
the population and, as such, it is a function of the share of income accruing to that segment of the
population. It also purports to capture elements of inequality aversion; however, the examples
presented here show that the indicator is somewhat limited in this regard.
4
For a thorough discussion of welfare measures and inequality measures, see Foster and Sen (1997), among others
in a vast literature.
4
III. Algebra: The Simplest Decomposition
A couple of simple algebraic observations allow one to do a decomposition of the
distributional and income factors affecting SP(x). First of all, assuming away “integer
problems,” then r = 0.4*n. Let s(x) denote the share of total income accruing to the bottom 40
percent of the distribution of income for income profile x. By definition, this share can be
written as:
() = [1 + 2 + ⋯ + ]/[1 + 2 + ⋯ + ]
One can then substitute the above expressions for r and s into the equation for SP(x):
[1 + 2 + ⋯ + ]
() = () ∗
(0.4 ∗ )
To simplify terms, let y(x) be the income per capita of the total population of a country
with income profile x, and note that the summation of xi’s divided by n is simply y(x). Then
SP(x) can be re-written as:
()
() = � � ∗ ()
0.4
Writing the shared prosperity indicator in this way makes explicit the “share of the pie”
and “size of the pie” characteristics of this measure. The indicator is very simple and tractable
and easy to calculate; however, it does not take into consideration every distributional
characteristic of the income profile, as noted in the previous section. 5 Differences in SP(x)
across countries or over time are based upon differences in the share of income going to the
bottom 40 percent and differences in income per capita.
One can express the percentage change in SP(x) between time t and t+1, in the traditional
way, and suppressing the “(x)” notation for simplicity:
+1
� = ��
� ∗ +1 − � � ∗ � / [� � ∗ ]
0.4 0.4 0.4
This expression then simplifies to a standard way of writing the percentage change:
5
Note that, written in this form, the SP(x) looks similar to a “Sen Index”: (1-g)y, where g is the gini coefficient. The
shared prosperity indicator is income per capita weighted by the share that goes to the bottom 40 percent.
5
[+1 ∗ +1 ]
� =
−1
[ ∗ ]
The above expression can be set equal to zero to derive a curve that can then be plotted to
show the relationship between “growing the pie” (yt+1/yt) and “sharing of the pie” (st+1/st):
+1 +1
= 1/( )
Plotting this curve allows one to show graphically the relationship between the two
parameters affecting the growth of shared prosperity (Figure 2). Points above the dotted curve
represent increasing shared prosperity. Clearly if both the income per capita ratio and the share
ratio are greater than one, then both the growth and sharing effects are positive and shared
prosperity must be increasing. Similarly, if both ratios are less than 1, then shared prosperity is
declining. There is also the possibility of growth and sharing moving in opposite directions.
Then the question is whether one ratio is larger than the other. The various possibilities are
displayed below in Figure 2.
Figure 2: Growth and Increased “Sharing”
5
yt+1/
yt
4.5
4
yt+1/yt >
3.5
1/ (st+1/st)
3 yt+1/yt and st+1/st > 1
2.5
yt+1/yt <
2
1/ (st+1/st)
1.5
1
(st+1/st) > 1/(yt+1/yt)
0.5 yt+1/yt and st+1/st < 1
(st+1/st) < 1/(yt+1/yt)
0
0 0.5 1 1.5 2
st+1/st
6
In addition, there are infinitely many combinations of changes in s and y. Starting at 1, if
SP increases by 0.5, or 50 percent, then there would be a curve – a kind of “iso-shared
prosperity” curve—that represents the combinations of ratios for y and s that add up to a 50
percent change. More generally, for a change a, there would be a curve described by:
+1 +1
= ( + 1)/( )
These points are illustrated in Figure 3 below. The dotted line shows the zero growth set of
points for shared prosperity, and the solid lines show the combination of income per capita
growth (ratio) and shared prosperity growth (ratio) that result in a 50 percent, 100 percent or -50
percent change in shared prosperity, respectively.
Figure 3: “Iso-Shared Prosperity” Curves
yt+1 9
/yt
8
7
6
5
+50%
4
3
2
1
-50%
0
0 0.5 1 1.5 st+1/st 2
One additional issue worth mentioning – even if it is fairly obvious—is that the
maximum value of s is 0.4, and this represents an income profile with perfect equality (i.e., a
Lorenz curve that lies flat on the 45 degree line in Figure 1). The minimum value of s is zero;
however, this would imply every individual in the bottom 40 percent has zero income. Neither
extreme has ever been experienced in modern economies. A more important point, however, is
that clearly over the long-run, growth of income per capita is the limitless factor that creates the
7
potential for ever increasing levels of shared prosperity. On the other hand, a permanently
declining share approaching zero clearly would be social unsustainable.
Another way of making the dynamics even simpler is to visualize the following
� represent the percentage
approximation for the percentage change in SP(x), where ̂ and
6
change in s and y, respectively :
� ≅ ̂ +
�
In this case, one can plot the percentage change in each component, and the zero percentage
change in SP is the downward sloping 45 degree line in Figure 4 below.
Figure 4: Growth and Increased “Sharing” – An Approximation
10
�
� > ̂ 8
̂ > 0
6
�>0
4
� < ̂
2
�
0
-10 0 10
-2 �
̂ >
̂ < 0 -4
�<0
-6
�
̂ <
-8
-10
Another straightforward and simple decomposition worth considering is to decompose
the sharing parameter, s, into the difference between the growth rate of the income per capita of
the bottom 40 percent and the income per capita of the total population. This comes directly
from the definition of s:
+1
̂ = � �−1
6
In continuous time, the log difference in shared prosperity is exactly equal to the sum of the log differences of the
share and the income per capita of the total population, as noted in Dollar, Kleineberg and Kraay (2013).
8
+1 +1
1 + ⋯ +
�+1 +1 �
1 + ⋯
�=
−1
1 + ⋯ +
� �
1 + ⋯ +
If we multiply the numerator and denominator by (n/r), we have:
+1 + ⋯ + +1
� � ∗ � 1 +1 +1 �
1 + ⋯
�=
−1
1 + ⋯ +
� � ∗ � �
1 + ⋯ +
Finally, if we use the notation and to represent the income per capita of the bottom
40 percent and the income per capita of the total population, respectively, then we can simplify
the above expression to the following:
+1
+1
� = [(
)/ ( )] −1
In other words, the percentage change in the share is equal to the ratio of the change in
income per capita of the bottom 40 percent (itself expressed as a ratio) to the change in the
income per capita of the total population (also expressed as a ratio). In discrete time, the
approximation would be the difference:
� r -
̂ = �n
When thinking of the shared prosperity indicator, there is a tendency to want to compare
the income growth of the bottom 40 percent with the income growth of the total population. With
this approximation, one can see clearly this sort of comparison can be reduced to a calculation of
the changing share over time that accrues to the bottom 40 percent. As noted in the figures
above, the share, s, is just one dimension of the indicator’s path over time.
A simple simulation illustrates the dynamics of increasing shared prosperity over time.
Consider a country with robust, but non-spectacular, growth in income per capita at about the
long-run growth of per capita income in high-income countries (about 1.9 percent). For
illustrative purposes, we assume that the country experiences a dramatic increase in the share of
income going to the bottom 40 percent—from 0.1 to 0.15 (or 50 percent). Finally, we consider a
long-run timeframe of 50 years. Table 1 displays the accounting for the increase in shared
prosperity for this country. The example illustrates the dominance of economic growth for
determining the path of shared prosperity over the long-term. In this example, roughly two-
thirds of the increase in shared prosperity is due to the increase in income per capita. In the
9
short-term, there could be a very substantial role for increased “sharing,” as has been the case in
a number of countries historically – as discussed in the next section.
Table 1: Long-run growth with increased “sharing”—an example
1950 2000 End/Start % Annualized
Ratio Change Compound
% Change
s 0.1 0.15 1.50 50 0.8
Total Income (Y) 2000 8800 4.40 340 3.0
Population (P) 50 85 1.70 70 1.1
Income per Capita 40 104 2.59 159 1.9
(y = Y/P)
SP Indicator:
s*Y/(.4*P) 10 38.8 3.88 288 2.8
Note: The End/Start Ratio for SP = 1.5*2.6 – the product of the ratios of income per capita
and the share of total income, as noted in the equations above. Also note that the
approximation by summing percentage change is much more precise when comparing
compound annualized growth rates than 50 year cumulative growth rates, which is to be
expected.
Another simulation can illustrate a phenomenon that concerns many in the development
community: “enclave” patterns of growth that do not benefit the bottom end of the income
distribution. Often these might be associated with a brief growth spurt of a decade or so.
Consider, for example, a hypothetical case of a country that is stagnated but experiences a
natural resource boom that results in about 5 percent annual growth. As a poor country, perhaps
a reasonable population growth would be about 1.8 percent. Now suppose that much of the new
growth attracts resources from traditional sectors that employ much of the bottom 40 percent, so
that there is a fairly dramatic drop in the share of total income accruing to that group: from 15
percent to 10 percent in a decade. In such a scenario (Table 2), the SP indicator could actually
decline despite the growth in income per capita of the total population.
Table 2: “Enclave” growth spurt simulation
2000 2010 End/Start % Annualized
Ratio Change Compound
% Change
s 0.15 0.1 0.67 -33 -4.0
Total Income (Y) 2000 3300 1.65 65 5.1
Population (P) 50 60 1.20 20 1.8
Income per 40 55 1.38 38 3.2
Capita (y = Y/P)
SP Indicator:
15 14 0.92 -8 -0.9
s*Y/(.4*P)
10
IV. Some Illustrative Data
We used data from the World Development Indicators to map episodes of changes in
shared prosperity. There are limited data available on the quintile shares of income by country.
We came up with a total of 34 developing countries with at least two data points for the income
share over the last 20 years. 7 We then used this (admittedly small) sample for a variety of
descriptive figures to display some of the empirical “regularities” that are relevant to the above
discussion of the shared prosperity indicator.
In the following graph (Figure 5), we plot the approximated decomposition of the
percentage change in shared prosperity for the same set of countries. In some cases, the
timespan between observations on shared prosperity is only five years; in other cases, it is
longer. The criterion was to use the two most recent observations of the share for that country.
To make the scatter points comparable across countries, we annualized the percentage change in
the growth of GDP per capita and the growth in the share of total income accruing to the bottom
40 percent. Note that – at least for this limited set of countries – the vast majority experienced
gains in shared prosperity: most data points are above the “negative” 45 degree line. There is a
wide variety of combinations of increased “sharing” and “growth” contributions to increasing
shared prosperity; however, many data points in this sample lie above the “positive” 45 degree
line, indicating a larger contribution from growth than from increased “sharing.” One would
certainly expect that over several decades, as noted in the simulation of the previous section;
however, it is interesting that this phenomenon shows up over shorter periods (e.g., 5 to 15
years). Admittedly, there is only a very limited set of countries in this analysis.
7
The timespan for data explored was between 1990 and 2010, to be precise. One can go directly to individual
country household surveys and calculate shares and income growth directly from there, and perhaps double the
sample of countries. The intent here is to show the diversity of recent experience – in particular, in terms of the
simple decomposition of factors.
11
Figure 5: Growth and changing shares of income in 34 countries.
10 �
8
6
4
2
�
0
-10 -5 0 5 10
-2
-4
-6
-8
-10
Source: Author’s calculations based on data from WDI. Constant local currency GDP per capita
was used for growth rates.
Using Povcalnet at the World Bank, we then searched for the average household income
for the countries in the above sample, based on household survey data. This reduced the sample
to 27, but we reproduced the decomposition graph above. The general pattern is fairly similar, as
can be seen in Figure 6 below. Note that some country data points with slower income growth
according to household surveys drop into the quadrant where sharing has a larger relative role
than pure growth.
12
Figure 6: Growth and changing shares of income in 27 countries,
using household survey data
10 �
8
6
4
2
�
0
-10 -5 0 5 10
-2
-4
-6
-8
-10
Source: Author’s calculations based on data from Povcalnet.
V. Discussion
The introduction made note of the fact that many of the bottom 40 percent in developing
countries have income levels below the poverty line in rich countries. Some numbers make the
point more explicit: with the exception of the Russian Federation, in all of the other “BRICs”,
the people in the bottom 40 percent have incomes below the average incomes of the bottom 5
percent of the population in the United States (Milanovic, 2012, Figure 7). Certainly, then,
promoting income growth of the bottom 40 percent is a worthwhile pursuit in even the more
advanced developing countries.
As noted above, the “simplest” algebraic decomposition should not be taken too literally.
There is a vast literature on the theory and evidence of the links between inequality and
economic growth. In addition, a constant negative value for ̂ is socially “unsustainable” in the
most brutal sense: if it persists, incomes of the bottom 40 percent shrink to zero. On the other
hand, the simple decomposition highlights the role of economic growth in the long-run. On the
positive end, the “sharing” runs into the limit of perfect equality and only growth can then drive
ever higher levels of prosperity. The simulations above illustrate how long-run compounding of
13
growth enhances shared prosperity even with there is a substantial “shift” in “sharing.” This is
probably the most useful element of the algebraic decomposition: it reveals that, to a certain
extent, the share is like a “shift” factor between steady states, while growth in income per capita
is more akin to a steady-state growth rate of shared prosperity. It is interesting to note that the
(admittedly small sample) empirical experience presented shows that, in many cases, growth has
a larger role to play than “sharing” in increasing this measure of shared prosperity – even over a
relatively short timeframe.
As mentioned above, a fully specified growth model would – in principle – make both the
share and the growth rate endogenous. On that basis, one might hypothesize that there is a
systematic positive or negative relationship between the two variables. There is, in fact, a vast
literature on both the theoretical 8 and empirical 9 bases for a relationship between inequality and
growth. In a well-known paper, Dollar and Kraay (2002) found empirical evidence that there is
no systematic relationship between income per capita growth and the share of the bottom 20
percent. In other words, the income per capita of the bottom quintile (on average) grew in line
with the income per capita of the total population. They used this finding to declare in the title,
“Growth Is Good for the Poor.” More recently, Dollar, Kleineberg and Kraay (2013) looked at
the income per capita of the bottom 40 percent, using additional data now available, and reached
a similar conclusion: “Growth Still Is Good for the Poor.” The original Dollar-Kraay paper
focused on bottom quintile since many national poverty lines tend to settle at the 20 percentile.
At the subnational level, Virmani (2007, 2008) has studied the relative performance of both
economic growth and poverty levels across India’s states. He finds that, among other structural
factors, the consumption share of the bottom 40 percent of a state’s population has a significant
effect on a state’s poverty rate. In other words, promoting shared prosperity is consistent with
reducing poverty: the World Bank Group’s twin goals, in this case, are complementary.
A key remaining issue is that the coverage, frequency and comparability of household
survey data need to improve if one is to do more detailed work on the empirics of shared
prosperity. This paper only presented some illustrative data for understanding how the basic
algebra works.
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8
See Asano (2012) for a recent model in this literature.
9
See, for example, Perotti (1996), Deininger and Squire (1998), and Banerjee and Duflo (2003).
14
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