Measuring inequality_ Using the Lorenz Curve and Gini Coefficient

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					  Measuring inequality: Using the Lorenz Curve
              and Gini Coefficient
1. Introduction                              land, houses as well as money) and the
Almost thirty years ago, the author of       distribution of national income (which is
this brief attended a lecture addressing     a flow since it is paid per week, month,
the economics of inequality. At the start,   year etc).
the class was invited to imagine the
implications of individual wealth being      The terms wealth and income should not
reflected in our personal height. Assume     be      used      interchangeably.     The
that by government decree, everyone          distribution of the stock of wealth will be
has to march past a fixed point over the     much more unequal than the flow of
period of one hour, starting with the        income in a given year. As we have
smallest people and ending with the          already seen in the case of the parade
largest. The parade would begin with all     of dwarves and giants, many people
the people who owed money. They              have     substantial    negative   wealth.
would march underground. Even after          Furthermore, studies based on wealth
20 minutes, marchers would be invisible      are a lot less common since it is harder
since they had no wealth.                    to measure wealth than it is income.

At the half way point, the parade would      The aim of this case study is to
comprise of dwarves, little more than a      introduce readers to two interlinked
few centimetres in height. Only 12           methods of measuring inequality: the
minutes before the end of the parade         Lorenz Curve and the Gini Coefficient.
would people start to be of an average       Both originate from the early years of
height and hence, of average wealth. In      the twentieth century: in 1905, Max
the last few minutes of the parade,          Otto Lorenz published a paper in an
marchers would evolve rapidly into           American statistical journal outlining the
giants, assuming heights of several          technique which was to bear his name.
metres. The heads of the last few            Corrado    Gini’s   index     of   income
participants would be invisible since        inequality   was      published     shortly
their height would be measured in            afterwards in 1914. However, it was the
kilometres rather than metres.               work on poverty and income inequality
                                             by Sir Tony Atkinson during the 1970s
The moral, of what seems to be a trivial     that led to the popular dissemination
example, is that despite increases in        and development of the original work of
living   standards,   wealth    is   not     Lorenz and Gini.
distributed evenly through society. The
vast majority is owned by the very few.      2. The Lorenz Curve
                                             Let us assume that we wish to construct
The degree to which the inequality           a Lorenz Curve to measure wealth
depicted in the parade of dwarves and        inequality. The standard framework can
giants is undesirable is a normative         be built up in four stages. First, we draw
issue. Commentators concerned with           a set of axes in which the cumulative
poverty levels in society will argue that    percentage of wealth is measured along
one of the major roles of government         the   y-axis     while   the    cumulative
should involve some degree of re-            percentage of households is measured
distribution. If a government is to play a   along the x-axis. Usually, the graph’s
role in such a process, then it needs to     axes are closed off to form a box.
be informed about the distribution of
wealth (which is a stock and can include     The second stage requires us to order
a range of assets that include shares,       the distribution from the smallest


                                                                                      1
through to the largest, thereby enabling
us to answer the following sequential
questions:
 (a) what proportion of wealth is owned
     by the poorest 10 percent of the
     population?
 (b) what proportion of wealth is owned
     by the poorest 20 percent of the
     population?
                                              In Figure Two, societal wealth remains
 (c) what proportion of wealth is owned
                                              unevenly distributed, but the poorer
     by the poorest 30 percent of the
                                              households are (on average) better off
     population?
                                              than they are under the first scenario.
This process continues until we reach
                                              One of the advantages of using the
the point where 100 per cent of wealth
                                              Lorenz Curve is that it provides a visual
is owned by 100 per cent of the
                                              representation of the information we
population. The third step is to assume
                                              wish to consider, in this case the
that we live in a truly equal society. If
                                              inequality of wealth prevailing in
this were to be the case, the
                                              society.
relationship would be such that as we
move along the x-axis, each 10 per cent
                                              We could superimpose several Lorenz
increment of households would own an
                                              Curves onto the same diagram to show
additional 10 per cent of wealth. In this
                                              changes in the way in which wealth has
case, the line we would draw would be a
                                              been distributed across society at
straight line emanating from the origin.
                                              various points in time. For example we
This is known as the line of absolute
                                              might wish to compare the years 1945,
equality and will have a slope of 45
                                              1975 and 2005 (see Figure Three).
degrees.

Finally, we can insert a line that is based
on the data set available to us. In this
case, the line will bow away from the
line of absolute equality. The more
unequal society is, the further it will
deviate away from the line of absolute
equality. It is this line which is known as
the Lorenz Curve.
                                              Even if the shape of the Lorenz Curve is
                                              not    changing    significantly,   poorer
                                              members of society may still be much
                                              better off in terms of what they can
                                              afford to buy. In other words, they are
                                              relatively no better off, but in terms of
                                              spending     power,    they     have   the
                                              opportunity to enjoy a wider range of
                                              luxury items. Commodities which were
In Figures One and Two, the line of
                                              considered to be luxuries fifty years ago
absolute equality is labelled OA.
                                              (for     example,      televisions    and
However, the Lorenz curve assumes a
                                              telephones) are now taken for granted
different shape in the two diagrams. In
                                              by most people.
Figure One, it can be seen that the
poorest sections of society command a
                                              3. The Gini Coefficient
very small proportion of the country’s
                                              The Gini Coefficient is a complementary
wealth.
                                              way of presenting information about

                                                                                      2
inequality. It is the ratio of the area       Chart 1: Gini Coefficient 1981-2005/6
between the Lorenz Curve and the line                      37



of absolute equality (numerator) and                       35


the whole area under the line of                           33

absolute equality (denominator). Based




                                              Percentage
on Figure Four, it can be seen that the
                                                           31




Gini Coefficient = C/0AB.                                  29



                                                           27



                                                           25
                                                            1979   1982   1985   1988   1991   1993/94 1996/97 1999/00 2002/03 2005/06


                                              Source: The effects of taxes and benefits on household
                                              income, 2005-06, Office for National Statistics.1


                                              Table 2 provides a cross sectional
                                              insight into income equality in fourteen
                                              countries plus the European Union. Note
The extreme values of the Gini                that there are variations in the base
Coefficient are 0 and 1. These are often      year. This is common when international
presented in statistical publications as      comparisons are being undertaken.
percentages. Hence, the corresponding
extreme values are 0% and 100%. The           Table 2: Distribution of Family Income
former implies perfect equality (in other     Country               Gini, %      Year
words, everyone in society has exactly        Argentina                 48.3     2006
the same amount of wealth) whereas            Australia                 35.2     1994
the latter implies total inequality in that   Belgium                   33.0     2000
one person has all the wealth and             El Salvador               52.4     2002
                                              European Union            31.6     2003
everyone else has nothing. Clearly,
                                              Finland                   26.9     2000
these two extremes are trivial; the key
                                              France                    26.7     2002
thing to bear in mind is that the lower
                                              Germany                   28.3     2000
the figure that Gini Coefficient takes
                                              India                     36.8     2004
(between 0% and 100%), the greater
                                              Japan                     38.1     2002
the degree of prevailing equality.
                                              Morocco                   40.0     2005
                                              Nigeria                   43.7     2003
The Gini Coefficient is an example of a
                                              Russia                    40.5     2005
summary statistic. In other words, it
                                              United Kingdom            36.0     1999
compresses a broader array of statistical
                                              United States             45.0     2004
information into a single figure. We can
                                              Yemen                     33.4     1998
present Gini Coefficient data in two main
                                              Source: CIA World Factbook2
ways, first as a time series trend and
second as a set of cross section figures.     Consider the following questions before
Let us consider each in turn.                 moving to the next section of this brief.
                                              (i) Which country is the most equal
Chart 1 presents the Gini Coefficient for          and least equal in terms of family
the UK since 1981. It is based on the              income distribution?
distribution of household disposable          (ii) If   you    were    undertaking      a
income. The rapid increase in inequality           comparative     analysis   of     the
during the late 1980s is attributed to a           fourteen countries listed in Table
widening of the gap in remuneration                One based on income distribution,
between skilled and unskilled workers,
together with income tax cuts and a
decline in the power of trade unions.         1
                                                    The full report can be downloaded at
Can you think of any reasons why there        http://www.statistics.gov.uk/downloads/theme_social/T
has been a small improvement in               axes_Benefits_2005-2006/Taxes_Benefits_2005_06.pdf
                                              2
                                                    The figures can be viewed at
inequality since 2000?                        https://www.cia.gov/library/publications/the-world-
                                              factbook/fields/2172.html

                                                                                                                                         3
     what additional information would          income, the further the Lorenz curve
     you find useful to help you with           deviates from the line of absolute
     your study (hint: you may choose           equality, the more concentrated it is. A
     to navigate yourself round    the          similar exercise could be undertaken
                   3
     CIA’s Factbook )                           from the point of view of employment.

4. Interpreting Lorenz Curves and               Researchers interested in the analysis of
Gini Coefficients                               competitive balance in sport4 use Lorenz
It was noted in the introduction that           Curves and Gini Coefficients to estimate
Lorenz Curves and Gini Coefficients can         the degree to which teams dominate
be used to gain insights into a range of        sporting competition.
cases in which inequality is seen as a
policy issue. It is inevitable that there       5. Drawing your own Lorenz Curve
will be considerable variation between          Let’s imagine that an industry that
countries since, to varying degrees,            produces plastic turnips is made up of
each will have different resource               ten firms. Each firm’s contribution to the
endowments and institutional norms              overall industry output in a given year is
which will have driven and constrained          as follows:
its historical, political and social
development.                                    Firm                  Units sold (millions)
                                                 A                                         25
It should be emphasised that the figures         B                                          4
in Table Two are not telling us which            C                                          3
country is the richest. The Gini                 D                                         12
Coefficients    are    simply     providing      E                                         17
information about how each country’s             F                                         30
income, large or small, is distributed           G                                         20
between families (noting of course that          H                                         17
the definition of a family unit will vary        I                                         12
from country to country). Furthermore,           J                                         10
the fact that two countries have the            Total                                     150
same Gini Coefficient does not mean
that their respective Lorenz Curves are         (i)   Draw the axes that you will use to
similarly shaped also. If the area C for              draw your Lorenz Curve and label
two countries (see Figure Four) is                    your axes appropriately.
identical, their Gini Coefficients will also    (ii) Insert the line of absolute equality.
be the same. However one income                       State precisely what this line means
distribution might be seen as less                    in this context.
desirable than the other (as a normative        (iii) Now derive a Lorenz Curve using
judgement).                                           the hypothetical outputs set out
                                                      above.
There is no reason why the two                  (iv) Are there any other measures that
measures cannot be applied to other                   can be used to gain an insight into
scenarios which also have a distributive              the degree of concentration in the
aspect to them. Perhaps the most                      plastic turnip industry?
common application is to the analysis of
industry structure. For example, we
could draw Lorenz Curves to measure
the distribution of output across firms
within an industry. As with wealth or
                                                4
                                                    See the case studies ‘Measuring the
3
 The homepage for the CIA Factbook is           competitiveness of sport: are the top teams
https://www.cia.gov/library/publications/the-   getting too strong?’ and ‘Is the Scottish Premier
world-factbook/                                 League less competitive than its English
                                                Counterpart?’

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