gini coefficient and lorenz curve report

Sunshine Group | Economic Development | March 4, 2013

Lorenz Curve & Gini Coefficient THEORY & APPLICATION

Table of Contents

INTRODUCTION ................................................................................. 3

Lorenz curve ......................................................................................... 4

Gini coefficient ..................................................................................... 4

THEORY ................................................................................................ 7

I. Overview of Inequality Metrics ...................................................... 7

Defining Income ................................................................................ 7

Some common metrics of measuring inequality ............................... 8

Properties of Inequality Metrics as in Lorenz Curve and Gini

coefficient ........................................................................................ 12

II. Lorenz Curve ............................................................................. 14

Definition: ....................................................................................... 14

How to construct a Lorenz Curve? .................................................. 14

Calculation ...................................................................................... 16

Properties......................................................................................... 18

Advantages ...................................................................................... 18

Limitations ....................................................................................... 19

III. Gini Coefficient ......................................................................... 21

Definition......................................................................................... 21

Ways to present Gini coefficient ...................................................... 21

Calculation ...................................................................................... 23

Advantages ...................................................................................... 23

Limitations ....................................................................................... 23

IV. Other uses of lorenz curve and gini coefficient .......................... 26

APPLICATION ................................................................................... 28

ZuXiang Wang and Russell Smyth ...................................................... 28

1. Structure of the article A hybrid method for creating Lorenz

Curve with an application to measuring world income inequality 28

2. Findings from Lorenz Curves for selected years (1950, 1960,

1970, 1980, 1990, 2000 and 2005) ................................................... 29

3. Findings from the trend of AG, its decomposition and average

PPP of the world .............................................................................. 33

4. Density distribution function: ................................................... 37

Other authors findings: ...................................................................... 37

1. A World Bank working paper by Branko Milanovic (2012) ....... 38

2. Branko Milanovic (2010b) ........................................................ 40

3. Solt (2010) ................................................................................. 41

4. Some discrepancies among economists all over the world: ...... 44

CONCLUSION .................................................................................... 46

INTRODUCTION

In contemporary time, many issues arising in accordance with development

in all corners of life, one of them is inequality happening all around the

world. Inequality can happen when there are discrepancies between

countries or even among groups or individuals within a country. Exactly

appearing in almost every sectors of society (economics, human

development, wealth distribution, education, health care) with clear

indications, this issue receives a great concern from public.

A child born in Swaziland is nearly 30 times more likely to die before the

age of five than a child born in Sweden. A typical Somalia person can live

50 years before he dies while Japanese people are famous for their great

life expectancy of over 84 years. 40% of school-age children in Africa do

not attend primary school mainly because of poverty while the US

government spent $12,743 per public school student on average. 400

richest Americans worth more than the GDP of Canada or Mexico. There

are numerous example of inequalities happening around the world relating

to social, racial, sexual, educational aspects, among individuals and groups

within a society as well as between countries

However, it is clear that all those problems are brought about by

Economic differences between the rich and the poor, which is the main

cause for the inequality in many other aspects such as education and health

care etc. Economic inequality varies between societies, historical periods,

economic structures and systems (for example, capitalism or socialism),

and between individuals' abilities to create wealth. The term can refer to

cross sectional descriptions of the income or wealth at any particular

period, and to the lifetime income and wealth over longer periods of time.

To measure economic inequality, income distribution deserves a much

more attention and effort of economists to do research, investigate and

invent a tools to measure it. To calculate income inequality, it provides 2

methods to apply:

The first one is Size distribution of income or Personal distribution

of income: one of the two methods to measure inequality in income.

Under this method, income of individuals and household are collected and

arranged in ascending order. The data, is then, divided among groups. Most

common method is to divide data in quintiles or deciles in percentage

form. Then it is determined that what percentage of total income is received

by each income group.

The second method is Functional distribution of income is a theory

explaining how income is divided between different groups involved in the

production process, specifically, the income earned by the "owners" of

various factors or steps in production. This income is determined by the

supply and demand for the end goods produced by each of them.

However, in reality, the Size distribution of income method is preferable

to use, and from it, the income inequality can be calculated and visualized

by Loren Curve and Gini coefficient which is the most prominent index

to measure the income inequality. In this assignment, we will only focus

on Lorenz Curve and Gini coefficient to measure inequality in income.

LORENZ CURVE

Inventor

Max Otto Lorenz (September 19, 1876 in Burlington, Iowa July 1, 1959

in Sunnyvale, California) was an American economist who developed

the Lorenz curve in 1905 to describe income inequalities. He published this

paper when he was a doctoral student at the University of Wisconsin

Madison. His doctorate (1906) was on 'The Economic Theory of Railroad

Rates' and made no reference to perhaps his most famous paper.

He was active in both publishing and teaching and was at various times

employed by the U.S. Census Bureau, the U.S. Bureau of Railway

Economics, the U.S. Bureau of Statistics and the U.S Interstate Commerce

Commission.

Publish time:

The term Lorenz curve seems first to have been used in 1912 in a

textbook The Elements of Statistical Method.

GINI COEFFICIENT

Author:

Corrado Gini (May 23, 1884 March 13, 1965) was an Italian statistician,

demographer and sociologist who developed the Gini coefficient, a

measure of the income inequality in a society. Gini's scientific work ran in

two directions: towards the social sciences and towards statistics. His

interests ranged well beyond the formal aspects of statisticsto the laws

that govern biological and social phenomena.

He was the president of the International Federation of Eugenics Societies

in Latin-language Countries, president of the Italian Sociological Society,

president of the Italian Statistical Society and other important roles in the

economic societies.

Publish time:

In 1912 in paper "Variability and Mutability" (Italian: Variabilit e

mutabilit)

After Max Lorenz and Gini, there have been many other scientists and

economists have applied LC and Gini coefficient for their business model

to measure inequality in income coming along with greater concern for

society matter. In contemporary time, the source or information as well as

techniques to do research make a big improvement in the quantity and

quality mutually, so existing other models with higher accuracy that done

by other researchers base on these theories. One of them is a research of

ZuXiang Wang and Russell Smyth namely A hybrid method for

creating Lorenz Curves with an application to measuring world

income inequality published in Department of Economics, Monash

University, Australia to act as a discussion topic for students.

In this article, a hybrid method is introduced by the two authors to create

efficient functional models for the Lorenz curve from the single-parameter

functional forms. A set of models are created and first tested using income

distribution data from the United States. The test show that the models

perform well. Then as an application, one of their best models is then used

to study world income inequality between 1950 and 2006.

To be more specific, they show clearly step by step with the beginning of

how to set up models to draw Lorenz Curve. Then, the authors used one of

this models, served it on the U.S. data to calculate the residuals for the

model and prove that is their models are better compared to the old ones in

criteria of accuracy. From that, they took the best model to measure the

inequality of the world. Finally, the results for the research are quite

interesting and at the same time they also proves that model they are

consistent with the original theory of the density distribution of Lorenz

Curve.

In this report, our group would like to summarize as much as possible the

basic knowledge of the theory, as well as show our outcome after

investigating the complementary article: A hybrid method for creating

Lorenz Curve with an application to measuring world income inequality

of Wang and Smyth in 2013.

THEORY

I. OVERVIEW OF INEQUALITY METRICS

The concept of inequality is distinct from that of poverty and fairness.

Income inequality metrics or income distribution metrics are used by social

scientists to measure the distribution of income, and economic inequality

among the participants in a particular economy, such as that of a specific

country or of the world in general. While different theories may try to

explain how income inequality comes about, income inequality metrics

simply provide a system of measurement used to determine the dispersion

of incomes.

Income distribution has always been a central concern of economic theory

and economic policy. Classical economists such as Adam Smith, Thomas

Malthus and David Ricardo were mainly concerned with factor income

distribution, that is, the distribution of income between the main factors of

production, land, labor and capital. It is often related to wealth distribution

although separate factors influence wealth inequality.

Defining Income

All of the metrics described below are applicable to evaluating the

distributional inequality of various kinds of resources. Here the focus is on

income as a resource. As there are various forms of "income", the

investigated kind of income has to be clearly described.

One form of income is the total amount of goods and services that a person

receives, and thus there is not necessarily money or cash involved. If a

subsistence farmer in Uganda grows his own grain, it will count as income.

Services like public health and education are also counted in. Often

expenditure or consumption (which is the same in an economic sense) is

used to measure income. The World Bank uses the so-called "living

standard measurement surveys" to measure income. These consist of

questionnaires with more than 200 questions. Surveys have been

completed in most developing countries.

Applied to the analysis of income inequality within countries, "income"

often stands for the taxed income per individual or per household. Here

income inequality measures also can be used to compare the income

distributions before and after taxation in order to measure the effects of

progressive tax rates.

Some common metrics of measuring inequality

Among the most common metrics used to measure inequality are the Gini

index (also known as Gini coefficient), the Theil index, and the Hoover

index.

Gini index

The range of the Gini index is between 0 and 1 (0% and 100%), where 0

indicates perfect equality and 1 (100%) indicates maximum inequality.

The Gini index is the most frequently used inequality index. The reason for

its popularity is that it is easy to understand how to compute the Gini index

as a ratio of two areas in Lorenz curve diagrams. As a disadvantage, the

Gini index only maps a number to the properties of a diagram, but the

diagram itself is not based on any model of a distribution process. The

"meaning" of the Gini index only can be understood empirically.

Additionally the Gini does not capture where in the distribution the

inequality occurs. As a result two very different distributions of income

can have the same Gini index.

20:20 Ratio

The 20:20 or 20/20 ratio compares how much richer the top 20% of

populations are to the bottom 20% of a given population, this can be more

revealing of the actual impact of inequality in a population, as it reduces

the effect on the statistics of outliers at the top and bottom and prevents the

middle 60% statistically obscuring inequality that is otherwise obvious in

the field. The measure is used for the United Nations Development

Program Human Development Indicators. The 20:20 ratio for example

shows that Japan and Sweden have a low equality gap, where the richest

20% only earn 4 times the poorest 20%, whereas in the UK the ratio is 7

times and in the US 8 times. Some believe the 20:20 ratio is a more useful

measure as it correlates well with measures of human development and

social stability including the index of child well-being, index of health and

social problems, population in prison, physical health, mental health and

many others.

Palma ratio

The Palma ratio is defined as the ratio of the richest 10% of the population's

share of gross national income divided by the poorest 40%'s share. It is

based on the work of Chilean economist Gabriel Palma who found that

middle class incomes almost always represent about half of gross national

income while the other half is split between the richest 10% and poorest

40%, but the share of those two groups varies considerably across

countries.

The Palma ratio addresses the Gini index's over-sensitivity to changes in

the middle of the distribution and insensitivity to changes at the top and

bottom, and therefore more accurately reflects income inequality's

economic impacts on society as a whole. Palma has suggested that

distributional politics pertains mainly to the struggle between the rich and

poor, and who the middle classes side with.

Hoover index

The Hoover index is the simplest of all inequality measures to calculate: It

is the proportion of all income which would have to be redistributed to

achieve a state of perfect equality.

In a perfectly equal world, no resources would need to be redistributed to

achieve equal distribution: a Hoover index of 0. In a world in which all

income was received by just one family, almost 100% of that income would

need to be redistributed (i.e., taken and given to other families) in order to

achieve equality. The Hoover index then ranges between 0 and 1 (0% and

100%), where 0 indicates perfect equality and 1 (100%) indicates

maximum inequality.

Theil index

A Theil index of 0 indicates perfect equality. A Theil index of 1 indicates

that the distributional entropy of the system under investigation is almost

similar to a system with an 82:18 distribution. This is slightly more unequal

than the inequality in a system to which the "80:20 Pareto principle"

applies. The Theil index can be transformed into an Atkinson index, which

has a range between 0 and 1 (0% and 100%), where 0 indicates perfect

equality and 1 (100%) indicates maximum inequality.

The Theil index is an entropy measure. As for any resource distribution

and with reference to information theory, "maximum entropy" occurs once

income earners cannot be distinguished by their resources, i.e. when there

is perfect equality. In real societies people can be distinguished by their

different resources, with the resources being incomes. The more

"distinguishable" they are, the lower is the "actual entropy" of a system

consisting of income and income earners. Also based on information

theory, the gap between these two entropies can be called "redundancy". It

behaves like negative entropy.

For the Theil index also the term "Theil entropy" had been used. This

caused confusion. As an example, Amartya Sen commented on the Theil

index: "given the association of doom with entropy in the context of

thermodynamics, it may take a little time to get used to entropy as a good

thing." It is important to understand that an increasing Theil index does not

indicate increasing entropy, instead it indicates an increasing redundancy

(decreasing entropy).

High inequality yields high Theil redundancies. High redundancy means

low entropy. But this does not necessarily imply that a very high inequality

is "good", because very low entropies also can lead to explosive

compensation processes. Neither does using the Theil index necessarily

imply that a very low inequality (low redundancy, high entropy) is "good",

because high entropy is associated with slow, weak and inefficient resource

allocation processes.

There are three variants of the Theil index. When applied to income

distributions, the first Theil index relates to systems within which incomes

are stochastically distributed to income earners, whereas the second Theil

index relates to systems within which income earners are stochastically

distributed to incomes. A third "symmetrized" Theil index is the arithmetic

average of the two previous indices. Interestingly, the formula of the third

Theil index has some similarity with the Hoover index (as explained in the

related articles). As in case of the Hoover index, the symmetrized Theil

index does not change when swapping the incomes with the income

earners. How to generate that third Theil index by means of a spreadsheet

computation directly from distribution data is shown below.

An important property of the Theil index which makes its application

popular is its decomposability into the between-group and within-group

component. For example, the Theil index of overall income inequality can

be decomposed in the between-region and within region components of

inequality, while the relative share attributable to the between-region

component suggests the relative importance of spatial dimension of income

inequality.

However, to the extent of this report, we just analyze the two main

inequality metrics: Lorenz Curve and Gini coefficient, which are presented

in the following parts.

Properties of Inequality Metrics as in Lorenz Curve and Gini

coefficient

In the economic literature on inequality four properties are generally

postulated that any measure of inequality should satisfy. These four

properties could be considered the major characteristics of them including

Lorenz Curve and Gini Coefficient.

Anonymity

This assumption states that an inequality metric does not depend on the

"labeling" of individuals in an economy and all that matters is the

distribution of income. For example, in an economy composed of two

people, Mr. Smith and Mrs. Jones, where one of them has 60% of the

income and the other 40%, the inequality metric should be the same

whether it is Mr. Smith or Mrs. Jones who has the 40% share. This property

distinguishes the concept of inequality from that of fairness where who

owns a particular level of income and how it has been acquired is of central

importance. An inequality metric is a statement simply about how income

is distributed, not about who the particular people in the economy are or

what kind of income they "deserve".

Scale independence

This property says that richer economies should not be automatically

considered more unequal by construction. In other words, if every person's

income in an economy is doubled (or multiplied by any positive constant)

then the overall metric of inequality should not change. Of course the same

thing applies to poorer economies. The inequality income metric should be

independent of the aggregate level of income.

Population independence

Similarly, the income inequality metric should not depend on whether an

economy has a large or small population. An economy with only a few

people should not be automatically judged by the metric as being more

equal than a large economy with lots of people. This means that the metric

should be independent of the level of population.

Transfer principle

The PigouDalton, or transfer principle, is the assumption that makes an

inequality metric actually a measure of inequality. In its weak form it says

that if some income is transferred from a rich person to a poor person, while

still preserving the order of income ranks, then the measured inequality

should not increase. In its strong form, the measured level of inequality

should decrease.

II. LORENZ CURVE

Definition:

Lorenz Curve is a graph depicting the variance of the size distribution of

income from perfect equality.

How to construct a Lorenz Curve?

The standard framework can be built up in four stages.

First, we draw a set of axes in which the number of income recipients are

plotted on the horizontal axis (or the x-axis), not in absolute terms but in

cumulative percentage while the vertical axis (or the y-axis) shows the

cumulative share of total income received by each percentage of

population. Usually, the graphs axes are closed off to form a square.

The second stage requires us to order the distribution from the smallest

through to the largest, thereby enabling us to answer the following

sequential questions:

(a) What proportion of income is owned by the poorest 10 percent of the

population?

(b) What proportion of income is owned by the poorest 20 percent of the

population?

(c) What proportion of income is owned by the poorest 30 percent of the

population?

This process continues until we reach the point where 100 per cent of

wealth is owned by 100 per cent of the population. The third step is to

assume that we live in a truly equal society. If this were to be the case, the

relationship would be such that the percentage of income received is

exactly equal to the percentage of income recipients. For example, as we

move along the x-axis, each 10 per cent increment of population would

own an additional 10 per cent of income. In this case, the diagonal line that

is drawn from the lower left corner (the origin) of the square to the upper

right corner is known as the line of absolute equality and will have a slope

of 45 degrees.

Finally, the most important step is to draw the Curve. 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. This line is known as the Lorenz

Curve. With increasing inequality, the Lorenz Curve starts to fall below the

diagonal in a loop that is always bowed out to the right of the diagram; it

cannot curve to the other way. The slope of the curve at any point is simply

the contribution of the population at that point to the cumulative share of

income. Because we have ordered income recipients from poorest to

richest, this marginal contribution cannot ever fall. This is the same as

saying that the Lorenz Curve can never get flatter as we move from left to

right. The overall distance between the diagonal line (the 450 line) and

the Lorenz Curve is indicative of the amount of inequality present in the

society that it represents. The greater the extent of the inequality, the

further the Lorenz Curve will be from the diagonal line.

Figure 1: How to construct a Lorenz Curve?

Calculation

The Lorenz curve can usually be represented by a function L(F), where F,

the cumulative portion of the population, is represented by the horizontal

axis, and L, the cumulative portion of the total wealth or income, is

represented by the vertical axis.

For a population of size n, with a sequence of values yi, i = 1 to n, that are

indexed in non-decreasing order (yi yi +1), the Lorenz curve is

the continuous piecewise linear function connecting the points (Fi, Li), i =

0to n, where F0 = 0, L0 = 0, and for i = 1to n:

Note that the statement that the Lorenz curve gives the portion of the wealth

or income held by a given portion of the population is only strictly true at

the points defined above, but not at the points on the line segments between

these points. For instance, in a population of 10 households, it doesn't make

sense to say that 45% of them earn a certain portion of the total. If the

population is modeled as a continuum then this subtlety disappears.

For a discrete probability function f(y), let yi, i = 1to n, be the points with

non-zero probabilities indexed in increasing order (yi < yi +1).

The Lorenz curve is the continuous piecewise linear function connecting

the points (Fi, Li), i = 0to n, where F0 = 0, L0 = 0, and for i = 1to n:

For a probability density function f(x) with the cumulative distribution

function F(x), the Lorenz curve L(F(x)) is given by:

For a cumulative distribution function F(x) with inverse x(F), the Lorenz

curve L(F) is given by:

The previous formula can still apply by generalizing the definition of x(F):

x(F1) = inf {y : F(y) F1}

Properties

A Lorenz curve always starts at (0,0) and ends at (1,1).

The Lorenz curve is not defined if the mean of the probability distribution

is zero or infinite.

The Lorenz curve for a probability distribution is a continuous function.

However, Lorenz curves representing discontinuous functions can be

constructed as the limit of Lorenz curves of probability distributions, the

line of perfect inequality being an example.

The information in a Lorenz curve may be summarized by the Gini

coefficient and the Lorenz asymmetry coefficient.

The Lorenz curve cannot rise above the line of perfect equality. If the

variable being measured cannot take negative values, the Lorenz curve:

- cannot sink below the line of perfect inequality,

- increasing and convex.

Advantages

First, the Lorenz Curve provides a visual presentation of the information

we wish to consider, in this case the inequality of the income distribution

prevailing in the society.

Second, we could superimpose the Lorenz Curves onto the same diagram

to show changes in the way in which income has been distributed across

society at various points in time.

In general, the advantage of using Lorenz Curve to show income inequality

is that the shape and the position of the curve can indicate the income

inequality well.

Limitations

Lorenz Curve is not a quantitative measure of inequality in income

distribution:

Despite the Lorenz Curves visual presentation in order to point out the

inequality of income, Lorenz Curve is not a quantitative measure of

inequality in income distribution. On the other hand, even when comparing

the Lorenz Curves between countries in a visual way, in many cases, we

cannot conclude which country has a higher level of inequality. If the

Lorenz Curves do not intersect, the curve that is further from the diagonal

line shows a higher level of income distribution inequality. However, in

case the Lorenz Curves intersect, it would be difficult to judge the

difference of income distribution. As shown in Figure 2, when the two

Lorenz Curves B and C cross, we need more information or additional

assumptions before we can determine which of the underlying economies

is more equal.

Figure 2: Four possible Lorenz Curves

The amount of inequality may be misleading:

If richer households are able to use their incomes more efficiently than

lower income households, the amount of inequality could be understated.

Lorenz curve ignores life cycle effects:

The measurement of income inequality with a Lorenz curve shows income

distribution only at a given time while an individuals income varies over

his lifetime, and this variation is not considered when analyzing inequality

using a Lorenz curve. For instance, the income of a sports man and of a

lecturer may be about the same over their lifetimes. But the income of the

lecturer may be spread over a number of years say for 40 years whereas

that of sports man may be realized in 10 years. Hence, the two incomes are

likely to be highly unequal in a given year

Not Based on Disposable Income:

The Lorenz curve is based on the data relating to money income rather than

disposable income. It does not take into consideration personal income

taxes, social security deductions, subsidies received by the poor families

etc. Moreover, the data are converted to a per capita basis to adjust for

differences in average family size within each quintile (5th) or decile (10th)

group of the population. As a consequence, smaller families may

sometimes be shown better off than large ones with greater incomes.

Does not consider age Differences: The construction of a Lorenz curve

does not consider the ages of the persons, who receives income. The

income of a young individual who enters jobs recently those in mid-career

and of old people who have retired are not the same. But the Lorenz curve

does not distinguish incomes by ages and reflects inequalities across all

ages. It is therefore not correct to group the incomes of the people

belonging to different age groups for measuring income inequality.

III. GINI COEFFICIENT

Definition

Gini coefficient is an aggregate numerical measure of income inequality

ranging from 0 (perfect equality, everyone in the society has exactly the

same amount of wealth) to 1 (perfect inequality, one person has all the

wealth and everyone else has nothing). It is measured graphically by

dividing the area between the perfect equality line and the Lorenz curve by

the total area lying to the right of the equality line in a Lorenz diagram.

The higher the value of the coefficient, the higher the inequality of income

distribution; the lower it is, the more equal the distribution of income.

Ways to present Gini coefficient

Gini coefficient is a summary statistic. We can present Gini Coefficient in

2 main ways:

Time series trend

The y-axis represents the percentage of Gini coefficient while the X-axis

represents the years the Gini index was calculated.

Cross section figures

The table gives Gini coefficient of 15 counties on different year base.

Figure 3: Time series trend

Figure 4 Cross section figures

Calculation

The Gini index is defined as a ratio of the areas on the Lorenz curve

diagram. The Gini index can be calculated through various mathematic

methods such as: integration, discrete probability function, cumulative

distribution function, relative mean difference, lognormal distribution, a

combination of techniques for interpolation, trapezoid estimation, or a

trick regression model (Ogwang 2000) and of course the hybrid method

of the two author Zuxiang Wang and Russell Smyth 2013.

Advantages

Gini coefficient has features that make it useful as a measure of dispersion

in a population, and inequalities in particular. It is a ratio analysis method

making it easier to interpret. It also avoids references to a statistical average

or position unrepresentative of most of the population, such as per capita

income or gross domestic product. For a given time interval, Gini

coefficient can therefore be used to compare diverse countries and different

regions or groups within a country; for example states, counties, urban

versus rural areas, gender and ethnic groups. Gini coefficients can be used

to compare income distribution over time, thus it is possible to see if

inequality is increasing or decreasing independent of absolute incomes.

Limitations

Although Gini coefficient quantifies the degree of inequality of income

distribution, but economists found that the Gini coefficient reflects only the

most general aspects of the distribution of income, in some cases, not

assess specific issues.

Different income distributions with the same Gini coefficient

Even when the total income of a population is the same, in certain

situations two countries with different income distributions can have the

same Gini index (e.g. cases when income Lorenz Curves cross).

Economies with similar incomes and Gini coefficients can have very

different income distributions

Extreme wealth inequality, yet low income Gini coefficient

A Gini index does not contain information about absolute national or

personal incomes. Populations can have very low income Gini indices, yet

simultaneously very high wealth Gini index. By measuring inequality in

income, the Gini ignores the differential efficiency of use of household

income. By ignoring wealth (except as it contributes to income) the Gini

can create the appearance of inequality when the people compared are at

different stages in their life.

Small sample bias sparsely populated regions more likely to have

low Gini coefficient

Gini index has a downward-bias for small populations. Counties or states

or countries with small populations and less diverse economies will tend to

report small Gini coefficients. For economically diverse large population

groups, a much higher coefficient is expected than for each of its regions.

The Gini coefficient measure gives different results when applied to

individuals instead of households, for the same economy and same income

distributions. If household data is used, the measured value of income Gini

depends on how the household is defined. When different populations are

not measured with consistent definitions, comparison is not meaningful.

Gini coefficient is unable to discern the effects of structural changes

in populations

Expanding on the importance of life-span measures, the Gini coefficient as

a point-estimate of equality at a certain time, ignores life-span changes in

income. Typically, increases in the proportion of young or old members of

a society will drive apparent changes in equality, simply because people

generally have lower incomes and wealth when they are young than when

they are old. Because of this, factors such as age distribution within a

population and mobility within income classes can create the appearance

of inequality when none exist taking into account demographic effects.

Thus a given economy may have a higher Gini coefficient at any one point

in time compared to another, while the Gini coefficient calculated over

individuals' lifetime income is actually lower than the apparently more

equal (at a given point in time) economy's. Essentially, what matters is not

just inequality in any particular year, but the composition of the distribution

over time.

Egalitarianism aspect:

Another limitation of Gini coefficient is that it is not a proper measure of

egalitarianism, as it is only measures income dispersion. For example, if

two equally egalitarian countries pursue different immigration policies, the

country accepting a higher proportion of low-income or impoverished

migrants will report a higher Gini coefficient and therefore may appear to

exhibit more income inequality.

Gini coefficient falls yet the poor get poorer, Gini coefficient rises yet

everyone getting richer

The income of poorest fifth of households can be lower when Gini

coefficient is lower, than when the poorest income bracket is earning a

larger percentage of all income. This is counter-intuitive and Gini

coefficient cannot tell what is happening to each income bracket or the

absolute income.

Inability to value benefits and income from informal economy affects

Gini coefficient accuracy

Some countries distribute benefits that are difficult to value. Countries that

provide subsidized housing, medical care, education or other such services

are difficult to value objectively, as it depends on quality and extent of the

benefit. In absence of free markets, valuing these income transfers as

household income is subjective. The theoretical model of Gini coefficient

is limited to accepting correct or incorrect subjective assumptions.

IV. OTHER USES OF LORENZ CURVE AND GINI COEFFICIENT

The Lorenz Curve and Gini coefficient can in theory be applied in any field

of science that studies a distribution. For example, in ecology the Lorenz

Curve has been used as a measure of biodiversity, where the cumulative

proportion of species is plotted against cumulative proportion of

individuals. In health, it has been used as a measure of the inequality of

health related quality of life in a population. In education, it has been used

as a measure of the inequality of universities. In chemistry it has been used

to express the selectivity of protein kinase inhibitors against a panel of

kinases. In engineering, it has been used to evaluate the fairness achieved

by Internet routers in scheduling packet transmissions from different flows

of traffic. In statistics, building decision trees, it is used to measure the

purity of possible child nodes, with the aim of maximizing the average

purity of two child nodes when splitting, and it has been compared with

other equality measures. The Gini coefficient is sometimes used for the

measurement of the discriminatory power of rating systems in credit risk

management.

APPLICATION

ZUXIANG WANG AND RUSSELL SMYTH

1. Structure of the article A hybrid method for creating

Lorenz Curve with an application to measuring world

income inequality

ZuXiang Wang and Russell Smyth has written a discussion paper

proposing a new method to create Lorenz Curves. They believe that their

method not also provide acceptable accuracy but also satisfies the

condition of the Lorenz Curve in terms of its density distribution.

First, in their paper, they suggest, propose and introduce the hybrid method

for building convex combination models for the Lorenz Curve and

demonstrate how to build Lorenz models from the forms generated.

Then, to test the performance of their proposed models, they use the data

of the US income distribution which previously also used by several other

economists for easier comparison. The results showed that their models

perform well with less residuals than other methods.

After that, they use one of their best models to apply on studying global

income inequality. Their results are investigated by our groups and will be

presented hereafter.

2. Findings from Lorenz Curves for selected years (1950,

1960, 1970, 1980, 1990, 2000 and 2005)

Figure 5: Lorenz curves for selected year

a) Comparing global inequality while two Aggregate Lorenz curves

separate (do not intersect each other)

As we can see from the chart above, from 1950 to 2005, there are many

fluctuations in the inequality all over the World. Normally, based on two

different Lorenz Curves, we can compare the inequality in different years.

For example, we can conclude from the graph that inequality in 1960 is

more than in 1950 (1960> 1950), 2005> 1950, 1960> 1970, 2000> 1980,

1960> 1990, 1960> 2005.

Take an example of the year 1950-1960: the inequality increases rapidly

Figure 6: Lorenz Curves for 1950 and 1960 (by Wang and Smyth)

Explanation: The vast majority of developing countries had lower

growth rates than high income OECD countries since high-income

countries became richer for the achievements from Industrial

Revolution while the poor remained poor. In some African countries

researched, there were even crisis in their political and economic

environment.

b) Comparing global inequality while two Aggregate Lorenz curves

intersect each other

However, there are some situations in which the two Lorenz Curves

intersect each other. In these cases, though we may be able to rank some

subsets contained in the entire set of all the aggregate Lorenz curves, we

cannot rank the entire set itself.

Take an example of the year 2000-2005:

Figure 7: Lorenz Curves for 2000 and 2005 (by Wang and Smyth)

- Increase in the proportion of Income in the poorest 40% of the

population (0-40th percentile)

Explanation: Improvement in global economic environment with

developing regions adopting macro economies and social policies.

Most developing and transition economies got high rapid GDP

growth and benefited from rapid expansion of World Trade and

also, easier access to global finance and rising migrant

remittances. More developing and transition economies began and

resumed the catching process.

- Increase in the proportion of Income in the richest 30% of the

population (70-100th percentile)

Explanation: In most OECD countries, the household income of the

top decile was growing faster than that of the bottom decile and the

total population due to an under-proportional increase of the real

wages compared to productivity, the over-proportional increase of

top management and superstar wages, and the unevenly distributed

capital income.

The accumulative Income increases in both the rich and the poor, so we

cannot conclude how the inequality changes from 2000-2005 by just

comparing the two Lorenz Curve as stated. Therefore, we will need another

component in order to find conclusion for this situation and that is the

Aggregate Gini Coefficient.

3. Findings from the trend of AG, its decomposition and

average PPP of the world

Figure 8: The trend of AG, its decomposition and average PPP of the

world.

Figure 8 graphs the trend in world inequality over the period 1950 to 2006. This

figure also represents the link between AG (Aggregate Gini) and the between

contribution, the within contribution plus residual and the average PPP income

per capita. The between contribution is the income inequality between different

countries; the within contribution expresses the economic inequality between

individuals or households in one country.

- The AG is high for almost all the years considered. There are 19, out of

57 years in which the AG was larger than 0.6. There are 36 years in which

AG was larger than 0.55 and there are 13 years in which the AG was less

than 0.5: this means the total inequality of the world (including the

inequality among individuals in a country and the inequality among

countries) is very high. The world as a whole has been suffering from high

inequality for a long time.

- The contribution of the between contribution is the main component of

the AG. There are 42 years in which the within contribution is less than

0.1 and 22 years in which the value is less than 0.05. The average within-

contribution across the period studied is only about 18% while the average

between contribution is about 64%. Therefore, the reason for the large

AG is not the within contribution, but the large income disparity between

countries. This means the inequality among countries are more visible

and contributes more to the total inequality of the world than the

inequality across individuals within each country does.

- From 1960 to 1966: Though it is larger than 0.2 for most of the 1st decade

considered, the within contribution (the intra-country income inequality)

falls in 1960 and kept low since then until 1966 at the expense of the rise

of inter-country income inequality during this period. There are several

reasons for this: First, from 1960, the vast majority of developing

countries had lower growth rates than high income OECD countries,

high-income countries became richer for the achievements from

Industrial Revolution. Second, before 1960, the authors only examined 4-

6 countries while since 1960, the authors increased the sample size to 10

countries and most of them are poor countries which may lead to the over-

increase in the proportion of poor countries and make inter-country

inequality increase.

- From 1966 to 1978: the within contribution increased but the AG

decreased, meaning the between contribution decreased and the rate of

this decrease must be large enough to offset the increase in the within

contribution. This can be explained by the stagflation in high income

countries. 1970s is the time of great inflation and stagnation in

industrialized countries. The first oil shock also happened in this period.

And also because of the break-down of the Bretton Woods fixed exchange

rate systems and the maturity in Japanese economy, high-income

countries became poorer and suffered from very low growth rates.

Therefore, the inter-country income inequality reduced during this

period. While in India, Indian people were enjoying their first

achievements after a long time being invaded by the English people and

having wars with their neighbor Pakistan.

- In 1978, the Aggregate Gini dropped from more than 0.5 to only 0.3 and

the within contribution (intra-country inequality) contributed more than

90% of this index. This happened not because of any social or political

reason but because of the sample size of the study. In this year, the authors

took only 4 countries into account and these countries are: Barbados,

Germany, Italy and UK. All these countries are medium to high income

countries in the West, there were hardly any income disparity between

these countries. The main component of the AG in this year was then the

inequality within each of 4 countries, which were also very low compared

to that of other developing countries.

- From 1978 to 1983, the between contribution (or inter-country income

inequality) was growing again, making the AG increase in total. This

period as we have known is the time of capital explosion with the freer

flows of capital around the world. The developed countries started to

grow fast and gained back what they lost during the stagflation period

thanks to the Keynesianism. On the other side of the world, there were the

sluggish growth performance in Latin America (following the debt crisis

and the neoliberal reforms), the decline in Eastern European/former

Soviet Union incomes (following the collapse of the Eastern Bloc and the

subsequent free market reform), and due to the disastrous economic

developments within many African economies.

- From 1983 to 2000, the Aggregate Gini index and its components

stabilized. It is because there were no significant economic shock during

this period or because the growth of many countries all over the world has

offset each other (the China gained success in their economic reforms

while the high-income countries also grew steadily) so the inter-country

income inequality remain unchanged.

- From 2000, there were an upward trend of within contribution and a

downward trend of the between contribution. This happened because of

many reasons. First, the intra-country inequality increase significantly

because of wage dispersion, technological change, demographic

changes, returns on capital, inheritance and initial inequality differences

and changes in income distribution policies and taxation policies.

According to recent empirical results, the most important reason for the

increasing income inequality in OECD was that in most of these countries

the household income of the top decile was growing faster than that of the

bottom decile and the total population. This increase in top incomes can

be mainly explained by an under-proportional increase of the real wages

compared to productivity, the over-proportional increase of top

management and superstar wages, and the unevenly distributed capital

income. The redistributive policies became weaker and could not offset

the growth in income inequality in this period. The rise of the top income

shares was the increase in business profits (i.e. by the relative high

dividends for shareholders) and top management salaries (including

bonuses and stock options), which suggests that changes in the income at

the very top explain most of the increase in income inequality in OECD

countries.

The downward trend of the between inequality happened because

of many reasons also: Improvement in global economic environment,

developing regions adopting proper macro economies and social

policies. Most developing and transition economies got high rapid GDP

growth and benefited from rapid expansion of World Trade and also,

easier access to global finance and rising migrant remittances. More

developing and transition economies began and resumed the catching

process.

4. Density distribution function:

The density distribution function explains why the 1960 Lorenz curve

represent the most inequality.

Figure 6 shows that the relationship of the corresponding densities is also

complex. The difference between the curves is attributable primarily to

the low-income portion of the distributions. The curve for 1960 is

different from the others, with a large population in the low-income

portion. The curve for 1950 has the most uniformly distributed population

among the seven distributions, and, therefore, suggests the least income

inequality. The curves for 1980 and 2000 can be regarded alike, and

similarly those for 1970, 1990 and 2005 can be seen as consisting of

another group. The curves in the latter group seem to have less positive

skew than those in the former.

OTHER AUTHORS FINDINGS:

Beside the application of Wang and Smyths method of measuring worlds

income inequality, many other economists have also conducted their own

studies and produced different findings. There are findings that have

similar results, supporting our two authors figures, but there are others that

are quite different from previous findings. We would like to list some of

them here to bring you a wider perspective on this issue:

1. A World Bank working paper by Branko Milanovic (2012):

Figure 9: Global Gini coefficient compared to the Ginis of selected

countries

In this research, Milanovic and his team found out that the Gini index is of

about 0.7 during the period from 1990 to 2010. Though there are some

small oscillations around the 0.7 level, the general trend is rather flat and

stable. In our authors research, the trend of the Aggregate Gini of the

world stabilized after 1983 and there is no huge change since then, which

partly confirm the trend in world income inequality. The level of the Gini

index is quite different, while in Milanovics research, the index is around

0.7, the index in Wang and Smyth study is around 0.6. Knowing that the 2

studies receive data from one source which is WIID version 2, but

according to Wang and Smyth, they do not use all the figures presented in

the WIID 2 for their accuracy adjustment purpose at the expense of taking

less countries into account, and this lead to quite a different result in the

two pieces of research. Despite all of that, we still come to a conclusion

that the level of inequality in income of the world since 1990 is high (0.6

or 0.7) and the trend is stabilizing.

Figure 10: Lorenz Curve for the global income distribution in 1988 and

2008 by Milanovic 2012

Milanovic from World Bank also presented his findings in Lorenz Curves

for the two years 1988 and 2008, and when compared to the Lorenz Curve

in 1990 and 2000 in Figure 5 of Wang and Smyth research, we can realize

some similarities: the Lorenz Curve of 1988 and 1990 are dominated by

the Curve of 2008 and 2000 in both studies. This again confirms the

discovery that the world income distribution is becoming more equal with

the significant rise in income of the middle portion.

2. Branko Milanovic (2010b)

Same Branko Milanovic but back to the year 2010, this author also

published his study on income inequality of the world but with a different

source of data. Therefore, once again there are differences with the Wang

and Smyth findings in terms of figures, however, the trend is similar in

some way, especially in inter-country income inequality or the between

contribution as written in the two articles.

As in weighted Gini coefficient from this chart, we can see that there were

a significant decrease in the inter-country income inequality during the

period from 2000 afterwards. This also happened in Wang and Smyths

line graph where the AG stabilize and the within contribution increase

significantly leading to the huge drop in Between contribution Gini index.

In terms of the figures, again, in these 2 graphs, there are discrepancies

between the 2 levels but the general idea is that they (inter-country

inequality indexes) are both high (above 0.5). In other words, the inequality

happened cross countries are high in general but positively quickly

reducing in recent years. The reasons for this have been mentioned above

in this report.

3. Solt (2010)

Solt dataset in 2010 gave certainly interesting findings in intra-country (or

inequality within each country) Gini index. Here are 6 graphs showing the

trends of intra-country income inequality Gini indexes. As can be seen

easily from these graphs, the intra-country inequality or the within

contribution Gini indexes increased in most parts of the world, from High-

income countries, European, Central Asian, East Asian, South Asian to

Caribbean developing countries while the Middle East and Africa are the

only two areas that has the within inequality reduced in the period from

1990 to 2005 but only with a slight reduction. When summing up these

data, we can conclude that the within contribution of the world Gini index

increased during this period. So, again the trend in Wang and Smyth

research consents with this findings: the income inequality happened

between groups/individuals within a country is increasing during the

period 1990 to 2005.

Before going to older findings from other economists, lets sum up what

we have known so far in this part. We have been looking at several papers

which all consent or at least support the trends in the Wang and Smyth

paper. Those similarities are:

- The Aggregate Gini index of the world income inequality is high and

stable since 1983.

- The income inequality across countries all over the word is high and since

2000 the index has decreased significantly.

- The income inequality happened within each country seems to increase

at the expense of the reduction in the inter-country income inequality from

1990 to 2005.

4. Some discrepancies among economists all over the world:

To be more objective, we would like to note that: Due to the differences in

methodologies and data sources, many other authors have produced

different findings, even contrary to these papers. The earlier the papers

were written, the more different they are from the pieces of study we have

mentioned above. There always be improvements in methodology and

quality of data sources. Since all papers we have mentioned above are

written recently (from 2010 to 2013), the availability of data sources are

hugely differ from what available therebefore.

Here we would like to present the various pathways of global income

inequality that some other economists have published. As can be seen from

this chart, there are many trends proposed by different authors, some of

them even contrast to the others. So, before any solid evidence and

completely thorough further research are conducted, all trends and figures

are for your own reference.

However, one thing that is in common from all studies available: the

world Gini index are always high above 0.5, which means there are

undeniable global inequality in income.

CONCLUSION

In the Introduction, we shortly reviewed the background of two theories of

Lorenz Curve and Gini coefficient on the inventors, the publish time and

the name of the paper.

In the Theory, we have reviewed the definitions, way to construct and

calculate the Lorenz Curve and Gini coefficient. It is important to know

that Gini coefficient is a very powerful tool in measuring income

inequality. While Lorenz Curve can show how the income is distributed

among individuals or among countries, the Gini coefficient can give a more

specific figure on the inequality, especially when it comes to the

comparison over time. There are ones limitations that are also the

advantages of the other. The two tools complete each other and help the

economists as well as the governments achieve the so-called income

equality.

The applications of two economists Wang and Smyth, though have some

limitations, still provide interesting and informative findings about the

trends and the degree of the global income inequality. We are happy to

have a chance to know more about Gini decomposition into inter- and intra-

country inequality and about the reasons behind the increase and decrease

of those contributions. Recent changes in the global income inequality

showed some positive signs though it still depends on the viewpoint of the

policy maker. Even though each economist has their own way to calculate

and conclude about the income inequality of the world, it is undeniable that

the inequality of the world is high and varies among countries.

The more time we spent with this topic, the more serious and important it

is that we realized. We hope that this small work beside helping us in our

own study, can show our eagerness to learn and our gratitude to our

beloved teacher.

Sunshine Group

K50CLC2

March 3, 2014.

gini coefficient and lorenz curve report

Documents

uses of lorenz curve

lorenz curves

present gini coefficient

world income inequality

overview of inequality

world bank

authors findings

table of contents introduction