Ranking income distributions using the social welfare approach

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Ranking income distributions using the social welfare approach the example of West Germany, 1950-1970
Milleville, Laura Albers
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viii, 59 leaves : illustrations ; 29 cm


Subjects / Keywords:
Income distribution -- Germany (West) ( lcsh )
Income distribution ( fast )
Germany (West) ( fast )
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )


Includes bibliographical references (leaves 58-59).
General Note:
Submitted in partial fulfillment of the requirements for the degree, Master of Arts, Department of Economics.
Statement of Responsibility:
by Laura Albers Milleville.

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University of Colorado Denver
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Full Text
Laura Albers Milleville
B.A., Valparaiso University, 1985
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Arts
Department of Economics

This thesis for the Master of Arts
degree by
Laura Albers Milleville
has been approved for the
Department of
Steven G. Medema
/ 2/5 ten

Milleville, Laura Albers (M.A., Economics)
Ranking Income Distributions Using the Social Welfare
Approach: The Example of West Germany, 1950-1970
Thesis directed by Professor W. James Smith
This thesis addresses the ranking of income
distributions using generalized Lorenz dominance. The
purpose of ranking income distributions is to establish
an ordinal ranking of the most preferable to least
preferable; be it over time, across countries, or before
versus after taxes. The conventional method is to rank
income distributions strictly by their degree of
inequality. However, the standard of living is also an
element of social welfare and therefore should be
included as a criterion in the ranking of income
This thesis surveys the conventional methods of
ranking income distributions but focuses on the method of
generalized Lorenz dominance. Generalized Lorenz curves
(ordinary Lorenz curves multiplied by the mean of the
corresponding income distribution) were used to rank West
German income distributions from 1950 to 1970. The
application of this method produces a complete ranking of
the income distributions and shows that social welfare in
West Germany unambiguously increased over the twenty-year

The form and content of this abstract are approved.
I recommend its publication.

1. INTRODUCTION................................... 1
Scope and Purpose...............................3
2. THE WEST GERMAN WELFARE STATE: 1950 1970.......6
Historical Background of the Social
Market Economy..................................7
West Germany's Economic Growth During the
1950s and 1960s................................10
West Germany as a Welfare State................14
Full Employment.............................14
Income Inequality and
Income Redistribution.......................16
The Future of the West German
Welfare State..................................21
THE SOCIAL WELFARE APPROACH.....................24
Conventional Measures of Income Inequality.... 24
The Atkinson Theorem...........................33
Generalized Lorenz Dominance...................35
INCOME DATA.................................... 40
Gini Coefficients..............................43
Lorenz Dominance...............................43
Generalized Lorenz Curves......................48

WORKS CITED...........................................58

3.1. The Lorenz curve................................26
3.2. Lorenz dominance................................27
3.3. The Gini coefficient............................29
3.4. Concave utility function........................34
3.5. Generalized Lorenz dominance....................38
4.1. Lorenz curves for West Germany,
1950 1970....................................45
4.2. Generalized Lorenz curves for West Germany,
1950 1970....................................49

4.1. Gini Coefficients and Corresponding Ranks.......44
4.2. Lorenz Curve Values and Corresponding
Mean Incomes for West Germany, 1950 1970......47

This thesis is a study of the distribution of income
in West Germany from 1950 to 1970. This time period is
particularly interesting because 1950 marked the
beginning of a pattern of stable and strong economic
growth which lasted into the 1970s. Studying growth and
income distribution in West Germany is interesting
because the economic system is a blend of market economy
and welfare state. The goal of growth is usually
associated with the former, while the goal of equality is
usually associated with the latter. This thesis
addresses how these two goals have been combined in
economic theory to track income inequality, growth, and
welfare between 1950 and 1970.
The conventional approach to ranking income
distributions is to examine summary measures of
inequality such as the Gini coefficient, variance, and/or
coefficient of variation. These measures attempt to
capture inequality or dispersion of the statistical
income distribution and are discussed in detail in
Chapter Three. The purpose of measuring income

inequality is to compare inequality among various
populations. Although the context for analyzing
inequality may vary (one may want to know the effect of
time or taxation on inequality or how inequality varies
among countries), the use of these measures nevertheless
assumes equality to be the single criterion for
preferring one distribution over another. This
preference for equality is commonly subsumed under the
title "the egalitarian principle."
The degree of inequality, however, is not the only
criterion by which to judge societal welfare. A
society's standard of living must certainly weigh in the
balance. To see this most clearly, suppose income were
distributed as equally in Indonesia as in Sweden. If
Sweden possesses ten times the mean income of Indonesia,
which country is better off? The conventional measures
would rank them equally because the conventional approach
does not account for differing levels of income but only
A second, related problem with conventional measures
of inequality is that they implicitly subsume social
welfare functions which may or may not accord well with
a society's values (Atkinson 1970). Because of these
implicit social welfare functions, the rankings of the
income distributions often vary with the measure used,

even though applied to the same statistical information.
It follows that the appropriateness of a measure in a
given situation is determined by the social welfare
function specified. As such, any particular conclusion
must necessarily be conditional, depending as it does on
the application of a particular social welfare function
when many are possible. Although the concept of social
welfare as essential to the measurement of income
inequality was discussed as early as 1920 by Hugh Dalton,
it is only in the last two decades that it has been
translated into techniques for ranking income
Scope and Purpose
The purpose of this thesis is to rank income
distributions in West Germany between 1950 and 1970. The
social welfare method of the generalized Lorenz curve
(which is the ordinary Lorenz curve multiplied by the
mean income of the corresponding income distribution)
will be applied to West German income data in the attempt
to draw a conclusion about the trend in social welfare
over the twenty-year period. The measurement of income
inequality has traditionally occurred in the study of
developing countries, most likely from the desire to
measure the extent of inequality as the price paid for

initial growth.1 Relatively little has been done in
analyzing time-series data for developed countries. West
Germany is analyzed in this thesis because of its dual
commitment to growth and equality as reflected in its
unique mix of welfare state and capitalism.
The economic system and economic events specific to
West Germany during the period 1950 to 1970 will be
discussed in full in Chapter Two. Chapter Three contains
a discussion of the conventional summary measures of
income inequality and explains the social welfare
approach to ranking income distributions. Chapter Four
contains a summary of the results of applying both Lorenz
dominance and generalized Lorenz dominance to West German
income data from 1950 through 1970. Chapter Five
presents the conclusions on trends in inequality and
social welfare and comments on the social welfare
Because the distribution of income is usually
studied for the purpose of analyzing and forming social
policy, the stance in this paper is that of a
disinterested policy-maker. Throughout the thesis it is
assumed that one is strictly interested in the situation
of the population as a whole and that one is therefore
not concerned with personal envy or any particular
person's place in the distribution, however influential

these may be in the actual political process of income
distribution and redistribution.

1950 THROUGH 1970
Since World War II, West Germany has operated as a
"market-based" economy. There are negative side-effects
of market capitalism, however; and severe inequality in
the distribution of income can result. Dissatisfaction
with these side effects has given rise to the modern
"welfare state" (Schnitzer 1972, 109). West Germany is
considered a welfare state, though not to the extent of
the Scandinavian countries, in that it is committed to an
equitable distribution of income, a minimum standard of
living for all citizens, and full employment as the most
important public policy goal. A welfare state
necessarily implies government intervention in the
economy (Schnitzer 1972, 32).
West Germany has been exceptionally successful in
fulfilling its goals as a welfare state while maintaining
high rates of employment and economic growth. In fact it
is the growth and full employment which have facilitated
the maintenance of welfare programs. West Germany's
accomplishments in social welfare are best understood by
examining the historical context which gave rise to the

current economic system and the historical and
demographic factors which have been conducive to the
success of the economic system. In recent years there
has been increasing concern over the ability to maintain
the welfare state in the future. While this chapter
focuses on the period corresponding to the quantitative
analysis in this thesis, 1950 to 1970, the concern over
the future of the welfare state is addressed to some
extent at the end of the chapter.
Historical Background of the "Social Market Economy"
A number of experiences led the Germans to their
current economic system of the "Social Market Economy."
Under Bismarck's leadership, the thirty-nine German
states which had been operating as independent economic
units were unified in the late 1800s. Even though the
Germans learned that their former barriers to trade had
hindered their economic development, they never accepted
"the invisible hand" philosophy which pervaded British
thought in the 1800s (Hallett 1973, 13-14). During
Bismarck's reign there was increasing discontent with
rising industrialization. Socialist movements gained
momentum, particularly among laborers. Bismarck
successfully headed off socialism by introducing
compulsory social insurance in 1881, national health

insurance in 1883, and workmen's compensation in 1884.
Most of these were the first of their kind in the world
(Schnitzer 1972, 111-12).
The Weimar Republic faced severe difficulties in
managing the economy after World War I. The financing of
war reparations created an astronomical increase in the
money supply and the resulting hyperinflation left many
middle class investors impoverished. Hyperinflation in
the 1920s was followed by depression in the early 1930s
during which the unemployment rate peaked at 20 percent.
The Weimar Republic's inability to manage the economy
ushered in the National Socialist party to power (Hallett
1973, 15-16).
By the time World War II ended in 1945, West Germany
was in ruins. A fifth of all housing had been destroyed,
millions of refugees were migrating from the. East,
several million women were left widowed, and the standard
of living was at a subsistence level (Hallett 1973, 25).
In addition to the physical suffering, the nation faced
the problem of establishing a new political and economic
system. A market economy had appeal, but many questioned
whether a market economy was capable of meeting the
immediate social needs of the country and whether it
would mean giving up the social care they had become
accustomed to since the 1880s (Braun 1990, 176-77).

West Germany was under pressure from the Allied
Control Council to establish a competitive market system,
and Marshall Plan aid was contingent upon it. The goal
was to define a system which was at the same time
competitive and decentralized enough to provide growth,
efficiency, and freedom and yet could be influenced by
the state to correct for market failures and provide a
minimum level of care for all citizens. Neoliberal
members of the Freiburg School proposed ideas which led
to the concept of the "Social Market Economy." This
concept was the product of a number of schools of
thought: Keynesian economics, liberal social and
political theory, Catholic social teaching, and a
tradition of local self government (Hallett 1973, 18,
22). The Social Market Economy was to be a competitive
market system supplemented by monetary policy to provide
stability, small decision-making groups to provide
decentralized guidance in industry (such as cooperatives
and chambers of commerce) and social welfare programs to
compensate for the negative effects of capitalism
(Hallett 1973, 20-22).
It should be noted that the Social Market Economy
was not intended to be a welfare state. In fact many
argued against the bureaucracy associated with welfare
states. The Social Market Economy was to provide social

security only to those who could not help themselves.
Although income inequality was to be moderated by
progressive taxation, it was to be accomplished through
relatively low marginal rates so as not to undermine
risk-taking (Braun 1990, 177-78). This economic system
has remained largely intact with the exception of adding
the use of fiscal policy to control cyclical
West Germanys Economic Growth
During the 1950s and 1960s
As mentioned at the beginning of the chapter, West
Germany's ability to maintain welfare programs has been
greatly facilitated by its strong economic growth. From
1949 to 1985 there was tremendous economic growth, but at
declining rates. In 1949 growth in industrial production
was at a standstill and by the year's end nearly two
million were unemployed. The combination of a currency
reform, the lifting of price controls, a new government
policy supporting full employment, Marshall Plan aid, and
the new market for West German exports created by the
outbreak of the Korean war in June of 1950 ushered in the
beginning of what was to become a long period of economic
growth (Braun 1990, 178-79; Smith 1983, 30).
The "economic miracle" can be attributed to a number
of factors. Despite the economic setbacks at the end of

the war, West Germany had a number of things working in
its favor. Located in what had been the most industrial
area of the Third Reich, it contained the Ruhr coal
fields, a few oil fields, the North Sea ports of Hamburg
and Bremen, and a large stock of mineral resources
(Schnitzer 1972, 24). It is estimated that fixed
industrial assets, even after dismantlement by the Allied
Control Council, were larger in the late 1940s than they
were in 1939 (Braun 1990, 146-49). Migrating refugees
from East Germany provided an unlimited supply of highly
skilled labor. In addition to these resources, West
Germany's economic growth was significantly furthered by
the Marshall Plan of 1947. Although the Marshall Plan
benefitted public utilities, transportation, and the iron
and steel industries the most, many businesses received
technical assistance fromU.S. economists, engineers, and
business consultants (Braun 1990, 153-54) There was
little inflation due to the stable wages resulting from
the general surplus of labor.
Even though these conditions were conducive to
economic growth and high employment, the concept and
economic policies of the Social Market Economy
contributed to the economic miracle in the 1950s. Tax
exemptions and incentives for business and industry were
designed to encourage work, savings, and investment.

Export incentives and housing subsidies were also offered
(Braun 1990, 179, 203; Schnitzer 1972, 11). During the
1950s real output grew by about 8 percent per year, which
was a higher rate than those found in other European
countries at that time. Growth rates in the 1960s, while
still very good, were somewhat lower, averaging 4.7
percent per year (Braun 1990, 168; Smith 1983, 33).
By the 1960s, unemployment rates had reached
phenomenally low levels by European and American
standards. The unemployment rate was 1.3 percent in
1960, .8 percent in 1964, and .7 percent in 1970
(Schnitzer 1972, 126-27). In the 1960s, the labor force
participation rate declined because of the changing age
structure of the population, an earlier standard
retirement age, and an increase in the amount of time
spent on education and training. Though not as
significant, the building of the Berlin Wall in 1961
halted the flow of skilled labor which had been a
valuable resource for the West German economy. This
shortage of labor resulted in the recruitment of
Gastarbeiter, or guestworkers, from Mediterranean
countries. Though not as skilled as the East Germans had
been, they were able to fill many vacancies for unskilled

labor. A consequence of the labor shortage was a shift
to more capital-intensive production methods (Braun 1990,
During the 1960s, the government continued to
promote economic growth through incentives for new small-
and medium-sized businesses. Funds generated from the
Marshall Plan aid which didn't need to be repaid were
placed in a special fund separate from the federal budget
(Smith 1983, 30). This fund was used to support small
businesses in the form of credits or grants, with greater
assistance given to businesses locating in regionally
depressed areas. From the time the fund was established
in 1967 to about 1971, over DM 20 billion were injected
into the economy. Federal and state governments also
tried to assist economically weak areas bordering
European Community countries (Schnitzer 1972, 96-97).
The combination of capital and guestworkers along with
the economic policies of the Social Market Economy
sustained the trend of relatively high rates of growth,
with the exception of a short recession in 1966-67. In
the 1960s attempts were also made to improve the
distribution of wealth and to strengthen the "social net"
(Braun 1990, 200), and it is to these actions that we now
turn our attention.

West Germany as a Welfare State
As mentioned earlier, a welfare state attempts to
correct the negative side effects of market capitalism.
The two major negative effects which West Germany has
tried to cure are the inequality in the distribution of
income and the irregularity in the level of employment,
both of which have stronger consequences for lower income
groups. West Germany has used a variety of policies to
counter these problems.
Full Employment
The policy goal of full employment aims to prevent
irregularities in employment and abates income inequality
at the same time. Maintaining full employment naturally
prevents the loss of personal income that usually occurs
in lower income groups during times of unemployment.
While income inequality is attributable somewhat to the
property income which the richer classes receive from
interest, rents, and dividends, full employment tends to
increase labor's share of national income, thereby
proportionately shifting income from property to labor,
which will tend to reduce inequality (Schnitzer 1972,
The Social Market Economy concept called for labor
market policies which supported the goals of full

employment and a more equal distribution of income.
Labor market policies include all measures which are
designed to affect the supply of and demand for labor.
Policies have included encouraging a transfer of labor to
more productive and better paying branches of industry
and providing vocational guidance and incentives for job
training, thereby improving the mobility of the labor
force (Schnitzer 1972, 111).
Also consistent with the Social Market Economy
concept, labor policies tend to be carried out in a
decentralized way. In addition to the regional labor
offices which coordinate unemployment insurance and
provide vocational guidance (Schnitzer 1972, 122),
employees and unions work closely with management in
arriving at optimal solutions to business problems. In
the early 1950s legislation was passed requiring that
one-third of the members on the supervisory boards of all
joint stock companies be representatives from labor
(Braun 1990, 215; Hallett 1973, 88). West Germany's
method of industrial relations has proved to be
successful in dealing with the problems of capitalism in
a humane way and in supporting the goals of full
employment and a more equal distribution of income
(Hallett 1973, 99). Though West Germany has not been

free of strikes, the occurrence of strikes between 1950
and 1970 was minimal by U.S. standards (Schnitzer 1972,
182) .
Income Inequality and Income Redistribution
Even with full employment, the market-determined
distribution of income still leaves a significant degree
of inequality. To counter the remaining inequality, the
welfare state affects the distribution of income through
progressive taxation and a system of transfer payments;
taxes reduce income and transfer payments increase
income. Although West Germany has a number of welfare
programs, only those which are cash benefits are
considered transfer payments (Schnitzer 1972, 36).
The transfer payment system has the dual objective
of minimizing income inequality and guaranteeing a
minimum standard of living for everyone (Schnitzer 1972,
35). While the transfer payment system is essentially a
redistribution of income from richer individuals to
poorer ones, some transfer payments do not have this
effect, such as the family allowance granted to all
families regardless of income (Schnitzer 1972, 137-38).
The extent to which the tax and transfer payment system
redistributes income from upper income groups to lower
ones is difficult to determine because of the fragmented
way in which transfer payments are financed. Because

half of the federal tax revenues comes from indirect
taxes, it is difficult to determine who is paying and how
much they are paying (Schnitzer 1972, 113, 127).
However, progressive income taxation does tend to counter
inequality because the effective rate of taxation
increases with the size of one's income. People with
higher incomes will have their incomes reduced by a
greater percentage than those with lower incomes
(Schnitzer 1972, 110).
Most social security expenditures are part of the
"social budget" which, though under the jurisdiction of
the federal government, is administered separately from
the federal budget. The social security system is
largely financed by payroll taxes shared by employees and
employers, but is supplemented by general revenues of the
federal budget (Schnitzer 1972, 93, 112-13). The
federal government's revenues generally come from
indirect taxation, the three main sources being the
value-added tax, tobacco tax, and petroleum tax. These
three sources provided over half the federal government's
tax revenue in 1971 (Schnitzer 1972, 113). The remainder
is made up of other indirect taxes and income and
corporate taxes which are divided among federal, state,
and local governments (Braun 1990, 196).

Transfer payments in West Germany in the early 1970s
included old-age and invalid pensions, unemployment
compensation, maternity grants, and family allowances
(Schnitzer 1972, 36). In 1971 old-age insurance or
pensions were basically self-financing. Contributions to
the pension funds are a percentage of income and one's
benefits generally depend on what one's contributions
were (Hallett 1973, 94). In 1957 legislation was passed
requiring that old-age and invalidity pensions increase
annually according to the average rise in earnings.
Although the federal government subsidizes this program
in order to provide increased benefits, pensions still
vary with how much one contributed to the fund. Upon
retirement, most pensioners receive one-half to two-
thirds of their working income (Hallett 1973, 96-97;
Braun 1990, 203-4).
The unemployment rate in West Germany was so low
during the 1960s that unemployment compensation was
almost non-existent. In 1970 benefits ranged from 40 to
90 percent of earnings and were paid from thirteen to
fifty-two weeks. Cash supplements were provided
according to the number of dependents, and when benefits
ran out one could apply for an extension (Schnitzer 1972,
119) .

As mentioned earlier, although the family allowance
does not redistribute income from the rich to the poor
(it essentially redistributes from those without children
to those with children), it is counted as a transfer
payment because it is a cash benefit in the social
budget. The allowance varies strictly with the number of
children one has (beyond one child), not one's income.
This benefit is however, designed to improve or "correct"
the differentials between family need and family income
(Schnitzer 1972, 120). Health insurance is another
social security program, though it is not considered a
transfer payment. National health insurance is generally
carried out through "health funds" which are financed by
equal contributions from employees and employers (Hallett
1973, 94). Benefits in 1970 included prescriptions,
hospitalization, maternity benefits, and payment of 50
percent of wages lost due to illness (Schnitzer 1972,
111) As in other countries, the health care system has
been plagued by rising costs from increased technology
and treatment, and an increase in longevity (Hallett
1973, 95-96).
Other welfare programs in 1970 included workmen's
compensation (in 1970 90 percent of the cost was covered
by employers) and war pensions and related assistance
such as care for orphans and scholarships, all of which

was funded by general government revenues (Schnitzer
1972, 119-22).
Even though it is difficult to determine the extent
to which the tax and transfer payment system creates a
more equal distribution of income, some have cited an
increase in equality over the 1950s and 1960s. Josef
Korner found that in ranking income distributions from
1950 to 1965 using Lorenz curves, inequality did diminish
over the fifteen year period (Schnitzer 1972, 133).
There are a number of economic phenomena which occurred
during that time period which could have resulted in the
shift of income from upper to lower income groups.
With the increase in industrial productivity there
was a shift from agricultural jobs to urban-industrial
ones. Because agricultural jobs were generally lower-
paying ones, this shift helped the lower income groups.
Full employment was a high-priority economic policy and
the low unemployment allowed an increase in real wages.
Labor received a larger portion of national income,
compared to income accruing to property and entrepreneurs
(Schnitzer 1972, 125-26). Another factor which could
have contributed to the trend toward greater equality is
the post-war economic incentives favoring small and
medium-sized business development and ownership of
housing. Ownership of property increased for all Germans

and significantly boosted the incomes of those at the
lower end of the income distribution (Schnitzer 1972,
West Germany's ability to support a welfare state
system during the 1950s and 1960s was facilitated by its
strong economic growth and full employment. Its strong
economic growth and full employment were, in turn, partly
attributable to Social Market Economy policies, but
demographic and external factors assisted considerably.
One may question whether the Social Market Economy would
have been as successful if these factors had not been
working in its favor.
The Future of the West German Welfare State
In recent years there has been increasing discussion
of "the crisis of the welfare state." Signs of crisis,
or "the welfare backlash," have been seen in a number of
countries. A study in the late 1970s showed that West
Germany experienced relatively little welfare backlash as
compared with other countries.2 This may be partly
attributable to the fact that for nearly a century
Germans have viewed public welfare as one of the
responsibilities of government and that Germany's growth
rate was strong and steady up to the time of the survey
(Brauns and Kramer 1989, 124, and Higgins 1981, 152).

During the late 1970s there was a deterioration in
the conditions conducive to economic growth. There was
increasing unemployment and, as in most countries,
inflation in the mid to late seventies. In the early
1980s, the growth in GNP slowed. Increased unemployment
created a greater burden on the welfare system (Brauns
and Kramer 1989, 142-43). Since 1983 a recovery has been
underway, but Germans no longer take permanent strong
economic growth for granted (Braun 1990, 169-70; Brauns
and Kramer 1989, 135-36).
With the government's resort to austerity measures
in the late 1970s and early 1980s, limits of the welfare
state have increasingly drawn attention (Brauns and
Kramer 1989, 127). Until the 1970s, the welfare system
was rarely criticized and was characterized by ever-
increasing expansion of its scope and level of benefits.
So far there has been no dismantlement of programs, but
the rates of growth in programs have been reduced. In
the late 1980s there were signs of renewed vigor in the
West German economy, indicating that the welfare state
would survive (Brauns and Kramer 1989, 138-40). However,
the current reunification of East and West Germany
introduces a new set of problems. The ramifications of
this historical event are as yet unknown but, given the
economic troubles East Germany is bringing to the table,

the economic adjustment process that will accompany
reunification may necessitate high levels of welfare
spending. Although West Germany has been dedicated to
maintaining high economic growth, it has also been
dedicated to a welfare state. The extent to which
Germany will trade off growth for entitlement welfare
remains to be seen.

As mentioned in Chapter One, the conventional
approach to measuring income inequality commonly utilizes
summary statistics that presuppose a social welfare
function. In this chapter, some of these methods are
reviewed and critiqued from the social welfare
perspective. The latter half of the chapter delineates
the social welfare approach and generalized Lorenz
dominance which will be used to analyze the income data
on West Germany.
Conventional Measures of Income Inequality
Some of the most commonly used measures of
inequality which are considered here include the Gini
coefficient, variance, coefficient of variation, and
relative mean deviation. Another common method of
evaluating inequality is the Lorenz curve, which is
graphical in nature as opposed to the statistical
measures just mentioned. Because the Lorenz curve is an
integral part of the social welfare theory explained
later, and because the Gini coefficient will be used for

comparative purposes in Chapter Four, these two methods
are explored in some depth.
The Lorenz curve analyzes the distribution of income
by arranging household incomes in ascending order and
then dividing this sequence into groups or "sizes," such
as quintiles or deciles. As shown in Figure 3.1, the
income recipients are plotted on the horizontal axis by
cumulative percentages. The percentage of income is
plotted along the vertical axis. Any point on the graph
represents the percentage income that the lowest 10
percent, 20 percent, 30 percent, etc., of households
receive. For example, if the poorest 30 percent of the
population receives 10 percent of the income, the point
would be plotted as (30,10). The diagonal is the line of
perfect equality. In this case, each household receives
the same level of income: ten percent of the households
receive ten percent of the income, twenty percent of the
households receive twenty percent of the income, and so
on. The larger the area between the Lorenz curve and the
line of perfect equality, the more unequal is the income
distribution the Lorenz curve represents.
Ranking income distributions by this method is
referred to as Lorenz dominance (see Figure 3.2). If
one Lorenz curve lies everywhere above a second, the
first is said to dominate the other. Thus the income

Percent of Population
Figure 3.1. The Lorenz curve.

Figure 3.2. Lorenz dominance. Lorenz
curve A dominates Lorenz curve B.

distribution associated with the dominating Lorenz curve
ranks highest. Use of summary statistics, such as those
mentioned earlier, will yield the same ranking. However,
if the Lorenz curves intersect, ranking is not possible
(Atkinson 1980, 40-41). These summary statistics are
used more frequently than Lorenz dominance for the very
reason that, due to their numerical nature, they give a
rank where no ranking would otherwise be possible in
situations of intersecting Lorenz curves. Of these
statistical measures, the Gini coefficient is probably
the most frequently used.
The Gini coefficient measures the area distance of
the Lorenz curve from the line of equality (see Figure
3.3), and therefore bears a close relationship to the
Lorenz curve. Mathematically,
^\i[y[yF(y)-\i$(y)]f(y)dy. (3.1)
2 Jo
The closer is the Gini coefficient to 1, the more unequal
is the income distribution. The closer is the Gini
coefficient to zero, the more equal is the income
distribution. It must be emphasized however that the
Gini coefficient cannot lay claim to welfare content
when Lorenz curves intersect (Atkinson, 1970).

Percent of Population
Figure 3.3. The Gini coefficient (a
graphical representation). The Gini
coefficient is area D divided by the
total area ABC.

Ranking by either the Lorenz curve or the Gini
coefficient assumes perfect equality to be the singular
social goal, regardless of the level of income or number
of income recipients (i.e., it supports the egalitarian
principle). Recall from Chapter One, however that the
level of income is also a component of social welfare.
Income should therefore be incorporated in determining
which distribution should be preferred over another.
Because there is a functional relationship between income
and welfare, the analysis of income distribution should
acknowledge and specify this relationship, with the
appropriateness of a measure in a given situation
depending on the relation assumed between welfare and
income (Dalton 1920, 348).
Dalton made some preliminary assumptions about
welfare: (1) economic welfare of different persons is
additive, (2) the relation of income to economic welfare
is the same for all members of the same community, and
(3) marginal economic welfare diminishes as income
increases (Dalton 1920, 349). From these assumptions,
Dalton deduced that for a given level of income, maximum
social welfare is achieved with perfect income equality.
Deriving from the same set of assumptions, the principle
of transfers states that inequality is diminished when a
transfer of income is made from a richer income-receiver

to a poorer one, provided the transfer is not so large as
to result in simply trading places, thereby leaving the
degree inequality unchanged (Dalton 1920, 351). Because
the principle of transfers diminishes inequality, the
selected measure of inequality should be sensitive to all
transfers made according to this principle (Dalton 1920,
354) .
Dalton showed that the standard deviation and mean
difference are sensitive to the principle of transfers
while the mean deviation is not. Dalton concluded that
based on the principle of transfers, the Lorenz curve is
an appropriate and preferable measure because it is
essentially equal to one half the relative mean
difference in addition to being graphically convenient
(Dalton 1920, 354, 361).
Half a century later, Anthony Atkinson (1970)
readdressed the implicit welfare functions contained in
measures of inequality (Atkinson 1970, 253, 262).
Atkinson showed that, although some measures may meet the
principle of transfers, they vary in their relative
sensitivity to transfers at various points in the
distribution (Atkinson 1970, 255). For example, if one
wanted to give more weight to transfers at the lower end
of the distribution than at the top, the coefficient of
variation would be an appropriate measure. The standard

deviation, previously approved by Dalton, presupposes a
social welfare function which gives more weight to
transfers made at the lower end of the distribution and
ceases to be concave at high incomes. In the case of the
Gini coefficient, weight is attached to transfers made at
the center of the distribution. The relative mean
deviation is sensitive only to transfers made across the
mean, not those made within each side of the mean
(Atkinson 1970, 256). These measures obviously assume
different welfare functions, some of which may or may not
accord with social values. It comes as no surprise that
empirically these summary measures give conflicting
rankings (Atkinson 1970, 262).
Atkinson states that the correct approach is to
explicitly specify the social welfare function one wishes
to employ and to determine the most suitable
corresponding measure of inequality. Because the social
welfare function could take on a wide variety of
normative forms, Atkinson suggests that attention be
directed at a function with minimal assumptions so as to
garner the widest support possible (Atkinson 1970, 257).
Atkinson formalized Dalton's initial assumptions
that the welfare function be an additively separable and
symmetric function of individual incomes (Atkinson 1970,
245) :

Wmf u(y) f(y) dy.
where U(y) is a level of utility attributed to each
income y, and the function U(y) is restricted to those
that are increasing and concave (U'> 0, U"< 0). This is
equivalent to evaluating social welfare as average
utility in society (Lambert 1989, 55). By assuming an
increasing and concave social welfare function, the
principle of transfers is guaranteed (Atkinson 1970,
249). A transfer made according to the principle of
transfers embraces two phenomena: (1) a reduction in
inequality, and (2) an increase in total social welfare
(see Figure 3.4).
The Atkinson Theorem
The Atkinson theorem states that if two Lorenz
curves do not intersect and the corresponding
distributions have equal mean incomes, we can conclude
without further specification of U(y), that the
distribution associated with the dominating Lorenz curve
will be preferred.3 More formally, for two income
distributions F(y) and F*(y) with equal means mF = mF*,
LF(p)zLF.(p) for all p e [0,1] =
fu(y)f(y)dyz fu(y)f*(y)dy. (33)

Figure 3.4. Concave utility function.
Net welfare increases from a transfer of
income from a higher income to a lower

for all functions U(y) such that U'(y)>0 and U" (y)<0
(Atkinson 1970, 247). If we are analyzing distributions
with different means, as is often the case, we can make
the same conclusion as long as the mean associated with
the dominating Lorenz curve is at least as great as the
mean associated with the more unequal Lorenz curve (i.e.,
mF > mF*) (Atkinson 1970, 247) .4
The Atkinson theorem does not allow us to rank
distributions with crossing Lorenz curves or to rank
those distributions where the dominating Lorenz is
associated with the lower mean income. The generalized
Lorenz curve, developed by Shorrocks (1983) improves our
ability to make judgments about distributions with these
problems. Consequently, it has proved to be more
successful than the Atkinson theorem in ranking income
Generalized Lorenz Dominance
Shorrocks considers two types of preferences in his
analysis: (1) the preference for a more equitable
distribution of income (equity preference), and (2) the
preference for higher real incomes (efficiency
preference). Consequently, the ranking of distributions
by the generalized Lorenz curve can be divided into two
parts: (1) establishing the degree of inequality (to

satisfy the equity preference), and (2) examining the
mean incomes of the distributions (to satisfy the
efficiency preference) (Shorrocks 1983, 3).
Shorrocks questioned whether one could specify a
social welfare function that would both produce
conclusive rankings and also hold widespread support
given the equity and efficiency preference. We begin
with a welfare function,
W=W(y) =W(y1, . ,yn) . (3.4)
where y. is the income of individual i and households are
assumed to be identical in all respects except for their
household incomes. The equity preference is captured by
requiring that W(.) is Schur-concave (Shorrocks 1983, 4) :
(S-concavity) Vf(By) ^W(y) for all
bistochasitc matrices B.
The efficiency preference can be captured by the
restriction that W(.) be a non-decreasing function of all
incomes (Shorrocks 1983, 5):
(monotonicity) W(ylf . .yn) is non-decreasing
£ y, i1/
The generalized Lorenz curve is simply a Lorenz
curve multiplied by the corresponding mean income:

GL(y,p) = mL(y,p) .
where y is ordered by increasing incomes (Shorrocks 1983,
6). Given the assumptions about the social welfare
function above, we can rank distributions by generalized
Lorenz dominance: let y and y' be two income
distributions and let W denote the set of welfare
functions that are non-decreasing and concave, then
W(y)^W(y') fox all W{.) eltiff (3.8)
GL{y,p)>.GL(yl,p) fox all p.
As shown in Figure 3.5, this simply means that if a
generalized Lorenz curve lies everywhere above the other
curve, it is said to dominate and therefore rank highest.
An unambiguous ranking using generalized Lorenz curves is
possible only if the generalized Lorenz curves do not
intersect (Shorrocks 1983, 6),6
It has been shown that generalized Lorenz dominance
allows us to do a number of things in the ranking of
distributions. It allows us to assume that both equality
and the level of income are important aspects of social
welfare and allows us to include these in the ranking of
income distributions. It also allows different "social
policy makers" to place varying weights on different

Percent of Population
Figure 3.5. Generalized Lorenz
dominance. Generalized Lorenz curve A
dominates generalized Lorenz curve B.

parts of the income distribution, and still arrive at the
same rankings, as long as their social welfare functions
meet the conditions of being Schur-concave and a non-
decreasing function of all incomes.
Generalized Lorenz dominance is applied to income
data on West Germany in the next chapter in the effort to
make conclusions regarding the trend in social welfare
over the twenty-year period.

In this chapter an age-old problem in examining
income distributions is encountered: finding comparable
and meaningful statistics. Any investigation of income
distribution faces several basic problems: (1) how income
should be defined, (2) over what period of time income
should be measured, and (3) how. the income unit should be
defined (Lambert 1989, 14; Sawyer 1976, 4). Because
income recipients in the upper deciles often have income
from property and investments (at least more so than in
the lower deciles), omitting these sources of income
would tend to underestimate inequality. If a country
with a strong national social policy is part of the
analysis, it would be important to include as many
welfare benefits as possible. Although there may be non-
cash benefits that yield welfare, it is significantly
more difficult to measure these (Sawyer 1976, 4-5).,
Most studies of income distribution measure annual
income. This prevents distortions which might otherwise
occur in observing incomes which are irregular throughout
the year (such as seasonal salaries) and has the

advantage of coinciding with national tax statistics. We
also get a clearer picture of welfare by looking at a
year than by looking at a shorter time unit, such as one
particular week.
The income-receiving unit is most commonly defined
as a household because the household is the primary
economic consumption unit (Sawyer 1976, 5) This unit of
measurement is not without problems; because size,
income, and needs vary considerably among households, so
will welfare (Poentinen and Uusitalo 1974, 4-5; Sawyer
1976, 6). One can partially overcome this problem of
comparability by employing equivalence scales which
attempt to account for differences in household size and
the ages of the dependents.7
Another problem is accounting for all groups of the
population. National statistics frequently exclude a
number of groups (prisoners, orphans, the homeless, and
non-taxpayers) because they are not included
systematically in the tax system (Sawyer 1976, 6) .
Despite the many advances in technology for data
collection and processing, it is evident that there still
exists a large gap between the real distribution of
income and the information we actually have about that
distribution. The information we do have, however
incomplete, still remains our best estimate of reality

and should not prevent us from making conclusions;
rather, it should remind us of the need for further
improvement in data collection and processing.
The statistical information used in this chapter was
obtained from a 1974 report by the German Institute for
Economic Research (DIW). The report covered income
distributions for 1950, 1955, 1960, 1964, 1968, and 1970.
In that report, survey data were used to establish a
structural base which would closely represent the entire
population. Because the surveys were voluntary and
answers were sometimes incomplete, this information was
supplemented by national statistics to provided the most
accurate representation possible (Goeseke and Bedau 1974,
13) .8 To the extent possible, the income figures
included all wage, entrepreneurial, and investment
income. They also included transfer payments such as
social security, government pensions, old age pensions,
unemployment and sickness benefits, child allowances,
public assistance, and rent subsidies (Goeseke and Bedau
1974, 53). All income figures were deflated with the
assistance of OECD Economic Surveys and are expressed in
1976 Deutsch marks.

Gini Coefficients
The Gini coefficients and their corresponding rank
for the observed years are shown in Table 4.1. Although
the difference in the Gini coefficient values are very
small, it appears that inequality decreased over the
first decade and increased over the second decade.
Recall, however, that the Gini coefficient assumes an
unspecified social welfare function which conflicts with
the condition that the measure be uniformally sensitive
to all transfers made according to the principle of
transfers. The Lorenz curve is consistent with our
desire for a concave utility function and is uniformally
sensitive to any transfers made from richer persons to
poorer persons.
Lorenz Dominance
Lorenz curves for the observed years were plotted
using cumulative percentages calculated from deciles (see
Figure 4.1). As one can see, the curves are very close
together and it is tempting to conclude that inequality
has not changed much over the twenty year period.
Statistical analysis would inform us whether the
differences in these curves are significant. However, we
can not apply such procedures in this situation because
microdata are required.9 A close inspection of

Table 4.1. Gini Coefficients and Corresponding Ranks.
Year Gini coefficient Rank
1950 .396 5
1955 .384 2
1960 .38 1
1964 .38 1
1968 .387 3
1970 .392 4
Source: Gerhard Goeseke and Klaus-Dietrich Bedau,
Verteiluna und Schichtuncr der Einkommen der Privaten
Haushalte in der Bundesreoublik Deutschland 1950 bis
1970. DIW Beitraege zur Strukturforschung, Heft Nr. 31.
(Berlin: Duncker und Humblot, 1974).

Percent of income (cumulative)
1950 1960 1968
1955 1964 1970
Figure 4.1. Lorenz curves for West Germany,
1950 1970.

the cumulative percentages in Table 4.2 reveals that the
Lorenz curves intersect in 40 percent of the fifteen
pairwise comparisons. On the basis of Lorenz dominance,
1960 ranks higher than 1955, and 1955 ranks higher than
1950. It is interesting to note that this ranking is
different from the ranking given by the Gini
coefficients. This is an example of how assuming
different welfare functions will give different results.
The Gini coefficient presupposes a social welfare
function which is relatively more sensitive to income
transfers made at the center of the income distribution,
whereas the Lorenz curve is uniformally sensitive to all
points in the distribution. This Lorenz ranking also
gives a result somewhat different from the study cited in
Chapter Two which indicated inequality clearly diminished
through at least 1967. This probably results from using
different data and is an example of the problem of
comparability mentioned earlier.
The Atkinson theorem gives the same ranking because
the real mean income in 1960 was greater than the real
mean income in 1955 which was in turn greater than the
real mean income in 1950. We are unable to completely
rank the remaining distributions because of their
intersecting Lorenz curves. Because generalized Lorenz
dominance often resolves this problem, we now

Table 4.2. Lorenz Curve Values and Corresponding Mean
Incomes for West Germany, 1950 1970.
Cumulative Percentages
1950 1955 1960 1964 1968 1970
2.0 2.3 2.4 2.5 2.6 2.4
5.4 5.7 6.0 6.1 6.2 5.9
10.1 10.4 10.7 10.9 10.9 10.5
16.1 16.4 16.7 16.9 16.8 16.3
23.4 23.8 24.1 24.2 23.9 23.4
32.1 32.6 32.9 33.0 32.4 31.9
42.4 43.1 43.4 43.4 42.6 42.1
54.9 55.8 56.0 55.9 54.9 54.5
70.7 71.2 71.8 71.6 70.8 70.7
100.0 100.0 100.0 100.0 100.0 100.0
Mean Incomes (per year, per household in 1976 DM)
1950 1955 1960 1964 1968 1970
9,489. 14,070. 18,469. 21,543. 23 ,907. 26,634.
Source: Calculated from deciles and nominal income found
in Gerhard Goeseke and Klaus-Dietrich Bedaur Verteiluncr
und Schichtuna der Einkommen der Privaten Haushalte in
der Bundesreoublik Deutschland 1950 bis 1970. DIW
Beitraege zur Strukturforschung, Heft Nr. 31. (Berlin:
Duncker und Humblot, 1974).

utilize this theory in the attempt to arrive at a more
complete ranking.
Generalized Lorenz Curves
The generalized Lorenz curves are shown in Figure
4.2. Because none of the curves intersect, a complete
ranking is possible, with the ranking being essentially
in chronological order. We can conclude unambiguously
that each year observed represents an increase in social
welfare over the previous observed year. Judging from
the Lorenz curves in Figure 4.1 which indicated
relatively little change in inequality over the twenty
years, it appears that the increase in social welfare is
largely attributable to the growth in mean income. This
result should come as no surprise given the status of the
West German economy during this time period. West
Germany experienced tremendous growth during the 1950s
and 1960s, with the exception of a mild recession in
1966-67. If the policy toward income redistribution and
transfer payments stayed relatively constant during this
time, it may explain why there was little change in the
degree of inequality.

Income in 1976 DM (in thousands)
0 JO 20 30 40 50 60 70 80 90 100
Percent of population (cumulative)
Figure 4.2. Generalized Lorenz curves for West
Germany, 1950 1970.

It is interesting to note that if we had not used
the level of income in our ranking criteria, we would
have instead ranked the distributions as being
essentially equally desirable. From this point of view,
generalized Lorenz dominance may be more accurate in
ranking distributions according to their overall

The purpose of this thesis is to rank income
distributions of West Germany between 1950 and 1970 in
terms of their social welfare and to perhaps detect a
trend. Generalized Lorenz dominance was used to rank the
income distributions because there are several drawbacks
to the conventional approach. First, the commonly used
summary statistics presuppose varying social welfare
functions which may not accord with social values.
Second, this approach assumes one is only interested in
ranking income distributions on the basis of their degree
of inequality.
Generalized Lorenz dominance, like the conventional
approach, acknowledges that there is a general social
preference for equality. However, generalized Lorenz
dominance differs in that it also claims that there is a
general social preference for higher incomes over lower
ones. If one considers the political debates over income
distribution in most industrial countries, this latter
view is probably a more accurate view of society's
collective preference.

While the social welfare function one chooses to
assume could take on many forms, the generalized Lorenz
dominance outlined in Chapter Three hinges on the twin
preferences for greater equality and higher mean incomes.
The assumptions made within this framework were made so
as to be minimal enough to command the widest support
possible, yet also allow a fairly thorough ranking of the
income distributions.
The equity preference is supported by the assumption
of diminishing marginal utility of income. Some may
question (a) whether diminishing marginal utility is in
fact the relation between income and utility and (b)
whether' this assumption accords with social values. A
number of empirical studies show that the diminishing
marginal utility of income is a fair representation of
reality. Utility of income was found to have a more or
less linear relationship to the cube root of income,
which is fairly consistent with diminishing marginal
utility of income (Rainwater, Rein, and Schwartz 1986,
26) Whether one thinks the assumption accords with
social values will most likely depend on one's place in
the income distribution (people at the top would tend to
object). But, if we maintain the stance of the
disinterested social observer, we should be indifferent.

The preference for higher incomes is supported by
the assumption that any addition of income to the
population represents an increase in social welfare
regardless of who the income recipient is. Few people
would argue with the basic assumption that, ceteris
paribus, more is preferred to less. However, this
assumption will probably tend to reap more criticism than
the equity-supporting assumption because many people,
even disinterested social observers, might disagree that
an addition of income to the richest person necessarily
represents an increase in social welfare.
Although generalized Lorenz dominance has partly
overcome the problem of hidden social welfare functions
by making them explicit, it has introduced a new hidden
concept. Generalized Lorenz dominance allows a trade-off
of growth for inequality, but does so without specifying
the trade-off of one for the other.
The power of generalized Lorenz dominance is that if
"social policy makers" assumed social welfare functions
which met the conditions of being Schur-concave and a
non-decreasing function of all incomes, but differed in
the weights they attached to different parts of the
distribution, they would all arrive at the same ranking
using generalized Lorenz dominance. If a social policy
maker, however, assumed a social welfare function which

did not meet the conditions of being Schur-concave or a
non-decreasing function of all incomes, they would arrive
at a different ranking and different conclusions. For
example, if one were strictly concerned with how the
income is distributed and not with how much is
distributed, then one would revert to using Lorenz
dominance. Atkinson himself, in retrospect, states that
perhaps inequality should be pursued in its own right,
and not as a means to some other goal (Atkinson 1980).
West Germany has demonstrated commitment to both
equality and higher incomes and therefore represents an
interesting case for study. If as a social observer one
accepts the assumptions outlined above, then we can use
generalized Lorenz dominance to rank income distributions
in order to draw a conclusion about the trend in social
welfare. The analysis in Chapter Four shows that for the
twenty-year period examined, we can conclude that social
welfare improved with each consecutive year. It also
showed that the increase in social welfare was largely
attributable to the increase in mean income rather than
to greater equality in the distribution of income,
perhaps indicating that West Germany is relatively more
committed to growth than to equality.
Although this study showed that inequality remained
relatively constant, other studies with other data have

shown that there was a move toward greater equality. It
would be interesting to compare this to other industrial
countries, both welfare states and non-welfare states, to
see how West Germany's supposed commitment to equality
compares. There is little doubt in my mind, however,
that the population as a whole was better off under the
distribution of income in 1970 than in 1950.

1. For an interesting discussion and counter-argument of
the traditional view of inequality as necessary for
growth via the Harrod-Domar growth model, see Todaro
(1981), 137-39.
2. This study by Wilensky (1976) was a rudimentary
attempt to rank by a number of criteria the extent to
which countries were experiencing welfare-tax backlash.
Higgins (1981) states that while the rankings should be
regarded with caution, those countries which ranked
highest have unquestionably shown greater conflict over
welfare spending and taxes (Higgins 1981, 151).
3. For a related but separate approach to rank-
dominance, see Saposnik (1981).
4. This parallels the condition developed in the area of
financial decision-making under uncertainty in which a
distribution f(y) will be preferred to another
distribution f*(y) according to the criterion that the
welfare function be increasing and concave if and only if
This is equivalent to the condition of non-intersecting
Lorenz curves (Atkinson 1970, 246-47). Atkinson also
shows that this is equivalent to the necessary and
sufficient condition that if one distribution can be
obtained from the other by a process of transfers from
those with higher incomes to those with lower incomes
(i.e., it passes the principle of transfers test), then
we can rank the distributions without further
specification of U(y). As mentioned earlier, the
2 [F(y) -F* (y) ] dy^O for all z, Qzzzy
A F(y) *F* (y) for some y,

assumption that U(y) is concave guarantees the principle
of transfers and thereby guarantees this condition being
met as well.
Ranking income distributions according to W is
technically the same as ranking probability distributions
f(y) according to expected utility. The assumption that
U(y) is concave is equivalent to assuming that a person
is risk-averse (Atkinson 1970, 245).
5. Shorrocks analyzed income distributions of 20
countries and discovered that Lorenz curves crossed in
108 of 190 pairwise comparisons, thus allowing
unambiguous rankings in only 40 percent of the cases.
Ranking by the Atkinson theorem yielded conclusive
results in only 28 percent of the 190 comparisons.
Ranking by generalized Lorenz dominance yielded
conclusive results in 84 percent of the 190 comparisons
(Shorrocks 1983, 8).
6. Sen's Axiom of Size Independence (Sen 1973, 1976)
shows that the size of the population does not affect the
level of social welfare. Generalized Lorenz dominance
can therefore be applied to populations of varying sizes.
7. Lambert (1989) proposes some specific scales for this
8. Survey data are known for their inferior quality
because of inaccurate representation of the population
resulting from poor sample sizes, erroneous answers,
incomplete information, etc. The authors of this report
relied heavily on national income statistics along with
the surveys to obtain the best data set possible, but
recognize that even this is an incomplete representation
of reality.
9. See Beach and Davidson (1983) for a theoretical
explanation and Bishop, Formby, and Smith (publication
forthcoming) for an empirical application of this

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