Economic Exclusion and Income Inequality
o
Ram´n J. Torregrosa
Universidad de Salamanca
JEL Classification: J1, J2
Abstract
In this paper is presented a theoretical model which allows to charac-
terize the Lorenz curve connecting income with the mass of population by
means of a production function. The model also permits to characterize
exclusion as a partition of the mass of population. In this trend, economic
exclusion determines the shape of the Lorenz curve in such way that eco-
nomic exclusion is given by the percentage of population which do not
earn income. Moreover, this framework allows to show that the higher
economic exclusion the higher the income inequality measured through
the Gini index.
1 Introduction
Social and economic exclusion have become in a recent concept in Social Sci-
ences. In this trend there is not much theoretical literature about this issue
which could help economists and sociologists to measure it. In some extent
social and economic exclusion is related with income inequality, enduring un-
employment, disable people and poverty. Since the point of view of economic
analysis we could say that economic exclusion entails the lack of participation of
some individuals in markets as a consequence of a gather of disabilities or a low
endowment of human capital (Atkinson,1998). Silver (1994) and Nasse (1992)
stated that despite of the difficulty of a clearly definition of the condition of
excluded, it could be said that those are these social categories involving mental
and health disabilities, indigence, drugadiction, delinquency, alienation and mal-
adjustion. In short, the basic characteristic of social and economic exclusion is
the impossibility of such individuals to participate in social and economic insti-
tutions (Weinberg and Ruano-Borbalan, 1993). In line with this idea Fern´ndeza
o
de C´rdoba and Torregrosa (2006) present a theoretical model where exclusion
is a consequence of the low endowment of agency on behalf of agents. An agency
is given by the stock of human and physic capital that allow agents to produce
efficient units of primary input. This concept is taken from Knight (1935). In
this trend, this model presents a theoretical concept of exclusion by means os
the idea of excluded agency. This framework allow to characterize the supply
of primary input as a function of the number (mass) of agents. Therefore, the
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equilibrium price of the primary input determines the sizes of agencies that
participates in markets, spliting the mass of individuals in those who supply
their primary input and those who are excluded from markets. Thus inclusion
and exclusion depend on the stock of agency of individuals in such a way that
the higher the price of the primary input the lower the size of agency taken in
the economy and the lower the mass of excluded agencies. Therefore, in this
paper we define the Lorenz curve from this set-up as the income attained by
each percentage of the mass of population. Moreover, the Gini index can be
readily assessed from the Lorenz curve as the area between this function an the
line of perfect equality. As we will see, economic exclusion is related with the
shape of the Lorenz curve in such a way that the mass of excluded agencies is
determined by the cutting point of the Lorenz curve and the horizontal axis.
This way the higher the exclusion the sharply the Lorenz curve. This effect also
moves the Lorenz curve far from the line of perfect equality and makes the Gini
index to increase, with the consequence that higher exclusion is related with
higher income inequality.
The paper is structured as follows: section two, presents the model where
agencies are defined, together with a definition of Social Plant and excluded
Social Plant. Those definitions are used in section three to show how to assess
the Lorenz curve and the Gini index, its properties and shapes regarding to the
excluded social plant. A simple example is attached in this section in face to
help the reader to understand the results. Finally section four is devoted to the
comments.
2 The model
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Let us follow the model proposed by Fern´ndez de C´rdoba and Torregrosa
(2006) composed of a population with measure 1, where the agents are char-
acterized by a stock h, interpreted as the size of the agency. h is distributed
trough a continuous mass function F, over the finite support Θ = [hp , hr ] ⊂ R+
of agency sizes, in such a way that hp and hr denote the lowest and largest
agency sizes respectively. Each agent transforms her stock h into a primary
input, or services, using a simple unitary technology s = h. The agents consume
the quantity s − l of their services, and offer the market the amount l as ser-
vices at the price w. The services offered to the market are the primary input
required to produce the commodity c. The preferences of an agent with agency
size h is represented by the utility function:
u(c, θ) = c + v(h − l),
agent’s budget constraint is:
c = wl; 0 ≤ l ≤ h.
First order conditions yield to v 0 (h − l) = w, so that the agent’s supply function
of the primary input can be written as a function of her agency size and the
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price of the primary input:
½
h − ϕ(w), for w > v 0 (h)
l(w, h) = (1)
0 , for w 0. = ϕ(K)dF (h) > 0.
Substituting in the RHS side of (8) we have
−ϕ0 (w) [ϕ(w) − ϕ0 (w)] dF (ϕ(w)),
which is positive because ϕ0 (w) 0. (9)
∂w
This property is not surprisingly because the higher the w the more agencies
participate in markets, that is ϕ(w) declines, rising the accumulate participation
due to the enhance of (ϕ(w), hr ).
Finally in face to obtain the Lorenz curve let us follow Gastwirth (1972) and
calculate the inverse of the mass function, that is, given
t = F (h) so that h ∈ Θ
h = F −1 (t) so that t ∈ [0, 1],
thus, substituting in (7) the Lorenz curve is given by
½
0 for t ∈ [0, F (ϕ(w)))
λ(t, w) =
y(t, w) for t ∈ [F (ϕ(w)), 1]
that is, in our model the Lorenz curve cuts horizontal axis in the Excluded
Social Plant (see equation 3), and can be witten in a reduced way as
λ(t, w) = max {0, y(t, w)} with t ∈ [0, 1].
5
1
0.9
0.8 L(w0)
0.7
0.6
0.5
0.4
0.3
0.2 L(w1)
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 1: Lorenz curve
Figure 1 depicts this fact for two different values of w. For instance, assuming
that w0 > w1 it is clear that the LEA for w0 is lower than the LEA for w1,
that is ϕ(w0) h
1 1
l(w, h) = 1
0 , for w < h
with h ∼ U [1, 3] , that is for 1 ≤ h ≤ 3 the density function is f (h) = 1 , and
¡ 2 ¢
the mass function is F (h) = h−1 . Therefore, given ¡
2 the price w ∈ 1 , 1 , the
¢ 3
1
supply of primary input until the agency of size h ∈ w , 3 is given by
Z µ ¶ ∙ ¸
1 h 1 1 h2 h 1
L(h, w) = h− ds = − + ,
2 1/w w 2 2 w 2w2
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and
1
Y (h, w) = [hw − 1]2 .
4w
¡1 ¢ ¡1 ¢
Notice that for h ∈ w , 3 the Excluded Social Plant is F (w) = 1 w − 1 . As
2
seen the Excluded Social Plant is inversely proportional to w.
Taking into account equation (7) the accumulated participation of agency of
¡1 ¢
size h ∈ w , 3 in the income can be written as
∙ ¸2
hw − 1
y(h, w) = .
3w − 1
Considering the mass function t = h−1 and assessing its inverse h = 2t + 1 with
£ ¡1 ¢ ¤ 2
t ∈ 1 w − 1 , 1 and substituting in y(h, w) we hold the Lorenz curve as
2
∙ ¸2
(2t + 1) w − 1
y(t, w) = .
3w − 1
Finally, the Gini index is given by
Z 1 ∙ ¸2
1 (2t + 1) w − 1 1
G(w) = − dt = ,
2 2 ( w −1)
1 1 3w − 1 6w
¡ ¢
for w ∈ 1 , 1 . As seen the Gini index is inversely proportional to w because
3
the higher the w the lower is the Excluded Social Plant.
4 Comments
In this paper the Lorenz curve have been obtained by means of a production
function from the supply of primary input. The particular way in which the
model is conceived allow to characterize the supply of primary input as a func-
tion of the mass of population. This feature allow to connect directly total
output with the mass of population generating the Lorenz curve as a function
which links the income and the mass of population. In addition, as the model
characterizes economic exclusion as a partition of the mass of population which
depends on the price of primary input and the size of agencies in hands of house-
holds. Economic exclusion appears as a flat section of the Lorenz curve, as a
consequence of that, the excluded social plant do not produce primary input for
market and thus do not earn any income. Thus, in the model, the shape of the
Lorenz curve depends on the mass of excluded social plant in such a way that
the higher the exclusion the higher the flat interval of the Lorenz curve. This
outcome suggest an easily way to measure economic exclusion by the simple
observation of the Lorenz curve in practice. Finally, as a consequence of this
feature arises a clear-cut result in our model: the higher the excluded social
plant the higher the Gini index. This result entails that income inequality is
directly proportional to economic exclusion.
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5 References
Atkinson, A. B. (1998). Economics of Poverty, Unemployment and Social Ex-
clusion. Poverty in Europe. Blackwell.
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Fern´ndez de C´rdoba, G. and R. J. Torregrosa (2006). Exclusion of Agen-
cies from Markets. Mimeo.
Gastwirth, J. L. (1972). The Estimation of the Lorenz Curve and Gini Index.
The Review of Economics ans Statistics 54.
Knight, F. (1935). The Ricardian Theory of Production and Distribution.
The Canadian Journal of Economics and Political Science 1.
Nasse, P. (1992). Exclus et Exclusions: Connaitre les populations comprende
les processus. Paris. Commissariat General du Plan.
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Silver, H. (1994). Social Exclusi´n and Social Solidarity: three paradigms.
International Labour Review.
Weinberg, A. y J. C. Ruano-Borbalan. (1993). Comprendre l´Exclusion.
Sciences Humaines, 28.
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