Economic Exclusion and Income Inequality

Economic Exclusion and Income Inequality Ram´on J. Torregrosa Universidad de Salamanca JEL Classification: J1, J2 Abstract In this paper is presented a theoretical model which allows to characterize 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 economic 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 Sciences. 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 unemployment, 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 maladjustion. In short, the basic characteristic of social and economic exclusion is the impossibility of such individuals to participate in social and economic institutions (Weinberg and Ruano-Borbalan, 1993). In line with this idea Fern´andez de C´ordoba 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 1

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

Let us follow the model proposed by Fern´andez de C´ordoba and Torregrosa (2006) composed of a population with measure 1, where the agents are characterized 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 services 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 2

price of the primary input: l(w, h) =

½

h − ϕ(w), for w > v 0 (h) 0 , for w < v 0 (h),

(1)

where ϕ = (v 0 )−1 with ϕ0 < 0. Let F (·) be the mass function and dF (·) the density function of agency sizes so that

F (h) =

Z

h

dF (s), hp

is the mass of agents with an agency of size lesser of equal to h, and

E(h) =

Z

h

sdF (s) hp

the measure of agencies with a size lesser of equal to h. Thus, F (hr ) = R hr Rh dF (s) = 1, and hpr sdF (s) = H, is the total endowment of agencies in hp this economy, the so-called Social Plant. A concept which will be useful for our purposes is that of Largest Excluded Agency (LEA). That is, the size of agency hLEA so that v0 (hLEA ) = w, in short hLEA = ϕ(w). In order to obtain the aggregate supply of the primary input in the economy let us focus in the case in which v 0 (hr ) ≤ w ≤ v0 (hp ). This is because this case splits the mass of population in two groups: excluded and included agencies. Thus, the aggregate supply of the primary input is given by: Z hr L(w) = (s − ϕ(w))dF (s), v 0 (hr ) ≤ w ≤ v0 (hp ). (2) ϕ(w)

Where

F (ϕ(w)) =

Z

ϕ(w)

dF (s),

(3)

hp

is the Excluded Social Plant, that is the measure of excluded agencies from markets and

E(ϕ(w)) =

Z

ϕ(w)

sdF (s)

(4)

hp

is the size of the excluded Social Plant. Thus exclusion of agents depends on w, the higher is w, the lesser is ϕ(w) and the lesser is the number of agencies excluded. 3

Finally, total income in the economy is given by w ∈ R+ ,

Y = wL;

Substituting (2), (3) and (4), total income in the economy can be written as Y (w) = w [(H − E(ϕ(w))) − ϕ(w)(1 − F (ϕ(w)))] .

3

(5)

The Lorenz curve and Gini index

The Lorenz curve is a function which relates the percentage of income held by each percentage of population. As it is well-known, a perfectly equal income distribution would be one in which every person has the same income. In this case, the bottom N % of population would always have N % of the income. This can be depicted by the line of perfect equality or the 45◦ line. The importance of that relationship is that most of the measures of income inequality are derived from it. In particular the Gini index, the most used measure of inequality. Therefore, in order to obtain the Lorenz curve we have to assess the level of income imputed to each point of the mass function F. Therefore, given w ∈ [v0 (hr ), v0 (hp )], the supply of primary input until the agency of size h ∈ (ϕ(w), hr ) is given by L(h, w) =

Z

h ϕ(w)

(s − ϕ(w))dF (s),

or L(h, w) = (E(h) − E(ϕ(w))) − ϕ(w)(F (h) − F (ϕ(w))). Thus, total income until the agency of size h ∈ (ϕ(w), hr ) is given by Y (h, w) = w [(E(h) − E(ϕ(w))) − ϕ(w)(F (h) − F (ϕ(w)))] .

(6)

Notice that, on the one hand, Y (hr , w) = Y (w) of equation (5), and on the other hand Y (ϕ(w), w) = 0, which means that income begins to be produced for that agencies whose size is at least ϕ(w). In short, what shows equation (6) is the income as a function of the sizes of included agencies. Therefore, dividing equations (6) and (5) we obtain the accumulated participation of agency of size h ∈ (ϕ(w), hr ) in the income y(h, w) =

Y (h, w) ∈ [0, 1]. Y (hr , w)

(7)

where y(hr , w) = 1 and y(ϕ(w), w) = 0. Other remarkable property of y(h, w), which will be useful here in after, is that it is increasing with respect to w. In 4

fact, deriving y(h, w) with respect to w taking into account equations (6) and (5) and operating, we have −ϕ0 (w) [E 0 (ϕ(w)) − ϕ0 (w)dF (ϕ(w))] [H − E(h) − ϕ(w)(1 − F (h))] ∂y(h, w) , = ∂w [H − E(ϕ(w)) − ϕ(w)(1 − F (ϕ(w)))]2 it is straightforward that sign

∂y(h, w) = sign {−ϕ0 (w) [E 0 (ϕ(w)) − ϕ0 (w)dF (ϕ(w))]} . ∂w

(8)

In order to meet the sign of the RHS side of (8) it is necessary to assess E 0 (·). Thus, departing from its definition and integrating by parts Z h Z h E(h) = tdF (t) = hF (h) − hp F (hp ) − F (t)dt, hp

hp

and taking derivatives under the integral sign, E 0 (h) = F (h) + hdF (h) − F (h) = hdF (h) > 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. Therefore ∂y(h, w) > 0. ∂w

(9)

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

L(w0)

0.8 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) < ϕ(w1). Thus the Lorenz curve for w1 is sharply that the Lorenz curve for w0. This fact, of course, has consequences on the Gini index: the higher the Excluded Social Plant the higher the Gini index. In fact, as it is seen in figure 1, the Gini index is the area between the line of perfect equality and the observed Lorenz curve. The lower the w, the higher the LEA and the higher the Excluded Social Plant F (ϕ(w)), which is the point where the Lorenz curve becomes zero. Thus the lower the w, the sharply the Lorenz curve and the higher the area between it and the perfect equality line. To show this feature in analytical terms let us write the Gini index as Z 1 1 G(w) = − y(t, w)dt. (10) 2 F (ϕ(w)) For sake of simplicity in calculations, let us define Z ∂P (t, w) P (t, w) = y(t, w)dt so that = y(t, w), ∂t thus, using this definition and the Barrow rule, (10) can be written as G(w) =

1 − [P (1, w) − P (F (ϕ(w)), w)] , 2 6

(11)

deriving G(w) with respect to w we hold ∙ ∙ ¸¸ dG(w) dP (1, w) dP (F (ϕ(w)), w) dP (F (ϕ(w)), w) 0 =− − dF (ϕ(w))ϕ (w) + . dw dw dt dw But, according to (11) and the properties of y(t, w) dP (F (ϕ(w)), w) = y(ϕ(w), w) = 0, dt substituting dG(w) dP (F (ϕ(w)), w) dP (1, w) = − , dw dw dw or d dG(w) = dw dw

Z

F (ϕ(w))

y(t, w)dt −

0

d dw

Z

1

y(t, w)dt,

0

which can be writen as a derivative under the integral sign dG(w) =− dw

Z

1 F (ϕ(w))

dy(t, w) dt, dw

which is negative due to (9). Therefore, the Gini index is inversely proportional to the price of primary input. This entails that as the excluded social plant is inversely proportional to the price of primary input, the Gini index is directly proportional to the excluded social plant. In short, the higher the economic exclusion the higher the income inequality.

3.1

Example

To help the reader to understand our former results, let us illustrate our calculations by means of a example. Let us assume that u(c, θ) = c + ln(h − l), thus, the agent’s supply function of the primary input is given by ½ h − w1 , for w > h1 l(w, h) = 0 , for w < h1 with h ∼ U [1, 3] , that is for 1 ≤ h ≤ 3 the density function is f (h) = 12 , and ¡ ¢ w ∈ 13 , 1 , the the mass function is F (h) = h−1 2 . Therefore, given ¡the price ¢ supply of primary input until the agency of size h ∈ w1 , 3 is given by µ ¸ ¶ ∙ Z 1 h 1 1 h2 h 1 L(h, w) = h− , ds = − + 2 1/w w 2 2 w 2w2 7

and Y (h, w) =

1 [hw − 1]2 . 4w

¢ ¢ ¡ ¡ Notice that for h ∈ w1 , 3 the Excluded Social Plant is F (w) = 12 w1 − 1 . As seen the Excluded Social Plant is inversely proportional to w. Taking ¡ into ¢ account equation (7) the accumulated participation of agency of size h ∈ w1 , 3 in the income can be written as ∙

hw − 1 y(h, w) = 3w − 1

¸2

.

Considering the function t = h−1 2 and assessing its inverse h = 2t + 1 with £1 ¡ 1 ¢ mass ¤ t ∈ 2 w − 1 , 1 and substituting in y(h, w) we hold the Lorenz curve as y(t, w) =

∙

(2t + 1) w − 1 3w − 1

¸2

.

Finally, the Gini index is given by ¸2 ∙ Z 1 1 (2t + 1) w − 1 1 dt = , G(w) = − 1 1 2 3w − 1 6w 2 ( w −1) ¢ ¡ for w ∈ 13 , 1 . As seen the Gini index is inversely proportional to w because 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 function 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 households. 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. 8

5

References

Atkinson, A. B. (1998). Economics of Poverty, Unemployment and Social Exclusion. Poverty in Europe. Blackwell. Fern´andez de C´ordoba, G. and R. J. Torregrosa (2006). Exclusion of Agencies 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. Silver, H. (1994). Social Exclusi´on 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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