Inequality of Life Chances and the Measurement of

Social Immobility�

Jacques Silber

Department of Economics, Bar-Ilan University,

52900 Ramat-Gan, Israel, Email: jsilber_2000@yahoo.com

Amedeo Spadaro

Paris School of Economics, 48 Bd Jourdan, 75014 Paris, France,

and

Universitat de les Illes Balears, ctra Valldemossa Km 7,5,

07122 Palma de Mallorca, Spain, email: spadaro@pse.ens.fr

January 2008

Abstract

This paper suggests applying tools of analysis that had been used previously in the �eld of oc-

cupational segregation and horizontal inequity measurement to the study of intergenerational social

immobility. The paper introduces the concept of �social immobility curve�and two indices of social

immobility are proposed that are extensions of the Theil and Gini indices of inequality. The paper

stresses also the need to make a distinction between concepts of �gross�and �net�social immobility

and hence to base the comparison of the degree of social immobility in two groups on the case where

the margins of the matrices corresponding to the two groups are the same. Finally this paper suggests

two measures of inequality in circumstances, derived also from the Theil and Gini indices. The paper

ends with an empirical illustration based on French and Israeli data.

Key Words : circumstances �equality of opportunity �France - Gini index � independence �

Israel - social immobility - Theil index.

JEL Classi�cation : D31 �D63

�This paper was written when Jacques Silber was visiting the Levy Economics Institute of Bard College. He wishes to

thank this Institute for its very warm hospitality. The authors wish to thank an anonymous referee as well as Marc Fleurbaey

for their extremely useful comments. They also acknowledge the help of Adi Lazar in preparing the revised version of this

paper. Amedeo Spadaro acknowledges �nancial support of the Spanish Government (SEJ2005-08783-C04-03/ECON) and

of the French Government (ANR BLAN06-2_139446).

1 Introduction

The topic of intergenerational mobility measurement has been increasingly popular among economists in

recent years (for interesting surveys, see, for example, Solon (1999) and Corak, Gustafsson and Österberg

(2004)). In a recent paper Van de gaer, Schokkaert and Martinez (2001) have made a distinction between

three meanings of intergenerational mobility, stressing respectively the idea of movement, the inequality

of opportunity and the inequality of life chances.

When mobility is viewed as movement, the goal is usually to measure the degree to which the position

of children is di¤erent from that of their parents. Those adopting such an approach tend to work with

transition matrices whose typical element pij refers to the probability that a child has position j, given

that his/her parents had position i. These matrices are square matrices and it should be clear that

mobility in this sense will be higher the smaller the value of the probabilities pii.

The second approach to mobility attempts to measure the degree to which the income prospects of

children are equalized, that is, do not depend on the social origin of the parents. What is often done in

such types of analyses, when comparing, for example, two groups of parents (e.g. two educational levels

or two occupations) is to check whether the distribution of the outcomes (incomes) of the children of

one group of parents stochastically dominates the distribution of the second category of parents (see, for

example, Peragine, 2004; Lefranc, Pistolesi and Trannoy 2006).

The third approach emphasizes the concept of inequality of life chances so that �the only thing that

matters here is that children get equal lotteries. The prizes do not matter.� (Van de gaer, Schokkaert

and Martinez 2001). Such an approach is particularly relevant when one analyzes the movement between

socio-economic categories which cannot always be ranked in order of importance.

In the present paper we focus most of our attention on the third approach and �rst propose an ordinal

approach to measuring inequality in life chances based on the use of what we call social immobility curves.

We then suggest two cardinal measures of social immobility that do not require the matrix to be analyzed

to be a square matrix. This allows us to study the transition from the original social category (educational

level or occupation) of the parents to the income class to which the children belong. But these measures

could also be applied to an analysis of the transition from one type of social category (e.g. educational

level of the parents) to another type of social category (profession of the children), assuming there is no

speci�c ordering of these categories.

In the second stage of the analysis we stress the importance of the marginal distributions when compar-

ing social immobility in two populations. More precisely we recommend neutralizing di¤erences between

the two populations concerned in the distribution of the parents by social category (e.g. occupation or

educational level) and in the income distribution of the children. We suggest borrowing ideas that have

appeared, for example, in the literature on occupational segregation or horizontal inequity measurement

so that a distinction can be made, when comparing two populations, between di¤erences in gross and

net social immobility, the latter concept assuming that the marginal distributions of the two populations

compared are identical.

1

In a third stage, borrowing some concepts from the literature on the equality of opportunity (e.g.

Roemer, 1998, Kolm, 2001, Ruiz-Castillo, 2003, and Villar, 2005), we de�ne an inequality in circumstances

curve and relate it to the social immobility curve previously presented.

Two empirical illustrations attempt then to illustrate the usefulness of the concepts previously de�ned.

The �rst set of data comes from a survey conducted in France in 1998 and allows us to measure the degree

of social immobility and of inequality in circumstances on the basis of information on the occupation of

fathers or mothers and of the income class to which sons or daughters belong. Using a second set of data,

based on a social survey conducted in Israel in 2003, we then measure social immobility and inequality

in circumstances by studying the transition from the educational level of the fathers to the income class

to which the children belong. Both illustrations con�rm that tools that had previously been used in the

�eld of occupational segregation and horizontal inequity measurement are also relevant for the study of

equality of opportunity. Concluding comments are given at the end of the paper.

2 Measuring Social Immobility

2.1 An ordinal approach to the measurement of social immobility: Drawing

�Social Immobility Curves�

Assume a data matrix M whose lines i correspond to the social origin of the individuals (e.g. occupation

or the educational level of the father or mother) and whose columns j correspond to the income brackets

to which these individuals belong. For example, Mij would give the number of individuals whose income

belongs to the income bracket j and whose father had educational level i.

If mij , mi: and m:j are de�ned as:

mij =MijPI

i=1

PJj=1Mij

(1)

mi: =XJ

j=1mij (2)

and

m:j =XI

i=1mij (3)

then perfect social mobility may be assumed to exist when the probability that an individual belongs

to a speci�c income bracket k is independent of his social origin h, that is, when mhk = (mh:m:k).

Assume we order the products (mi:m:j) of the elements (mi:) and (m:j) by increasing values of the

ratios (mij

mi:m:j). The shares mij are similarly ordered by increasing values of the ratios

mij

mi:m:j. By

plotting the cumulative values of the elements (mi:m:j) on the horizontal axis and the cumulative values

of the shares (mij) on the vertical axis one obtains a �social immobility curve�. This curve is in fact

what is known in the literature as a relative concentration curve and clearly its slope is non-decreasing.

2

In the speci�c case when mij = mi:m:j for each i and j, this curve will become the diagonal line going

from (0; 0) to (1; 1). Such a "social immobility curve" is evidently an illustration of what Chakravarty

and Silber (2007) have called a "dependence curve".

Assume now thatmfh

mf:m:h>

mfk

mf:m:kand

mlk

ml:m:k>

mlh

ml:m:land that two �transfers�of size � take

place, one from mfh to mfk and one from mlk to mlh. The combination of these two transfers implies

that there was no change in the marginal shares. These two �margin-preserving� transfers have been

called "independence-increasing margins-preserving composite transfer (IIMPCT)" by Chakravarty and

Silber who have proven the link between IIMPCT1 and the concept of independence domination. The

latter is evidently checked via the "social immobility curves".

2.2 Cardinal Measures of Social Mobility

Chakravarty and Silber (2007) derived the link between the ordering of dependence curves and the values

taken by dependence indices which have the following properties. They obey the property of symmetry

(implying, for example, that the exact label of the social status of the fathers or the sons does not

matter) and their value decreases when there is a IIMPCT. Chakravarty and Silber also stress that the

two following indices, which are derived from the Theil and Gini indices, belong indeed to this family of

indices.

2.3 De�ning two indices of social immobility

The �rst index we have selected is directly related to Theil´s (1967) famous indices. The latter may be

interpreted as comparing �prior probabilities�with �posterior probabilities�. In our case the �prior prob-

abilities�would be the products (mh:m:k) while the �posterior probabilities�would be the proportions

mhk. Such a formulation of the Theil index leads then to the following index of social immobility:

Tsim =XI

i=1

XJ

j=1(mi:m:j)ln

�mi:m:j

mij

�(4)

Note that this index will be equal to 0 when there is perfect independence between the social origins

and the income brackets.1This type of transfer has been mentioned before in the literature under the name of "marginal preserving swap" (MPS).

To the best of our knowledge most of the previous works have dealt however with the idea of an ordering on discrete bivariate

distributions (e.g. Tchen, 1980), the derivation of a measure of dependence for pairs of random variables with continuous

distribution functions (e.g. Schweizer and Wol¤, 1981) or the concept of (positive quadrant) dependence in two-way tables

with ordered margins (e.g. Bartolucci et al., 2001). The assumption that the margins may be ordered was also made by

Epstein and Tanny (1980), Atkinson and Bourguignon (1982) and Dardanoni and Lambert (2001). In such a case use can

be made of evaluation functions called L-superadditive (see, Marshall and Olkin, 1979) which approve such MPS. We are

indeed very thankful to an anonymous referee for drawing our attention to some of these studies. We want however to stress

that our analysis does not require that the margins can be ordered. Our paper derives for each cell of a contingency table

a variable measuring the degree of dependence between its corresponding margins and this variable is then the criterion of

ordering, as shown by Chakravarty and Silber (2007).

3

Another possibility is to use a Gini-related index, as suggested originally by Flückiger and Silber

(1994). As stressed also by Silber (1989a) the Gini index may be also used to measure the degree

of dissimilarity between a set of �prior probabilities� and a set of �posterior probabilities�. In the

case of inequality measurement the �prior probabilities� are the population shares and the �posterior

probabilities�the income shares. Here the Gini-related index of social immobility will be expressed as

Gsim = [: : : (mi:m:j) : : :]0G[: : :mij :::] (5)

where [: : : (mi:m:j) : : :]0 is a row vector giving the �prior probabilities�corresponding to the various

(I � J) cells (i; j) while [: : :mij :::] is a column vector giving the �posterior probabilities� (the actual

probabilities) for these cells. Note that, as indicated in Silber (1989a), the elements of these row and

column vectors have both to be ranked by decreasing ratiosmij

mi:m:j. The operator G in (5), called the

G-matrix (see, Silber, 1989b), is a (I � J) by (I � J) square matrix whose typical element gpq is equal to0 if p = q, to +1 if p > q and to �1 if p < q.Note that the index Gsim will also be equal to zero when all �prior probabilities�(mi:m:j) are equal

to the �posterior probabilities�mij .

3 Comparing two social mobility matrices

Assume now that we want to compare two social mobility matrices {mij} and {vij}. Such matrices may

refer to two di¤erent periods or to two population subgroups at a given time, such as ethnic groups,

regions, etc. . . On the basis of each of these two matrices we could compute social immobility indices such

as those de�ned in expressions (4) and (5) and conclude in which case social immobility is higher. This

may however be too hasty a way to draw �rm conclusions, as will now be shown.

Assume the social mobility matrices {mij} and {vij} correspond to two di¤erent time periods. The

point is that social mobility may vary over time for various reasons. First there may have been a change

in the distribution of parents by social origin. Second there may have been a change in the income

distribution of the children. These two possibilities correspond evidently to a variation in one of the

margins of the social mobility matrix. There may however be a third reason for a variation in the degree

of social mobility. Even if there was no change over time in the margins of the social mobility matrix, there

may have been a change in the degree of independence between the rows (social origin of the parents)

and columns (shares of various income brackets) of this matrix. This distinction between the impact of a

change in the margins of a matrix and a change in the �internal structure�of this matrix (the degree of

independence between the lines and columns of a matrix) was pointed out by Karmel and MacLachlan

(1988) in the framework of an analysis of changes over time in occupational segregation by gender2 . In

the Appendix we show how it is possible to apply the methodology proposed by Karmel and MacLachlan

2The idea of making a distinction between changes in the "internal structure" and in the margins of a matrix is not a

novel idea. Thus in the literature on social mobility there is a long tradition of making a di¤erence between what is called

exchange and structural mobility. We thank Marc Fleurbaey for stressing this point to us.

4

when analyzing changes over time in the degree of social mobility (or when comparing the degree of social

mobility of two population subgroups). Such a methodology uses an algorithm originally proposed by

Deming and Stephan (1940)3 . In addition it will be shown (as was already stressed by Deutsch, Flückiger

and Silber 2007, in the framework of occupational segregation analysis) that it is possible to generalize

this methodology by applying also the concept of Shapley decomposition (see, Chantreuil and Trannoy,

1999, Shorrocks, 1999, and Sastre and Trannoy, 2002). A simple presentation of this approach is given

in the Appendix.

4 Measures of social immobility versus measures of inequality

in circumstances

Given that the lines of the matrix to be analyzed correspond to the �social origin� of the individual

and the columns to the income class to which he/she belongs one can adopt the terminology used in

the literature on equality of opportunity and call the lines �types� or �circumstances�. Under certain

conditions it is also possible to call the columns �levels of e¤ort�, although such an extension implies

quite strong assumptions concerning the link between income and e¤ort.

Adopting Kolm�s (2001) ideas let us therefore de�ne the inequality in circumstances as the weighted

average of the inequalities within each �income class�(�e¤ort level�), the weights being the population

shares of the various income classes. We do however measure inequality the way Kolm (2001) suggested by

comparing the average level of income for a given level of e¤ort with what he calls the �equal equivalent�

level of income for this same level of e¤ort. We measure inequality within a given income class (�e¤ort

level�) by comparing the distribution of the �actual shares�mij

m:jfor each income class j with what could

be considered as the �expected shares�mi:

1= mi:.

Using one of Theil�s inequality measures this leads to the following measure of inequality in circum-

stances within income class j:

Tj =XI

i=1(mi:)ln

�mi:

mij=m:j

�(6)

The Theil measure of overall inequality in circumstances Tcirc would then be de�ned as

Tcirc =XJ

j=1(m:j)Tj =

XJ

j=1(m:j)

XI

i=1(mi:)ln

�mi:

mij=m:j

�, Tcirc =

XJ

j=1

XI

i=1(mi:m:j)ln

�mi:m:j

mij

�(7)

3The distinction between variations in the "margin" and in the "internal structure" of a matrix has also a long tradition

in mathematical statistics. In this literature the term "copula" rather than that of "internal structure" is used. We thank an

anonymous referee for drawing our attention to this point. Note also that the algorithm proposed by Deming and Stephan

is not the only one which can be used. Little and Wu (1991) mention three other methods.

5

which is in fact identical to the measure Tsim of social immobility suggested in (4).

Let us now measure inequality in circumstances on the basis of the Gini index. Here again we will

measure inequality within a given income class (�e¤ort level�) by comparing the distribution of the

�actual shares�mij

m:jfor each income class j with what could be considered as the �expected shares�

mi:

1= mi:. Using the Gini-matrix which was de�ned in (5) we derive the following measure of inequality

within income class (�e¤ort level�) j:

Gj = [: : : (mi:) : : :]0G[: : :

mij

m:j:::] =

[: : : (mi:) : : :]0G[: : :mij : : :]

m:j(8)

where the two vectors (of length I) on both sides of the G-matrix in (8) are ranked by decreasing

values of the ratios (mij

mi:).

To derive an overall Gini index of inequality of circumstances Gcirc we will have to weight the indices

given in (8) by the weights of the income classes j. We should however remember that in de�ning such

an overall within groups Gini inequality index the sum of the weights will not be equal to 1 because each

weight will in fact be equal to (m:j)2 in the same way as in the traditional within groups Gini index the

weights are equal to the product of the population and income shares. We therefore end up with

Gcirc =XJ

j=1

(m:j)2

m:j[: : : (mi:) : : :]

0G[: : :mij :::] =XJ

j=1(m:j)[: : : (mi:) : : :]

0G[: : :mij :::]

, Gcirc =XJ

j=1[: : : ; (m:j)(mi:); : : :]

0G[: : :mij :::] (9)

Note that the formulation for Gcirc in (9) is not identical to that of Gsim in (5). To see the di¤erence

between these two formulations the following graphical interpretation may be given. As was done when

drawing a social immobility curve put respectively on the horizontal and vertical axes the expected

shares (mi:) and the actual shares (mij

m:j), starting with income class 1 and ranking both sets of shares

by increasing ratiosmij

mi:m:j. Then do the same for income class 2 and continue with the other classes

until you end up with income class J . What we have then obtained is a curve which could be called an

�inequality in circumstances�curve which comprises J sections, one for each income class. Clearly the

slope of this curve is not always non-decreasing. It is non-decreasing within each income class but the

curve reaches the diagonal each time we end with an income class.

We should however note that the shares used to draw such an �inequality in circumstances curve�are

the same as that used in constructing a social immobility curve (compare both sides of the G-matrix in

(5) and (9)). In drawing the curve measuring inequality in circumstances we have simply �reshu­ ed�the

sets of shares used in drawing a social immobility curve. Rather than ranking both sets of shares (on one

hand the cumulative shares (mi:m:j), the cumulative shares (mij) on the other hand) by increasing values

of the ratiosmij

mi:m:jworking with all the (I � J) shares, we have �rst collected the shares corresponding

6

to the �rst (poorest) income class and ranked them by increasing ratiosmi1

mi:m:1and then did the same

successively for all income classes.

An illustration of the di¤erence between an �inequality in circumstances�curve and a social immobility

curve is given in Figure 1 which will be analyzed in the empirical section. Note that whereas the Index

Gcirc is equal to twice the area lying between the �inequality in circumstances�curve and the diagonal,

the index Gsim is equal to twice the area lying between the social immobility curve and the diagonal.

The area lying between the inequality in circumstances curve and the social immobility curve may then

be considered as a measure of the degree of overlap between the various income classes in terms of the

gaps between the �expected�and �actual�shares.

5 The empirical analysis

5.1 The data sources

We have analyzed two sets of data.

The �rst set is a survey of 2000 individuals conducted in France by Thomas Piketty in the year

1998 with �nancial support from the McArthur Foundation (see Piketty, 1999). The data set contains

65 variables and includes income, many socio-demographic characteristics and answers to questions on

social, political, ethical, and cultural issues.

Two types of variables were drawn from this database. To measure the social origin of the parents we

used information on the profession of either the father or the mother4 . The social status of the children�s

generation was measured via their monthly income classi�ed in eight categories5 .

On the basis of these two variables we built the social mobility matrix {mij} de�ned previously whose

lines i refer to the profession of either the father or the mother, depending on the case, and whose

columns j correspond to the income group to which the individual (the child, either the son or the

daughter, depending on the case examined) belongs.

The second data set we worked with is the Social Survey that was conducted in Israel in 2003. The

social origin of the individual was measured via the highest educational certi�cate or degree the father of

the individual had received6 . The social status of the children was measured via the total gross income

of all members of the household to which the individual belonged, whatever the source of the income

(work, pensions, support payments, rents, etc.). Ten income classes were distinguished7 . On the basis

of these two variables we built a social mobility matrix {mij} whose de�nition is similar to that de�ned

4Eight professions were distinguished: farmer (i.e. head of agricultural enterprise), businessman, store owner or �artisan�,

manager or independent professional, technician or middle rank manager, employee, blue collar worker, including salaried

persons working in agriculture, not working outside the household, retired.5The range of the income classes is given in Tables 3-A and 3-B.6Seven educational categories were distinguished: Elementary school completion, Secondary school completion without a

baccalaureate, Baccalaureate certi�cate, Post-secondary level with a non-academic certi�cate, obtained a BA, an academic

or a similar certi�cate, holds a MA, a MD or a similar certi�cate, has a PhD or a similar diploma.7The range of the income classes is given in Table 3-C.

7

previously for the French data, the educational of the father replacing the profession of the parents.

We then analyzed di¤erences in the degree of social mobility between three groups: the individuals

whose father was born in Asia or Africa, those whose father was born in Europe or America and those

whose father was born in Israel8 .

5.2 The results of the empirical investigation

5.2.1 Measuring Social Immobility

The French data. The results of the analysis are reported in Table 1 which examines six di¤erent

comparisons. In each comparison we give the value of the Gini social immobility index and decompose

the di¤erence between the values taken by this index in two di¤erent cases into the three components

mentioned previously. For each index and component we also give con�dence intervals based on a boot-

strap analysis. It is easy to observe that in all six tables (1-A to 1-F) the two Gini indices of social

immobility which are compared are always signi�cantly di¤erent one from the other. Moreover each of

the components in all the six tables is always signi�cantly di¤erent from zero.

The �rst striking result is that the degree of social immobility is quite higher when comparing fathers

and daughters (Gini social immobility index equal to 0.202) or fathers and sons (Gini index equal to 0.193)

than when comparing mothers and sons (Gini index equal to 0.143) or even mothers and daughters (Gini

index equal to 0.166). We also observe when comparing, for example, the degree of social immobility from

fathers to sons with that from mothers to sons that the di¤erence (lower degree of social immobility in

the latter case) is even much higher once we control for the margins. In other words the di¤erence in the

degree of �net social immobility� is in this case much higher (0.140 rather than 0.049 for the di¤erence

in �gross social immobility�). Note also that this impact of the margins is usually mainly one which is

related to di¤erences in the occupational structure of the parents.

In Figure 1 we have drawn the �Gross Social Immobility Curve� for the case where the transition

analyzed is that from Father�s Occupation to Daughter�s Income.

The Israeli data. The results of the analysis are presented in Table 2 with again bootstrap con�dence

intervals. Note that here also in the three comparisons which are made the Gini indices of social immobility

that are compared are signi�cantly di¤erent one from the other and each of the components is always

signi�cantly di¤erent from zero.

The most striking result here is certainly the fact that social immobility is much higher among those

who were born in Asia or Africa (Gini index of 0.233) than among those born in Europe or America (Gini

index of 0.124) or even among those born in Israel (Gini index of 0.147).8Note that in both empirical illustrations, despite the fact that we have data on "prizes" (incomes), we ignore these

values. We are well aware of the fact that the value of the �prizes� matter (see, Van de gaer et al., 2001). Since the

methodological section of this paper emphasized the concept of "dependence" between margins which are non ordered

categorical variables, we preferred, in a �rst stage, to concentrate our analysis on the measurement of inequality of life

chances. In future work we hope to integrate the "prizes" in the analysis, an extension which should allow us to measure

not only inequality in life chances but also inequality of opportunities.

8

The results are even more striking when we compare �gross�with �net�immobility. When looking at

the results given in Table 2-A we observe that whereas the �gross di¤erence�between the Gini indices of

social immobility of those born in Asia or Africa and those born in Europe or America is equal to 0.110,

the �net di¤erence�(net of changes in the margins) is equal to 0.310. Note also that this impact of the

margins is essentially an impact of di¤erences in the education levels of (the fathers of) the two groups

compared.

Quite similar conclusions may be drawn when comparing individuals whose father was born in Asia

or Africa and in Israel. (Table 2-B).

In Figure 2 we compare �Gross�and �Net Social Immobility Curves�. The �Gross�curves are drawn

for two groups, corresponding respectively to the individuals whose father was born in Europe or America

(EA) and Asia or Africa (AA). The �Net Social Immobility Curve�was drawn on the basis of a matrix

which has the margins of the matrix of those born in Europe or America but the �internal structure�of

the matrix of those born in Asia or Africa. This evidence shows how bigger the gap in social immobility

is, when comparing the EA and AA groups, once we control for the margins.

5.2.2 Inequality in Circumstances

The French data. In Table 3 we give two examples of the use of the Theil or Gini Indices of Inequality

in Circumstances. The �rst example refers to French data where the data analyzed are those relative to

the transition from the occupation of the fathers to the income class to which the daughters belong (see,

Table 3-A and 3-B). First we may note (see Table 3-A) that, as expected, the sum of the contributions

of the various Income Classes to the overall Theil index of inequality in circumstances is indeed equal to

this latter index. Second, according to both the Theil and the Gini index, the two income classes where

inequality in circumstances is highest are the income classes corresponding respectively to the income

ranges 4,000FF to 5,999FF and 20,000FF or more. As far as the highest relative contributions of the

income classes to the overall Theil or Gini indices of inequality in circumstances are concerned (remember

that the weights of the income classes are not the same for the Theil and the Gini index) the results

are quite similar. In both cases the highest relative contribution is that of the richest income class. The

second highest is that of the class with an income range of 4,000FF to 5,999FF for the Theil index and

that of the classes 4,000FF to 5,999FF as well as 12,000FF to 14,999FF (the results are almost identical

for these two income classes) for the Gini index.

As mentioned previously we have drawn in Figure 1 the �Gross Social Immobility Curve�for the case

where the transition analyzed is that from Father�s Occupation to Daughter�s Income. On this same

graph we have plotted what was called previously a �Curve of Inequality in Circumstances�. The large

area lying between both curves indicates clearly that the gap between the �expected� shares (mi:m:j)

and the �actual�shares mij is not a function of income, hence the great degree of overlapping between

the income classes when the ranking is based on the ratio of actual over expected shares.

9

The Israeli data. The second illustration is based on Israeli data concerning individuals whose father

was born in Asia or Africa (see, Tables 3-C). Unfortunately we could not compute Theil indices because

some of the cells in the data matrix were empty. For the Gini index (see, Table 3-C) the two income

classes with the highest values of this index are those corresponding to the income ranges NIS 3,001

to 4,000 and NIS 4,001 to 5,000. We may also observe that according to the Gini index (see, Table

3-C), the two income classes that contribute most to the overall value of the Gini index of inequality in

circumstances are those corresponding to the income ranges from NIS 7,001 to 9,000 and 9,001 to 12,000.

6 Concluding Comments

This paper suggested �rst applying to the study of intergenerational social immobility tools of analysis

that had been used previously in the �eld of occupational segregation and horizontal inequity. The

concept of �social immobility curve�was introduced and two indices of social immobility, related to the

Theil and Gini indices, were proposed. The empirical results con�rmed the need to make a distinction

between concepts of �gross�and �net�social immobility. Finally this paper suggested also two measures

of inequality in circumstances, derived again from the Theil and Gini indices. Whereas the Theil index of

inequality in circumstances turned out to be identical to the Theil index of social immobility we showed

that the Gini index of inequality in circumstances did not measure the same thing as the Gini index of

Social Immobility and these di¤erences were indeed con�rmed by the empirical illustration.

7 Appendix: Decomposing variations over time in the extent of

social mobility

The purpose of this Appendix is to show how it is possible to derive the speci�c contributions to the

overall variation in the extent of social immobility of changes in the income distribution of the individuals

(the children) and in the distribution of the parents by social category (these two changes correspond to

variations in the margins of the social mobility matrix) as well as in the �pure�variation in the extent

of social immobility (changes in the degree of independence between the lines and columns of the social

mobility matrix itself).

Neutralizing the speci�c impacts of changes in the margins of the social mobility matrix.

The methodology to be used to isolate the speci�c e¤ects of changes in the margins is borrowed from

the literature on the measurement of occupational segregation (see, Karmel and McLachlan, 1988) and

is extended here to the measurement of variations in the extent of social mobility.

Assume that the population under scrutiny is divided into I categories of social origin and J income

brackets. We can now de�ne a matrix {qij} in such a way that its typical element qij is equal to the

product (mi:m:j) of the margins i and j of the matrix {mij}. Clearly if mij is equal to qij for all i and

10

j, there is independence between the social origin (the lines of the matrix {mij} ) and the income of

the individuals (the columns of the matrix {mij} ). If mij is not equal to qij for at least some i andj,

the rows and the columns are at least partially dependent and such a link between the income of an

individual and his/her social origin should help us measure the extent of social immobility.

Let us now assume that we wish to decompose the variation over time in the extent of social mobility.

In order to do so we will adopt a technique originally proposed by Deming and Stephan (1940) and used

by Karmel and McLachlan (1988). The idea, when comparing two matrices of proportions {mij} and

{vij}, is to build a third matrix {sij} which would have, for example, the internal structure of the matrix

{mij} but the margins of the matrix {vij}. To derive {sij}, one multiplies �rst all the cells (i; j) of {mij}

by the ratios (vi:mi:

) where vi: and mi: are respectively the horizontal margins of the matrices {vij} and

{mij}. Call {rij} the matrix you derived after such a multiplication. Multiply now all the cells (i; j) of

this matrix {rij} by the ratios (v:jr:j) where v:j and r:j are now the vertical margins of the matrices {vij}

and {rij}. Call {uij} the new matrix you just derived. If you renew this procedure several times, the

matrix you derive will quickly converge, as shown by Deming and Stephan (1940), to a matrix {sij} that

has the margins of the matrix {vij} but, in a way, the internal structure of the matrix {mij}. In other

words the degree of social immobility corresponding to the matrix {sij} is identical to that corresponding

to the original matrix {mij} but the matrix {sij} has the same �income distribution of the individuals�

and the same �structure of the social origin of these individuals�as that of the matrix {vij}. One could

naturally have proceeded in the reverse order by starting with the matrix {vij} and ending up with a

matrix {wij} that would have the margins of the matrix {mij} but the internal structure of the matrix

{vij}.

Decomposing variations over time in the extent of social mobility. We have just shown how

the �move� from the matrix {mij} to the matrix {vij} included really two stages: one in which the

margins were changed and one in which the internal structure of the matrix was modi�ed. Let �SM =

SM(v) � SM(m) refer to the overall variation in the extent of social mobility. �SM may in fact be

expressed as the sum of a contribution C�ma of di¤erences in the margins and a contribution C�is of

di¤erences in the internal structure of the two social mobility matrices compared. Then, applying the

idea of a Nested Shapley decomposition (see, Sastre and Trannoy, 2002), we show that it is possible to

further decompose this contribution C�ma into the sum of the contributions Ch and Ct of the horizontal

and vertical margins.

Using Shapley�s decomposition the contribution C�ma of a change in the margins to the overall

variation �SM may be written as:

C�ma = (1=2)f(�ma) + (1=2)[f(�ma;�is)� f(�is)] (10)

Similarly the contribution C�is of the change in the internal structure of the matrix to the overall

variation �SM will be expressed as:

11

C�is = (1=2)f(�is) + (1=2)[f(�ma;�is)� f(�ma)] (11)

It is easy to observe that C�ma + C�is = �SM

Using the de�nitions of the matrices {sij} and {wij} given previously we derive that the contributions

C�ma and C�is may be also written as

C�ma = (1=2)[SM(s)� SM(m)] + (1=2)f[SM(v)� SM(m)]� [SM(w)� SM(m)]g

, C�ma = (1=2)f[SM(s)� SM(m)] + [SM(v)� SM(w)]g (12)

C�is = (1=2)[SM(w)� SM(m)] + (1=2)f[SM(v)� SM(m)]� [SM(s)� SM(m)]g

, C�is = (1=2)f[SM(w)� SM(m)] + [SM(v)� SM(s)]g (13)

Applying the idea of a Nested Shapley decomposition (see, Sastre and Trannoy, 2002) we can now

further decompose the contribution C�ma. We de�ned previously the matrix {rij} which was obtained by

multiplying the cells (i; j) of {mij} by the ratios (vi:mi:

) where vi: and mi: are respectively the horizontal

margins of the matrices {mij} and {vij}. Let us now call {nij} a matrix which is obtained by multiplying

the cells (i; j) of {mij} by the ratios (v:jm:j

) where v:j and m:j are respectively the vertical margins of

the matrices {mij} and {vij}. Using again the idea of Shapley decomposition applied this time to the

di¤erence Dma1 de�ned as

Dma1 = [SM(s)� SM(m)] (14)

we may write that

Dma1 = g(�h;�t) (15)

where �h and �t refer respectively to the changes in the horizontal and vertical margins. The

contributions C�h1 and C�t1 to the di¤erence Dma1 may be then de�ned as

C�h1 = (1=2)g(�h) + (1=2)f[g(�h;�t)]� g(�t)g (16)

C�t1 = (1=2)g(�t) + (1=2)f[g(�h;�t)]� g(�h)g (17)

Using the de�nitions of {rij} and {nij} given previously, these contributions C�h1 and C�t1 may be

also expressed as

C�h1 = (1=2)[SM(r)� SM(m)] + (1=2)f[SM(s)� SM(m)]� [SM(n)� SM(m)]g

12

, C�h1 = (1=2)f[SM(r)� SM(m)] + [SM(s)� SM(n)]g (18)

Similarly

C�t1 = (1=2)[SM(n)� SM(m)] + (1=2)f[SM(s)� SM(m)]� [SM(r)� SM(m)]g

, C�t1 = (1=2)f[SM(n)� SM(m)] + [SM(s)� SM(r)]g (19)

Let us now decompose in a similar way the di¤erence Dma2 de�ned as

Dma2 = [SM(v)� SM(w)] (20)

We now de�ne two additional matrices {cij} and {fij} de�ned as follows. The elements cij of the

matrix {cij} are obtained by multiplying the elements vij of the matrix {vij} by the ratios (wi:vi:) where

wi: and vi: refer to the horizontal margins of the matrices {wij} and {vij}. Similarly the elements fij

of the matrix {fij} are obtained by multiplying the elements vij of the matrix {vij} by the ratios (w:jv:j)

where w:j and v:j refer to the vertical margins of the matrices {wij} and {vij}. We may therefore de�ne

the contributions C�h2 and C�t2 to the di¤erence Dma2 as

C�h2 = (1=2)[SM(v)� SM(c)] + (1=2)f[SM(v)� SM(w)]� [SM(v)� SM(f)]g

, C�h2 = (1=2)f[SM(v)� SM(c)] + [SM(f)� SM(w)]g (21)

Similarly

C�t2 = (1=2)[SM(v)� SM(f)] + (1=2)f[SM(v)� SM(w)]� [SM(v)� SM(c)]g

, C�t2 = (1=2)f[SM(v)� SM(f)] + [SM(c)� SM(w)]g (22)

Combining now (12), (14), (18), (19), (20), (21) and (22) we conclude that the contribution C�ma

may be written as

C�ma = Ch + Ct (23)

where

Ch = (1=4)f[(SM(r)� SM(m)) + (SM(s)� SM(n))] + [(SM(v)� SM(c)) + (SM(f)� SM(w)]g (24)

13

Ct = (1=4)f[(SM(n)�SM(m))+ (SM(s)�SM(r))] + [(SM(v)�SM(f)) + (SM(c)�SM(w))]g (25)

Combining (24) and (25) we observe that, as expected,

Ch + Ct = (1=2)f[(SM(s)� SM(m)] + [SM(v)� SM(w)]g = C�ma (26)

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Figure 1: (Gross) Social Immobility Curve versus Curve of Inequality inCircumstances. France: Father’s Occupation versus Daughter’s Income

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

cumulative values of the “a priori” probabilities qij(case of independence between the rows i and the columns j)

of belonging to social origin i and to income group j

cum

ulat

ive 

valu

es o

f the

 cor

resp

ondi

ng a

ctua

l pro

babi

litie

s pi

j.

Gross Social Immobility Inequality in Circumstances

23

Figure 2: Gross and Net Social Immobility Curves. Israel: Fathers born inEurope or America (EA) versus Fathers born in Asia or Africa (AA)

24