There are intuitive appeals to the notion that the components of a vector are "less spread out" or "more nearly equal" then the components of a vector . For instance, we may be interested in examining whether the coefficient of variation or another similar indicator is lower for a distribution of income than another distribution of income . Those notions arise in a variety of contexts, and they can be made precise in a number of ways.
But in remarkably many cases the appropriate precise statement is " is majorized by ".
Some of these cases are reviewed here. We will show that depending on the situation, several conditions become equivalent. While some of them may be quite easy to implement, others may not be implementable so easily. However may make them quite appealing.
In subsequent discussion we will consider the problem of ranking income distributions in a society or different societies from different perspectives. Therefore by social state in a community of n-persons, we will mean an income distribution where is the income of person , with (the strict inequality) for at least one , .
We assume that no ambiguity arises in the connection with the definitions of income, income receiving unit (here individual) and reference period over which income is observed. For a fixed , , the set of all income distributions is , the nonnegative orthant of the Euclidean n-space with the origin deleted, where is the set of positive integers. Thus, for any , .
Throughout the presentation we will adopt the following notation. For any two elements of , the coordinate-wise ordering , is denoted by . For any write for increasing rearrangement of , that is, . Similarly, let denote the components of in decreasing order, and let denote the decreasing rearrangement of .
A function is concave if
for all and for all , .
The superscript "n" in denotes the dimension of the domain of the function.
Clearly a strictly concave function is necessarily concave, but the converse is not true. for example, a linear function is concave but not strictly so. If we draw the graph of a (strictly) concave function. We find that this graph possesses the following geometric feature the straight line joining any two points on the graph lies (entirely below) on or below the graph.
A function is (strictly) convex if is (strictly) concave.
A function is quasiconcave if
for all and for all , .
Observe that quasiconcavity is a weaker condition than concavity. Any (strictly) concave function will be (strictly) quasiconcave, but not vice-versa. For instance, defined by is quasiconcave but not concave. for a two coordinated strictly quasiconcave function, definition 3 tells that if but so that and are on the same contours then , .
That is, the line joining and lies entirely above the contour containing and . Thus, strictly quasiconcave functions are those quasiconcave functions whose contours do not contain any flat section. On the other hand, strictly concave functions are those concave functions whose graphs do not contain any flat section.
A function is (strictly) quasiconvex if is (strictly) quasiconcave.
A square matrix of order is said to be a doubly stochastic or bistochastic matrix if all its entries are non-negative and each of its rows and columns sums to one.
Thus the matrix is bistochastic if
and
We rewrite the two above conditions in a more compact form as:
where is the n-coordinate vector of ones. Thus, 1 is a characteristic root of corresponding to the characteristic vector .
If the matrices and are bistochastic, then the product is bistochastic.
Since and have nonnegative elements, one sees directly from matrix product that also has nonnegative elements. Also
A matrix is a permutation matrix if it is bistochastic and has exactly one positive entry in each row and each column.
is bistochastic matrix but not a permutation matrix.
is a permutation matrix.
A function is symmetric if for all , where is a permutation matrix of order .
The function is a symmetric function
(symmetric)
(not symmetric)
where (not symmetric)
A function is S-concave if for all , where is any bistochastic matrix of order . For strict S-concavity of , the weak inequality is to be replace by a strict inequality whenever is not a permutation of .
A function is called (strictly) S-convex if (strictly) S-concave.
All S-concave and S-convex functions are symmetric. Symmetry and quasiconcavity of a function imply S-concavity, but the converse is not true. Thus, for a symmetric function, the idea of S-concavity is a generalization of the idea of quasiconcavity.
If , and the function:
is called entropy of . is strictly s-concave.
The function is strictly S-concave on , the strictly positive part of .
The function , , is strictly S-concave on .
The functions and are S-concave, but not strictly so.
All these examples are illustrations of the following proposition:
If is an interval and is (strictly) concave then
is (strictly) S-concave on .
The examples given above are all additive. Two nonadditive examples of strictly S-concave functions are the following:
where .
Consider a population of individuals where is the income of person , . Order the individuals from the poorest to the richest to obtain . Now plot the points , where , and . is the total income of the poorest persons under in the population. Join these points by line segments to obtain a curve connecting the origin with the point . The curve is called the Lorenz curve of the distribution . Notice that if the total wealth is uniformly distributed in the population, then the Lorenz curve is a straight line, which is called the line of equality. With unequal distribution the curves will always begin and end in the same points as the an equal distribution, but they will be "lower" in the middle, and the rule of interpretation is that the closer is the curve to the line of equality the more equal is the corresponding distribution.
Let represent the income of individuals similarly, let be another distribution. Than according to the idea of Lorenz is at least as equal as if
for all . Equivalently we say that weakly Lorenz dominates . That is, Lorenz dominates in the weak sense if the Lorenz curve of is nowhere below that of . We say that strictly Lorenz dominates , what we write , if (L) holds with the additional restriction that there will be strict inequality for at least some . means that is more equal than . is the condition that the Lorenz curve of is nowhere below that of and at some places (at least) it is above the Lorenz curve of . If () holds with the additional restriction that , then we say that is (strictly) majorized by .
In figure fig2 both the distributions and are strictly Lorenz dominated by distribution . (Note that in this figure A is the line of equality). It is clear that the ordering of income distributions generated by the Lorenz curve comparison is a quasi-ordering, since if the Lorenz curves cross neither can be said to be preferred by the Lorenz domination criterion. We cannot rank distributions and by this criterion. Obviously, while the weak Lorenz domination satisfies both reflexivity and transitivity, the strict domination meets only transitivity. However, none of them is a complete ordering.
In order to interpret the Lorenz curve ordering from alternative perspectives, the following definition is also needed.
For any , we say that is obtained from through a progressive transfer if
where .
That is, and are identical except for a positive transfer of income from the rich person to the poor person .
A remarkable consequence of the strict Lorenz domination relation was proved by Dasgupta, Sen and Starrett.
Let and be two distributions of the same amount of total income over a given population size, that is, , where . Then the following conditions are equivalent:
for all strictly S-concave social welfare functions .
What theorem 1 tells is that if the Lorenz curves of the distributions and with the same mean do not cross, then the distribution closer to the line of equality will represent higher level of social welfare for any social welfare function provided that it is strictly S-concave. We interpret as a social welfare function because it is defined on income distributions of the society and, as we show below, the value of increases if there is a rank preserving progressive transfer from a rich to a poor. The theorem also shows that if two curves intersect the ranking of the distributions cannot be unambiguous by social welfare functions: we can get two strictly S-concave social welfare functions that will rank the distributions in different directions.
The proof of the theorem 1 proceeds by using certain well-known results of Hardy,Littlewood and Polya on mathematical inequalities. For , where , the following conditions are equivalent Note_1 :
For all , , with at least one such that ()
If is not a permutation of , there exists a bistochastic matrix such that
For any strictly concave real valued ,
can be transformed into by a finite sequence of transformations of the form
for and , with if .
Since , the first condition in the Hardy, Littlewood, Polya theorem means .
Condition says that each income in is a mixture of the income on . Condition means that is regarded as more equal than by the symmetric utilitarian social welfare functions as long as the identical utility function is strictly concave. finally, condition tells us that the distribution is obtained from the distribution by a finite sequence of rank preserving transformations transferring income from the rich to the poor.
Suppose . Now by the Hardy, Littlewood, Polya theorem implies that there must be a bistochastic matrix such that . It follows that for any strictly S-concave . Since an s-concave function is symmetric, we have and .
Since in the Hardy, Littlewood, Polya theorem conditions and are equivalent, not implies not . That is, if not , then for some strictly concave , .
Now, given strict concavity of , we can regard as a strictly S-concave function. Therefore, it followed that if not , then for some strictly S-concave real valued , . That is, not implies not . Hence .
Theorem 1 is very valuable. It gives strong justification to accept strict S-concavity as an equitable principle. Strict S-concavity of a social welfare functions equivalent to the condition that a rank preserving transfer of income from a rich person to a poor person increases welfare.
Theorem 1 is applicable to comparison of income distributions of a given total over a given population size. However, international comparisons usually involve different population sizes, as do intertemporal comparisons for the same country.
For instance, we may be interested in comparing the social welfares of Italy and Belgium or of Italy at present and Italy ten years ago the problem of ranking with different means will be discussed later. For ranking distributions with the same mean over different population sizes. Dasgupta, Sen and Starrett proposed an axiom, which they called the symmetry axiom for population. According to this axiom, if an income distribution is replicated times, then the aggregate welfare of the replicated distribution is simply times the welfare of the actual distribution.
For all , , where , each .
Set and be two distributions of income with the same mean over population sizes and respectively. That is, and , where . Then the following statements are equivalents:
, where is any arbitrary social welfare function that satisfies strict S-concavity and SAP, and whose domain is , .
Let () denote the () fold replication of (). Clearly, , , . Now, the Lorenz curve is population replication invariant. Consequently, and have the same Lorenz curves as and respectively.
Therefore, implies that . Hence by theorem 1, for all strictly S-concave social welfare functions , . But the welfare function satisfies SAP. Therefore, and . Hence .
This part of the theorem follows from a construction similar to the above, again using theorem 1.
Thus, theorem 2 suggests that under SAP and strict S-concavity, Lorenz domination with the same mean income implies a higher average welfare even for the case of variable population sizes.
Advantages of theorems 1 and 2:
Consider theorem 1. There can be infinitely many strictly S-concave social welfare functions. Therefore, it will be extremely difficult to check whether one function has more value than another for two distributions of income of a given total. Because if there are strictly S-concave welfare functions, we have to make comparisons, and the value of is unboundly large. However, if condition holds, that is, if strictly Lorenz dominates , or is strictly majorized by , then we can be sure that will have a higher value than for any strictly S-concave social welfare function. Computation of values of individual functions is not necessary. A similar explanation holds for theorem 2.
We now discuss some extensions of the theorems 1 and 2 to the case of variable mean incomes. It would be reasonable to assume that if one distribution has both a higher mean and a higher Lorenz curve than another distribution , then social welfare under is higher than under . This will be true if is increasing and strictly S-concave. But even now the ranking of income distributions with differing means is limited because we need both a higher Lorenz curve and higher mean for a clear verdict.
The ability of Lorenz curves to provide unambiguous ranking of distributions with differing means improves substantially if we extend the concept of Lorenz curve to generalized Lorenz curve (Shorrocks, 1983). The generalized Lorenz curve is constructed by scaling up the Lorenz curve by the mean income. formally, the generalized Lorenz curve of is a plot of against , , and at the ordinate of the curve is .
We say that an income distribution weakly dominates another distribution in the generalized Lorenz sense if
for all . strictly generalized Lorenz dominates , what we write , if (GL) holds with the additional restriction that there will be strict inequality for at least one . Thus, means that the generalized Lorenz curve of is nowhere below and at some places (at least) it is strictly inside that of .
The following theorem of Shorrocks (1983) explains the role of generalized Lorenz curves in ranking distributions with different means.
Let and be two distributions of income over a given population size , that is, . Then the following conditions are equivalent:
for all increasing, strictly S-concave social welfare functions .
Thus unlike the Lorenz criterion, the generalized Lorenz criterion takes explicitly into account the sizes of the distributions.
The proof of the theorem 3 relies on a result of Marshall and Olkin (1979).
Marshall and Olkin showed that for , the following conditions are equivalent:
is strictly supermajorized by , that is,
for all , with strict inequality for at least one .
For any strictly concave, increasing real valued ,
There exists a doubly stochastic matrix such that , that is, the i coordinate of is i coordinate of , with for at least one .
The first condition in the Marshall Olkin theorem is that . According to condition , the distribution is regarded as better than distribution by the symmetric utilitarian social welfare function, where the utility function is strictly concave and increasing. Finally, condition shows how incomes in can be transformed into incomes .
Suppose . now, by the Marshall and Olkin theorem there exists a doubly stochastic matrix such that , that is, the i coordinate of is at least as large as the i coordinate of , with strict inequality for at least one . Let be strictly S-concave and increasing. Then by increasingness of , . Again strict s-concavity of implies that .By symmetry of , we have and . thus implies Note_3 that
Since in the Marshall and Olkin theorem, conditions and are equivalent, not implies not. Thus, if not , the for some increasing, strictly concave , .
Now, given increasingness and strict concavity of , we can regard as an increasing, strictly S-concave function. Thus it follows that if not , then for some increasing, strictly S-concave real valued , . That is, not implies not . Hence implies .
Theorem 3 indicates that an unambiguous ranking of income distributions by all increasing, strictly S-concave social welfare functions can be obtained if and only if the generalized Lorenz curves do not intersect. Clearly, this latter condition will apply if one of the distributions has both a higher mean and higher Lorenz curve. But it will also be satisfied in other cases if the higher mean is sufficient to offset the lower part of the Lorenz. Thus, the Generalized Lorenz curves are likely to intersect less often than the Lorenz curves.
In figure fig2, is preferred to both and . The larger mean of is sufficient to compensate for whatever differences in the income distribution may exist. The generalized Lorenz domination has similar advantages as the Lorenz domination with additional flexibility of covering the case of different mean incomes.
,
We can extend theorem 3 to the variable population case. For this we assume additionally that the social welfare function satisfies SAP.
Let and be two income distributions over population sizes and respectively, that is, and .
Then the following statements are equivalent:
, where is any real valued social welfare function that satisfies increasingness, strict S-concavity and SAP, and whose domain is , .
The social desire for more equitable distribution may come into conflict with generalized Lorenz criterion. To see this, increase the income of the richest person. Then total income increases, as does income inequality. Thus, a welfare improvement according to the generalized Lorenz domination may be compatible with a inequality increase. We therefore look for alternative monotonicity condition (or efficiency preferences). (In economics preference for higher income is known for efficiency preference. The conflict of efficiency with equity that we have just indicated is called equity - efficiency trade-off) As a first suggestion we argue that welfare should increase if all incomes are raised equiproportionally. Formally, a welfare function satisfies scale improvement condition if for all for any ,
In (SIC) total income increases more as increases and the distribution of relative incomes remain unaltered. We have preference for higher incomes keeping all inequality indicators that are homogeneous of degree zero (e.g. the coefficient of variation) unchanged. (Note that the coefficient of variation decreases under a transfer of income from a rich to a poor.)
We then have:
For , the following conditions are equivalent:
and weakly Lorenz dominates ( for short)
for all social welfare functions that satisfy scale improvement condition and S-concavity.
Suppose and . Define . Since satisfies the scale improve condition . By construction, the Lorenz curve of and coincide Note_5 . Hence implies . But we also have . Therefore, by S-concavity of , we get . Hence Note_6 .
Let us define by where and is S-concave on .
Clearly, satisfies the scale improvement condition and S-concavity. Now, for implies . by choosing . implies . But is arbitrary S-concave Note_7 function and the distributions and have the same mean (equal to ). Hence by the weak version of theorem 1, , from which we get .
An alternative to scale improvement condition (SIC) is incremental improvement condition. Formally, satisfies incremental improvement condition if for all , for all ,
Thus, as the value of increases in (IIC) total income increases, and the distribution of absolute income differentials remains unchanged. The incremental efficiency condition corresponds to a preference for higher income keeping all inequality indicators that remain unaltered under equal absolute additions in incomes (e.g., the standard deviation). Note that in (SIC) absolute differences of the form get widened.
For , the following conditions are equivalent:
and for all where is the height of the generalized Lorenz curve of the distribution at population proportion .
for all social welfare functions that satisfy S-concavity and the incremental improvement condition.
Define . Since satisfies the incremental improvement condition . Furthermore, . for any . this shows that for all , where is the height of the Lorenz curve at the population proportion . Hence by S-concavity of , , which shows Note_8 that .
Suppose that is given by , where and is such that .
is assumed to be S-concave on . Clearly defined above is S-concave and meets the incremental improvement condition. Now, for , implies . By choosing , implies . Alternatively by choosing , we get . Since the distributions and have the same mean , and is any S-concave function we conclude that for all . Now, in the case of distributions with common mean the generalized Lorenz ranking. The above inequality is therefore same as for all . the result now follows from this inequality.
Thus Theorem Theorem7 can be used to test whether any pair of income distributions and can be ranked using social welfare functions satisfying (IIC) and S-concavity. One way of doing the comparison is to calculate for each distribution of those values and the mean incomes.
Moyes (1987) offered an alternative interpretation of the theorem Theorem7. He referred to the generalized Lorenz curve of the distribution as the absolute Lorenz curve of . The absolute Lorenz curve coincides with the horizontal axis when all incomes are equal. The absolute Lorenz curve of is decreasing with for and increasing for where is such that .
It is easy to see that measures the vertical distance between the line of equality and the at the population proportion . Thus, it represents the income shortfall of the bottom proportion of the population, expressed as a fraction of the aggregate population size, from the equal income distribution .
Theorem Theorem7 can now be restated as:
Theorem 6*: Let be arbitrary. Then the following conditions are equivalent:
for all that are S-concave and satisfy the incremental improvement condition.
and for all .
Larry Temkin (1993) suggested that we may think of the inequality simply as an aggregate of individual complaints experienced by individuals located in disadvantaged by positions in the income distribution. The idea of complaint arises two related questions: who has a complaint? What is the point of reference for a complaint?
Temkin considered a number of possibilities but focussed attention mainly on the following. Everyone but the person with the highest income has a legitimate complaint and everyone has the same reference point, the highest income.
For any , define the cumulation of complaints
This concept can be used to draw the cumulative complaint contour () of a distribution , formed by joining the points . The is increasing and concave. If lies on or above then distribution does not exhibit less complaint inequality than distribution . Formally,
For any , weakly dominates by complaint inequality if for all .
This ordering is related to the weak generalized Lorenz ordering.
Let be arbitrary. Then the following statements are equivalent:
dominates by complaint inequality
weakly generalized Lorenz dominates
Suppose dominates by complaint inequality. Then
This is equivalent to
which means that .
This part of the proof can be obtained by simply reverse calculations.
Theorem Theorem7b shows a new type of ranking principle that embodies the comparison of income distributions in terms of complaints.
The quasi ordering we consider next is more demanding as it requires that all pairwise inequality be less in distribution than in distribution for the former to be declared as less unequal than the latter. We say that distribution dominates distribution in absolute differentials, which we write if
for all .
This simply means that the differences, between two individual incomes taken in nondecreasing order are less in than in .
This quasiordering considered by Marshall and Olkin may be considered as a suitable inequality criterion, since it turns out to be a sufficient condition for the Lorenz quasiordering.
Let and be two distributions of a given total over a given population size , that is , where , are arbitrary. Then implies , but the converse is not true, unless .
Given with , suppose holds. Note that we can rewrite condition (AD) as
for . Since , that is, , and since is nonincreasing in (from (6)), there exists , , such that for , , .
Therefore , and , . Since , the last inequalities can be rewritten as , . Thus, for each , the sum of the largest of the is dominated by . Hence to prove that the converse implication is false for , take , . We have but does not hold.
Hence is sufficient but not necessary for .
In many theoretical as well as practical situations we are confronted with the necessity of making a prediction about a decision maker's preference (choice) between given pairs of uncertain prospects without having any knowledge of the decision maker's utility function.
Problems of this type can be solved using the theory of stochastic dominance. This approach corresponds closely to the problem of ranking income distributions in terms of inequality and welfare.
We suppose that income distributions are defined on the continuum are represented by a distribution function defined on Note_9 . is the proposition of population with income less than or equal to . , and is increasing. is assumed to be continuously differentiable. The continuous function , the derivative of , is the income density function.
Given two income distribution functions and defined on , we say that is preferred to by the first order stochastic dominance criterion if
for all , with strict inequality for at least one .
Let : Set of all continuously differentiable increasing utility functions on . That is,
is preferred to by the first order stochastic dominance criterion if and only if
for all .
By definition
Integrating by parts, we get
given that for all and for all , with strict inequality for at least one , we must have
The proof of the converse is by contradiction. Suppose that exists , such that for with for at least one , and for .
Consider the following utility function
where are arbitrary.
Observe continuity of at . Now,
is continuous at . Note also positivity of .
Then for this utility function by virtue of (7)
Now .
Therefore is positive. Similarly,
Note next that .
Therefore, we can choose sufficiently large to make the first term on the right hand side of (8) quite large negatively so that the right hand side of (8) as a whole become negative. Thus, there exists a utility function for which becomes smaller than . Hence the result.
Interpretation: We can rewrite first order stochastic dominance alternatively as
for all with for at least one . Thus, the cumulative proportion of persons with income greater than or equal to is not lower under than under , and for at least one this inequality is strict. Therefore preference for higher income is than main distinguishing characteristic for first order dominance. Hence the theorem says that is preferred to by the first order stochastic dominance criterion if and only if is regarded as better than by two utilitarian rule for any increasing utility function.
Given two income distributions and defined on , we say that is preferred to by the second order stochastic dominance criterion if
for all with for at least one .
Let
That is, is the set of all increasing, strictly concave and twice continuously differentiable utility functions defined on .
Given two income distribution functions and defined on , is preferred to by the second order stochastic dominance criterion if and only if
for all .
Integration by parts yields
From proof of theorem 9, we know that
From (9) and (10) we get
Now, given that for all , with for at least one , it must be the case that . Further, given , , the sum of the two expressions on the right hand side of (11) is positive. Hence
To prove the converse, let
Suppose that there exists , such that for with for at least one , and for .
Now, consider the utility function
Note continuity if at . Now
Note that and also observe continuity of at . ( for is given by
and is continuous at .
Then for the utility function considered above, we have by virtue of (11),
The first term on the right hand side of (12) is nonnegative, while the second term is negative. Observe that is decreasing in . Therefore, the magnitude of the second term can be made as large as possible negatively by choosing a sufficiently large.
Thus, there exists a utility function in for which becomes negative. Hence the result.
Theorem 10 demonstrates that is preferred to if and only if is preferred to by the utilitarian rule, where the utility function is increasing (showing preference for higher income) and strictly concave (showing preference for equity).
Each of these statements also corresponds to the condition that the generalized Lorenz curve of dominates that .
To state these equivalent conditions rigorously, let us define Lorenz and generalized Lorenz curves for a continuum of population. As before we assume that the income of an individual is a random variable defined on the interval . is the income distribution function. The mean income is
The proportional share of total income enjoyed by "agents" with income less than or equal yo is
The derivative of with respect to is
The positivity of this derivative implies that like , rises at rises from to . Also , . is called that first moment distribution function of income.
The Lorenz curve associated with the continuous type income distribution considered here is a graph of against , as rises from to . Often we can eliminate between them and can express explicitly as a function of . Since , using (13) we get
which is positive for positive incomes.
Again
Which is also positive. Therefore, the Lorenz curve is increasing and concave. Note the generalized Lorenz curve if is simply the plot of against , as goes from to .
We can now state equivalence of theorem 10 with generalize Lorenz domination rigorously as follows.
Given two income distribution functions and defined on , the following conditions are equivalent:
for all , with for at least one .
for all increasing, strictly concave utility function .
for all , with for at least one .
The following result drops out as a corollary to Theorem Theorem11.
Let and be two income distributions on I such that . Then the following conditions are equivalent:
for all strictly concave utility function .
for all , with for at least one , .
The stochastic dominance conditions defined above can be extended to the n degree stochastic dominance rule. Of two income distribution functions and defined on , the former is said to dominate the latter by the n degree stochastic dominance rule if
for all , with for at least one , where
and . We can now have the following theorem:
Let and be two income distribution functions on the interval . Then the following conditions are equivalent:
dominates by the n degree stochastic dominance rule.
for all , where where is continuously differentiable times Note_10 , all odd order derivatives of through are positive, and all its even order derivatives are .
For condition in Theorem Theorem12a means that is preferred to by the utilitarian rule that approves of equity (), efficiency () and the diminishing transfers principle that put more emphasis to income transfers at the lower end of the distribution ().
In order to illustrate the idea of stochastic dominance numerically, let . Then
for all , where is the number of individuals having income less than or equal to in .
Let . and are given below.
Two example of utility functions, that are members of , are:
, ;
, .
Rather than representing a discrete a distribution by means of its distribution function, we may "appeal" to its inverse attribution function. Let denote the inverse associated with the distribution function , where
For any in the range of , is defined as the minimum income in the domain of such that .
The Lorenz curve associated with the distribution having distribution function can now be defined as
where .
We can restate the definition of Lorenz domination and generalized Lorenz domination using this definition of the Lorenz curve.
Let . The distribution function is defined by
The inverse function is defined by
Hence,
We can find the generalized Lorenz curve of the distribution by multiplying by .
An indicator of polarization is a measure of the extent of decline of the middle class. Two characteristics that are regarded as being intrinsic to the notion of polarization are nondecreasing spread and nondecreasing spread, a movement of incomes from the middle position to the tails of the income distribution makes the distribution at least as polarized as before. In other words, as the distribution becomes more spread out from the middle position, polarization does not decrease. On the other hand, nondecreasing bipolarity means that a clustering of incomes below or above the median leads to a distribution at least as polarized as before. Equivalently a reduction of gaps between any two incomes above or below median does not reduce polarization. Thus, polarization involves both an inequality-like component, the nondecreasing spread criterion, which does not reduce both inequality and polarization, and an equality-like component, the clustering or bunching principle, which does not lower polarization, while not increasing any inequality indicator that satisfies the transfer principle, a requirement under which inequality is nondecreasing for a transfer of income from rich to a poor. Thus, polarization and inequality are two different concepts, although there is a nice complementarity between them.
We now address the problem of ranking two distributions of income using alternative indicators of polarization. An indicator of polarization is a real valued function defined on , that is, where all income distributions in are ordered nondecreasingly. For any , gives the level of polarization associated with distribution . For any , we write for the median value of . If is odd, is observation in . If is even the average of the and observations in is taken as the median of . We write and for the subvectors of that include for and respectively, where . Thus, for any , if is even and if is odd.
For , , is said to be obtained from by a simple increment if for some and for all . We write to indicate this. Given that and are nondecreasingly ordered, the transformation allows only rank-preserving increments. Further, if is obtained from by a progressive transfer we write to denote this. Note that only rank-preserving transfers are allowed under the operation .
A polarization indicator can be homogeneous of degree zero or invariant under equal absolute changes in all incomes. An indicator is called relative or absolute according as it satisfies the former or the latter. A polarization indicator whether relative or absolute, should satisfy the following postulates:
Nondecreasing Spread (): If , where , are selected through anyone of the following cases,
, ;
, ;
, ,
then .
Nondecreasing Bipolarity (): If , where are related through anyone of the following cases,
, ;
, ;
, ,
then .
Symmetry: for all permutation matrices of order .
is a monotonicity principle. Since rank-preserving increments (reductions) in incomes above (below) the median widens the distribution, polarization should not go down. That is, greater distancing between the groups below and above the median should not make the distribution less polarized.
Nondecreasing Spread
is a bunching or clustering principle. Because a rank-preserving egalitarian (that is, equitable) transfer on the same side of the median brings the individuals closer to each other, polarization should be nondiminishing. As an equitable transfer demands that inequality should not reduce.
explicitly establishes that inequality and polarization are two nonidentical concepts.
Nondecreasing
Bipolarity
demands that polarization should remain unchanged if we trade a reordering of incomes. Thus, any characteristic other than income e.g., the names of the individuals is irrilevant to the measurement of polarization. One implication of is that a polarization indicator can be defined directly on ordered incomes (as we have done).
To understand these properties more explicity, let . Then , , . Let . Then , , since . Next suppose . Then , . Clearly, because unit of income has been transferred from the person with units of income to the one with income units. Similarly, . Note that the transfers are rank-preserving.
We want to develop a ranking of alternative income distributions in terms of absolute polarization indicators. (The relative case can be done analogously.)
The ranking relies on the absolute polarization curve. The absolute polarization curve of any income distribution shows for any population proportion, how far the total income enjoyed by that proportion, expressed as a fraction of the population size, is from the corresponding income that would receive under the hypothetical situation where everybody enjoys the median income. For any , the absolute polarization curve () is a plot of against , and against , .
Note that the ordinate at involves the income level . Now, if is odd, is one of the incomes in the distribution. However, for even , is not in , we define the ordinate at , since in polarization measurement, the median income is the reference income level. For the distributions , the ordinates of the absolute polarization curve are and the corresponding population proportions are .
For a typical income distribution , up to , the absolute polarization curve decreases monotonically, at it coincides with the horizontal axis and then it decreases monotonically. If is an equal distribution, then the curve becomes the horizontal axis itself. For a distribution , where is unequal but is equal, the curve runs along the horizontal axis up to , and starting from it rises gradually. Similarly, if is such that is unequal but is equal, the curve is decreasing up to after which it runs along the horizontal axis.
Given any two income distributions , is said to weakly dominate with respect to absolute polarization, which we write if the absolute polarization curve of is nowhere below that of .
Let be arbitrary. Then the following statements are equivalent:
;
for all absolute polarization indicators that satisfy , and .
In proving the theorem we assume that is odd. A similar proof will run for the case when is even.
Suppose that . Define . Since absolute polarization curve is invariant under equal absolute change in all incomes, the absolute polarization curve of is same as that of . Therefore, is same as . Note also that .
Now, by we have
Which in view of implies that
where . (14) means that is weakly majorized by . This is equivalent to the condition that , where is doubly superstochastic matrix of order (Marshall and Olkin, 1979). Now, is doubly superstochastic if there exists a bistochastic matrix of order such that for all , . Hence , which means that the i coordinate of is greater than or equal to the corresponding coordinate of . Since all incomes are ordered, cannot be a permutation of except itself. Relationship between and is then given by exactly one of the following:
, ;
, ;
, ;
, .
Condition means that is obtained from by increasing some incomes above the median. If , then is deducible from by a sequence of transfers, transferring income from rich to poor (note . Thus, is the condition that is obtained from by some progressive transfers in the region above the median. If holds, then is obtained from by a sequence of spread increasing movements away from the median and a sequence of progressive transfers for persons above the median.
From we can next establish that is related to by either one of the condition that parallel or . When , holds along with . Then by translation invariance of , . If , then one of the () and one of the four (first three) conditions showing relationship between and must hold simultaneously. This means that in either case, the overall distribution can be acquired from the corresponding distribution through spread augmenting movements away from the median and/or some equalizing transfers on the same side of the media. Since satisfies and , we have . Note that is symmetric because it has been defined on ordered distribution. As is absolute index, . Hence .
If , we define . Then the absolute polarization curve of and coincide so that is same as . Furthermore . The rest of the proof is analogous to the employed earlier and hence omitted.
Consider the polarization indicator
This indicator satisfies , and . It is also an absolute index. Thus, for , which in turn implies that .
Theorem Theorem12b indicates that an unambiguous ranking of income distribution over a given population size by all symmetric, absolute polarization indices satisfying and can be obtained if and only if their absolute polarization curves do not intersect.
Cross-population comparisons
Interpopulation comparisons of polarization usually involve different population sizes, as do intertemporal comparisons. For ranking distributions with different population sizes assume that polarization indices satisfy Population Principle () that is, for all , where , each . Then we have the following theorem.
Let , be arbitrary. Then the following conditions are equivalent:
, where () is an absolute polarization index that satisfies , , and .
In order to rank distribution by relative polarization indicators, define relative polarization curve of as the plot of against , and of against , . Then we can have counterparts to theorem Theorem12b and Theorem13.
, where , , , is an example of an absolute polarization indicator.
A relative indicator can be , where , , .
is a relative polarization indicator, where is the Gini coefficient of . By multiplying it with , we get an absolute indicator (Wolson index).
A person's feeling of deprivation arises out of the comparison of his situation with those of better off persons. The notion of individual deprivation originating in the work of Runciman Note_11 accommodates the view that an individual's assessment of a social state depends on his situation compared with the situations of individuals more favorably treated than him. He argued that the deprivation of an individual in a social state is the extent of the difference between the desired situation and that of the person desiring it.
Thus, the absolute deprivation felt by an individual with income relative to person's income , , can be taken as their income differential, as a fraction of :
increasing in and decreasing in .
Now, an individual with income is deprived of all higher incomes . Therefore, the total absolute deprivation felt by this person is
By definition, the most well off person is never deprived, and the deprivation of the worst off person is maximal.
We say that there is no more absolute deprivation in than in , which we write iffor all .
Instead of focussing on absolute income differentials, we may view deprivation as arising from relative losses and letmeasure the relative deprivation of individual in situation . We will say that there is more relative deprivation in than in , which we write if for all .
and possess many interesting properties:
They are decreasing in .
They are independent of incomes smaller than .
An increase in any income higher than increases them.
They decrease under a rank preserving income transfer from someone with income higher than to someone with income lower than .
They remain unaffected when a rank preserving income transfer takes place between two persons with incomes higher or lower than .
While is homogeneous of degree zero in all incomes, is translation invariant.
While decreases under equal absolute augmentation in all incomes, increases or decreases on multiplication of all incomes by according as or .
In section 6 we showed that absolute differential ordering is a sufficient condition for Lorenz ordering if mean income is fixed. We have a similar theorem for .
The following statement is true but not its converse unless : for all , implies , where is the strictly positive part of .
Suppose , which by definition is equivalent to for . A sufficient condition for (15) to hold is that every term within brackets on the right hand side is greater than or equal to the corresponding term on the left hand side, which "??on??" decomposition is equivalent to
Now, a sufficient condition for (16) to hold is that , for all , which is since as .
To see that the converse is not true, take , where , . Then but not .
To develop a sufficient condition for , we define the relative differentials ordering as follows. We say that situation dominates in relative differentials which we write iffor all . Thus, the ratio of incomes arranged in nondecreasing order must be less or equal to in than in .
The following statement is true but not its converse unless : for all ,, implies , where is the strictly positive part of .
Suppose , which is equivalent to
for . A sufficient condition for (17) to hold is that
This is verified as soon as , , which is .
To prove the converse take again , , where . Then but not .
the following theorem shows the relation between absolute deprivation ordering and Lorenz ordering.
Let , with be such that . Then for , we have which we rewrite aswhich in view of , gives . Next, for , we get from which we have
Since , adding () on the left hand (right hand) side of (18), we get which gives
Assume that the result holds for all . We prove it for all . Then we havefor all . now from we getwhich gives
Therefore,
Adding times the left hand side of (20) for with the left hand side of (21) and the corresponding expression of the right hand side of (20) with the right hand side of (21), we getwhich on cancellation of from both sides along with (20) is what we wanted to show.
Thus the result holds for all , since is arbitrary it follows that .
To show that the converse is not true, take and . Then holds but not .
We now wish to develop an ordering consistent with the ordering . For this we say that given , , is obtained from by a mean preserving transformation if , . (Note that the progressive transfer principle is a mean preserving transformation).
Person is called a donor, recipient on unaffected according as is positive, negative or zero. Since , there is at least one donor and one recipient. Note also that . A mean preserving transformation is called globally equity oriented if for each , , , where is the set of persons in whose incomes are at least as large as , and is equal to the number of persons in . To explain this, let . Then globally equity orientation demands that the amount received by person under the mean preserving transformation is not smaller than the average receipt of persons richer than him.
A similar explanation holds for we will say that a deprivation indicator is globally equity oriented if , where is obtained from through a mean preserving globally equity oriented transformation. We can now prove the following theorem.
Let , , where , be arbitrary, then the following conditions are equivalent:
for all that satisfy symmetry and globally equity orientation.
Suppose . Then which on rearrangement gives from which we get
This means that the mean preserving transformation is globally equity oriented. Hence . is symmetric because it is defined on ordered distributions.
To prove the converse consider the deprivation indicator
This indicator satisfies symmetry and global equity orientation. Hence for , which in turn implies .
The following theorems are easy extensions of theorem Theorem16.
Let , be arbitrary. Then the following conditions are equivalent:
for all absolute deprivation indices that satisfy symmetry and global equity orientation.
Let , be arbitrary. Then the following conditions are equivalent:
for all absolute deprivation indices () defined on () that satisfy symmetry and global equity orientation and population principle.
The absolute deprivation curve associated with the distribution is defined as the plot of against the cumulative proportion of population , where . If incomes are equally distributed, then there is no feeling of deprivation by any person. In this case the curve coincides with the horizontal axis.
To study the slope and curvature of the absolute deprivation curve rigorously, suppose that the income distribution is represented by a distribution function . Then for any arbitrary income , the absolute deprivation function becomes .
We observe that
Therefore, the curve is decreasing.
Since may be positive, zero or negative, no definite conclusion can be drawn regarding the curvature of curve.
Then we can restate theorem Theorem16 in terms of two curves. Restatements of theorem Theorem17, Theorem18 and Theorem19 are also possible.
Recall the definition of :
We regard the complement of to as the satisfaction function of the person with income . The function is
We can interpret as follows. Since person (in the ordered distribution ) is not deprived of incomes , he may be regarded as being satisfied if he compares hi income with these lower incomes. Therefore, the first term in is based on the truncated distribution . Next, since this person is deprived of the incomes , to eliminate his feeling of deprivation about these incomes we replace each of these incomes by . Thus, this approach ignores actual information on incomes of persons richer than but counts them in with his income level .
Given that in addition to person there are persons with income and since all these persons are treated identically, we simply add the term to for arriving at . Therefore, for each , is based on the censored income distribution .
possesses the following properties:
It is increasing in . (Note that since are ordered, only rank preserving increments are allowed.)
It does not get affected by a rank preserving increase in any income greater than .
It increases if any income lower than increases.
A rank preserving income transfer from someone with income greater than to someone with income less than increases .
A rank preserving transfer of income between two persons on the same side of does not change .
It is linearly homogeneous in income.
It increases under equal absolute increase in all income.
We say that there is no less satisfaction in than in , which we write , iffor all .
The following theorem shows that the satisfaction ordering is a sufficient condition for the weak generalized Lorenz ordering.
Let be arbitrary. Then implies, but is not implied by .
means
for all . Putting in (23) we get .
Thus,
hold for . Assume that () holds for all . That is,
for all . We show that it holds for all . The ordering for gives
Assume in (25) and multiply both sides of the resulting expression by to get
Adding the left-hand (right-hand) side of (26) to the left-hand (right-hand) side of (27) we get
which gives
Thus,
holds for all . Since is arbitrary, it follows that holds.
To see that the converse is not true, take and . (in fact, ) holds but not .
If the income distribution is represented by a distribution function , then for any income , the satisfaction function becomes
Thus, the satisfaction curve of which plots against , , is increasing. Given that , no definite conclusion can be drawn regarding curvature of curve.
If is doubly stochastic and nonsingular, then satisfies , .
implies and implies .
Doubly stochastic matrices need not to be nonsingular. Note if a doubly stochastic matrix is nonsingular and is its inverse, need not have nonnegative elements, so that may not be doubly stochatic.
If and are both doubly stochastic, then is a permutation matrix.
If is a doubly stochastic such that , then is a permutation matrix.