Introduction and Motivation

There are intuitive appeals to the notion that the components of a vector $x$ are "less spread out" or "more nearly equal" then the components of a vector $y$. For instance, we may be interested in examining whether the coefficient of variation or another similar indicator is lower for a distribution of income $x$ than another distribution of income $y$. 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 "$x$ is majorized by $y$".

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.

Notion, Definition and Preliminaries

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 MATH where $x_{i}\geq 0$ is the income of person $i$, with (the strict inequality) $>$ for at least one $i$, $1\leq i\leq n$.

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 $n\geq 1$, $n\in \U{2115} $, the set of all income distributions is $D^{n}$, the nonnegative orthant of the Euclidean n-space $\U{211d} ^{n}$ with the origin deleted, where $\U{2115} $ is the set of positive integers. Thus, for any $x\in D^{n}$, MATH.

Throughout the presentation we will adopt the following notation. For any two elements $x,y$ of $D^{n}$, the coordinate-wise ordering $x_{i}\leq y_{i}$, $i=1,2,...,n$ is denoted by $x\leq y$. For any $x\in D^{n}$ write MATH for increasing rearrangement of $x$, that is, MATH. Similarly, let MATH denote the components of $x$ in decreasing order, and let MATH denote the decreasing rearrangement of $x$.

Definition

A function MATH is concave if

MATH

for all MATH and for all $x,y\in D^{n}$, $x\neq y$.

Remark

The superscript "n" in $f^{n}$ 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.

Definition

A function MATH is (strictly) convex if $-f^{n}$ is (strictly) concave.

Definition

A function MATH is quasiconcave if

MATH

for all MATH and for all $x,y\in D^{n}$, $x\neq y$.

Observe that quasiconcavity is a weaker condition than concavity. Any (strictly) concave function will be (strictly) quasiconcave, but not vice-versa. For instance, MATH defined by $f^{1}=z^{3}$ is quasiconcave but not concave. for a two coordinated strictly quasiconcave function, definition 3 tells that if MATH but $x\neq y$ so that $x$ and $y $ are on the same contours then MATH, MATH.

That is, the line joining $x$ and $y$ lies entirely above the contour containing $x$ and $y$. 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.

Definition

A function MATH is (strictly) quasiconvex if $-f^{n}$ is (strictly) quasiconcave.

Definition

A square matrix of order $n$ 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 $n\times n$ matrix MATH is bistochastic if

MATH

and

MATH

We rewrite the two above conditions in a more compact form as:

MATH

where MATH is the n-coordinate vector of ones. Thus, 1 is a characteristic root of $P$ corresponding to the characteristic vector $e$.

Proposition

If the matrices $P_{1}$ and $P_{2}$ are bistochastic, then the product $P=P_{1}P_{2}$ is bistochastic.

Proof

Since $P_{1}$ and $P_{2}$ have nonnegative elements, one sees directly from matrix product that $P$ also has nonnegative elements. Also MATH

Definition

A matrix is a permutation matrix if it is bistochastic and has exactly one positive entry in each row and each column.

Example

MATH is bistochastic matrix but not a permutation matrix.

MATH is a permutation matrix.

Definition

A function MATH is symmetric if MATH for all $x\in D^{n}$, where $P$ is a permutation matrix of order $n$.

Example

The function MATH is a symmetric function

Example

MATH (symmetric)

MATH (not symmetric)

MATH where $a\neq b\neq c$ (not symmetric)

Definition

A function MATH is S-concave if MATH for all $x\in D^{n}$, where $B$ is any bistochastic matrix of order $n$. For strict S-concavity of $f^{n}$, the weak inequality is to be replace by a strict inequality whenever $xB$ is not a permutation of $x$.

Definition

A functionMATH is called (strictly) S-convex if $-f^{n}$ (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.

Example

$(i)$ If $p_{i}\geq 0$, $i=1,2,...,n$ and MATH the function:

MATH

is called entropy of $p$. $H\left( p\right) $ is strictly s-concave.

$(ii)$ The function MATH is strictly S-concave on $D_{+}^{n}$, the strictly positive part of $D^{n}$.

$(iii)$ The function MATH, $r<1$, $r\neq 0$ is strictly S-concave on $D_{+}^{n}$.

$(iv)$ The functions MATH and MATH are S-concave, but not strictly so.

All these examples are illustrations of the following proposition:

Proposition

If MATH is an interval and $g$ is (strictly) concave then

MATH

is (strictly) S-concave on $I^{n}$.

The examples given above are all additive. Two nonadditive examples of strictly S-concave functions are the following:

Example

$(v)$ MATH

$(vi)$ MATH where MATH.

The Lorenz Domination

Consider a population of $n$ individuals where $x_{i}$ is the income of person $i$, $i=1,...,n$. Order the individuals from the poorest to the richest to obtain MATH. Now plot the points MATH, where $k=0,1,...,n$, $S_{0}=0$ and MATH. MATH is the total income of the poorest $k$ persons under $x$ in the population. Join these points by line segments to obtain a curve connecting the origin with the point $\left( 1,1\right) $. The curve is called the Lorenz curve of the distribution $x$. 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 MATH represent the income of $n$ individuals similarly, let MATH be another distribution. Than according to the idea of Lorenz MATH is at least as equal as MATH if

MATH

for all $k=1,2,...,n$. Equivalently we say that $x$ weakly Lorenz dominates $y$. That is, $x$ Lorenz dominates $y$ in the weak sense if the Lorenz curve of $x$ is nowhere below that of $y$. We say that $x$ strictly Lorenz dominates $y$, what we write $xLy$, if (L) holds with the additional restriction that there will be strict inequality for at least some $k<n$. $xLy$ means that $x$ is more equal than $y$. $xLy$ is the condition that the Lorenz curve of $x$ is nowhere below that of $y$ and at some places (at least) it is above the Lorenz curve of $y$. If $xWLy$ ($xLy$) holds with the additional restriction that MATH, then we say that $x$ is (strictly) majorized by $y$.

In figure fig2 both the distributions $z$ and $y$ are strictly Lorenz dominated by distribution $x$. (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 $y$ and $z$ 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.


Figure

In order to interpret the Lorenz curve ordering from alternative perspectives, the following definition is also needed.

Definition

For any $y\in D^{n}$, we say that $x$ is obtained from $y$ through a progressive transfer if

MATH

where $c>0$.

That is, $x$ and $y$ are identical except for a positive transfer of income from the rich person $i$ to the poor person $i$.

A remarkable consequence of the strict Lorenz domination relation was proved by Dasgupta, Sen and Starrett.

Theorem (Dasgupta, Sen and Starrett, 1973)

Let $x$ and $y$ be two distributions of the same amount of total income over a given population size, that is, $x,y\in D^{n}$, where MATH. Then the following conditions are equivalent:

  1. $xLy$

  2. MATH for all strictly S-concave social welfare functions MATH.

What theorem 1 tells is that if the Lorenz curves of the distributions $x$ and $y$ 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 $W^{n}$ as a social welfare function because it is defined on income distributions of the society and, as we show below, the value of $W^{n}$ 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 $x,y\in D^{n}$, where MATH, the following conditions are equivalent Note_1 :

  1. For all $k\leq n$, MATH, with at least one $k<n$ such that MATH ($xLy$)

  2. If $\overline{x}$ is not a permutation of $\overline{y}$, there exists a bistochastic matrix $Q$ such that MATH

  3. For any strictly concave real valued $U$, MATH

  4. $\overline{y}$ can be transformed into $\overline{x}$ by a finite sequence of transformations of the form

    MATH

    for $i<j$ and $c^{\alpha }>0$, with MATH if $k\neq i,j$.

Since MATH, the first condition in the Hardy, Littlewood, Polya theorem means $xLy$.

Condition $(b)$ says that each income in $\overline{x}$ is a mixture of the income on $\overline{y}$. Condition $(c)$ means that $x$ is regarded as more equal than $y$ by the symmetric utilitarian social welfare functions as long as the identical utility function $U$ is strictly concave. finally, condition $(d)$ tells us that the distribution $\overline{x}$ is obtained from the distribution $\overline{y}$ by a finite sequence of rank preserving transformations MATH transferring income from the rich to the poor.

Proof of theorem 1

MATH

Suppose $xLy$. Now by the Hardy, Littlewood, Polya theorem $xLy$ implies that there must be a bistochastic matrix $Q$ such that MATH. It follows that for any strictly S-concave MATH MATH. Since an s-concave function is symmetric, we have MATH and MATH.

MATH

Since in the Hardy, Littlewood, Polya theorem conditions $(a)$ and $(c)$ are equivalent, not $(a)$ implies not $(c)$. That is, if not $xLy$, then for some strictly concave $U$, MATH.

Now, given strict concavity of $U$, we can regard MATH as a strictly S-concave function. Therefore, it followed that if not $xLy$, then for some strictly S-concave real valued $W^{n}$, MATH. That is, not $(i)$ implies not $(ii)$. Hence MATH.

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 $k$ times, then the aggregate welfare of the replicated distribution is simply $k$ times the welfare of the actual distribution.

Axiom (Symmetry Axiom for Population (SAP))

For all $x\in D^{n}$, MATH, where MATH, each $x^{i}=x$.

Theorem (Dasgupta, Sen and Starrett)

Set $y^{1}$ and $y^{2}$ be two distributions of income with the same mean over population sizes $n_{1}$ and $n_{2}$ respectively. That is, $y^{1}\in D^{n_{1}}$ and $y^{2}\in D^{n_{2}}$, where MATH. Then the following statements are equivalents:

  1. $y^{1}Ly^{2}$

  2. MATH, where $W^{n_{i}}$ is any arbitrary social welfare function that satisfies strict S-concavity and SAP, and whose domain is $D^{n_{i}}$, $i=1,2$.

Proof

MATH

Let $y^{3}$ ($y^{4}$) denote the $n_{2}$ ($n_{1}$) fold replication of $y^{1} $ ($y^{2}$). Clearly, $y^{3}$, MATH, MATH. Now, the Lorenz curve is population replication invariant. Consequently, $y^{3}$ and $y^{4}$ have the same Lorenz curves as $y^{1}$ and $y^{2}$ respectively.

Therefore, $y^{1}Ly^{2}$ implies that $y^{3}Ly^{4}$. Hence by theorem 1, for all strictly S-concave social welfare functions $W^{n_{1}n_{2}}$, MATH. But the welfare function satisfies SAP. Therefore, MATH and MATH. Hence MATH.

MATH

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 $m$ strictly S-concave welfare functions, we have to make $\binom{m}{2}$ comparisons, and the value of $m$ is unboundly large. However, if condition $(ii)$ holds, that is, if $x$ strictly Lorenz dominates $y$, or $x$ is strictly majorized by $y$, then we can be sure that $x$ will have a higher value than $y$ for any strictly S-concave social welfare function. Computation of values of individual functions is not necessary. A similar explanation holds for theorem 2.

The Generalize Lorenz Domination

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 $x\in D^{n}$ has both a higher mean and a higher Lorenz curve than another distribution $y\in D^{n}$, then social welfare under $x$ is higher than under $y$. This will be true if MATH 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 $x\in D^{n}$ is a plot of MATH against $\frac{k}{n}$, $k=0,1,...,n$, and at $k=0$ the ordinate of the curve is $0$.

We say that an income distribution $x\in D^{n}$ weakly dominates another distribution $y\in D^{n}$ in the generalized Lorenz sense if

MATH

for all $k=0,1,...,n$. $x$ strictly generalized Lorenz dominates $y$, what we write $xGLy$, if (GL) holds with the additional restriction that there will be strict inequality for at least one $k\leq n$. Thus, $xGLy$ means that the generalized Lorenz curve of $x$ is nowhere below and at some places (at least) it is strictly inside that of $y$.

The following theorem of Shorrocks (1983) explains the role of generalized Lorenz curves in ranking distributions with different means.

Theorem (Shorrocks)

Note_2

Let $x$ and $y$ be two distributions of income over a given population size $n$, that is, $x,y\in D^{n}$. Then the following conditions are equivalent:

  1. $xGLy$

  2. MATH for all increasing, strictly S-concave social welfare functions MATH.

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 $x,y\in D^{n}$, the following conditions are equivalent:

  1. $x$ is strictly supermajorized by $y$, that is,

    MATH

    for all $k\leq n$, with strict inequality for at least one $k$.

  2. For any strictly concave, increasing real valued $U$,

    MATH

  3. There exists a doubly stochastic matrix $D$ such that MATH, that is, the i$^{\text{th}}$ coordinate of $\overline{x}$ is $\geq $ i$^{\text{th}}$ coordinate of $\overline{y}D$, with $>$ for at least one $i$.

The first condition in the Marshall Olkin theorem is that $xGLy$. According to condition $(b)$, the distribution $x$ is regarded as better than distribution $y$ by the symmetric utilitarian social welfare function, where the utility function is strictly concave and increasing. Finally, condition $(c)$ shows how incomes in $\overline{y}$ can be transformed into incomes $\overline{x}$.

Proof of theorem 3

MATH

Suppose $xGLy$. now, by the Marshall and Olkin theorem there exists a doubly stochastic matrix $D$ such that MATH, that is, the i$^{\text{th}}$ coordinate of $\overline{x}$ is at least as large as the i$^{\text{th}}$ coordinate of $\overline{y}D$, with strict inequality for at least one $i$. Let MATH be strictly S-concave and increasing. Then by increasingness of $W^{n}$, MATH. Again strict s-concavity of $W^{n}$ implies that MATH.By symmetry of $W^{n}$, we have MATH and MATH. thus $xGLy$ implies Note_3 that MATH

MATH

Since in the Marshall and Olkin theorem, conditions $(a)$ and $(b)$ are equivalent, not$(a)$ implies not$(b)$. Thus, if not $xGLy$, the for some increasing, strictly concave $U$, MATH.

Now, given increasingness and strict concavity of $U$, we can regard MATH as an increasing, strictly S-concave function. Thus it follows that if not $xGLy$, then for some increasing, strictly S-concave real valued $W^{n}$, MATH. That is, not $(i)$ implies not $(ii)$. Hence $(ii)$ implies $(i)$.

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, $x^{3}$ is preferred to both $x^{1}$ and $x^{2}$. The larger mean of $x^{3}$ 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.


Figure
$x^{3}GLx^{2}$, $x^{3}GLx^{1}$

We can extend theorem 3 to the variable population case. For this we assume additionally that the social welfare function satisfies SAP.

Theorem (Shorrocks)

Note_4

Let $y^{1}$ and $y^{2}$ be two income distributions over population sizes $n_{1}$ and $n_{2}$ respectively, that is, $y^{1}\in D^{n_{1}}$ and $y^{2}\in D^{n_{2}}$.

Then the following statements are equivalent:

  1. $y^{1}GLy^{2}$

  2. MATH, where $W^{n_{i}}$ is any real valued social welfare function that satisfies increasingness, strict S-concavity and SAP, and whose domain is $D^{n_{i}}$, $i=1,2$.

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 MATH satisfies scale improvement condition if for all $x\in D^{n}$ for any $k\geq 1$,

MATH

In (SIC) total income increases more as $k$ increases and the distribution of relative incomes MATH 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:

Theorem (Shorrocks)

For $x,y\in D^{n}$, the following conditions are equivalent:

  1. MATH and $x$ weakly Lorenz dominates $y$ ($xWLy$ for short)

  2. MATH for all social welfare functions MATH that satisfy scale improvement condition and S-concavity.

Proof

MATH

Suppose MATH and $xWly$. Define MATH. Since $W^{n}$ satisfies the scale improve condition MATH. By construction, the Lorenz curve of $y $ and $y^{\prime }$ coincide Note_5 . Hence $xWLy$ implies $xWLy^{\prime }$. But we also have MATH. Therefore, by S-concavity of $W^{n}$, we get MATH. Hence Note_6 MATH.

MATH

Let us define MATH by MATH where $m\geq 0$ and $f$ is S-concave on $D^{n}$.

Clearly, $W^{n}$ satisfies the scale improvement condition and S-concavity. Now, for $x,y\in D^{n}$ MATH implies MATH. by choosing MATH. MATH implies MATH. But $f$ is arbitrary S-concave Note_7 function and the distributions MATH and MATH have the same mean (equal to $1$). Hence by the weak version of theorem 1, MATH, from which we get $xWLy$.

An alternative to scale improvement condition (SIC) is incremental improvement condition. Formally, MATH satisfies incremental improvement condition if for all $x\in D^{n}$, for all $k\geq 0$,

MATH

Thus, as the value of $k$ increases in (IIC) total income increases, and the distribution of absolute income differentials $x_{i}-x_{j}$ 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 $x_{i}-x_{j}$ get widened.

Theorem (Shorrocks)

For $x,y\in D^{n}$, the following conditions are equivalent:

  1. MATH and MATH for all $k\leq n$ where MATH is the height of the generalized Lorenz curve of the distribution $z$ at population proportion $\frac{k}{n}$.

  2. MATH for all social welfare functions MATH that satisfy S-concavity and the incremental improvement condition.

Proof

MATH

Define MATH. Since $W^{n}$ satisfies the incremental improvement condition MATH. Furthermore, MATH. MATH for any $\frac{k}{n}$. this shows that MATH for all $k\leq n$, where MATH is the height of the Lorenz curve at the population proportion $\frac{k}{n}$. Hence by S-concavity of $W^{n}$, MATH, which shows Note_8 that MATH.

MATH

Suppose that MATH is given by MATH, where $m\geq 0$ and $\theta >0$ is such that MATH.

$f$ is assumed to be S-concave on $D^{n}$. Clearly $W^{n}$ defined above is S-concave and meets the incremental improvement condition. Now, for $x,y\in D^{n}$, MATH implies MATH. By choosing MATH, MATH implies MATH. Alternatively by choosing $m=0$, we get MATH. Since the distributions MATH and MATH have the same mean $\theta $, and $f$ is any S-concave function we conclude that MATH for all $k\leq n$. Now, in the case of distributions with common mean the generalized Lorenz ranking. The above inequality is therefore same as MATH for all $k\leq n$. the result now follows from this inequality.

Thus Theorem Theorem7 can be used to test whether any pair of income distributions $x$ and $y$ can be ranked using social welfare functions satisfying (IIC) and S-concavity. One way of doing the comparison is to calculate MATH 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 MATH as the absolute Lorenz curve of $x$. The absolute Lorenz curve coincides with the horizontal axis when all incomes are equal. The absolute Lorenz curve MATH of $x$ is decreasing with $k$ for MATH and increasing for $k^{\ast }<k\leq n$ where $k^{\ast }$ is such that MATH.

It is easy to see that MATH measures the vertical distance between the line of equality $a\left( x\right) $ and the MATH at the population proportion $\frac{k}{n}$. Thus, it represents the income shortfall of the bottom $\frac{k}{n}$ proportion of the population, expressed as a fraction of the aggregate population size, from the equal income distribution $a\left( x\right) $.

Theorem Theorem7 can now be restated as:

Theorem 6*: Let $x,y\in D^{n}$ be arbitrary. Then the following conditions are equivalent:

  1. MATH for all MATH that are S-concave and satisfy the incremental improvement condition.

  2. MATH and MATH for all $k=0,1,...,n$.

The Complaint Ordering

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.

Definition

For any $x\in D^{n}$, define the cumulation of complaints

MATH

This concept can be used to draw the cumulative complaint contour ($CCC$) of a distribution $x$, formed by joining the points MATH. The $CCC$ is increasing and concave. If MATH lies on or above MATH then distribution $x$ does not exhibit less complaint inequality than distribution $y$. Formally,

For any $x,y\in D^{n}$, $x$ weakly dominates $y$ by complaint inequality if MATH for all $k=0,1,...,n$.

This ordering is related to the weak generalized Lorenz ordering.

Theorem (Cowell and Ebert, 2002)

Let $x,y\in D^{n}$ be arbitrary. Then the following statements are equivalent:

  1. $x$ dominates $y$ by complaint inequality

  2. MATH weakly generalized Lorenz dominates MATH

Proof

MATH

Suppose $x$ dominates $y$ by complaint inequality. Then

MATH

This is equivalent to

MATH

which means that MATH.

MATH

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.


Figure

The Absolute Differential Ordering

The quasi ordering we consider next is more demanding as it requires that all pairwise inequality be less in distribution $x$ than in distribution $y$ for the former to be declared as less unequal than the latter. We say that distribution $x$ dominates distribution $y$ in absolute differentials, which we write $xADy$ if

for all $i=1,2,...,n-1$.

This simply means that the differences, between two individual incomes taken in nondecreasing order are less in $x$ than in $y$.

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.

Theorem

Let $x$ and $y$ be two distributions of a given total over a given population size $n$, that is $x,y\in D^{n}$, where MATH, are arbitrary. Then $xADy$ implies $xWLy$, but the converse is not true, unless $n=2$.

Proof

Given $x,y\in D^{n}$ with MATH, suppose $xADy$ holds. Note that we can rewrite condition (AD) as

MATH

for $i=1,2,...,n-i$. Since MATH, that is, MATH, and since MATH is nonincreasing in $i$ (from (6)), there exists $k$, $1\leq k\leq n$, such that MATH for $i=1,2,...,k$, MATH, $i=k+1,...,n$.

Therefore MATH, $j=1,2,...,k$ and MATH, $j=k+1,...,n$. Since MATH, the last $\left( n-k\right) $ inequalities can be rewritten as MATH, $j=k+1,...,n$. Thus, for each $k$, the sum of the $k$ largest of the $\overline{x}_{i}$ is dominated by MATH. Hence $xWLy$ to prove that the converse implication is false for $n>2$, take MATH, MATH. We have $xLy$ but $xADy$ does not hold.

Hence $AD$ is sufficient but not necessary for $WL$.

Stochastic Dominance

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 $F$ defined on MATH Note_9 . $F\left( t\right) $ is the proposition of population with income less than or equal to $t$. MATH, MATH and $F$ is increasing. $F$ is assumed to be continuously differentiable. The continuous function $f$, the derivative of $F$, is the income density function.

Definition

Given two income distribution functions $F$ and $G$ defined on $I$, we say that $F$ is preferred to $G$ by the first order stochastic dominance criterion if

MATH

for all $t\in I$, with strict inequality $<$ for at least one $t\in I$.

Let $W_{1}$: Set of all continuously differentiable increasing utility functions on $I$. That is, MATH

Theorem

$F$ is preferred to $G$ by the first order stochastic dominance criterion if and only if

MATH

for all $U\in W_{1}$.

Proof

By definition

MATH

Integrating by parts, we get

MATH

given that MATH for all $t$ and MATH for all $t$, with strict inequality $<$ for at least one $t$, we must have

MATH

The proof of the converse is by contradiction. Suppose that exists $x_{k}\in I$, $x_{k}>x_{1}$ such that MATH for MATH with $>$ for at least one $t$, and MATH for MATH.

Consider the following utility function

MATH

where $a>b>0$ are arbitrary.

Observe continuity of $U$ at $x_{k}$. Now,

MATH

$U^{\prime }$ is continuous at $x_{k}$. Note also positivity of $U^{\prime }$.

Then for this utility function by virtue of (7)$.$

MATH

Now MATH.

Therefore MATH is positive. Similarly,

MATH

Note next that MATH.

Therefore, we can choose $a$ 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 $U$ for which MATH becomes smaller than MATH. Hence the result.

Interpretation: We can rewrite first order stochastic dominance alternatively as

Proof

MATH

for all $t\in I$ with $>$ for at least one $t$. Thus, the cumulative proportion of persons with income greater than or equal to $t$ is not lower under $F$ than under $G$, and for at least one $t$ this inequality is strict. Therefore preference for higher income is than main distinguishing characteristic for first order dominance. Hence the theorem says that $F$ is preferred to $G$ by the first order stochastic dominance criterion if and only if $F$ is regarded as better than $G$ by two utilitarian rule for any increasing utility function.

Definition

Given two income distributions $F$ and $G$ defined on $I$, we say that $F$ is preferred to $G$ by the second order stochastic dominance criterion if

MATH

for all $t\in I$ with $<$ for at least one $t\in I$.

Let MATH

That is, $W_{2}$ is the set of all increasing, strictly concave and twice continuously differentiable utility functions defined on $I$.

Theorem

Given two income distribution functions $F$ and $G$ defined on $I$, $F$ is preferred to $G$ by the second order stochastic dominance criterion if and only if

MATH

for all $U\in W_{2}$.

Proof

Integration by parts yields

MATH

From proof of theorem 9, we know that

MATH

From (9) and (10) we get

MATH

Now, given that MATH for all $t\in I$, with $<$ for at least one $t\in I$, it must be the case that MATH. Further, given $U^{\prime }>0$, MATH, the sum of the two expressions on the right hand side of (11) is positive. Hence

MATH

To prove the converse, let

MATH

Suppose that there exists $x_{k}\in I$, $x_{k}>x_{1}$ such that MATH for MATH with $>$ for at least one $t$, and MATH for MATH.

Now, consider the utility function

MATH

Note continuity if $U$ at $x_{k}$. Now

MATH

Note that $U^{\prime }>0$ and also observe continuity of $U^{\prime }$ at $x_{k}$. (MATH for MATH is given by MATH

MATH

MATH and $U^{\prime \prime }$ is continuous at $x_{k}$.

Then for the utility function considered above, we have by virtue of (11),

MATH

The first term on the right hand side of (12) is nonnegative, while the second term is negative. Observe that MATH is decreasing in $a$. 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 $W_{2}$ for which MATH becomes negative. Hence the result.

Theorem 10 demonstrates that $F$ is preferred to $G$ if and only if $F$ is preferred to $G$ 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 $G$.

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 MATH. $F$ is the income distribution function. The mean income is MATH

The proportional share of total income enjoyed by "agents" with income less than or equal yo MATH is MATH

The derivative of $F_{1}$ with respect to $t$ isMATH

The positivity of this derivative implies that like $F$, $F_{1}$ rises at $t$ rises from $x_{1}$ to $x_{n}$. Also MATH, MATH. $F_{1}$ 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 $F_{1}$ against $F$, as $t$ rises from $x_{1}$ to $x_{n}$. Often we can eliminate $t$ between them and can express $F$ explicitly as a function of $F$. Since MATH, using (13) we getMATH

which is positive for positive incomes.

AgainMATH

Which is also positive. Therefore, the Lorenz curve is increasing and concave. Note the generalized Lorenz curve if $F$ is simply the plot of MATH against $F\left( t\right) $, as $t$ goes from $x_{1}$ to $x_{n}$.

We can now state equivalence of theorem 10 with generalize Lorenz domination rigorously as follows.

Theorem

Given two income distribution functions $F$ and $G$ defined on $I$, the following conditions are equivalent:

  1. MATH

    for all $t\in I$, with $<$ for at least one $t$.

  2. MATH

    for all increasing, strictly concave utility function MATH.

  3. MATH

    for all $t\in I$, with $>$ for at least one $t\in I$.

The following result drops out as a corollary to Theorem Theorem11.

Corollary

Let $F$ and $G$ be two income distributions on I such that MATH. Then the following conditions are equivalent:

  1. MATH

    for all strictly concave utility function MATH.

  2. MATH

    for all $t\in I$, with $>$ for at least one $t\in I$, $t<x_{n}$.

The stochastic dominance conditions defined above can be extended to the n$^{\text{th}}$ degree stochastic dominance rule. Of two income distribution functions $F$ and $G$ defined on $I$, the former is said to dominate the latter by the n$^{\text{th}}$ degree stochastic dominance rule if

MATH

for all $t\in I$, with $<$ for at least one $t$, whereMATH

and MATH. We can now have the following theorem:

Theorem

Let $F$ and $G$ be two income distribution functions on the interval $I$. Then the following conditions are equivalent:

  1. $F$ dominates $G$ by the n$^{\text{th}}$ degree stochastic dominance rule.

  2. MATH

    for all $U\in W_{n}$, where MATHwhere $U$ is continuously differentiable $n$ times Note_10 , all odd order derivatives of $U$ through $n$ are positive, and all its even order derivatives are MATH.

For $n=3$ condition $\left( ii\right) $ in Theorem Theorem12a means that $F$ is preferred to $G$ by the utilitarian rule that approves of equity (MATH), efficiency ($U^{\prime }>0$) and the diminishing transfers principle that put more emphasis to income transfers at the lower end of the distribution (MATH).

In order to illustrate the idea of stochastic dominance numerically, let $x\in D^{m}$. Then MATH

for all $z\in \U{211d} ^{1}$, where MATH is the number of individuals having income less than or equal to $z$ in $x$.

Example

Let MATH. $F_{1},F_{2}$ and $F_{3}$ are given below.

$F_{1}$ $F_{2}$ $F_{3}$
$-\infty <z<1$ $0$ $0$ $0$
$1\leq z<5$ $\frac{1}{3}$ $\frac{z-1}{3}$ MATH
$5\leq z<6$ $\frac{2}{3}$ $\frac{2z-6}{3}$ MATH
$6\leq z<\infty $ $1$ $\frac{2z-12}{3}$ MATH

Example

Two example of utility functions, that are members of $W_{n}$, are:

  1. MATH, $0<\gamma <1$;

  2. MATH, $\alpha >0$.

Rather than representing a discrete a distribution by means of its distribution function, we may "appeal" to its inverse attribution function. Let MATH denote the inverse associated with the distribution function $F$, whereMATH

For any $p$ in the range of $F$, MATH is defined as the minimum income $z$ in the domain of $F$ such that MATH.

The Lorenz curve associated with the distribution having distribution function $F$ can now be defined asMATH

where $0\leq p\leq 1$.

We can restate the definition of Lorenz domination and generalized Lorenz domination using this definition of the Lorenz curve.

Example

Let MATH. The distribution function $F$ is defined byMATH

The inverse function $F^{-1}$ is defined byMATH

Hence,MATH

We can find the generalized Lorenz curve of the distribution by multiplying MATH by $5$.

Polarization Ordering

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 $T^{n}$ defined on $\overline{D}^{n}$, that is, MATH where all income distributions in $\overline{D}^{n}$ are ordered nondecreasingly. For any MATH, MATH gives the level of polarization associated with distribution $\overline{x}$. For any MATH, we write MATH for the median value of $\overline{x}$. If $n$ is odd, MATH is MATH observation in $\overline{x}$. If $n$ is even the average of the MATH and MATH observations in $\overline{x}$ is taken as the median of $x$. We write $\overline{x}_{-}$ and $\overline{x}_{+}$ for the subvectors of $\overline{x}$ that include $\overline{x}_{i}$ for $i<\overline{n}$ and $i>\overline{n}$ respectively, where $\overline{n}=n+1$. Thus, for any $x\in D^{n}$, MATH if $n$ is even and MATH if $n$ is odd.

For $\overline{x}$, MATH, $\overline{x}$ is said to be obtained from $\overline{y}$ by a simple increment if MATH for some $j$ and MATH for all $j\neq i$. We write MATH to indicate this. Given that $\overline{x}$ and $\overline{y}$ are nondecreasingly ordered, the transformation $C$ allows only rank-preserving increments. Further, if $\overline{x}$ is obtained from $\overline{y}$ by a progressive transfer we write MATH to denote this. Note that only rank-preserving transfers are allowed under the operation $E$.

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 MATH whether relative or absolute, should satisfy the following postulates:

Nondecreasing Spread ($NS$): If MATH, where MATH, are selected through anyone of the following cases,

  1. MATH, MATH;

  2. MATH, MATH;

  3. MATH, MATH,

then MATH.

Nondecreasing Bipolarity ($NB$): If MATH, where MATH are related through anyone of the following cases,

  1. MATH, MATH;

  2. MATH, MATH;

  3. MATH, MATH,

then MATH.

Symmetry: MATH for all permutation matrices of order $n$.

$NS$ 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.


Figure
Nondecreasing Spread

$NB$ 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.

$NB$ explicitly establishes that inequality and polarization are two nonidentical concepts.


Figure
Nondecreasing Bipolarity

$SM$ 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 $SM$ is that a polarization indicator can be defined directly on ordered incomes (as we have done).

To understand these properties more explicity, let MATH. Then MATH, MATH, MATH. Let MATH. Then MATH, MATH, since MATH. Next suppose MATH. Then MATH, MATH. Clearly, MATH because $1$ unit of income has been transferred from the person with $5$ units of income to the one with $2$ income units. Similarly, MATH. 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 MATH, the absolute polarization curve ($APC$) is a plot of MATH against $\frac{k}{n}$, MATH and MATH against $\frac{k}{n}$, MATH.

Note that the ordinate at MATH involves the income level MATH. Now, if $n$ is odd, MATH is one of the incomes in the distribution. However, for even $n$, MATH is not in $x$, we define the ordinate at MATH, since in polarization measurement, the median income is the reference income level. For the distributions MATH, the ordinates of the absolute polarization curve are MATH and the corresponding population proportions are MATH.

For a typical income distribution $\overline{x}$, up to MATH, the absolute polarization curve decreases monotonically, at MATHit coincides with the horizontal axis and then it decreases monotonically. If $x$ is an equal distribution, then the curve becomes the horizontal axis itself. For a distribution $\overline{x}$, where $\overline{x}_{+}$ is unequal but MATH is equal, the curve runs along the horizontal axis up to MATH, and starting from MATH it rises gradually. Similarly, if $\overline{x}$is such that $\overline{x}_{-}$ is unequal but MATHis equal, the curve is decreasing up to MATH after which it runs along the horizontal axis.

Given any two income distributions MATH, $\overline{x}$ is said to weakly dominate $\overline{y}$ with respect to absolute polarization, which we write MATH if the absolute polarization curve of $\overline{x}$ is nowhere below that of $\overline{y}$.

Theorem

Let MATH be arbitrary. Then the following statements are equivalent:

  1. MATH;

  2. MATH for all absolute polarization indicators MATH that satisfy $NS$, $NB$ and $SM$.

Proof

In proving the theorem we assume that $n$ is odd. A similar proof will run for the case when $n$ is even.

MATH

Suppose that MATH. Define MATH. Since absolute polarization curve is invariant under equal absolute change in all incomes, the absolute polarization curve of $\overline{z}$is same as that of $\overline{y}$. Therefore, MATH is same as MATH. Note also that MATH.

Now, by MATH we haveMATH

Which in view of MATH implies that MATH

where MATH. (14) means that $\overline{x}_{+}$ is weakly majorized by $\overline{z}_{+}$. This is equivalent to the condition that MATH, where MATH is doubly superstochastic matrix of order $n-\overline{n}$ (Marshall and Olkin, 1979). Now, $S$ is doubly superstochastic if there exists a bistochastic matrix MATH of order $n-\overline{n}$ such that $s_{ij}\geq m_{ij}$ for all $i$, $j$. Hence MATH, which means that the i$^{\text{th}}$ coordinate of $\overline{x}_{+}$ is greater than or equal to the corresponding coordinate of $\overline{z}_{+}M$. Since all incomes are ordered, $\overline{z}_{+}M$ cannot be a permutation of $\overline{z}_{+}$ except $\overline{z}_{+}$ itself. Relationship between $\overline{x}_{+}$ and $\overline{z}_{+}$ is then given by exactly one of the following:

  1. MATH, MATH;

  2. MATH, MATH;

  3. MATH, MATH;

  4. MATH, MATH.

Condition $\left( a\right) $ means that $\overline{x}_{+}$ is obtained from $\overline{z}_{+}$ by increasing some incomes above the median. If MATH, then $\overline{z}_{+}M$ is deducible from $\overline{z}_{+}$ by a sequence of transfers, transferring income from rich to poor (note MATH. Thus, $\left( b\right) $ is the condition that $\overline{x}_{+}$ is obtained from $\overline{z}_{+}$ by some progressive transfers in the region above the median. If $\left( c\right) $ holds, then $\overline{x}_{+}$is obtained from $\overline{z}_{+}$ by a sequence of spread increasing movements away from the median and a sequence of progressive transfers for persons above the median.

From MATH we can next establish that $\overline{x}_{-}$ is related to $\overline{z}$ by either one of the condition that parallel MATH or MATH. When MATH, MATH holds along with $\left( d\right) $. Then by translation invariance of $T^{n} $, MATH. If MATH, then one of the MATH (MATH) and one of the four (first three) conditions showing relationship between $\overline{x}_{-}$ and $\overline{z}_{-}$ must hold simultaneously. This means that in either case, the overall distribution $\overline{x}$ can be acquired from the corresponding distribution $\overline{z}$ through spread augmenting movements away from the median and/or some equalizing transfers on the same side of the media. Since $T^{n}$ satisfies $NS$ and $NB$, we have MATH. Note that $T^{n}$ is symmetric because it has been defined on ordered distribution. As $T^{n}$ is absolute index, MATH. Hence MATH.

If MATH, we define MATH. Then the absolute polarization curve of $\overline{t}$ and $\overline{x}$ coincide so that MATH is same as MATH. Furthermore MATH. The rest of the proof is analogous to the employed earlier and hence omitted.

MATH

Consider the polarization indicatorMATH

This indicator satisfies $NS$, $NB$ and $SM$. It is also an absolute index. Thus, MATH for $1\leq k\leq n$, which in turn implies that MATH.

Theorem Theorem12b indicates that an unambiguous ranking of income distribution over a given population size by all symmetric, absolute polarization indices satisfying $NS$ and $NB$ 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 ($PP$) that is, for all $x\in D^{n}$, MATH where MATH, each $x^{i}=x$. Then we have the following theorem.

Theorem

Let MATH, MATH be arbitrary. Then the following conditions are equivalent:

  1. MATH

  2. MATH, where $T^{m}$ ($T^{n}$) is an absolute polarization index that satisfies $NS$, $NB$, $SM$ and $PP$.

In order to rank distribution by relative polarization indicators, define relative polarization curve of MATH as the plot of MATH against $\frac{k}{n}$, MATH and of MATH against $\frac{k}{n}$, MATH. Then we can have counterparts to theorem Theorem12b and Theorem13.

Example

  1. MATH, where $f^{\prime }\geq 0$, MATH, MATH, is an example of an absolute polarization indicator.

  2. A relative indicator can be MATH, where $f^{\prime }\geq 0$, MATH, MATH.

  3. MATH is a relative polarization indicator, where MATH is the Gini coefficient of $\overline{x}$. By multiplying it with MATH, we get an absolute indicator (Wolson index).

Deprivation Ordering

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 $\overline{x}_{i}$ relative to $j^{th}$ person's income $\overline{x}_{j}$, MATH, can be taken as their income differential, as a fraction of $n$:MATH

$d_{ij}$ increasing in $\overline{x}_{j}$ and decreasing in $\overline{x}_{i} $.

Now, an individual with income $\overline{x}_{i}$ is deprived of all higher incomes MATH. Therefore, the total absolute deprivation felt by this person isMATH

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 $\overline{x}$ than in $\overline{y}$, which we write MATH ifMATHfor all $i=1,2,...,n$.

Instead of focussing on absolute income differentials, we may view deprivation as arising from relative losses and letMATHmeasure the relative deprivation of individual $i$ in situation $\overline{x} $. We will say that there is more relative deprivation in $\overline{x}$ than in $\overline{y}$, which we write MATH if MATHfor all $i=1,2,...,n$.

$ADP$ and $RDP$ possess many interesting properties:

  1. They are decreasing in $\overline{x}_{i}$.

  2. They are independent of incomes smaller than $\overline{x}_{i}$.

  3. An increase in any income higher than $\overline{x}_{i}$ increases them.

  4. They decrease under a rank preserving income transfer from someone with income higher than $\overline{x}_{i}$ to someone with income lower than $\overline{x}_{i}$.

  5. They remain unaffected when a rank preserving income transfer takes place between two persons with incomes higher or lower than $\overline{x}_{i} $.

  6. While $RDP$ is homogeneous of degree zero in all incomes, $ADP$ is translation invariant.

  7. While $RDP$ decreases under equal absolute augmentation in all incomes, $ADP$ increases or decreases on multiplication of all incomes by $c>0$ according as $c>1$ or $c<1$.

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 $ADP$.

Theorem

The following statement is true but not its converse unless $n=2$: for all MATH, MATH implies MATH, where $\overline{D}^{n}$ is the strictly positive part of $D^{n}$.

Proof

Suppose MATH, which by definition is equivalent to MATHfor $k=1,2,...,n-1$. 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 toMATH

Now, a sufficient condition for (16) to hold is that MATH, for all $i=j,j-1,...,k$, which is since as MATH.

To see that the converse is not true, take MATH, where $p<a$, MATH. Then MATH but not MATH.

To develop a sufficient condition for $RDP$, we define the relative differentials ordering as follows. We say that situation $\overline{x}$ dominates $\overline{y}$ in relative differentials which we write MATH ifMATHfor all $i=1,2,...,n-1$. Thus, the ratio of incomes arranged in nondecreasing order must be less or equal to in $\overline{x}$ than in $\overline{y}$.

Theorem

The following statement is true but not its converse unless $n=2$: for all $\overline{x}$,MATH,MATH implies MATH, where $\overline{D}^{n}$ is the strictly positive part of $D^{n}$.

Proof

Suppose MATH, which is equivalent to MATH

for $k=1,2,...,n$. A sufficient condition for (17) to hold is that MATH

This is verified as soon as MATH, $i=j,j-1,...,k$, which is MATH.

To prove the converse take again MATH, MATH, where $p<a$. Then MATH but not MATH.

the following theorem shows the relation between absolute deprivation ordering and Lorenz ordering.

Theorem

Let $\overline{x}$,MATH be arbitrary, where MATH. Then MATH implies MATH, but the converse is not true for $n>2$.

Proof

Let $\overline{x}$,$\overline{y}$ with MATH be such that MATH. Then MATHfor $i=1$, we have MATHwhich we rewrite asMATHwhich in view of MATH, gives MATH. Next, for $i=2$, we get MATHfrom which we haveMATH

Since MATH, adding MATH (MATH) on the left hand (right hand) side of (18), we get MATHwhich givesMATH

Assume that the result holds for all $i\leq k-1$. We prove it for all $i\leq k$. Then we haveMATHfor all $r=1,2,...,k-1$. now from MATH we getMATHwhich givesMATH

Therefore,MATH

Adding $\left( n-k\right) $ times the left hand side of (20) for $r=k-1 $ 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 getMATHwhich on cancellation of $n-k+1$ from both sides along with (20) is what we wanted to show.

Thus the result holds for all $1\leq k\leq n-1$, since $k$ is arbitrary it follows that MATH.

To show that the converse is not true, take MATH and MATH. Then MATH holds but not MATH.

We now wish to develop an ordering consistent with the ordering $ADP$. For this we say that given $\overline{x}$, MATH, $\overline{y}$ is obtained from $\overline{x}$ by a mean preserving transformation $b\in \U{211d} ^{n}$ if MATH, MATH. (Note that the progressive transfer principle is a mean preserving transformation).

Person $i$ is called a donor, recipient on unaffected according as $b_{i}$ is positive, negative or zero. Since MATH, there is at least one donor and one recipient. Note also that $b\leq \overline{x}$. A mean preserving transformation is called globally equity oriented if for each $i$, $1\leq i\leq n$, MATH, where $S_{i}$ is the set of persons in $\overline{x}$ whose incomes are at least as large as $\overline{x}_{i}$, and MATH is equal to the number of persons in $S_{i}$. To explain this, let MATH. Then globally equity orientation demands that the amount received by person $i$ under the mean preserving transformation $b$ is not smaller than the average receipt of persons richer than him.

A similar explanation holds for MATH we will say that a deprivation indicator MATH is globally equity oriented if MATH, where $\overline{y}$ is obtained from $\overline{x}$ through a mean preserving globally equity oriented transformation. We can now prove the following theorem.

Theorem

Let $\overline{x}$, MATH, where MATH, be arbitrary, then the following conditions are equivalent:

  1. MATH

  2. MATH for all MATH that satisfy symmetry and globally equity orientation.

Proof

Suppose MATH. Then MATHwhich on rearrangement gives MATHfrom which we getMATH

This means that the mean preserving transformation MATH is globally equity oriented. Hence MATH. MATH is symmetric because it is defined on ordered distributions.

To prove the converse consider the deprivation indicator MATH

This indicator satisfies symmetry and global equity orientation. Hence MATH for $2\leq k\leq n$, which in turn implies MATH.

The following theorems are easy extensions of theorem Theorem16.

Theorem

Let $\overline{x}$, MATH be arbitrary. Then the following conditions are equivalent:

  1. MATH

  2. MATH for all absolute deprivation indices MATH that satisfy symmetry and global equity orientation.

Theorem

Let MATH, MATH be arbitrary. Then the following conditions are equivalent:

  1. MATH

  2. MATH for all absolute deprivation indices $d^{m}$ ($d^{n}$) defined on $\overline{D}^{m}$ ($\overline{D}^{n}$) that satisfy symmetry and global equity orientation and population principle.

The absolute deprivation curve associated with the distribution $\overline{x} $ is defined as the plot of MATH against the cumulative proportion of population $\frac{i}{n}$, where $i=1,2,...,n$. 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 MATH. Then for any arbitrary income MATH, the absolute deprivation function $ADP$ becomes MATH.

We observe that MATH

Therefore, the curve is decreasing.

Since MATH may be positive, zero or negative, no definite conclusion can be drawn regarding the curvature of $ADP$ curve.

Then we can restate theorem Theorem16 in terms of two curves. Restatements of theorem Theorem17, Theorem18 and Theorem19 are also possible.

Satisfaction Ordering

Recall the definition of $DP$:MATH

We regard the complement MATH of MATH to MATH as the satisfaction function of the person with income $\overline{x}_{i}$. The function MATH is MATH

We can interpret MATH as follows. Since person $i$ (in the ordered distribution $\overline{x}$) is not deprived of incomes MATH, he may be regarded as being satisfied if he compares hi income $\overline{x}_{i}$ with these lower incomes. Therefore, the first term in MATH is based on the truncated distribution MATH. Next, since this person is deprived of the incomes MATH, to eliminate his feeling of deprivation about these $\left( n-i\right) $ incomes we replace each of these $\left( n-i\right) $ incomes by $\overline{x}_{i}$. Thus, this approach ignores actual information on incomes of persons richer than $i$ but counts them in with his income level $\overline{x}_{i}$.

Given that in addition to person $i$ there are $\left( n-i\right) $ persons with income $\overline{x}_{i}$ and since all these persons are treated identically, we simply add the term MATH to MATH for arriving at MATH. Therefore, for each $i$, MATH is based on the censored income distribution MATH.

MATH possesses the following properties:

  1. It is increasing in $\overline{x}_{i}$. (Note that since MATH are ordered, only rank preserving increments are allowed.)

  2. It does not get affected by a rank preserving increase in any income greater than $\overline{x}_{i}$.

  3. It increases if any income lower than $\overline{x}_{i}$ increases.

  4. A rank preserving income transfer from someone with income greater than $\overline{x}_{i}$ to someone with income less than $\overline{x}_{i}$ increases MATH.

  5. A rank preserving transfer of income between two persons on the same side of $\overline{x}_{i}$ does not change MATH.

  6. It is linearly homogeneous in income.

  7. It increases under equal absolute increase in all income.

We say that there is no less satisfaction in $\overline{x}$ than in $\overline{y}$, which we write MATH, ifMATHfor all $i=1,2,...,n$.

The following theorem shows that the satisfaction ordering is a sufficient condition for the weak generalized Lorenz ordering.

Theorem

Let MATH be arbitrary. Then MATH implies, but is not implied by MATH.

Proof

MATH means MATH

for all $1\leq i\leq n$. Putting $i=1$ in (23) we get MATH.

Thus,MATH

hold for $i\U{a7}=1$. Assume that () holds for all $i\leq k-1$. That is,

MATH

for all $1\leq i\leq k-1$. We show that it holds for all $1\leq i\leq k$. The $SF$ ordering for $i=k$ gives MATH

Assume $i=k-1$ in (25) and multiply both sides of the resulting expression by $(n-k)$ to get MATH

Adding the left-hand (right-hand) side of (26) to the left-hand (right-hand) side of (27) we get MATH

which gives MATH

Thus, MATH

holds for all $1\leq i\leq k$. Since $1\leq k\leq n$ is arbitrary, it follows that MATH holds.

To see that the converse is not true, take MATH and MATH. MATH (in fact, MATH) holds but not MATH.

If the income distribution is represented by a distribution function MATH, then for any income MATH, the satisfaction function $SF$ becomes MATH

Thus, the satisfaction curve of $\overline{x}$ which plots MATH against $\frac{i}{n}$, $i=1,2,...,n$, is increasing. Given that MATH, no definite conclusion can be drawn regarding curvature of $SF$ curve.

Some Properties of doubly stochastic matrix

Theorem

If $P$ is doubly stochastic and nonsingular, then $Q=P^{-1}$ satisfies $e\alpha =e$, MATH.

Proof

$eP=e$ implies $e=eP^{-1}=eQ$ and MATH implies MATH.

Doubly stochastic matrices need not to be nonsingular. Note if a doubly stochastic matrix $P$ is nonsingular and $Q$ is its inverse, $Q$ need not have nonnegative elements, so that $Q$ may not be doubly stochatic.

Theorem

If $P$ and $P^{-1}=Q$ are both doubly stochastic, then $P$ is a permutation matrix.

Theorem

If $P$ is a doubly stochastic such that $P^{-1}=P^{\prime }$, then $P$ is a permutation matrix.