# Variable Population Inequality Comparisons

When we deal with variable populations we find that the problem of ‘fractions versus whole numbers’ encountered in the measurement of poverty carries over also to the measurement of inequality. This is far from surprising: to recall from Sect. 5.2, properties such as Replication Invariance are concerned with population proportions, while properties such as Upper Pole Monotonicity are concerned with absolute population size. The conflict between these two ways of viewing population size is manifested in the following elementary impossibility result, stated and proved in Subramanian (2010) and Subramanian (2011a):

Proposition 5.5 There exists no anonymous inequality measure T. X ! R which satisfies Upper Pole Monotonicity (Axiom UPM), Replication Invariance (Axiom RI*) and Weak Upper-Bound Normalization (Axiom WUBN).

Proof Let x be any positive scalar, and let a,b, c and d be four income vectors such that a = (0,0,....,0, x), b = (0, 0, ...., 0, 0, x), c = (0, 0, ...., 0, 0, x, x) and d = (0, 0, ...., 0, x, x, ...., x), with n(c) = n(b) + 1, n(b) = n(a) + 1, and n(d) = n(a)n(c). Then, d is an n(c)-replication of a and an n(a)-replication of c, so that, by Axiom RI, I(d) = I(a), I(d) = I(c), and therefore I(a) = I(c); and I(c) < I(b) by Axiom UPM, whence I(a) < I(b), which, however, is contradicted by I(a) > I(b), as dictated by Axiom WUBN.^

The result above is reflected in the fact that each of the inequality measures C2, T, G and Ak presented in Sect. 5.2 satisfies Axiom UPM and RI* while violating Axiom WUBN, and each of the inequality measures C2*, T*, G* and Ak* satisfies Axiom UBN and UPM while violating Axiom RI*. This suggests an inherent tension between Replication Invariance and Upper-Bound Normalization, as is, indeed, confirmed by the following result, stated and proved in Subramanian (2011a):

Proposition 5.6 There exists no proper, anonymous and scale-invariant measure of inequality I: X ! R which satisfies Replication Invariance (Axiom RI*) and Weak Upper-Bound Normalization (Axiom WUBN).

Proof Let x be any positive scalar, and a, b, c, d, e and f be six income vectors such that a = (0, x), b = (0, 0, x), c = (0, 0, 0, x), d = (0, 0, 0, 2x), e = (0, 0, x, x) and f = (0, 0, 0, 0 x, x, x, x). By Axiom WUBN, I(a) > I(b) > I(c), whence I(a) > I(c); noting that d = 2c, Scale Invariance (Axiom SI) requires that I(d) = I(c) and hence (since I(a) > I(c)), I(a) > I(d)). Further, I(d) > I(e) by Axiom T*, whence, given I(a) > I(d), one must also have I(a) > I(e). Since f is a 2-replication of e and a 4-replication of a, Axiom RI* dictates that I(f) = I(e) and I(f) = I(a), whence I(e) = I(a) which, however, is contradicted by I(a) > I (e), as deduced earlier.^ From a wholly pragmatic point of view, inequality measurement without Replication Invariance is hard to conceive of: one would have to dispense with such devices of comparison as Stochastic Dominance and Lorenz Dominance, which are foundational aspects of inequality measurement as it is ‘standardly’ practiced, if one were to renounce Replication Invariance. Upper Bound Normalization is also a practically useful property in an inequality index: it permits one to express the extent of inequality in any general n-person distribution in terms of the share of the poorer of two individuals in a classic—and easily comprehended—two-person cake-sharing problem. (The equivalence between n-person inequality measures and two-person shares is dealt with in Subramanian (2002a) and Shorrocks (2005).) If —from these pragmatic considerations of manipulability and interpretability—one wished to retain the properties of Replication Invariance and Upper-Bound Normalization, then one would have to be prepared to sacrifice certain other properties of an inequality measure. Propositions 5.5 and 5.6 suggest that one may have to give up the variable population property of Upper Pole Monotonicity and the fixed population property of Transfer in this cause. It turns out, as it happens, that there does exist a ‘threshold’ inequality measure—namely one which satisfies the Weak Transfer but not the Transfer Axiom—which fulfills the requirements of Replication Invariance and Upper-Bound Normalization, while violating Upper Pole Monotonicity. This result, which is discussed in Subramanian (2011a), is reflected in the following Proposition:

Proposition 5.7 There exists a ‘threshold’ inequality measure I: X ! R which satisfies Replication Invariance (Axiom RI*) and Upper-Bound Normalization (Axiom UBN).

Proof Consider the inequality measure D: X ! R which, for all x 2 X, is given by: Where the incomes in the vector have been arranged in non-descending order.

Note first, in view of (5.1), that for any extremal distribution x 2 X, D(x) = 1, which establishes that D satisfies Axiom UBN. Next, for any ordered n-vector of incomes x, let y be a k-fold replication of x, where k is any positive integer. Then, given (5.1), one can (Since. obviously, p(y) = p(x)) = D(x), as required to establish Axiom RI*. That D is not violative of ‘equity-consciousness’ is clear from the fact that D resorts to a weighting structure in which the ith poorest person’s income is weighted by the (n +1 - i)th poorest person’s income, which ensures a non-increasing scheme of weights and, therefore, the fulfilment by D of at least Weak Transfer. More formally, let x and y be two ordered n vectors of income with the same mean p, and suppose the antecedents of the Transfer and the Weak Transfer Axioms, as stated in Sect. 5.2, to be satisfied. It can be verified that D(x) - D(y) = (28/np2) (xn + j _ j - xn + j _ k) > 0, since xn + j _ j - xn + j _ k >0 (which follows from the fact that incomes have been arranged in non-decreasing order), which is what is required to establish Weak Transfer. (However the regular Transfer axiom may be violated: if it should turn out that xn + j - j = xn + j - k, then one would have D (x) = D(y), a case where Weak Transfer, but not Transfer, is satisfied.)^

It may be added that the index D, by virtue of being normalized, lends itself to interpretation in terms of the simplest and most familiar representation of inequality one can think of—the share of the poorer person in the division of a cake of fixed size between two individuals. To see what is involved—the reader is also referred, in this connection, to Shorrocks (2005) and Subramanian (2002a, 2010, 2011a)— consider the following. For any n-person ordered income vector x with mean p and inequality value D, construct what may be called a dichotomously allocated equivalent distribution (DAED), which is the two person non-decreasingly ordered income vector x* = (x*v x*2 with the feature that its mean p* is the mean p of x, and its inequality value D* is the inequality value D of x. p* = p and D* = D entail, respectively [given (5.1)], that x*j + x*2 = 2p and

1 - x*j + x*2/p = D. Solving for x*j and x*2 If we designate by rD[= x/x + x?] the share of the poorer of the two individuals in the DAED x*, then, in view of (5.2), one obtains the following expression for rD in terms of the inequality index D: This relationship is of considerable value in interpreting the ‘meaning’ of the inequality measure. Thus, if in some actual situation involving an n-person distribution the extent of inequality as measured by D should be of the order of 0.25, then this is ‘equivalent’—in view of (5.3)—to a situation in which the poorer of two persons in a two-person distribution of a cake receives 25% of the cake. The utility of this ‘interpretational advantage’ must, of course, be set off against the fact that a measure such as D is not a ‘proper’, but only a ‘threshold’, measure of inequality, and it does not satisfy the property of Upper Pole Monotonicity.

In a general way, Proposition 5.7 suggests the existence of a trade-off amongst competing properties of an inequality index. How the trade-off is resolved must depend on the value system of the practitioner. What Proposition 5.7 does do is to indicate that a trade-off cannot be avoided.