Variable Population Poverty Comparisons (Written with Subbu Subramanian)

Nicole Hassoun

Motivation Debates about poverty relief and foreign aid

often hinge on claims about how many poor people there are in the world. Good measures of poverty are essential for addressing the world poverty problem.

Populations Change!

Since populations change, it is important to have poverty indexes that work well in variable population contexts. Much of the canonical work on poverty measurement considers desirable properties of poverty indices in the context of populations of fixed size.

Thesis: This paper questions Replication Invariance, a

standard axiom that is often assumed in the poverty measurement literature to allow some variable population comparisons.

Significance:

Replication Invariance (RI) is the purportedly unexceptionable requirement that the extent of poverty in a situation should remain the same if

the population is replicated any number of times. Virtually all real-valued measures of poverty in the literature

incorporate within themselves the headcount ratio which satisfies RI.

RI underlies Lorenz Dominance, Generalized Lorenz Dominance, and Stochastic Dominance.

Outline

Discuss the axioms upon which we will rely. Present three preliminary inconsistency results. Consider which of the axioms used in generating

the results should be rejected and provide reason to question RI.

Conclude

Anonymity

We will assume that poverty assessments do not depend on the personal identities of individuals.

A B C D

Fixed Population Axioms

The Monotonicity Axiom (M) states that, other things equal, an increase in a poor person’s income should reduce poverty.

Continued…

The (Very Weak) Transfer Axiom (T) demands that, ceteris paribus, a mutually rank-preserving transfer of income from a non-poor person to a poor person that keeps the former non-poor, should reduce poverty.

Variable Population Axioms

Replication Invariance (RI) requires the extent of poverty to remain unchanged by any k-fold replication of a population.

Continued…

Weak Poverty Growth (WPG) requires that, in a situation where there is at least one non-poor person and all the poor have the same income, the addition of another poor person with this same income ought to increase poverty.

Continued…

Weak Population-Focus says poverty is not reduced by changes in the non-poor population which leave the income distribution amongst the poor unchanged.

Continued…

The Population-Focus Axiom requires an addition to the non-poor population to leave the extent of poverty unchanged (and not just to leave it un-reduced).

Proposition 1:

It is impossible to satisfy RI, WPG and WPF.

A B C

Proposition 1:

By WPG, B has less poverty than C.

B C

WPG: in a situation where there is at least one non-poor person and all the poor have the same income, the addition of another poor person with this same income ought to increase poverty.

Proposition 1:

By RI, C has the same amount of poverty as A.

C A

RI: The extent of poverty remains unchanged by any k-fold replication of a population.

Proposition 1: By transitivity B has less poverty than A

B A

But by WPF, it is not the case that B has less poverty than A. (On WPF, poverty is not reduced by changes in the non-poor population which leave the income distribution amongst the poor unchanged.)

Proposition 2:

It is impossible to satisfy M, RI, and WPF.

A B C

Proposition 2:

By M, B has less poverty than A.

On M, an increase in a poor person’s income should reduce poverty.

B A

Proposition 2:

By RI, A has the same amount of poverty as C.

On RI the extent of poverty is unchanged by any k-fold replication of a population.

A C

Proposition 2:

By transitivity, B has less poverty than C.

But by WPF, it is not the case that B has less poverty than C. (On WPF, poverty is not reduced by changes in the non-poor population which leave the income distribution amongst the poor unchanged.)

B C

Proposition 3:

It is impossible to satisfy RI, T, and PF.

A B C D E

Proposition 3:

By PF, A has the same amount of poverty as B and B has the same amount of poverty as C, so by transitivity A has the same amount of poverty as C.

On PF, an addition to the non-poor population to leave the extent of poverty unchanged.

A B C

Proposition 3: By T, C has less poverty than D.

On T, a mutually rank-preserving transfer of income from a non-poor person to a poor person that keeps the former non-poor, should reduce poverty.

C D

Proposition 3:

By PF, D has the same amount of poverty as E.

On PF an addition to the non-poor population to leave the extent of poverty unchanged.

D E

Proposition 3:

By transitivity A has less poverty than E.

A E

Recall that:A has the same amount of poverty as C.C has less poverty than D.D has the same amount of poverty as E.(A=C<D=E) -> (A<E)

Proposition 3:

But by RI, A has the same amount of poverty as E:

On RI the extent of poverty is unchanged by any k-fold replication of a population.

A E

Remark

As Sen’s (1976) seminal work in the area indicates, the quest for income-responsive and distribution-sensitive poverty measures was motivated by the failure of the headcount ratio to satisfy fixed-population axioms like Monotonicity and Transfer.

However, in a variable populations context, the headcount ratio is the archetypal Replication Invariance-satisfying measure, and this note has shown that, when we employ a population-focus axiom, it is impossible to hold Replication Invariance and Monotonicity (Proposition 2) or Transfer (Proposition 3).

Evaluation of the Axioms:

If some sort of population-focus axiom is accepted, then one either has to reject Replication Invariance or each of the following: Weak Poverty Growth, Monotonicity, and Transfer.

Why Population Focus?

Why might one wish to accept some form of population-focus axiom? For the same reason, we would argue, that one may wish to accept an income-focus axiom.

Just as increasing the income of a rich person should not reduce poverty, increasing the number of rich should not reduce poverty.

Monotonicity

Monotonicity embodies, in the context of poverty measurement, what Broome (2003) calls ‘the principle of personal good’. The principle requires that one endorse any change which is good for someone and not bad for anyone. This is unexceptionable.

A Reason to Reject Transfer?

Perhaps one could argue distributions amongst the poor may be more or less desirable because of the way it came about, but that fact may not change our judgment about how much poverty there is in a situation.

Of course, our argument will only be stronger if this is not the case.

Against Weak Poverty Growth? One might reject Weak Poverty Growth because

it seems to suggest that poverty declines when the poor are removed from a population.

However, this worry is misplaced because it overlooks the distinction between a measure of how much poverty there is in a situation and a metric for guiding policy decisions.

Replication Invariance?

On the other hand, it is not immediately obvious that there is some inherently indispensable ethical appeal attaching to Replication Invariance. So one should probably reject Replication Invariance. And yet, as we noted, Replication Invariance occupies a central place in all of distributional, welfare, and poverty analysis.

Conclusion

Briefly, then, it would appear that variable populations constitute a simple but undeniable headache for poverty comparisons.

The Future

Further research on poverty measurement in variable population contexts is pressing and important.

Thank you!

The Non-Poverty Growth Axiom (NPG) stipulates that poverty should decline with the addition of a non-poor person to the population. Proponents might argue that poverty should decline with a

decline in the headcount ratio. Alternately, perhaps poverty declines if the redistributive capacity to relieve it is, in principle, enhanced.