## Abstract

This paper proposes classes of intertemporal poverty measures which take into account both the debilitating impact of prolonged spells in poverty and the mitigating effect of periods of affluence on subsequent poverty. The weight assigned to the level of poverty in each time period depends on the length of the preceding spell of poverty or of non-poverty. The proposed classes of intertemporal poverty measures are quite general and allow for a range of possible judgements as to the overall impact on a poor period of preceding spells of poverty or affluence. We discuss the properties of the proposed classes of measures and axiomatically characterize these measures.

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## Notes

Hoy and Zheng (2011) would rank these cases differently. However, the motivation for doing so is very different from that here. They explicitly consider poverty early in life to be more damaging than poverty later on. We have no such assumption here.

Zheng (2011) proposes several classes of measures. Here, and for the rest of the paper, we are concerned mainly with his Newtonian poverty measure (p. 10).

Our notion of income smoothing is based on (Morduch (1995), p. 104) where households can smooth income by “making conservative production and employment choices and diversifying economic activities.”

Note that here, and throughout the paper, we are referring to Foster (2009)’s total intertemporal poverty measure, not his chronic poverty measure. Foster (2009) defines a poverty duration cut-off line as the minimum proportion of time periods a person must be poor in order to be deemed chronically poor; individuals who are in poverty for a proportion of periods less than this threshold are considered transiently poor. Foster (2009)’s total intertemporal poverty measure is obtained by choosing a poverty duration cut-off line of zero.

Net income can be thought of as consumption, but then our interpretation of the measure has to change in line with this. If we assume consumption as our primitive, then the mitigating impact of affluent periods reflects non-consumption smoothing mechanisms since presumably any consumption smoothing is already reflected in the consumption vector.

For example, \(p_{t}\) could be any static poverty measure from the literature, such as a normalized poverty gap. In fact, with some minor amendments, our results will go through for a more general definition, where \(p_{t}\in \mathbb R _{+}.\)

If both \(\alpha =0\) and \(\beta =0\), provided \(p_{t}\) is a normalized poverty gap, the measure reduces to the simple average of static poverty measures advocated by Foster (2009).

The approach of effectively censoring the income in each time period at the poverty line is common in the literature on intertemporal poverty measurement and is also adopted by Bossert et al. (2012) and Mendola et al. (2011), among others. However, this leads to a discontinuity in the measure at the poverty line. For a continuous measure see Hoy and Zheng (2011).

See Dutta et al. (2011) for a more general stucture for poverty mitigation.

Note that explictly incorporating the non-income dimensions is not feasible in this context, since all we observe is the ex-post income distribution for the individual across time.

A natural counter-argument of course is that the proportion of poverty which is mitigated remains the same, regardless of the poverty level. This possible criticism is somewhat reminiscent of the charge often made against relative inequality measures, which register no change in inequality when all incomes are increased by the same proportion. Those who regard absolute differences in income to be important with respect to inequality would reject such measures.

Note that the summation in \(P_{A}\) is over only those periods where \( p_{t}\ne 0\). This is a technical requirement since in our paper \(n_{t}\) (the number of immediately preceding affluent periods) is not defined when \( p_{t}=0\).

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## Acknowledgments

We are extremely grateful to two anonymous referees and to Buhong Zheng, whose comments have led to substantial improvements in the paper. We have also profited from discussing this work with Sabina Alkire, Conchita D’Ambrosio, Partha Dasgupta, Roger Hartley, David Hulme, Ravi Kanbur, Ajit Mishra, Simon Peters and seminar participants at Manchester, Oxford and York. We are grateful for their comments.

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## Appendix: Proofs

### Appendix: Proofs

###
*Proof of Proposition 1:*

We concentrate on the “only if” part of the proof, as it is immediate to verify that \(P_{R}\) satisfies the axioms stated in Proposition 1.

Suppose that \(P\) satisfies single period equivalence (Axiom 1), time decomposability (Axiom ), normalization (Axiom 3), constant-relative poverty mitigation (Axiom ) and relative poverty intensification (Axiom ). We need to show that for any time period \(T\in \mathbb N \) and any poverty profile \(\mathbf p \in [0,1]^{T}\) we have \(P( \mathbf p )=P_{R}(\mathbf p )\) to for an exogenously determined \(\alpha , \beta \ge 0\) and \(\theta \ge 1\). So, take any \(T\in \mathbb N \) and any poverty profile \(\mathbf p \in [0,1]^{T}\).

Suppose \(T=1\). In this case single period equivalence holds and \(P(\mathbf p )=P(p)=P_{R}(p)\) for some \(p\in (0,1]\). Hence, \(P(\mathbf p )=P_{R}( \mathbf p )\) follows.

Assume now that \(T>1\). If \(\mathbf p =\mathbf 0 ^{T}\), then by normalization we obtain \(P(\mathbf p )=P(\mathbf 0 ^{T})=0=P_{R}(\mathbf 0 )\). Thus, \(P( \mathbf p )=P_{R}(\mathbf p )\) follows.

Next we proceed by induction on the number of poor periods, when \(\mathbf p\ne 0 ^{T}\). Consider first the case where there is exactly one poor period. Recall that we represent a \(T\)-period poverty profile \(\mathbf p= (p_{1},\ldots ,p_{s},\ldots p_{T})\), where \(\forall t\ne s,\) \(p_{t}=0\) and \( p_{s}=1\) as \(\mathbf e _{s}^{T}\). Thus \(\mathbf p= p\cdot \mathbf e _{s}^{T}\) would stand for a \(T\)-period poverty profile \(\mathbf p= (p_{1},\ldots ,p_{s},\ldots p_{T})\), where \(\forall t\ne s,\) \(p_{t}=0\) and \(p_{s}=p\).

Without loss of generality, we can write \(\mathbf p =p\cdot \mathbf e _{t}^{T}\) where \(p>0\) and \(t\in \left\{ 1,\ldots ,T\right\} \). Thus \(P( \mathbf p )=P(p\cdot \mathbf e _{t}^{T})\). Then

Since \(k_{t}=1\), \(t=(1+n_{t})\) and \(\forall \) \(i\ne t\), \(p_{i}=0\) we can write (2) as

Thus we obtain \(P(\mathbf p )=P_{R}(\mathbf p )\).

Suppose now that \(P(\hat{\mathbf{p }})=P_{R}(\hat{\mathbf{p }})\) whenever \(\hat{\mathbf{p }}\) contains \(m\) poor periods, for some \(m\in \left\{ 1,\ldots ,T-1\right\} \). Let \(\mathbf p \in [0,1]^{T}\) be any poverty profile, such that the number of poor periods is \(m+1\). Let \(t\in \left\{ 2,\ldots ,T\right\} \) be such that the final poor period is period \(t\). Thus \(t=\max \{s:2\le s\le T,p_{s}>0\}\). From time decomposability we derive

Now \(t\) being the final poor period means that by normalization we get

Let \(s\ne t\) be maximal with \(p_{s}>0\). So \(s\) is the last poor period prior to \(t\). Suppose \(s\ne t-1\). Then by time decomposability we obtain

Further, \(P(p_{s+1},\ldots ,p_{t})=P(p_{t}\cdot \mathbf e _{t-s}^{t-s})\) since \(p_{i}=0\) for all \(i\in \left\{ s+1,\ldots ,t-1\right\} \). Applying single period equivalence, time decomposability, normalization and constant-relative poverty mitigation and noting that \(k_{t}=1\) and \( n_{t}=t-s-1\), we obtain

Now consider the case when \(s=t-1\). Then using relative poverty intensification we get

Using time decomposability, single period equivalence and noting that \( n_{t}=0\), (6) can be written as

Now \((p_{1},\ldots ,p_{s})\) contains \(m\) poor periods. Thus, by the induction hypothesis, we have

Substituting (8) and (5) into (4) (for the \(s\ne t-1\) case) or substituting (8) into (7) (for the \(s=t-1\) case) yields

Further, substituting (9) into (3) we obtain

Finally, since \(p_{i}=0\) for all \(i\in \left\{ s+1,\ldots ,t-1\right\} \) and all \(i\in \left\{ t+1,\ldots ,T\right\} \), we have

This concludes the proof for the case of \(m+1\) poor periods, and by induction it follows that \(P(\mathbf p )\,{=}\,P_{R}(\mathbf p )\) for any poverty profile \(\mathbf p \). This completes the proof of Proposition .\(\square \)

###
*Proof of Proposition 2*

We demonstrate that axioms single period equivalence (Axiom 1), time decomposability (Axiom ), normalization (Axiom ), absolute poverty mitigation (Axiom ), constant-relative poverty mitigation (Axiom 4) and relative poverty intensification (Axiom 5) are independent by presenting a separate poverty measure that satisfies all the axioms except one. We do this one at a time for each of the five axioms.

Consider a poverty profile \(\mathbf p \in [0,1]^{T}\). Then the following measure violates single period equivalence (Axiom 1) but satisfies the other axioms.

The next measure violates time decomposability (Axiom ) but satisfies the other axioms.

A measure which violates normalization (Axiom ) but satisfies the other axioms is given by

A measure which violates constant-relative poverty mitigation (Axiom 4) but satisfies the others is

A measure which violates relative poverty intensification (Axiom 5) and satisfies the rest is as follows.

\(\square \)

###
*Proof of Proposition 3*

The proof is similar to that of Proposition and is omitted. \(\square \)

###
*Proof of Proposition 4*

We concentrate on the “only if” part of the proof, as it is immediate to verify that \(P_{A}\) satisfies the axioms stated in Proposition 4.

Suppose that \(P\) satisfies single period equivalence (Axiom 1), time decomposability (Axiom ), normalization (Axiom ), relative poverty intensification (Axiom ), absolute poverty mitigation (Axiom 7), and monotonic poverty mitigation (Axiom 8). We need to show that for any time period \(T\in \mathbb N \) and any poverty profile \(\mathbf p \in [0,1]^{T}\) we have \(P( \mathbf p )=P_{A}(\mathbf p )\) for a monotonically increasing function \(f: \mathbb R _{+}\rightarrow \mathbb R _{+}\) such that \(f(0)=0\) and \(f(n_{t}+1)\ge f(n_{t})\).

Take any \(T\in \mathbb N \) and any poverty profile \(\mathbf p \in [0,1]^{T}\). Suppose \(T=1\). Note that \(k_{1}=1\) and \(n_{1}=0\). By absolute poverty mitigation, we have

By construction let \(f(n_{t})=th(t)\), \(t=n_{t}+1\). Given \(h(t)\in \mathbb R _{+}\) and \(t\in \left\{ 1,\ldots ,T\right\} , f: \mathbb Z _{+}\rightarrow \mathbb R _{+}\). When \(t=1\), it implies \(f(0)=h(1)=0\). Due to single period equivalence and \(k_{1}=1\) we can write Eq. (10) as

Assume now that \(T>1\). If \(\mathbf p =\mathbf 0 ^{T}\), then by normalization we obtain \(P(\mathbf p )=P(\mathbf 0 ^{T})=0=P_{A}(\mathbf p )\) . Thus, \(P(\mathbf p )=P_{A}(\mathbf p )\).

Next, we proceed by induction on the number of poor periods when \(\mathbf p\ne 0 ^{T}\). Consider the case where there is exactly one poor period. Without loss of generality, we can write \(\mathbf p =p\cdot \mathbf e _{t}^{T}\) where \(p\in (0,1]\) and \(t\in \left\{ 1,\ldots ,T\right\} \). Thus \( P(\mathbf p )=P(p\cdot \mathbf e _{t}^{T})\). If \(t=1\) then, applying single period equivalence, time decomposability and normalization, and noting that \( k_{1}=1\) and \(f(0)=0\), yields

For \(t>1\) applying time decomposability and normalization, we obtain

To show that \(f(n_{t})\) is monotonic, consider another profile \(P(p\cdot \mathbf e _{t-1}^{T})\). Applying (11) we get

By monotonic poverty mitigation it must be the case that \(P(p\cdot \mathbf e _{t-1}^{T})\ge P(p\cdot \mathbf e _{t}^{T})\). Comparing (11) and (12) and noting that \(n_{t}=n_{t-1}+1\), we can show \(f(n_{t})\ge f(n_{t-1})\). Applying single period equivalence in (11) and noting that \(k_{t}=1\), we can show

Since \(p_{i}=0\) for all \(i\ne t\), we can write (13) as

Suppose now that \(P(\hat{\mathbf{p }})=P_{A}(\hat{\mathbf{p }})\) whenever \( \hat{\mathbf{p }}\) contains \(m\) poor periods, for some \(m\in \left\{ 1,\ldots ,T-1\right\} \). Let \(\mathbf p \in [0,1]^{T}\) be any poverty profile, such that the number of poor periods is \(m+1\). Let \(t\in \left\{ 2,\ldots ,T\right\} \) be such that the final poor period is period \(t\). Thus \(t=\max \{s:2\le s\le T,p_{s}>0\}\). From time decomposability we derive

Now \(t\) being the final poor period means that by normalization we get

Let \(s\ne t\) be maximal with \(p_{s}>0\). So \(s\) is the last poor period prior to \(t\). Suppose \(s\ne t-1\). Then by time decomposability we obtain

Further, \(P(p_{s+1},\ldots ,p_{t})=P(p_{t}\cdot \mathbf e _{t-s}^{t-s})\) since \(p_{i}=0\) for all \(i\in \left\{ s+1,\ldots ,t-1\right\} \). Using (14) and noting that \(k_{t}=1\) and \(n_{t}=t-s-1\), we obtain

Substituting (17) in (16) we get

Now consider the case when \(s=t-1\). Using relative poverty intensification we get

Applying time decomposability, single period equivalence and using (11 ) we can write (19) as

Note that in (20) \(f(n_{1})=0\), since \(n_{1}=0\) by definition. Now \( (p_{1},\ldots ,p_{s})\) contains \(m\) poor periods. By the induction hypothesis, we have

Substituting (21) into (18) (for the \(s\ne t-1\) case) or substituting (21) into (20) (for the \(s=t-1\) case) yields

Further, substituting (22) into (15) we obtain

Finally, since \(p_{i}=0\) for all \(i\in \left\{ t+1,\ldots ,T\right\} \), we have

This concludes the proof for the case of \(m+1\) poor periods, and by induction it follows that \(P(\mathbf p )=P_{A}(\mathbf p )\) for any poverty profile \(\mathbf p \). It therefore follows that \(P=P_{A}\), which completes the proof of Proposition .\(\square \)

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Dutta, I., Roope, L. & Zank, H. On intertemporal poverty measures: the role of affluence and want.
*Soc Choice Welf* **41**, 741–762 (2013). https://doi.org/10.1007/s00355-012-0709-8

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DOI: https://doi.org/10.1007/s00355-012-0709-8