Abstract
We characterize axiomatically a new index of urban poverty that i) captures aspects of the incidence and distribution of poverty across neighborhoods of a city, ii) is related to the Gini index and iii) is consistent with empirical evidence that living in a high poverty neighborhood is detrimental for many dimensions of residents’ well-being. Widely adopted measures of urban poverty, such as the concentrated poverty index, may violate some of the desirable properties we outline. Furthermore, we show that changes of urban poverty within the same city are additively decomposable into the contribution of demographic, convergence, re-ranking and spatial effects. We collect new evidence of heterogeneous patterns and trends of urban poverty across American metro areas over the last 35 years.
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Let X and Y be k × k matrices. The Hadamard product X ⊙Y is defined as the k × k matrix with the (i, j)-th element equal to xijyij.
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Acknowledgments
We are grateful to conference participants at RES 2018 meeting (Sussex), LAGV 2018 (Aix en Provence) and ECINEQ 2019 (Paris) and to two anonymous reviewers for valuable comments. The usual disclaimer applies. Replication code for this article is accessible from the authors’ web-pages. This article forms part of the research project The Measurement of Ordinal and Multidimensional Inequalities (grant ANR-16-CE41-0005-01) of the French National Agency for Research and the NORFACE project IMCHILD: The impact of childhood circumstances on individual outcomes over the life-course (grant INTER/NORFACE/16/11333934/IMCHILD) of the Luxembourg National Research Fund (FNR), whose financial support is gratefully acknowledged.
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Appendices
Appendix A: Proofs
1.1 A.1 Proof of Theorem 1
We will prove the theorem making use of a sequence of lemmas that will highlight the role of the different axioms in the derivation of the final result.
Lemma 1
Let \(\mathcal {A} \in {\Omega }\), ζ ∈ [0, 1), and z ≥ 1, UP(.; ζ) satisfies AGG and INV-S if and only if there exist a continuous function \(A:[0,1]^{2}\rightarrow \mathbb {R}_{+}\) and a function \(h:[0,1]\rightarrow \mathbb {R}\) continuous in (0,1) with h(0) = 0 such that:
with \(\bar {N}_{0}:=0\).
Proof
The proof combines the effect of AGG with INV-S by deriving a functional restriction on the class of weighting functions \(w_{i} \left (\frac {N_{1}}{N},\ldots ,\frac {N_{i}}{N},\ldots ,\frac {N_{n}}{N}\right )\) that appear in the definition of AGG. We leave to the reader to verify that the index in Eq. 4 satisfies AGG and INV-S, here we focus on the proof of the (only if) part of the statement in the lemma.
First recall that, given AGG, we can write
where \(A:[0,1]^{2}\rightarrow \mathbb {R}_{+}\) and \(w_{i}: {\Delta }_{n}\rightarrow \mathbb {R}\) satisfy the conditions specified in AGG.
Let z ≥ 1, we apply INV-S. Note that because of the definition of INV-S, the scaling component \(A\left (\frac {P}{N}, \frac {\bar {P}_{z}}{\bar {N}_{z}}\right )\) of Eq. 5 is not affected by splitting operations. Thus INV-S only affects the component \({\sum }_{i=1}^{z} \frac {N_{i}}{N} \cdot \left (\frac {P_{i}}{N_{i}} - \zeta \right ) \cdot {w_{i}} \left (\frac {N_{1}}{N}, \ldots , \frac {N_{i}}{N}, \ldots , \frac {N_{n}}{N}\right )\).
We construct the proof in two steps. We first derive the restrictions on the function w1(.) and then in a recursive manner we derive also the restrictions on all the other functions wi(.) for i = 2, 3,…,n.
We first note that the function wi(.) does not depend on ζ, and then we set ζ such that for a given \(\mathcal {A} \in {\Omega }\) we have that n = z. Note that for any \(\mathcal {A} \in {\Omega }\) there exist values of ζ such that n = z, for instance this is the case if we let ζ = 0.
Step 1. Suppose that n = z = 2, and assume that \(\frac {P_{1}}{N_{1}} >\zeta \), while \(\frac {P_{2}}{N_{2}}=\zeta \). Apply repeatedly the splitting operations over the neighborhood indexed by i = 2. Because of the invariance requirement in INV-S and the specification in Eq. 5, if we denote by \(\hat {n}_{i}:=\frac {N_{i}}{{N}}\) we obtain that \(\hat {n}_{1} \left (\frac {P_{1}}{N_{1}} - \zeta \right ) \cdot w_{1} (\hat {n}_{1}, 1-\hat {n}_{1}) = \hat {n}_{1} \left (\frac {P_{1}}{N_{1}} - \zeta \right ) \cdot w_{1} (\hat {n}_{1}, \hat {n}_{2}, \hat {n}_{3}, \ldots , \hat {n}_{z})\) with \(\hat {n}_{2} + \hat {n}_{3} + {\ldots } + \hat {n}_{z} = 1-\hat {n}_{1}\), this result holds for all z = n ≥ 2. Recalling that \(\frac {P_{1}}{N_{1}} - \zeta >0\), we then obtain
with \(\hat {n}_{2} + \hat {n}_{3} +\ldots +\hat {n}_{z}=1-\hat {n}_{1}\), for all z = n ≥ 2 and \(\hat {n}_{1}\in (0,1)\). We can thus define the function \(h:[0,1]\rightarrow \mathbb {R}\) such that \(h(\hat {n}):=\hat {n} w_{1} (\hat {n}, 1-\hat {n})\). It then follows that by definition
for all z = n ≥ 2 and \(\hat {n}_{1}\in (0,1)\). Given that by AGG \(\hat {n}_{1} w_{1}(\hat {n}_{1},1-\hat {n}_{1})\) is continuous for \(\hat {n}_{1}\in (0,1)\) then h(.) is continuous on (0,1).
Step 2. Let z = n = 1, and assume to split into two neighborhoods the neighborhood 1 where \(\frac {P_{1}}{N_{1}}-\zeta >0\), then one obtains two neighborhoods of relative sizes \(\hat {n}_{1}\) and \(1-\hat {n}_{1}\). INV-S then implies that \(w_{1}(1) = \hat {n} w_{1} (\hat {n},1-\hat {n}) + (1- \hat {n}) w_{2} (\hat {n},1-\hat {n})\). Let h(1) := w1(1), then one obtains for z = n = 2, \((1-\hat {n}) w_{2}(\hat {n},1-\hat {n})=w_{1}(1)-\hat {n} w_{1} (\hat {n},1-\hat {n})\), that is \((1-\hat {n}) w_{2} (\hat {n},1-\hat {n}) = h(1)-h(\hat {n})\), in other words \((1-\hat {n}) w_{2}(\hat {n},1-\hat {n})=h(\hat {n}+(1-\hat {n})) - h(\hat {n})\). This gives the definition of w2(.) for z = n = 2.
The argument could be further generalized. Let z = n = 2, assume that \(\frac {P_{1}}{N_{1}}>\zeta \), while \(\frac {P_{2}}{N_{2}}=\zeta \). Then split neighborhood 1 of relative size \(\hat {n}\) into two neighborhoods of relative sizes respectively \(\hat {n}_{1}\) and \(\hat {n}_{2}\) such that \(\hat {n}_{1} + \hat {n}_{2}=\hat {n}\), and, either leave neighborhood 2 unaffected, or split it into many others. According to INV-S it follows that \(\hat {n} \left (\frac {P_{1}}{N_{1}} - \zeta \right ) \cdot w_{1} (\hat {n},1-\hat {n})=\hat {n}_{1} \left (\frac {P_{1}}{N_{1}} - \zeta \right ) w_{1}(\hat {n}_{1},\hat {n}_{2},\hat {n}_{3},\ldots ,\hat {n}_{z^{\prime }}) + \hat {n}_{2} \left (\frac {P_{1}}{N_{1}} - \zeta \right ) w_{2}(\hat {n}_{1},\hat {n}_{2},\hat {n}_{3},\ldots , \hat {n}_{z^{\prime }})\) where \(z^{\prime } = n\geq 3\) and \(\hat {n}_{3}+\ldots + \hat {n}_{z^{\prime }} = 1-\hat {n}=1-\hat {n}_{1}-\hat {n}_{2}\).
That is, \((\hat {n}_{1}+\hat {n}_{2}) \cdot w_{1} (\hat {n}_{1} + \hat {n}_{2}, 1-\hat {n}_{1}-\hat {n}_{2})=\hat {n}_{1}w_{1}(\hat {n}_{1},\hat {n}_{2},\hat {n}_{3},\ldots ,\hat {n}_{z^{\prime }}) + \hat {n}_{2} w_{2}(\hat {n}_{1},\hat {n}_{2},\hat {n}_{3},\ldots ,\hat {n}_{z^{\prime }})\). Recalling that \(\hat {n}_{1} w_{1} (\hat {n}_{1},\hat {n}_{2},\hat {n}_{3},\ldots ,\hat {n}_{z}) = h(\hat {n}_{1})\) for all z = n ≥ 2 and \(\hat {n}_{1}\in (0,1)\), one obtains
for all \(z^{\prime } = n\geq 2\) and \(\hat {n}_{1} + \hat {n}_{2}\in (0,1],\hat {n}_{1},\hat {n}_{2}\in (0,1)\).
By replicating the same logic and splitting into three neighborhoods the first one, then one can derive the definition of w3(.) from
for all \(z^{\prime } = n\geq 3\) and \(\hat {n}_{1}+\hat {n}_{2}+\hat {n}_{3}\in (0,1],\hat {n}_{1},\hat {n}_{2},\hat {n}_{3}\in (0,1)\).
We can then obtain in general that \(\frac {N_{i}}{\bar {N}_{z}} \cdot w_{i} \left (\frac {N_{1}}{\bar {N}_{z}},\ldots , \frac {N_{i}}{\bar {N}_{z}},\ldots , \frac {N_{z}}{\bar {N}_{z}}\right ) =h \left (\frac {\bar {N}_{i}}{\bar {N}_{z}}\right ) - h\left (\frac {\bar {N}_{i-1}}{\bar {N}_{z}}\right )\) for i = 1, 2,…,z and z = n where \(\frac {\bar {N}_{0}}{\bar {N}_{z}}:=0\) and h(0) := 0. If z = n then we have that \(\bar {N}_{z}=N\), leading to
for i = 1, 2,…,n where \(\frac {\bar {N}_{0}}{N}:=0\) and h(0) := 0.
As pointed out the function wi(.) does not depend on ζ, therefore even if it is derived under the assumption that ζ is such that z = n, the specification also holds for any ζ ∈ [0, 1), and therefore for any z ≤ n, provided that z ≥ 1 as required in the definition of AGG. □
Lemma 2
Let \(\mathcal {A}\in {\Omega }\), ζ ∈ [0, 1), and z ≥ 1,UP(.; ζ) satisfies AGG, INV-S, INV-T if and only if there exist a continuous functions \(A:[0,1]^{2}\rightarrow \mathbb {R}_{+}\) and \(\beta _{0},\gamma _{0}\in \mathbb {R}\) such that:
with \(\bar {N}_{0}:=0\).
Proof
We take the result from Lemma 1 and investigate the implications on the specification of UP(.; ζ) generated by further imposing INV-T. We leave to the reader to check that the obtained specification of UP(.; ζ) satisfies all axioms, here we focus on the “only if” part of the lemma.
For z = 1, INV-T does not hold. Note that when z = 1 the specification of UP(.; ζ) in the lemma is consistent with the one derived in Lemma 1 where h(1) = β0 if z = n = 1. While the specification in the lemma for h(.) that is valid also when z = 1 < n, will be obtained in the next general part of the proof.
We set z ≥ 2 and consider the transfers involved in the definition of INV-T. Note that with z = 2, the axiom is satisfied by construction given that it involves two transfers of population taking place in opposite directions and therefore their effects cancel out leading to the initial configuration \(\mathcal {A}\).
Without loss of generality we assume that there are z ≥ 2 neighborhoods with highly concentrated poverty with \(\frac {P_{i}}{N_{i}} \geq \zeta \) and such that their population size is equal, that is Ni = N0 for i = 1, 2,…,z. It follows that their relative population size within this set of neighborhoods is \(\frac {N_{i}}{\bar {N}_{z}} = \frac {1}{z}\), with \(\frac {\bar {N}_{i}}{\bar {N}_{z}} = \frac {i}{z}\).
Moreover, we consider first the case where ζ ∈ [0, 1) is such that for a given \(\mathcal {A}\in {\Omega }\) we have z = n ≥ 2.
Consider the effect of the combined transfers of population in INV-T, and apply them to the specification derived in Lemma 1. Note that these transfers take place among neighborhoods in {1, 2,…,z} and do not affect the components \(A\left (\frac {P}{N}, \frac {\bar {P}_{z}}{\bar {N}_{z}}\right )\) and \(\left [h\left (\frac {\bar {N}_{i}}{N}\right ) - h \left (\frac {\bar {N}_{i-1}}{N}\right )\right ]\) but only the distributions of \(\left (\frac {P_{i}}{N_{i}} - \zeta \right )\). The application of the transfers in INV-T leads to the following condition
for ε > 0, satisfying the conditions specified in INV-T for all i, j ∈{1, 2,…,z − 1} with z = n ≥ 2.
It then follows that \(\left [h\left (\frac {j+1}{z}\right ) - h\left (\frac {j}{z}\right )\right ] - \left [h\left (\frac {j}{z}\right ) - h\left (\frac {j-1}{z}\right )\right ] = \left [h\left (\frac {i+1}{z}\right ) - h\left (\frac {i}{z}\right )\right ] - \left [h\left (\frac {i}{z}\right ) - h\left (\frac {i-1}{z}\right )\right ]\) for all i, j ∈{1, 2,…,z − 1}, and for all z = n ≥ 2. Thus, we have that \(\left [h\left (\frac {i+1}{z}\right ) - h\left (\frac {i}{z}\right )\right ] - \left [h\left (\frac {i}{z}\right ) - h\left (\frac {i-1}{z}\right )\right ]\) does not depend on i but eventually only on 1/z. In general, there exists a function g(1/z) such that
for all i ∈{1, 2,…,z − 1}, all z = n ≥ 2.
To simplify the exposition, denote \(\frac {1}{z}=\sigma \) and let f(i) := h(iσ) for a fixed σ. We then have
for all i ∈{1, 2,…,z − 1}, all z = n ≥ 2.
Let d(i) := f(i) − f(i − 1) for i ∈{1, 2,…,z − 1} with by construction f(0) = h(0) = 0. Thus, d(1) + d(2) + … + d(i) = f(i) − f(0) = f(i). We then have
for i ∈{1, 2,…,z − 1}, that leads to d(j) = d(1) + (j − 1)g(σ) for i ∈{1, 2,…,z} and all z = n ≥ 2.
Thus, \(f(i)={\sum }_{j=1}^{i}d(j) = {\sum }_{j=1}^{i}d(1) +(j-1)g (\sigma )\), with f(1) = d(1). It follows that
for i ∈{1, 2,…,z}, z = n ≥ 2. Recalling that f(i) := h(i/z) we have
for i ∈{1, 2,…,z}, z = n ≥ 2. Given that i and z can take any pair of natural number values such that i ≤ z = n, and that h(0) = 0 by construction, then the above formula is a functional equation that allows to specify the value of the function h(.) for all rational numbers in [0, 1].
Let i = z, we then obtain
note that the value of h(1) is constant and therefore independent from z, rearranging it then follows that for any z = n ≥ 2 it holds that
Recalling the derivation of \(h\left (\frac {i}{z}\right )\) and inserting \(h\left (\frac {1}{z}\right )\) one obtains
for i ∈{1, 2,…,z}, z = n ≥ 2.
Note that \(\frac {i}{z}\) are unaffected if both i and z = n are replicated r times for r ∈{1, 2, 3, 4,…}, we then obtain \(h\left (\frac {i}{z} \right ) = h\left (\frac {ri}{rz}\right ) = h(1) \frac {ri}{rz} - \frac {ri(rz-ri)}{2}g \left (\frac {1}{rz}\right ) =h(1) \frac {i}{z}-r^{2} \frac {i(z-i)}{2}g \left (\frac {1}{rz}\right )\) for all r. As a result it should hold that
for all r, z ∈{2, 3, 4,…} with z = n, thus, \(g\left (\frac {1}{rz}\right ) = \frac {1}{r^{2}}g \left (\frac {1}{z}\right )\). By switching z with r one obtains \(g\left (\frac {1}{rz}\right ) = \frac {1}{z^{2}}g \left (\frac {1}{ r}\right )\). As a result \(\frac {1}{r^{2}}g \left (\frac {1}{z}\right ) = \frac {1}{z^{2}}g \left (\frac {1}{r}\right )\) for all r, z ∈{2, 3, 4,…}, that is, \(z^{2}g \left (\frac {1}{z}\right ) =r^{2}g \left (\frac {1}{r}\right )\) for all r, z. Thus, there exists a constant \(\gamma _{0}\in \mathbb {R}\) such that \(z^{2}g\left (\frac {1}{z}\right ) =-\gamma _{0}\), leading to \(g\left (\frac {1}{z}\right ) =-\frac {\gamma _{0}}{z^{2}}\) for all z ∈{2, 3, 4,…}. By substituting into the definition of \(h\left (\frac {i}{z}\right )\) and letting \(h(1) =\beta _{0}\in \mathbb {R}\) it follows that \(h\left (\frac {i}{z}\right ) =\beta _{0}\frac {i}{z}+\frac {\gamma _{0}}{2} \frac {i}{z} \frac {(z-i)}{z}\). Note that we have derived the specification of h(.) under the assumption that z = n ≥ 2, then replacing z with n we obtain
Recall that \(\frac {i}{n}\) by construction could be any rational number in (0, 1], with h(0) = 0 already set in Lemma 1. Given that the set of rational numbers is dense in (0, 1] and that h(.) is continuous in that interval the result could be extended to all real numbers in [0, 1], with h(0) = 0. Recalling that \(\frac {\bar {N}_{i}}{N} =\frac {i}{n}\) we can then write more generally
Consider the weighting function \(\left [ h\left (\frac {\bar {N}_{i}}{N}\right ) - h\left (\frac {\bar {N}_{i-1}}{N}\right )\right ]\) from Lemma 1, it can then be specified as:
By substituting into the specification of \(UP(\mathcal {A};\zeta )\) in Lemma 1, one obtains the results presented in this lemma.
Recall that we have derived the result under the assumption that for ζ ∈ [0, 1) we have z = n ≥ 2 and note that the function h(.) does not depend on ζ. In order to extend the result to all cases where n ≥ z ≥ 2 it needs to be checked that the obtained functional form for h(.) allows to satisfy INV-T also when z < n. Note that, as in Eq. 7, the application of the transfers in INV-T when n > z ≥ 2 requires that the following condition
has to be satisfied for all i, j ∈{1, 2,…,z − 1}, for n > z ≥ 2, for ε > 0. That is, after substituting for the derived specification of \(h\left (\frac {\bar {N}_{i}}{N}\right ) - h\left (\frac {\bar {N}_{i-1}}{N}\right )\), it should be verified that
Note that \(\left [h\!\left (\frac {\bar {N}_{i}}{N}\right ) - h\!\left (\frac {\bar {N}_{i-1}}{N}\right )\right ] - \left [h\left (\frac {\bar {N}_{i+1}}{N}\right ) - h\!\left (\frac {\bar {N}_{i}}{N}\right ) \right ]\) equals \(\beta _{0} \frac {N_{i}}{N} + \frac {\gamma _{0}}{2} \frac {N_{i}}{N} \!\left (\frac {(N-\bar {N}_{i})-\bar {N}_{i-1}}{N}\right )\! -\) \(\beta _{0} \frac {N_{i+1}}{N} - \frac {\gamma _{0}}{2} \frac {N_{i+1}}{N} \left (\frac {(N-\bar {N}_{i+1})-\bar {N}_{i}}{N}\right )\), and recall that according to INV-T Ni = Nj = Ni+ 1 = Nj+ 1 it then follows that
This is similarly the case if we consider the neighborhood with index j. As a result INV-T holds also for n > z ≥ 2.
To complete the exposition we consider the case where n > z = 1. In this case INV-T cannot be applied, however we have already derived the required specifications for function h(.) from the previous steps of the proof. □
Lemma 3
Let \(\mathcal {A}\in {\Omega }\), ζ ∈ [0, 1), UP(.; ζ) satisfies AGG, INV-S, INV-T, INV-PL, MON, TRAN and NOR if and only if there exist β, γ ≥ 0 such that:
with \(\bar {N}_{0}:=0\), if z ≥ 1, otherwise \(UP(\mathcal {A};\zeta )=0\).
Proof
We consider the result from Lemma 2 and investigate the implications on the specification of UP(.; ζ) generated by further imposing INV-PL, MON, TRAN and NOR. We leave to the reader to check that the obtained specification of UP(.; ζ) satisfies all axioms, here we focus on the “only if” part of the lemma. Recall that, if z ≥ 1, then according to Lemma 2 it is possible to write
We first consider INV-PL(i). By applying the scale component λ > 0 one obtains that from \(\mathcal {A}\) to \(\mathcal {A}^{\prime }\) the values \(\left (\frac {P}{N}, \frac {\bar {P}_{z}}{\bar {N}_{z}}, \frac {P_{i}}{N_{i}}, \zeta \right )\) are scaled to \(\left (\lambda \frac {P}{N}, \lambda \frac {\bar {P}_{z}}{\bar {N}_{z}}, \lambda \frac {P_{i}}{N_{i}}, \lambda \zeta \right ) \), it then follows that \(UP(\mathcal {A};\zeta )=UP(\mathcal {A}^{\prime }; \lambda \zeta )\) if and only if \(A\left (\frac {P}{N}, \frac {\bar {P}_{z}}{\bar {N}_{z}}\right ) = \lambda A\left (\lambda \frac {P}{N}, \lambda \frac {\bar {P}_{z}}{\bar {N}_{z}}\right ) \geq 0\).
We take into account two cases, first when ζ = 0 and then when ζ ∈ (0, 1).
Case 1: ζ = 0. In this case holds only INV-PL(i). If ζ = 0 then \(\frac {P}{N}=\frac {\bar {P}_{z}}{\bar {N}_{z}}\). By applying INV-PL(i) it then follows that \(A \left (\frac {P}{N}, \frac {P}{N}\right ) = \lambda A\left (\lambda \frac {P}{N}, \lambda \frac {P}{N}\right )\) for λ > 0. Let \(\lambda \frac {P}{N}=1\), we obtain \(A \left (\frac {P}{N}, \frac {P}{N}\right ) = \frac {A(1,1)}{\frac {P}{N}}\), letting A(1, 1) := K ≥ 0, then \(A\left (\frac {P}{N}, \frac {P}{N}\right ) = \frac {K}{\frac {P}{N}}\).
Case 2: ζ ∈ (0, 1]. In this case \(\frac {P}{N} \leq \zeta \leq \frac {\bar {P}_{z}}{\bar {N}_{z}} \leq 1\). INV-PL(i) holds if and only if \(A\left (\frac {P}{N}, \frac {\bar {P}_{z}}{\bar {N}_{z}}\right ) = \lambda A\left (\lambda \frac {P}{N}, \lambda \frac {\bar {P}_{z}}{\bar {N}_{z}}\right ) \geq 0\) for λ > 0. Moreover, according to INV-PL(ii) one obtains \(A\left (\frac {P}{N}, \frac {\bar {P}_{z}}{\bar {N}_{z}}\right ) = A\left (\frac {P}{N}, \frac {\bar {P}_{z}}{\bar {N}_{z}} + \theta \right )\) where \(\frac {P}{N}<\zeta \leq \frac {\bar {P}_{z}}{\bar {N}_{z}}\). By INV-PL(ii) it follows that there exists a continuous function H(.) such that \(A\left (\frac {P}{N}, \frac {\bar {P}_{z}}{\bar {N}_{z}}\right ) := H\left (\frac {P}{N}\right )\) whenever \(\frac {P}{N} < \zeta \leq \frac {\bar {P}_{z}}{\bar {N}_{z}}\). Note that \(\frac {P}{N} = \frac {\bar {P}_{z}}{\bar {N}_{z}} \frac {\bar {N}_{z}}{N} + \left (1-\frac {\bar {N}_{z}}{N}\right ) \zeta ^{\prime }\) for \(\zeta ^{\prime } < \zeta \), where \(\zeta ^{\prime }\) denotes the average poverty incidence of the neighborhood with incidence below ζ. By letting \(\bar {N}_{z} \rightarrow 0\) and \({\zeta }^{\prime } \rightarrow {\zeta }\) one obtains the case where \(\frac {P}{N} \rightarrow \zeta \). We then have that \(A\left (\frac {P}{N}, \frac {\bar {P}_{z}}{\bar {N}_{z}}\right ) = H\left (\frac {P}{N}\right )\) for \(\frac {P}{N} < \frac {\bar {P}_{z}}{\bar {N}_{z}}\). Thus, by INV-PL(i) we have \(\lambda A \left (\lambda \frac {P}{N}, \lambda \frac {\bar {P}_{z}}{\bar {N}_{z}}\right ) = \lambda H \left (\lambda \frac {P}{N} \right ) = H\left (\frac {P}{N}\right ) = A\left (\frac {P}{N}, \frac {\bar {P}_{z}}{\bar {N}_{z}}\right )\) for \(\frac {P}{N}<\frac {\bar {P}_{z}}{\bar {N}_{z}}\) and for λ > 0 such that \(\lambda \frac {\bar {P}_{z}}{\bar {N}_{z}}\leq 1\). Suppose that there exists \(\bar {c}\) such that \(\lambda \frac {P}{N}=\bar {c}\), then \(\lambda H\left (\lambda \frac {P}{N} \right ) = H\left (\frac {P}{N}\right )\) implies that \(\frac {\bar {c}H \left (\bar {c}\right )}{\frac {P}{N}} = H\left (\frac {P}{N}\right )\). By letting \(\bar {c} H (\bar {c}) := K\geq 0\) one obtains that \(A\left (\frac {P}{N}, \frac {\bar {P}_{z}}{\bar {N}_{z}}\right ) = H\left (\frac {P}{N}\right ) = \frac {K}{\frac {P}{N}}\) for \(\frac {P}{N} < \frac {\bar {P}_{z}}{\bar {N}_{z}}\).
Thus, we have that both
for \(\frac {P}{N} < \frac {\bar {P}_{z}}{\bar {N}_{z}} \leq 1\) and for \(\frac {P}{N} = \frac {\bar {P}_{z}}{\bar {N}_{z}} \leq 1\) that identify all possible ranges of values of the arguments of A(.).
By applying the result to the specification in Lemma 2, and letting β := β0K and γ := γ0K/2 one obtains
We now investigate the effects of MON and TRAN. According to MON considering that \(UP(\mathcal {A}^{\prime }; \zeta ) = {\sum }_{i=1}^{z} \left (\frac {\nu P_{i}}{\nu P} - \zeta \frac {N_{i}}{\nu P} \right ) \cdot \left [\beta + \gamma \left (\frac {(N-\bar {N}_{i}) - \bar {N}_{i-1}}{N}\right )\right ]\) for ν > 1, it should hold \(UP(\mathcal {A}^{\prime }; \zeta ) - UP(\mathcal {A}; \zeta )\geq 0\), this requires that
for all ν > 1, all ζ ∈ [0, 1), all \(\mathcal {A}\in {\Omega }\). Rearranging the condition, it implies that
for all ν > 1, all ζ ∈ [0, 1), all \(\mathcal {A}\in {\Omega }\).
This is the case if and only if \({\sum }_{i=1}^{z} \frac {N_{i}}{N} \cdot \left [\beta +\gamma \left (\frac {(N-\bar {N}_{i})-\bar {N}_{i-1}}{N}\right )\right ] \geq 0\) for all \(\mathcal {A}\in {\Omega }\). This condition depends on the value of z ≥ 1, and in particular, because of the construction of the weighting function wi(.) that satisfies INV-S, the condition depends only on \(\frac {\bar {N}_{z}}{N}\). In fact, in this case, because of INV-S, without loss of generality, one can consider distributions with two neighborhoods and z = 1. In this case \(N_{1} = \bar {N}_{1} = \bar {N}_{z}\) and recall that \(\bar {N}_{0}=0\). After substituting, one obtains the condition
for all \(\frac {\bar {N}_{z}}{N}\in (0,1]\). Letting \(\frac {\bar {N}_{z}}{N} =1\), that is if z = n, it follows that a necessary condition for MON to hold is β ≥ 0. Moreover, letting \(\frac {\bar {N}_{z}}{N}\rightarrow 0\), the additional derived necessary condition is β + γ ≥ 0, because otherwise, if γ < −β for sufficiently small values of \( \frac {\bar {N}_{z}}{N}\) is possible to violate the condition in Eq. 10. Both necessary conditions β ≥ 0 and β + γ ≥ 0 turn out to be sufficient for Eq. 10 to hold for all \(\frac {\bar {N}_{z}}{N}\in (0,1]\).
We consider now the restrictions required by axiom TRAN. First we consider the case where, because of the transfer, the poverty incidence in neighborhood j does not fall below ζ, that is \(P_{j}^{\mathcal {A}^{\prime }}/{N}_{j}^{\mathcal {A}^{\prime }}\geq \zeta \).
Recall moreover, that according to TRAN the considered transfer does not affect the ranking of the neighborhoods. Consider Eq. 9 and note that according to TRAN, only Pi and Pj are modified by the transfer, it should then be verified that
for j > i, with j ≤ z, and ε > 0. Thus
with \(\bar {N}_{j} - \bar {N}_{i}>0\) and \(\bar {N}_{j-1} - \bar {N}_{i-1}>0\), which implies
as a result should hold γ ≥ 0.
Note that the condition should hold for all i < j ≤ z with z ≤ 2 and therefore letting z = n, should hold for all i < j ≤ n.
We consider now the case where \({P}_{j}^{\mathcal {A}^{\prime }}/{N}_{j}^{\mathcal {A}^{\prime }}<\zeta \), with j = z, by applying TRAN, it follows that
where \(0\leq \varepsilon ^{\prime } \leq \varepsilon \) with \(\varepsilon ^{\prime } = {\min \limits } \{\varepsilon ,P_{j}-\zeta N_{j} \}\). The condition can then be simplified as
Recalling that \(\frac {(N - \bar {N}_{i}) - \bar {N}_{i-1}}{N} > \frac {(N-\bar {N}_{j}) - \bar {N}_{j-1}}{N}\) for j > i, that \(\varepsilon ^{\prime } \leq \varepsilon \), and that β ≥ 0, then γ ≥ 0 is sufficient to verify that \(UP(\mathcal {A}^{\prime };\zeta ) \geq UP(\mathcal {A};\zeta )\).
Thus, γ ≥ 0 is necessary and sufficient for TRAN to hold. By combining with the parametric restrictions derived by applying MON one obtains β ≥ 0, and γ ≥ 0.
All derivations illustrated so far consider the case where z ≥ 1. Note that for\(\mathcal {A}\in {\Omega }\) and given ζ ∈ (0, 1) it is possible to take into account also configurations where \(\frac {P_{i}}{N_{i}}<\zeta \) for all neighborhoods i. In this case the value of the index is derived by considering axiom NOR. For all these configurations the value of the index coincides with the infimum of the index taken over all the other possible configurations in \(\mathcal {A}\) where z ≥ 1. Consider the obtained derivation of UP for z ≥ 1, where
with β, γ ≥ 0, note that the first term in the summation \(\frac {P_{i} - \zeta N_{i}}{P}\geq 0\) is non-increasing in i, and that the term \(\frac {(N-\bar {N}_{i}) - \bar {N}_{i-1}}{N}\) is also non-increasing in i. It follows that, given that β, γ ≥ 0 also \(\beta +\gamma \left (\frac {(N-\bar {N}_{i}) - \bar {N}_{i-1}}{N}\right )\) is non-increasing in i. The summation in Eq. 11 is then minimized, for each z ≥ 1 if the terms \(\frac {P_{i} - \zeta N_{i}}{P}\) are equalized. Given that \(\frac {P_{i}-\zeta N_{i}}{P}\geq 0\) then the minimum for each z ≥ 1 is obtained for Pi − ζNi = 0 for all i ≤ z. It follows that in this case UP = 0.
Thus, by NOR the value of the index is 0 when \(\frac {P_{i}}{N_{i}}<\zeta \) for all i.
To complete the proof we rearrange the specification of UP(.; ζ) in Eq. 11. We can rewrite:
Note that
in fact \({\sum }_{i=1}^{z} N_{i} \left (\bar {N}_{z} - \bar {N}_{i} - \bar {N}_{i-1}\right ) = {\sum }_{i=1}^{z} N_{i} \bar {N}_{z} - {\sum }_{i=1}^{z} N_{i} (\bar {N}_{i} + \bar {N}_{i-1}) = (\bar {N}_{z})^{2} - {\sum }_{i=1}^{z} (\bar {N}_{i} - \bar {N}_{i-1}) (\bar {N}_{i} + \bar {N}_{i-1})\) where \({\sum }_{i=1}^{z} (\bar {N}_{i} - \bar {N}_{i-1}) (\bar {N}_{i} + \bar {N}_{i-1}) = {\sum }_{i=1}^{z} (\bar {N}_{i})^{2} - (\bar {N}_{i-1})^{2} = (\bar {N}_{z})^{2}\). It then follows that the term \({\sum }_{i=1}^{z} \left (\frac {P_{i}-\zeta N_{i}}{P}\right ) \cdot \left (\frac {(\bar {N}_{z} - \bar {N}_{i}) - \bar {N}_{i-1}}{\bar {N}_{z}} \right )\) simplifies to \({\sum }_{i=1}^{z} \frac {P_{i}}{P} \cdot \left (\frac {(\bar {N}_{z} - \bar {N}_{i}) - \bar {N}_{i-1}}{\bar {N}_{z}}\right )\). Thus, we obtain:
□
In order to complete the proof of the Theorem one has to link the result in Lemma 3 with the Gini index formula \(G(\mathcal {A};\zeta )\). Next lemma provides this link.
Lemma 4
Let \(\mathcal {A}\in {\Omega }\), ζ ∈ [0, 1), and z ≥ 1, then
Proof
The Gini index G(.; ζ) can be written as follows:
Thus
We now develop the first term appearing in squared brackets in Eq. 12, denoted \(\max \limits \) in short-hand notation, to show that it can written as a function of the rank weights. First, let develop the double summations term as follows:
After subtracting \({\sum }_{i=1}^{z} \frac {N_{i}}{\bar {N}_{z}} \frac {P_{i}}{N_{i}}\) we obtain
As a result \(\bar {P}_{z}G(\mathcal {A};\zeta )={\sum }_{i=1}^{z} P_{i} \left (\frac {\bar {N}_{z} - \bar {N}_{i}}{\bar {N}_{z}} - \frac {\bar {N}_{i-1}}{\bar {N}_{z}}\right )\), after dividing both sides by P we obtain the result in the lemma. □
By substituting from Lemma 4 into the specification of Lemma 3 in Eq. 8 for z ≥ 1, we obtain the specification of UP(.; ζ) in the Theorem for z ≥ 1 :
To complete the proof we show that all axioms are independent, meaning that it is possible to derive alternative functional forms for UP(.; ζ) by dropping one of the axioms and considering all the others.
Drop NOR: consider Eq. 3 for z ≥ 1 and set UP(.; ζ) = k < 0 in all other cases.
Drop TRAN: consider Eq. 3 with γ = − 1 and β = 0 for z ≥ 1, and set \(UP(.;\zeta )=-\sup \{\frac {{\bar {N}_{z}}}{N}\frac {\bar {P}_{z}}{P}\cdot G(\mathcal {A};\zeta )+\frac {N{-\bar {N}_{z}}}{N}{\frac {\bar {P}_{z}-\zeta {\bar {N}_{z}}}{P}}:\mathcal {A}\in {\Omega } \) with z ≥ 1} in all other cases.
Drop MON: consider Eq. 3 with γ = 0 and β = − 1 for z ≥ 1, and set UP(.; ζ) = − 1 in all other cases.
Drop INV-PL: consider Eq. 3 multiplied by P/N for z ≥ 1, and set UP(.; ζ) = 0 in all other cases.
Drop INV-T: consider
with \({\bar {N}_{0}:=0}\) for z ≥ 1, and set UP(.; ζ) = 0 in all other cases.
Drop INV-S: consider
for z ≥ 1, and set UP(.; ζ) = 0 in all other cases.
Drop AGG: consider
for z ≥ 1, and set UP(.; ζ) = 0 in all other cases. QED.
1.2 A.2 Proof of Corollary 4
Proof
Let \(p_{i}=\frac {P_{i}}{N_{i}}\) and \(s_{i}=\frac {N_{i}}{N}\) denote the poverty incidence and population share of neighborhood i, respectively.
Let \(\mathbf {p} = \left (p_{1},\ldots ,p_{n}\right )^{T}\) be the n × 1 vector of neighborhood poverty rates sorted in decreasing order and \(\mathbf {s} = \left (s_{1}, \ldots , s_{n}\right )^{T}\) be the n × 1 vector of the corresponding population shares. A urban poverty configuration is fully identified by the pair (s, p), and is used interchangeably. Let 1n being the n × 1 vector with each element equal to 1, P is the n × n skew-symmetric matrix:
where \(\bar {p}\) is the overall poverty rate in the city. The elements of P are the n2 relative pairwise differences between the neighborhood poverty incidences as ordered in p. Let \(\mathbf {S}=\mathit {diag}\left \{\mathbf {s}\right \}\) be the n × n diagonal matrix with diagonal elements equal to the population shares in s, and G be a n × n G-matrix (a skew-symmetric matrix whose diagonal elements are equal to 0, with upper diagonal elements equal to − 1 and lower diagonal elements equal to 1) (Silber 1989). The Gini index of urban poverty is expressed in matrix form:
where the matrix \(\tilde {\mathbf {G}}=\mathbf {S}\mathbf {G}\mathbf {S}\) is the weighting G-matrix, a generalization of the G-matrix introduced by Mussini and Grossi (2015) to add weights in the calculation of the Gini index. The change in urban poverty from t to \(t^{\prime }\) is measured by the difference between the Gini index in \(t^{\prime }\) and the Gini index in t:
Equation 18 can be broken down into three components by applying the matrix approach used in Mussini and Grossi (2015) and in Mussini (2017). The three components separate the contributions of changes in neighborhood population shares, ranking of neighborhoods and disparities between neighborhood poverty rates. Let \(\mathbf {s}_{t|t^{\prime }}\) stand for the n × 1 vector of the t population shares arranged by the decreasing order of the corresponding \(t^{\prime }\) poverty rates. Let \(\lambda = \bar {p}_{t^{\prime }}/\bar {p}_{t^{\prime }|t}\) be the ratio of the actual \(t^{\prime }\) overall poverty rate to the fictitious \(t^{\prime }\) overall poverty rate which is the weighted average of \(t^{\prime }\) poverty rates where the weights are the corresponding population shares in t. After defining \(\mathbf {S}_{t|t^{\prime }} = \mathit {diag} \left \{\mathbf {s}_{t|t^{\prime }}\right \}\), the Gini index of \(t^{\prime }\) neighborhood poverty rates calculated by using the t neighborhood population shares is
where \(\tilde {\mathbf {G}}_{t|t^{\prime }} = \mathbf {S}_{t|t^{\prime }}\mathbf {G}\mathbf {S}_{t|t^{\prime }}\) is the weighting G-matrix obtained by using the neighborhood population shares in t instead of those in \(t^{\prime }\). In Eq. 22, the multiplication of \({\mathbf {P}}_{t^{\prime }}^{T}\) by λ ensures that the pairwise differences between the \(t^{\prime }\) neighborhood poverty incidences are divided by \(\bar {p}_{t^{\prime }|t}\) instead of \(\bar {p}_{t^{\prime }}\). By adding and subtracting \(G\left (\mathbf {s}_{t|t^{\prime }},\mathbf {p}_{t^{\prime }}\right )\) in Eq. 21, the contribution to ΔUP due to changes in neighborhood population shares can be separated from that attributable to changes in disparities between neighborhood poverty rates:
where \(\mathbf {W} = \tilde {\mathbf {G}}_{t^{\prime }} - \lambda \tilde {\mathbf {G}}_{t|t^{\prime }}\). Component W measures the effect of changes in neighborhood population shares. A positive value of W indicates that the weights assigned to more unequal pairs of neighborhoods are larger in \(t^{\prime }\) than in t, increasing urban poverty from t to \(t^{\prime }\). A negative value of W indicates that the weights assigned to more unequal pairs of neighborhoods are smaller in \(t^{\prime }\) than in t, reducing urban poverty.
The difference enclosed within square brackets on the right-hand side of Eq. 23 can be additively split into two components: one component measuring the re-ranking of neighborhoods, a second component measuring the change in disparities between neighborhood poverty rates. Let \(\mathbf {p}_{t^{\prime }|t}\) be the n × 1 vector of \(t^{\prime }\) neighborhood poverty rates sorted in decreasing order of the respective t neighborhood poverty rates, and B be the n × n permutation matrix re-arranging the elements of \(\mathbf {p}_{t^{\prime }}\) to obtain \(\mathbf {p}_{t^{\prime }|t}\), that is \(\mathbf {p}_{t^{\prime }|t} = \mathbf {B}\mathbf {p}_{t^{\prime }}\). Matrix \(\mathbf {P}_{t^{\prime }|t} = \left (1/\bar {p}_{t^{\prime }|t}\right ) \left (\mathbf {1}_{n}\mathbf {p}^{T}_{t^{\prime }|t}-\mathbf {p}_{t^{\prime }|t}\mathbf {1}^{T}_{n}\right )\) contains the n2 relative pairwise differences between the neighborhood poverty rates as arranged in \(\mathbf {p}_{t^{\prime }|t}\). The concentration index of the \(t^{\prime }\) neighborhood poverty rates sorted by the t neighborhood poverty rates, calculated by using the t population shares, is defined as follows:
By using permutation matrix B, the concentration index \(C\left (\mathbf {s}_{t},\mathbf {p}_{t^{\prime }|t}\right )\) can be re-written as a function of \(\mathbf {P}_{t^{\prime }}\) instead of \(\mathbf {P}_{t^{\prime }|t}\). Since \(\mathbf {P}_{t^{\prime }|t}=\mathbf {B}\lambda \mathbf {P}_{t^{\prime }}\mathbf {B}^{T}\), the concentration index \(C\left (\mathbf {s}_{t}, \mathbf {p}_{t^{\prime }|t}\right )\) expressed as a function of \(\mathbf {P}_{t^{\prime }}\) becomes
By adding \(C\left (\mathbf {s}_{t},\mathbf {p}_{t^{\prime }|t}\right )\) as expressed in Eq. 24 and subtracting it as expressed in Eq. 25 to the difference enclosed within square brackets on the right-hand side of Eq. 23, we obtain
where \(\mathbf {R}=\tilde {\mathbf {G}}_{t|t^{\prime }}-\mathbf {B}^{T}\tilde {\mathbf {G}}_{t}\mathbf {B}\) and \(\mathbf {D}=\mathbf {P}_{t^{\prime }|t}-\mathbf {P}_{t}\). Component R measures the effect of re-ranking of neighborhoods from t to \(t^{\prime }\) and its contribution to the change in urban poverty is always non-negative. The nonzero elements of R indicate the pairs of neighborhoods which have re-ranked from t to \(t^{\prime }\).
Component D measures the effect of disproportionate changes in neighborhood poverty rates. The generic (i, j)-th element of D compares the relative difference between the t poverty rates of the neighborhoods in positions j and i in pt with the relative difference between the \(t^{\prime }\) poverty rates of the same two neighborhoods in \(\mathbf {p}_{t^{\prime }|t}\). A positive (negative) value of D indicates that relative disparities in neighborhood poverty rates have increased (decreased) from t to \(t^{\prime }\), increasing (reducing) urban poverty. If all neighborhood poverty rates have changed in the same proportion from t to \(t^{\prime }\), then D = 0.
Given Eqs. 23 and 26, a three-term decomposition of ΔUP is obtained:
Since component D would not reveal changes in neighborhood poverty rates if all neighborhood poverty rates changed in the same proportion, this component is split into two further terms: one measuring the change in the city poverty rate, the second measuring the changes in disparities between neighborhood poverty rates by assuming that the city poverty rate remains the same from t to \(t^{\prime }\). Let c stand for the change in the city poverty rate by assuming that neighborhood population shares are unchanged from t to \(t^{\prime }\):
Let \({\mathbf {p}}_{t^{\prime }|t}^{c} = \mathbf {p}_{t} + c\mathbf {p}_{t}\) be the vector of neighborhood poverty rates we would observe in \(t^{\prime }\) if every neighborhood poverty rate changed by proportion c. This implies that \({\bar {p}}_{t^{\prime }|t}^{c} = \bar {p}_{t^{\prime }|t}\). Vector \(\mathbf {p}_{t^{\prime }|t}\) can be expressed as
where the elements of vector \({\mathbf {p}}_{t^{\prime }|t}^{\delta }\) are the element-by-element differences between vectors \(\mathbf {p}_{t^{\prime }|t}\) and \({\mathbf {p}}_{t^{\prime }|t}^{c}\). Since \({\mathbf {p}}_{t^{\prime }|t}^{c}=\mathbf {p}_{t}+c\mathbf {p}_{t}\), \(\mathbf {p}_{t^{\prime }|t}\) can be re-written as
where the elements of \(\mathbf {p}^{e}_{t^{\prime }|t}\) account for disproportionate changes in neighborhood poverty rates from t to \(t^{\prime }\), as \(\mathbf {p}^{e}_{t^{\prime }|t}\) would equal pt if there were no disproportionate changes in neighborhood poverty rates. Given equations above, matrix \(\mathbf {P}_{t^{\prime }|t}\) can be written as
Since matrix D in Eq. 27 is obtained by subtracting Pt from \(\mathbf {P}_{t^{\prime }|t}\), D can be re-written as
By replacing D in Eq. 27 with its expression in Eq. 31, the decomposition of the change in urban poverty becomes
□
1.3 A.3 Proof of Corollary 5
Proof
Building on the Rey and Smith (2013) spatial decomposition of the Gini index and the spatial decomposition of the change in inequality in Mussini (2020), ΔUP, W, R and E can be broken down into spatial components. Let Nt be the n × n spatial weights matrix having its (i, j)-th entry equal to 1 if and only if the (i, j)-th element of Pt is the relative difference between the poverty rates of two neighborhoods that are spatially close, otherwise the (i, j)-th element of Nt is 0. Using the Hadamard product,Footnote 1 the relative pairwise differences between the poverty rates of neighborhoods that are spatially close can be selected from Pt:
For each pair of neighborhoods, the relative difference between their \(t^{\prime }\) poverty rates in \(\mathbf {P}^{e}_{t^{\prime }|t}\) has the same position as the relative difference between their t poverty rates in Pt. Thus, Nt also selects the relative pairwise differences between neighbors from \(\mathbf {P}^{e}_{t^{\prime }|t}\):
Since \(\mathbf {E} = \mathbf {P}^{e}_{t^{\prime }|t}-\mathbf {P}_{t}\), the Hadamard product between Nt and E is a matrix with nonzero elements equal to the elements of E pertaining to neighborhoods that are spatially close:
Let \(\mathbf {N}_{t^{\prime }}\) be the n × n spatial weights matrix having its (i, j)-th entry equal to 1 if and only if the (i, j)-th element of \(\mathbf {P}_{t^{\prime }}\) is the relative difference between the poverty rates of two neighborhoods that are spatially close, otherwise the (i, j)-th element of \(\mathbf {N}_{t^{\prime }}\) is 0. The Hadamard product of \(\mathbf {N}_{t^{\prime }}\) and \(\mathbf {P}_{t^{\prime }}\) is the matrix
The nonzero elements of \(\mathbf {P}_{N,t^{\prime }}\) are the relative pairwise differences between the \(t^{\prime }\) poverty rates of neighborhoods that are in spatial proximity.
The decomposition of the change in the neighborhood component of urban poverty is obtained by replacing \(\mathbf {P}_{t^{\prime }}\) and E in Eq. 32 with \(\mathbf {P}_{N,t^{\prime }}\) and EN, respectively:
Let Jn be the matrix with diagonal elements equal to 0 and extra-diagonal elements equal to 1, the matrix with nonzero elements equal to the relative pairwise differences between the \(t^{\prime }\) poverty rates of neighborhoods that are not in spatial proximity is
The matrix selecting the elements of E pertaining to the pairs of neighborhoods that are not spatially close is
The decomposition of the change in the non-neighborhood component of urban poverty is obtained by replacing \(\mathbf {P}_{t^{\prime }}\) and E in Eq. 32 with \(\mathbf {P}_{nN,t^{\prime }}\) and EnN, respectively:
Given Eqs. 40 and 37, the spatial decomposition of the change in urban poverty is
□
Appendix B: Additional results
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Andreoli, F., Mussini, M., Prete, V. et al. Urban poverty: Measurement theory and evidence from American cities. J Econ Inequal 19, 599–642 (2021). https://doi.org/10.1007/s10888-020-09475-2
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DOI: https://doi.org/10.1007/s10888-020-09475-2