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An Inter-temporal Relative Deprivation Index

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The paper provides an axiomatic characterization of a new class of relative deprivation indices. Relative deprivation is the feeling that an individual experiences when she compares herself with someone who is better off. We believe that individuals not only take care of their relative position with respect to others but also of their relative position with respect to their own past. Therefore, we introduce a history-regarding reference group, while in the traditional relative deprivation framework the reference group is only other-regarding. The new index is sensitive to the proximity of transfers in the reference groups: an individual may feel more deprived if an increase in achievements occurs close or far to her current position. The new index is illustrated with an application to EU countries.

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    The OECD modified scale assigns value of 1 to the household head, of 0.5 to each additional adult member and of 0.3 to each child in the household.


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We are extremely grateful to Casilda Lasso de la Vega, Conchita D’Ambrosio, Buhong Zheng, Lars Osberg, Claudio Zoli, the participants at the 19th international meeting of the Society for Social Choice and Welfare, at the 31st General IARIW Conference, at the IV GRASS workshop and at the IV Meeting of ECINEQ for their interesting suggestions and comments. We are also extremely grateful to two anonymous referees for the set of comments that helped to improve the paper. The general disclaimer applies.

Author information

Correspondence to Chiara Gigliarano.


Annex 1: Formal Statement of the Axioms Used in Sect. 3

Axiom A1.1

(Continuity) \(A({{\mathbf {I}}^{n}})\) is continuous on \(\cup _{n \in {\mathcal {A}}} {{\mathcal {I}}^n}\).

Axiom A1.2

(Monotonicity) For any \({{\mathbf {I}}^{n}}\), \({\tilde{\mathbf{I}}^{n}} \in {{\mathcal {I}}^{n}}\), if \({\tilde{\mathbf{I}}^{n}}\) is obtained from \({{\mathbf {I}}^{n}}\) by adding \(\gamma \in {{\mathbb {R}}_{+}}\) to a generic element \(I^{i}\in\) \({{\mathbf {I}}^{n}}\), then \(A({{\mathbf {I}}^{n}})\le A({\tilde{\mathbf{I}}^{n}})\).

Axiom A1.3

(Independence) For any \({{\mathbf {I}}^{n}}=(I^{1}, \ldots , I^{i}, \ldots , I^{n})\), and for any \({\tilde{\mathbf{I}}^{n}}=(\tilde{I}^{1}, \ldots , \tilde{I}^{i}, \ldots , \tilde{I}^{n})\), \({{\mathbf {I}}^{n}}, {\tilde{\mathbf{I}}^{n}} \in {{\mathcal {I}}^{n}}\), if \(A(I^{1}, \ldots , I^{i}, \ldots , I^{n}) = A(\tilde{I}^{1}, \ldots , \tilde{I}^{i}, \ldots , \tilde{I}^{n})\) and \(I^{i} = \tilde{I}^{i}\) for some i, then \(A(I^{1}, \ldots , \theta , \ldots , I^{n}) = A(\tilde{I}^{1}, \ldots , \theta , \ldots , \tilde{I}^{n})\) for any \(\theta \in {{\mathbb {R}}_{+}}\).

Axiom A1.4

(Anonimity) Given any permutation \(\pi\) of \(N\), \(A({{\mathbf {I}}^{n}})= A(I^{\pi (1)}, I^{\pi (2)}, \ldots , I^{\pi (i)}, \ldots , I^{\pi (n)})\).

Axiom A1.5

(Population Proportionality) For any \(\zeta \in {\mathbb {N}}\): \(A({{\mathbf {I}}^{n}}) =A(\underbrace{{{\mathbf {I}}^{n}}, \ldots , {{\mathbf {I}}^{n}}}_{\zeta times})\).

Axiom A1.6

(Transfer Principle) For any \({{\mathbf {I}}^{n}}, {\tilde{\mathbf{I}}^{n}}\in {{\mathcal {I}}^{n}}\), such that \({{\mathbf {I}}^{n}}=(I^{1}, \ldots , I^{i}, \ldots , I^{j}, \ldots , I^{n}), {\tilde{\mathbf{I}}^{n}}=(I^{1}, \ldots , I^{i}-\epsilon , \ldots , I^{j}+\epsilon , \ldots , I^{n})\), and \(I^{i}\le I^{j}\), then: \(A({{\mathbf {I}}^{n}})\le A({\tilde{\mathbf{I}}^{n}})\), for any \(\epsilon >0\).

Annex 2: Proofs of the Propositions

Proof of Proposition 2.1


By Continuity, Monotonicity and Independence, from Theorem 5.5 in Fishburn (1970) it follows that

$$\begin{aligned} I^{i}(D^{i})=g^{-1}\left( \sum _{d_{k}\in D^{i}}g_{k}(d_{k})\right) \end{aligned}$$

where \(g\) and \(g_{k}\) are continuous and strictly increasing, and \(k=1, 2, \ldots , m\). By Normalization, \(g(0)=g_{k}(0)=0\). By Focus we can restrict the sum only to positive comparisons:

$$\begin{aligned} I^{i}(D^{i})=g^{-1}\left( \sum _{d_{k}>0}g_{k}(d_{k})\right) . \end{aligned}$$

By Reference-group replication:

$$\begin{aligned} I^{i}(D^{i}) = I^{i}( \underbrace{D^{i}, \ldots , D^{i}}_{\zeta \, times} ) \end{aligned}$$

Similarly to Proposition 2 in Ebert and Moyes (2002), it follows that:

$$\begin{aligned} I^{i}(D^{i})=g^{-1} \left( \frac{1}{\zeta }\left( \sum _{d_{k}>0}g_{k}(d_{k})\right) \right) \end{aligned}$$

Without loss of generality, we choose \(\zeta =m\).

We can split the sum into the two components other-regarding and history-regarding:

$$\begin{aligned} I^{i}(D^{i})=g^{-1}\left( \sum _{\delta _{j}>0} f_{j}(\delta _{j})+\sum _{\eta _{t}>0} f_{t}(\eta _{t})\right) \end{aligned}$$

By Anonymity \(f_{j}=f\) for each \(j\in R^{i}_{O}\):

$$\begin{aligned} I^{i}(D^{i})=g^{-1}\left( \sum _{\delta _{j}>0} f(\delta _{j})+\sum _{\eta _{t}>0} f_{t}(\eta _{t})\right) . \end{aligned}$$


Proof of Proposition 2.2


Close (Far) Transfer Principle is equivalent to assume that \(I^{i}(D^{i})\) is strictly Schur-concave (convex), see Theorem A.4 of Marshall and Olkin (1979) Chapt. 3, p. 57, Axiom 2. By Marshall and Olkin (Marshall and Olkin (1979), theorem C.1.a., p.64) this condition is equivalent to \(g_{t}(\cdot )\) and \(f_{t}(\cdot )\) being strictly concave (convex).\(\square\)

Proof of Proposition 3.1


By Continuity, Monotonicity and Independence from Theorem 5.5 in Fishburn (1970), it follows that:

$$\begin{aligned} A({{\mathbf {I}}^{n}}) =g^{-1}\left( \sum _{i=1}^n f_{i}\left( I^{i}\right) \right) \end{aligned}$$

where \(g\) and \(f_{i}\) are continuous and strictly increasing for any \(i=1, \ldots , n\). By Population Proportionality:

$$\begin{aligned} A({{\mathbf {I}}^{n}}) =A(\underbrace{{{\mathbf {I}}^{n}}, \ldots , {{\mathbf {I}}^{n}}}_{\zeta times}) \end{aligned}$$


$$\begin{aligned} A({{\mathbf {I}}^{n}})=g^{-1}\left( \frac{1}{\zeta }\sum _{i=1}^n f_{i}\left( I^{i}\right) \right) . \end{aligned}$$

Without loss of generality, we choose \(\zeta =n\) and \(g^{-1}\) as the identity function. By Anonimity \(f_{i}=f\) for each \(i=1, 2, \ldots , n\). By Kolm (1976) and Marshall and Olkin (1979), Tranfer Principle insures that \(f(\cdot )\) is convex.\(\square\)

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Ceriani, L., Gigliarano, C. An Inter-temporal Relative Deprivation Index. Soc Indic Res 124, 427–443 (2015). https://doi.org/10.1007/s11205-014-0791-7

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  • Relative deprivation
  • Inter-temporal measurement
  • Income distribution

JEL Classification

  • I32
  • D31
  • D63
  • D71
  • D81