Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

An Inter-temporal Relative Deprivation Index

  • 311 Accesses

Abstract

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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. 1.

    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.

References

  1. Bossert, W., D’Ambrosio, C., & Peragine, V. (2007). Deprivation and social exclusion. Economica, 74(296), 777–803.

  2. Bossert, W., Chakravarty, S. R., & D’Ambrosio, C. (2012). Poverty and time. Journal of Economic Inequality, 10(2), 145–162.

  3. Chakraborty, A., Pattanaik, P. K., & Xu, Y. (2008). On the mean of squared deprivation gaps. Economic Theory, 34, 181–187.

  4. Chakravarty, S. R. (1997). Relative deprivation and satisfaction orderings. Keio Economic Studies, 34, 17–31.

  5. Chakravarty, S. R. (2007). A deprivation-based axiomatic characterization of the absolute Bonferroni index of inequality. Journal of Economic Inequality, 5, 339–351.

  6. Chakravarty, S. R., & Chattopadhyay, N. (1994). An ethical index of relative deprivation. Research on Economic Inequality, 5, 231–240.

  7. Chakravarty, S. R., & D’Ambrosio, C. (2006). The measurement of social exclusion. Review of Income and Wealth, 52(3), 377–398.

  8. Davis, J. A. (1959). A formal interpretation of the theory of relative deprivation. Sociometry, 22(4), 280–296.

  9. Ebert, U., & Moyes, P. (2000). An axiomatic characterization of Yitzhaki’s index of individual deprivation. Economics Letters, 68, 263–270.

  10. Ebert, U., & Moyes, P. (2002). A simple axiomatization of the Foster Greer and Thorbecke poverty orderings. Journal of Public Economic Theory, 4(4), 455–473.

  11. Esposito, L. (2010). Upper boundedness for the measurement of relative deprivation. Review of Income and Wealth, 56(3), 632–639.

  12. Eurostat. (2010). Combating poverty and social exclusion. A statistical portrait of the European Union 2010. European Union: Eurostat Statistical Books.

  13. Ferrer-i-Carbonell, A. (2005). Income and well-being: An empirical analysis of the comparison income effect. Journal of Public Economics, 89, 997–1019.

  14. Fields, G. S. (2007) Income mobility. Cornell University ILR collection working papers.

  15. Fishburn, P. (1970). Utility theory for decision making. New York: Wiley.

  16. Kolm, S. C. (1976). Unequal inequalities II. Journal of Economic Theory, 13, 82–111.

  17. Marshall, A. W., & Olkin, I. (1979). Inequalities: Theory of majorization and its applications. San Diego: Academic Press.

  18. Mendola, D., & Busetta, A. (2012). The importance of consecutive spells of poverty: A path-dependent index of longitudinal poverty. Review of Income and Wealth, 58(2), 355–374.

  19. Mukerjee, D. (2001). Measuring multidimensional deprivation. Mathematical Social Sciences, 42, 233–251.

  20. Ok, E. A., & Masatlioglu, Y. (2007). A theory of (relative) discounting. Journal of Economic Theory, 137, 214–245.

  21. Paul, S. (1991). An index of relative deprivation. Economics Letters, 36, 337–341.

  22. Podder, N. (1996). Relative deprivation, envy and economic inequality. Kyklos, 3, 353–376.

  23. Runciman, W. (1966). Relative deprivation and social justice. London, UK: Routledge.

  24. Stouffer, S., Suchman, E., De Vinney, L., Star, S., & Williams, R. (1949). The American soldier: Adjustments during army life 1. Princeton: Princeton University Press.

  25. Subramanian, S., & Majumdar, M. (2002). On measuring deprivation adjusted for group disparities. Social Choice and Welfare, 19, 265–280.

  26. Verme, P. (2011) Happiness, deprivation and the alter ego. In J. Deutsch, & J. Silber (Eds.), The measurement of individual well-being and group inqualities: Essays in memory of Z. M. Berrebi, chapter 2. London: Routledge.

  27. Yi, R., Gatchalian, K. M., & Bickel, W. K. (2006). Discounting of past outcomes. Experimental and Clinical Psychopharmacology, 14(3), 311–317.

  28. Yitzhaki, S. (1979). Relative deprivation and the gini coefficient. Quarterly Journal of Economics, 32, 321–324.

Download references

Acknowledgments

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.

Appendices

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

Proof

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 \(j\in R^{i}_{O}\):

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

\(\square\)

Proof of Proposition 2.2

Proof

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

Proof

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

$$\begin{aligned} A({{\mathbf {I}}^{n}}) =g^{-1}\left( \sum _{i=1}^n f_{i}\left( I^{i}\right) \right) \end{aligned}$$
(7)

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

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

Therefore:

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

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.\(\square\)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

  • Relative deprivation
  • Inter-temporal measurement
  • Income distribution

JEL Classification

  • I32
  • D31
  • D63
  • D71
  • D81