## Abstract

This paper aims to fill some gaps in the literature concerning the sensitivity of segregation measures. We examine the definitions of regressive and progressive movement, and formally describe the requirements for these movements. As a result of this analysis, we relax a strong assumption established in the literature regarding these movements. Since these measures increase with regressive movement, we are interested in analyzing how the measures’ sensitivities vary as a function of the position of the strata. This analysis allows us to establish how the segregation measures behave with an increase in the number of people in one category in a stratum. We analyze these concepts in terms of the Gini index and the class of additively decomposable measures and analyze the sensitivity of the index of dissimilarity to regressive movements. Data from a national household survey are used to illustrate the results found in the paper.

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## Notes

- 1.
Theoretically, \(X_j \) and \(Y_j\) can be nonnegative real numbers (and not only nonnegative integers), and

*X*and*Y*can be positive real numbers (and not only positive integers). At this point, we prefer the more restrictive presentation, since it facilitates the use of the concepts in applied works. For the definitions of regressive and progressive movements, we relax assumptions \(X_j,Y_j\in {\mathbb {N}}\cup \{0\}\) for all*j*and \(X,Y\in {\mathbb {N}} \). - 2.
For the reader interested in this discussion, the Hutchens (2004) and Chakravarty and Silber (2007) notation adapted in this paper is (

*a*) \(Y_h=Y_i=Y_h^*=Y_i^*>0\), (*b*) \(X_h/Y_h<X_i/Y_i\), (*c*) \(X_h^*=X_h-X\theta \), and \(X_i^*=X_i+X\theta \) for \(0<X\theta \le X_h\), and (*d*) \(X_j^*=X_j\), and \(Y_j^*=Y_j\) for \(j\ne h,i\). In all these items, \(X_j^*\) and \(Y_j^*\) for all*j*refer to the situation after the regressive movement. - 3.
These values are the largest multiple of 1 /

*X*less than or equal to \((x_iy_h-x_hy_i)/(y_h+y_i)\) for a progressive movement in the*X*category and the largest multiple of 1 /*Y*less than or equal to \((x_iy_h-x_hy_i)/(x_h+x_i)\) for a progressive movement in the*Y*category. - 4.
Most authors sort the strata not by their relative participation but by \(X_j/(X_j+Y_j)\). It is not difficult to demonstrate that this arrangement and (2) are equivalent.

- 5.
Hutchens (2004) presents a slightly different version, with \(c=1-\varepsilon \).

- 6.
These values for \(\theta \) are \(0<\theta \le (y_i-x_i)\) if the relative participation in both cases is less than one and \(0<\theta \le (x_h-y_h)\) if greater than one. These ranges ensure that the relative participation of each case (

*h*and*i*)’ in the new distribution are still greater/less than one. For \(\theta \) larger than these values, the index of dissimilarity increases with the regressive movement proportionally to \(\theta \) minus the maximum value indicated [\((y_i-x_i)\) or \((x_h-y_h)\)]. - 7.
The new distribution considering the increase of \(\lambda X\) people in the

*i*th stratum is \({\mathbf {X}}^i_\lambda =\left[ {\begin{matrix} X_1&{}X_2&{}\ldots &{}X_i+\lambda X&{} \ldots &{}X_{J-1}&{}X_J\\ Y_1&{}Y_2&{}\ldots &{}Y_i&{} \ldots &{}Y_{J-1}&{}Y_J \end{matrix}}\right] \) and the total of*X*type people in the new distribution is \((1+\lambda )X\).

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## Acknowledgements

We are grateful to an anonymous referee for making excellent suggestions during the review process. We also acknowledge financial support from the Brazilian National Council for Technological and Scientific Development (CNPq - *Conselho Nacional de Desenvolvimento Científico e Tecnológico*).

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## Appendix

### Appendix

### Proof of Theorem 1:

Without loss of generality, consider a regressive movement in the *X* category according to Definition 1 (*a*1)-(*d*1), generating a distribution \({\mathbf {X}}^*\) from \({\mathbf {D}}\). If \(\varOmega ({\mathbf {D}})=\sum _{j=1}^J g(x_j,y_j)\), define \(h({\mathbf {D}},\theta ):=\varOmega ({\mathbf {X}}^*)-\varOmega ({\mathbf {D}})=g(x_h-\theta ,y_h)+g(x_i+\theta ,y_i)-g(x_h,y_h)-g(x_i,y_i)\), \(0<\theta \le x_h\). If \(h_\theta ({\mathbf {D}},\theta )>0\), we conclude the proof. We have \(h_\theta ({\mathbf {D}},\theta )=g_x(x_i+\theta ,y_i)-g_x(x_h-\theta ,y_h)\). Note that \(g_{xx}(x,y)>0\) implies \(g_{x}(x=x_i+\theta ,y={\bar{y}})>g_{x}(x=x_h-\theta ,y={\bar{y}})\) whenever \(x_i+\theta >x_h-\theta \). Hence, if \(y_h=y_i={\bar{y}}\), the desired result is straight, and homogeneity of degree 1 of \(g(x_j,y_j)\) is not necessary. However, if \(g(x_j,y_j)\) is homogeneous of degree 1 and \(g_{xx}(x_j,y_j)>0\), \(g_{x}(x_j,y_j)\) is an increasing function of \(x_j\) and an increasing function of \(x_j/y_j\). Thus, \(h_\theta ({\mathbf {D}},\theta )>0\) and \(\varOmega ({\mathbf {X}}^*)> \varOmega ({\mathbf {D}})\), for sufficiently small \(\theta \) whenever necessary. The result for any strict increasing transformation of \(\sum _{j=1}^J g(x_j,y_j)\) is direct because the sign of the derivative does not change. \(\square \)

### Proof of Proposition 1:

Let us consider a regressive movement in the *X* category. Similar analysis for movements in the *Y* category or for progressive movements achieve the same results *mutatis mutandis*. For the Gini index, consider a regressive movement between strata (*h*) and (*i*) with \(i=h+k\). Then, \(G(\hat{{\mathbf {X}}}^*)-G(\hat{{\mathbf {D}}})=\theta (y_{(h)} + y_{(h+k)} +2\sum _{j=h+1}^{h+k-1} y_{(j)})>0\). Hutchens (2004) demonstrates that the general measure satisfies Property 1 if \(y_h=y_i\). For the less restrictive case, we have \(F(u)=u\), \(g(x_j,y_j)=y_j[(x_j/y_j)^{1-\varepsilon }-1]/[\varepsilon (\varepsilon -1)]\) is homogeneous of degree 1 in \((x_j,y_j)\), and \(g_{xx}(x_j,y_j)=y_j^\varepsilon x_j^{-\varepsilon -1}>0\) for \(j=h,i\), \(\theta \in (0,x_h)\). Thus, from Theorem 1, \(I_\varepsilon ({\mathbf {D}})\) satisfies Property 1. \(\square \)

### Proof of Theorem 2:

**Sufficiency** (contradiction): Suppose that \(\varOmega ({\mathbf {D}})\) is a differentiable function that satisfies Property 1, \(\lambda >0\), \(\delta [\varOmega ({\mathbf {X}}_\lambda ^i)]<0\) for some *i*, and the \({\mathbf {D}}\) element of the set \({\overline{D}}=\{{\mathbf {D}}\in D_J: x_j=y_j \text{ for } \text{ all } j\}\). Therefore, \(\varOmega ({\mathbf {X}}_\lambda ^i)<\varOmega ({\mathbf {D}})\), a contradiction, because \(\varOmega ({\mathbf {D}})=0\) is the minimum value of the measure. **Necessity**: Suppose that \(\varOmega ({\mathbf {D}})\) is a differentiable function that satisfies Property 1 and \({\mathbf {D}}\notin {\overline{D}}\). Then, there is at least one *i* such that \(x_{(i)}/y_{(i)}<x_{(i+1)}/y_{(i+1)}\) for some \(1\le i<J\). Define the set \(\xi =\{i: |x_{(i)}/y_{(i)}-x_{(i+1)}/y_{(i+1)}|>0, \, 1\le i < J\}\) and the range \(0<\lambda \le \min \{x_{(i+1)}y_{(i)}/y_{(i+1)}-x_{(i)}:i\in \xi \}\ne \emptyset \), which ensures that an increase of \(\lambda X\) people in any stratum \(i\in \xi \) does not change the strata ordering and decreases the distance between the relative participation in the *i*th and (\(i+1\))th strata. Thus, \(\delta [\varOmega ({\mathbf {X}}_\lambda ^i)]<0\) for \(i\in \xi \). \(\square \)

### Proof of Theorem 3:

Assume \(\varOmega ({\mathbf {D}})\) is a differentiable function satisfying Property 1 and \({\mathbf {D}}\) in \(\overline{{\overline{D}}}=\{{\mathbf {D}}\in D_J: x_j\ne y_j \text{ for } \text{ some } j, \text { and} \, y_j>0 \,\text { for all } j \}\). Thus, considering the strata sorted by nondecreasing order according to relative participation, we have strict inequality for at least one *j*, and \(x_{(1)}/y_{(1)}<x_{(J)}/y_{(J)}<\infty \). We know that for all measures \(\varOmega ({\mathbf {D}})\) satisfying Property 1, we have \(\delta [\varOmega ({\mathbf {X}}_{\lambda }^h)]<\delta [\varOmega ({\mathbf {X}}_{\lambda }^i)]\) whenever \(h<i\) and \(\lambda >0\) is sufficiently small (if necessary, take the limit \(\lambda \downarrow 0\)). In particular, for \(h=1\), by Theorem 2 and because \(\delta [\varOmega ({\mathbf {X}}_{\lambda }^j)]\) is an increasing function in *j*, we have \(\delta [\varOmega ({\mathbf {X}}_{\lambda }^1)]<0\). Let us show that \(\delta [\varOmega ({\mathbf {X}}_{\lambda }^J)]>0\). Suppose not; then, \(\delta [\varOmega ({\mathbf {X}}_{\lambda }^J)]\le 0\) and \(\lambda >0\). This increase increases the difference between the *J*th stratum’s relative participation in relation to the other strata, so we must have \(\varOmega ({\mathbf {X}}_\lambda ^J)>\varOmega ({\mathbf {D}})\), a contradiction with \(\delta [\varOmega ({\mathbf {X}}_{\lambda }^J)]\le 0\). Therefore, \(\delta [\varOmega ({\mathbf {X}}_{\lambda }^J)]> 0\). Thus, there is a \(1<j^*<J\) associated with the relative participation value *R* of \(x_{j^*}/y_{j^*}\) such that \(i<j^* \Rightarrow \delta [\varOmega ({\mathbf {X}}_{\lambda }^i)]<0\) and \(j^*<i\le J \Rightarrow \delta [\varOmega ({\mathbf {X}}_{\lambda }^i)]> 0\) whenever \({\mathbf {D}}\in \overline{{\overline{D}}}\). \(\square \)

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### Cite this article

Botassio, D.C., Hoffmann, R. Measuring gender segregation.
*Econ Theory Bull* **8, **25–47 (2020). https://doi.org/10.1007/s40505-019-00162-8

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### Keywords

- Segregation
- Segregation measurement
- Sensitivity
- Pigou–Dalton principle
- Gender segregation

### JEL Classification

- D63
- J01
- J16
- J21