## Abstract

We develop a normative approach to the measurement of inequality of opportunity. That is, we measure inequality of opportunity by the welfare gain obtained in moving from the actual income distribution to the optimal income distribution of the total available income. Our study brings together the main approaches in the literature: we axiomatically characterize social welfare functions, we obtain prominent allocation rules as their optima, and we derive familiar classes of inequality of opportunity measures. Our analysis captures moreover the key philosophical distinctions in the literature: ex post versus ex ante compensation, and liberal versus utilitarian reward.

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## Acknowledgments

We thank Francisco Ferreira, Marc Fleurbaey, Jean-Jacques Herings, Erwin Ooghe, Dirk Van de gaer and audiences in Aix-en-Provence (Aix-Marseille Université), Bari (Fifth Meeting of the Society for the Study of Economic Inequality (ECINEQ)), Leuven (KU Leuven), Louvain-la-Neuve (Université catholique de Louvain) and Lund (Thirteenth Meeting of the Society for Social Choice and Welfare) for useful comments.

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## Appendix

### Appendix

###
*Proof of Proposition 1*

Let *W* be a social welfare function that satisfies ex post Pigou-Dalton and liberal Pigou-Dalton.

Assume to the contrary that market incomes are not additively separable. That is, there exist *i* and *j* in *C* and *k* and *l* in *R* such that *m*_{ik} − *m*_{il} ≠ *m*_{jk} − *m*_{jl}. Let *X* be an income distribution such that *x*_{ik} = *x*_{jk} = (*m*_{ik} + *m*_{jk})/2 and *x*_{il} = *x*_{jl} = (*m*_{il} + *m*_{jl})/2. Let \(X^{\prime }\) be an income distribution such that \({x}_{ik}^{\prime } + {x}_{il}^{\prime } = x_{ik} + x_{il}\), \({x}_{ik}^{\prime } - {x}_{il}^{\prime } = m_{ik} - m_{il}\), \({x}_{jk}^{\prime } + {x}_{jl}^{\prime } = x_{jk} + x_{jl}\) and \({x}_{jk}^{\prime } - {x}_{jl}^{\prime } = m_{jk} - m_{jl}\) with \(X^{\prime }\) and *X* coinciding everywhere else. Ex post Pigou-Dalton implies \(W(X)>W(X^{\prime })\), whereas liberal Pigou-Dalton implies \(W(X^{\prime })>W(X)\). We have a contradiction. □

The following two lemmas are used throughout the proofs. A progressive transfer is a transfer of income from a richer to a poorer individual such that the one that starts out with less money does not end up with more than the other. We say that a function is Pigou-Dalton consistent if its value increases as a result of a progressive transfer. See Olkin and Marshall (1979, pp. 10–12) for the first lemma and Dasgupta et al. (1973, p. 183) for the second lemma.

###
**Lemma 1**

For all vectors *a* and *b* in \(\mathbb {R}^{n}\), *a* is obtained from *b* by a finite sequence of progressive transfers and permutations if and only if *a* = *b**B* for some *n* × *n* bistochastic matrix *B*.

###
**Lemma 2**

Each symmetric and Pigou-Dalton consistent function \(f: \mathbb {R}^{n} \rightarrow \mathbb {R}\) is Schur-concave.

###
*Proof of Theorem 1*

It is easy to verify that the specified social welfare function satisfies the axioms in the case of additively separable market incomes. We focus on the reverse implication.

Let *W* be a social welfare function that satisfies the axioms.

By monotonicity and continuity, there exists a strictly increasing and continuous function \(\hat {f}: \mathbb {R}^{cr} \rightarrow \mathbb {R}\) such that, for each *X* in \(\mathbb {R}^{c\times r}\), we have \(W(X) = \hat {f}((x_{ik})_{(i,k) \in C \times R})\). Translation invariance implies that, for all *x* and \(x^{\prime }\) in \(\mathbb {R}^{cr}\) and each real number *λ*, we have \(\hat {f}(x) \geq \hat {f}(x^{\prime })\) if and only if \(\hat {f}(x + \lambda 1_{cr}) \geq \hat {f}(x^{\prime } + \lambda 1_{cr})\), i.e., \(\hat {f}\) is a translatable function. Let *f* be the function \(f:\mathbb {R}^{cr} \rightarrow \mathbb {R}\) such that, for each vector (*x*_{ik})_{(i,k)∈C×R}, we have \(f((x_{ik} - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }})_{(i,k) \in C \times R})=\hat {f}((x_{ik})_{(i,k) \in C \times R})\). It follows that, for each *X* in \(\mathbb {R}^{c\times r}\), we have \(W(X)=f(x_{11} - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , x_{ik} - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }}, \ldots , x_{cr} - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\). The function *f* is strictly increasing, continuous and translatable since \(\hat {f}\) is strictly increasing, continuous and translatable.

Next, we show that *f* is symmetric. Let *X* and \(X^{\prime }\) be income distributions such that the vector \(({x}_{11}^{\prime } - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , {x}_{ik}^{\prime } - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }}, \ldots , {x}_{cr}^{\prime } - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\) is obtained from the vector \((x_{11} - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , x_{ik} - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }}, \ldots , x_{cr} - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\) by a switch of two components. First, assume the switch is between the components corresponding to individuals (*i*,*k*) and (*j*,*k*). Note that \(m_{ik} - \bar {m}_{i\boldsymbol {\cdot }} = m_{jk} - \bar {m}_{j\boldsymbol {\cdot }}\) by additive separability of *M*. Because the same value (\(m_{ik} - \bar {m}_{i\boldsymbol {\cdot }} = m_{jk} - \bar {m}_{j\boldsymbol {\cdot }}\)) is subtracted from the incomes *x*_{ik} and *x*_{jk}, the switch is equivalent to a switch of these incomes. Hence, \(W(X) = W(X^{\prime })\) by ex post symmetry. Second, assume the switch is between the components corresponding to individuals (*i*,*k*) and (*i*,*l*). This switch is equivalent to a switch of the subsidies *x*_{ik} − *m*_{ik} and *x*_{il} − *m*_{il}. Hence, we have \(W(X) = W(X^{\prime })\) by liberal symmetry. Third, assume the switch is between the components corresponding to individuals (*i*,*k*) and (*j*,*l*). Let *Y* be the income distribution such that \((y_{11} - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , y_{ik} - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }}, \ldots , y_{cr} - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\) is obtained from the vector \((x_{11} - m_{11}+\bar {m}_{1\boldsymbol {\cdot }}, \ldots , x_{ik} - m_{ik}+\bar {m}_{i\boldsymbol {\cdot }}, \ldots , x_{cr} - m_{cr}+\bar {m}_{c\boldsymbol {\cdot }})\) by a switch of the components corresponding to individuals (*i*,*l*) and (*j*,*l*). Using the same reasoning as above, by ex post symmetry, we have *W*(*X*) = *W*(*Y* ). Let \(Y^{\prime }\) be the income distribution such that \(({y}_{11}^{\prime } - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , {y}_{ik}^{\prime } - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }}, \ldots , {y}_{cr}^{\prime } - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\) is obtained from the vector \((y_{11} - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , y_{ik} - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }}, \ldots , y_{cr} - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\) by a switch of the components corresponding to individuals (*i*,*k*) and (*i*,*l*). Using the same reasoning as above, by liberal symmetry, we have \(W(Y)=W(Y^{\prime })\). The vector \(({x}_{11}^{\prime } - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , {x}_{ik}^{\prime } - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }}, \ldots , {x}_{cr}^{\prime } - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\) is obtained from the vector \(({y}_{11}^{\prime } - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , {y}_{ik}^{\prime } - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }}, \ldots , {y}_{cr}^{\prime } - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\) by a switch of the components corresponding to individuals (*i*,*l*) and (*j*,*l*). Using the same reasoning as above, by ex post symmetry, we have \(W(X^{\prime }) = W(Y^{\prime })\). Thus, we obtain \(W(X)=W(X^{\prime })\).

Finally, we show that *f* is strictly Schur-concave. Since *f* is symmetric, it suffices to show that *f* is Pigou-Dalton consistent. Let *X* and \(X^{\prime }\) be income distributions such that the vector \(({x}_{11}^{\prime } - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , {x}_{ik}^{\prime } - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }}, \ldots , {x}_{cr}^{\prime } - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\) is obtained from the vector \((x_{11} - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , x_{ik} - m_{ik}+\bar {m}_{i\boldsymbol {\cdot }}, \ldots , x_{cr} - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\) by a single progressive transfer. First, assume the transfer is from the component corresponding to individual (*i*,*k*) to the component corresponding to individual (*j*,*k*). Because the same value is subtracted from the incomes *x*_{ik} and *x*_{jk}, this transfer is equivalent to a progressive transfer of income between (*i*,*k*) and (*j*,*k*). Hence, \(W(X) > W(X^{\prime })\) by ex post Pigou-Dalton. Second, assume the transfer is from the component corresponding to individual (*i*,*k*) to the component corresponding to individual (*i*,*l*). This transfer is equivalent to a progressive transfer from the subsidy *x*_{ik} − *m*_{ik} to the subsidy *x*_{il} − *m*_{il}. Hence, we have \(W(X) > W(X^{\prime })\) by liberal Pigou-Dalton. Third, assume the transfer is from the component corresponding to individual (*i*,*k*) to the component corresponding to (*j*,*l*). Let *Y* be the income distribution such that \((y_{11} - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , y_{ik} - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }}, \ldots , y_{cr} - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\) is obtained from the vector \((x_{11} - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , x_{ik} - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }}, \ldots , x_{cr} - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\) by a switch of the components corresponding to individuals (*i*,*l*) and (*j*,*l*). Using the same reasoning as above, by ex post symmetry, we have *W*(*X*) = *W*(*Y* ). Let \(Y^{\prime }\) be the income distribution such that \(({y}_{11}^{\prime } - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , {y}_{ik}^{\prime } - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }}, \ldots , {y}_{cr}^{\prime } - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\) is obtained from the vector \((y_{11} - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , y_{ik} - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }}, \ldots , y_{cr} - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\) by a transfer from the component corresponding to (*i*,*k*) to the component corresponding to (*i*,*l*). Using the same reasoning as above, by liberal Pigou-Dalton, we have \(W(Y^{\prime })>W(Y)\). The vector \(({x}_{11}^{\prime } - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , {x}_{ik}^{\prime } - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }}, \ldots , {x}_{cr}^{\prime } - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\) is obtained from the vector \(({y}_{11}^{\prime } - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , {y}_{ik}^{\prime } - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }}, \ldots , {y}_{cr}^{\prime } - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\) by a switch of the components corresponding to individuals (*i*,*l*) and (*j*,*l*). Using the same reasoning as above, by ex post symmetry, we have \(W(X^{\prime })=W(Y^{\prime })\). Thus, we obtain \(W(X^{\prime })>W(X)\). □

###
*Proof of Proposition 2*

Let *W* be a social welfare function that satisfies liberal Pigou-Dalton and ex ante symmetry.

Assume to the contrary that market incomes are not additively separable. That is, there exist *i* and *j* in *C* and *k* and *l* in *R* such that *m*_{ik} − *m*_{il} ≠ *m*_{jk} − *m*_{jl}. Let *X* be an income distribution such that there exist positive real numbers *α* and *β* such that *x*_{i⋅} = *m*_{i⋅} + *α*1_{r} and *x*_{j⋅} = *m*_{j⋅} + *β*1_{r} with \(\min \limits _{k \in R}x_{ik} > \max \limits _{k \in R} x_{jk}\). Let \(X^{\prime }\) be the income distribution obtained from *X* by switching the *i* th and the *j* th rows of *X*. Let \(X^{\prime \prime }\) be the income distribution obtained from \(X^{\prime }\) by an income transfer of an amount *𝜖* between individuals (*i*,*k*) and (*i*,*l*) that corresponds to a progressive transfer in their subsidies. Moreover, let \(\epsilon < \min \limits _{k \in R}x_{ik} - \max \limits _{k \in R} x_{jk}\), which implies that \(\min \limits _{k \in R}{x}_{jk}^{\prime \prime } > \max \limits _{k \in R} {x}_{ik}^{\prime \prime }\). Let \(X^{\prime \prime \prime }\) be the income distribution obtained from \(X^{\prime \prime }\) by switching the *i* th and the *j* th rows of \(X^{\prime \prime }\).

We have \(W(X)=W(X^{\prime })\) by ex ante symmetry, \(W(X^{\prime }) < W(X^{\prime \prime })\) by liberal Pigou-Dalton and \(W(X^{\prime \prime })=W(X^{\prime \prime \prime })\) by ex ante symmetry. Hence, \(W(X)< W(X^{\prime \prime \prime })\). However, we have \(W(X)> W(X^{\prime \prime \prime })\) by liberal Pigou-Dalton. We have a contradiction. □

###
*Proof of Theorem 4*

It is easy to verify that the specified social welfare function satisfies the axioms. We focus on the reverse implication.

Let *W* be a social welfare function that satisfies the axioms. By monotonicity, continuity and ex post aggregation, there exist a strictly increasing and continuous function \(h : \mathbb {R}^{r} \rightarrow \mathbb {R}\) and strictly increasing and continuous functions \(f_{1},f_{2}, \ldots , f_{r}: \mathbb {R}^{c} \rightarrow \mathbb {R}\) such that, for each *X* in \(\mathbb {R}^{c \times r}\), we have *W*(*X*) = *h*(*f*_{1}(*x*_{⋅1}),*f*_{2}(*x*_{⋅2}),…,*f*_{r}(*x*_{⋅r})). Using uniform utilitarian symmetry, we can define strictly increasing and continuous functions \(\hat {g} : \mathbb {R}^{r} \rightarrow \mathbb {R}\) and \(\hat {f}: \mathbb {R}^{c} \rightarrow \mathbb {R}\) such that, for each *X* in \(\mathbb {R}^{c\times r}\), we have \(W(X) = \hat {g} (\hat {f} (x_{\boldsymbol {\cdot }1}), \hat {f}(x_{\boldsymbol {\cdot }2}),\ldots , \hat {f}(x_{\boldsymbol {\cdot }r}))\).

Translation invariance implies that, for all *x* and \(x^{\prime }\) in \(\mathbb {R}^{c}\) and each real number *λ*, we have \(\hat {f}(x) \geq \hat {f}(x^{\prime })\) if and only if \(\hat {f}(x + \lambda 1_{c}) \geq \hat {f}(x^{\prime } + \lambda 1_{c})\), i.e., \(\hat {f}\) is a translatable function. Hence, there exist a strictly increasing and continuous function \(\psi : \mathbb {R} \rightarrow \mathbb {R}\) and a unit-translatable function \(f : \mathbb {R}^{c} \rightarrow \mathbb {R}\) such that \(\hat {f} = \psi \circ f\). Define the strictly increasing and continuous function \(g:\mathbb {R}^{c} \rightarrow \mathbb {R}\) such that, for each (*t*_{1},*t*_{2},…,*t*_{r}) in \(\mathbb {R}^{r}\), we have \(g(t_{1}, t_{2},\ldots , t_{r}) = \hat {g}(\psi (t_{1}), \psi (t_{2}), \ldots , \psi (t_{r}))\). It follows that, for each *X* in \(\mathbb {R}^{c\times r}\), we have *W*(*X*) = *g*(*f*(*x*_{⋅1}),*f*(*x*_{⋅2}),…,*f*(*x*_{⋅r})). Note that by translation invariance, *g* is translatable. The function *f* is strictly increasing and continuous because *g* and \(\hat {f}\) are strictly increasing and continuous. Moreover, *f* is symmetric by ex post symmetry, Pigou-Dalton consistent by ex post Pigou-Dalton, and hence strictly Schur-concave.

Next we show that *W* is a strictly increasing function of \(\frac {1}{r}{\sum }_{k \in R} f(x_{\boldsymbol {\cdot } k})\). Let *X* and \(X^{\prime }\) be income distributions in \(\mathbb {R}^{c\times r}\). Let *Y* and \(Y^{\prime }\) in \(\mathbb {R}^{c\times r}\) be income distributions such that *y*_{ik} = *f*(*x*_{⋅k}) and \({y}_{ik}^{\prime } = f({x}_{\boldsymbol {\cdot } k}^{\prime })\) for each (*i*,*k*) in *C* × *R*. We have *W*(*Y* ) = *g*(*f*(*f*(*x*_{⋅1})1_{c}),*f*(*f*(*x*_{⋅2})1_{c}),…,*f*(*f*(*x*_{⋅r})1_{c})). Since *f* is unit-translatable, we have *W*(*Y* ) = *g*(*f*(*x*_{⋅1}) + *α*,*f*(*x*_{⋅2}) + *α*,…,*f*(*x*_{⋅r}) + *α*) and \(W(Y^{\prime })=g(f({x}_{\boldsymbol {\cdot } 1}^{\prime }) + \alpha , f({x}_{\boldsymbol {\cdot } 2}^{\prime })+ \alpha , \ldots , f({x}_{\boldsymbol {\cdot } r}^{\prime })+ \alpha )\), where *α* = *f*(0 × 1_{c}). Next, let *Z* and \(Z^{\prime }\) be income distributions such that \(z_{ik} = \frac {1}{r} {\sum }_{k \in R} f(x_{\boldsymbol {\cdot } k})\) and \({z}_{ik}^{\prime } = \frac {1}{r} {\sum }_{k \in R} f({x}_{\boldsymbol {\cdot } k}^{\prime })\) for each (*i*,*k*) in *C* × *R*. By monotonicity, \(W(Z) \geq W(Z^{\prime })\) if and only if \(\frac {1}{r} {\sum }_{k \in R} f(x_{\boldsymbol {\cdot } k}) \geq \frac {1}{r} {\sum }_{k \in R} f({x}_{\boldsymbol {\cdot } k}^{\prime })\). By uniform utilitarian Pigou-Dalton, *W*(*Y* ) = *W*(*Z*) and \(W(Y^{\prime }) = W(Z^{\prime })\). Hence, \(W(Y) \geq W(Y^{\prime })\) if and only if \(\frac {1}{r} {\sum }_{k \in R} f(x_{\boldsymbol {\cdot } k}) \geq \frac {1}{r} {\sum }_{k \in R} f({x}_{\boldsymbol {\cdot } k}^{\prime })\). We have already established that \(W(X) \geq W(X^{\prime })\) if and only if \(W(Y) \geq W(Y^{\prime })\). Therefore, \(W(X) \geq W(X^{\prime })\) if and only if \(\frac {1}{r} {\sum }_{k \in R} f(x_{\boldsymbol {\cdot } k}) \geq \frac {1}{r} {\sum }_{k \in R} f({x}_{\boldsymbol {\cdot } k}^{\prime })\). It follows that there exists a strictly increasing and continuous function \(F:\mathbb {R} \rightarrow \mathbb {R}\) such that \(W(X) = F(\frac {1}{r}{\sum }_{k \in R}f(x_{\boldsymbol {\cdot } k}))\) for each *X* in \(\mathbb {R}^{c\times r}\). □

###
*Proof of Theorem 5*

It is easy to verify that the specified social welfare function satisfies the axioms. We focus on the reverse implication.

Let *W* be a social welfare function that satisfies the axioms. First, we show that, for each *X* and \(X^{\prime }\) in \(\mathbb {R}^{c \times r}\), if \({\sum }_{k \in R} x_{ik} = {\sum }_{k \in R} {x}_{ik}^{\prime }\) for each *i* in *C*, then we have \(W(X) = W(X^{\prime })\). Let *Y* be the income distribution obtained from *X* such that for each individual (*i*,*k*), we have \(y_{ik} = \bar {x}_{i\boldsymbol {\cdot }}\) and let \(Y^{\prime }\) be the income distribution obtained from \(X^{\prime }\) such that for each individual (*i*,*k*), we have \({y}_{ik}^{\prime } = {\bar {x}}_{i\boldsymbol {\cdot }}^{\prime }\). By utilitarian reward, we have *W*(*X*) = *W*(*Y* ) and \(W(X^{\prime }) = W(Y^{\prime })\). By construction, \(Y=Y^{\prime }\) and hence \(W(X)=W(X^{\prime })\).

Furthermore, if \({\sum }_{k \in R} x_{ik} \geq {\sum }_{k \in R} {x}_{ik}^{\prime }\) for each *i* in *C* with at least one inequality holding strictly, then we have \(W(X) > W(X^{\prime })\). This follows using monotonicity and the reasoning above. It follows that there exists a strictly increasing function \(f:\mathbb {R}^{c} \rightarrow \mathbb {R}\) such that, for each *X* in \(\mathbb {R}^{c \times r}\), we have \(W(X) = f(\bar {x}_{1\boldsymbol {\cdot }}, \bar {x}_{2\boldsymbol {\cdot }}, \ldots , \bar {x}_{c\boldsymbol {\cdot }})\). The function *f* is continuous by continuity, symmetric by ex ante symmetry and translatable by translation invariance.

Next, we show that *f* is strictly Schur-concave. Let *X* and \(X^{\prime }\) in \(\mathbb {R}^{c \times r}\) be such that the vector \((\bar {x}_{1\boldsymbol {\cdot }}, \bar {x}_{2\boldsymbol {\cdot }}, \ldots , \bar {x}_{c\boldsymbol {\cdot }} )\) is obtained from the vector \(({\bar {x}}_{1\boldsymbol {\cdot }}^{\prime }, {\bar {x}}_{2\boldsymbol {\cdot }}^{\prime }, \ldots , {\bar {x}}_{c\boldsymbol {\cdot }}^{\prime })\) by a progressive transfer. Let *Y* be an income distribution such that \(y_{i \boldsymbol {\cdot }} = \bar {x}_{i\boldsymbol {\cdot }} 1_{r}\) for each *i* in *C*, and let \(Y^{\prime }\) be an income distribution such that \({y}_{i \boldsymbol {\cdot }}^{\prime } = {\bar {x}}_{i\boldsymbol {\cdot }}^{\prime } 1_{r}\) for each *i* in *C*. Utilitarian reward implies that *W*(*X*) = *W*(*Y* ) and \(W(X^{\prime })=W(Y^{\prime })\). Ex ante Pigou-Dalton implies that \(W(Y) > W(Y^{\prime })\). Hence, we have \(W(X) > W(X^{\prime })\). That is, *f* is a Pigou-Dalton consistent function. Since it is also symmetric, by Lemma 2, *f* is Schur-concave. □

###
*Proof of Proposition 4*

Let *W* be a social welfare function that satisfies monotonicity, continuity and translation invariance.

(i) Let *W* satisfy, in addition, ex post compensation and liberal reward, and let market incomes in *M* be additively separable (Theorem 1). Let *X* be an income distribution in \(\mathbb {R}^{c \times r}\). First, to find Ξ(*X*), we look for the optimal income distribution *Y* such that *W*(*Y* ) = *W*(*X*). Since *Y* is optimal, we have \(y_{ik}=m_{ik} - \bar {m}_{i \boldsymbol {\cdot }} + \bar {Y}\) for each (*i*,*k*) in *C* × *R* by Proposition 3(i). By Theorem 1, \(W(Y)= f(\bar {Y},\ldots ,\bar {Y})\) and \(W(X)=f(x_{11} - m_{11}+\bar {m}_{1\boldsymbol {\cdot }}, \ldots , x_{ik} - m_{ik}+\bar {m}_{i\boldsymbol {\cdot }}, \ldots , x_{cr} - m_{cr}+\bar {m}_{c\boldsymbol {\cdot }})\). Since *W*(*Y* ) = *W*(*X*), we have \(f(\bar {Y},\ldots ,\bar {Y})=f(x_{11} - m_{11}+\bar {m}_{1\boldsymbol {\cdot }}, \ldots , x_{ik} - m_{ik}+\bar {m}_{i\boldsymbol {\cdot }}, \ldots , x_{cr} - m_{cr}+\bar {m}_{c\boldsymbol {\cdot }})\). Hence, \(\bar {Y}= \xi (x_{11} - m_{11}+\bar {m}_{1\boldsymbol {\cdot }}, \ldots , x_{ik} - m_{ik}+\bar {m}_{i\boldsymbol {\cdot }}, \ldots , x_{cr} - m_{cr}+\bar {m}_{c\boldsymbol {\cdot }})\), where *ξ* is the equally distributed equivalent income associated with *f*. Since \(\bar {Y}={\Xi }(X)\), we obtain \(I(X)=\bar {X} - \xi (x_{11} - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , x_{ik} - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }}, \ldots , x_{cr} - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\), i.e., \(I(X)=J(x_{11} - m_{11} + \bar {m}_{1\boldsymbol {\cdot }}, \ldots , x_{ik} - m_{ik} + \bar {m}_{i\boldsymbol {\cdot }}, \ldots , x_{cr} - m_{cr} + \bar {m}_{c\boldsymbol {\cdot }})\), where *J* is the KAS inequality measure associated with *f* in Eq. 1. The proofs of (ii) and (iii) are similar and are therefore omitted.

(iv) Let *W* satisfy, in addition, ex post compensation and uniform utilitarian reward (Theorem 4). Let *X* be an income distribution in \(\mathbb {R}^{c \times r}\). Let *Y* be an income distribution in \(\mathbb {R}^{c \times r}\) such that *y*_{ik} = *α*_{k}1_{c} and *f*(*α*_{k}1_{c}) = *f*(*x*_{⋅k}) for each *k* in *R*. By Proposition 3(iv), *Y* is optimal and by Theorem 4, *W*(*X*) = *W*(*Y* ). For each *k* in *R*, we have *α*_{k} = *ξ*(*x*_{⋅k}), where *ξ* is the equally distributed equivalent income associated with *f*. Using \({\Xi }(X)=\bar {Y}={\sum }_{k \in R} \alpha _{k}/r={\sum }_{k \in R} \xi (x_{\boldsymbol {\cdot }k})/r\) and \(X={\sum }_{k \in R}\bar {x}_{x_{\boldsymbol {\cdot }k}}/r\), we obtain \(I(X) = \bar {X} - {\Xi }(X) = {\sum }_{k \in R} [\bar {x}_{\boldsymbol {\cdot } k} - \xi (\bar {x}_{\boldsymbol {\cdot } k})]/r \), i.e., \(I(X)={\sum }_{k \in R} J(x_{\boldsymbol {\cdot }k})/r\), where *J* is the KAS inequality measure associated with *f* in Eq. 4.

(v) Let *W* satisfy, in addition, ex ante compensation and utilitarian reward (Theorem 5). Let *X* be an income distribution in \(\mathbb {R}^{c\times r}\). Again, we look for an optimal distribution *Y* such that *W*(*Y* ) = *W*(*X*). Since *Y* is optimal, we have \(\bar {y}_{i\boldsymbol {\cdot }}=\bar {y}_{j\boldsymbol {\cdot }}\) for all circumstance groups *i* and *j* in *C* by Proposition 3(v). By Theorem 5, \(W(Y)=f(\bar {Y},\ldots , \bar {Y})\) and \(W(X)=f(\bar {x}_{1\boldsymbol {\cdot }}, \bar {x}_{2\boldsymbol {\cdot }},\ldots , \bar {x}_{c\boldsymbol {\cdot }})\). Since *W*(*Y* ) = *W*(*X*), we have \(f(\bar {Y},\ldots , \bar {Y})=f(\bar {x}_{1\boldsymbol {\cdot }}, \bar {x}_{2\boldsymbol {\cdot }},\ldots , \bar {x}_{c\boldsymbol {\cdot }})\). Hence, \(\bar {Y}=\xi (\bar {x}_{1\boldsymbol {\cdot }}, \bar {x}_{2\boldsymbol {\cdot }},\ldots , \bar {x}_{c\boldsymbol {\cdot }})\), where *ξ* is the equally distributed equivalent income associated with *f*. We obtain that \(I(X)=\bar {X}-\xi (\bar {x}_{1\boldsymbol {\cdot }}, \bar {x}_{2\boldsymbol {\cdot }},\ldots , \bar {x}_{c\boldsymbol {\cdot }})\), i.e., \(I(X)=J(\bar {x}_{1\boldsymbol {\cdot }}, \bar {x}_{2\boldsymbol {\cdot }},\ldots , \bar {x}_{c\boldsymbol {\cdot }})\), where *J* is the KAS inequality measure associated with *f* in Eq. 5. □

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Bosmans, K., Öztürk, Z.E. Measurement of inequality of opportunity: A normative approach.
*J Econ Inequal* **19**, 213–237 (2021). https://doi.org/10.1007/s10888-020-09468-1

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DOI: https://doi.org/10.1007/s10888-020-09468-1