Measurement of inequality of opportunity based on counterfactuals

Abstract

The theoretical literature on inequality of opportunity formulates basic properties that measures of inequality of opportunity should have. Standard methods for the measurement of inequality of opportunity determine the inequality in counterfactual outcome distributions that are constructed by statistical methods. We show that, when standard parametric procedures are used to construct the counterfactuals, the choice of inequality measurement method and the statistical specification interact to determine whether the resulting measure of inequality of opportunity satisfies the basic properties.

Appendices

Appendices

1.1 A Proofs

1.2 A.1 Notation

Define, for all individuals $l \in N$ the following vectors:

$$\begin{aligned} \mathbf {x}_{\mathbf {l}}^{C}&= \left[ c_{l,1}-\mu _{C1} \ldots c_{l,d^{C}}-\mu _{Cd^{C}} \right] ^{\prime }, \\ \mathbf {x}_{\mathbf {l}}^{E}&= \left[ e_{l,1}-\mu _{E1} \ldots e_{l,d^{E}}-\mu _{Ed^{E}} \right] ^{\prime }, \\ \mathbf {x}_{\mathbf {l}}^{C \overline{C}}&= \left[ c_{l,1}-\overline{c}_{1} \ldots c_{l,d^{C}}-\overline{c}_{d^{C}} \right] ^{\prime }, \\ \mathbf {x}_{\mathbf {l}}^{E \overline{E}} &= \left[ e_{l,1}-\overline{e}_{1} \ldots e_{l,d^{E}}-\overline{e}_{d^{E}} \right] ^{\prime }. \end{aligned}$$

Hence, $\mathbf {x}_{\mathbf {l}}^{C}$$(\mathbf {x}_{\mathbf {l}}^{C \overline{C}})$ is the $d^{C}$-dimensional vector of the deviation of circumstances of individual l from their mean (reference) values, and $\mathbf {x}_{\mathbf {l}}^{E}$$(\mathbf {x}_{\mathbf {l}}^{E \overline{E}})$ is the $d^{E}$-dimensional vector of the deviation of his efforts from their mean (reference) values. Next, define the $(d^{C}+d^{E})$-dimensional vectors

$$\begin{aligned} \mathbf {x}_{\mathbf {l}}^{CE}&= \left[ c_{l,1}-\mu _{C1} \ldots c_{l,d^{C}}-\mu _{Cd^{C}} \,\, e_{l,1}-\mu _{E1} \ldots e_{l,d^{E}}-\mu _{Ed^{E}} \right] ^{\prime }, \\ \mathbf {x}_{\mathbf {l}}^{C \overline{E}} &= \left[ c_{l,1}-\mu _{C1} \ldots c_{l,d^{C}}-\mu _{Cd^{C}} \,\, \overline{e}_{1}-\mu _{E1} \ldots \overline{e}_{d^{E}}-\mu _{Ed^{E}} \right] ^{\prime }, \\ \mathbf {x}_{\mathbf {l}}^{\overline{C}E}&= \left[ \overline{c}_{1}-\mu _{C1} \ldots \overline{c}_{d^{C}}-\mu _{Cd^{C}} \,\, e_{l,1}-\mu _{E1} \ldots e_{l,d^{E}}-\mu _{Ed^{E}} \right] ^{\prime }, \\ \mathbf {x}_{\mathbf {l}}^{0E}&= \left[ 0 \ldots 0 \,\, e_{l,1}-\overline{e}_{1} \ldots e_{l,d^{E}}-\overline{e}_{d^{E}} \right] ^{\prime }, \\ \mathbf {x}_{\mathbf {l}}^{C0}&= \left[ c_{l,1}-\overline{c}_{1} \ldots c_{l,d^{C}}- \overline{c}_{d^{C}} \,\, 0 \ldots 0 \right] ^{\prime }, \end{aligned}$$

and the n-dimensional vectors $\mathbf {E_{i}} = \left[ 0 \ldots 0 \,\, 1 \,\, 0 \ldots 0 \right] ^{\prime }$ and ${\iota } = \left[ 1 \ldots 1 \right] ^{\prime }$, such that $\mathbf {E_{i}}$ has zeros everywhere, except for its i-element which equals one and all elements in ${\iota }$ are equal to one.

1.3 A.2 Proof of Proposition 2

Proof of Lemma 1

Define $\widetilde{\mathbf {Y}} = \mathbf {Y}+ \delta (\mathbf {E_{j}}-\mathbf {E_{i}})$, which is the vector $\mathbf {Y}$ after a Pigou–Dalton transfer of an amount $\delta$ from observation j to i. After the transfer we estimate the equation in deviational form

$$\begin{aligned} \widetilde{\mathbf {Y}}^{D} = \widetilde{\mathbf {Y}}- {\iota } \mu _{Y} = \mathbf {X}^{A} \widetilde{{\beta }}^{A}+ \widetilde{\mathbf {U}}^{A}. \end{aligned}$$

(A.1)

For the least squares estimate $\widetilde{\mathbf {b}}^{A}=\left( \mathbf {X}^{A\prime } \mathbf {X}^{A} \right) ^{-1} \mathbf {X}^{A \prime } \widetilde{\mathbf {Y}}^{D}$ we obtain

$$\begin{aligned} \left( \mathbf {X}^{A\prime } \mathbf {X}^{A} \right) ^{-1} \mathbf {X}^{A \prime } \mathbf {Y}^{D} + \left( \mathbf {X}^{A\prime } \mathbf {X}^{A} \right) ^{-1} \mathbf {X}^{A \prime } \left( \mathbf {E_{j}}-\mathbf {E_{i}} \right) , \end{aligned}$$

from which the expression in the Lemma follows immediately. $\square$

Lemma A1

Under the assumption of linearity and using the least squares estimator, the counterfactuals (6)–(10) and (12)–(16) become

$$\begin{aligned} y^{c1}_l= \mu _{Y}+\mathbf {x}_{\mathbf {l}}^{C \prime } \mathbf {b}^{C}, \end{aligned}$$

(A.2)

$$\begin{aligned} y^{c2}_l= \mu _{Y}+y_{l}^{D}-\mathbf {x}_{\mathbf {l}}^{E \overline{E} \prime } \mathbf {b}^{E}, \end{aligned}$$

(A.3)

$$\begin{aligned} y^{c3}_l= \mu _{Y}+\mathbf {x}_{\mathbf {l}}^{C \overline{E} \prime } \mathbf {b}^{CE}, \end{aligned}$$

(A.4)

$$\begin{aligned} y^{c4}_l= \mu _{Y}+y_{l}^{D}-\mathbf {x}_{\mathbf {l}}^{0E \prime } \mathbf {b}^{CE}, \end{aligned}$$

(A.5)

$$\begin{aligned} y_l^{c5}= \mu _Y + \frac{1}{\left| N_{l \cdot } \right| } \sum _{k \in N_{l \cdot }} \mathbf {x}_{\mathbf {k}}^{CE \prime } \mathbf {b}^{CE} \end{aligned}$$

(A.6)

$$\begin{aligned} y^{EO1}_l= \mu _{Y}+\mathbf {x}_{\mathbf {l}}^{E \prime } \mathbf {b}^{E}, \end{aligned}$$

(A.7)

$$\begin{aligned} y^{EO2}_l= \mu _{Y}+y_{l}^{D}-\mathbf {x}_{\mathbf {l}}^{C \overline{C}\prime } \mathbf {b}^{C}, \end{aligned}$$

(A.8)

$$\begin{aligned} y^{EO3}_l= \mu _{Y}+\mathbf {x}_{\mathbf {l}}^{\overline{C}E \prime } \mathbf {b}^{CE}, \end{aligned}$$

(A.9)

$$\begin{aligned} y^{EO4}_l= \mu _{Y}+y_{l}^{D}-\mathbf {x}_{\mathbf {l}}^{C0 \prime } \mathbf {b}^{CE}. \end{aligned}$$

(A.10)

$$\begin{aligned} y_l^{E05}= \mu _Y + \frac{1}{\left| N_{\cdot l} \right| } \sum _{k \in N_{\cdot l}} \mathbf {x}_{\mathbf {k}}^{CE \prime } \mathbf {b}^{CE} \end{aligned}$$

(A.11)

Proof of Lemma A1

Equations (A.2), (A.4), (A.6), (A.7), (A.9) and (A.11) are straightforward. The others are only slightly more complicated. We prove (A.3); the proof of the others is analogous. From (7), with the linear specification, and, for all individuals $l \in N$, $\mathbf {x}_{\mathbf {l}}^{\overline{E} \mu _{E}} = \left[ \overline{e}_{1}-\mu _{E1} \ldots \overline{e}_{d^{E}}-\mu _{Ed^{E}} \right] ^{\prime }$, we have

$$\begin{aligned} y_{l}^{c2}&= \mu _{Y}+\mathbf {x}^{\overline{E} \mu _{E} \prime } \mathbf {b}^{E} +\widehat{u}_{l}^{R} \\&= \mu _{Y}+\mathbf {x}^{\overline{E} mu _{E} \prime } \mathbf {b}^{E} +y_{l}^{D}-\mathbf {x}_{\mathbf {l}}^{E \prime } \mathbf {b}^{E} \\&= \mu _{Y}+y_{l}^{D}-\left( \mathbf {x}_{\mathbf {l}}^{E \prime }- \mathbf {x}^{\overline{E} \mu _{E} \prime }\right) \mathbf {b}^{E} \\&= \mu _{Y}+y_{l}^{D}-\mathbf {x}_{\mathbf {l}}^{E\overline{E} \prime } \mathbf {b}^{E}, \end{aligned}$$

which is expression (A.3). $\square$

Using this Lemma, that the covariance matrix of the variables in $\mathbf {X}^{A}$, $\mathbf {S}^{A}=\frac{1}{n} \mathbf {X}^{A \prime } \mathbf {X}^{A}$, and adding a tilde to denote the counterfactuals after the transfer, it is easy to obtain the following expressions for the effect of the Pigou–Dalton transfers on the counterfactuals.

Lemma A2

The change in the estimated counterfactuals of a Pigou–Dalton transfer $\delta$ from observation i to j is

$$\begin{aligned} \widetilde{y}^{c1}_{l}-y^{c1}_{l}= \frac{\delta }{n} \mathbf {x}_{\mathbf {l}}^{C \prime } (\mathbf {S}^{C})^{-1} (\mathbf {x}_{\mathbf {j}}^{C}-\mathbf {x}_{\mathbf {i}}^{C}), \end{aligned}$$

(A.12)

$$\begin{aligned} \widetilde{y}^{c2}_{l}-y^{c2}_{l}= \widetilde{y}^{D}_{l}-y^{D}_{l} - \frac{ \delta }{n} \mathbf {x}_{\mathbf {l}}^{E \overline{E}\prime } (\mathbf {S}^{E})^{-1} (\mathbf {x}_{j}^{E} - \mathbf {x}_{\mathbf {i}}^{E}), \end{aligned}$$

(A.13)

$$\begin{aligned} \widetilde{y}^{c3}_{l}-y^{c3}_{l}= \frac{\delta }{n}\mathbf {x}_{\mathbf {l}}^{C \overline{E} \prime } (\mathbf {S}^{CE})^{-1} (\mathbf {x}_{\mathbf {j}}^{CE} - \mathbf {x}_{\mathbf {i}}^{CE}), \end{aligned}$$

(A.14)

$$\begin{aligned} \widetilde{y}^{c4}_{l}-y^{c4}_{l}= \widetilde{y}^{D}_{l}-y^{D}_{l} - \frac{ \delta }{n} \mathbf {x}_{\mathbf {l}}^{0E \prime } (\mathbf {S}^{CE})^{-1} (\mathbf {x}_{\mathbf {j}}^{CE}-\mathbf {x}_{\mathbf {i}}^{CE}), \end{aligned}$$

(A.15)

$$\begin{aligned} \widetilde{y}^{c5}_{l}-y^{c5}_{l}= \frac{\delta }{n} \frac{1}{\left| N_{l \cdot } \right| } \sum _{k \in N_{l \cdot }} \mathbf {x}_{\mathbf {k}}^{CE \prime } (\mathbf {S}^{CE})^{-1}(\mathbf {x}_{\mathbf {j}}^{CE}-\mathbf {x}_{\mathbf {i}}^{CE}), \end{aligned}$$

(A.16)

$$\begin{aligned} \widetilde{y}^{EO1}_{l}-y^{EO1}_{l}= \frac{\delta }{n} \mathbf {x}_{\mathbf {l}}^{E \prime } (\mathbf {S}^{E})^{-1} (\mathbf {x}_{\mathbf {j}}^{E}-\mathbf {x}_{\mathbf {i}}^{E}), \end{aligned}$$

(A.17)

$$\begin{aligned} \widetilde{y}^{EO2}_{l}-y^{EO2}_{l}= \widetilde{y}^{D}_{l}-y^{D}_{l} - \frac{ \delta }{n} \mathbf {x}_{\mathbf {l}}^{C \overline{C} \prime } (\mathbf {S}^{C})^{-1} (\mathbf {x}_{\mathbf {j}}^{C}-\mathbf {x}_{\mathbf {i}}^{C}), \end{aligned}$$

(A.18)

$$\begin{aligned} \widetilde{y}^{EO3}_{l}-y^{EO3}_{l}= \frac{\delta }{n} \mathbf {x}_{\mathbf {l}}^{\overline{C} E \prime } (\mathbf {S}^{CE})^{-1} (\mathbf {x}_{\mathbf {j}}^{CE}-\mathbf {x}_{\mathbf {i}}^{CE}), \end{aligned}$$

(A.19)

$$\begin{aligned} \widetilde{y}^{EO4}_{l}-y^{EO4}_{l}= \widetilde{y}^{D}_{l}-y^{D}_{l} - \frac{ \delta }{n} \mathbf {x}_{\mathbf {l}}^{C0 \prime } (\mathbf {S}^{CE})^{-1} (\mathbf {x}_{\mathbf {j}}^{CE}-\mathbf {x}_{\mathbf {i}}^{CE}). \end{aligned}$$

(A.20)

$$\begin{aligned} \widetilde{y}^{EO5}_{l}-y^{EO5}_{l}= \frac{\delta }{n} \frac{1}{\left| N_{\cdot l} \right| } \sum _{k \in N_{\cdot l}} \mathbf {x}_{\mathbf {k}}^{CE \prime } (\mathbf {S}^{CE})^{-1}(\mathbf {x}_{\mathbf {j}}^{CE}-\mathbf {x}_{\mathbf {i}}^{CE}), \end{aligned}$$

(A.21)

Proof of Proposition 2

Due to the similarity of the proofs, we first prove parts (a) and (f), followed by parts (b) and (g), (c) and (h), and, finally, parts (d) and (i).

Consider part (a) of the Proposition and Eq. (A.12). Observe that, $\sum _{l=1}^{n} x^{C}_l= 0$, such that, from (A.12), $\sum _{l=1}^{n} \widetilde{y}^{c1}_{l} = \sum _{l=1}^{n} y^{c1}_{l}$. The mean of the counterfactual has not changed, and there is no need to normalize the counterfactual distribution to study the effects of a Pigou–Dalton transfer.

Take $\mathbf {x}_{\mathbf {j}}^{C}=\mathbf {x}_{\mathbf {i}}^{C}$. From (A.12), $\widetilde{y}_{l}^{c1}={y}_{l}^{c1}$, such that the counterfactual has not changed. Hence, UR is satisfied. Take $\mathbf {x}_{\mathbf {j}}^{E}=\mathbf {x}_{\mathbf {i}}^{E}$. Under C1M, $(\mathbf {S}^{C})^{-1}>0$ and $c_{i1}>c_{j1}$. From (A.12) we immediately have that the value of the counterfactual decreases (increases) for those observations for which $x_{l}^{C} > (<) 0$, i.e. for which $c_{l1} > (<) \mu _{C1}$. Hence, $I(\mathbf {Y}^{c1})$ decreases and the measure satisfies COM.

Consider part (f) of the Proposition, and Eq. (A.17). Observe that, $\sum _{l=1}^{n} x^{E}_l=0$, such that, from (A.17), $\sum _{l=1}^{n} \widetilde{y}^{EO1}_{l} = \sum _{l=1}^{n} y^{EO1}_{l}$. The mean of the counterfactual has not changed, and there is no need to normalize the counterfactual distribution to study the effects of a Pigou–Dalton transfer.

Take $\mathbf {x}_{\mathbf {j}}^{E}=\mathbf {x}_{\mathbf {i}}^{E}$. From (A.17), $\widetilde{y}_{l}^{EO1}=y_{l}^{EO1}$, such that the counterfactual has not changed. However, the Pigou–Dalton transfer decreases the inequality in the income vector $\mathbf {Y}$, such that $I(\mathbf {Y})$ decreases, and thus $I(\mathbf {Y})-I \left( \mathbf {Y}^{EO1} \right)$ decreases. Hence COM is satisfied. Take $\mathbf {x}_{\mathbf {j}}^{C}=\mathbf {x}_{\mathbf {i}}^{C}$. In case $d^{E}=1$, $(\mathbf {S}^{E})^{-1} > 0$. Let $e_{i1}>e_{j1}$. From (A.17), we have that for those observations for which $e_{l1} > (<) \mu _{E1}$, the counterfactual decreases (increases). Hence the transfer decreases inequality in $\mathbf {Y}^{EO1}$. However, it also decreases the inequality in $\mathbf {Y}$ in a different manner, and the effect on $I(\mathbf {Y})-I \left( \mathbf {Y}^{EO1} \right)$ is ambiguous, such that UR is not even satisfied with one-dimensional effort.

Consider part (b) of the Proposition and Eq. (A.13). We have

$$\begin{aligned} \frac{\sum _{l=1}^{n} \widetilde{y}_{l}^{c2}}{n} = \frac{\sum _{l=1}^{n} {y} _{l}^{c2}}{n} - \frac{\delta }{n} (\mu _{\mathbf {E}}- \overline{\mathbf {e}})^{\prime } (\mathbf {S}^{E})^{-1} (\mathbf {x}_{\mathbf {j}}^{E}-\mathbf {x}_{\mathbf {i}}^{E}). \end{aligned}$$

(A.22)

Take $\mathbf {x}_{\mathbf {j}}^{E}=\mathbf {x}_{\mathbf {i}}^{E}$. From (A.22), the mean of the counterfactual has not changed. We then see from (A.13) that $\widetilde{y}^{c2}_{l}-y^{c2}_{l}= \widetilde{y}^{D}_{l}-y^{D}_{l}$, such that the inequality in the counterfactual declines and the measure satisfies COM. Take $\mathbf {x}_{\mathbf {j}}^{C}=\mathbf {x}_{\mathbf {i}}^{C}$. If $\overline{\mathbf {e}} = \mu _{\mathbf {E}}$, from (A.22), the mean of the counterfactual has not changed, and no normalization of the counterfactual is necessary to analyze the consequences of the Pigou–Dalton transfer. When $d^{E}=1$, $(\mathbf {S}^{E})^{-1} >0$, and assuming that $e_{i1}>e_{j1}$, from (A.13) the change in the counterfactual is larger (smaller) than the change in the actual income distribution for those with $e_{l1} > (<) \mu _{E1}$, and inequality of opportunity changes. Hence the measure does not even satisfy UR under E$\mu$ with one-dimensional efforts.

Consider part (g) of the Proposition and Eq. (A.18). We have

$$\begin{aligned} \frac{\sum _{l=1}^{n} \widetilde{y}_{l}^{EO2}}{n} = \frac{\sum _{l=1}^{n} {y} _{l}^{EO2}}{n} - \frac{\delta }{n} (\mu _{\mathbf {C}}- \overline{\mathbf {c}})^{\prime } (\mathbf {S}^{C})^{-1} (\mathbf {x}_{\mathbf {j}}^{C}-\mathbf {x}_{\mathbf {i}}^{C}). \end{aligned}$$

(A.23)

Take $\mathbf {x}_{\mathbf {j}}^{C}=\mathbf {x}_{\mathbf {i}}^{C}$. From (A.23), the mean of the counterfactual has not changed. We then see, from (A.18) that $\widetilde{y}^{EO2}_{l}-y^{EO2}_{l}= \widetilde{y}^{D}_{l}-y^{D}_{l}$, such that the change in the counterfactual is identical to the change in the actual income distribution: in both distributions we get a Pigou–Dalton transfer from i to j. However, nothing guarantees that the change in $I(\mathbf {Y})-I(\mathbf {Y}^{EO2})$ will be the same, such that the measure does not satisfy UR. Take $\mathbf {x}_{\mathbf {j}}^{E}=\mathbf {x}_{\mathbf {i}}^{E}$. If $\overline{\mathbf {c}}=\mu _{\mathbf {C}}$, from (A.23), the mean of the counterfactual has not changed, and no normalization of the counterfactual is necessary to analyze the consequences of the Pigou–Dalton transfer. With this reference value, with one-dimensional circumstances such that $(\mathbf {S}^{C})^{-1}>0$, and assuming that $c_{i1}>c_{j1}$ (i.e. under C1M), from (A.18), we have that the change in the counterfactual is larger (smaller) than the change in the actual distribution for those with $c_{l1} > (<) \mu _{C1}$. However, this does not guarantee that $I(\mathbf {Y})-I(\mathbf {Y}^{EO2})$ decreases, and so even under these assumptions COM need not be satisfied. If $\overline{\mathbf {c}} \ne \mu _{\mathbf {C}}$, it follows from (A.23) that the mean of the counterfactual has changed. Under C1M, $c_{i1}>c_{j1}$, such that, if $\overline{c}_{1} < (>)\mu _{C1}$, the mean increases (decreases), and this can counter the effect on inequality of opportunity that arises from the fact that, assuming that $c_{i1}>c_{j1}$, from (A.18), we have that the change in the counterfactual is larger (smaller) than the change in the actual distribution for those with $c_{l1} > (<) \overline{c}_{1}$. A similar issue occurs in the following cases if the mean of the counterfactual changes; for that reason, we focus on cases where the mean remains constant.

The counterfactuals in (c), (d), (e), (h), (i) and (j) of the Proposition rely on estimates of $\mathbf {b}^{CE}$, such that $\mathbf {S}^{CE}$, the estimated covariance matrix of circumstances and efforts plays a role. Define the following matrix

$$\begin{aligned} (\mathbf {S}^{CE})^{-1}= \left[ \begin{array}{cc} \mathbf {A}^{CC} &{} \mathbf {A}^{CE} \\ \mathbf {A}^{EC} &{} \mathbf {A}^{EE} \end{array} \right] . \end{aligned}$$

In case circumstances and efforts are not correlated, their covariance is zero, and $\mathbf {S}^{CE}$ is block diagonal. The inverse of a block-diagonal matrix is also block diagonal, such that, if efforts and circumstances are not correlated, $\mathbf {A}^{CE}=(\mathbf {A}^{EC})^{\prime }$ contains only zeros.

Consider part (c) of the Proposition and Eq. (A.14). Observe,

$$\begin{aligned} \frac{\sum _{l=1}^{n} \widetilde{y}_{l}^{c3}}{n} = \frac{\sum _{l=1}^{n} {y} _{l}^{c3}}{n} + \frac{\delta }{n} (\overline{\mathbf {e}}-\mu _{\mathbf {E}})^{\prime } \left[ \mathbf {A}^{EC} (\mathbf {x}_{\mathbf {j}}^{C}-\mathbf {x}_{\mathbf {i}}^{C}) + \mathbf {A}^{EE} (\mathbf {x}_{\mathbf {j}}^{E}-\mathbf {x}_{\mathbf {i}}^{E}) \right] . \end{aligned}$$

(A.24)

Take $\overline{\mathbf {e}}=\mu _{\mathbf {E}}$. The mean of the counterfactual has not changed. First, with $\mathbf {x}_{\mathbf {j}}^{C}=\mathbf {x}_{\mathbf {i}}^{C}$, from (A.14),

$$\begin{aligned} \widetilde{y}_{l}^{c3} = {y}_{l}^{c3} + \frac{\delta }{n} \mathbf {x}_{\mathbf {l}}^{C \prime } \mathbf {A}^{CE} \left[ \mathbf {x}_{\mathbf {j}}^{E} - \mathbf {x}_{\mathbf {i}}^{E} \right] . \end{aligned}$$

If, in addition, $\mathbf {A}^{CE}=\mathbf {0}$, we get $\widetilde{y}_{l}^{c3} = {y}_{l}^{c3}$: the transfer has no effect on the counterfactual $\mathbf {Y}^{c3}$. Hence, in this case, the measure satisfies UR. Second, with $\mathbf {x}_{\mathbf {j}}^{E}=\mathbf {x}_{\mathbf {i}}^{E}$, from (A.14),

$$\begin{aligned} \widetilde{y}_{l}^{c3} = {y}_{l}^{c3} + \frac{\delta }{n} \mathbf {x}_{\mathbf {l}}^{C \prime } \mathbf {A}^{CC} \left[ \mathbf {x}_{\mathbf {j}}^{C} - \mathbf {x}_{\mathbf {i}}^{C} \right] , \end{aligned}$$

(A.25)

If, in addition $d^{C}=1$ such that $A^{CC}>0$, and $c_{i1}>c_{j1}$, from (A.25), the counterfactual for those observations for which $c_{l1} > (<) \mu _{C1}$ decrease (increase), such that $I(\mathbf {Y}^{c3})$ decreases and the measure satisfies COM.

Take $\overline{\mathbf {e}} \ne \mu _{\mathbf {E}}$, $\mathbf {A}^{EC}=\mathbf {0}$ and $\mathbf {x}_{\mathbf {j}}^{E}=\mathbf {x}_{\mathbf {i}}^{E}$. From (A.24), the mean of the counterfactual has not changed, and, from (A.14), we obtain again (A.25), and the same conclusion follows: under C1M the measure satisfies COM.

Consider part (h) of the Proposition and Eq. (A.19). Observe,

$$\begin{aligned} \frac{\sum _{l=1}^{n} \widetilde{y}_{l}^{EO3}}{n} = \frac{\sum _{l=1}^{n} {y} _{l}^{EO3}}{n} + \frac{\delta }{n} (\overline{\mathbf {c}}-\mu _{\mathbf {C}})^{\prime } \left[ \mathbf {A}^{CC} (\mathbf {x}_{\mathbf {j}}^{C}-\mathbf {x}_{\mathbf {i}}^{C}) + \mathbf {A}^{CE} (\mathbf {x}_{\mathbf {j}}^{E}-\mathbf {x}_{\mathbf {i}}^{E}) \right] . \end{aligned}$$

(A.26)

Take $\overline{\mathbf {c}}=\mu _{\mathbf {C}}$. The mean of the counterfactual has not changed. First, with $\mathbf {x}_{\mathbf {j}}^{E}=\mathbf {x}_{\mathbf {i}}^{E}$, from (A.19),

$$\begin{aligned} \widetilde{y}_{l}^{EO3}= {y}_{l}^{EO3} + \frac{\delta }{n} \mathbf {x}_{\mathbf {l}}^{E \prime } \mathbf {A}^{EC} \left[ \mathbf {x}_{\mathbf {j}}^{C}-\mathbf {x}_{\mathbf {i}}^{C} \right] . \end{aligned}$$

If, in addition, $\mathbf {A}^{EC}=\mathbf {0}$, we get $\widetilde{y}_{l}^{EO3}= {y}_{l}^{EO3}$: the transfer has no effect on the counterfactual. However, $I(\mathbf {Y})$ falls, hence $I(\mathbf {Y})-I(\mathbf {Y}^{EO3})$ decreases, and the measure satisfies COM. Second, with $\mathbf {x}_{\mathbf {j}}^{C}=\mathbf {x}_{\mathbf {i}}^{C}$, from (A.19),

$$\begin{aligned} \widetilde{y}_{l}^{EO3}= {y}_{l}^{EO3} + \frac{\delta }{n} \mathbf {x}_{\mathbf {l}}^{E \prime } \mathbf {A}^{EE} \left[ \mathbf {x}_{\mathbf {j}}^{E}-\mathbf {x}_{\mathbf {i}}^{E} \right] . \end{aligned}$$

(A.27)

If, in addition $d^{E}=1$ such that $A^{EE}>0$, and $e_{i1}>e_{j1}$, from (A.27), we have that the counterfactual for those observations for which $e_{l1}>(<)\mu _{E1}$ decreases (increases). Hence the transfer decreases inequality in $\mathbf {Y}^{EO3}$. However, since the inequality in $\mathbf {Y}$ decreases in a different way, the effect on $I(\mathbf {Y})-I(\mathbf {Y}^{EO3})$ cannot be determined and the measure does not even satisfy UR in the one-dimensional case.

Take $\overline{\mathbf {c}} \ne \mu _{\mathbf {C}}$, $\mathbf {A}^{CE}=0$ and $\mathbf {x}_{\mathbf {j}}^{C}=\mathbf {x}_{\mathbf {i}}^{C}$. From (A.26), the mean of the counterfactual has not changed, and, from (A.14), we obtain again (A.27), and the same conclusion follows: the measure does not satisfy UR even in the one-dimensional case.

Consider part (d) of the Proposition and Eq. (A.15). Observe,

$$\begin{aligned} \frac{\sum _{l=1}^{n} \widetilde{y}_{l}^{c4}}{n} = \frac{\sum _{l=1}^{n} {y} _{l}^{c4}}{n} - \frac{\delta }{n} (\mu _{\mathbf {E}} - \overline{\mathbf {e}})^{\prime } \left[ \mathbf {A}^{EC} (\mathbf {x}_{\mathbf {j}}^{C}-\mathbf {x}_{\mathbf {i}}^{C}) + \mathbf {A}^{EE} (\mathbf {x}_{\mathbf {j}}^{E}-\mathbf {x}_{\mathbf {i}}^{E}) \right] . \end{aligned}$$

(A.28)

Take $\overline{\mathbf {e}}=\mu _{\mathbf {E}}$. The mean of the counterfactual has not changed. First, with $\mathbf {x}_{\mathbf {j}}^{C}=\mathbf {x}_{\mathbf {i}}^{C}$, from (A.15),

$$\begin{aligned} \widetilde{y}_{l}^{c4} -{y}_{l}^{c4} = \widetilde{y}_{l}^{D} -{y}_{l}^{D} - \frac{\delta }{n} \mathbf {x}_{\mathbf {l}}^{E \prime } \mathbf {A}^{EE} (\mathbf {x}_{\mathbf {j}}^{E}-\mathbf {x}_{\mathbf {i}}^{E}). \end{aligned}$$

(A.29)

With this reference value, with one-dimensional efforts such that $A^{EE}>\mathbf {0}$, and assuming that $e_{i1}>e_{j1}$, from (A.29), the change in the counterfactual is larger (smaller) than the change in the actual income distribution for those with $e_{l1} > (<) \mu _{E1}$, and inequality of opportunity changes. Hence the measure does not satisfy UR. Second, with $\mathbf {x}_{\mathbf {j}}^{E}=\mathbf {x}_{\mathbf {i}}^{E}$, from (A.15),

$$\begin{aligned} \widetilde{y}^{c4}_{l}-y^{c4}_{l}= \widetilde{y}^{D}_{l}-y^{D}_{l} - \frac{ \delta }{n} \mathbf {x}_{\mathbf {l}}^{E \prime } \mathbf {A}^{EC} (\mathbf {x^{C}_{j}}-\mathbf {x^{C}_{i}}). \end{aligned}$$

(A.30)

If, in addition, $\mathbf {A}^{EC}=\mathbf {0}$, $\widetilde{y}_{l}^{c4} -{y}_{l}^{c4} = \widetilde{y}_{l}^{D} -{y}_{l}^{D}$, such that inequality in the counterfactual decreases and the measure satisfies COM.

Take $\overline{\mathbf {e}} \ne \mu _{\mathbf {E}}$, $\mathbf {A}^{EC}= \mathbf {0}$ and $\mathbf {x}_{\mathbf {j}}^{E}=\mathbf {x}_{\mathbf {i}}^{E}$. From (A.28), the mean of the counterfactual has not changed, and, from (A.15), we obtain $\widetilde{y} ^{c4}_{l}-y^{c4}_{l}= \widetilde{y}^{D}_{l}-y^{D}_{l}$, meaning that inequality decreased. Hence the measure satisfies COM.

Consider part (i) of the Proposition and Eq. (A.20). Observe

$$\begin{aligned} \frac{\sum _{l=1}^{n} \widetilde{y}_{l}^{EO4}}{n} = \frac{\sum _{l=1}^{n} {y} _{l}^{EO4}}{n} - \frac{\delta }{n} (\mu _{\mathbf {C}}-\overline{\mathbf {c}})^{\prime } \left[ \mathbf {A}^{CC} (\mathbf {x}_{\mathbf {j}}^{C}-\mathbf {x}_{\mathbf {i}}^{C}) + \mathbf {A}^ {CE} (\mathbf {x}_{\mathbf {j}}^{E}-\mathbf {x}_{\mathbf {i}}^{E}) \right] . \end{aligned}$$

(A.31)

Take $\overline{\mathbf {c}}=\mu _{\mathbf {C}}$. The mean of the counterfactual has not changed. First, with $\mathbf {x}_{\mathbf {j}}^{E}=\mathbf {x}_{\mathbf {i}}^{E}$, from (A.20),

$$\begin{aligned} \widetilde{y}^{EO4}_{l}-y^{EO4}_{l}= \widetilde{y}^{D}_{l}-y^{D}_{l} - \frac{\delta }{n} \mathbf {x}_{\mathbf {l}}^{C \prime } \mathbf {A}^{CC} (\mathbf {x}_{\mathbf {j}}^{C}-\mathbf {x}_{\mathbf {i}}^{C}). \end{aligned}$$

(A.32)

If in addition $d^{C}=1$ such that $A^{CC}>0$, and $c_{i1}>c_{j1}$, by A.32, the change in the counterfactual for those observations for which $c_{l1} > (<) \mu _{C1}$ is larger (smaller) than the change in the actual income distribution. Hence inequality in the counterfactual increases more than the inequality in the actual income distribution. However, this does not guarantee that $I(\mathbf {Y})-I(\mathbf {Y}^{EO4})$ decreases, and so even under these assumptions COM need not be satisfied. Second, with $\mathbf {x}_{\mathbf {j}}^{C}=\mathbf {x}_{\mathbf {i}}^{C}$, from (A.20),

$$\begin{aligned} \widetilde{y}^{EO4}_{l}-y^{EO4}_{l}= \widetilde{y}^{D}_{l}-y^{D}_{l} - \frac{\delta }{n} \mathbf {x}_{\mathbf {l}}^{C \prime } \mathbf {A}^{CE} (\mathbf {x}_{\mathbf {j}}^{E}-\mathbf {x}_{\mathbf {i}}^{E}). \end{aligned}$$

If, in addition, $\mathbf {A}^{CE}=\mathbf {0}$, from (A.20), $\widetilde{y}^{EO4}_{l}-y^{EO4}_{l} = \widetilde{y}^{D}_{l}-y^{D}_{l}$, and the change in the counterfactual is identical to the change in the actual income distribution: in both distributions we get a Pigou–Dalton transfer from i to j. However, nothing guarantees that the change in $I(\mathbf {Y}) - I(\mathbf {Y}^{EO4})$ will be the same, such that the measure does not satisfy UR.

Take $\overline{\mathbf {c}} \ne \mu _{\mathbf {C}}$, $\mathbf {A}^{CE}=\mathbf {0}$ and $\mathbf {x}_{\mathbf {j}}^{C}=\mathbf {x}_{\mathbf {i}}^{C}$. From (A.31), the mean of the counterfactual has not changed, and, from (A.15), we obtain again $\widetilde{y}^{EO4}_{l}-y^{EO4}_{l} = \widetilde{y}^{D}_{l}-y^{D}_{l}$. Again, the change in the counterfactual is identical to the change in the actual income distribution: in both distributions we get a Pigou–Dalton transfer from i to j. However, nothing guarantees that the change in $I(\mathbf {Y}) - I(\mathbf {Y}^{EO4})$ will be the same, such that the measure does not satisfy UR.

Consider part (e) of the Proposition and Eq. (A.16). Observe,

$$\begin{aligned} \frac{\sum _{l=1}^n \widetilde{y}^{c5}_{l}}{n} = \frac{\sum _{l=1}^n y^{c5}_{l}}{n} + \frac{\delta }{n} \frac{1}{n} \sum _{l=1}^n \frac{1}{\left| N_{l \cdot } \right| } \sum _{k \in N_{l \cdot }} \mathbf {x}_{\mathbf {k}}^{CE \prime } (\mathbf {S}^{CE})^{-1}(\mathbf {x}_{\mathbf {j}}^{CE}-\mathbf {x}_{\mathbf {i}}^{CE}). \end{aligned}$$

Since $\sum _{l=1}^n \frac{1}{\left| N_{l \cdot } \right| } \sum _{k \in N_{l \cdot }} \mathbf {x}_{\mathbf {k}}^{CE \prime } = \mathbf {0}$, the $d^C + d^E$-dimensional null vector, the mean of the distribution has not changed. First, with $\mathbf {x}_{\mathbf {j}}^{C}=\mathbf {x}_{\mathbf {i}}^{C}$ and COR0 from (A.16),

$$\begin{aligned} \widetilde{y}^{c5}_{l} = y^{c5}_{l} + \frac{\delta }{n} \frac{1}{\left| N_{l \cdot } \right| } \sum _{k \in N_{l \cdot }} \mathbf {x}_{\mathbf {k}}^{E \prime } \mathbf {A}^{EE} (\mathbf {x}_{\mathbf {j}}^{E}-\mathbf {x}_{\mathbf {i}}^{E}). \end{aligned}$$

The i-th element in the vector $\frac{1}{\left| N_{l \cdot } \right| } \sum _{k \in N_{l \cdot }} \mathbf {x}_{\mathbf {k}}^{E \prime }$ is $\frac{1}{\left| N_{l \cdot } \right| } \sum _{k \in N_{l \cdot }} e_{ki} - \mu _{Ei}$, the difference between the average value of effort i of those that have the same circumstance as l and the average value of effort i in the population. Under COR0, the distribution of efforts conditional on circumstances is independent of the value of circumstances and equals the unconditional distribution. Hence $\frac{1}{\left| N_{l \cdot } \right| } \sum _{k \in N_{l \cdot }} \mathbf {x}_{\mathbf {k}}^{E \prime }$ is equal to the $d^E$-dimensional null vector, $\widetilde{y}^{yc5}_{l} = y^{yc5}_{l}$ and the measure satisfies UR.

Second, with $\mathbf {x}_{\mathbf {j}}^{E}=\mathbf {x}_{\mathbf {i}}^{E}$, from (A.16),

$$\begin{aligned} \widetilde{y}^{c5}_{l} = y^{c5}_{l} + \frac{\delta }{n} \frac{1}{\left| N_{l \cdot } \right| } \sum _{k \in N_{l \cdot }} \mathbf {x}_{\mathbf {k}}^{CE \prime } \left[ \begin{array}{c} \mathbf {A}^{CC} \\ \mathbf {A}^{EC} \end{array} \right] (\mathbf {x}_{\mathbf {j}}^{C}-\mathbf {x}_{\mathbf {i}}^{C}). \end{aligned}$$

Under COR0, this reduces to

$$\begin{aligned} \widetilde{y}^{c5}_{l} = y^{c5}_{l} + \frac{\delta }{n} \mathbf {x}_{\mathbf {l}}^{C \prime } A^{CC} (\mathbf {x}_{\mathbf {j}}^{C}-\mathbf {x}_{\mathbf {i}}^{C}). \end{aligned}$$

With one-dimensional circumstances, therefore,

$$\begin{aligned} \widetilde{y}^{c5}_{l} = y^{c5}_{l} + \frac{\delta }{n} \left[ c_{l,1} - \mu _{C1} \right] (S^C)^{-1} (c_{j,1}^{C}- c_{i,1}^{C}), \end{aligned}$$

and the variance of the one-dimensional circumstance, $S^C > 0$. C1M implies that $c_{j,1}^{C} < c_{i,1}^{C}$. Hence we get $\widetilde{y}^{c5}_{l} < y^{c5}_{l}$ for those with $c_{l,1} > \mu _{C1}$, which by C1M is for those that have a higher than average income. Similarly, $\widetilde{y}^{c5}_{l} > y^{c5}_{l}$ for those with $c_{l,1} < \mu _{C1}$, which by C1M is for those that have a lower than average income. Hence we the transfer reduced inequality; COM is satisfied.

Consider part (i) of the Proposition and Eq. (A.21). Observe,

$$\begin{aligned} \frac{\sum _{l=1}^n \widetilde{y}^{EO5}_{l}}{n} = \frac{\sum _{l=1}^n y^{EO5}_{l}}{n} + \frac{\delta }{n} \frac{1}{n} \sum _{l=1}^n \frac{1}{\left| N_{\cdot l} \right| } \sum _{k \in N_{\cdot l}} \mathbf {x}_{\mathbf {k}}^{CE \prime } (\mathbf {S}^{CE})^{-1}(\mathbf {x}_{\mathbf {j}}^{CE}-\mathbf {x}_{\mathbf {i}}^{CE}). \end{aligned}$$

Since $\sum _{l=1}^n \frac{1}{\left| N_{\cdot l} \right| } \sum _{k \in N_{\cdot l}} \mathbf {x}_{\mathbf {k}}^{CE \prime } = \mathbf {0}$, the $d^C + d^E$-dimensional null vector, the mean of the distribution has not changed. First, with $\mathbf {x}_{\mathbf {j}}^{C}=\mathbf {x}_{\mathbf {i}}^{C}$ and COR0, from (A.21),

$$\begin{aligned} \widetilde{y}^{EO5}_{l} = y^{EO5}_{l} + \frac{\delta }{n} \mathbf {x}_{\mathbf {l}}^{E \prime } \mathbf {A}^{EE} (\mathbf {x}_{\mathbf {j}}^{E}-\mathbf {x}_{\mathbf {i}}^{E}). \end{aligned}$$

Hence, under COR0 and with one-dimensional effort,

$$\begin{aligned} \widetilde{y}^{EO5}_{l} = y^{EO5}_{l} + \frac{\delta }{n} \left[ e_{l1} - \mu _{E1} \right] (\mathbf {S}^{EE})^{-1} (\mathbf {x}_{\mathbf {j}}^{E}-\mathbf {x}_{\mathbf {i}}^{E}). \end{aligned}$$

and it cannot be guaranteed that the change in inequality is equal to the change in inequality in the actual income distribution. Hence UR is not satisfied.

Second, with $\mathbf {x}_{\mathbf {j}}^{E}=\mathbf {x}_{\mathbf {i}}^{E}$, from (A.21),

$$\begin{aligned} \widetilde{y}^{EO5}_{l} = y^{EO5}_{l} + \frac{\delta }{n} \frac{1}{\left| N_{\cdot l} \right| } \sum _{k \in N_{\cdot l}} \mathbf {x}_{\mathbf {k}}^{CE \prime } \left[ \begin{array}{c} \mathbf {A}^{CC} \\ \mathbf {A}^{EC} \end{array} \right] (\mathbf {x}_{\mathbf {j}}^{C}-\mathbf {x}_{\mathbf {i}}^{C}). \end{aligned}$$

Under COR0, this reduces to

$$\begin{aligned} \widetilde{y}^{EO5}_{l} = y^{EO5}_{l} + \frac{\delta }{n} \frac{1}{\left| N_{\cdot l} \right| } \sum _{k \in N_{\cdot l}} \mathbf {x}_{\mathbf {k}}^{C \prime } \mathbf {A}^{CC} (\mathbf {x}_{\mathbf {j}}^{C}-\mathbf {x}_{\mathbf {i}}^{C}). \end{aligned}$$

The i-th element in the vector $\frac{1}{\left| N_{\cdot l} \right| } \sum _{k \in N_{\cdot l}} \mathbf {x}_{\mathbf {k}}^{C \prime }$ is $\frac{1}{\left| N_{\cdot l} \right| } \sum _{k \in N_{\cdot l}} c_{ki} - \mu _{Ci}$, the difference between the average value of circumstance i of those that have the same effort as l and the average value of circumstance i in the population. Under COR0, the distribution of circumstances conditional on efforts is independent of the value of effort and equals the unconditional distribution. Hence $\frac{1}{\left| N_{\cdot l} \right| } \sum _{k \in N_{\cdot l}} \mathbf {x}_{\mathbf {k}}^{C \prime }$ is equal to the $d^C$-dimensional null vector, $\widetilde{y}^{EO5}_{l} = y^{EO5}_{l}$, and the measure satisfies COM. $\square$

1.4 A.3 Proof of Proposition 3

Proof of Lemma 2

Define $\widetilde{\mathbf {Y}}=\mathbf {Y}+ \delta (\mathbf {E_{j}}-\mathbf {E_{i}})$, which is the vector $\mathbf {Y}$ after a Pigou–Dalton transfer of an amount $\delta$ from observation j to i. After the transfer, for $A \in \left\{ C,E,CE \right\}$, we estimate the equation in deviational form

$$\begin{aligned} log(\widetilde{\mathbf {Y}})- {\iota } \mu _{log(\widetilde{Y})} = \mathbf {X}^{A} \widetilde{{\alpha }}^{A}+ \widetilde{\mathbf {V}}^{A}. \end{aligned}$$

(A.33)

Observe that

$$\begin{aligned} log(\widetilde{\mathbf {Y}})=log(\mathbf {Y}) + \left( log(y_{j}+ \delta ) -log(y_{j}) \right) \mathbf {E_{j}} + \left( log(y_{i}- \delta ) -log(y_{i}) \right) \mathbf {E_{i}}, \end{aligned}$$

and that

$$\begin{aligned} \mu _{log(\widetilde{Y})}=\mu _{log(Y)}+ \frac{1}{n} \left[ log(y_{j}+ \delta ) -log(y_{j}) + log(y_{i}- \delta ) -log(y_{i}) \right] , \end{aligned}$$

such that the least squares estimate,

$$\begin{aligned} \widetilde{\mathbf {a}}^{A}= \left( \mathbf {X}^{A\prime } \mathbf {X}^{A} \right) ^{-1} \mathbf {X}^{A\prime } \left[ log(\widetilde{\mathbf {Y}})-{\iota } \mu _{log(\widetilde{Y})} \right] , \end{aligned}$$

(A.34)

can be written as

$$\begin{aligned} \widetilde{\mathbf {a}}^{A}= \mathbf {a}^{A} + \left( \mathbf {X}^{A\prime } \mathbf {X}^{A} \right) ^{-1} \mathbf {X}^{A \prime } \cdot \\&\left[ \mathbf {E_{j}} \left( log(y_{j}+ \delta ) -log(y_{j}) \right) + \mathbf {E_{i}} \left( log(y_{i}- \delta ) -log(y_{i}) \right) \right. \\&\left. (1/n) {\iota } \left( log(y_{j}+ \delta ) -log(y_{j}) + log(y_{i}- \delta ) -log(y_{i}) \right) \right] . \end{aligned}$$

As $\mathbf {S}^{A}=\frac{1}{n} \mathbf {X}^{A \prime } \mathbf {X}^{A}$, $\mathbf {X}^{A \prime } \mathbf {E_{j}} = \mathbf {x}_{\mathbf {j}}^{A}$, $\mathbf {X}^{A \prime } \mathbf {E_{i}} = \mathbf {x}_{\mathbf {i}}^{A}$ and $(1/n) \mathbf {X}^{A \prime } {\iota } = 0$, we obtain the expression in the Lemma. $\square$

Lemma A3

Under the assumption of loglinearity and using the least squares estimator, the counterfactuals (6)–(10) and (12)–(16) are defined by

$$\begin{aligned} log(y^{c1}_l)= \mu _{log(Y)} + \mathbf {x}_{\mathbf {l}}^{C \prime } \mathbf {a}^{C}, \end{aligned}$$

(A.35)

$$\begin{aligned} log(y^{c2}_l)= log(y_{l})- \mathbf {x}_{\mathbf {l}}^{E \overline{E} \prime } \mathbf {a}^{E}, \end{aligned}$$

(A.36)

$$\begin{aligned} log(y^{c3}_l)= \mu _{log(Y)} + \mathbf {x}_{\mathbf {l}}^{C \overline{E} \prime } \mathbf {a}^{CE}, \end{aligned}$$

(A.37)

$$\begin{aligned} log(y^{c4}_l)= log(y_{l})- \mathbf {x}_{\mathbf {l}}^{0E \prime } \mathbf {a}^{CE}, \end{aligned}$$

(A.38)

$$\begin{aligned} log(y^{c5}_l)= \mu _{log(Y)} + \frac{1}{\left| N_{l \cdot } \right| } \sum _{k \in N_{l \cdot }} \mathbf {x}_{\mathbf {k}}^{CE \prime } \mathbf {a}^{CE}, \end{aligned}$$

(A.39)

$$\begin{aligned} log(y^{EO1}_l)= \mu _{log(Y)} + \mathbf {x}_{\mathbf {l}}^{E \prime } \mathbf {a}^{E}, \end{aligned}$$

(A.40)

$$\begin{aligned} log(y^{EO2}_l)= log(y_{l})-\mathbf {x}_{\mathbf {l}}^{C \overline{C} \prime } \mathbf {a}^{C}, \end{aligned}$$

(A.41)

$$\begin{aligned} log(y^{EO3}_l)= \mu _{log(Y)} + \mathbf {x}_{\mathbf {l}}^{\overline{C}E \prime } \mathbf {a}^{CE}, \end{aligned}$$

(A.42)

$$\begin{aligned} log(y^{EO4}_l)= log(y_{l})- \mathbf {x}_{\mathbf {l}}^{C0 \prime } \mathbf {a}^{CE}, \end{aligned}$$

(A.43)

$$\begin{aligned} log(y^{EO5}_l)= \mu _{log(Y)} + \frac{1}{\left| N_{\cdot l} \right| } \sum _{k \in N_{\cdot l}} \mathbf {x}_{\mathbf {k}}^{CE \prime } \mathbf {a}^{CE}. \end{aligned}$$

(A.44)

Using Lemma 2, it is straightforward to prove Lemma A4.

Lemma A4

Define $\varDelta _{j}=log(y_{j}+\delta )-log(y_{j})$and $\varDelta _{i}=log(y_{i}-\delta )-log(y_{i})$. The change in the estimated counterfactuals of a Pigou–Dalton transfer $\delta$from observation i to j is

$$\begin{aligned} log(\widetilde{y}^{c1}_{l})-log(y^{c1}_{l})= \frac{1}{n} \left[ \varDelta _{j} + \varDelta _{i} \right] + \frac{1}{n} \mathbf {x}_{\mathbf {l}}^{C \prime } (\mathbf {S}^{C})^{-1} \left[ \mathbf {x}_{\mathbf {j}}^{C} \varDelta _{j} + \mathbf {x}_{\mathbf {i}}^{C} \varDelta _{i} \right] , \end{aligned}$$

(A.45)

$$\begin{aligned} log(\widetilde{y}^{c2}_{l})-log(y^{c2}_{l})&= log(\widetilde{y}_{l})-log(y_{l}) - \frac{1}{n} \mathbf {x}_{\mathbf {l}}^{E\overline{E} \prime } (\mathbf {S}^{E})^{-1}\nonumber \\&\quad \times \left[ \mathbf {x}_{\mathbf {j}}^{E} \varDelta _{j} + \mathbf {x}_{\mathbf {i}}^{E} \varDelta _{i} \right] , \end{aligned}$$

(A.46)

$$\begin{aligned} log(\widetilde{y}^{c3}_{l})-log(y^{c3}_{l})&= \frac{1}{n} \left[ \varDelta _{j} + \varDelta _{i} \right] + \frac{1}{n} \mathbf {x}_{\mathbf {l}}^{C \overline{E} \prime } (\mathbf {S}^{CE})^{-1}\nonumber \\&\quad\times \left[ \mathbf {x}_{\mathbf {j}}^{CE} \varDelta _{j} + \mathbf {x}_{\mathbf {i}}^{CE} \varDelta _{i} \right] , \end{aligned}$$

(A.47)

$$\begin{aligned} log(\widetilde{y}^{c4}_{l})-log(y^{c4}_{l})&= log(\widetilde{y}_{l})-log(y_{l}) - \frac{1}{n} \mathbf {x}_{\mathbf {l}}^{0E \prime } (\mathbf {S}^{CE})^{-1}\nonumber \\&\quad\times \left[ \mathbf {x}_{\mathbf {j}}^{CE} \varDelta _{j} + \mathbf {x}_{\mathbf {i}}^{CE} \varDelta _{i} \right] , \end{aligned}$$

(A.48)

$$\begin{aligned} log(\widetilde{y}^{c5}_{l}) - log(y^{c5}_l)&= \frac{1}{n} \left[ \varDelta _{j} + \varDelta _{i} \right] + \frac{1}{n} \frac{1}{\left| N_{l \cdot } \right| } \sum _{k \in N_{l \cdot }} \mathbf {x}_{\mathbf {k}}^{CE \prime } (\mathbf {S}^{CE})^{-1}\nonumber \\&\quad\times \left[ \mathbf {x}_{\mathbf {j}}^{CE} \varDelta _{j} + \mathbf {x}_{\mathbf {i}}^{CE} \varDelta _{i} \right] , \end{aligned}$$

(A.49)

$$\begin{aligned} log(\widetilde{y}^{EO1}_{l})-log(y^{EO1}_{l})&= \frac{1}{n} \left[ \varDelta _{j} + \varDelta _{i} \right] + \frac{1}{n} \mathbf {x}_{\mathbf {l}}^{E \prime } (\mathbf {S}^{E})^{-1}\nonumber \\&\quad\times \left[ \mathbf {x}_{\mathbf {j}}^{E} \varDelta _{j} + \mathbf {x}_{\mathbf {i}}^{E} \varDelta _{i} \right] , \end{aligned}$$

(A.50)

$$\begin{aligned} log(\widetilde{y}^{EO2}_{l})-log(y^{EO2}_{l})&= log(\widetilde{y}_{l})-log(y_{l}) - \frac{1}{n} \mathbf {x}_{\mathbf {l}}^{C \overline{C} \prime } (\mathbf {S}^{C})^{-1}\nonumber \\&\quad\times \left[ \mathbf {x}_{\mathbf {j}}^{C} \varDelta _{j} + \mathbf {x}_{\mathbf {i}}^{C} \varDelta _{i} \right] , \end{aligned}$$

(A.51)

$$\begin{aligned} log(\widetilde{y}^{EO3}_{l})-log(y^{EO3}_{l})&= \frac{1}{n} \left[ \varDelta _{j} + \varDelta _{i} \right] + \frac{1}{n} \mathbf {x}_{\mathbf {l}}^{\overline{C}E \prime } (\mathbf {S}^{CE})^{-1}\nonumber \\&\quad\times \left[ \mathbf {x}_{\mathbf {j}}^{CE} \varDelta _{j} + \mathbf {x}_{\mathbf {i}}^{CE} \varDelta _{i} \right] , \end{aligned}$$

(A.52)

$$\begin{aligned} log(\widetilde{y}^{EO4}_{l})-log(y^{EO4}_{l})&= log(\widetilde{y}_{l})-log(y_{l}) - \frac{1}{n} \mathbf {x}_{\mathbf {l}}^{C0 \prime } (\mathbf {S}^{CE})^{-1}\nonumber \\&\quad\times \left[ \mathbf {x}_{\mathbf {j}}^{CE} \varDelta _{j} + \mathbf {x}_{\mathbf {i}}^{CE} \varDelta _{i} \right] , \end{aligned}$$

(A.53)

$$\begin{aligned} log(\widetilde{y}^{EO5}_{l}) - log(y^{EO5}_l)&= \frac{1}{n} \left[ \varDelta _{j} + \varDelta _{i} \right] + \frac{1}{n} \frac{1}{\left| N_{\cdot l} \right| } \sum _{k \in N_{\cdot l}} \mathbf {x}_{\mathbf {k}}^{CE \prime } (\mathbf {S}^{CE})^{-1}\nonumber \\&\quad\times \left[ \mathbf {x}_{\mathbf {j}}^{CE} \varDelta _{j} + \mathbf {x}_{\mathbf {i}}^{CE} \varDelta _{i} \right] . \end{aligned}$$

(A.54)

Proof of Proposition 3

The left hand side in the equations of Lemma A4 give the percentage change in the estimated counterfactual. First, observe that, in (A.45), (A.47), (A.49), (A.50) (A.52) and (A.54), the first term is the same for all observations, and so has no effect on the inequality of the counterfactual, provided a relative measure of inequality is used. The problem to sign the effect of the transfer on the inequality measure is that the second term will be different for different observations, and will never disappear (not even when the transfer is between individuals having the same circumstances and/ or efforts - except, when, in addition they have the same income level, but in that case there is no Pigou–Dalton transfer). Hence, these measures satisfy neither COM nor UR. Second, observe that in (A.46), (A.48) (A.51) and (A.53), the first term, $log(\widetilde{y}_{l})-log(y_{l})$, is zero for all observations, except for i and j. Again, however, the problem is that the other term never vanishes, making it impossible to assess the effect of the transfer on the inequality in the counterfactual income distributions. Hence we obtain Proposition 3. $\square$

B Alternative proportional transfer principles

Let $y_{i}$ be individual i’s income before the transfer, $y_{j}$ individual j’s income before the transfer, $\widetilde{y}_{i}$ individual i’s income after the transfer, and $\widetilde{y}_{j}$ individual j’s income after the transfer. Throughout we require

$$\begin{aligned} y_{i}>\widetilde{y}_{i} \ge \widetilde{y}_{j}>y_{j}, \end{aligned}$$

such that the transfer goes from individual i to individual j, and also after the transfer i has at least as much income as j. Different proportional transfer principles impose different conditions on the transfers.

The Factor Proportional Transfer Principle (this paper) requires that, with $A>1$,

$$\begin{aligned} \widetilde{y}_{i}=\frac{y_{i}}{A} \text { and } \widetilde{y}_{j}=y_{j} \cdot A. \end{aligned}$$

(B.1)

The Proportional Transfer Principle (Fleurbaey and Michel 2001, p. 4) requires that, with $\delta >0$,

$$\begin{aligned} \widetilde{y}_{i}=y_{i} (1-\delta ) \text { and } \widetilde{y}_{j}=y_{j} (1+\delta ). \end{aligned}$$

(B.2)

The Proportional Ex-Post Transfer Principle (Fleurbaey and Michel 2001, p. 4) requires that, with $\delta >0$,

$$\begin{aligned} \widetilde{y}_{i}=\frac{y_{i}}{1+\delta } \text { and } \widetilde{y}_{j}= \frac{y_{j}}{1-\delta }. \end{aligned}$$

(B.3)

Assuming that the transfer described in the principle is desirable (because it decreases inequality), the following Proposition formulates the logical relationship between the three transfer principles.

Proposition C

The Proportional Transfer Principle is stronger than the Factor Proportional Transfer Principle, which is stronger than the Proportional Ex-Post Transfer Principle.

Proof of Proposition C

• (a) Comparison of (B.2) and (B.1). Consider the case where the transfer implies the same transfer in favor of the poor individual, i.e. $A=(1+\delta )$. In that case, the income after transfer for the rich person under (B.1) is higher than the income of the rich person under (B.2), as

  $$\begin{aligned} 1-\delta ^{2}=(1-\delta )(1+\delta )<1 \Longleftrightarrow \frac{y_{i}}{A}= \frac{y_{i}}{1+\delta }>y_{i} (1-\delta ). \end{aligned}$$

  Hence, all transfers that are acceptable under (B.1) are also acceptable under (B.2), but the reverse does not hold true.

• (b) Comparison of (B.1) and (B.3). Consider the case where the transfer implies the same transfer in favor of the poor individual, i.e. $A \prime =1/(1-\delta )$. In that case, the income after transfer for the rich person under (B.3) is higher than the income of the rich person under (B.1), as

  $$\begin{aligned} 1-\delta ^{2}=(1-\delta )(1+\delta )<1 \Longleftrightarrow \frac{y_{i}}{1+\delta }>(1-\delta ) y_{i}= \frac{y_{i}}{A \prime }. \end{aligned}$$

  Hence, all transfers that are acceptable under (B.3) are also acceptable under (B.1), but the reverse does not hold true.

$\square$