Consider a setting with two individuals. We single out the envy that one individual experiences towards the other by replacing the latter’s preference relation with the flat preference \({\bar{R}}\). We propose axioms that require the envy measure to react to simple changes in the bundles of the two individuals. The axioms bear on the envy measure E, but only directly impose properties on the elementary envy measure corresponding to E.
Betweenness requires the elementary envy that individual i experiences towards individual j to decrease if i’s bundle improves or j’s bundle worsens according to i’s preferences. In terms of i’s preferences, the new bundles lie ‘in between’ the original bundles. Figure 1 depicts an example where betweenness implies that \(E(x_i,x_j,R_i,{\bar{R}}) > E(x_i',x_j',R_i,{\bar{R}})\).
Betweenness For all individuals i and j in \({\mathcal {N}}\), all bundles \(x_{i}\), \(x_{j}\), \(x'_{i}\) and \(x'_{j}\) in X and each preference relation \(R_{i}\) in \({\mathcal {R}}\) such that \(x_{j} \mathop {P_i} x_{i}\), we have that \(x_j \mathop {R_i} x_j'\), \(x_j' \mathop {R_i} x_i'\) and \(x_i' \mathop {R_i} x_i\) imply \(E(x_i,x_j,R_i,{\bar{R}}) \ge E(x_i',x_j',R_i,{\bar{R}})\) with strict inequality holding whenever \(x_j \mathop {P_i} x_j'\) or \(x_i' \mathop {P_i} x_i\).
We emphasize an implication of betweenness. Let \(u_i\) be an arbitrary utility representation of the preference relation \(R_i\). Betweenness implies that the elementary envy \(E(x_i,x_j,R_i,{\bar{R}})\) experienced by individual i towards j can be expressed as a function only of the utility levels \(u_i(x_i)\) and \(u_i(x_j)\). That is, if \(u_i(x_i)=u_i(x'_i)\) and \(u_i(x_{j}) = u_i(x'_{j})\) (as depicted in Fig. 2), then \(E(x_i,x_j,R_i,{\bar{R}}) = E(x_i',x_j',R_i,{\bar{R}})\). Note that the second and third Feldman–Kirman measures are in this functional form and satisfy betweenness.
The next axiom captures the idea of gauging elementary envy by the radial distance between bundles. Consider two approaches. The first approach, as adopted in the Chaudhuri and Diamantaras–Thomson measures, measures the elementary envy experienced by i towards j using the factor \(\lambda _{ij}\) by which j’s bundle has to be shrunk in order for i to stop envying j. That is, \(\lambda _{ij}\) is such that \(x_i \mathop {I_i} \lambda _{ij} x_j\), and the higher \(\lambda _{ij}\), the lower the elementary envy of i. The second approach measures the elementary envy experienced by i towards j using the factor \(\kappa _{ij}\) by which i’s bundle has to be expanded in order for i to stop envying j. That is, \(\kappa _{ij}\) is such that \(\kappa _{ij} x_i \mathop {I_i} x_j\), and the higher \(\kappa _{ij}\), the higher the elementary envy of i. The two approaches are a priori equally appealing, but yield conflicting results.Footnote 10 To see this, consider the social states \(s = (x_i , x_j, R_i, {\bar{R}})\) and \(s' = (x'_i , x'_j, R_i, {\bar{R}})\) in Fig. 2. The first approach implies \(E(s) < E(s')\) because \(\lambda _{ij} > \lambda '_{ij}\), whereas the second approach implies \(E(s) > E(s')\) because \(\kappa _{ij} > \kappa '_{ij}\).Footnote 11
We do not make a choice among the two conflicting approaches. Instead, we formulate an axiom that is sufficiently weak to be consistent with both. The axiom only considers the cases where the bundles of the envied and envious are proportional to each other and says that a decrease of the radial distance between these two bundles reduces elementary envy. In these cases \(\lambda _{ij} = 1/\kappa _{ij}\) and the two above approaches coincide.
Proportionality For all individuals i and j in \({\mathcal {N}}\), all bundles \(x_{i}\), \(x_{j}\), \(x'_{i}\) and \(x'_{j}\) in X such that \(\kappa x_i = x_j\) and \(\kappa ' x_i' = x_j'\) and all preference relations \(R_{i}\) and \(R'_{i}\) in \({\mathcal {R}}\) such that \(x_j \mathop {P_i} x_i\) and \(x_j' \mathop {P_i}' x_i'\), we have that \(\kappa \ge \kappa '\) implies \(E(x_i,x_j,R_i,{\bar{R}}) \ge E(x_i',x_j',R_i',{\bar{R}})\) with strict inequality holding if and only if \(\kappa > \kappa '\).
However, betweenness and proportionality are incompatible: there is no envy measure that satisfies both axioms. Consider the social states \(s = (x_i,x_j,R_i,{\bar{R}})\) and \(s' = (x'_i,x'_j,R_i,{\bar{R}})\) in Fig. 3. Betweenness implies \(E(s) > E(s')\), whereas proportionality implies \(E(s) < E(s')\) because \(\kappa _{ij} < \kappa '_{ij}\).Footnote 12
We treat betweenness as essential and therefore weaken proportionality. The following axiom requires all bundles to be proportional to a predetermined reference bundle r. Later we will argue that the axiom may be regarded as a minimum weakening of proportionality that is compatible with betweenness.
r -Proportionality There is a bundle r in X such that the following holds. For all individuals i and j in \({\mathcal {N}}\), all bundles \(x_{i}\), \(x_{j}\), \(x'_{i}\) and \(x'_{j}\) in X proportional to r and such that \(\kappa x_i = x_j\) and \(\kappa ' x_i' = x_j'\) and all preference relations \(R_{i}\) and \(R'_{i}\) in \({\mathcal {R}}\) such that \(x_j \mathop {P_i} x_i\) and \(x_j' \mathop {P_i}' x_i'\), we have that \(\kappa \ge \kappa '\) implies \(E(x_i,x_j,R_i,{\bar{R}}) \ge E(x_i',x_j',R_i',{\bar{R}})\) with strict inequality holding if and only if \(\kappa > \kappa '\).
Before proceeding, we define the ‘ray’ utility representation. Samuelson (1977), Pazner (1979), and Fleurbaey and Maniquet (2011), among others, have used this utility presentation in social welfare analysis with ordinal and non-comparable preferences.Footnote 13 Let \(\rho \) be a predetermined reference bundle in X. The ray utility level attained by individual i with bundle \(x_i\) and preference relation \(R_i\) is equal to the real number \(u_{\rho }(x_i, R_i)\) such that i is indifferent between the fraction \(u_{\rho }(x_i, R_i)\) of the bundle \(\rho \) and his own bundle \(x_i\). Figure 4 provides an illustration. Formally, for each preference relation \(R_i\) in \({\mathcal {R}} {\setminus } \{{\bar{R}}\}\) and each bundle \(x_i\) in X, let \(u_{\rho }(x_i, R_i)\) be the real number such that \(x_{i} \mathop {I_i} u_{\rho }(x_i,R_i) \rho \). Let \(u_{\rho }(x_i, {\bar{R}})\) be equal to a positive constant for each bundle \(x_i\) in X. Clearly, the function \(u_{\rho }( \, \cdot \, , R_i)\) is a utility representation of the preference relation \(R_i\).
The following proposition says that an envy measure satisfies betweenness and r-proportionality if and only if it measures the elementary envy experienced by individual i towards j by the ratio of i’s ray utility levels associated with j’s and i’s bundles. Moreover, the reference bundle \(\rho \) that determines the ray utility representation must be chosen such that \(\rho = r\).
Proposition 2
Let E be an envy measure that satisfies anonymity. Then E satisfies betweenness and r-proportionality if and only if there exists a strictly increasing function \(f : \mathbb {R}\rightarrow \mathbb {R}\) such that, for all individuals i and j in \({\mathcal {N}}\), all bundles \(x_{i}\) and \(x_{j}\) in X and each preference relation \(R_i\) in \({\mathcal {R}}\) such that \(x_j \mathop {P_i} x_i\), we have
$$\begin{aligned} E(x_i,x_j,R_i,{\bar{R}}) = f\left( \frac{u_{r}(x_j, R_i)}{u_{r}(x_i, R_i)}\right) . \end{aligned}$$
Proof
It is easy to see that the stated envy measure satisfies betweenness and r-proportionality. We focus on the reverse implication.
Let r be a given bundle in X. Let i and j be individuals in \({\mathcal {N}}\), let \(x_{i}\), \(x'_{i}\), \(x_j\) and \(x_j'\) be bundles in X and let \(R_{i}\) and \(R'_i\) be preference relations in \({\mathcal {R}}\) such that \(x_j \mathop {P_i} x_i\) and \(x_j' \mathop {P'_i} x_i'\). We have to show that
$$\begin{aligned} E\left( x_i,x_j,R_i,{\bar{R}}\right) \ge E\left( x'_i,x'_j,R'_i,{\bar{R}}\right) \end{aligned}$$
(1)
if and only if
$$\begin{aligned} \frac{u_{r}\big (x_{j}, R_i\big )}{u_{r}\big (x_{i}, R_i\big )} \ge \frac{u_{r}\big (x'_{j}, R'_i\big )}{u_{r}\big (x'_{i}, R'_i\big )} . \end{aligned}$$
(2)
Then there exists a strictly increasing function f as stated. Note that f does not depend on i and j by anonymity.
Let \(y_i\), \(y'_i\), \(y_j\) and \(y_j'\) be bundles in X proportional to r and such that \(y_i \mathop {I_i} x_i\), \(y'_i \mathop {I'_i} x'_i\), \(y_j \mathop {I_i} x_j\) and \(y'_j \mathop {I'_i} x'_j\). Such bundles exist since \(R_{i}\) and \(R'_i\) are continuous and strictly monotonic. Let \(\kappa \) and \(\kappa '\) be such that \(\kappa y_i= y_j\) and \(\kappa ' y'_i= y_j'\).
Suppose that Eq. (1) holds. We have to show that Eq. (2) holds as well. By betweenness, we have \(E(x_i,x_j,R_i,{\bar{R}}) = E(y_i,y_j,R_i,{\bar{R}})\) and \(E(x_i',x_j',R'_i,{\bar{R}}) = E(y'_i,y'_j,R'_i,{\bar{R}})\). Hence, we obtain \(E(y_i,y_j,R_i,{\bar{R}}) \ge E(y'_i,y'_j,R'_i,{\bar{R}})\). If \(\kappa < \kappa '\), then \(E(y_i,y_j,R_i,{\bar{R}}) < E(y'_i,y'_j,R'_i,{\bar{R}})\) by r-proportionality. Hence, it must be that \(\kappa \ge \kappa '\). From the definition of \(u_r\), it follows that \(\kappa = u_{r}(y_j, R_i)/u_{r}(y_i, R_i)\) and \(\kappa ' = u_{r}(y'_j, R'_i)/u_{r}(y'_i, R'_i)\). Since \(u_{r}(x_j, R_i)/u_{r}(x_i, R_i) = u_{r}(y_j, R_i)/u_{r}(y_i, R_i)\) and \(u_{r}(x'_j, R'_i)/u_{r}(x'_i, R'_i) = u_{r}(y'_j, R'_i)/u_{r}(y'_i, R'_i)\), we obtain Eq. (2).
Now, suppose that Eq. (2) holds. We have to show that Eq. (1) holds as well. Equation (2) implies that \(\kappa = u_{r}(y_j, R_i)/u_{r}(y_i, R_i) \ge \kappa ' = u_{r}(y'_j, R'_i)/u_{r}(y'_i, R'_i)\). Since \(\kappa \ge \kappa '\), we have \(E(y_i,y_j,R_i,{\bar{R}}) \ge E(y'_i,y'_j,R'_i,{\bar{R}})\) by r-proportionality. Using betweenness, we obtain Eq. (1). \(\square \)
The measure of elementary envy in Proposition 2 shares with the second and third Feldman–Kirman measures that it depends on the utility distance between the bundles of the envious and the envied. However, the utility representation used is not an arbitrary choice as in those measures. Rather, the radial distance idea inherent in the Chaudhuri and Diamantaras–Thomson measures singles out the ray utility representation.
Note that, for a given individual i, the criterion in Proposition 2 provides a complete ranking of all social states of the form \((x_i,x_j,R_i,{\bar{R}})\). This means that any further strengthening of r-proportionality in the direction of proportionality will either lead to conflicts or is already implied by the combination of r-proportionality and betweenness. In other words, r-proportionality minimally weakens proportionality while ensuring compatibility with betweenness.