Preferences and Coalitions

Abstract

Redistribution of income is assumed to require the vote of at least 50 % of all individuals with income and each of them insists to be a winner from the redistribution. When inequality is to be increased in the interest of some, coalition partners must be found by compensation schemes. Compensation minimization is shown to lead to coalition partners being either a connected or a disconnected income group. When inequality reaches certain critical levels, disconnection becomes unavoidable. For one-parametric income distributions, the critical levels are denoted as bifurcation points.

Values of bifurcation points are computed numerically for several one-parametric distributions. For income distributions with two or more parameters the bifurcation point is replaced by a bifurcation function.

Appendix: Bifurcation Parameters for One-Parametric Lorenz Curves—General Case

In general, the med-min function neither is continuous nor need it have two monotonicity segments for parametric classes of Lorenz curves. The first of these complications is addressed by replacing the maximum with the supremum and the second is taken care of by modifying the parameter set. When the modified parameter set is empty, a bifurcation parameter does not exist. The bifurcation parameter and the modified parameter set are stated as

$$\displaystyle\begin{array}{rcl} \vartheta _{\mathit{bif }}& =& \mathit{argsup}_{\vartheta \in \varTheta _{=}}\;L_{\vartheta }^{{\prime}}(0.5) - L_{\vartheta }^{{\prime}}(0) {}\\ \varTheta _{=}& =& \{\vartheta \in \varTheta \|\,\exists \,\vartheta ^{{\ast}}\in \varTheta -\{\vartheta \}\mbox{ with }L_{\vartheta ^{{\ast} }}\mbox{ Lorenz-dominating }L_{\vartheta }\mbox{ and } {}\\ & & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad L_{\vartheta ^{{\ast}}}^{{\prime}}(0.5) - L_{\vartheta ^{{\ast} }}^{{\prime}}(0) \leq L_{\vartheta }^{{\prime}}(0.5) - L_{\vartheta }^{{\prime}}(0)\}. {}\\ \end{array}$$

The supremum-argument is understood as argument leading to the maximum, if the maximum exists or, else, as

$$\displaystyle{\vartheta _{\mathit{bif }} =\lim _{i\rightarrow \infty }\,\vartheta _{i}\mbox{ with }\lim _{i\rightarrow \infty }L_{\vartheta _{i}}^{{\prime}}(0.5) - L_{\vartheta _{ i}}^{{\prime}}(0) = \mathit{sup}_{\vartheta \in \varTheta }\;L_{\vartheta }^{{\prime}}(0.5) - L_{\vartheta }^{{\prime}}(0).}$$

The supremum over the differences between median income and minimum income always exists since the median income is bounded by twice the average income. Thus, $L_{\vartheta }^{{\prime}}(0.5) - L_{\vartheta }^{{\prime}}(0) \leq 2$ for all $\vartheta \in \varTheta$. An alternative condition for a value $\vartheta _{\mathit{bif }}$ qualifying as bifurcation value is that for all $\vartheta \in \varTheta$ with $\vartheta \neq \vartheta _{\mathit{bif }}$ exists $\vartheta ^{{\ast}} \in \varTheta$ between $\vartheta$ and $\vartheta _{\mathit{bif }}$ such that $L_{\vartheta ^{{\ast}}}^{{\prime}}(0.5) - L_{\vartheta ^{{\ast}}}^{{\prime}}(0) > L_{\vartheta }^{{\prime}}(0.5) - L_{\vartheta }^{{\prime}}(0)$.