Taxation and poverty

Abstract

We explore the implications of four natural axioms in taxation: continuity (small changes in the data of a taxation problem should not lead to large changes in the tax allocation), equal treatment of equals (agents with the same pre-tax incomes pay equal taxes), consistency (the way in which a group allocates a tax burden is immune to secessions of taxpayers) and composition down (an increase in the tax burden is handled according to agents’ current post-tax incomes). The combination of the four axioms characterizes a large family of rules, which we call generalized equal-sacrifice rules, encompassing the so-called equal-sacrifice rules (such as the flat tax), as well as constrained equal-sacrifice rules (such as the head tax), and exogenous poverty-line rules (such as the leveling tax, and some of its possible compromises with the previous ones).

Appendix

1.1 The tightness of the characterization result

The rules presented next are all chosen to be homogeneous, which hence also shows the tightness of Theorem 2.

• 1. A rule that satisfies all properties except for equal treatment of equals is a lexicographic dictatorship.

• 2. A rule that satisfies all properties except for composition down is the so-called Talmud rule (introduced by Aumann and Maschler 1985).

• 3. A rule that satisfies all properties except for consistency is a rule which coincides with the head tax for pairs of agents, and the leveling tax otherwise.

• 4. A rule that satisfies all properties except for continuity is the fully regressive tax, which orders agents according to the size of their incomes, and taxes the poorest agents first. An agent with a high income only pays a tax if all poorer agents have fully exhausted their incomes. Agents with equal incomes are taxed equally.

• 5. A rule that satisfies all properties except balance is the rule that always taxes everybody their entire income.

1.2 Some generalized equal-sacrifice rules and their representations

Assume, for ease of notation, that, for each $\left( N,c,E\right) \in \mathcal {D}$, the claims vector is such that $c_1\le c_2\le \dots \le c_n$, where n denotes the cardinality of N.

• The leveling tax Let $\Lambda =\{\{x\}:x\in \mathbb {R}_{+}\}$.³⁰ Then, for each $\left( N,c,E\right) \in \mathcal {D}$, let k be the smallest non-negative integer in $\{0,\ldots ,n\}$ such that $E\le \left( \left( \sum _{i=1}^{k}c_{i} \right) +(n-k)c_{k+1} \right)$. Let $\lambda =\frac{E-\left( \sum _{i=1}^{k}c_{i}\right) }{n-k}$. It follows that, for each $i\in N$ such that $c_i \le \lambda$, $L_i\left( N,c,E\right) =c_i$. Furthermore, for each $c_i > \lambda$, $L_i\left( N,c,E\right) =\lambda$.

• The flat tax Let $\Lambda =\{[0,+\infty )\}$ and $\Gamma =\{g_{\lambda }\}$, where $g_{\lambda }:[0,+\infty )\rightarrow \mathbb {R}\cup \{-\infty \}$ is such that, for each $x\in [0,+\infty )$, $g_{\lambda }(x)=\log (x)$. Then, for each $\left( N,c,E\right) \in \mathcal {D}$, and each pair $i,j\in N$, $g_{\lambda }(c_i)-g_{\lambda }(F_i\left( N,c,E\right) )=g_{\lambda }(c_j)-g_{\lambda }(F_j\left( N,c,E\right) )$.

• The head tax Let $\Lambda =\{[0,+\infty )\}$ and $\Gamma =\{g_{\lambda }\}$, where $g_{\lambda }:[0,+\infty )\rightarrow \mathbb {R}\cup \{-\infty \}$ such that, for each $x\in [0,+\infty )$, $g_{\lambda }(x)=x$. Then, for each $\left( N,c,E\right) \in \mathcal {D}$, and each pair $i,j\in N$, $g_{\lambda }(c_i)-g_{\lambda }(H_i\left( N,c,E\right) )<g_{\lambda }(c_j)-g_{\lambda }(H_j\left( N,c,E\right) )$ implies that $H_i\left( N,c,E\right) =0$.

• Equal-sacrifice rules We proceed as with the flat tax, but considering a generic utility function with the properties stated at the definition of this family (not necessarily the logarithmic function, as we considered for the flat tax). More precisely, let $\Lambda =\{[0,+\infty )\}$ and $\Gamma =\{g_{\lambda }\}$, where $g_{\lambda }:[0,+\infty )\rightarrow \mathbb {R}\cup \{-\infty \}$ such that, for each $x\in [0,+\infty )$, $g_{\lambda }(x)=u(x)$. Then, for each $\left( N,c,E\right) \in \mathcal {D}$, and each pair $i,j\in N$, $g_{\lambda }(c_i)-g_{\lambda }(ES^{u}_i\left( N,c,E\right) )=g_{\lambda }(c_j)-g_{\lambda }(ES^{u}_j\left( N,c,E\right) )$.

• Constrained equal-sacrifice rules We proceed as with the head tax, but considering a generic utility function with the properties stated at the definition of this family (not necessarily the identity function, as we considered for the head tax). More precisely, let $\Lambda =\{[0,+\infty )\}$ and $\Gamma =\{g_{\lambda }\}$, where $g_{\lambda }:[0,+\infty )\rightarrow \mathbb {R}\cup \{-\infty \}$ such that, for each $x\in [0,+\infty )$, $g_{\lambda }(x)=u(x)$. Then, for each $\left( N,c,E\right) \in \mathcal {D}$, and each pair $i,j\in N$, $g_{\lambda }(c_i)-g_{\lambda }(CES^{u}_i\left( N,c,E\right) )<g_{\lambda }(c_j)-g_{\lambda }(CES^{u}_j\left( N,c,E\right) )$ implies that $CES^{u}_i\left( N,c,E\right) =0$.

• Compromises between the flat tax and the leveling tax We concentrate on the rule illustrated at Fig. 4, i.e., leveling tax until all agents are guaranteed a certain income ($\alpha$) and then flat tax. Let $\Lambda =\{\{x\},[\alpha ,+\infty ):0\le x<\alpha \}$, i.e., degenerate brackets from 0 to $\alpha$ and a unique non-degenerate bracket $[\alpha ,+\infty )$.³¹ Now, let $\Gamma =\{g_{\alpha }\}$, where $g_{\alpha }:[\alpha ,+\infty )\rightarrow \mathbb {R}\cup \{-\infty \}$ such that, for each $x\in [\alpha ,+\infty )$, $g_{\alpha }(x)=\log (x-\alpha )$. Then, for each $\left( N,c,E\right) \in \mathcal {D}$, let $k_{\alpha }$ denote the smallest integer number for which $c_{k_{\alpha }}<\alpha$.³² Then, we distinguish two cases:

  Case 1 $E\le \left( \left( \sum _{i=1}^{k_{\alpha }}c_{i} \right) +(n-k_{\alpha })c_{k+1} \right)$. In this case, let k be the smallest non-negative integer in $\{0,\ldots ,k_{\alpha }\}$ such that $E\le \left( \left( \sum _{i=1}^{k}c_{i} \right) +(n-k)c_{k+1} \right)$. Let $\lambda =\frac{E-\left( \sum _{i=1}^{k}c_{i}\right) }{n-k}$. It follows that, for each $i\in N$ such that $c_i \le \lambda$, $S_i\left( N,c,E\right) =c_i$. Furthermore, for each $c_i > \lambda$, $S_i\left( N,c,E\right) =\lambda =L_i\left( N,c,E\right)$.

  Case 2 $E> \left( \left( \sum _{i=1}^{k_{\alpha }}c_{i} \right) +(n-k_{\alpha })c_{k+1} \right)$. In this case, let $\lambda =[\alpha ,+\infty )$. It follows that, for each $i\in N$ such that $c_i \le \lambda$, $S_i\left( N,c,E\right) =c_i$. Furthermore, for each pair $i,j\in N$, such that $\min \{c_i,c_{j}\} > \alpha$, $g_{\alpha }(c_i)-g_{\alpha }(S_i\left( N,c,E\right) )=g_{\alpha }(c_j)-g_{\alpha }(S_j\left( N,c,E\right) )$.

• Poverty-line rules Let $\Lambda =\{\{x\},[\alpha _{k} , \beta _{k}):k \in K;\, x\in \mathbb {R}_{+}{\setminus }\cup _{k\in K}[\alpha _{k} , \beta _{k})\}$, i.e., besides the intervals $[\alpha _{k} , \beta _{k})$, all points lying outside those intervals are also considered in this partition as degenerate brackets. Let $\Gamma =\{g_{k}:k\in K\}$, where, for each $k\in K$, $g_{k}:[\alpha _{k},\beta _{k})\rightarrow \mathbb {R}\cup \{-\infty \}$. Then, for each $\left( N,c,E\right) \in \mathcal {D}$, let $k_{\alpha }$ denote the smallest integer number for which $c_{k_{\alpha }}<\alpha _{1}$.

  Case 1 $E\le \left( \left( \sum _{i=1}^{k_{\alpha }}c_{i} \right) +(n-k_{\alpha })c_{k+1} \right)$. In this case, let k be the smallest non-negative integer in $\{0,\ldots ,k_{\alpha }\}$ such that $E\le \left( \left( \sum _{i=1}^{k}c_{i} \right) +(n-k)c_{k+1} \right)$. Let $\lambda =\frac{E-\left( \sum _{i=1}^{k}c_{i}\right) }{n-k}$. It follows that, for each $i\in N$ such that $c_i \le \lambda$, $S_i\left( N,c,E\right) =c_i$. Furthermore, for each $c_i > \lambda$, $S_i\left( N,c,E\right) =\lambda =L_i\left( N,c,E\right)$.

  Case 2 $E> \left( \left( \sum _{i=1}^{k_{\alpha }}c_{i} \right) +(n-k_{\alpha })c_{k+1} \right)$. In this case, several subcases can be considered. Before presenting them, we need to introduce some notation. For each $k\in K$, let $N_{k}^{c}=\{i\in N: \alpha _{k}< c_{i}\le \beta _{k}\}$ and $n_{k}^{c}$ denote its cardinality. Furthermore, let $N_{\infty }^{c}\equiv \{i\in N: \beta _{\sup K}< c_{i}\}$ and $n_{\infty }^{c}$ denote its cardinality.

  Case 2.1 $E\le \sum _{i\in \{1,\dots ,k_{\alpha }\}\cup N_{1}^{c}}c_i +(n-k_{\alpha }-n_{1}^{c})\beta _1$. In this subcase, let $\lambda =[\alpha _{1},\beta _{1})$. It follows that, for each $i\in N$ such that $c_i \le \lambda$, $S_i\left( N,c,E\right) =c_i$. Furthermore, for each pair $i,j\in N$, such that $\min \{c_i,c_{j}\} > \alpha _{1}$, $g_{1}(\min \{c_i,\beta _{1}\})-g_{1}(S_i\left( N,c,E\right) )<g_{1}(\min \{c_j,b_{1}\})-g_{1}(S_j\left( N,c,E\right) )$ implies that $S_i\left( N,c,E\right) =\alpha _{1}$. Case 2.2 $\sum _{i\in \{1,\dots ,k_{\alpha }\}\cup N_{1}^{c}}c_i +(n-k_{\alpha }-n_{1}^{c})\beta _1<E\le \sum _{i\in \{1,\dots ,k_{\alpha }\}\cup N_{1}^{c}}c_{i}+(n-k_{\alpha }-n_{1}^{c}) \alpha _{2}$. In this subcase, let $k_{1}$ be the smallest non-negative integer in $\{k_{\alpha },\dots ,n\}$ such that $\hat{E}_{1}=E-\sum _{i\in \{1,\dots ,k_{\alpha }\}\cup N_{1}^{c}}c_i +(n-k_{\alpha }-n_{1}^{c})\beta _1\le (\sum _{i=1}^{k_{1}}c_{i})+(n-k_{1})c_{k_{1}+1}$. Let $\lambda =\beta _{1}+\frac{\hat{E}_{1}-\left( \sum _{i=1}^{k_{1}}c_{i}\right) }{n-k_{1}}\in (\beta _{1},\alpha _{2})$. It follows that, for each $i\in N$ such that $c_i \le \lambda$, $S_i\left( N,c,E\right) =c_i$. Furthermore, for each $c_i > \lambda$, $S_i\left( N,c,E\right) =\lambda =L_i\left( N,c,E\right)$. Case 2.3 $\sum _{i\in \{1,\dots ,k_{\alpha }\}\cup N_{1}^{c}}c_{i}+(n-k_{\alpha }-n_{1}^{c}) \alpha _{2}<E\le \sum _{i\in \{1,\dots ,k_{\alpha }\}\cup N_{1}^{c}}c_i +(n-k_{\alpha }-n_{1}^{c})\beta _2$. In this subcase, let $\lambda =[\alpha _{2},\beta _{2})$. It follows that, for each $i\in N$ such that $c_i \le \lambda$, $S_i\left( N,c,E\right) =c_i$. Furthermore, for each pair $i,j\in N$, such that $\min \{c_i,c_{j}\} > \alpha _{2}$, $g_{2}(\min \{c_i,\beta _{2}\})-g_{2}(S_i\left( N,c,E\right) )<g_{2}(\min \{c_j,b_{2}\})-g_{2}(S_j\left( N,c,E\right) )$ implies that $S_i\left( N,c,E\right) =\alpha _{2}$.

  $\cdots$

  The remaining cases are similarly obtained.