Rethinking the Pfähler–Lambert decomposition to analyse real-world personal income taxes

Abstract

We provide a generalization and adaptation of the decomposition methodology by Pfähler (Bull Econ Res 42:121–129, 1990) and Lambert (The distribution and redistribution of income, 1st edn, 1989, The distribution and redistribution of income, 3rd edn, 2001), designed to assess the redistributive effect of personal income taxation. In particular, we generalize the methodology to several deductions, allowances, schedules or tax credits, making it suitable for real-world complex tax structures, especially dual income taxes. Additionally, we avoid the problem of sequentiality on the measurement of partial redistributive effects and also take into account the re-ranking effects of tax treatments not related to income. Finally we illustrate the utility of the methodology by carrying out an empirical analysis for the 2007 Spanish Personal Income Tax, which meant a shift from a quasi-comprehensive to a semi-dual income tax.

Appendix

We take Eq. 2 (Lambert’s version) as a starting point:

$$\begin{aligned} {\Pi }_{\mathrm{Lambert}}^{\mathrm{RS}} =\frac{\overline{B-S} }{\overline{Y-S} }\hat{{\Pi }}_{{B,B-S}}^{\mathrm{RS}} -\frac{\bar{S}}{\overline{Y-S} }\Bigg ( {\frac{\overline{Y-A} }{\bar{B}}\hat{{\Pi }}_{{Y,Y-A}}^{\mathrm{RS}} +\frac{\overline{Y-D} }{\bar{B}}\hat{{\Pi }}_{{Y,Y-D}}^{\mathrm{RS}} }\Bigg ) \end{aligned}$$

(2)

The first step is generalizing the split between allowance and deductions to $n$ allowances and deductions (for simplicity we only use the word deductions, D):

$$\begin{aligned} {\Pi }_{\mathrm{Lambert}}^{\mathrm{RS}}&= \frac{\overline{B-S} }{\overline{Y-S} }\hat{{\Pi }}_{{B,B-S}}^{\mathrm{RS}} -\frac{\bar{S}}{\overline{Y-S} }\Bigg ( \frac{\overline{Y-D_1 } }{\bar{B}}\hat{{\Pi }}_{{Y,Y-D}_1 }^{\mathrm{RS}} +\frac{\overline{Y-D_2 } }{\bar{B}}\hat{{\Pi }}_{{Y,Y-D}_2 }^{\mathrm{RS}} \nonumber \\&+\cdots +\frac{\overline{Y-D_{n} } }{\bar{B}}\hat{{\Pi }}_{{Y,Y-D}_{n} }^{\mathrm{RS}} \Bigg ) \nonumber \\&= \frac{\overline{B-S} }{\overline{Y-S} }\hat{{\Pi }}_{{B,B-T}}^{\mathrm{RS}} -\frac{\bar{S}}{\overline{Y-S} }\mathop \sum \limits _{{i}=1}^{n} \frac{\overline{Y-D_{i} } }{\bar{B}}\hat{{\Pi }}_{{Y,Y-D}_{i} }^{\mathrm{RS}} \end{aligned}$$

(12)

Using the same procedure, we now generalize the first term to l different tax schedules, assuming that the overall tax liability is a sum of partial tax liabilities ($S=S_1 +S_2 +\cdots +S_\mathrm{l} )$:

$$\begin{aligned} {\Pi }_{\mathrm{Lambert}}^{\mathrm{RS}} =\frac{\overline{B-S} }{\overline{Y-S} }\mathop \sum \limits _{{i}=1}^{l} \frac{\overline{B-S_{i} } }{\overline{B-S} }\hat{{\Pi }}_{{B,B-S}_{i} }^{\mathrm{RS}} -\frac{\bar{S}}{\overline{Y-S} }\mathop \sum \limits _{{i}=1}^{n} \frac{\overline{Y-D_{i} } }{\bar{B}}\hat{{\Pi }}_{{Y,Y-D}_1}^{\mathrm{RS}} \end{aligned}$$

(13)

Since this equation does not provide the effect of tax credits, we include it at the end of the equation just by adding the redistribution induced by the tax credits expressed by a partial Reynolds–Smolensky index:

$$\begin{aligned} {\Pi }_{\mathrm{Lambert} + \mathrm{tax~credits}}^{\mathrm{RS}}&= \frac{\overline{B-S} }{\overline{Y-S} }\mathop \sum \limits _{{i}=1}^{l} \frac{\overline{B-S_{i} } }{\overline{B-S} }\hat{{\Pi }}_{{B,B-S}_{i} }^{\mathrm{RS}} -\frac{\bar{S}}{\overline{Y-S} }\mathop \sum \limits _{{i}=1}^{n} \frac{\overline{Y-D_{i} } }{\bar{B}}\hat{{\Pi }}_{{Y,Y-D}_1 }^{\mathrm{RS}}\nonumber \\&+\hat{{\Pi }}_{{ Y-S,Y-S+C}}^{\mathrm{RS}} \end{aligned}$$

(14)

Now we also generalize the new term for $m$ tax credits, as we did with deductions and schedules:

$$\begin{aligned}&{\Pi }_{\mathrm{Lambert} + \mathrm{tax~ credits}}^{\mathrm{RS}}=\nonumber \\&\frac{\overline{B-S} }{\overline{Y-S} }\mathop \sum \limits _{{i}=1}^{l} \frac{\overline{B-S_{i} } }{\overline{B-S} }\hat{{\Pi }}_{{B,B-S}_{ i} }^{\mathrm{RS}} -\frac{\bar{S}}{\overline{Y-S} }\mathop \sum \limits _{{i}=1}^{n} \frac{\overline{Y-D_{i} } }{\bar{B}}\hat{{\Pi }}_{{Y,Y-D}_1 }^{{RS}}\nonumber \\&\quad +\mathop \sum \limits _{{i}=1}^{m} \frac{\overline{Y-S-C_{i} } }{\overline{Y-T} }\hat{{\Pi }}_{{Y-S,Y-S+C}_{ i} }^{\mathrm{RS}} \end{aligned}$$

(15)

Since Lambert’s formula does not include the re-ranking effect we just add it to obtain the Reynolds–Smolensky index:

$$\begin{aligned} {\Pi }^{\mathrm{RS}}&= {\Pi }_{\mathrm{Lambert} + \mathrm{tax~credits}}^{\mathrm{RS}} -R=\frac{\overline{B-S} }{\overline{Y-S} }\mathop \sum \limits _{{ i}=1}^{l} \frac{\overline{B-S_{i} } }{\overline{B-S} }\hat{{\Pi }}_{{B,B-S}_{i} }^{\mathrm{RS}}\nonumber \\&-\frac{\bar{S}}{\overline{Y-S} }\mathop \sum \limits _{{i}=1}^{n} \frac{\overline{Y-D_{i} } }{\bar{B}}\hat{{\Pi }}_{{Y,Y-D}_1 }^{\mathrm{RS}} \nonumber \\&+\mathop \sum \limits _{{i}=1}^{m} \frac{\overline{Y-S+C_{i} } }{\overline{Y-T} }\hat{{\Pi }}_{{Y-S,Y-S+C}_{ i} }^{\mathrm{RS}} -R \end{aligned}$$

(16)

which is actually Eq. 8.