The elasticity of taxable income of high earners: evidence from Hungary

Abstract

The paper studies how high-income taxpayers responded to the introduction of the “extraordinary tax on individuals” in Hungary in 2007. The study is based on a panel of tax returns containing information on 10 % of tax filers from 2005 and three subsequent years. We estimate the elasticity of taxable income with respect to the marginal net-of-tax rate and find that the taxable income of Hungarian high earners is moderately responsive to taxation: the estimated elasticity is about 0.24. We also find evidence for a sizeable income effect. The estimated effect is not caused by income shifting.

Appendices

Appendix A: Tables

See Tables 5, 6 and 7.

Table 5 Robustness analysis 1. Income limits of the sample

Table 6 Robustness analysis 2. Time period 2005–2007

Table 7 Regression results for groups by income source

Appendix B: The derivation of the income effect

In the main specifications we followed Bakos et al. (2008) in the operationalization of the income effect. Their solution is a slight variant of that by Gruber and Saez (2002), the difference being a minor step of approximation. In explaining the difference we somewhat extend the exposition of Bakos et al. (2008).

The starting point of the theory is an optimizing agent who has an income supply function $y=y((1-\tau ),R)$, where $\tau$ is the marginal tax rate and R is virtual income. Virtual income is defined, in a non-linear tax schedule, by the expression $y-T\left( y \right) =R+\left( {1-\tau } \right) y$, where $T(y)$ is the total tax due at income level $y$. The agent’s response to a tax change can be written as:

$$\begin{aligned} \mathrm{d}y=-\frac{\partial y}{\partial (1-\tau )}\mathrm{d}\tau +\frac{\partial y}{\partial R}\partial R. \end{aligned}$$

Introducing the uncompensated tax price elasticity $\beta ^{u}=[(1-\tau )/y][\partial y/\partial (1-\tau )]$, the income effect $\phi =\left( {1-\tau } \right) \partial y/\partial R$ and the compensated tax price elasticity $\beta =\beta ^{u}-\phi$ we obtain:

$$\begin{aligned} \frac{\mathrm{d}y}{y}=-\beta \frac{\mathrm{d}\tau }{1-\tau }+\phi \frac{\mathrm{d}R-y\mathrm{d}\tau }{y\left( {1-\tau } \right) }. \end{aligned}$$

Most studies estimate this equation in a log-log specification, replacing $\mathrm{d}y/y$ by $\hbox {log}(y_2 /y_1)$ and $(-\mathrm{d}\tau /(1-\tau ))$ by $\log [\left( {1-\tau _2} \right) /(1-\tau _1)]$. Before Gruber and Saez (2002) the income effect was mostly assumed to be zero. They, in contrast, did include the income effect in the estimation by approximating the last term $(\mathrm{d}R-y\mathrm{d}\tau )/\left( y\left( {1-\tau } \right) \right)$ with $\hbox {log}[\left( y_2 -T_2 \left( {y_2} \right) \right) /(y_1 -T_1 \left( {y_1} \right) )]$. As they note in a footnote on page 10 they use the approximation $y\left( {1-\tau } \right) \approx y-T(y)$ to obtain this form. We can thus reconstruct their derivation as follows:

$$\begin{aligned} \frac{\mathrm{d}R-y\mathrm{d}\tau }{y\left( {1-\tau } \right) }\approx \frac{\mathrm{d}[y-T\left( y \right) ]}{y\left( {1-\tau } \right) }\approx \frac{\mathrm{d}[y-T\left( y \right) ]}{y-T(y)}\approx \log \left[ {\frac{y_2 -T_2 \left( {y_2} \right) }{y_1 -T_1 \left( {y_1} \right) }} \right] . \end{aligned}$$

The first equation uses the fact that $\mathrm{d}R-y\mathrm{d}\tau$ is equal to the change in tax liability with changing tax rates and a constant income $y$; the second step makes the approximation in the denominator described above; while the last step is just a logarithmic approximation. Thus the income effect, equal in its original form to the change in after-tax income divided by after-tax income minus virtual income, is approximated by the percentage change in after-tax income.

Bakos et al. (2008) use the same approximation $y\left( {1-\tau } \right) \approx y-T(y)$ to justify a different estimated equation. They reach the following income effect:

$$\begin{aligned} \frac{\mathrm{d}R-y\mathrm{d}\tau }{y\left( {1-\tau } \right) }\approx \log \left[ {\frac{\left( y_2 -T_2 \left( {y_2} \right) \right) /y_2}{\left( y_1 -T_1 \left( {y_1 } \right) \right) /y_1}} \right] =\mathrm{d}\hbox {log}\left( {1-\hbox {AETR}} \right) . \end{aligned}$$

We can derive the approximation of Bakos et al. (2008) “backwards,” i.e., starting from the resulting form and reaching the original expression, as follows:

$$\begin{aligned} \mathrm{d}\log \left( {\frac{y-T(y)}{y}} \right)&= \mathrm{d}\log \left( {\frac{R+y-y\tau }{y}} \right) =\frac{\mathrm{d}R+\mathrm{d}y-\mathrm{d}y\tau -y\mathrm{d}\tau }{R+y-y\tau }-\frac{\mathrm{d}y}{y}\\&\approx \frac{\mathrm{d}R+\mathrm{d}y-\tau \mathrm{d}y-y\mathrm{d}\tau }{y(1-\tau )}-\frac{\mathrm{d}y}{y}=\frac{\mathrm{d}R-y\mathrm{d}\tau }{y\left( {1-\tau } \right) }. \end{aligned}$$

Here the first equality follows from the definition of virtual income $R$; the second equality follows from total differentiation; the third step uses, in the denominator, the same approximation that Gruber and Saez (2002) also use $(R+y-y\tau =y-T(y)\approx y(1-\tau ))$; while the last step is just a subtraction.

The difference between both approximations is slight. Total differentiation in the derivation of Bakos et al. means that in that step they allow $y$ to change as well as the tax rates. In contrast, the first step in the derivation of Gruber and Saez (2002), as reconstructed here, holds exactly only if income is constant; it is an approximation if income changes.

Clearly, both approximations are legitimate. In this paper we choose the approximation of the income effect as derived by Bakos et al. for two reasons. First, while this form is reached using a crucial approximating step Gruber and Saez (2002) also use, it appears to us that altogether it is reached after fewer approximating steps. Second, we find it esthetically appealing to measure both the substitution and the income effect by the change of an easily interpretable tax rate, i.e., by the change of the marginal and average net-of-tax rate, respectively.