Distributional and welfare effects of Germany’s year 2000 tax reform: the context of savings and portfolio choice

Abstract

This paper empirically investigates distributional and welfare effects of Germany’s year 2000 tax reform in the context of the household savings decision and the allocation of wealth to a portfolio of assets. The reform is simulated in an ex ante behavioural microsimulation approach. Behavioural responses resulting from changes in capital income taxation are estimated in an empirical demand model for household savings and asset allocation, which is applied to German survey data. Significant reductions in tax rates result in income gains for most of the households. Gains are found to be greater for households in higher tax brackets, whereby income inequality increases, in particular in East Germany. Moreover, households increase savings and alter the structure of asset demand as a result of shifts in relative asset prices. As a result, utility losses reduce welfare effects for almost all households, in particular for households with greater savings.

Appendices

Appendix 1: Reform

See Fig. 2.

Fig. 2

Personal income tax rates and allowances over the time frame of the reform. Notes Tax rates exclude solidarity surcharge of 5.5 %. Source: Own illustration. Figures from Bundesministerium der Finanzen (2004)

Appendix 2: Methodology

1.1 Budget and price elasticities

In the linearized QUAIDS, the budget elasticity for asset j in levels follows from Eq. (2):

$$\begin{aligned} \eta _{ij} \equiv \frac{\partial A_{ij}}{\partial A_{il}} \frac{A_{il}}{A_{ij}} = 1 +\left( \beta _{1j} + 2\beta _{2j}\ln \left( A_{il}/P^{*}\right) \right) /s_{ij} \end{aligned}$$

(3)

The uncompensated price elasticity for the levels of asset j, w.r.t. price of good k, is:

$$\begin{aligned} \varepsilon ^{\,u}_{ijk} \equiv \frac{\partial A_{ij}}{\partial p_{jk}} \frac{p_{jk}}{A_{ij}} = - \delta _{jk} + \gamma _{jk} / s_{ij} - \left( \beta _{1j} + 2\beta _{2j}\ln \left( A_{il}/P^{*}\right) \right) \, \bar{s}_{k} / s_{ij} \end{aligned}$$

(4)

where $\bar{s}_{k}$ is the average share of asset k and $\delta _{jk}$ is the Kronecker delta, i.e. $\delta _{jk}=1$ if $j=k$ and $\delta _{jk}=0$ if $j \ne k$. By the Slutsky equation, the compensated price elasticity follows as:

$$\begin{aligned} \varepsilon ^{\,c}_{ijk} \equiv \varepsilon ^{\,u}_{ijk} + s_{ik} \eta _{ij} = - \delta _{jk} + \gamma _{jk}/s_{ij} + s_{ik} + \left( \beta _{1j} + 2\beta _{2j}\ln \left( A_{il}/P^{*}\right) \right) \, \left( s_{ik} - \bar{s}_{k}\right) /s_{ij} \end{aligned}$$

(5)

For the sake of interpretation, compensated (after-tax) rate-of-return elasticities, rather than price elasticities, will be presented. They follow from the price elasticities as:

$$\begin{aligned} \varepsilon ^{\,(r)\,c}_{ijk} =-\varepsilon ^{\,c}_{ijk} \, \frac{\widetilde{r}^\mathrm{net}_{j}}{1+\widetilde{r}^\mathrm{net}_{j}} \end{aligned}$$

(6)

where $\widetilde{r}^{\mathrm{net}}_{j} = r^{\mathrm{gro}}_{j}(1-t_{ij})-\pi _{i}$ is the after-tax real rate of return to asset $j, r^{\mathrm{gro}}_{j}$ the respective pre-tax return, $\pi _{i}$ is the inflation rate relevant for household i, and $t_{ij}$ the marginal tax rate on income from asset j, which is simulated in the income taxation module. The uncompensated rate-of-return elasticity follows accordingly from Eq. (4).

1.2 Indices of inequality

The general entropy index, GE($-$1), is defined as:

$$\begin{aligned} I_{\mathrm{GE}} = \frac{1}{2n} \sum _{i=1}^{n} \left( \frac{\bar{x}}{x_i}-1\right) \end{aligned}$$

(7)

The Gini coefficient is defined as:

$$\begin{aligned} I_{\mathrm{Gini}} = \sum _{i=1}^{n} \frac{x_i}{n\bar{x}} \frac{(2i-n-1)}{n} \end{aligned}$$

(8)

The Theil index, GE(1), is defined as:

$$\begin{aligned} I_{\mathrm{Theil}}= \frac{1}{n} \sum _{i=1}^{n} \mathrm{log}\left( \frac{x_i}{\bar{x}}\right) \, \frac{x_i}{\bar{x}} \end{aligned}$$

(9)

1.3 Approximating the compensating variation

Following the welfare-concept definition, the compensating variation is defined as (see Deaton and Muellbauer 1980, pp. 184–190):

$$\begin{aligned} \mathrm{CV}_{i} = c\left( u^{1}_{i},p^{1}_{i}\right) - c\left( u^{0}_{i},p^{1}_{i}\right) \end{aligned}$$

(10)

where $c(u^{1}_{i},p^{1}_{i})$ is the cost function for expenditures of household i to gain the post-reform utility level at post-reform prices, and $c(u^{0}_{i},p^{1}_{i})$ is the respective cost function to gain the pre-reform utility level at post-reform prices. If this difference is strictly greater than zero, the household is better off after the reform in money-metric welfare terms.

As differences in utility are not observed, the CV needs to be approximated with the help of estimates for the compensated price elasticities in Eq. (5). A second-order Taylor expansion of $c(u^{0},p^{1})$ around ($u^{0},p^{0}$) yields (see Deaton and Muellbauer 1980, p.174, or an application in Banks et al. 1996):

$$\begin{aligned} c(u^{0},p^{1})\approx & {} c(u^{0},p^{0}) + \sum _{j}\frac{\partial c(u^{0},p^{0})}{\partial p^{0}_{j}}(p^{1}_{j}-p^{0}_{j})\nonumber \\&+\,\frac{1}{2} \, \sum _{j} \sum _{k}{\frac{\partial ^2 c(u^{0},p^{0})}{\partial p^{0}_{j} \partial p^{0}_{k}}(p^{1}_{j}-p^{0}_{j})(p^{1}_{k}-p^{0}_{k})} \end{aligned}$$

(11)

Applying the fact that the first derivative of the cost function equals Hicksian demand (Mas-Colell et al. 1995, pp. 67–75):

$$\begin{aligned} \frac{\partial c\left( u^{0},p^{0}\right) }{\partial p^{0}_{j}}=h_{j}(u^{0},p^{0})=q^{0}_{j} \end{aligned}$$

(12)

it follows from Eq. (11) that

$$\begin{aligned} c\left( u^{0},p^{1}\right)\approx & {} c\left( u^{0},p^{0}\right) + \sum _{j}q^{0}_{j}(p^{1}_{j}-p^{0}_{j})\nonumber \\&+\,\frac{1}{2} \, \sum _{j} \sum _{k}{\frac{\partial h_{j}\left( u^{0},p^{0}\right) }{\partial p^{0}_{k}}\left( p^{1}_{j}-p^{0}_{j}\right) \left( p^{1}_{k}-p^{0}_{k}\right) } \end{aligned}$$

(13)

where $h_{j}(u^{0},p^{0})$ denotes pre-reform Hicksian demand for asset j.

Rewriting the definition of the CV for a compound income and price change in Eq. (10) and applying the fact that $c(u^{1},p^{1})=y^{1}$, yields:

$$\begin{aligned} \mathrm{CV} = y^{1} -y^{0} -c\left( u^{0},p^{1}\right) + c\left( u^{0},p^{0}\right) \end{aligned}$$

(14)

Plugging Eq. (13) into Eq. (14) and rearranging, it follows that:

$$\begin{aligned} \widehat{\mathrm{CV}} \approx y^{1}- y^{0}- \sum _{j}{p^{0}_{j}q^{0}_{j} \left( \frac{p^{1}_{j}-p^{0}_{j}}{p^{0}_{j}}\right) \left( 1 + \frac{1}{2} \sum _{k}{\widehat{\varepsilon }^{\,c}_{jk} \, \frac{p^{1}_{k}-p^{0}_{k}}{p^{0}_{k}}} \right) } \end{aligned}$$

(15)

where $\widehat{\varepsilon }^{\,c}_{jk}$ is an estimate for the compensated price elasticity of asset j w.r.t. price of asset k. Note that these are average elasticities over all households. Their application in the welfare measure implies the assumption of equal social utility weights for all households (see Banks et al. 1996). In simulations with log-linear utility, this approximation performed accurately in case the differentials in pre- and post-reform prices are of similar size and the same sign for all assets. In case, the variation in the differentials is not too large, the approximation error appeared to be acceptable. For further simulations on the approximation error, also see Banks et al. (1996).

Equation (15) contains only variables that are observed or that have been estimated, while all utility terms have been replaced. In case demand is completely inelastic for all assets, there are no distortionary effects, i.e.  $\widehat{\varepsilon }^{\,c}_{jk} = 0 \, \forall \, j,k = 1,\ldots ,J$, and the CV reduces to the income changes added to the changes in expenditures for constant demand resulting from the price shifts: $\widetilde{\mathrm{CV}} \approx y^{1} - y^{0} - \sum \nolimits _{j}{p^{0}_{j}q^{0}_{j} \left( \frac{p^{1}_{j} - p^{0}_{j}}{p^{0}_{j}}\right) }$. This is denoted in the literature as first-order approximation to the welfare measure (Banks et al. 1996). Generally, there is a trade-off regarding accuracy between such a first-order approximation and a second-order approximation of the form in Eq. (11). The latter is, on the one hand, found to produce lower approximation error in specific empirical applications (Banks et al. 1996). On the other hand, it gives rise to potential imprecision, or even bias, from the estimation of substitution elasticities in demand, which is not needed for first-order approximations. In Sect. 6, implications for further research are directed to the investigation into which approximation is the most appropriate for the application at hand.

Appendix 3: Results

See Tables 4 and 5.

Table 4 Compensated rate-of-return elasticities on unconditional asset demand levels (from the unconstrained estimation on the pooled 1998 and 2003 data)

Table 5 OLS regression of welfare effects on savings ratio