Revenue and welfare effects of financial sector VAT exemption

Abstract

This paper provides an analysis of revenue and welfare effects associated with a VAT exemption of financial services, which is common among OECD countries. We follow a general equilibrium approach that considers effects of repealing the VAT exemption not only on consumer demand and intermediate-input demand for financial services, but takes account also of the VAT distortion of labor supply. We derive formal expressions for revenue and welfare effects, which can be quantified with a minimum of information about behavioral effects. Using VAT statistics as well as national accounts, we provide quantitative estimates of the effects of repealing the VAT exemption in Germany. Our baseline estimate indicates that tax revenues would increase by some €1.7 billion or 1.3 % of VAT revenues (excluding import turnover tax). Provided these revenue gains are used to finance a reduction in the VAT rate or in other distortive labor taxes our results indicate a modest welfare gain of about €1 billion, or 0.04 % of GDP.

Appendix

1.1 A.1 Producer price effects of repealing the VAT exemption

With perfect competition, the producer price equals unit cost and obeys

$$\begin{aligned} p_j = \sum_{i=1}^n a_{ij}\bigl(1+ (1-\alpha ) (1-I_j )\tau_i \bigr)p_i+b_j, \end{aligned}$$

where I _j =1 if the sector j is subject to tax or zero-rated, and I _j =0 if the sector is exempt. Therefore, the producer price of a sector depends on the input prices p _i and input coefficients a _ij as well as on the tax rates τ _i and the input tax refund. b _j is the per-unit labor input in sector j, as above the wage rate is set to unity.

For sectors other than n, deductibility ensures that producer prices can be determined without taking account of tax effects. For these sectors, we have

$$p_j = \sum_{i=1}^n a_{ij} p_i + b_j \quad \forall j \neq n. $$

For sector n, before exemption is repealed

$$\begin{aligned} p_n = \sum_{i=1}^{n-1} a_{in} p_i \bigl(1+ (1-\alpha )\tau_i\bigr) + {a}_{nn} p_n + {b}_n. \end{aligned}$$

(15)

This would suggest that taxes on inputs paid by sector n matter for the output price of this sector to the extent that the components are necessary as inputs. Yet, depending on the substitution elasticity, input price effects might be compensated by changes of intermediate input demand. This requires us to allow for changes in the intermediate input coefficients. However, according to the Envelope theorem, the changes in the input quantities sum up to zero, i.e., the changes in the technical input coefficients for intermediate inputs and for labor can be disregarded for small price changes. Therefore, we can use the case with fixed input coefficients as a first approximation. After repealing exemption, the price in sector n is

$$p_n' = \sum_{i=1}^{n} a_{in} p_i' + {b}_n. $$

Using (15), we subtract the price under exemption

$$\begin{aligned} p_n'-p_n = \sum _{i=1}^{n} a_{in} \bigl(p_i' -p_i \bigr) - (1-\alpha ) \sum_{i=1}^{n-1} a_{in} p_i \tau_i . \end{aligned}$$

For sectors j≠n, we have

$$\begin{aligned} p_j'-p_j = \sum _{i=1}^{n} a_{ij} \bigl(p_i' -p_i \bigr). \end{aligned}$$

Translating into value based input coefficients

$$\begin{aligned} \widehat{p}_1 =& \sum_{i=1}^n \widetilde{a}_{i1}\widehat{p}_i, \\ \vdots& \\ \widehat{p}_{n-1} =& \sum_{i=1}^n \widetilde{a}_{in-1}\widehat{p}_i, \\ \widehat{p}_n =& \sum_{i=1}^{n} \widetilde{a}_{in}\widehat{p}_i - { (1-\alpha )}\sum _{i=1}^{n-1}\widetilde{a}_{in} \tau_i, \end{aligned}$$

with $\widetilde{a}_{ij}=\frac{a_{ij} p_{i}}{p_{j}}$.

With the transpose of the input-coefficient-matrix (excluding row n and column n)

$$\begin{aligned} \widetilde{\mathsf{A}}^T_{n-1\times n-1}= \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \widetilde{a}_{11} & \widetilde{a}_{21} &\cdots & \widetilde{a}_{n-11} \\ \widetilde{a}_{12} & & & \\ \vdots &\ddots & &\vdots \\ \widetilde{a}_{1n-1} &\widetilde{a}_{2n-1} &\cdots &\widetilde{a}_{n-1n-1} \end{array} \right ] \end{aligned}$$

we can solve for the vector of the relative price changes. Rewriting the system of equations of relative price changes for n sectors in vector notation:

$$\begin{aligned} \bigl[ \mathsf{I}_{n-1\times n-1}-\widetilde{\mathsf{A}}^T_{n-1\times n-1} \bigr] \left [ \begin{array}{c} \widehat{p}_1\\ \vdots\\ \widehat{p}_{n-1} \end{array} \right ] - \left [ \begin{array}{c} \widetilde{a}_{n1}\\ \vdots\\ \widetilde{a}_{nn-1} \end{array} \right ] \widehat{p}_n =& 0, \\ \left [ \begin{array}{c} \widehat{p}_1\\ \vdots\\ \widehat{p}_{n} \end{array} \right ] =& - { (1-\alpha )} \sum_{i=1}^{n-1} \widetilde{a}_{in} \tau_i, \end{aligned}$$

where I _i×j is a i times j identity matrix.

$$\begin{aligned} &\Leftrightarrow\quad \underbrace{\left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \mathsf{I}_{n-1\times n-1}-\widetilde{\mathsf{A}}^T_{n-1\times n-1}& & & -\widetilde{a}_{n1} \\ & & & \vdots \\ & & & -\widetilde{a}_{nn-1} \\ -\widetilde{a}_{1n} &\cdots& -\widetilde{a}_{n-1n} & 1-\widetilde{a}_{nn} \\ \end{array} \right ]}_{\widetilde{\mathsf{A}}^T_{n\times n}} \left [ \begin{array}{c} \widehat{p}_1\\ \vdots\\ \widehat{p}_n \end{array} \right ]\\ &\hphantom{\Leftrightarrow\quad}\quad = \left [ \begin{array}{c} 0\\ \vdots\\ 0\\ - { (1-\alpha )} \sum_{i=1}^{n-1}\widetilde{a}_{in}\tau_i \end{array} \right ], \end{aligned}$$

where the first set of equations above represent rows 1 to n−1 of the above system—the last equation represents row n. Solving for the vector of relative price changes:

$$\begin{aligned} \Leftrightarrow \quad \left [ \begin{array}{c} \widehat{p}_1\\ \vdots\\ \widehat{p}_n \end{array} \right ] = \bigl(\mathsf{I}_{n\times n}-\widetilde{\mathsf{A}}_{n\times n}^T \bigr)^{-1} \left [ \begin{array}{c} 0\\ \vdots\\ 0\\ - { (1-\alpha )} \sum_{i=1}^{n-1}\widetilde{a}_{in}\tau_i \end{array} \right ], \end{aligned}$$

where the entry in the vector on the right-hand side captures hidden input taxes.

1.2 A.2 Determination of total output change in the financial industry

Using value based input coefficients

$$\begin{aligned}{} [ \mathsf{I}-\widetilde{\mathsf{A}} ] \left [ \begin{array}{c} d (p_1 X_1 )\\ \vdots\\ d (p_n X_{n} ) \end{array} \right ] = \left [ \begin{array}{c} d (p_1 x_1 ) \\ \vdots\\ d (p_n x_{n} ) \end{array} \right ]. \end{aligned}$$

If we follow the above assumption that substantial price effects are only obtained with regard to sector n, we have

$$\begin{aligned} \left [ \begin{array}{c} d (p_1 X_1 )\\ \vdots\\ d (p_n X_{n} ) \end{array} \right ] = [ \mathsf{I}-\widetilde{\mathsf{A}} ]^{-1} \left [ \begin{array}{c} p_1 \frac{\partial h_1}{\partial q_n} d q_n \\ \vdots\\ h_n d p_n + p_n \frac{\partial h_n}{\partial q_n} d q_n \end{array} \right ]. \end{aligned}$$

Using Slutsky symmetry,

$$\begin{aligned} \left [ \begin{array}{c} d (p_1 X_1 )\\ \vdots\\ d (p_n X_{n} ) \end{array} \right ] = [ \mathsf{I}-\widetilde{\mathsf{A}} ]^{-1} \left [ \begin{array}{c} p_1 \frac{\partial h_n}{\partial q_1} d q_n \\ \vdots\\ h_n d p_n + p_n \frac{\partial h_n}{\partial q_n} d q_n \end{array} \right ]. \end{aligned}$$

Rearranging terms, we see that the output changes are linear functions of the price changes.

$$\begin{aligned} \left [ \begin{array}{c} d (p_1 X_1 )\\ \vdots\\ d (p_n X_{n} ) \end{array} \right ] = [ \mathsf{I}-\widetilde{\mathsf{A}} ]^{-1} \left [ \begin{array}{c} (\frac{h_n}{1+\tau_1} ) \epsilon_{n1} d q_n \\ \vdots\\ h_n d p_n + (\frac{h_n}{1+\tau_n} ) \epsilon_{nn} d q_n \end{array} \right ]. \end{aligned}$$

In the case where all taxes are equal τ _i =τ,∀i<n, except for τ _n =0, initially, and where a tax τ _n =τ is introduced, we obtain the changes in the value of gross outputs

$$\begin{aligned} \left [ \begin{array}{c} d (p_1 X_1 )\\ \vdots\\ d (p_n X_{n} ) \end{array} \right ] = p_n h_n [ \mathsf{I}-\widetilde{\mathsf{A}} ]^{-1} \left [ \begin{array}{c} (\frac{ \epsilon_{n1}}{1+\tau} ) (\widehat{p}_n + \tau ) \\ \vdots\\ \widehat{p}_n + \epsilon_{nn} (\widehat{p}_n + \tau ) \end{array} \right ]. \end{aligned}$$

The last element of this vector gives the output change in the exempted sector evaluated at pretax prices.

1.3 A.3 Table of sector-specific VAT rates

See Table 2.

Table 2 Average specific VAT rates in Germany for 2007