Tax rates as strategic substitutes

Abstract

This paper analytically derives conditions under which the slope of the tax-reaction function is negative in a classical tax competition model. If countries maximize welfare, a negative slope (reflecting strategic substitutability) occurs under relatively mild conditions. The strategic tax response is crucial for understanding tax competition games, as well as the welfare effects of partial tax unions (whereby a subset of countries coordinate their tax rates). Indeed, contrary to earlier findings that have assumed strategic complementarity in tax rates, we show that partial tax unions might reduce welfare under strategic substitutability.

Appendix: Deriving tax coefficients

This appendix derives the tax coefficients $\eta _{k,i}$ and $\eta _{r,i}$ defined in Eqs. (3) and (4) in Sect. 2.1.

1.1 Decentralization

Under decentralization, we first differentiate Eq. (1) to show how a change in the tax rate in country $i$ affects the capital stock in countries $i$ and $j$

$$\begin{aligned} \frac{\partial k_{i}}{\partial t_{i}}&=\frac{1}{f_{i}^{\prime \prime }(.)}\left[ 1+\frac{\partial \rho }{\partial t_{i}}\right] ,\end{aligned}$$

(17)

$$\begin{aligned} \frac{\partial k_{j}}{\partial t_{i}}&=\frac{1}{f_{j}^{\prime \prime }(.)}\left[ \frac{\partial \rho }{\partial t_{i}}\right] \quad \hbox { for } \quad j \ne i . \end{aligned}$$

(18)

From Eq. (2) it follows that the total size of the capital stock is fixed, therefore

$$\begin{aligned} s_{i}\frac{\partial k_{i}}{\partial t_{i}}=-\sum _{j = 1, j \ne i}^n s_{j}\frac{\partial k_{j}}{\partial t_{i}}. \end{aligned}$$

(19)

Combining Eqs. (17)–(19) we obtain

$$\begin{aligned} 0 > \frac{\partial \rho }{\partial t_{i}}=\frac{-s_{i}/f_{i}^{\prime \prime }(.)}{\sum _{j = 1}^n s_{j}/f_{j}^{\prime \prime }(.)} \equiv -\eta _{r,i}^{D} > -1, \end{aligned}$$

(20)

filling this in Eq. (17) gives

$$\begin{aligned} \frac{\partial k_{i}}{\partial t_{i}}=\frac{1}{f_{i}^{\prime \prime }(.)}\left\{ \frac{\sum _{j = 1, j \ne i}^n \left[ s_{j}/f_{j}^{\prime \prime }(.)\right] }{\sum _{j = 1}^n \left[ s_{j}/f_{j}^{\prime \prime }(.)\right] } \right\} \equiv -\eta _{k,i}^{D}<0. \end{aligned}$$

(21)