Safe haven or earnings stripping rules: a prisoner’s dilemma?

Abstract

Multinational firms use internal debt financing to shift profits from high-tax to low-tax countries. Therefore, governments restrict the deductibility of interest expenses by applying thin-capitalization rules (TCRs). TCRs fall into two main categories: safe haven rules (SHR) and earnings stripping rules (ESR). We analyze the optimal TCR choice in a two-country tax competition model. We show that unilateral replacement of SHR by ESR imposes a negative profit shifting externality on the other country. This effect can explain the recently observed switch from SHR to ESR in many countries. However, ESR may be a dominant strategy even when SHR is socially optimal, i.e., the observed policies of ESR implementation may indicate a prisoner’s dilemma.

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Fig. 1

Notes

  1. 1.

    ESRs are sometimes applied if some debt-to-equity ratio is exceeded, i.e., together with an SHR. This is the case in the USA (Gresik et al. 2017).

  2. 2.

    For a further description and overview of thin-capitalization rules, see Ambrosanio and Caroppo (2005) and Dourado and de la Feria (2008).

  3. 3.

    While the results of Hong and Smart (2010) point to a positive impact of tax havens on high-tax countries’ welfare, Slemrod and Wilson (2009) find that tax havens worsen the welfare of non-haven jurisdictions. In the model of Slemrod and Wilson (2009), the existence of tax havens leads to a wasteful expenditure of resources by both firms (that engage in tax planning) and governments (that implement tax enforcement policies).

  4. 4.

    Furthermore, Mardan (2017) shows that countries with low financial development are more likely to allow positive internal interest deductions.

  5. 5.

    Internal debt financing is usually restricted by the agency and bankruptcy costs of debt. Due to these costs, investments are at least partly equity financed even in the absence of TCRs. We show in an extension in Sect. 7 that the inclusion of these costs does not affect the results.

  6. 6.

    In Eq. (1), we assume that the interest rate on internal debt is the true interest rate r. However, MNEs may also distort this interest rate to additionally shift profits to the tax haven through transfer price manipulation (see, e.g., Mardan 2017; Gresik et al. 2017). We consider interest rate manipulation in an extension in Sect. 7.

  7. 7.

    We consider the implications of an endogenous tax rate in an extension in Sect. 7.

  8. 8.

    In the following analysis, we drop the superscript when no ambiguity arises.

  9. 9.

    Note that the deviation considered above is such that \(k=k^*\) only to ensure that the foreign government’s first-order condition remains unaffected. We do not claim that \(k=k^*\) must hold in the ESSH equilibrium. It is, however, sufficient to show that there exists a deviation from the symmetric SH equilibrium, which makes the deviating country better-off, without triggering a response from the other country, to prove that keeping SHR cannot be a best response.

  10. 10.

    As in the benchmark model, we assume that the concealment costs are not tax deductible.

  11. 11.

    I would like to thank one of the referees for pointing out the problem of uneven tax revenue distribution.

  12. 12.

    When there is no risk of confusion, we drop the superscript \(^{SH}\) for clarity.

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Appendices

Appendix A: Derivation of \({\tilde{z}}^{SH}\) and \({\tilde{z}}^{ES}\)

To derive \({\tilde{z}}^{SH},\) note that in a symmetric equilibrium with \(t=t^*,\) we have \(k^{SH}=k^{*SH}={\overline{k}}, {\tilde{z}}^{SH}={\tilde{z}}^{*SH}\). Therefore, we can simplify Eqs. (11a) and (11b):

$$\begin{aligned} \frac{{\text{ d }}k}{{\text{ d }}z}\Bigg |_{z=z^*}= & {} \frac{r}{2f^{\prime \prime }({\overline{k}})}<0, \end{aligned}$$
(A.1)
$$\begin{aligned} \frac{\hbox {d}r}{{\text{ d }}z}\Bigg |_{z=z^*}= & {} \frac{-r}{2z}<0. \end{aligned}$$
(A.2)

Thus, in a symmetric equilibrium with interior solution Eq. (14) simplifies to

$$\begin{aligned} \frac{{\text{ d }}T^{SH}}{d\delta }=\left[ r{\overline{k}}(1-t)+({\tilde{z}}^{SH}-1)r\frac{r}{2f^{\prime \prime }({\overline{k}})}+{\overline{k}}\left( {\tilde{z}}^{SH}(1-t)-1\right) \frac{-r}{2{\tilde{z}}^{SH}}\right] \frac{{\text{ d }}z}{d\delta }=0. \end{aligned}$$
(A.3)

Note that due to the homogeneity of degree \(\alpha \) of the production function, its first derivative is homogeneous of degree \(\alpha -1\). Therefore, it satisfies \((\alpha -1)f^{\prime }({\overline{k}})=f^{\prime \prime }({\overline{k}}){\overline{k}}\). We use this equation to express \(f^{\prime \prime }({\overline{k}})\) as \(f^{\prime \prime }({\overline{k}})=(\alpha -1)f^{\prime }({\overline{k}})/{\overline{k}}=(\alpha -1)r{\tilde{z}}^{SH}/{\overline{k}}\). Inserting this expression in Eq. (A.3), we get

$$\begin{aligned} \frac{{\text{ d }}T^{SH}}{d\delta }=\left[ 2(1-t)+\frac{({\tilde{z}}^{SH}-1)}{(\alpha -1){\tilde{z}}^{SH}}-(1-t)+\frac{1}{{\tilde{z}}^{SH}}\right] \frac{r{\overline{k}}}{2}\frac{{\text{ d }}z}{d\delta }=0. \end{aligned}$$
(A.4)

In equilibrium, the term in brackets must equal zero. Simplification of Eq. (A.4) gives Equation (14’) in the main text.

To derive \({\tilde{z}}^{ES},\) we use Eqs. (A.1), (A.2), and (13b). In a symmetric situation, the first-order condition (15) becomes

$$\begin{aligned} \frac{{\text{ d }}T^{ES}}{\hbox {d}\epsilon }&=\left[ r{\overline{k}}(1-t(1-{\tilde{\epsilon }}))+({\tilde{z}}^{ES}-1)r\frac{r}{2f^{\prime \prime }({\overline{k}})}+{\overline{k}}\left( {\tilde{z}}^{ES}(1-t(1-{\tilde{\epsilon }}))-1\right) \frac{-r}{2{\tilde{z}}^{ES}}\right] \frac{{\text{ d }}z}{\hbox {d}\epsilon }\nonumber \\&\quad -t\left[ f({\overline{k}})-{\tilde{z}}^{ES}r{\overline{k}}-\frac{t(1-{\tilde{\epsilon }})}{C^{\prime \prime }(0)}\right] =0. \end{aligned}$$
(A.5)

We use (12b), as well as the homogeneity of the production function to express \(f^{\prime \prime }({\overline{k}})\) as \(f^{\prime \prime }({\overline{k}})=(\alpha -1)f^{\prime }({\overline{k}})/{\overline{k}}=(\alpha -1)r{\tilde{z}}^{ES}/{\overline{k}}\). This leads to the following expression

$$\begin{aligned}&-\left[ (1-t(1-{\tilde{\epsilon }}))+\frac{({\tilde{z}}^{ES}-1)}{(\alpha -1){\tilde{z}}^{ES}}+\frac{1}{{\tilde{z}}^{ES}}\right] \frac{t{\tilde{z}}^{ES}r{\overline{k}}}{2(1-t(1-{\tilde{\epsilon }}))}\nonumber \\&\quad =t\left[ f({\overline{k}})-{\tilde{z}}^{ES}r{\overline{k}}-\frac{t(1-{\tilde{\epsilon }})}{C^{\prime \prime }(0)}\right] . \end{aligned}$$
(A.6)

We use Eq. (8a) to express \({\tilde{z}}^{ES}r{\overline{k}}\) as \(f^{\prime }({\overline{k}}){\overline{k}}\). Furthermore, using the homogeneity of the production function, we have \(f^{\prime }({\overline{k}}){\overline{k}}=\alpha f({\overline{k}})\). Rearranging terms, we get

$$\begin{aligned}&-\left[ 1+(1-t(1-{\tilde{\epsilon }}))+\frac{({\tilde{z}}^{ES}-1)(2-\alpha )}{(\alpha -1){\tilde{z}}^{ES}}\right] \nonumber \\&\quad =\frac{2(1-t(1-{\tilde{\epsilon }}))}{\alpha f({\overline{k}})}\left[ (1-\alpha )f({\overline{k}})-\frac{t(1-{\tilde{\epsilon }})}{C^{\prime \prime }(0)}\right] . \end{aligned}$$
(A.7)

Further simplification of Eq. (A.7) gives Equation (15’) in the main text.

Appendix B: Proof of Proposition 1

To compare the equilibrium tax revenues, we first compute the revenues in an SHR equilibrium \({\widetilde{T}}^{SH}\). Note that in a symmetric equilibrium, there is no profit shifting \(\sigma ^{SH}=0\), and no capital mobility \(k^{SH}=k^{*SH}={\overline{k}}\). Hence, we can write \({\widetilde{T}}^{SH}\) as

$$\begin{aligned} {\widetilde{T}}^{SH}=t\left[ f({\overline{k}})-{\tilde{z}}^{SH}r{\overline{k}}\right] +\frac{({\tilde{z}}^{SH}-1)}{{\tilde{z}}^{SH}}{\tilde{z}}^{SH}r{\overline{k}}. \end{aligned}$$
(B.1)

We use Eq. (7a) and the homogeneity of the production function to express \({\tilde{z}}^{SH}r{\overline{k}}\) as \(f^{\prime }({\overline{k}}){\overline{k}}=\alpha f({\overline{k}})\). Lastly, we use Equation (14’) to simplify (B.1):

$$\begin{aligned} {\widetilde{T}}^{SH}= & {} t(1-\alpha )f({\overline{k}})+\frac{({\tilde{z}}^{SH}-1)}{{\tilde{z}}^{SH}}\alpha f({\overline{k}})\nonumber \\= & {} t(1-\alpha )f({\overline{k}})+\frac{(1-\alpha )(2-t)}{(2-\alpha )}\alpha f({\overline{k}})\nonumber \\= & {} \frac{2(1-\alpha )}{(2-\alpha )}f({\overline{k}})\left[ t+\alpha (1-t)\right] . \end{aligned}$$
(B.2)

Analogously, we can use \(\sigma ^{ES}=0\) and \(k^{ES}=k^{*ES}={\overline{k}}\) to write the equilibrium tax revenues under ESR as

$$\begin{aligned} {\widetilde{T}}^{ES}=t(1-{\tilde{\epsilon }})\left[ f({\overline{k}})-{\tilde{z}}^{ES}r{\overline{k}}\right] +\frac{({\tilde{z}}^{ES}-1)}{{\tilde{z}}^{ES}}{\tilde{z}}^{ES}r{\overline{k}}. \end{aligned}$$
(B.3)

Following the same steps as in the previous case and using Equation (15’), we can rewrite the equilibrium tax revenues \({\widetilde{T}}^{ES}\) as

$$\begin{aligned} {\widetilde{T}}^{ES}=\frac{2(1-\alpha )}{(2-\alpha )}f({\overline{k}})\left[ t(1-{\tilde{\epsilon }})+\alpha (1-t(1-{\tilde{\epsilon }}))\left( 1+\frac{1-\alpha }{\alpha }-\frac{t(1-{\tilde{\epsilon }})}{\alpha f({\overline{k}})C^{\prime \prime }(0)}\right) \right] .\!\!\!\!\nonumber \\ \end{aligned}$$
(B.4)

Using Eqs. (B.2) and (B.4), one can directly compare the equilibrium tax revenues. After some calculations, we can show that

$$\begin{aligned} {\widetilde{T}}^{ES}\gtreqless {\widetilde{T}}^{SH} \Leftrightarrow \frac{1}{C^{\prime \prime }(0)}\lesseqgtr \frac{(1-\alpha )f({\overline{k}})}{t(1-{\tilde{\epsilon }})}\frac{1-t}{1-t(1-{\tilde{\epsilon }})}. \end{aligned}$$
(B.5)

Denote the critical level of profit shifting in Eq. (B.5) as \(\overline{C^{\prime \prime }}\). Then, we have

$$\begin{aligned} {\widetilde{T}}^{ES}\gtreqless {\widetilde{T}}^{SH} \Leftrightarrow C^{\prime \prime }(0)\gtreqless \overline{C^{\prime \prime }}, \end{aligned}$$
(B.6)

which is the condition from Proposition 1. To show that \(\overline{C^{\prime \prime }}>\widehat{C^{\prime \prime }}\), note that at \(\widehat{C^{\prime \prime }}\), the equilibrium EMTRs are equal and \({\widetilde{T}}^{ES}<{\widetilde{T}}^{SH}\). Since \({\widetilde{T}}^{SH}\) is independent of \(C^{\prime \prime }(0),\) while, according to Eq. (B.4), \({\widetilde{T}}^{ES}\) is increasing in \(C^{\prime \prime }(0),\) it must hold true that \({\widetilde{T}}^{ES}={\widetilde{T}}^{SH}\) at \(\overline{C^{\prime \prime }}>\widehat{C^{\prime \prime }}\). \(\square \)

Appendix C: Proof of Proposition 3

Suppose that both governments apply SHR. Then, the MNE’s optimal investment and transfer pricing are described by Eqs. (23a)-(23d). First, we derive the comparative statics effects of a change in the SHR rule \(\delta \). We totally differentiate Eqs. (23a), (23b), (23d) and the capital market clearing condition \(k+k^*=2{\overline{k}}\) with respect to \(k,k^*,r,i,i^*\) and \(\delta \):Footnote 12

$$\begin{aligned} f^{\prime \prime }(k){\text{ d }}k&=q_rdr+q_idi+q_{\delta }d\delta , \end{aligned}$$
(C.1a)
$$\begin{aligned} f^{\prime \prime }(k^*){\text{ d }}k^*&=q^*_rdr+q^*_{i^*}di^*, \end{aligned}$$
(C.1b)
$$\begin{aligned} {\text{ d }}k+{\text{ d }}k^*&=0, \end{aligned}$$
(C.1c)
$$\begin{aligned} C_i^{\prime \prime }(di-dr)&=0, \end{aligned}$$
(C.1d)
$$\begin{aligned} C_i^{*\prime \prime }(di^*-dr)&=0. \end{aligned}$$
(C.1e)

Note that, according to (C.1d) and (C.1e), \(di=di^*=dr\). Thus, we can solve for \({\text{ d }}k\) and dr, which are given by

$$\begin{aligned} \frac{{\text{ d }}k}{d\delta }&=-\frac{{\text{ d }}k^{*}}{d\delta }=\frac{(q^*_{r}+q^*_{i^*})q_{\delta }}{(q_{r}+q_i)f^{*\prime \prime }+(q^*_{r}+q^*_{i^*})f^{\prime \prime }}, \end{aligned}$$
(C.2a)
$$\begin{aligned} \frac{\hbox {d}r}{d\delta }&=\frac{-q_{\delta }f^{*\prime \prime }}{(q_{r}+q_i)f^{*\prime \prime }+(q^*_{r}+q^*_{i^*})f^{\prime \prime }}. \end{aligned}$$
(C.2b)

Next, we derive \(q_r, q_i\) and \(q_{\delta }\). Using the definition of q from Eq. (23a), we get

$$\begin{aligned} q_r+q_i&=\frac{1-\delta C_i^{\prime }}{1-t}-\frac{\delta (t-C_i^{\prime })}{1-t}=\frac{1-\delta t}{1-t}\equiv z^{SH}, \end{aligned}$$
(C.3a)
$$\begin{aligned} q_{\delta }&=-\frac{ti-C_i}{1-t}<0. \end{aligned}$$
(C.3b)

We use (C.3a) to simplify the comparative statics. Dropping the superscript \(^{SH}\), we get

$$\begin{aligned} \frac{{\text{ d }}k}{d\delta }&=-\frac{{\text{ d }}k^{*}}{d\delta }=\frac{z^{*}q_{\delta }}{zf^{*\prime \prime }+z^*f^{\prime \prime }}>0, \end{aligned}$$
(C.4a)
$$\begin{aligned} \frac{\hbox {d}r}{d\delta }&=\frac{-q_{\delta }f^{*\prime \prime }}{zf^{*\prime \prime }+z^*f^{\prime \prime }}>0. \end{aligned}$$
(C.4b)

Consider now the symmetric SHR situation, in which the governments maximize the tax revenues (4), where the interest costs rb are replaced by ib. We again rearrange the tax revenues to an expression similar to Eq. (9a). After some manipulation, we can show that \(T^{SH}\) is given by

$$\begin{aligned} T^{SH}=t\left[ f(k)-q^{SH}k-\sigma \right] +(q^{SH}-r-\delta C_i(i-r))k. \end{aligned}$$
(C.5)

Note that the EMTR is now given by \((q^{SH}-r-\delta C_i(i-r))/r\) and is decreasing in the degree of interest rate manipulation. The home government’s first-order condition with respect to \(\delta \) is given by

$$\begin{aligned}&\frac{{\text{ d }}T^{SH}}{d\delta }=k(1-t)q_{\delta }+(q-r-\delta C_i)\frac{{\text{ d }}k}{d\delta }+k\left( (q_r+q_i)(1-t)-1\right) \frac{\hbox {d}r}{d\delta }\nonumber \\&\quad -C_ik\le 0,\quad \delta \ge 0,\quad \frac{{\text{ d }}T^{SH}}{d\delta }\delta =0. \end{aligned}$$
(C.6)

Evaluating the above first-order condition in a symmetric equilibrium with \(t=t^*, k=k^*={\overline{k}}, \delta =\delta ^*\) and using (C.4a), (C.4b), we get

$$\begin{aligned} \left[ (q-r-\delta C_i)\frac{1}{2f^{\prime \prime }({\overline{k}})}+\frac{{\overline{k}}(1-t)}{2}+\frac{{\overline{k}}}{2z}\right] q_{\delta }-C_i{\overline{k}}=0. \end{aligned}$$
(C.7)

Note that due to the homogeneity of the production function, we can express \(f^{\prime \prime }({\overline{k}})\) as \((\alpha -1)f^{\prime }({\overline{k}})/{\overline{k}}=(\alpha -1)q/{\overline{k}}\). Thus, (C.7) simplifies to

$$\begin{aligned} \left[ \frac{(q-r-\delta C_i)}{(\alpha -1)q}+(1-t)+\frac{1}{z}\right] =\frac{2C_i}{q_{\delta }}. \end{aligned}$$
(C.8)

To present the equilibrium EMTR in a way comparable to the EMTR under ESR, we add and subtract \(1+(q-r-\delta C_i)/q\) from the left-hand side of Eq. (C.8). Simplifying the resulting expression leads to

$$\begin{aligned} \frac{{\tilde{q}}^{SH}-r-{\tilde{\delta }} C_i}{{\tilde{q}}^{SH}}=\frac{1-\alpha }{2-\alpha }\left[ 2-t+\left( \frac{{\tilde{q}}^{SH}-r-{\tilde{\delta }} C_i}{{\tilde{q}}^{SH}}-\frac{z^{SH}-1}{z^{SH}}\right) -\frac{2C_i}{q_{\delta }}\right] . \end{aligned}$$
(C.9)

Note that the term in parentheses on the right-hand side in Eq. (C.9) is larger than \(-1\), while the last term in brackets is positive. Therefore, the right-hand side of (C.9) is positive irrespective of the degree of interest rate manipulation and the equilibrium EMTR is also positive. On the other hand, the equilibrium EMTR under ESR is zero for a sufficiently small value of \(C^{\prime \prime }(0)\). Therefore, there exists a positive \(\widehat{C^{\prime \prime }}\), such that \(({\tilde{q}}^{SH}-r-{\tilde{\delta }} C_i)/{\tilde{q}}^{SH}>({\tilde{z}}^{ES}-1)/{\tilde{z}}^{ES}\) for \(C^{\prime \prime }(0)<\widehat{C^{\prime \prime }}\). To determine \(\widehat{C^{\prime \prime }}\), we equate the right-hand sides of Equations (15’) and (C.9). Denoting again the equilibrium \({\tilde{\epsilon }}\) associated with \(\widehat{C^{\prime \prime }}\) as \({\widehat{\epsilon }}\), we get after some manipulation

$$\begin{aligned} (1-\alpha )f({\overline{k}})= & {} \frac{t(1-{\widehat{\epsilon }})}{\widehat{C^{\prime \prime }}}+\frac{\alpha f({\overline{k}})}{2(1-t(1-{\widehat{\epsilon }}))}\nonumber \\&\times \left[ \left( \frac{{\tilde{q}}^{SH}-r-{\tilde{\delta }} C_i}{{\tilde{q}}^{SH}}-\frac{z^{SH}-1}{z^{SH}}\right) -\frac{2C_i}{q_{\delta }}-t{\widehat{\epsilon }}\right] .\quad \end{aligned}$$
(C.10)

The equilibrium tax revenues in the symmetric SHR case are given by

$$\begin{aligned} {\widetilde{T}}^{SH}&=t\left[ f({\overline{k}})-{\tilde{q}}^{SH}{\overline{k}}\right] +\frac{({\tilde{q}}^{SH}-r-{\tilde{\delta }} C_i)}{{\tilde{q}}^{SH}}{\tilde{q}}^{SH}{\overline{k}}\nonumber \\&=t(1-\alpha )f({\overline{k}})+\frac{({\tilde{q}}^{SH}-r-{\tilde{\delta }} C_i)}{{\tilde{q}}^{SH}}\alpha f({\overline{k}}). \end{aligned}$$
(C.11)

The equilibrium tax revenues \({\widetilde{T}}^{ES}\) are determined by Eq. (B.3). A comparison of (C.11) and (B.3) shows that \(({\tilde{q}}^{SH}-r-{\tilde{\delta }} C_i)/{\tilde{q}}^{SH}>({\tilde{z}}^{ES}-1)/{\tilde{z}}^{ES}\) is sufficient for \({\widetilde{T}}^{SH}>{\widetilde{T}}^{ES}\). Thus, the symmetric SHR case leads to higher equilibrium tax revenues if \(C^{\prime \prime }(0)<\widehat{C^{\prime \prime }}\).

Consider now the incentives of the local governments to deviate from an SHR equilibrium in the case \(C^{\prime \prime }(0)<\widehat{C^{\prime \prime }}\). Suppose the home government deviates to ESR and sets the earnings stripping rule at \(\epsilon ^D,\) where \(\epsilon ^D\) is such that the user cost of capital remains unchanged, i.e., \(z^{ES}(\epsilon ^D)r={\tilde{q}}^{SH}\). This deviation does not cause the MNE to change its capital demand in either country. Thus, the interest rate r also remains unaffected. Furthermore, the foreign government’s first-order condition is also unchanged, and thus, it does not react to the policy change by the home government. Using the definitions of \(z^{ES}\) and \(q^{SH},\) we can solve for \(\epsilon ^D\):

$$\begin{aligned} \epsilon ^D=\frac{{\tilde{\delta }}(ti-C_i)}{t{\tilde{q}}^{SH}}. \end{aligned}$$
(C.12)

Note that \(\epsilon ^D\) equates \(z^{ES}r\) to \({\tilde{q}}^{SH}\) and is therefore more restrictive than \({\widehat{\epsilon }},\) because by definition \((z^{ES}({\widehat{\epsilon }})-1)/z^{ES}({\widehat{\epsilon }})=({\tilde{q}}^{SH}-r-{\tilde{\delta }} C_i)/{\tilde{q}}^{SH},\) i.e., \(z^{ES}({\widehat{\epsilon }})r={\tilde{q}}^{SH}-z^{ES}({\widehat{\epsilon }}){\tilde{\delta }}C_i<{\tilde{q}}^{SH}\). Thus, \(\epsilon ^D<{\widehat{\epsilon }}\).

Denote the home tax revenues after deviation by \(T^D\). They are given by

$$\begin{aligned} T^D&=t(1-\epsilon ^D)\left[ f({\overline{k}})-{\tilde{q}}^{SH}{\overline{k}}-\sigma ^{ES,SH}\right] +\frac{({\tilde{q}}^{SH}-r)}{{\tilde{q}}^{SH}}{\tilde{q}}^{SH}{\overline{k}}\nonumber \\&={\widetilde{T}}^{SH}-t\epsilon ^D(1-\alpha )f({\overline{k}})-t(1-\epsilon ^D)\sigma ^{ES,SH}+{\tilde{\delta }}C_i{\overline{k}}. \end{aligned}$$
(C.13)

We use Eq. (17) to express \(\sigma ^{ES,SH}\) as \(-t\epsilon ^D/C^{\prime \prime }(0)\). Additionally, we substitute \((1-\alpha )f({\overline{k}})\) using (C.10). Thus, we can rearrange \(T^D\) to get

$$\begin{aligned} T^D=&{\widetilde{T}}^{SH}+t^2\epsilon ^D\left[ \frac{1-\epsilon ^D}{C^{\prime \prime }(0)}-\frac{1-{\widehat{\epsilon }}}{\widehat{C^{\prime \prime }}}+\frac{\alpha f({\overline{k}}){\widehat{\epsilon }}}{2(1-t(1-{\widehat{\epsilon }}))}\right] \nonumber \\&+\left\{ {\tilde{\delta }}C_i{\overline{k}}-\frac{t\epsilon ^D\alpha f({\overline{k}})}{2(1-t(1-{\widehat{\epsilon }}))}\left[ \left( \frac{{\tilde{q}}^{SH}-r-{\tilde{\delta }} C_i}{{\tilde{q}}^{SH}}-\frac{z^{SH}-1}{z^{SH}}\right) -\frac{2C_i}{q_{\delta }}\right] \right\} . \end{aligned}$$
(C.14)

The terms in brackets in the first row of Eq. (C.14) are positive in the case of \(C^{\prime \prime }(0)<\widehat{C^{\prime \prime }}\) due to \(\epsilon ^D<{\widehat{\epsilon }}\). In order to determine the sign of the terms in the second row, we first use the definitions of \(q^{SH}, q_{\delta }\) and \(z^{SH}\) from Eqs. (23a), (C.3a) and (C.3b). Thus, we can simplify the term in brackets in the following way:

$$\begin{aligned}&\left( \frac{{\tilde{q}}^{SH}-r-{\tilde{\delta }} C_i}{{\tilde{q}}^{SH}}-\frac{z^{SH}-1}{z^{SH}}\right) -\frac{2C_i}{q_{\delta }}\nonumber \\&\quad =\left( 1-\frac{r+{\tilde{\delta }} C_i}{{\tilde{q}}^{SH}}-1+\frac{1}{z^{SH}}\right) +\frac{2C_i(1-t)}{(ti-C_i)}\nonumber \\&\quad =\frac{\left( {\tilde{q}}^{SH}-z^{SH}(r+{\tilde{\delta }} C_i)\right) }{{\tilde{q}}^{SH}z^{SH}}+\frac{2C_i(1-t)}{(ti-C_i)}\nonumber \\&\quad =\frac{r-{\tilde{\delta }}(ti-C_i)-(1-{\tilde{\delta }}t)(r+{\tilde{\delta }}C_i)}{{\tilde{q}}^{SH}z^{SH}(1-t)}+\frac{2C_i(1-t)}{(ti-C_i)}\nonumber \\&\quad ={\tilde{\delta }}t\frac{r-i+{\tilde{\delta }}C_i}{{\tilde{q}}^{SH}z^{SH}(1-t)}+\frac{2C_i(1-t)}{(ti-C_i)}. \end{aligned}$$
(C.15)

Using (C.15) and the value of \(\epsilon ^D,\) given by Eq. (C.12), we can simplify (C.14) to

$$\begin{aligned} T^D&={\widetilde{T}}^{SH}+t^2\epsilon ^D\left[ \frac{1-\epsilon ^D}{C^{\prime \prime }(0)}-\frac{1-{\widehat{\epsilon }}}{\widehat{C^{\prime \prime }}}+\frac{\alpha f({\overline{k}}){\widehat{\epsilon }}}{2(1-t(1-{\widehat{\epsilon }}))}\right] \nonumber \\&\quad +\frac{{\tilde{\delta }}{\overline{k}}}{1-t(1-{\widehat{\epsilon }})}\left[ t{\widehat{\epsilon }}C_i+\frac{(ti-C_i){\tilde{\delta }}t}{2{\tilde{q}}^{SH}z^{SH}(1-t)}\left( i-r-{\tilde{\delta }}C_i\right) \right] >{\widetilde{T}}^{SH}.\quad \end{aligned}$$
(C.16)

The first term in brackets in the second row of (C.16) is positive. The second term is also positive because interest manipulation is only done when the benefits \(t(i-r)b\) (the reduction in the tax liability) exceed the costs \(C_i(i-r)b\). Thus, \(t(i-r)-C_i>0\). It is immediate to see that \(i-r>t(i-r)>C_i>{\tilde{\delta }}C_i\).

Therefore, the home government can increase its tax revenues by deviating from SHR to ESR without affecting the other government’s first-order condition. Thus, keeping SHR cannot be a best response to the foreign government’s strategy and home has the incentive to deviate. Due to symmetry, the foreign government has the same incentive. If the home government deviates unilaterally, it will set the ESR rule \({\tilde{\epsilon }}^{ES, SH}\) at a level different from \(\epsilon ^D\) if and only if this results in higher equilibrium tax revenues than \(T^D\). Therefore, \({\widetilde{T}}^{ES,SH}\ge T^D>{\widetilde{T}}^{SH}\). Together with our previous result, this leads to \({\widetilde{T}}^{ES,SH}>{\widetilde{T}}^{SH}>{\widetilde{T}}^{ES}\) for \(C^{\prime \prime }(0)<\widehat{C^{\prime \prime }}\). \(\square \)

Appendix D: Proof of Proposition 4

To prove Proposition 4, we begin by performing the comparative statics analysis of the MNE’s profit maximization problem, analogously to the analysis of Sect. 3.4. The effects of z on the capital demands \(k,k^*\) and the interest rate r are again given by Eqs. (11a) and (11b). The partial derivates of \(z^{SH}\) and \(z^{ES}\) with respect to \(\delta \) and \(\epsilon \) are, however, different. They are given by

$$\begin{aligned} \frac{\partial z^{SH}}{\partial \delta }=-\frac{t-\tau }{1-t}<0, \end{aligned}$$
(D.1)
$$\begin{aligned} \frac{\partial z^{ES}}{\partial \epsilon }=-\frac{(t-\tau )z^{ES}}{1-t(1-\epsilon )-\tau \epsilon }. \end{aligned}$$
(D.2)

The comparative statics analysis of the optimal transfer price’s reaction to changes in \(\delta \) and \(\epsilon \) shows the following results:

$$\begin{aligned} \frac{\hbox {d}\sigma ^{SH}}{d\delta }=0, \quad \frac{\hbox {d}\sigma ^{ES}}{\hbox {d}\epsilon }=-\frac{t-\tau }{C^{\prime \prime }}<0. \end{aligned}$$
(D.3)

Consider first the two symmetric situations. In the case of SHR, each government maximizes over \(\delta \) its tax revenues, given by (26a). The first-order condition is

$$\begin{aligned}&\frac{{\text{ d }}T^{SH}}{d\delta }=\left[ rk(1-t)+(z-1)r\frac{{\text{ d }}k}{{\text{ d }}z}+k\left( z(1-t)-1\right) \frac{\hbox {d}r}{{\text{ d }}z}\right] \frac{\hbox {d} z}{d \delta }\nonumber \\&\quad -\tau rk\le 0,\quad \delta \ge 0,\quad \frac{{\text{ d }}T^{SH}}{d\delta }\delta =0. \end{aligned}$$
(D.4)

Solving (D.4) following the steps outlined in Appendix A, we get

$$\begin{aligned} \frac{{\tilde{z}}^{SH}-1-{\tilde{\delta }}\tau }{{\tilde{z}}^{SH}}=\frac{1-\alpha }{2-\alpha }\left[ 2-t+\frac{2\tau (1-t)}{(t-\tau )}\right] . \end{aligned}$$
(D.5)

It is immediate to see that for \(\tau =0\), the solution collapses to Equation (14’) of the main model. The equilibrium tax revenues can be derived as in Appendix B and equal

$$\begin{aligned} {\widetilde{T}}^{SH}=\frac{2(1-\alpha )}{(2-\alpha )}f({\overline{k}})\left[ t+\alpha \left( 1-t+\frac{\tau (1-t)}{t-\tau }\right) \right] . \end{aligned}$$
(D.6)

Turning to the case of symmetric ESR, the first-order condition of the home government is

$$\begin{aligned} \frac{{\text{ d }}T^{ES}}{\hbox {d}\epsilon }= & {} \left[ rk(1-t(1-\epsilon )-\tau \epsilon )+[(z-1)r-\tau \epsilon f^{\prime }(k)]\frac{{\text{ d }}k}{{\text{ d }}z}\right. \nonumber \\&\left. +k\left( z(1-t(1-\epsilon )-\tau \epsilon )-1\right) \frac{\hbox {d}r}{{\text{ d }}z}\right] \frac{\hbox {d} z}{d \epsilon }\nonumber \\&-(t-\tau )\left[ f(k)-zrk-\sigma \right] -\tau (f(k)-\sigma )-[t(1-\epsilon )\nonumber \\&+\,\tau \epsilon ]\frac{\hbox {d}\sigma }{\hbox {d}\epsilon }\le 0,\quad \epsilon \ge 0,\quad \frac{{\text{ d }}T^{ES}}{\hbox {d}\epsilon }\epsilon =0. \end{aligned}$$
(D.7)

The solution procedure for (D.7) in a symmetric equilibrium is again outlined in Appendix A and leads to the following expression

$$\begin{aligned} \frac{{\tilde{z}}^{ES}(1-\tau {\tilde{\epsilon }})-1}{{\tilde{z}}^{ES}}=&\frac{1-\alpha }{2-\alpha }\left[ 1-\tau {\tilde{\epsilon }}+\left( 1-t(1-{\tilde{\epsilon }})-\tau {\tilde{\epsilon }}\right) \right. \nonumber \\&\left. \left( 1+2\left( \frac{1-\alpha }{\alpha }-\frac{t(1-{\tilde{\epsilon }})+\tau {\tilde{\epsilon }}}{\alpha f({\overline{k}})C^{\prime \prime }(0)}+\frac{\tau }{\alpha (t-\tau )}\right) \right) \right] .\end{aligned}$$
(D.8)

We use Eq. (D.8) to derive the symmetric equilibrium tax revenues \({\widetilde{T}}^{ES}\), following the steps from Appendix B. The result is

$$\begin{aligned}&{\widetilde{T}}^{ES}=\frac{2(1-\alpha )}{(2-\alpha )}f({\overline{k}})\left[ t(1-{\tilde{\epsilon }})+\alpha (1-t(1-{\tilde{\epsilon }})-\tau {\tilde{\epsilon }})\right. \nonumber \\&\quad \left. \left( 1+\frac{1-\alpha }{\alpha }-\frac{t(1-{\tilde{\epsilon }})+\tau {\tilde{\epsilon }}}{\alpha f({\overline{k}})C^{\prime \prime }(0)}+\frac{\tau }{\alpha (t-\tau )}\right) \right] . \end{aligned}$$
(D.9)

Now, we can begin with the proof of Proposition 4. First, we derive the critical value \(\widehat{C^{\prime \prime }}\) which equates \({\tilde{z}}^{SH}\) to \({\tilde{z}}^{ES}\). Denote the value of \({\tilde{\epsilon }}\) associated with it as \({\widehat{\epsilon }}\). Using the definitions of \({\tilde{z}}^{SH}\) and \({\tilde{z}}^{ES}\) from Eq. (24), we can show that \({\widehat{\epsilon }}={\tilde{\delta }}/{\tilde{z}}^{SH}\). Plugging this expression on the left-hand side of (D.8), we see that at \(\widehat{C^{\prime \prime }}\), the left-hand sides of (D.8) and (D.5) coincide. Hence, the right-hand sides must also be equal. Equating these right-hand sides and solving for \(\widehat{C^{\prime \prime }}\), we get

$$\begin{aligned} \frac{1}{\widehat{C^{\prime \prime }}}=\frac{tf({\overline{k}})}{t(1-{\widehat{\epsilon }})+\tau {\widehat{\epsilon }}}\left[ \frac{(1-\alpha )}{(t-\tau )}+\frac{\alpha {\widehat{\epsilon }}}{2(1-t(1-{\widehat{\epsilon }})-\tau {\widehat{\epsilon }})}\right] . \end{aligned}$$
(D.10)

The EMTR \({\tilde{z}}^{ES}\) is increasing in \(C^{\prime \prime }(0)\). Hence, for values of \(C^{\prime \prime }(0)\) greater (smaller) than \(\widehat{C^{\prime \prime }}\), \({\tilde{z}}^{ES}\) is greater (smaller) than \({\tilde{z}}^{SH}\).

Note that at \(\widehat{C^{\prime \prime }}\), we must have \({\widetilde{T}}^{SH}>{\widetilde{T}}^{ES}\) because the EMTRs are the same, while the effective tax on economic rent under SHR is strictly greater: \(t>t(1-\epsilon )+\tau \epsilon \). Therefore, a critical value \(\overline{C^{\prime \prime }}>\widehat{C^{\prime \prime }}\) is required such that the equilibrium tax revenues are the same. This critical value can be found by equating the right-hand sides of Eq. (D.6) and (D.9). Denoting the value of \(C^{\prime \prime }(0)\) associated with this condition as \(\overline{C^{\prime \prime }}\), we get

$$\begin{aligned} \frac{1}{\overline{C^{\prime \prime }}}=\frac{(1-\alpha )f({\overline{k}})t(1-t)}{(t-\tau )[1-t(1-{\tilde{\epsilon }}-\tau {\tilde{\epsilon }})][t(1-{\tilde{\epsilon }}+\tau {\tilde{\epsilon }})]}. \end{aligned}$$
(D.11)

Because \({\widetilde{T}}^{SH}\) is independent of \(C^{\prime \prime }(0)\), while \({\widetilde{T}}^{ES}\) is increasing in it, the coordinated outcome is SHR if \(C^{\prime \prime }(0)<\overline{C^{\prime \prime }}\) and ESR, otherwise.

Consider now the decentralized choice of TCR. As in the main model, the first-order conditions of the local governments are too difficult to analyze. However, we show that a high degree of transfer price manipulation exists, where both governments can benefit by unilaterally deviating away from a symmetric SHR case to ESR.

Suppose the two countries are in a symmetric SHR equilibrium and home decides to deviate to ESR. Assume that home sets \(\epsilon ={\widehat{\epsilon }}\) such that \(z^{ES}({\widehat{\epsilon }})={\tilde{z}}^{SH}\). This deviation leaves the first-order conditions of the foreign government unchanged. Thus, the foreign government keeps \(z^{*SH}={\tilde{z}}^{*SH}\) as its best response to the policy of the home country. However, the policy change incentivizes the MNE to change its transfer price. The optimal transfer price, in this case, is given by

$$\begin{aligned} C^{\prime }(\sigma ^{ES,SH})=t(1-\epsilon )+\tau \epsilon -t^*. \end{aligned}$$
(D.12)

Using Taylor expansion around \(\epsilon =0\), we can approximate \(\sigma ^{ES,SH}\) as

$$\begin{aligned} \sigma ^{ES,SH}({\widehat{\epsilon }})\approx \sigma ^{ES,SH}(\epsilon =0)+\frac{\hbox {d}\sigma ^{ES,SH}}{\hbox {d}\epsilon }\left( {\widehat{\epsilon }}-0\right) =-\frac{(t-\tau ){\widehat{\epsilon }}}{C^{\prime \prime }(0)}<0. \end{aligned}$$
(D.13)

The tax revenues resulting from the deviation \({\widehat{T}}\) are, thus, given by

$$\begin{aligned} {\widehat{T}}=&[t(1-{\widehat{\epsilon }})+\tau {\widehat{\epsilon }}]\left[ f({\overline{k}})-{\tilde{z}}^{SH}r{\overline{k}}-\sigma ^{ES,SH}\right] \nonumber \\&+({\tilde{z}}^{SH}-1)r{\overline{k}}-\tau {\widehat{\epsilon }}(f({\overline{k}})-\sigma ^{ES,SH})\nonumber \\ =&[t(1-{\widehat{\epsilon }})+\tau {\widehat{\epsilon }}]\left[ f({\overline{k}})-{\tilde{z}}^{SH}r{\overline{k}}-\sigma ^{ES,SH}\right] +\frac{({\tilde{z}}^{SH}-1)}{\tilde{z^{SH}}}r{\tilde{z}}^{SH}{\overline{k}}-\frac{\tau {\tilde{\delta }}}{{\tilde{z}}^{SH}}f({\overline{k}})\nonumber \\&+\tau {\widehat{\epsilon }}\sigma ^{ES,SH}\nonumber \\ =&\,[t(1-{\widehat{\epsilon }})+\tau {\widehat{\epsilon }}]\left[ f({\overline{k}})-{\tilde{z}}^{SH}r{\overline{k}}-\sigma ^{ES,SH}\right] +\frac{({\tilde{z}}^{SH}-1-\tau {\tilde{\delta }})}{\tilde{z^{SH}}}\alpha f({\overline{k}})\nonumber \\&-\frac{\tau {\tilde{\delta }}}{{\tilde{z}}^{SH}}(1-\alpha )f({\overline{k}}) +\tau {\widehat{\epsilon }}\sigma ^{ES,SH}\nonumber \\ =&{\widetilde{T}}^{SH}-{\widehat{\epsilon }}(t-\tau )(1-\alpha )f({\overline{k}})-\tau {\widehat{\epsilon }}(1-\alpha )f({\overline{k}})-\sigma ^{ES,SH}[t(1-{\widehat{\epsilon }})\nonumber \\&+\tau {\widehat{\epsilon }}-\tau {\widehat{\epsilon }}].\nonumber \\&={\widetilde{T}}^{SH}-{\widehat{\epsilon }}t(1-\alpha )f({\overline{k}})-t(1-{\widehat{\epsilon }})\sigma ^{ES,SH}. \end{aligned}$$
(D.14)

Using the approximate value of \(\sigma ^{ES,SH}\) and the definition of \(\widehat{C^{\prime \prime }}\), we get

$$\begin{aligned} {\widehat{T}}&\approx {\widetilde{T}}^{SH}-{\widehat{\epsilon }}t(1-\alpha )f({\overline{k}})+t(1-{\widehat{\epsilon }})\frac{(t-\tau ){\widehat{\epsilon }}}{C^{\prime \prime }(0)}\nonumber \\&={\widetilde{T}}^{SH}-{\widehat{\epsilon }}t(t-\tau )\left[ \frac{(1-\alpha )f({\overline{k}})}{t-\tau }-\frac{(1-{\widehat{\epsilon }})}{C^{\prime \prime }(0)}\right] \nonumber \\&={\widetilde{T}}^{SH}+{\widehat{\epsilon }}t(t-\tau )\left[ \frac{t{\widehat{\epsilon }}\alpha f({\overline{k}})}{2(1-t(1-{\widehat{\epsilon }}))}-\frac{t(1-{\widehat{\epsilon }})+\tau {\widehat{\epsilon }}}{t\widehat{C^{\prime \prime }}}+\frac{(1-{\widehat{\epsilon }})}{C^{\prime \prime }(0)}\right] . \end{aligned}$$
(D.15)

The first term in brackets in the last line of (D.15) is positive, while the sum of the remaining terms in the brackets has an undetermined sign. If the sum of these two terms is positive, then \({\widehat{T}}>{\widetilde{T}}^{SH}\). The negative second term in brackets is declining in \(\tau \) and is most negative, when \(\tau \) achieves its highest value, i.e., when \(\tau =t\). Then, the negative term becomes \(1/\widehat{C^{\prime \prime }}\). Thus, when \(\tau \) is as high as possible, the last row of (D.15) is positive for \(C^{\prime \prime }(0)<(1-{\widehat{\epsilon }})\widehat{C^{\prime \prime }}=(1-{\tilde{\delta }}/{\tilde{z}}^{SH})\widehat{C^{\prime \prime }}\). In this case, the deviation pays off for the home country. In the asymmetric equilibrium, the home country sets \(\epsilon \) equal to \({\tilde{\epsilon }}^{ES,SH}\) that differs from \({\widehat{\epsilon }}\) if and only if this results in higher tax revenues. Therefore, we have \({\widetilde{T}}^{ES,SH}\ge {\widehat{T}}>{\widetilde{T}}^{SH}\) for \(C^{\prime \prime }(0)<(1-{\tilde{\delta }}/{\tilde{z}}^{SH})\widehat{C^{\prime \prime }}\). Moreover, we already showed that for these values of \(C^{\prime \prime }(0)\), it holds \({\widetilde{T}}^{SH}>{\widetilde{T}}^{ES}\). Thus, it must hold \({\widetilde{T}}^{ES,SH}>{\widetilde{T}}^{SH}>{\widetilde{T}}^{ES}\). \(\square \)

Appendix E: Proof of Proposition 5

To prove Proposition 5, we first start with the profit maximization problem of the MNE under the consideration of the costs of internal debt \(C_b(b/k)rk\). These costs are assumed to not be deductible from the tax base of the respective subsidiary, and the after-tax profit of the MNE is

$$\begin{aligned} \Pi= & {} f(k)-rk-t\left[ f(k)-rb-\sigma \right] -C(\sigma )-C_b\left( \frac{b}{k}\right) rk \nonumber \\&+\, f(k^*)-rk^*-t^*\left[ f(k^*)-rb^*+\sigma \right] -C_b\left( \frac{b^*}{k^*}\right) rk^*. \end{aligned}$$
(E.1)

The SHR and ESR constraints are given by Eqs. (2) and (3). Profit maximization in the case of symmetric SHR results in the following conditions:

$$\begin{aligned} f^{\prime }(k^{SH})&=z^{SH}r, \quad \text {where}\quad z^{SH}\equiv \frac{1-\delta t+C_{b}(\delta )}{1-t}, \end{aligned}$$
(E.2a)
$$\begin{aligned} f^{\prime }(k^{*SH})&=z^{*SH}r, \quad \text {where}\quad z^{*SH}\equiv \frac{1-\delta ^* t+C_{b}(\delta ^*)}{1-t}, \end{aligned}$$
(E.2b)
$$\begin{aligned} C^{\prime }(\sigma ^{SH})&=t-t^*. \end{aligned}$$
(E.2c)

Thus, cost of internal debt affects only the EMTR, but not the optimal transfer price \(\sigma ^{SH}\).

Under a symmetric ESR scenario, we get the following profit-maximizing conditions:

$$\begin{aligned} f^{\prime }(k^{ES})&=z^{ES}r, \quad \text {where}\quad z^{ES}\equiv \frac{1+C_{b}(b/k)-(b/k)C_b^{\prime }(b/k)}{1-t(1-\epsilon )-\epsilon C_b^{\prime }(b/k)}, \end{aligned}$$
(E.3a)
$$\begin{aligned} f^{\prime }(k^{*ES})&=z^{*ES}r, \quad \text {where}\quad z^{*ES}\equiv \frac{1+C_{b}(b^*/k^*)-(b^*/k^*)C_b^{\prime }(b^*/k^*)}{1-t(1-\epsilon ^*)-\epsilon ^* C_b^{\prime }(b^*/k^*)}, \end{aligned}$$
(E.3b)
$$\begin{aligned} C^{\prime }(\sigma ^{SH})&=t(1-\epsilon )-t^*(1-\epsilon ^*)+\epsilon C_b^{\prime }(b/k)-\epsilon ^* C_b^{\prime }(b^*/k^*). \end{aligned}$$
(E.3c)

We see that, in this case, the costs of internal debt affect the optimal transfer price as well. The reason is that these costs impact the net marginal benefits of internal debt, which ultimately determines the effective tax rate on economics rents. To see that, rewrite the home government’s tax revenues from Eqs. (9a) and (9b) to include the costs of internal debt. The resulting expressions are

$$\begin{aligned} T^{SH}&=t\left[ f(k)-z^{SH}rk-\sigma \right] +(z^{SH}-1)rk-C_b(b/k)rk. \end{aligned}$$
(E.4a)
$$\begin{aligned} T^{ES}&=[t(1-\epsilon )+\epsilon C_b^{\prime }(b/k)]\left[ f(k)-z^{ES}rk-\sigma \right] +(z^{ES}-1)rk-C_b(b/k)rk. \end{aligned}$$
(E.4b)

According to (E.4b), the effective tax on economic rents in the second case is \(t(1-\epsilon )+\epsilon C_b^{\prime }(b/k)\).

We proceed by deriving the equilibrium values of \(\delta \) and \(\epsilon \) in the symmetric games of stage 2. Under SHR, the home government maximizes (E.4a), taking into account the MNE’s behavior described by (E.2a)-(E.2c). The equilibrium \({\tilde{z}}^{SH}\) is given by

$$\begin{aligned} \frac{{\tilde{z}}^{SH}-1-C_b\left( {\tilde{\delta }}^{SH}\right) }{{\tilde{z}}^{SH}}=\frac{1-\alpha }{2-\alpha }\left[ 2-t-\frac{2C_b^{\prime }({\tilde{\delta }}^{SH})}{\partial z^{SH}/\partial \delta }\right] . \end{aligned}$$
(E.5)

The symmetric ESR case is now slightly more complicated. The reason is that \(z^{ES}\) is not only a function of \(\epsilon \) but also of b / k, where both b and k are endogenous parameters. To simplify the remaining analysis, denote \(\delta ^{ES}\equiv b^{ES}/k^{ES}\) and \(\theta (\epsilon ,\delta ^{ES})\equiv [t(1-\epsilon )+\epsilon C_b^{\prime }(\delta ^{ES})]\). Thus, we can rewrite

$$\begin{aligned} z^{ES}(\epsilon ,\delta ^{ES})=\frac{1+C_b(\delta ^{ES})-\delta ^{ES}C_b^{\prime }(\delta ^{ES})}{1-\theta (\epsilon ,\delta ^{ES})}. \end{aligned}$$
(E.6)

Furthermore, the optimal transfer price is now defined by \(C^{\prime }(\sigma ^{ES})=\theta -\theta ^*\). Hence, we have \(\sigma (\theta (\epsilon ,\delta ^{ES}),\theta ^{*}(\epsilon ^*,\delta ^{*ES}))\). Lastly, the debt level is \(rb=\epsilon (f(k)-\sigma )\). Noting that r and k are functions of \(z^{ES}\), we get

$$\begin{aligned} r\left( z^{ES}(\epsilon ,\delta ^{ES})\right) \delta ^{ES}=\epsilon \left[ \frac{f\left[ k\left( z^{ES}(\epsilon ,\delta ^{ES}))\right) \right] }{k(z^{ES}(\epsilon ,\delta ^{ES}))}-\frac{\sigma (\epsilon ,\delta ^{ES})}{k(z^{ES}(\epsilon ,\delta ^{ES}))}\right] . \end{aligned}$$
(E.7)

Equation (E.7) defines implicitly \(\delta ^{ES}\) as a function of \(\epsilon \). Thus, by choosing \(\epsilon \), the government effectively also defines a debt-to-capital ratio \(\delta ^{ES}\). Define the inverse function of \(\delta ^{ES}(\epsilon )\) as \(\epsilon \equiv \varepsilon (\delta ^{ES})\). Then, we have \(d\theta /d\delta =-\varepsilon ^{\prime }(t-C_b^{\prime })+\varepsilon C_b^{\prime \prime }\). We assume \(d\theta /d\delta <0\), which must be the case when the ESR is binding and an increase of \(\epsilon \) by the government is profitable to the firm, such that (E.7) is satisfied.

We can continue by maximizing the tax revenues (E.4b) over \(\delta ^{ES}\):

$$\begin{aligned} \max _{\delta ^{ES}}\,\theta (\delta ^{ES},\varepsilon (\delta ^{ES}))\left[ f(k)-z^{ES}rk-\sigma (\theta (\delta ^{ES},\varepsilon (\delta ^{ES})))\right] +(z^{ES}-1)rk-C_b(\delta ^{ES})rk. \end{aligned}$$
(E.8)

Noting that in a symmetric equilibrium, the effect of \(\theta \) on the optimal transfer price is \(\partial \sigma ^{ES}/\partial \theta =1/C^{\prime \prime }(0)\), we can simplify the first-order condition to the following expression:

$$\begin{aligned}&\frac{{\tilde{z}}^{ES}-1-C_b\left( {\tilde{\delta }}^{ES}\right) }{{\tilde{z}}^{ES}}=\frac{1-\alpha }{2-\alpha }\left[ 2-\theta -\frac{2}{\partial z^{ES}/\partial \delta }\left( C_b^{\prime }({\tilde{\delta }}^{ES})\right. \right. \nonumber \\&\quad \left. \left. -\frac{\hbox {d}\theta /d\delta }{r{\overline{k}}}\left( (1-\alpha )f({\overline{k}})-\frac{\theta }{C^{\prime \prime }(0)}\right) \right) \right] . \end{aligned}$$
(E.9)

As in the main model, more aggressive transfer price manipulation (lower value of \(C^{\prime \prime }(0)\)) lowers the effective tax rate \({\tilde{z}}^{ES}\) and vice versa. To find the level of transfer price manipulation that equates \({\tilde{z}}^{ES}\) and \({\tilde{z}}^{SH}\) at \({\tilde{\delta }}^{ES}={\tilde{\delta }}^{SH}\), we equate the right-hand sides of (E.5) and (E.9) and denote the value of \(C^{\prime \prime }(0)\) as \(\widehat{C^{\prime \prime }}\) and the equilibrium \(\theta \) as \({\widehat{\theta }}\) to get:

$$\begin{aligned} (1-\alpha )f({\overline{k}})=\frac{{\widehat{\theta }}}{\widehat{C^{\prime \prime }}}+\frac{({\widehat{\theta }}-t)r{\overline{k}}\frac{\partial z^{ES}}{\partial \delta }}{2\frac{\hbox {d}\theta }{d\delta }}. \end{aligned}$$
(E.10)

For values of \(C^{\prime \prime }(0)\) above \(\widehat{C^{\prime \prime }}\), we have \({\tilde{z}}^{ES}>{\tilde{z}}^{SH}\) and vice versa. Furthermore, we note that at \(\widehat{C^{\prime \prime }}\), the tax revenues in the symmetric SHR case are higher than in the symmetric ESR case. Since \(\widehat{C^{\prime \prime }}\) defines \({\tilde{z}}^{ES}={\tilde{z}}^{SH}\) at \({\tilde{\delta }}^{ES}={\tilde{\delta }}^{SH}\), we can use (E.4a) and (E.4b) to express the difference of the symmetric equilibrium tax revenues as

$$\begin{aligned} \left[ {\widetilde{T}}^{SH}-{\widetilde{T}}^{ES}\right] _{C^{\prime \prime }(0)=\widehat{C^{\prime \prime }}}=(t-{\widehat{\theta }})(1-\alpha )f({\overline{k}})>0. \end{aligned}$$
(E.11)

Since \({\widetilde{T}}^{SH}\) is independent of \(C^{\prime \prime }(0)\) while \({\tilde{z}}^{ES}\) is increasing in it, the tax revenue difference is declining in \(C^{\prime \prime }(0)\), such that \({\widetilde{T}}^{SH}-{\widetilde{T}}^{ES}>0\) for \(C^{\prime \prime }(0)<{\widehat{C}}^{\prime \prime }\).

We move now to the decentralized case. To prove Proposition 5, it suffices to show that a deviation from the symmetric SHR case exists that makes the deviating country better-off without affecting the first-order condition of the country that keeps SHR. Suppose that \(C^{\prime \prime }(0)<\widehat{C^{\prime \prime }}\) and consider a deviation from the symmetric SHR equilibrium by the home country that sets \(\delta ^{ES,SH}_D={\tilde{\delta }}^{SH}\) such that \(z^{ES,SH}_D={\tilde{z}}^{SH}\), where the subscript D denotes the case of deviation. Under this deviation, we get \(\theta (\delta ^{ES,SH}_D)={\widehat{\theta }}\), as defined in Eq. (E.10). This deviation does not affect the first-order condition of the foreign government and, thus, leaves the MNE’s investment decisions unchanged. In this case, the tax revenues of the home country become

$$\begin{aligned} T_D&={\widehat{\theta }}\left[ f({\overline{k}})-{\tilde{z}}^{SH}r{\overline{k}}-\sigma ^{ES,SH}\right] +({\tilde{z}}^{SH}-1)r{\overline{k}}-r{\overline{k}}C_b({\tilde{\delta }}^{SH})\nonumber \\&={\widetilde{T}}^{SH}-(t-{\widehat{\theta }})(1-\alpha )f({\overline{k}})-{\widehat{\theta }}\sigma ^{ES,SH}. \end{aligned}$$
(E.12)

To evaluate (E.12), we need the value of \(\sigma ^{ES,SH}\). It is given by the maximization of the MNE’s profit in the case ESSH, which results in the first-order condition \(C^{\prime }(\sigma ^{ES,SH})=\theta -t\). The value of \(\sigma ^{ES,SH}\) can be approximated using Taylor expansion as

$$\begin{aligned} \sigma ^{ES,SH}({\widehat{\theta }})\approx \sigma ^{ES,SH}(\theta =t)+\frac{\hbox {d}\sigma ^{ES,SH}}{d\theta }\left( {\widehat{\theta }}-t\right) =\frac{{\widehat{\theta }}-t}{C^{\prime \prime }(0)}<0. \end{aligned}$$
(E.13)

Using (E.10), (E.12) and (E.13), we get

$$\begin{aligned} T_D&\approx {\widetilde{T}}^{SH}-(t-{\widehat{\theta }})(1-\alpha )f({\overline{k}})+{\widehat{\theta }}\frac{{\widehat{\theta }}-t}{C^{\prime \prime }(0)}\nonumber \\&={\widetilde{T}}^{SH}-(t-{\widehat{\theta }})\left[ (1-\alpha )f({\overline{k}})-\frac{{\widehat{\theta }}}{C^{\prime \prime }(0)}\right] \nonumber \\&={\widetilde{T}}^{SH}-(t-{\widehat{\theta }})\left[ \frac{({\widehat{\theta }}-t)r{\overline{k}}\frac{\partial z^{ES}}{\partial \delta }}{2\frac{\hbox {d}\theta }{d\delta }}+\frac{{\widehat{\theta }}}{\widehat{C^{\prime \prime }}}-\frac{{\widehat{\theta }}}{C^{\prime \prime }(0)}\right] >{\widetilde{T}}^{SH}. \end{aligned}$$
(E.14)

Hence, our discussion of the symmetric cases and Eq. (E.14) together leads to \(T_D>{\widetilde{T}}^{SH}>{\widetilde{T}}^{ES}\). The home government sets \(\delta ^{ES,SH}\) at a value different from \(\delta ^{ES,SH}_D\) if and only if \({\widetilde{T}}^{ES,SH}>T_D\). Thus, we have \({\widetilde{T}}^{ES,SH}>{\widetilde{T}}^{SH}>{\widetilde{T}}^{ES}\). \(\square \)

Appendix F: Endogenous Tax Rates

To derive the equilibrium tax rates, we first calculate the effects of \(t,\delta \), and \(\epsilon \) on \(z^i\) and \(\sigma ^i\) for \(i=SH,ES\). Consider first the partial derivatives of \(z^{i}\) with respect to \(t,\delta \), and \(\epsilon \), derived from Eqs. (27) and (28a):

$$\begin{aligned}&\frac{\partial z^{SH}}{\partial t}=\frac{(1-\delta )(1-\rho )}{(1-t)^2}>0,\quad \frac{\partial z^{SH}}{\partial \delta }=-\frac{t(1-\rho )}{1-t}<0, \end{aligned}$$
(F.1)
$$\begin{aligned}&\frac{\partial z^{ES}}{\partial t}=\frac{(1-\epsilon )(1-\rho )}{[1-t(1-\epsilon (1-\rho ))]^2}>0,\quad \frac{\partial z^{ES}}{\partial \epsilon }=-\frac{t(1-\rho )(1-t\rho )}{[1-t(1-\epsilon (1-\rho ))]^2}<0.\nonumber \\ \end{aligned}$$
(F.2)

The effects of t and \(\delta \) on \(\sigma ^{SH}\) and t and \(\epsilon \) on \(\sigma ^{ES}\) are derived from Eqs. (7c) and (28c):

$$\begin{aligned} \frac{\hbox {d} \sigma ^{SH}}{d t}=\frac{1}{C^{\prime \prime }(\sigma )}>0,&\frac{\hbox {d} \sigma ^{SH}}{d \delta }=0, \end{aligned}$$
(F.3)
$$\begin{aligned} \frac{\hbox {d} \sigma ^{ES}}{d t}=\frac{1-\epsilon (1-\rho )}{C^{\prime \prime }(\sigma )}>0,&\frac{\hbox {d} \sigma ^{ES}}{d \epsilon }=-\frac{t(1-\rho )}{C^{\prime \prime }(\sigma )}<0. \end{aligned}$$
(F.4)

The home government’s first-order conditions in a symmetric ESR game are given by

$$\begin{aligned} \frac{{\text{ d }}T^{ES}}{\hbox {d}t}= & {} \frac{{\text{ d }}T^{ES}}{{\text{ d }}z}\frac{{\text{ d }}z^{ES}}{\hbox {d}t}+(1-\epsilon (1-\rho ))\left[ f(k)-z^{ES}rk-\sigma ^{ES}-t\frac{\hbox {d}\sigma ^{ES}}{\hbox {d}t}\right] \le 0,\nonumber \\&t\in [0,1],\quad t\frac{{\text{ d }}T^{ES}}{\hbox {d}t}=0, \end{aligned}$$
(F.5)
$$\begin{aligned} \frac{{\text{ d }}T^{ES}}{\hbox {d}\epsilon }= & {} \frac{{\text{ d }}T^{ES}}{{\text{ d }}z}\frac{{\text{ d }}z^{ES}}{\hbox {d}\epsilon }-t(1-\rho )\left[ f(k)-zrk-\sigma \right] -t(1-\epsilon (1-\rho ))\frac{\hbox {d}\sigma }{\hbox {d}\epsilon }\le 0,\nonumber \\&\epsilon \ge 0,\quad \frac{{\text{ d }}T^{ES}}{\hbox {d}\epsilon }\epsilon =0, \end{aligned}$$
(F.6)

where

$$\begin{aligned} \frac{{\text{ d }}T^{ES}}{{\text{ d }}z^{ES}}= & {} rk[1-t(1-\epsilon (1-\rho ))]+(z-1)r\frac{{\text{ d }}k}{{\text{ d }}z}\\&+\,k\left[ z(1-t(1-\epsilon (1-\rho )))-1\right] \frac{\hbox {d}r}{{\text{ d }}z}. \end{aligned}$$

Suppose that there is an interior solution. We solve (F.6) for \({\text{ d }}T^{ES}/{\text{ d }}z\) and insert the resulting expression in (F.5). Simplification of (F.5) leads to

$$\begin{aligned} \frac{{\text{ d }}T^{ES}}{\hbox {d}t}= & {} \frac{\rho }{z^{ES}}\left[ f(k)-z^{ES}rk-\sigma ^{ES}-\frac{t(1-\epsilon (1-\rho ))}{C^{\prime \prime }(\sigma )}\right] =0. \end{aligned}$$
(F.7)

In a symmetric situation, the solution of (F.7) is Eq. (31). According to (F.7), the last two terms in (F.6) cancel out and the solution to (F.6) is given by \({\text{ d }}T^{ES}/{\text{ d }}z=0\). The solution of this expression for \({\tilde{z}}^{ES}\) follows the same steps as the ones outlined in Appendix A. The result is given by Eq. (32).

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Kalamov, Z.Y. Safe haven or earnings stripping rules: a prisoner’s dilemma?. Int Tax Public Finance 27, 38–76 (2020). https://doi.org/10.1007/s10797-019-09545-w

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Keywords

  • Thin-capitalization rule
  • Safe haven rule
  • Earnings stripping rule
  • Profit shifting

JEL Classification

  • F23
  • H25
  • H7