Can fiscal equalisation mitigate tax competition? Ad valorem and residence-based taxation in a federation

Abstract

In this paper, we revisit the combined effect of horizontal and vertical tax externalities in a federal context, extending the theoretical framework of Keen and Kotsogiannis (Am Econ Rev 92(1):363–370, 2002) by allowing for ad valorem and residence-based taxation. When taxes are levied ad valorem rather than per-unit firstly, we find the interaction between both types of externalities is more ambiguous than commonly understood. As a result, and contrary to earlier findings, fiscal equalisation mechanisms such as the representative transfer system (RTS) fail to fully internalise the tax externalities. Given these limitations, we derive the conditions under which a standard RTS will either: (1) at least nudge politicians in the right direction; (2) realise no welfare gains at all; (3) considerably overshoot the second-best efficiency mark causing welfare loss. Lastly, we find that when taxation is residence-based rather than source-based, a different kind of competition emerges where tax cuts are aimed at stimulating outward factor flows, rather than attracting inward flows.

Appendices

Appendix 1: Derivation of Lemma 1

After plugging in the profit effect (8), and in symmetric equilibrium, the optimisation problem set out in Sect. 4 yields the same general expression for the marginal rate of substitution as in the unitary case derived in Sect. 3, so that

$$\begin{aligned} \frac{\varGamma _{G}}{\lambda }=-\frac{\left( L_{\mathrm{S}}\frac{\partial {\bar{w}}}{\partial t}-L_{\mathrm{D}}\frac{\partial w}{\partial t}\right) }{\left( L_{\mathrm{S}}w+t L_{\mathrm{S}}\frac{\partial w}{\partial t}+t w\frac{\partial L_{\mathrm{S}}}{\partial {\bar{w}}}\frac{\partial {\bar{w}}}{\partial t}\right) } \equiv \mathrm{MCPF}_{i}. \end{aligned}$$

(32)

Plugging in (7), and since $\frac{\partial {\bar{w}}}{\partial t}=\left( 1-\tau \right) \frac{\partial w}{\partial t}-w=\frac{w\left( n\varepsilon -(n-1)\eta \right) }{n\left( \eta -\varepsilon \right) }$, we get a different expression here

$$\begin{aligned} \frac{\varGamma _{G}}{\lambda }=-\frac{\left( \frac{\left( n\varepsilon -(n-1)\eta \right) }{n\left( \eta -\varepsilon \right) }-\frac{\eta }{n(1-\tau )\left( \eta -\varepsilon \right) }\right) }{\left( 1+t\frac{\eta }{n(1-\tau )\left( \eta -\varepsilon \right) }+\frac{t\eta }{(1-\tau )}\frac{\left( n\varepsilon -(n-1)\eta \right) }{n\left( \eta -\varepsilon \right) }\right) }, \end{aligned}$$

(33)

with the gross wage, and labour supply and demand dropping out because of symmetry, and keeping in mind that $\tau =t+T$. Reworking yields

$$\begin{aligned} \frac{\varGamma _{G}}{\lambda }=\frac{1}{\left( 1+\frac{-(1-t-T)\eta +\eta -t\eta -t\eta \left( n\varepsilon -(n-1)\eta \right) }{(1-\tau )\left( n\varepsilon -(n-1)\eta \right) -\eta }\right) }, \end{aligned}$$

(34)

so that, finally, we obtain

$$\begin{aligned} \frac{\varGamma _{G}}{\lambda }=\frac{1}{\left( 1-\frac{t\eta \left( n\varepsilon -(n-1)\eta \right) -T\eta }{(1-\tau )\left( n\varepsilon -(n-1)\eta \right) -\eta }\right) }\equiv \mathrm{MCPF}_{i}, \end{aligned}$$

(35)

which returns as expression (15) in Lemma 1.

Appendix 2: Proof of Proposition 1

Consider the unitary optimisation outcome derived in Sect. 3, where the unitary MCPF is derived maximising (9) subject to (10), which boils down to setting $\frac{\partial W(\tau ,G,G^F)}{\partial \tau }=0$ in symmetric equilibrium, with $\varGamma _{G}=\varGamma _{G}^F$ following from the first-order conditions on public provision. If, evaluated at the Nash equilibrium where $\tau =(t^{*}+T^{*})$, we have that $\frac{\partial W(\tau ,G,G^F)}{\partial \tau }|_{\tau =t^{*}+T^{*}}\lessgtr 0$, state taxation diverges from the second-best optimum, requiring up- or down-wards adjustment moving to second-best efficiency. This latter inequality can then be rewritten as

$$\begin{aligned} \mathrm{MCPF}_{i}(t^{*},T^{*})=\frac{\varGamma _{G}}{\lambda }\lessgtr \frac{1}{\left( 1-\frac{\tau \eta \varepsilon }{(1-\tau )\varepsilon -\eta }\right) }=\mathrm{MCPF}(\tau =t^{*}+T^{*}) , \end{aligned}$$

(36)

so that the condition for inefficiency in (36) boils down to the exact same trade-off described in the main text, and the second-best optimum is only attained when $\mathrm{MCPF}(\tau ^*)=\mathrm{MCPF}_{i}(t^{*},T^{*})$. As soon as the LHS of (36) outweighs the RHS, positive externalities result in undertaxation. Looking at the denominators in (36), and using (35), this is the case when

$$\begin{aligned} \frac{t{}^{*}\eta \left( n\varepsilon -(n-1)\eta \right) -T^{*}\eta }{(1-\tau )\left( n\varepsilon -(n-1)\eta \right) -\eta }>\frac{\tau \eta \varepsilon }{(1-\tau )\varepsilon -\eta }. \end{aligned}$$

(37)

Reworking then yields

$$\begin{aligned} \frac{\tau \left( n\varepsilon -(n-1)\eta \right) -T^{*}\left( n\varepsilon -(n-1)\eta \right) -T^{*}}{(1-\tau )\left( n\varepsilon -(n-1)\eta \right) -\eta }>\frac{\tau \varepsilon }{(1-\tau )\varepsilon -\eta }, \end{aligned}$$

(38)

which, after some more manipulation can be written as

$$\begin{aligned} \frac{n\tau \varepsilon -t{}^{*}(n-1)\eta -T^{*}(1+n\varepsilon )}{n(1-\tau )\left( \varepsilon -\eta \right) +(1-\tau )\eta -\eta }>\frac{n\tau \varepsilon }{n\left( (1-\tau )\varepsilon -\eta \right) }, \end{aligned}$$

(39)

or, arriving at expression (16) of Proposition 1,

$$\begin{aligned} \frac{n\tau \varepsilon -t{}^{*}(n-1)\eta -T^{*}(1+n\varepsilon )}{n\left( (1-\tau )\varepsilon -\eta )\right) +\left( n-1\right) \tau \eta }>\frac{n\tau \varepsilon }{n\left( (1-\tau )\varepsilon -\eta \right) }. \end{aligned}$$

(40)

$\square$

Appendix 3: Proof of Corollary 1

Keeping in mind that $\varepsilon <0$ and $\eta >0$, the denominators and the numerators on both sides of (16) will always be negative. A higher equilibrium labour supply elasticity $\eta$ then brings the denominator on the LHS down in absolute value, and this through $\left( n-1\right) \tau \eta$ , thus pushing up the bias. Turning to the numerators, we see that $\eta$ brings about the same effect, but via $t{}^{*}(n-1)\eta$. The equilibrium labour demand elasticity $\varepsilon$ on the other hand pulls the perceived MCPF down through $T^{*}(1+n\varepsilon )$. Lastly, as the state tax rate $t^{*}$ accounts for a smaller share of the total tax rate $\tau$, this latter effect comes out reinforced. All four points combined give us Corollary 1. $\square$

Appendix 4: Proof of Proposition 2

Using the federal budget constraint (12) evaluated in symmetric equilibrium, and under the assumption that state governments can see through the part of the federal budget constraint which concerns their own state, (17) can be written as

$$\begin{aligned} W(t,T,\tau )= & {} v\left( w(1-\tau ),\pi (w)\right) \nonumber \\&+\varGamma \left( twL_{\mathrm{S}}(w(1-\tau ))+\omega (\tau ),\frac{1}{n}T\left( \sum _{i}^n L_{\mathrm{D}}(w)w-\sum _{i}^n\omega _{i}(\tau )\right) \right) \nonumber \\ \end{aligned}$$

(41)

As explained in the main text, state public provision is augmented with the equalisation grant in (41), so that $G=twL_{\mathrm{S}}(w(1-\tau ))+\omega (\tau )$, whilst the sum total of equalisation grants comes in as additional federal expenditures. Totally differentiating (41) with respect to t, we can then write up the equivalent of (18) here as follows

$$\begin{aligned} \frac{\partial W}{\partial t}= & {} \lambda \left[ L_{\mathrm{S}}\left( (1-\tau )\frac{w'}{n}-w\right) -L_{\mathrm{D}}\frac{w'}{n}\right] \nonumber \\&+\varGamma _{G}\left[ L_{\mathrm{S}}w+tL_{\mathrm{S}}^{'}\left( (1-\tau )\frac{w'}{n}-w\right) w+tL_{\mathrm{S}}\frac{w'}{n}+\frac{\partial \omega }{\partial t}\right] \nonumber \\&+T\varGamma _{G^{F}}\left[ (wL_{\mathrm{D}}^{'}\frac{w'}{n}+L_{\mathrm{D}}\frac{w'}{n})\right] =0, \end{aligned}$$

(42)

which reduces to (23) in the main text, and where we prove in Appendix 5 that $\sum _{j}^{n}\frac{\partial \omega _{j}}{\partial t}=0$, so that equalisation does not give rise to vertical externalities. In order to filter out the externalities subsequently, we subtract (42) from (20) as in Sect. 4, to obtain

$$\begin{aligned} \frac{\partial W}{\partial t}= & {} \left[ -\lambda \tau L_{\mathrm{S}}+t^{*}\varGamma _{G}\left( L_{\mathrm{S}}^{'}(1-\tau )+L_{\mathrm{S}}\right) +T^{*}\varGamma _{G^{F}}\left( wL_{\mathrm{D}}^{'}+L_{\mathrm{D}}\right) \right] \nonumber \\&\times \left( 1-\frac{1}{n}\right) w'-\varGamma _{G}\frac{\partial {\hat{\omega }}}{\partial t}, \end{aligned}$$

(43)

where, contrary to (21), the vertical externality expressed by $\varGamma _{G^{F}}\left( wL_{\mathrm{D}}^{'}+L_{\mathrm{D}}\right)$ is now of degree $\left( 1-\frac{1}{n}\right)$ as well, and the incentive effect of equalisation $\frac{\partial \omega }{\partial t}$ drives states to increase their tax rates as it pulls (43) down. Setting (43) to zero and solving for $\frac{\partial \omega }{\partial t}$, we then obtain the prerequisite for any equalisation grant to internalise all externalities as

$$\begin{aligned} \frac{\partial {\hat{\omega }}}{\partial t}= & {} \frac{1}{\varGamma _{G}}\left[ -\lambda \tau L_{\mathrm{S}}+t^{*}\varGamma _{G}\left( L_{\mathrm{S}}^{'}(1-\tau )+L_{\mathrm{S}}\right) \right. \nonumber \\&\left. +T^{*}\varGamma _{G^{F}}\left( wL_{\mathrm{D}}^{'}+L_{\mathrm{D}}\right) \right] \left( 1-\frac{1}{n}\right) w'. \end{aligned}$$

(44)

With (44) plugged into (42), the latter will indeed be equal to (20) set to zero, imposing second-best efficiency. $\square$

Appendix 5: Proof of federal equalisation neutrality

Starting from the sum total of equalisation grants included in (41) above, we get

$$\begin{aligned} \sum _{j}^{n}\omega _{j}=\sum _{j}^{n}{\bar{t}}({\bar{B}}-B_{j}). \end{aligned}$$

(45)

Differentiating (45) with respect to $t_{i}$ yields

$$\begin{aligned} \sum _{j}^{n}\frac{\partial \omega _{j}}{\partial t_{i}}=\frac{\partial {\bar{t}}}{\partial t_{i}}({\bar{B}}-B_{i})+{\bar{t}}\left( \frac{\partial {\bar{B}}}{\partial t_{i}}-\frac{\partial B_{i}}{\partial t_{i}}\right) +\sum _{j\ne i}^{n}\left[ \frac{\partial {\bar{t}}}{\partial t_{i}}({\bar{B}}-B_{j})+{\bar{t}}\left( \frac{\partial {\bar{B}}}{\partial t_{i}}-\frac{\partial B_{j}}{\partial t_{i}}\right) \right] , \end{aligned}$$

(46)

which, evaluated in symmetric equilibrium where ${\bar{B}}=B_{i}$ and ${\bar{t}}=\frac{\sum _{i}^{n}t_{i}L_{\mathrm{S}_{i}}w}{\sum _{i}^{n}L_{\mathrm{S}_{i}}w}=t$ for all i, reduces to

$$\begin{aligned} \sum _{j}^{n}\frac{\partial \omega _{j}}{\partial t_{i}}=t\left( \frac{\partial {\bar{B}}}{\partial t_{i}}-\frac{\partial B_{i}}{\partial t_{i}}\right) +\sum _{j\ne i}^{n}t\left( \frac{\partial {\bar{B}}}{\partial t_{i}}-\frac{\partial B_{j}}{\partial t_{i}}\right) . \end{aligned}$$

(47)

Focusing first on the first term between brackets on the right-hand side of (47), we have that

$$\begin{aligned} \left( \frac{\partial {\bar{B}}}{\partial t_{i}}-\frac{\partial B_{i}}{\partial t_{i}}\right) =\left[ \frac{1}{n}\left( \frac{\partial (L_{\mathrm{S}_{i}}w)}{\partial t_{i}}+\frac{\partial \left( \sum _{j\ne i}^{n}L_{\mathrm{S}_{j}}w\right) }{\partial t_{i}}\right) -\frac{\partial (L_{\mathrm{S}_{i}}w)}{\partial t_{i}})\right] , \end{aligned}$$

(48)

so that

$$\begin{aligned} \left( \frac{\partial {\bar{B}}}{\partial t_{i}}-\frac{\partial B_{i}}{\partial t_{i}}\right)= & {} \left\{ \frac{1}{n}\left( -L_{\mathrm{S}}^{'}w^{2}+n\left[ L_{\mathrm{S}}^{'}\left( (1-\tau )\frac{w'}{n}\right) w+L_{\mathrm{S}}\frac{w'}{n}\right] \right) \right. \nonumber \\&\left. -\left[ L_{\mathrm{S}}^{'}\left( (1-\tau )\frac{w'}{n}-w\right) w+L_{\mathrm{S}}\frac{w'}{n}\right] \right\} , \end{aligned}$$

(49)

or

$$\begin{aligned} \left( \frac{\partial {\bar{B}}}{\partial t_{i}}-\frac{\partial B_{i}}{\partial t_{i}}\right) =\left( \frac{n-1}{n}\right) L_{\mathrm{S}}^{'}w^{2}, \end{aligned}$$

(50)

which also defines (25) in the main text. Similarly, examining the second term between brackets on the right-hand side of (47), we have that

$$\begin{aligned} \left( \frac{\partial {\bar{B}}}{\partial t_{i}}-\frac{\partial B_{j}}{\partial t_{i}}\right)= & {} \frac{-L_{\mathrm{S}}^{'}w^{2}}{n}+\left[ L_{\mathrm{S}}^{'}\left( (1-\tau )\frac{w'}{n}\right) w+L_{\mathrm{S}} \frac{w'}{n}\right] \nonumber \\&-\left[ L_{\mathrm{S}}^{'}\left( (1-\tau )\frac{w'}{n}\right) w+L_{\mathrm{S}}\frac{w'}{n}\right] , \end{aligned}$$

(51)

so that

$$\begin{aligned} \left( \frac{\partial {\bar{B}}}{\partial t_{i}}-\frac{\partial B_{j}}{\partial t_{i}}\right) =\frac{-L_{\mathrm{S}}^{'}w^{2}}{n}. \end{aligned}$$

(52)

Plugging both (50) and (52) into (47), then yields

$$\begin{aligned} \sum _{j}^{n}\frac{\partial \omega _{j}}{\partial t_{i}}=t\left( \left( \frac{n-1}{n}\right) L_{\mathrm{S}}^{'}w^{2}\right) +t(n-1)\left( \frac{-L_{\mathrm{S}}^{'}w^{2}}{n}\right) \equiv 0. \end{aligned}$$

(53)

$\square$

Driving this result is the effect of a tax hike by a single state i on the overall federal per capita tax base $\sum _{j}^{n}\frac{\partial {\bar{B}}}{\partial t_{i}}$, which is identical to the change in all state tax bases $\sum _{j}^{n}\frac{\partial \left( L_{\mathrm{S}_{j}}w\right) }{\partial t_{i}}$ due to the same tax increase in $t_{i}$. More specifically, because ${\bar{B}}$ is used as the equalisation reference, and because $\sum _{j}^{n}\frac{\partial {\bar{B}}}{\partial t_{i}}=\sum _{j}^{n}\frac{\partial \left( L_{\mathrm{S}_{j}}w\right) }{\partial t_{i}}$ in symmetric equilibrium, equalisation grants will have no impact on the overall per capita federal revenue used for equalisation and will not induce vertical externalities as a result.

Appendix 6: Proof of Corollary 2

Plugging in (6), we can write (26) as

$$\begin{aligned} \frac{\partial {\hat{\omega }}}{\partial t}= & {} \left( \frac{\eta }{\eta -\varepsilon }\right) t^{*}L_{\mathrm{S}}^{'}w^{2}+\left( \frac{\eta }{\eta -\varepsilon }\right) \nonumber \\&\times \left\{ \left( 1-\frac{1}{n}\right) \left( t^{*}L_{\mathrm{S}}-\tau \frac{\lambda }{\varGamma _{G}}L_{\mathrm{D}}+T^{*}\frac{\varGamma _{G^{F}}}{\varGamma _{G}}\right. \right. \nonumber \\&\left. \times \left. \left[ L_{\mathrm{S}}^{'}\left( (1-\tau )-\frac{w}{w'}\right) w+L_{\mathrm{S}}\right] \right) \frac{w}{(1-\tau )}\right\} , \end{aligned}$$

(54)

where we have also used the federal budget constraint (12) evaluated in symmetric equilibrium, and the fact that

$$\begin{aligned} \sum _{i}^{n}\frac{\partial \left( L_{\mathrm{S}_{i}}w\right) }{\partial t_{i}}= & {} \left[ L_{\mathrm{S}}^{'}\left( (1-\tau )\frac{w'}{n}-w\right) w+L_{\mathrm{S}}\frac{w'}{n}\right] +(n-1)\nonumber \\&\times \left[ L_{\mathrm{S}}^{'}\left( (1-\tau )\frac{w'}{n}\right) w+L_{\mathrm{S}}\frac{w'}{n}\right] , \end{aligned}$$

(55)

or,

$$\begin{aligned} \sum _{i}^{n}\frac{\partial \left( L_{\mathrm{S}_{i}}w\right) }{\partial t_{i}}=L_{\mathrm{S}}^{'}\left( (1-\tau )w'-w\right) w+L_{\mathrm{S}}w'. \end{aligned}$$

(56)

Assuming now that $\lambda =\varGamma _{G}=\varGamma _{G^{F}}$, and using (25), then allows us to write (54) as

$$\begin{aligned} \frac{\partial {\hat{\omega }}}{\partial t}=\left( \frac{\eta }{\eta -\varepsilon }\right) \left\{ \frac{\partial \omega }{\partial t}+\left( \frac{n-1}{n}\right) T^{*}\left[ L_{\mathrm{S}}^{'}\left( (1-\tau )-\frac{w}{w'}\right) w\right] \frac{w}{(1-\tau )}\right\} . \end{aligned}$$

(57)

Since $nG^{F}=TnL_{\mathrm{S}}w$ and $G=tL_{\mathrm{S}}w$, (57) becomes

$$\begin{aligned} \frac{\partial {\hat{\omega }}}{\partial t}=\left( \frac{\eta }{\eta -\varepsilon }\right) \left\{ \frac{\partial \omega }{\partial t}+\left( \frac{n-1}{n}\right) \frac{t_{i}wG^{F}}{G}\left[ L_{\mathrm{S}}^{'}\left( (1-\tau )-\frac{w}{w'}\right) \right] \frac{w}{(1-\tau )}\right\} , \end{aligned}$$

(58)

which, after some further manipulation gives us

$$\begin{aligned} \frac{\partial {\hat{\omega }}}{\partial t}=\left( \frac{\eta }{\eta -\varepsilon }\right) \frac{\partial \omega }{\partial t}\left\{ 1+\frac{G^{F}}{G(1-\tau )}\left( (1-\tau )-\frac{w}{w'}\right) \right\} , \end{aligned}$$

(59)

so that

$$\begin{aligned} \frac{\partial {\hat{\omega }}}{\partial t}=z\left[ \frac{\partial \omega }{\partial t}\right] , \end{aligned}$$

(60)

where the factor z can be written as

$$\begin{aligned} z=\left( \frac{\eta }{\eta -\varepsilon }\right) \left\{ 1+\frac{G^{F}}{G(1-\tau )}\left( (1-\tau )-\frac{w}{w'}\right) \right\} . \end{aligned}$$

(61)

Hence, using (6), we arrive at

$$\begin{aligned} z=\left( \frac{\eta }{\eta -\varepsilon }\right) \left\{ 1+\frac{G}{G}^{F}-\frac{G}{G}^{F}\frac{\left( \eta -\varepsilon \right) }{\eta }\right\} , \end{aligned}$$

(62)

or,

$$\begin{aligned} z=\left( \frac{\eta }{\eta -\varepsilon }\right) \left\{ 1+\frac{G}{G}^{F}\frac{\varepsilon }{\eta }\right\} , \end{aligned}$$

(63)

which, as noted above, could only be derived under the condition that $\lambda =\varGamma _{G}=\varGamma _{G^{F}}$. But this condition never holds since taxation is distortionary, so that even in the unitary optimum where $\varGamma _{G^F}=\varGamma _{G}$, because $\frac{\varGamma _{G^F}}{\lambda }=\frac{\varGamma _{G}}{\lambda }$, we still have that $\varGamma _{G^{(F)}}>\lambda$ since $\frac{\varGamma _{G^F}}{\lambda }=\frac{\varGamma _{G}}{\lambda }=\frac{1}{\left( 1-\frac{\tau \eta \varepsilon }{(1-\tau )\varepsilon -\eta }\right) } >1$.$\square$

Appendix 7: Residence-based tax on mobile capital

Let $S_{i}(\rho _{i})$ be the savings of citizens in state i, invested at a net interest rate of $\rho _{i}=(1-\tau _{i})r$ with $\tau _{i}$ the residence-based tax on savings in state i, and $K_{i}(r)$ the capital applied to production in state i at the market-clearing interest rate $r(\varvec{\tau })$. The latter follows from the equilibrium on the capital market

$$\begin{aligned} \sum _{i}^{n}S_{i}(\rho _{i}(\tau _{i},\varvec{\tau }))=\sum _{i}^{n}K_{i}(r(\varvec{\tau })) \end{aligned}$$

(64)

which, because of residence-based taxation, implies that investors relocate their savings until capital earns the same gross returns across states. Utility in state i is then given by

$$\begin{aligned} U_{i}(C_{i1},C_{i2}{}_{i},G_{i},G_{i}^{F})=u_{i}(C_{i1},C_{i2})+\varGamma (G_{i},G_{i}^{F}), \end{aligned}$$

(65)

where, as in Keen and Kotsogiannis (2002, 2004), $C_{i1}$ defines consumption in period 1 as the difference between endowed income e and savings

$$\begin{aligned} C_{i1}=e-S_{i}, \end{aligned}$$

(66)

and $C_{i2}$ is equal to accrued savings in period 2 plus rents $\pi _{i}$ from state production

$$\begin{aligned} C_{i2}=(1+\rho _{i})S_{i}+\pi _{i}, \end{aligned}$$

(67)

with rents given by

$$\begin{aligned} \pi _{i}=f_{i}(K_{i})-rK_{i}, \end{aligned}$$

(68)

and $u_i(.)$ and g(.) having the same properties as in the main text. As in Brülhart and Jametti (2006), first period consumption could also be defined as foreign investment, at a normalised net interest rate of zero, and period 2 consumption as domestic investment. Consumers then maximise utility given (66), (67) and (68), so that savings $S(\rho _{i})$ are implicitly defined by $\frac{\partial u_{i}}{\partial C_{i2}}(.)(1+\rho )-\frac{\partial u_{i}}{\partial C_{i1}}=0$, and assumed increasing in $\rho _i$. Using the federal budget constraint (12) evaluated in symmetric equilibrium, indirect utility is then given by

$$\begin{aligned} W(t,T,\tau )= & {} v\left( (1-\tau )r(\tau ),\pi \left( r(\tau )\right) \right) \nonumber \\&+\varGamma \left( trS((1-\tau )r(\tau )),\frac{T}{n}\sum _{i}^{n}r(\tau )K(r(\tau ))\right) \end{aligned}$$

(69)

which is the exact equivalent of (41), but here in the context of capital taxation. Totally differentiating (69) with respect to t then yields

$$\begin{aligned} \frac{\partial W}{\partial t}= & {} \lambda \left[ S\left( (1\!-\!\tau )\frac{r'}{n}-r\right) \!-\! K\frac{r'}{n}\right] \!+\!\varGamma _{G}\left[ Sr+tS'\left( (1-\tau )\frac{r'}{n}-r\right) r+tS\frac{r'}{n}\right] \nonumber \\&+T\varGamma _{G^{F}}\left[ rK'\frac{r'}{n}+K\frac{r'}{n}\right] , \end{aligned}$$

(70)

with $\frac{\partial W}{\partial t}=0$ characterising the equilibrium tax rate $t^{*}$. Differentiating (69) with respect to a common tax rate $t=t_{c}$ as before, we get

$$\begin{aligned} \frac{\partial W}{\partial t_{c}}= & {} \lambda \left[ S\left( (1-\tau )r'-r\right) -Kr'\right] +\varGamma _{G}\left[ Sr+tS'\left( (1-\tau )r'-r\right) r+tSr'\right] \nonumber \\&+ T\varGamma _{G^{F}}\left[ rK'r'+Kr'\right] .\end{aligned}$$

(71)

In order to filter out the externalities, we subtract (70) set to zero from (71) as in Sect. 4 with $\frac{\partial W}{\partial t_{c}}|_{t_{c}=t\text {*}}$, to obtain

$$\begin{aligned} \frac{\partial W}{\partial t_{c}}-\frac{\partial W}{\partial t}=\left[ -\lambda \tau S+t^{*}\varGamma _{G}\left( S'(1-\tau )+S\right) +T^{*}\varGamma _{G^{F}}\left( rK'+K\right) \right] \left( 1-\frac{1}{n}\right) r', \end{aligned}$$

(72)

which, since $r'>0$, gives rise to exactly the same externalities as in Corollary 2. What this implies is that we can always switch from a capital to a labour income setting, without loss of generality regarding the deduced externalities.

Appendix 8: Source-based tax on mobile capital

Let $K_{i}(r_{i})$ now be the capital applied to production in state i at a gross rate of return $r_{i}$, and $S_{i}(\rho )$ the savings of citizens in state i invested at a market-clearing net interest rate of $\rho =(1-\tau _{i})r_{i}$, with $\tau _{i}$ the source-based tax on capital applied in state i. The net return $\rho (\varvec{\tau })$ to investment follows from the equilibrium on the capital market defined by

$$\begin{aligned} \sum _{i}^{n}S_{i}(\rho (\varvec{\tau }))=\sum _{i}^{n}K_{i}(r_i(\tau _{i},\varvec{\tau })), \end{aligned}$$

(73)

which, because of source-based taxation, implies that investors relocate their savings until capital earns the same net returns across states. Taking the total differential with respect to $\tau _{i}$ of (73) yields

$$\begin{aligned} \sum _{j\ne i}^{n}\left( \frac{\partial \left( K_{j}(r_{j})\right) }{\partial r_{j}}\frac{\partial r_{j}}{\partial \tau _{i}}\right) +\frac{\partial K_{i}(r_{i})}{\partial r_{i}}\frac{\partial r_{i}}{\partial \tau _{i}}=\sum _{i}^{n}\left( \frac{\partial \left( S_{i}(\rho )\right) }{\partial \rho }\right) \frac{\partial \rho }{\partial \tau _{i}}. \end{aligned}$$

(74)

Rewriting net wages in terms of the gross wage and solving further then gives us

$$\begin{aligned} \sum _{j\ne i}^{n}\left( K'_{j}\rho _{t}\left( \!1\!-\!\tau _{i}\!\right) ^{-1}\right) d\tau _{i}\!+\!K'_{i}\left( \rho _{t}\left( 1\!-\!\tau _{i}\right) ^{-1}\!+\!\rho \left( 1\!-\! \tau _{i}\right) ^{-2}\right) d\tau _{i} \!=\!\left( \!\sum _{i}^{n}S'_{i}(\rho )\!\right) d\rho ,\nonumber \\ \end{aligned}$$

(75)

so that, factoring out and bringing over, we arrive at

$$\begin{aligned} \frac{\partial \rho }{\partial \tau _{i}}=\frac{K'_{i}\rho \left( 1-\tau _{i}\right) ^{-2}}{\left( \sum _{i}^{n}S'_{i}(\rho )\right) -\sum _{i}^{n}\left( K'_{i}\left( 1-\tau _{i}\right) ^{-1}\right) }, \end{aligned}$$

(76)

which, given symmetry and since $\frac{\partial \tau _{i}}{\partial t_{i}}=1$, becomes

$$\begin{aligned} \frac{\partial \rho }{\partial \tau _{i}}=\frac{\partial \rho }{\partial t}=\frac{\rho \varepsilon }{n\left( 1-\tau \right) \left( \eta -\varepsilon \right) }<0, \end{aligned}$$

(77)

with $\eta =S'\frac{\rho }{S}>0$ the savings elasticity and $\varepsilon =K'\frac{r}{K}<0$ the capital demand elasticity in state i in the symmetric equilibrium. Totally differentiating (73) with respect to the common tax rate T subsequently, and again given symmetry, we find that

$$\begin{aligned} \frac{\partial \rho }{\partial T}\equiv \rho '=n\frac{\partial \rho }{\partial t}, \end{aligned}$$

(78)

which also defines

$$\begin{aligned} r_{t,\tau }=\rho _{t,\tau }\left( 1-\tau \right) ^{-1}+\rho \left( 1-\tau _{i}\right) ^{-2}>0, \end{aligned}$$

(79)

and, $\forall j \ne i$,

$$\begin{aligned} \frac{\partial r_j}{\partial t_i}=\rho _{t}\left( 1-\tau \right) ^{-1}<0. \end{aligned}$$

(80)

Utility in state i is then given by

$$\begin{aligned} U_{i}(C_{i1},C_{i2}{}_{i},G_{i},G_{i}^{F})=u_{i}(C_{i1},C_{i2})+\varGamma (G_{i},G_{i}^{F}), \end{aligned}$$

(81)

of which the arguments are the same as in Appendix 7. Consumers maximise utility given (66), (67) and (68), so that savings $S(\rho )$ are implicitly defined by $\frac{\partial u_{i}}{\partial C_{i2}}(.)(1+\rho )-\frac{\partial u_{i}}{\partial C_{i1}}=0$. Using the federal budget constraint (12) evaluated in symmetric equilibrium, indirect utility is now given by

$$\begin{aligned} W(t,T,\tau )= & {} v\left( (1-\tau )r(\tau ),\pi \left( r(\tau )\right) \right) \nonumber \\&+\varGamma \left( trK(\rho (\tau )(1-\tau _{i})^{-1}),\frac{T}{n}\sum _{i}^{n}r(\tau )S(\rho (\tau ))\right) , \end{aligned}$$

(82)

which is the source-based counterpart of (69). Now, since $\frac{\partial r_i}{\partial t_i}\ne \frac{\partial r_j}{\partial t_i}$, and using (79) and (80) as a result, we know that

$$\begin{aligned} \frac{\partial \left( \sum _{i}^{n}rS\right) }{\partial t}= & {} \left[ rS'\left( \frac{\rho '}{n}\right) +S\left( \frac{\rho '}{n\left( 1-\tau _{i}\right) }+\frac{\rho }{\left( 1-\tau _{i}\right) ^{2}}\right) \right] \nonumber \\&+(n-1)\left[ rS'\frac{\rho '}{n}+S\frac{\rho '}{n\left( 1-\tau _{i}\right) }\right] ,\end{aligned}$$

(83)

or,

$$\begin{aligned} \frac{\partial \left( \sum _{i}^{n}rS\right) }{\partial t}=rS'\left( \rho '\right) +S\left( \frac{\rho '}{\left( 1-\tau \right) }+\frac{\rho }{\left( 1-\tau \right) ^{2}}\right) , \end{aligned}$$

(84)

so that, totally differentiating (82) with respect to t, we obtain

$$\begin{aligned} \frac{\partial W}{\partial t}= & {} \lambda \left[ S\frac{\rho '}{n}-K\left( \frac{\rho '}{n\left( 1-\tau \right) } +\frac{\rho }{\left( 1-\tau \right) ^{2}}\right) \right] \nonumber \\&+\varGamma _{G}\left[ Kr+t\left( K'r+K\right) \left( \frac{\rho '}{n\left( 1-\tau \right) }+\frac{\rho }{\left( 1-\tau \right) ^{2}}\right) \right] \nonumber \\&+T\varGamma _{G^{F}}\left[ S'\left( \frac{\rho '}{n}\right) r+S\left( \frac{\rho '}{n\left( 1-\tau \right) }+\frac{\rho }{n\left( 1-\tau \right) ^{2}}\right) \right] , \end{aligned}$$

(85)

with $\frac{\partial W}{\partial t}=0$ characterising the equilibrium tax rate $t^{*}$. Differentiating (82) with respect to a common tax rate $t=t_{c}$ subsequently, we get

$$\begin{aligned} \frac{\partial W}{\partial t_{c}}= & {} \lambda \left[ S\rho '-K\left( \frac{\rho '}{\left( 1-\tau \right) } +\frac{\rho }{\left( 1-\tau \right) ^{2}}\right) \right] \nonumber \\&+\varGamma _{G}\left[ Kr+t\left( K'r+K\right) \left( \frac{\rho '}{\left( 1-\tau \right) }+\frac{\rho }{\left( 1-\tau \right) ^{2}}\right) \right] \nonumber \\&+T\varGamma _{G^{F}}\left[ S'\rho 'r+S\left( \frac{\rho '}{\left( 1-\tau \right) }+\frac{\rho }{\left( 1-\tau \right) ^{2}}\right) \right] . \end{aligned}$$

(86)

In order to filter out the externalities, we subtract (85) set to zero from (86) as in Sect. 4, with $\frac{\partial W}{\partial t_{c}}|_{t_{c}=t\text {*}}$, to obtain

$$\begin{aligned} \frac{\partial W}{\partial t}= & {} \left\{ -\tau \lambda S\!+\!t^{*}\varGamma _{G}\left[ rK'\!+\!K\right] \!+\!T^{*}\varGamma _{G^{F}}\left[ rS'\left( 1-\tau \right) +S\left( 1+\frac{\rho }{\rho '\left( 1-\tau \right) }\right) \right] \right\} \nonumber \\&\frac{\left( 1-\frac{1}{n}\right) \rho '}{\left( 1-\tau \right) }, \end{aligned}$$

(87)

or, using (77),

$$\begin{aligned} \frac{\partial W}{\partial t}= & {} \left\{ -\tau \lambda S+t^{*} \varGamma _{G}\left[ rK'\!+\!K\right] \!+\!T^{*}\varGamma _{G^{F}}\left[ rS'\left( 1-\tau \right) +S\frac{\eta }{\varepsilon }\right] \right\} \nonumber \\&\times \left( 1-\frac{1}{n}\right) \frac{\rho '}{\left( 1-\tau \right) }, \end{aligned}$$

(88)

We thus find that in the case of source-based taxation, both the vertical and horizontal externalities still have an ambiguous sign, whilst the horizontal externality is composed differently. The profit effects will now have a positive effect on non-resident welfare, as $\frac{\partial \rho }{\partial t}<0$ pulls down the rate of return $r_{j\ne i}$ to be paid by firms in other states. Inversely, and again because $\frac{\partial \rho }{\partial t}<0$, the effect on tax revenues collected in other states will now also have a negative side.

Appendix 9: Equalisation and source-based taxation: Corollary 4

To compare equalisation in the source-based scenario derived in Appendix 8 with the residence-based setting, we consider the same equalisation mechanism (22) we had before

$$\begin{aligned} \omega _{i}={\bar{t}}({\bar{B}}-B_{i}), \end{aligned}$$

(89)

where now ${\bar{t}}=\frac{\sum _{i}^{n}t_{i}K_{i}r_{i}}{\sum _{i}^{n}K_{i}r_{i}}$ is the average state government tax rate. $B_{i}$ is the fiscal capacity of state i, captured by its tax base $K_{i}r_{i}$, whilst ${\bar{B}}=\frac{\sum _{i}^{n}K_{i}r_{i}}{n}$ is the benchmark fiscal capacity, being the average federation-wide tax base. Deriving the equalisation grant w.r.t. $t_{i}$ gives us

$$\begin{aligned} \frac{\partial \omega _{i}}{\partial t_{i}}=\frac{\partial {\bar{t}}}{\partial t_{i}}({\bar{B}}-B_{i})+{\bar{t}}\left( \frac{\partial {\bar{B}}}{\partial t_{i}}-\frac{\partial B_{i}}{\partial t_{i}}\right) . \end{aligned}$$

(90)

Similar to the symmetric residence-based setup, we again have that a tax hike in state i only influences the equalisation grant through a change in the actual fiscal capacities, as ${\bar{B}}=B_{i}$ in symmetric equilibrium, so that

$$\begin{aligned} \frac{\partial \omega }{\partial t}=t\left[ \frac{1}{n}\frac{\partial \left( \sum _{i}^{n}K_{i}r\right) }{\partial t_{i}}-\frac{\partial (K_{i}r)}{\partial t_{i}}\right] , \end{aligned}$$

(91)

or,

$$\begin{aligned} \frac{\partial \omega }{\partial t}=t\left[ \frac{1}{n}\left( \frac{\partial (K_{i}r)}{\partial t_{i}}+\frac{\partial \left( \sum _{j\ne i}^{n}K_{j}r\right) }{\partial t_{i}}\right) -\frac{\partial (K_{i}r)}{\partial t_{i}})\right] , \end{aligned}$$

(92)

from which, using (79) and (80) and reworking, we obtain

$$\begin{aligned} \frac{\partial \omega }{\partial t}= & {} t\left[ \frac{1}{n}\left( \left[ \frac{rK'\rho }{\left( 1-\tau \right) ^{2}}+\frac{K\rho }{\left( 1-\tau \right) ^{2}}\right] +n\left[ \frac{rK'\rho '}{n\left( 1-\tau \right) }+\frac{K\rho '}{n\left( 1-\tau \right) }\right] \right) \right] \nonumber \\&-t\left[ rK'\left( \frac{\rho '}{n\left( 1-\tau \right) }+\frac{\rho }{\left( 1-\tau \right) ^{2}}\right) +K\left( \frac{\rho '}{n\left( 1-\tau \right) }+\frac{\rho }{\left( 1-\tau \right) ^{2}}\right) \right] .\qquad \end{aligned}$$

(93)

Solving further then yields

$$\begin{aligned} \frac{\partial \omega }{\partial t}=t\left( \left( \frac{1}{n}-1\right) \frac{rK'\rho }{\left( 1-\tau \right) ^{2}}+\left( \frac{1}{n}-1\right) \frac{K\rho }{\left( 1-\tau \right) ^{2}}\right) , \end{aligned}$$

(94)

and finally, since $\rho =(1-\tau )r$,

$$\begin{aligned} \frac{\partial \omega }{\partial t}=-t\left( \frac{n-1}{n}\right) \left( \frac{rK'r+Kr}{\left( 1-\tau \right) ^{2}}\right) \gtreqless 0 , \end{aligned}$$

(95)

or, in terms of a general production factor P, with $P_D$ the supply of the taxed production factor per state, and p the market price,

$$\begin{aligned} \frac{\partial \omega }{\partial t}=-t\left( \frac{n-1}{n}\right) \left( \frac{P_D'p^2+P_D}{\left( 1-\tau \right) ^{2}}\right) \gtreqless 0, \end{aligned}$$

(96)

which returns as (31) in Corollary 4. $\square$

Appendix 10: Proof of Proposition 3

To derive the conditions for an efficiency-enhancing equalisation grant, we introduce the marginal equalisation effects to (88) following the same approach set out in Appendix 4, so that now

$$\begin{aligned} \frac{\partial W}{\partial t}= & {} \left\{ -\tau \lambda S+t^{*}\varGamma _{G}\left[ rK'+K-\frac{\partial {\hat{\omega }}}{\partial t}\right] +T^{*}\varGamma _{G^{F}}\left[ rS'\left( 1-\tau \right) +S\frac{\eta }{\varepsilon }\right] \right\} \nonumber \\&\left( 1-\frac{1}{n}\right) \frac{\rho '}{\left( 1-\tau \right) }. \end{aligned}$$

(97)

Setting (97) to zero and solving for $\frac{\partial {\hat{\omega }}_{i}}{\partial t_{i}}$ yields

$$\begin{aligned} \frac{\partial {\hat{\omega }}}{\partial t}= & {} \frac{1}{\varGamma _{G}}\left\{ -\tau \lambda S+t^{*}\varGamma _{G}\left[ rK'+K\right] +T^{*}\varGamma _{G^{F}}\left[ rS'\left( 1-\tau \right) +S\frac{\eta }{\varepsilon }\right] \right\} \nonumber \\&\times \left( 1-\frac{1}{n}\right) \frac{\rho '}{\left( 1-\tau \right) }, \end{aligned}$$

(98)

which, plugged in, reduces (85) to (86) set to zero, ensuring second-best efficiency. Using (77) and rewriting, we then get

$$\begin{aligned} \frac{\partial {\hat{\omega }}}{\partial t}= & {} \left( \frac{\varepsilon }{\eta -\varepsilon }\right) \left( \frac{n-1}{n}\right) \left\{ -\frac{\lambda }{\varGamma _{G}}\tau S+t\left[ rK'+K\right] \right. \nonumber \\&\left. +T^{*}\frac{\varGamma _{G^{F}}}{\varGamma _{G}}\left[ rS'\left( 1-\tau \right) +S\frac{\eta }{\varepsilon }\right] \frac{r}{\left( 1-\tau \right) }\right\} , \end{aligned}$$

(99)

from which, introducing the marginal equalisation effect of the conventional grant given by (95), we obtain

$$\begin{aligned} \frac{\partial {\hat{\omega }}}{\partial t}=-\left( \frac{\varepsilon }{\eta -\varepsilon }\right) \left\{ \frac{\partial \omega _{i}}{\partial t_{i}}+\left( \frac{n-1}{n}\right) \left( \frac{\lambda }{\varGamma _{G}}\tau S-T^{*}\frac{\varGamma _{G^{F}}}{\varGamma _{G}}\left[ S\eta +S\frac{\eta }{\varepsilon }\right] \right) \frac{r}{\left( 1-\tau \right) }\right\} . \end{aligned}$$

(100)

Rewriting (100) in terms of a production factor P with p the market-clearing price, and simplifying further, we finally arrive at

$$\begin{aligned} \frac{\partial {\hat{\omega }}}{\partial t}=\left( \frac{\varepsilon }{\varepsilon -\eta }\right) \left\{ \frac{\partial \omega _{i}}{\partial t_{i}}+\left( \frac{n-1}{n}\right) \left[ \frac{\lambda }{\varGamma _{G}}\tau P_{s}-T^{*}\frac{\varGamma _{G^{F}}}{\varGamma _{G}}\frac{\eta }{\varepsilon }\left( 1+\varepsilon \right) P_{s}\right] \frac{p}{\left( 1-\tau \right) }\right\} , \end{aligned}$$

(101)

as expressed by (30) in Proposition 3. To compare (101) with the residence-based case, we start from expression (26) of Proposition 2 where

$$\begin{aligned} \frac{\partial {\hat{\omega }}}{\partial t}= & {} \frac{1}{\varGamma _{G}}\left\{ \left[ -\tau \lambda L_{\mathrm{D}}+t^{*}\varGamma _{G}\left( L_{\mathrm{S}}^{'}(1-\tau )w+L_{\mathrm{S}}\right) \right] +T^{*}\varGamma _{G^{F}}\left[ L_{\mathrm{D}}^{'}w+L_{\mathrm{D}}\right] \right\} \nonumber \\&\left( 1-\frac{1}{n}\right) w'. \end{aligned}$$

(102)

Since we know from (4) that $\sum _{i}^{n}\frac{\partial \left( L_{\mathrm{D}_{i}}w\right) }{\partial t_{i}}=\sum _{i}^{n}\frac{\partial \left( L_{\mathrm{S}_{i}}w\right) }{\partial t_{i}}$, and $\sum _{i}^{n}\frac{\partial \left( L_{\mathrm{D}}w\right) }{\partial t}=n(L_{\mathrm{D}}\frac{w'}{n}+L'_{D}\frac{w'}{n}w)=L_{\mathrm{D}}w'+L'_{D}w'w$ in symmetric equilibrium, we also have that

$$\begin{aligned} \left[ L_{\mathrm{D}}^{'}w+L_{\mathrm{D}}\right] w'=L_{\mathrm{S}}^{'}\left( (1-\tau )w'-w\right) w+L_{\mathrm{S}}w', \end{aligned}$$

(103)

where $\sum _{i}^{n}\frac{\partial \left( L_{\mathrm{S}_{i}}w\right) }{\partial t_{i}}=L_{\mathrm{S}}^{'}\left( (1-\tau )w'-w\right) w+L_{\mathrm{S}}w'$ follows from (56). Applying this latter equivalence to the right-hand side of (102), we obtain

$$\begin{aligned} \frac{\partial {\hat{\omega }}}{\partial t}= & {} \frac{1}{\varGamma _{G}}\left\{ -\tau \lambda L_{\mathrm{D}}+t^{*}\varGamma _{G}\left( L_{\mathrm{S}}^{'}(1-\tau )w+L_{\mathrm{S}}\right) \right. \nonumber \\&\left. +T^{*}\varGamma _{G^{F}}\left[ L_{\mathrm{S}}^{'}\left( (1-\tau )-\frac{w}{w'}\right) w+L_{\mathrm{S}}\right] \right\} \left( 1-\frac{1}{n}\right) w', \end{aligned}$$

(104)

which, using (6), can be rewritten as

$$\begin{aligned} \frac{\partial {\hat{\omega }}}{\partial t}= & {} \frac{\eta }{\left( \eta -\varepsilon \right) }\left\{ -\tau \frac{\lambda }{\varGamma _{G}}L_{\mathrm{S}}+t^{*}\left( L_{\mathrm{S}}^{'}(1-\tau )w\right) +t^{*}L_{\mathrm{S}} \right. \nonumber \\&\left. +T^{*}\frac{\varGamma _{G^{F}}}{\varGamma _{G}}\left[ L_{\mathrm{S}}^{'}\left( (1-\tau )-\frac{w}{w'}\right) w+L_{\mathrm{S}}\right] \right\} \frac{\left( 1-\frac{1}{n}\right) w}{(1-\tau )}, \end{aligned}$$

(105)

or, since $\frac{\partial \omega _{i}}{\partial t_{i}}=t_{i}\left( \frac{n-1}{n}\right) L_{\mathrm{S}}^{'}w^{2}$, as

$$\begin{aligned} \frac{\partial {\hat{\omega }}}{\partial t}= & {} \left( \frac{\eta }{\eta -\varepsilon }\right) \left\{ \frac{\partial \omega _{i}}{\partial t_{i}}+\left( \frac{n-1}{n}\right) \left( t^{*}L_{\mathrm{S}}-\tau \frac{\lambda }{\varGamma _{G}}L_{\mathrm{D}}\right. \right. \nonumber \\&\left. \left. +T^{*}\frac{\varGamma _{G^{F}}}{\varGamma _{G}}\left[ L_{\mathrm{S}}^{'}\left( (1-\tau )-\frac{w}{w'}\right) w+L_{\mathrm{S}}\right] \right) \frac{w}{(1-\tau )}\right\} . \end{aligned}$$

(106)

which, rewritten in terms of a production factor P with market price p, yields

$$\begin{aligned} \frac{\partial {\hat{\omega }}}{\partial t}= & {} \left( \frac{\eta }{\eta -\varepsilon }\right) \left\{ \frac{\partial \omega _{i}}{\partial t_{i}}+\left( \frac{n-1}{n}\right) \left[ \left( t^{*}-\tau \frac{\lambda }{\varGamma _{G}}\right) P_{s}\right. \right. \nonumber \\&\left. \left. +T^{*}\frac{\varGamma _{G^{F}}}{\varGamma _{G}}P_{s}\left( 1+\varepsilon \right) \right] \frac{p}{(1-\tau )}\right\} . \end{aligned}$$

(107)

which accounts for (29) in Proposition 3. $\square$