Self-enforcing capital tax coordination

Abstract

Capital tax competition is known to result in inefficiently low tax rates and an undersupply of public goods. The provision of public goods and with it the welfare of all countries can be enhanced via tax coordination. Based on the standard Zodrow-Mieszkowski-Wilson tax-competition model this paper analyzes the conditions under which tax coordination by a group of countries is self-enforcing. In our analytical framework, there always exists a rather small stable tax coalition. For some subset of the parameter space the grand coalition is stable, even if the total number of countries is large. If the stable coalition is small, it is not very effective in mitigating the inefficiency of the non-cooperative Nash equilibrium. The ineffectiveness is increasing in the total number of countries.

Appendices

Appendix

1.1 Appendix A: Proof of (13) and (14)

The coalition’s first-order condition is given by

$$\begin{aligned} (1+\varepsilon ) t_c \frac{\mathop {\partial }k_c}{\mathop {\partial }t_c} + (\bar{k} - k_c) \frac{\mathop {\partial }r_c}{\mathop {\partial }t_c} + \varepsilon k_c =0, \end{aligned}$$

where $r_c := a - b \bar{k} - \Theta _c$, $k_c := \bar{k} - \frac{t_c}{b} + \frac{\Theta _c}{b}$ and $\Theta _c := \frac{m t_c + \sum _{j \in F} t_j}{n}$ or equivalently

$$\begin{aligned} -m (t_c -\Theta _c) -(n-m) (1+\varepsilon ) t_c + \varepsilon (n b \bar{k} - n t_c + n \Theta _c) =0.\end{aligned}$$

(25)

The first-order condition of the fringe country (8) can be written as

$$\begin{aligned} (1+\varepsilon ) t_f \frac{\mathop {\partial }k_f}{\mathop {\partial }t_f} + (\bar{k} - k_f) \frac{\mathop {\partial }r_f}{\mathop {\partial }t_f} + \varepsilon k_f =0, \end{aligned}$$

(26)

where $r_f := a - b \bar{k} - \Theta _f$, $k_f := \bar{k} - \frac{t_f}{b} + \frac{\Theta _f}{b}$ and $\Theta _f := \frac{t_f + \sum _{j \ne f} t_j}{n}$. Accounting for $r_f$, $k_f$ and $\Theta _f$ turns (26) into

$$\begin{aligned} -(t_f -\Theta _f) -(n-1) (1+\varepsilon ) t_f + \varepsilon (n b \bar{k} - n t_f + n \Theta _f) =0. \end{aligned}$$

(27)

Since all fringe countries are alike, a necessary condition for a symmetric equilibrium is $t_i=t_f$ for all $i \in F$ and thus

$$\begin{aligned} \Theta _c = \Theta _f = \frac{m t_c + (n-m) t_f}{n} =: \Theta . \end{aligned}$$

(28)

Solving (25) and (27) with respect to $t_c$ and $t_f$ yields

$$\begin{aligned} t_c^*= & {} \frac{\{m (n+1) - m^2 + n [n(1+2 \varepsilon )-1-\varepsilon ]\}\varepsilon \bar{k}b}{(n-m) (1+\varepsilon ) [m(1+\varepsilon ) + (n-1) (1+2 \varepsilon )]}>0, \end{aligned}$$

(29)

$$\begin{aligned} t_f^*= & {} \frac{[n^2(1+2 \varepsilon )-m^2-m(n\varepsilon - 1)] \varepsilon \bar{k} b}{(n-m) (1 + \varepsilon ) [m(1+\varepsilon ) + (n-1) (1+2 \varepsilon )]}>0. \end{aligned}$$

(30)

Substracting $W^c(t_c, t_f, m)$ from $W^f(t_c, t_f, m)$ yields

$$\begin{aligned} W^f(t_c, t_f, m) - W^c(t_c, t_f, m)= \frac{(t_c-t_f)\left[ 2 m \varepsilon (t_f-t_c) + n (t_c + t_f + 2 t_c \varepsilon - 2 b \varepsilon \bar{k}) \right] }{2bn}. \end{aligned}$$

(31)

Next, we insert $t_c^*$ and $t_f^*$ from (29) and (30) in (31) to obtain after some rearrangement of terms

$$\begin{aligned} W^f(t_c^*, t_f^*, m) - W^c(t_c^*, t_f^*, m)= \frac{(m-1) n b \varepsilon ^2 \bar{k}^2 \left[ (m+1) n (1+ 2 \varepsilon ) - 2m \varepsilon \right] }{2(n-m)^2 \left[ m(1+\varepsilon ) + (n-1) (1+2\varepsilon ) \right] ^2}>0. \end{aligned}$$

(32)

Substracting $W^c(t_o, t_o, m) = \left( a - \frac{b \bar{k}}{2} + t_o \varepsilon \right) \bar{k}$ from $W^c(t_c, t_f, m)$ yields

$$\begin{aligned}&W^c(t_c, t_f, m) - W^c(t_o, t_o, m) \nonumber \\&\quad = \frac{m^2 (t_c - t_f)^2 + 2 mn (t_c-t_f) (t_f + \varepsilon t_c) + n^2 \left[ t_f^2 - t_c^2 (1+2 \varepsilon ) - 2 b t_o \varepsilon \bar{k} + 2 t_c \varepsilon (t_f + b \bar{k}) \right] }{2b n^2}. \end{aligned}$$

(33)

Next, we insert $t_c^*$ and $t_f^*$ from (29) and (30) and $t_o = \frac{n \varepsilon \bar{k} b}{(n-1) (1+ \varepsilon )}$ in (33) to obtain after some rearrangement of terms

$$\begin{aligned}&W^c(t_c^*, t_f^*, m) - W^c(t_o, t_o, m) \nonumber \\&\quad = \frac{m^2 (n+1) +(n-1) (nm+m-n) + \left[ m^2 (3n+1) +(n-1)(5mn-3n+3n^2) \right] \varepsilon }{2 (n-m) (n-1) (1 + \varepsilon ) \left[ m (1 + \varepsilon ) + (n-1) (1 + 2 \varepsilon )\right] ^2} \nonumber \\&\quad + \frac{ \left[ 2 m^2 n +(n-1) (6mn -2n)\right] \varepsilon ^2}{2 (n-m) (n-1) (1 + \varepsilon ) \left[ m (1 + \varepsilon ) + (n-1) (1 + 2 \varepsilon )\right] ^2} >0. \end{aligned}$$

(34)

(32) and (34) establishes $W^f(t_c^*, t_f^*, m)> W^c(t_c^*, t_f^*, m) > W^c(t_o, t_o, m) \equiv w_o$. $\square$

1.2 Appendix B: Proof of (17)

Making use of (29), (30), (1), (4) and $k_o= \bar{k}$ in $X(k) +(\bar{k}-k) r$ we get after rearrangement of terms

$$\begin{aligned}&X(k_f^*) +(\bar{k}-k^*_f) r^* - \left[ X(k_o) + \underbrace{(\bar{k}-k_o)}_{=0} r_o \right] \\&= \frac{\left[ 2 n^2 - m^2 +m + (4n^2+m^2 -m-2m n) \varepsilon \right] b (m-1) m \varepsilon ^2 \bar{k}^2}{2 (n-m)^2 (1+\varepsilon ) \left[ m(1+ \varepsilon ) +(n-1) (1+2 \varepsilon ) \right] ^2}>0, \end{aligned}$$

(35)

$$\begin{aligned}&X(k_c^*) +(\bar{k}-k^*_c) r^* - \left[ X(k_o) + \underbrace{(\bar{k}-k_o)}_{=0} r_o\right] \\&= \frac{\left[ m^2+n - 2 n^2 -m(n+1) +(n+m+nm- 4n^2-m^2)\varepsilon \right] b (m-1) \varepsilon ^2 \bar{k}^2 }{2 (n-m) (1+\varepsilon ) \left[ m(1+ \varepsilon ) +(n-1) (1+2 \varepsilon ) \right] ^2}<0. \end{aligned}$$

(36)

1.3 Appendix C: Proof of Proposition 1

Differentiation of $t_c^* = \mathcal{T}^c(m)$ and $t_f^* = \mathcal{T}^f(m)$ from (29) and (30) with respect to m yields

$$\begin{aligned} \frac{{\mathrm {d}}\mathcal{T}^c(m)}{{\mathrm {d}}m}= \mathcal{T}_m^c= & {} \frac{\left[ n + \varepsilon (2n - 1)\right] \left[ m^2 + n(n-1) + n \varepsilon (n + 2m -2) \right] b \varepsilon \bar{k}}{(n-m)^2 (1+ \varepsilon ) \left[ m (1+ \varepsilon ) + (n-1)(1+2\varepsilon ) \right] ^2}>0, \end{aligned}$$

(37)

$$\begin{aligned} \frac{{\mathrm {d}}\mathcal{T}^f(m)}{{\mathrm {d}}m} = \mathcal{T}_m^f= & {} \frac{(n \varepsilon +1) \left[ (2m-1) n + (2n (2m-1) -m^2) \varepsilon \right] b \varepsilon \bar{k} }{(n-m)^2 (1+ \varepsilon ) \left[ m (1+ \varepsilon ) + (n-1)(1+2\varepsilon ) \right] ^2}>0. \end{aligned}$$

(38)

Define the welfare functions

$$\begin{aligned} W^c(t_c, t_f, m):= & {} X \left( \bar{k} - \frac{t_c}{b} + \frac{\Theta }{b} \right) - \left( a - \bar{k} b -\Theta + t_c \right) \cdot \left( \bar{k} - \frac{t_c}{b} + \frac{\Theta }{b} \right) + \left( a - \bar{k} b - \Theta \right) \bar{k} \\&+ (1+\varepsilon ) t_c \left( \bar{k} - \frac{t_c}{b} + \frac{\Theta }{ b} \right) \\= & {} \frac{ m^2 (t_c-t_f)^2 + 2mn (t_c-t_f) (t_f + \varepsilon t_c) }{2 b n^2} \\&+ \frac{n^2 \left[ t_f^2 - t_c^2 (1+ 2 \varepsilon ) +(2 \alpha - b \bar{k})\, b \bar{k} + 2 t_c \varepsilon (t_f + b \bar{k}) \right] }{2bn^2}, \end{aligned}$$

(39)

$$\begin{aligned} W^f(t_c, t_f, m):= & {} X \left( \bar{k} - \frac{t_f}{b} + \frac{\Theta }{b} \right) - \left( a - \bar{k} b -\Theta + t_f \right) \cdot \left( \bar{k} - \frac{t_f}{b} + \frac{\Theta }{b} \right) + \left( a - \bar{k} b - \Theta \right) \bar{k} \nonumber \\&+ (1+\varepsilon ) t_c \left( \bar{k} - \frac{t_f}{b} + \frac{\Theta }{ b} \right) \nonumber \\= & {} \frac{m^2 (t_c - t_f)^2 + 2 m n (t_c - t_f) t_f (1 + \varepsilon ) + b n^2 \bar{k} (2 \alpha + 2 t_f \varepsilon - b \bar{k})}{2bn^2}. \end{aligned}$$

(40)

Total differentiation of $\mathcal{W}^c(m) := W^c(\mathcal{T}^c(m), \mathcal{T}^f(m), m)$ and $\mathcal{W}^f(m) := W^f(\mathcal{T}^c(m), \mathcal{T}^f(m), m)$ yields

$$\begin{aligned} \frac{{\mathrm {d}}\mathcal{W}^c(m)}{{\mathrm {d}}m}= & {} \underbrace{W^c_{t_c}}_{=0} \cdot \mathcal{T}^c_m + W^c_{t_f} \cdot \mathcal{T}^f_m + W^c_m, \end{aligned}$$

(41)

$$\begin{aligned} \frac{{\mathrm {d}}\mathcal{W}^f(m)}{{\mathrm {d}}m}= & {} W^f_{t_c} \cdot \mathcal{T}^c_m + W^f_{t_f} \cdot \mathcal{T}^f_m + W^f_m, \end{aligned}$$

(42)

where

$$\begin{aligned} W^c_{t_f}= & {} \frac{(n-m) \left[ m(t_c-t_f) + n(t_f +t_c \varepsilon )\right] }{bn^2}, \end{aligned}$$

(43)

$$\begin{aligned} W^c_m= & {} \frac{(t_c-t_f) \left[ m(t_c-t_f) + n(t_f + t_c \varepsilon ) \right] }{bn^2}, \end{aligned}$$

(44)

$$\begin{aligned} W^f_{t_c}= & {} \frac{m \left[ m (t_c-t_f) +n t_f(1+\varepsilon ) \right] }{b n^2}, \end{aligned}$$

(45)

$$\begin{aligned} W^f_{t_f}= & {} \frac{m^2 (t_f-t_c) + m n (t_c -2 t_f) (1+ \varepsilon ) + b n^2 \varepsilon \bar{k}}{b n^2}, \end{aligned}$$

(46)

$$\begin{aligned} W^f_m= & {} \frac{(t_c-t_f) \left[ m(t_c-t_f) + n t_f (1+\varepsilon ) \right] }{bn^2}. \end{aligned}$$

(47)

Finally, we insert $t_c^* =\mathcal{T}^c(m)$ and $t_f^*=\mathcal{T}^f(m)$ from (29) and (30) in (46) to obtain after rearrangement of terms

$$\begin{aligned} W^f_{t_f}= & {} \frac{(n-m-1) \left[ n + \varepsilon (2n-m) \right] \bar{k}}{(n-m) \left[ m(1+ \varepsilon ) + (n-1) (1+2\varepsilon ) \right] }>0. \end{aligned}$$

(48)

Observe that $t_c^*> t_f^*>0$ implies $W^c_{t_f}>0$ in (43), $W^c_m>0$ in (44), $W^f_{t_c}>0$ in (45) and $W^f_m>0$ in (47). Using this information together with $\mathcal{T}_m^c>0$, $\mathcal{T}_m^f>0$ from (37), (38) and $W^f_{t_f}>0$ from (48) in (41) and (42) establishes $\frac{{\mathrm {d}}\mathcal{W}^c(m)}{{\mathrm {d}}m}>0$ and $\frac{{\mathrm {d}}\mathcal{W}^f(m)}{{\mathrm {d}}m}>0$. $\square$

1.4 Appendix D: Feasibility constraint on the parameter space

Throughout the paper we restrict our attention to feasible economies, i.e. to those economies that exhibit non-negative rates of return on capital. As mentioned in footnote 9, in the social optimum the rate of return on capital is positive, if and only if

$$\begin{aligned} \frac{a}{b \bar{k}} > 1 - \frac{\varepsilon }{1+2 \varepsilon }. \end{aligned}$$

(49)

Next, consider coalition-fringe equilibria.²⁶ Inserting $\mathcal{T}^c (m)$ and $\mathcal{T}^f (m)$ from (13) into (5) and solving for $r=0$ establishes that in the coalition-fringe equilibrium the rate of return on capital is positive, if and only if

$$\begin{aligned} \frac{a}{b \bar{k}} > \frac{n (1+2 \varepsilon )[(2n-1) \varepsilon +n-1] - m^2(1+ 2 \varepsilon ) - m [(2n-1) \varepsilon ^2+(n-3) \varepsilon -1]}{(n-m)(1+2 \varepsilon )[(n-1) (1+2 \varepsilon ) + (1+\varepsilon )m]} =: Q(m). \end{aligned}$$

(50)

Differentiation of Q(m) with respect to m yields after rearrangement of terms

$$\begin{aligned} Q_m = \frac{\varepsilon [n(1+2\varepsilon ) - \varepsilon ][(2m-1) n (1+2 \varepsilon ) -m^2 \varepsilon ]}{(n-m)^2(1+ \varepsilon )[(n-1) (1+2 \varepsilon ) + (1+\varepsilon )m]^2}. \end{aligned}$$

(51)

Observe that $Q_m>0$ for all $m \in [1, n-1]$. Hence, Q(m) attains its largest value at $n-1$, formally

$$\begin{aligned} Q(n-1) = \frac{2(n-1)+(n^2+5n-5) \varepsilon +(2n^2+n-1) \varepsilon ^2}{(n-1)(2 \varepsilon ^2+5\varepsilon +2)}. \end{aligned}$$

(52)

Finally, verify that

$$\begin{aligned} Q(n-1) -1-\frac{\varepsilon }{1+2 \varepsilon } = \frac{\left[ (n-1)(2+3 \varepsilon - \varepsilon ^2)+ n^2(1+2\varepsilon )^2\right] \varepsilon }{(n-1)(1+2\varepsilon )(3\varepsilon ^2 +5 \varepsilon +2)}>0. \end{aligned}$$

(53)

In view of (49)–(53) an economy is feasible if the parameters $(a, b , \bar{k},n, \varepsilon )$ satisfy

$$\begin{aligned} \frac{a}{b \bar{k}} > \frac{2(n-1)+(n^2+5n-5) \varepsilon +(2n^2+n-1) \varepsilon ^2}{(n-1)(2 \varepsilon ^2+5\varepsilon +2)}. \end{aligned}$$

(54)

The non-negativity of the rate of return on capital for Example 1 is shown in Figs 8.

Fig. 8

Interest rate (Example 1)

1.5 Appendix E: Proof of (20)–(23)

Making use of $\hat{w} = (1+\varepsilon ) a \bar{k} - b \bar{k}^2 \left( \frac{1}{2} +\varepsilon \right)$ and inserting $t_c^*$ and $t_f^*$ from (29) and (30) in $W^f(t_c, t_f, m)$ yields after rearrangement of terms

$$\begin{aligned}&\mathcal{W}^f(n-1) - \hat{w} = a \varepsilon \bar{k} \end{aligned}$$

(55)

$$\begin{aligned}&+ \frac{\left[ n^3 \varepsilon (1+ 3 \varepsilon + 2 \varepsilon ^2 ) + n^2 \varepsilon (1+ 3 \varepsilon + 4 \varepsilon ^2) +4 n (2+9 \varepsilon + 13 \varepsilon ^2 + 5 \varepsilon ^3) -2(4+18 \varepsilon + 25 \varepsilon ^2 + 9 \varepsilon ^3)\right] b \varepsilon \bar{k}^2}{2(n-1) (1+ \varepsilon )(2+ 3 \varepsilon )^2}. \end{aligned}$$

(56)

Setting $\mathcal{W}^f(n-1) - \hat{w}$ equal to zero and solving for $\frac{a}{b \bar{k}}$ yields after rearrangement of terms

$$\begin{aligned} \frac{a}{b \bar{k}}= & {} \frac{8(n-1) +\left[ n^2(n+1) +36(n-1) \right] \varepsilon +\left[ 3n^2(n+1) +52n -50 \right] \varepsilon ^2}{2(n-1) (1+ \varepsilon )(2+ 3 \varepsilon )^2}\\+ & {} \frac{\left[ 2n^2(n+2) +20n -18 \right] \varepsilon ^3}{2(n-1) (1+ \varepsilon )(2+ 3 \varepsilon )^2}=:F(\varepsilon , n). \end{aligned}$$

(57)

It is then straightforward to show that

$$\begin{aligned} \hat{w} - \mathcal{W}^f(n-1)\gtreqless 0 \quad \iff \quad \frac{a}{b \bar{k}} \gtreqless F(\varepsilon , n). \end{aligned}$$

(58)

Differentiation of $F(\varepsilon ,n )$ with respect to $\varepsilon$ and n yields

$$\begin{aligned} F_{\varepsilon }= & {} \frac{(1+\varepsilon )^2 (2+5 \varepsilon )n^3+ (2n^2+8n-8) +(9n^2+28n-20)\varepsilon }{2(n-1)(1+\varepsilon )^2(2+\varepsilon )^3}\\+ & {} \frac{(24n^2+8n+8) \varepsilon ^2 +(19n^2-16n +24) \varepsilon ^3}{2(n-1)(1+\varepsilon )^2(2+\varepsilon )^3}>0, \end{aligned}$$

(59)

$$\begin{aligned} F_n= & {} \frac{(n^3-n^2-n) +\left[ 3(n^3-n^2-n)-1 \right] \varepsilon +\left[ 2 n^3-n^2- 4 n \right] \varepsilon ^2}{(n-1)^2(1+\varepsilon )(2+3\varepsilon )^2}>0. \end{aligned}$$

(60)

In addition, the function F has the properties

$$\begin{aligned} F(n, 0)= & {} 1,\quad \lim _{n \rightarrow \infty } F(\varepsilon , n) = \infty ,\qquad (\text{E}7) \\ \lim _{\varepsilon \rightarrow \infty } F(\varepsilon , n)= & {} \frac{n^3 + 2 n^2 +10 n -9}{9(n-1)}=: \bar{F} (n)>0,\qquad (\text{E}8) \end{aligned}$$

and it holds $\bar{F}_n (n) = \frac{2 n^3 -n^2 - 1-4n}{9(n-1)^2} >0$ for $n \ge 2$.

Appendix F: Proof of Proposition 3

The proof of Proposition 3 follows from Table 1 and is supplemented by Fig. 9 which illustrates the function $\tilde{M}(\varepsilon , n)$ for $n \in [10, 200]$ and $\varepsilon \in [0,2]$. Analogous figures can be generated for $\varepsilon >2$.

Fig. 9

The function $\tilde{M} (\varepsilon , n)$