Decentralization with porous borders: public production in a federation with tax competition and spillovers

Abstract

We analyze the strategic interaction of regional and federal governments using a model that includes fiscal externalities in the form of inter-regional capital tax competition and technical externalities in the form of inter-regional spillovers. The federal government aims to correct for these inefficiencies using a transfer system. If the regional governments are policy leaders (such that federal policy is set conditional on regional choices), they will internalize both fiscal and technical externalities but free-ride on the transfer system. Efficiency can be achieved by introducing a second transfer scheme that is independent of regional public production. If the federal government sets its policy first and can commit itself to it, the outcome is efficient only if matching grants are used that are financed outside of the transfer system.

Appendix

1.1 Benchmark cases

The solution to the maximization problem (13) can be found by substituting the definition for $G_{i}$ into the utility function and form the Lagrangian:

$$\begin{aligned} \max _{\{c_{1\ldots n}\}, \{g_{1\ldots n}\}} L= & {} U_{i} \left( c_{i}, g_{i} + \beta \sum _{\begin{array}{c} m=1 \\ m\ne i \end{array}}^n g_{m}\right) + \sum _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^n U_{j} \left( c_{j}, g_{j} + \beta \sum _{\begin{array}{c} m=1 \\ m\ne j \end{array}}^n g_{m} \right) \\&- \lambda \sum _{i=1}^n \left( f(k_{i}) - c_{i} - g_{i}\right) . \end{aligned}$$

Taking the derivatives w.r.t. to $c_i$ and $g_i$ (which produces 2n first-order conditions) and rearranging them leads to the following n optimality conditions:

$$\begin{aligned} \frac{\partial U_{i}}{\partial c_{i}} = \frac{\partial U_{i}}{\partial G_{i}} \frac{\partial G_{i}}{\partial g_{i}} + \sum _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^n \frac{\partial U_{j}}{\partial G_{j}} \frac{\partial G_{j}}{\partial g_{i}} \quad \forall \,\, i . \end{aligned}$$

(A.1)

At the optimum, the change in the marginal utility of private consumption has to equal the change in the marginal utility of public good consumption for each region. Imposing symmetry, defining the marginal rate of substitution between public and private consumption as $\mathrm{MRS}_i \equiv {\frac{\partial U_{i}}{\partial G_{i}}}/{\frac{\partial U_{i}}{\partial c_{i}}}$ and suppressing the subscript yields expression (14). Finally, substituting $\partial U/\partial c=1/c$ and $\partial U/\partial G=1/G$ and equating with (14) leads to the optimal tax given by (16).

The Nash equilibrium is derived by rearranging the first-order condition from (20) to:

$$\begin{aligned} \hbox {MRS}_{i} \equiv \frac{\frac{\partial U}{\partial G_{i}}}{\frac{\partial U}{\partial c_{i}}}=- \frac{\frac{\partial c_{i}}{\partial t_{i}}}{\frac{\partial G_{i}}{\partial t_{i}}}. \end{aligned}$$

(A.2)

The marginal effect of taxes on consumption (income) is:

$$\begin{aligned} \frac{\partial c_{i}}{\partial t_{i}} = - f^{\prime }{}^{\prime }{}(k_{i}) \frac{\partial k_{i}}{\partial t_{i}} k_{i} + \frac{\partial R}{\partial t_{i}}{\bar{k}}_{i}. \end{aligned}$$

(A.3)

The effect on income is separated into the effect on wage income (first term) and capital income (second term), both of which are negative. Increasing taxes increases public consumption at the cost of private consumption. The derivative of public good consumption with respect to the tax rate is given by:

$$\begin{aligned} \frac{\partial G_{i}}{\partial t_{i}} = k_i + t_i \frac{\partial k_{i}}{\partial t_{i}} +\beta \sum _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^n\frac{\partial t_{j} k_{j} }{\partial t_{i}}. \end{aligned}$$

(A.4)

The first term is the mechanical increase in tax revenue if the tax rate is marginally raised, holding capital fixed. The second term is negative and captures the outflow of capital to other regions in response to a marginal tax increase in region i. This negative cost is however dampened by positive spillovers from other regions (third term). Substituting (A.3) and (A.4) into (A.2) yields:

$$\begin{aligned} \hbox {MRS}^\mathrm{NE}_{i} = \frac{f^{\prime }{}^{\prime }{}(k_{i}) \frac{\partial k_{i}}{\partial t_{i}} k_{i} - \frac{\partial R}{\partial t_{i}}{\bar{k}}_{i}}{k_{i} + \frac{\partial k_{i}}{\partial t_{i}} t_{i} +\beta \sum _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^n \frac{\partial k_{j}}{\partial t_{i}} t_{j}}. \end{aligned}$$

(A.5)

Imposing symmetry and using the capital market clearing conditions (10), we obtain the MRS which defines the Nash equilibrium in (21).

1.2 Decentralized leadership

1.2.1 Net single transfer system: Proof of Proposition 1

The optimality condition from solving (30) is given by:

$$\begin{aligned} \frac{\partial U_{i}}{\partial G_{i}} \frac{\partial G_{i}}{\partial t_{i}} = - \frac{\partial U_{i}}{\partial c_{i}} \frac{\partial c_{i}}{\partial t_{i}}. \end{aligned}$$

(A.6)

The marginal change in private consumption is

$$\begin{aligned} \frac{\partial c_{i}}{\partial t_{i}} = - f^{\prime }{}^{\prime }{}(k_{i}) \frac{\partial k_{i}}{\partial t_{i}} k_{i} + \frac{\partial R}{\partial t_{i}}{\bar{k}}_{i} =-k, \end{aligned}$$

(A.7)

where the last equality follows from substituting (10).

The marginal change in public consumption for region i is

$$\begin{aligned} \frac{\partial G_{i}}{\partial t_{i}}&= \frac{\partial t_{i} k_{i}}{\partial t_{i}} + \frac{\partial s_{i}}{\partial t_i} +\beta (n-1) \left( \frac{\partial t_{m} k_{m} }{\partial t_{i}} + \frac{\partial s_{m}}{\partial t_{i}} \right) \nonumber \\&= \frac{\partial t_{i} k_{i}}{\partial t_{i}} +\beta (n-1) \frac{\partial t_{m} k_{m} }{\partial t_{i}} + (1-\beta ) \left( \frac{n-1}{n}\right) \left( \frac{\partial t_{m} k_{m}}{\partial t_{i}} - \frac{\partial t_{i} k_{i} }{\partial t_{i}}\right) \nonumber \\&= \frac{1+\beta (n-1)}{n} \left( \frac{\partial t_{i} k_{i}}{\partial t_{i}} + (n-1) \frac{ \partial t_{m} k_{m} }{\partial t_{i}}\right) \nonumber \\&= \frac{1+(n-1) \beta }{n}k, \end{aligned}$$

(A.8)

where we used (29) (second line), collected terms (third line) and substituted the capital market clearing conditions (10) (fourth line). The change in mobile capital in region i and regions $m\ne i$ as a response to the change in $t_{i}$ cancels out in equilibrium due to the federal transfer setting:

$$\begin{aligned} \frac{\partial t_{i} k_{i}}{\partial t_{i}} + (n-1) \frac{\partial t_{m} k_{m}}{\partial t_{i}} = k + \frac{n-1}{n} \frac{1}{f''(k)} t - \frac{n-1}{n} \frac{1}{f''(k)} t = k\, . \end{aligned}$$

What remains is the additional positive tax revenue effect to region i given by k. Expression (31) follows directly from substituting (A.8) and (A.7) into (A.6) $\square$

1.2.2 Net transfer system including private income redistribution

In stage 2, the maximization problem of the federal government is given by:

$$\begin{aligned} \max _{\{G_{1\ldots n}\}, \{s_{1\ldots n}\}, \{\tau _{1\ldots n}\}, \{g_{1\ldots n}\}, \{c_{1\ldots n}\}} \quad \sum _{i=1}^n U_{i} (c_{i}, G_{i}) \qquad&\text {s.t.} \qquad G_{i}=g_{i} + \beta \sum _{\begin{array}{c} m=1 \\ m\ne i \end{array}}^n g_{m} \nonumber \\&\sum _{i=1}^n s_{i} = 0 \nonumber \\&\sum _{i=1}^{n} \tau _i = 0\nonumber \\&g_{i} = t_{i} k_{i} + s_{i} \nonumber \\&c_{i}= f(k_{i}) -\frac{\partial f(k_{i})}{\partial k_{i}} k_{i} + \bar{k_{i}} R + \tau _i. \end{aligned}$$

(A.9)

Substituting the regional governments’ and households’ budget constraints, the Lagrangian and the first-order conditions can be written as

$$\begin{aligned} \max _{\{s_{1\ldots n}\}, \{\tau _{1\ldots n}\}}&\quad L= \sum _{i=1}^n U_{i} \bigg (f(k_{i}) -\frac{\partial f(k_{i})}{\partial k_{i}} k_{i} + \bar{k_{i}} R + \tau _i , t_{i} k_{i} + s_{i} + \beta \sum _{\begin{array}{c} m=1 \\ m\ne i \end{array}}^n t_{m} k_{m} + s_{m} \bigg ) \nonumber \\&\quad \qquad + \lambda \sum _{i=1}^n s_{i} + \mu \sum _{i=1}^{n} \tau _i \end{aligned}$$

(A.10)

$$\begin{aligned}&\frac{\partial L}{\partial s_{i}} = \frac{\partial U}{\partial G_{i}} + \beta \sum _{\begin{array}{c} m=1 \\ m\ne i \end{array}}^n \frac{\partial U}{\partial G_{m}} + \lambda = 0 \quad \forall i \end{aligned}$$

(A.11)

$$\begin{aligned}&\frac{\partial L}{\partial \tau _i} = \frac{\partial U}{\partial c_{i}} + \mu = 0 \quad \forall i \end{aligned}$$

(A.12)

$$\begin{aligned}&\frac{\partial L}{\partial \lambda } = \sum _{i=1}^n s_{i} = 0 \end{aligned}$$

(A.13)

$$\begin{aligned}&\frac{\partial L}{\partial \mu } = \sum _{i=1}^n \tau _{i} = 0 . \end{aligned}$$

(A.14)

Using (A.11) for regions i and $j\ne i$, combining and imposing symmetry yields (33). Equation (34) is derived in the same manner using (A.12) for regions i and j. Finally, the reaction function for private consumption transfers can be found using (34) and the federal budget constraint (A.14):

$$\begin{aligned}&f(k_{i}) -\frac{\partial f(k_{i})}{\partial k_{i}} k_{i} + \bar{k_{i}} R + \tau _i = f(k_{j}) -\frac{\partial f(k_{j})}{\partial k_{j}} k_{j} + \bar{k_{j}} R + \tau _j \end{aligned}$$

(A.15)

$$\begin{aligned}&\tau _i = - (n-1) \tau _j \quad \text {or} \quad \tau _j = - \frac{1}{n-1} \tau _i . \end{aligned}$$

(A.16)

Using (A.14) and (9) in (A.15) and differentiating with respect to $t_i$ yields:

$$\begin{aligned}&f^{\prime }{}^{\prime }{}(k_{i}) \frac{\partial k_{i}}{\partial t_{i}} (\bar{k}_{i} - k_{i}) - {\bar{k}}_{i} + \frac{\partial \tau _i}{\partial t_i} = f^{\prime }{}^{\prime }{}(k_{j}) \frac{\partial k_{j}}{\partial t_{i}} (\bar{k}_{j} - k_{j}) - \frac{1}{n-1} \frac{\partial \tau _i}{\partial t_i} \end{aligned}$$

Substituting (10) and solving for $\frac{\partial \tau _i}{\partial t_i}$ and using it in (A.16) as well yields the reaction function (35).

Proof of Proposition 2

In the first stage, regional governments solve:

$$\begin{aligned}&\max _{G_i,g_i, c_i, t_{i}} \quad U(c_{i}, G_i) \quad \text {s.t.} \quad&G_{i}= g_i + \beta \sum _{\begin{array}{c} m=1 \\ m\ne i \end{array}}^n g_m \\&g_i= t_{i} k_{i} + s_{i} \\&c_i = f(k_{i}) - f'(k_{i}) k_{i} +\bar{k_{i}} R - \tau _i \\&\text {with} \quad s_i=s_i(t_i, t_j); \quad s_j= s_j(t_i, t_j); \tau _i=\tau _i(t_i, t_j). \end{aligned}$$

Solving the model results in the same optimality condition as in the Nash equilibrium, given by (A.2). The change in private consumption is given by

$$\begin{aligned} \frac{\partial c_{i}}{\partial t_{i}}&= - f^{\prime }{}^{\prime }{}(k_{i}) \frac{\partial k_{i}}{\partial t_{i}} k_{i} + \frac{\partial R}{\partial t_{i}}{\bar{k}}_{i} - \frac{\partial \tau _i}{\partial t_i} = - \frac{1}{n} k, \end{aligned}$$

(A.17)

where the last equality follows from substituting (10) and (35). The change in public consumption with respect to $t_i$ is equivalent to (A.8). Using (A.17) and (A.8) in (A.2) leads to (36). $\square$

1.2.3 Gross transfer system(s)

The federal government chooses both T and the region-specific transfers $s_i$ to maximize total welfare. After substituting the budget constraints, the Lagrangian can be written as:

$$\begin{aligned} \max _{\{s_{1\ldots n}\},T } \quad L= & {} \sum _{i=1}^n U_{i} \left( f(k_{i}) -\frac{\partial f(k_{i})}{\partial k_{i}} k_{i} + \bar{k_{i}} R - T ; \,\, t_{i} k_{i} + s_{i} + \beta \sum _{\begin{array}{c} m=1 \\ m\ne i \end{array}}^n t_{m} k_{m} + s_{m} \right) \\&+ \,\,\lambda \left( \sum _{i=1}^n s_{i} - n T \right) \end{aligned}$$

where $\lambda$ is the Lagrangian multiplier of the federal budget constraint. The first-order conditions are given by:

$$\begin{aligned}&\frac{\partial L}{\partial s_{i}} = \frac{\partial U}{\partial G_{i}} + \beta \sum _{\begin{array}{c} m=1 \\ m\ne i \end{array}}^n \frac{\partial U}{\partial G_{m}} + \lambda = 0 \quad \forall i \end{aligned}$$

(A.18)

$$\begin{aligned}&\frac{\partial L}{\partial T} = - \sum _{\begin{array}{c} i=1 \end{array}}^n \frac{\partial U}{\partial c_{i}} - \lambda n = 0 \end{aligned}$$

(A.19)

$$\begin{aligned}&\frac{\partial L}{\partial \lambda } = \sum _{i=1}^n s_{i} - n T = 0. \end{aligned}$$

(A.20)

Using (A.18) for i and $j \ne i$, combining and imposing symmetry yields (37). Combining (A.18) and (A.19) yields (38).

Taking the partial derivative of the first-order conditions with respect to $t_i$ (which is a parameter from the point of view of the central government) gives

$$\begin{aligned}&\big (1 + \beta (n-1)\big ) U_{GG} \frac{\partial G_i}{\partial t_{i}} = 0 \quad \forall i \end{aligned}$$

(A.21)

$$\begin{aligned}&- U_{CC} \frac{\partial c_i}{\partial t_{i}} = 0 \end{aligned}$$

(A.22)

$$\begin{aligned}&\frac{\partial s_{i} }{\partial t_i} = n \frac{\partial T }{\partial t_i} - (n-1) \frac{\partial s_{j} }{\partial t_i}, \end{aligned}$$

(A.23)

where $U_{GG} \equiv \frac{\partial ^2 U}{\partial G_{i}^2}$ and $U_{CC} \equiv \frac{\partial ^2 U}{\partial c_{i}^2}$. Equation (A.21) can be rewritten such that $\frac{\partial G_i}{\partial t_{i}} = \frac{\partial G_j}{\partial t_{i}}$ (differentiating (37) with respect to $t_i$ would be an alternative to derive the same expression):

$$\begin{aligned}&\frac{\partial t_i k_{i}}{\partial t_{i}} + \frac{\partial s_{i}}{\partial t_{i}} +\beta \sum _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^n \bigg (\frac{\partial \ t_{j} k_{j} }{\partial t_{i}} + \frac{\partial s_{j}}{\partial t_{i}}\bigg ) = \frac{\partial t_j k_{j}}{\partial t_{i}} + \frac{\partial s_{j}}{\partial t_{i}} + \beta \bigg ( \frac{\partial t_i k_{i}}{\partial t_{i}} + \frac{\partial s_{i}}{\partial t_{i}}\bigg ) + \beta \sum _{\begin{array}{c} m=1 \\ m\ne j \end{array}}^n \bigg (\frac{\partial \ t_{m} k_{m} }{\partial t_{i}} + \frac{\partial s_{m}}{\partial t_{i}}\bigg ) \\&\frac{\partial t_i k_{i}}{\partial t_{i}} + \frac{\partial s_{i}}{\partial t_{i}} +\beta (n-1) \bigg (\frac{\partial \ t_{j} k_{j} }{\partial t_{i}} + \frac{\partial s_{j}}{\partial t_{i}}\bigg ) = \big (1+\beta (n-2)\big ) \bigg (\frac{\partial t_j k_{j}}{\partial t_{i}} + \frac{\partial s_{j}}{\partial t_{i}}\bigg ) + \beta \bigg ( \frac{\partial t_i k_{i}}{\partial t_{i}} + \frac{\partial s_{i}}{\partial t_{i}}\bigg ) \\&(1-\beta ) \bigg (\frac{\partial t_i k_{i}}{\partial t_{i}} + \frac{\partial s_{i}}{\partial t_{i}}\bigg ) = (1-\beta ) \bigg (\frac{\partial t_j k_{j}}{\partial t_{i}} + \frac{\partial s_{j}}{\partial t_{i}}\bigg ), \end{aligned}$$

where symmetry is applied in the second line. Substituting (A.23), we get:

$$\begin{aligned} (1-\beta ) \frac{\partial s_{j}}{\partial t_{i}} = (1-\beta ) \bigg ( n \frac{\partial T }{\partial t_i} - (n-1) \frac{\partial s_{j} }{\partial t_i} + \frac{\partial t_i k_{i}}{\partial t_{i}} - \frac{\partial t_j k_{j}}{\partial t_{i}}\bigg ). \end{aligned}$$

Solving for $\frac{\partial s_{i} }{\partial t_i}$ and $\frac{\partial s_{j}}{\partial t_{i}}$, results in the reaction functions

$$\begin{aligned}&(1-\beta ) \frac{\partial s_{i} }{\partial t_i} = (1-\beta ) \left( \frac{n-1}{n} \bigg ( \frac{\partial t_j k_{j}}{\partial t_{i}} - \frac{\partial t_i k_{i}}{\partial t_{i}} \bigg ) + \frac{\partial T }{\partial t_i} \right) \\&(1-\beta ) \frac{\partial s_{j}}{\partial t_{i}} = (1-\beta ) \left( \frac{\partial T }{\partial t_i} + \frac{1}{n} \bigg ( \frac{\partial t_i k_{i}}{\partial t_{i}} - \frac{\partial t_j k_{j}}{\partial t_{i}}\bigg ) \right) \, , \end{aligned}$$

which simplifies to (39) for $\beta <1$.

In stage 1, the regional governments maximize regional utility incorporating the reaction of the federal policy to their tax increase:

$$\begin{aligned} \max _{G_i, g_i, c_i, t_{i}} \quad U(c_{i}, G_{i}) \qquad \text {s.t.} \qquad&G_{i}= g_i + \beta \sum _{\begin{array}{c} m=1 \\ m\ne i \end{array}}^n g_m \\&g_i = t_{i} k_{i} + s_{i} \\&c_i = f(k_{i}) - f'(k_{i}) k_{i} +\bar{k_{i}} R - T \\&t_i \ge 0\\ \text {with} \qquad \qquad&s_i=s_i(t_i, t_j); \quad s_j= s_j(t_i, t_j); \quad T=T(t_i, t_j). \end{aligned}$$

Corollary 1

In stage 2, the maximization problem of the federal government using a gross transfer system as defined by part i, is:

$$\begin{aligned} \max _{\{G_{1\ldots n}\}, \{s_{1\ldots n}\}, \{T_{1\ldots n}\}, \{g_{1\ldots n}\}, \{c_{1\ldots n}\}} \quad \sum _{i=1}^n U_{i} (c_{i}, G_{i}) \qquad \text {s.t.} \qquad&G_{i}=g_{i} + \beta \sum _{\begin{array}{c} m=1 \\ m\ne i \end{array}}^n g_{m} \nonumber \\&\sum _{i=1}^n s_{i} = \sum _{i=1}^{n} T_i \nonumber \\&g_{i} = t_{i} k_{i} + s_{i} \nonumber \\&c_{i}= f(k_{i}) -\frac{\partial f(k_{i})}{\partial k_{i}} k_{i} + \bar{k_{i}} R - T_i. \end{aligned}$$

(A.24)

The Lagrangian and the FOCs are:

$$\begin{aligned} \max _{\{s_{1\ldots n}\}, \{T_{1\ldots n}\}} \quad L&= \sum _{i=1}^n U_{i} \bigg (f(k_{i}) -\frac{\partial f(k_{i})}{\partial k_{i}} k_{i} + \bar{k_{i}} R - T_i , t_{i} k_{i} + s_{i} \nonumber \\&\quad + \beta \sum _{\begin{array}{c} m=1 \\ m\ne i \end{array}}^n t_{m} k_{m} + s_{m} \bigg ) + \lambda \left( \sum _{i=1}^n s_{i} - \sum _{i=1}^{n} T_i \right) \end{aligned}$$

(A.25)

$$\begin{aligned} \frac{\partial L}{\partial s_{i}}&= \frac{\partial U}{\partial G_{i}} + \beta \sum _{\begin{array}{c} m=1 \\ m\ne i \end{array}}^n \frac{\partial U}{\partial G_{m}} + \lambda = 0 \quad \forall i \end{aligned}$$

(A.26)

$$\begin{aligned} \frac{\partial L}{\partial T_i}&= - \frac{\partial U}{\partial c_{i}} - \lambda = 0 \quad \forall i \end{aligned}$$

(A.27)

$$\begin{aligned} \frac{\partial L}{\partial \lambda }&= \sum _{i=1}^n s_{i} - \sum _{i=1}^{n} T_i = 0 . \end{aligned}$$

(A.28)

Combining will lead to the same optimality conditions as in Sect. 4.1.2, i.e., (33) and (34).

The problem for part ii, after substituting the budget constraints, is given by:

$$\begin{aligned} \max _{\{s_{1\ldots n}\}, \{\tau _{1\ldots n}\},T } \quad L&= U_{i} \left( f(k_{i}) -\frac{\partial f(k_{i})}{\partial k_{i}} k_{i} + \bar{k_{i}} R + \tau _i -T, t_{i} k_{i} + s_{i} + \beta \sum _{\begin{array}{c} m=1 \\ m\ne i \end{array}}^n t_{m} k_{m} + s_{m}\right) \nonumber \\&\quad + \sum _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^n U_{j} \left( f(k_{j}) -\frac{\partial f(k_{j})}{\partial k_{j}} k_{j} + \bar{k_{j}} R + \tau _j -T, t_{j} k_{j} + s_{j} + \beta \sum _{\begin{array}{c} m=1 \\ m\ne j \end{array}}^n t_{m} k_{m} + s_{m} \right) \nonumber \\&\quad + \lambda \left( \sum _{i=1}^n s_{i} - nT\right) + \mu \left( \sum _{i=1}^{n} \tau _i\right) \end{aligned}$$

(A.29)

The first-order conditions are given by:

$$\begin{aligned}&\frac{\partial L}{\partial s_{i}} = \frac{\partial U}{\partial G_{i}} + \beta \sum _{\begin{array}{c} m=1 \\ m\ne i \end{array}}^n \frac{\partial U}{\partial G_{m}} + \lambda = 0 \quad \forall i \end{aligned}$$

(A.30)

$$\begin{aligned}&\frac{\partial L}{\partial T} = - \sum _{\begin{array}{c} i=1 \end{array}}^n \frac{\partial U}{\partial c_{i}} - \lambda n = 0 \end{aligned}$$

(A.31)

$$\begin{aligned}&\frac{\partial L}{\partial \tau _i} = \frac{\partial U}{\partial c_{i}} + \mu = 0 \quad \forall i \end{aligned}$$

(A.32)

$$\begin{aligned}&\frac{\partial L}{\partial \lambda } = \sum _{i=1}^n s_{i} - n T = 0 \end{aligned}$$

(A.33)

$$\begin{aligned}&\frac{\partial L}{\partial \mu } =\sum _{i=1}^{n} \tau _i = 0. \end{aligned}$$

(A.34)

Combining leads to:

$$\begin{aligned}&U^{\prime }{}(G_{i}) = U ^{\prime }{}(G_{j})&\Rightarrow G_{i} = G_{j}&\forall \,\, i, \, j, \end{aligned}$$

(A.35)

$$\begin{aligned}&U^{\prime }{}(c_{i}) = U ^{\prime }{}(c_{j})&\Rightarrow c_{i} = c_{j}&\forall \,\, i, \, j , \end{aligned}$$

(A.36)

$$\begin{aligned}&\frac{\frac{\partial U}{\partial G_{i}}}{\frac{\partial U}{\partial c_{i}}} \equiv MRS_i = \frac{1}{1 + (n-1) \beta } \ \end{aligned}$$

(A.37)

in symmetry. Using (A.30) for region i and j results in (A.35). Likewise, using (A.32) for i and j leads to (A.36). Combining (A.30) with (A.31) establishes (A.37).

The federal government chooses transfers $s_i$ and $\tau _i$ such that the marginal utilities of private and public consumption across regions are equalized, which implies equal private and public good consumption across regions in symmetry. These two conditions suffice to establish an efficient solution as demonstrated in 4.1.2. The same is true if we concentrate on the first-order condition for T instead. Given that this problem has one redundant instrument and that regions are ex ante symmetric, the regional tax rates $t_i=t_j=t$ are not identified in this equilibrium. If regional taxes are doubled, then the transfers would simply be cut in half, leaving public and private consumption unchanged as defined by (A.35)-(A.37). Without loss of generality, we can therefore set $t_i$ to zero for all regions, or to any other value.

1.3 Centralized leadership

Starting in stage 2, the regional governments’ problem is given by

$$\begin{aligned} \max _{G_i, g_i, c_i, t_{i}} U_i&\left( c_{i}, G_i\right) \\ \qquad \text {s. t. } \,\,&g_{i} = \frac{t_{i}k_{i}}{1-\alpha } \\&c_{i}= f(k_{i}) -\frac{\partial f(k_{i})}{\partial k_{i}} k_{i} + \bar{k_{i}} R - T\\&G_i = g_{i} + \beta \sum _{\begin{array}{c} m=1 \\ m\ne i \end{array}}^n g_{m}. \end{aligned}$$

The optimality condition from combining the first-order conditions results in (A.2). When regional governments maximize regional utility, the marginal effect of taxes on consumption (income) is equivalent to the effect in the Nash game:

$$\begin{aligned} \frac{\partial c_{i}}{\partial t_{i}} = - f^{\prime }{}^{\prime }{}(k_{i}) \frac{\partial k_{i}}{\partial t_{i}} k_{i} + \frac{\partial R}{\partial t_{i}}{\bar{k}}_{i}. \end{aligned}$$

(A.38)

The marginal effect of taxes on public consumption is given by:

$$\begin{aligned} \frac{\partial G_{i}}{\partial t_{i}} = \frac{1}{1-\alpha } \bigg (k_i + t_i \frac{\partial k_{i}}{\partial t_{i}} +\beta \sum _{\begin{array}{c} m=1 \\ m\ne i \end{array}}^n\frac{\partial t_{m} k_{m} }{\partial t_{i}} \bigg ). \end{aligned}$$

(A.39)

Using (A.38) and (A.39) in the MRS (A.2) gives

$$\begin{aligned} MRS_{i}^\mathrm{CE} = \frac{f^{\prime }{}^{\prime }{}(k_{i}) \frac{\partial k_{i}}{\partial t_{i}} k_{i} - \frac{\partial R}{\partial t_{i}}{\bar{k}}_{i}}{\frac{1}{1-\alpha } [k_{i} + \frac{\partial k_{i}}{\partial t_{i}} t_{i}] + \beta \sum _{\begin{array}{c} m=1 \\ m\ne i \end{array}}^n \frac{\partial k_{m}}{\partial t_{i}} t_{m}}. \end{aligned}$$

(A.40)

In symmetry and incorporating the reaction of the capital market given by (10), (A.40) becomes the optimality condition for the regions’ problem:

$$\begin{aligned} MRS^\mathrm{CE} = \frac{1-\alpha }{1 + (1-\beta ) (\frac{n-1}{n}) \frac{1}{f''(k)} \frac{t}{k}}. \end{aligned}$$

(A.41)

In stage 1, substituting all budget constraints (households, regional and federal governments) and the definition for $G_i$, the federal government chooses $\alpha$ to maximize

$$\begin{aligned} \max _{\begin{array}{c} \alpha \end{array}} \, \sum _{i=1}^n&U_i\left( f(k_{i}) -\frac{\partial f(k_{i})}{\partial k_{i}} k_{i} + \bar{k_{i}} R - \frac{\alpha }{n} \sum _{i=1}^n g_{i} ; \, \, \frac{t_{i}k_{i}}{1-\alpha } + \beta \sum _{\begin{array}{c} m=1 \\ m\ne i \end{array}}^n \frac{t_{m}k_{m}}{1-\alpha } \right) , \end{aligned}$$

(A.42)

which is

$$\begin{aligned} \begin{aligned}&\frac{\partial U_{i}}{\partial c_{i}} \sum _{\begin{array}{c} i=1 \end{array}}^n \bigg ( -\frac{1}{n} \frac{t_{i}k_{i}}{1 -\alpha } - \frac{\alpha }{n} \frac{t_{i}k_{i}}{(1 -\alpha )^2} \bigg ) + \frac{\partial U_{i}}{\partial G_{i}} \bigg (\frac{t_{i}k_{i}}{(1-\alpha )^2} + \beta \sum _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^n \frac{t_{j}k_{j}}{(1-\alpha )^2}\bigg ) \\&\quad + \sum _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^n \frac{\partial U_{j}}{\partial c_{j}} \sum _{\begin{array}{c} j=1 \end{array}}^n \bigg ( -\frac{1}{n} \frac{t_{j}k_{j}}{1-\alpha } - \frac{\alpha }{n} \frac{t_{j}k_{j}}{(1-\alpha )^2}\bigg )+ \sum _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^n \frac{\partial U_{j}}{\partial G_{j}} \frac{t_{j}k_{j}}{(1-\alpha )^2} \\&\quad + \beta \sum _{\begin{array}{c} m=1 \\ m \ne j \end{array}}^n \frac{t_{m}k_{m}}{(1-\alpha )^2} = 0. \\ \end{aligned} \end{aligned}$$

Multiplying by $\frac{1}{(1-\alpha )^2}$, simplifying, imposing symmetry and collecting terms gives us in symmetry:

$$\begin{aligned} MRS^\mathrm{CE} = \frac{1}{1 + (n-1) \beta } = MRS^\mathrm{PO} \, . \end{aligned}$$

(A.43)

which is equivalent to the social optimum in (14).

Proof of Proposition 4

(i) When treating $s_i$ as a lump-sum transfer rather than a matching grant, the transfer drops out of regions’ first-order conditions, and the resulting equilibrium is characterized by expression (21) and thus identical to the Nash equilibrium.

(ii) In order to induce Pareto-efficiency, the federal government sets the matching rate $\alpha$ such that (A.41) and (A.43) are equalized:

$$\begin{aligned} \frac{1-\alpha }{1 + (1-\beta ) (\frac{n-1}{n}) \frac{1}{f''(k)} \frac{t}{k}} = \frac{1}{1+(n-1) \beta }. \end{aligned}$$

Solving for $\alpha$ results in

$$\begin{aligned} \alpha ^\mathrm{CE}&= \frac{n-1}{n}\cdot \frac{ n \beta + (1-\beta ) \frac{1}{-f''(k)} \frac{t}{k} }{ 1 + (n-1)\beta } \nonumber \\&= \frac{ (n-1) \beta + (1-\beta ) \epsilon }{ 1 + (n-1)\beta }, \end{aligned}$$

(A.44)

with $\epsilon \equiv \frac{n-1}{n}\frac{1}{-f''(k_i) }\frac{t}{k}$

(iii) The derivative of the matching rate w.r.t. the spillover parameter is given by:

$$\begin{aligned} \frac{\partial \alpha ^\mathrm{CE}}{\partial \beta } = \frac{(n-1)(1 + \frac{1}{f''(k)} \frac{t}{k})}{(1 + ( n - 1) \beta )^2 }. \end{aligned}$$

(A.45)

Provided that $n>1$, it follows that for $\frac{1}{|f''(k)|} \frac{t}{k} < 1$, $\partial \alpha ^\mathrm{CE}/\partial \beta >0$, and vice versa. $\square$