Tobin meets Oates: solidarity and the optimal fiscal federal structure

Abstract

We explore the implications for the optimal degree of fiscal decentralization when people’s preferences for goods and services—which classic treatments of fiscal federalism (Oates in Fiscal federalism, 1972) place in the purview of local governments—exhibit specific egalitarianism (Tobin in J. Law Econ. 13(2): 263–277, 1970), or solidarity. We find that a system in which the central government provides a common minimum level of the publicly provided good, and local governments are allowed to use their own resources to provide an even higher local level, performs better from an efficiency perspective relative to all other systems analyzed for a relevant range of preferences over solidarity.

Appendices

Appendix A: Formal characterization of five fiscal federal systems

1.1 A.1 Centralization

Under this system, taxing power is solely in the hands of the central government. The central government imposes a common tax function and gives a common grant to each region. Spending on the publicly provided good is the same across regions because the only source of funding is the uniform central grant. Regions have no decision-making power in this system; once the central tax function and central grant are set, the levels of both goods are determined.

To simplify the analysis we assume a proportional tax on income, ϕ(ω _i )=tω _i , where t is the tax rate and is the same for all regional governments. Private consumption c _i is equal to after-tax income (1−t)ω _i . We define g as the common level of publicly provided good realized in each region. Note that the variance in public spending e is equal to zero in this case.

The decision variables of the government are the tax rate t and the common level of publicly provided good g for all regions. By the balanced budget restrictions, for every level of t, unique levels of publicly provided good g and privately provided good c _i are generated. Hence the set of allocations attainable through the centralized system can be parametrized by t. Assume that the central government chooses tax rate and publicly provided good levels that are not second-best Pareto-dominated (that is, points in the locus between P and R in Figs. 1–5). This means that, given utility levels $\bar{u}_{j}$ for regions j≠i, the following problem is solved: choose t and g such that

(9)

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If the problem has interior solutions—as is the case with the Cobb–Douglas utility functions used in the simulations—the first-order condition for region i is

$$ \sum_{j = 1}^{n} \alpha_{j} \frac{\partial u_{j}}{\partial c_{j}}\frac{\omega_{j}}{\sum_{j = 1}^{n} \omega_{j}} = \frac{1}{n}\sum _{j = 1}^{n} \alpha_{j} \frac{\partial u_{j}}{\partial g}. $$

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This condition states that a weighted average of the regions’ marginal contributions of the privately provided good, where the weights are each region’s relative share of total wealth, is equal to the average marginal contribution of the publicly provided good to social welfare. In general, this will differ from the Pareto optimality condition in Eq. (7) and the centralized system will lead to inefficient outcomes.

1.2 A.2 Decentralization

Under this system the regions are free to tax themselves and independently set an appropriate level of public expenditure g _i . In choosing their actions, they are aware of the interdependence of their decisions and try to anticipate each other’s behavior.

We model this case as a simultaneous game with expenditures on the publicly provided good as strategic variables. In order to find the Nash equilibrium we compute the best response function of region i to a given level g _j of the other region’s public expenditure for each region j. The equilibrium is the solution of the following maximization problem: choose c _i and g _i so as to solve

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If all solutions are interior the first-order condition for region i is

$$ \frac{\partial u_{i}}{\partial c_{i}} = \frac{\partial u_{i}}{\partial g_{i}} + \frac{\partial u_{i}}{\partial e} \frac{\partial e}{\partial g_{i}} = \frac{\partial u_{i}}{\partial g_{i}} + \frac{2}{n}(g_{i} - \bar{g})\frac{\partial u_{i}}{\partial e}. $$

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The marginal contributions of c _i and g _i to i’s utility are equalized. What is ignored by the region is the impact of expenditures on the publicly provided good g _i through e on other regions’ utilities (as required in the Pareto optimality condition in Eq. (7)).

1.3 A.3 Voluntary transfers

Under this system, each regional government has complete freedom of choice over both goods. In addition, each can set interregional transfers from region i to j, s _ij , which are voluntary transfers to solidarity. Thus, each regional government chooses g _i , c _i and s _ij (for j≠i),²⁰ taking all other variables as given, so as to solve the following maximization problem:

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The Nash equilibrium is obtained by solving simultaneously the n systems of necessary conditions.

Assuming interior solutions for c _i and g _i , and noting that $\frac{\partial e}{\partial g_{i}} = \frac{2}{n}(g_{i} - \bar{g})\frac{\partial u_{i}}{\partial e}$, we get the following first-order necessary condition for region i:

$$ \frac{\partial u_{i}}{\partial c_{i}} = \frac{\partial u_{i}}{\partial g_{i}} + \frac{\partial u_{i}}{\partial e} \frac{\partial e}{\partial g_{i}} = \frac{\partial u_{i}}{\partial g_{i}} + \frac{2}{n}(g_{i} - \bar{g})\frac{\partial u_{i}}{\partial e}. $$

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The marginal contributions of c _i and g _i to i’s utility are equalized, but, because each region acts independently to maximize its own utility, region i does not take into account the impact of g _i through e on other regions’ utilities as required in Eq. (7) for a Pareto optimum.²¹

1.4 A.4 Guaranteed minimum

In this model the central government finances a uniform, minimum expenditure on the publicly provided good through a uniform grant and the regions are then free to tax themselves if they wish to spend more than the centrally funded minimum. We model this as a sequential process in which, in the first stage, the central government sets a common tax rate t for all regions. To keep the analysis very general we do not assume a specific objective for the central government, but rather obtain results for all possible values of t and, therefore, any possible objective of the central government. The revenue is equally distributed so that the grant to any one region is equal to $\frac{1}{n}t\sum_{j = 1}^{n} \omega_{j} = \frac{t}{n}\varOmega$. This grant sets up a minimum level of the publicly provided good in all regions.

At a later stage, knowing the central tax rate and the corresponding grant, the regions are free to choose a higher level of the publicly provided good by raising additional revenue through local taxes. The second phase is modeled as a simultaneous game with the regions as players. The strategic variables are the levels of the privately provided good, c _i , and the locally financed publicly provided good, gr _i ≥0. The level of the ith region’s publicly provided good is $g_{i} = gr_{i} + \frac{t}{n}\varOmega$.

Given the value of the central government’s strategic variable (the tax rate t) and taking the values of the other regions’ strategic variables as given, the ith regional government chooses c _i and gr _i so as to solve

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The Lagrangian expression for this problem is

$$L(c_{i},gr_{i},\lambda_{i}) = u_{i} \biggl(c_{i},gr_{i} + \frac{t}{n}\varOmega,e\biggr) - \lambda_{i}\bigl(c_{i} + gr_{i} - (1 - t) \omega_{i}\bigr) $$

and taking the first derivatives we obtain the Kuhn–Tucker first-order necessary conditions:

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If the minimum level guaranteed by the central government is below the level that the regional government would like to provide, we will have an interior solution for gr _i . Assuming also an interior solution for c _i , from (25) and (26) we obtain

$$ \frac{\partial u_{i}}{\partial c_{i}} = \frac{\partial u_{i}}{\partial g_{i}} + \frac{2}{n}(g_{i} - \bar{g})\frac{\partial u_{i}}{\partial e}. $$

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At the margin, the decision of allocating resources between the privately and publicly provided goods is identical to the decision in the cases of decentralization and voluntary transfers. Continuing to assume an interior solution for c _i , if the minimum level guaranteed by the central government is equal to or above the level that the regional government would like to provide, there will be no local provision of the publicly provided good, and, thus, we will have a corner solution with gr _i =0. In this case, from (25) and (26) we obtain

$$ \frac{\partial u_{i}}{\partial c_{i}} \ge\frac{\partial u_{i}}{\partial g_{i}} + \frac{2}{n}(g_{i} - \bar{g})\frac{\partial u_{i}}{\partial e} $$

(28)

and the central government guaranteed minimum level, because it takes low-spending regions beyond where they would be on their own, may result in an allocation that gets closer (or equal) to the optimal allocation because the inequality in (28) may approach (or equal) the condition in Eq. (7).

1.5 A.5 Matching grants

In this model regions are free to tax themselves to set an appropriate level of locally financed public expenditure, gr _i , and the central government provides a matching grant zgr _i , where z∈[0,1] is the matching rate.

We model a sequential game with the central government as a Stackelberg leader. The central government chooses a matching rate z and a tax rate t that balances the budget $z\sum_{i = 1}^{n} gr_{i} = t\sum_{i = 1}^{n} \omega_{i}$.

At a later stage, knowing the tax rate and the corresponding subsidy, the regions decide on the level of locally financed public expenditure gr _i ≥0. The second phase is modeled as a simultaneous game with the regions as players. The strategic variables are the levels of private consumption, c _i , and the locally financed public expenditures, gr _i . The ith region’s total publicly provided good is g _i =(1+z)gr _i .

Given the value of the central government’s strategic variables z and t and taking the values of the other regions’ strategic variables as given, the ith region chooses c _i and gr _i so as to solve

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Assuming interior solutions for c _i and gr _i , we get the first-order necessary condition for region i:

$$ \frac{\partial u_{i}}{\partial c_{i}} = \frac{\partial u_{i}}{\partial gr_{i}} + \frac{\partial u_{i}}{\partial e}\frac{\partial e}{\partial gr_{i}} = \frac{\partial u_{i}}{\partial g_{i}} + \frac{2}{n}(g_{i} - \bar{g})\frac{\partial u_{i}}{\partial e}. $$

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Again, the marginal contributions of c _i and gr _i to i’s utility are equalized, but region i does not take into account the impact of expenditures on the publicly provided good gr _i through e on other regions’ utilities (as required in Eq. (7) derived from the first-order conditions for a Pareto optimum).

Appendix B: Formal derivation of the measure of efficiency loss

To measure the inefficiency of a given allocation $((\bar{c}_{1},\bar{g}_{1}),(\bar{c}_{2},\bar{g}_{2}),\ldots,(\bar {c}_{n},\bar{g}_{n}))$ we take the vector of utilities $(\bar{u}_{1},\bar{u}_{2},\ldots,\bar{u}_{n})$, where $\bar{u}_{i} = u_{i}(\bar{c}_{i},\bar{g}_{i})$, and find the minimum amount of resources necessary to attain these utility levels.²² Formally, we solve the following problem:

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Let $\omega(\bar{u})$ denote the level of resources that solves this problem, and let $\bar{\omega} = \sum_{i = 1}^{n} \bar{c}_{i} + \sum_{i = 1}^{n} \bar{g}_{i}$ denote the level of resources utilized by the given allocation. Then the difference $\bar{\omega} - \omega(\bar{u})$ is a measure of the resources wasted by the allocation $((\bar{c}_{1},\bar{g}_{1}),(\bar{c}_{2},\bar{g}_{2}),\ldots,(\bar {c}_{n},\bar{g}_{n}))$ and the efficiency loss is defined as $\frac{\bar{\omega} - \omega (\bar{u})}{\bar{\omega}}$.

For the three systems where the central government can choose different values for its parameter, we calculate the efficiency loss to be the minimum value of the index for all parameter values. For the other two systems, the efficiency loss is calculated at the unique allocation under the system.