## 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.

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## Notes

Preferences for specific egalitarianism are quite different from preferences for a more equal distribution of income. The former seeks a more egalitarian distribution of the goods and services that determine the opportunity to earn income and prosper, while preferences for a more equal distribution of income focus on outcomes and may result in policies that introduce perverse incentives for work, saving and investment. Anand (2002) emphasizes the differential incentives associated with income inequality and health inequality and argues that aversion to inequality in health is likely to be greater than aversion to inequality in income.

The history and development of school finance reform in states in the United States provide additional evidence that people care about inequalities in the provision of elementary and secondary education. Beginning with a constitutional challenge to the system of education funding in California in the late 1960s that led to two California State Supreme Court decisions in favor of the plaintiffs, state legislatures around the country have taken action aimed at reducing inequities in funding and spending of local school districts. There is ample evidence that a court ruling that a state’s existing school finance system is unconstitutional results in a sizable equalizing effect on education spending per pupil across school districts. See, for example, Card and Payne (2002).

We find support in the literature for a concern for equity or justice, both in surveys and experimental work (see Konow, 2003, for a comprehensive discussion of the literature on justice).

See Oates’s decentralization theorem (Oates, 1972, p. 35).

Garcia-Milà and McGuire (2004) introduce this formalization of inequality in the public provision of certain goods and services across regions. The authors argue that preferences over equality in provision of publicly provided goods might stem from a desire to bring or hold a country together after an upheaval or from a desire to provide access to essential publicly provided goods to all residents of the country.

While variance accords well with Tobin’s idea of egalitarianism, it potentially violates monotonicity of preferences in that an extra unit of the publicly provided good helicopter-dropped on a high-wealth region could potentially result in a decrease in utility. This troubling possibility is not very likely, however, because it requires unrealistically strong preferences for solidarity, so strong that the marginal utility of own consumption of the publicly provided good is negative.

Note that while we equate expenditures on the publicly provided good to units of the publicly provided good, the basic construct of our paper could apply in a model where the production function for the publicly provided good incorporates local conditions.

We considered other measures of inequality, including the Atkinson inequality index and the Gini coefficient. We chose the variance measure because, as shown above, the derivative of the variance with respect to the publicly provided good is a tractable function of the relevant variables, whereas the derivatives for these other measures are complicated and difficult to interpret.

If we allowed for differential treatment of the regions, by choosing the appropriate region-specific tax rates, matching-grant rates, or publicly provided goods, the central government could implement any allocation on the Pareto frontier.

Uniform provision is challenged by Besley and Coate (2003), for example. We employ the uniformity assumption here because it is a not-implausible system due to information and political constraints. In addition, our focus on a concern for equality makes it natural to posit a common tax function and undifferentiated levels of the publicly provided good for the centralized system. Finally, these assumptions provide easy comparison with much of the literature and, in particular, with the work of Oates.

As mentioned above, we believe it is interesting to explore outcomes when the central government is constrained to treat all regions equally. If we allowed the matching rate to vary by region, the central government could choose a set of differentiated matching rates that would achieve a Pareto-optimal allocation.

This idea of

*n*-replicas was introduced by Debreu and Scarf (1963) when they generalized Edgeworth’s famous result about the shrinking of the contract curve towards the competitive allocation as the size of the economy becomes large.The transformation is \(\mu= ( 1 - \frac{1000}{1000 + \gamma\bar{e}} ) \times100\).

See Appendix B for a formal definition of the efficiency loss measure.

Others have arrived at the guaranteed minimum system, but have done so with models that in essence override local preferences. For example, Musgrave and Musgrave (1989) discuss the national objective of ensuring minimum levels of provision for merit goods. Gramlich (1985) examines the decentralized setting of cash welfare benefits in the 50 US states. His analysis leads him to take a “somewhat paternalistic position” (p. 43) in which he advocates for a federally guaranteed minimum level, with optional state supplementation. His point is that, without the federally financed minimum level, state governments, acting on their preferences, will choose benefit levels that are not high enough to raise families out of poverty. Our approach is very different in that the guaranteed minimum system arises from local preferences as a means of overcoming the market failure associated with the public good solidarity.

The point at which the efficiency curves for centralization and decentralization cross shifts to the left at a decreasing speed as the number of replicas increases.

We have explored large federations, as large as

*n*=10,000. The dominance of the guaranteed minimum system prevails.We thank an anonymous referee for this suggestion.

We assume

*s*_{ ii }=0.Inter-regional transfers will be positive only under reasonable and intuitive conditions. It can be shown that if region

*i*sends transfers to region*j*then necessarily*g*_{ i }>*g*_{ j }and \(g_{j} < \bar{g}\).This inefficiency measure is a simple instance (because there is a single commodity) of Debreu’s (1951) coefficient of resource utilization.

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## Acknowledgements

We wish to thank conference participants at the University of Kentucky IFIR and CESIfo Conference on New Directions in Fiscal Federalism; seminar participants at the Federal Reserve Bank of Chicago; participants at the Workshop on Taxation, Public Provision and the Future of the Nordic Welfare Model under the auspices of the Labour Institute for Economic Research, Helsinki, Finland; Andreu Mas-Colell, Wallace Oates, Efraim Sadka, the editor, and two anonymous referees for very useful comments. Guy Arie provided exceedingly helpful research assistance. Calsamiglia acknowledges support from the Spanish Ministerio de Educación y Ciencia (SEJ2006-09993). Garcia-Milà acknowledges support from CREI and from the Spanish Ministerio de Educación y Ciencia (SEJ2007-64340).

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## 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

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

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

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

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*),^{Footnote 20} taking all other variables as given, so as to solve the following maximization problem:

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*:

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.^{Footnote 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 *i*th 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 *i*th regional government chooses *c*
_{
i
} and *gr*
_{
i
} so as to solve

The Lagrangian expression for this problem is

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

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

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

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 *i*th 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 *i*th region chooses *c*
_{
i
} and *gr*
_{
i
} so as to solve

Assuming interior solutions for *c*
_{
i
} and *gr*
_{
i
}, we get the first-order necessary condition for region *i*:

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.^{Footnote 22} Formally, we solve the following problem:

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.

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Calsamiglia, X., Garcia-Milà, T. & McGuire, T.J. Tobin meets Oates: solidarity and the optimal fiscal federal structure.
*Int Tax Public Finance* **20**, 450–473 (2013). https://doi.org/10.1007/s10797-012-9233-0

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DOI: https://doi.org/10.1007/s10797-012-9233-0