Productive effects of public spending, spillovers, and optimal matching grant rates

The paper asks how a central government should set an optimal matching grant rate for local public expenditure when that expenditure has dual characteristics: it provides local public good benefits (consumption effects) and functions as a public input that raises productivity (production effects). The study extends traditional matching-grant analyses that focus only on public goods and consumption spillovers by introducing productive effects and production spillovers. It examines two settings—immobile private capital and perfectly mobile private capital—to understand how consumption spillovers (β) and production spillovers (γ) interact with productivity effects and, when relevant, tax competition, to determine optimal matching rates. The motivation stems from the difficulty of cleanly separating local public goods from local public inputs (e.g., roads, education, environmental quality) and from mixed empirical evidence on the presence and magnitude of productivity spillovers across regions. The purpose is to derive conditions and comparative statics for optimal grant rates that internalize interjurisdictional externalities and fiscal externalities, offering guidance for policies where public spending has both consumption and production impacts.

The study builds on several strands of literature. Early work on matching grants and voluntary provision of public goods (e.g., Boadway et al., 1989; Feldstein, 1980, 1987; Warr, 1982; Andreoni and Bergstrom, 1996; Roberts, 1992; Glazer and Konrad, 1993; Kirchsteiger and Puppe, 1997; Akai and Ihori, 2002) did not incorporate tax competition with mobile capital. Tax competition models (Wilson, 1986; Zodrow and Mieszkowski, 1986; Bjorvatn and Schjelderup, 2002) show underprovision of public goods due to fiscal externalities from capital mobility, in addition to free-riding from consumption spillovers. Zodrow and Mieszkowski (1986) model public inputs that raise marginal productivity of capital; later critiques and refinements (Noiset, 1995; Matsumoto, 1998) show under- or overprovision can arise depending on production technology; Keen and Marchand (1997) distinguish public goods versus public inputs and argue public inputs may be overprovided relative to public goods. Ogawa (2006) analyzes optimal matching grants accounting for consumption spillovers and tax competition but not for local public inputs. Empirical studies of productivity spillovers from public investment are mixed: early work (Holtz-Eakin, 1994; Holtz-Eakin and Schwartz, 1995) finds little evidence, while later work (Pereira and Roca-Sagalés, 2003) supports their relevance. The paper systematizes interjurisdictional externalities into four types: (i) consumption spillovers; (ii) tax externalities from capital taxation; (iii) public input fiscal externalities via attraction of mobile factors; and introduces (iv) production spillovers, whereby local public inputs raise productivity in other jurisdictions. Prior models typically consider public goods and public inputs separately; this paper considers a good embodying both characteristics and both spillovers.

The study develops a theoretical model with n identical jurisdictions. Each jurisdiction i has one immobile resident with utility u(x_i, G_i), where G_i = g_i + β Σ_{j≠i} g_j captures consumption spillovers (β ∈ [0,1]). Local spending g_i serves as a public good and a public input. The production side initially assumes immobile private capital and an aggregate production function f(K_i*), with diminishing returns, where K_i* = g_i + γ Σ_{j≠i} g_j captures production spillovers (γ ∈ [0,1]). Jurisdictions finance g_i with a lump-sum tax z_i and a central matching grant s_i = m g_i; residents also pay a uniform lump-sum tax h for the central budget. Central government’s budget: Σ_i s_i = n h. Two-stage game: Stage 1, the center chooses (m, h); Stage 2, jurisdictions choose (z_i, g_i) taking (m, h) as given. Jurisdictions maximize u subject to budget constraints; the FOC links marginal utility trade-offs to the net marginal cost of g_i incorporating the match rate. The social planner maximizes Σ_i u_i subject to aggregate resource feasibility Σ_i x_i + Σ_i g_i = Σ_i f(K_i*). Comparing jurisdictional FOCs to the planner condition yields the optimal uniform matching rate m that decentralizes the Pareto optimum in a symmetric equilibrium. For immobile capital, the optimal matching rate is m = [β(n − 1)]/{1 + β(n − 1) + [(γ − β)(n − 1) f′(K)]/[1 + β(n − 1)]}, where f′(K) is the marginal productivity of the public input component. This nests special cases: with no productive effect (f′=0), m reduces to Ogawa’s (2006) expression m₀ = β(n−1)/[1+β(n−1)]; with β=1 it matches Boadway et al. (1989). Comparative statics show ∂m/∂β > 0 when 1 − f′(K)[1 + γ(n − 1)] > 0. The analysis then extends to perfectly mobile private capital. Output in i is y_i = f(k_i, K^g), with capital perfectly mobile and after-tax returns equalized: f_k(k_i, K^g) − t_i = r. Jurisdictions finance g_i using a capital tax t_i and the matching grant: t_i k_i + s_i = g_i, with s_i = m g_i; the center sets h to balance the matching budget. Jurisdictions choose t_i to maximize resident utility. The FOC can be written (in elasticity form) as 1 + E(1 − β) over (1 − m) minus f_{kG}(k, K)[1 + E(1 − γ)] equals ∂G/∂x_i, where E = (∂k/∂t)(t/k) is the elasticity of private capital with respect to the jurisdiction’s capital tax and can be positive or negative because public input provision can attract capital, while higher taxes repel it; production spillovers (γ) mitigate the attraction effect. The planner condition analogous to the immobile case implies an optimal matching rate m that depends on β, γ, f_{kG}, and E. The paper provides expressions (22) and (22′) showing how the optimal m with productive effects (m_γ) relates to the baseline without such effects (m₀) and to Wildasin’s (1989) m_δ term. Consistency checks show that when E = 0 (capital immobile) the expression collapses to the earlier immobile-capital m, and when f_{kG} = 0 it reduces to Ogawa (2006). The model assumes symmetry and uses a conventional Nash behavior in the policy stage for tractability.

• With immobile private capital: The optimal matching grant m increases with the degree of consumption spillover β (∂m/∂β > 0). The “productive effect” (difference between m with productive characteristics, m_γ, and without, m₀) depends on the relative size of production spillovers γ to consumption spillovers β: m_γ > m₀ when γ > β; m_γ < m₀ when γ < β; m_γ = m₀ when γ = β. Intuition: productive effects create an additional benefit from higher local spending funded by taxes; large production spillovers shift benefits to other regions, requiring a higher matching rate to internalize them, while small γ reduces the need for matching relative to the pure public good case. - The direct, positive effect of β on m dominates the indirect productive effect via β that would otherwise reduce m; hence optimal m remains positive under standard conditions on marginal productivity (m_i > 0 when f′(K) > 0 and stability conditions hold). - When γ = 0, higher marginal productivity f′(K) reduces the optimal m relative to the normal public-good case (because productive gains mitigate underprovision driven by consumption spillovers). - With perfectly mobile private capital: The sign of the productive effect on m depends on both γ versus β and on the elasticity E of capital with respect to the tax rate (relative to δ, the elasticity holding public inputs fixed). Specifically, m_γ is lower than m₀ when E > δ and γ < β, or when E < δ and γ > β; m_γ is higher than m₀ when γ = 1 and E > δ (unless β = 1). - Proposition-level insights: (i) The productive effect on m depends on whether production spillovers exceed consumption spillovers (γ > β) in the immobile-capital case; (ii) The direct β effect raising m dominates the productive effect via β that lowers m; (iii) With mobile capital, the productive effect depends jointly on E versus δ and on γ versus β (Proposition 3); (iv) When 1 ≥ γ ≥ β and E > 0, if γ = 1 > β, then m_γ ≥ m₀, but if 1 > γ = β, then m_γ < m₀ (Proposition 4). - Table 1 summarizes multiple parameter regions (immobile vs mobile capital, ranges of E, γ, and β) indicating when m_γ is greater than, equal to, or less than m₀.

The results directly address the central question of how productive effects and interjurisdictional spillovers shape optimal central matching grants for local spending that serves both as a public good and a public input. By embedding both consumption and production spillovers into a unified model, the paper shows that optimal matching is not monotone in productive impacts alone; rather, it hinges on the relative magnitudes of production versus consumption spillovers and, under tax competition, on how public input provision affects capital mobility. This clarifies when central governments should use higher matching rates to internalize positive externalities (large production spillovers) and when productive effects reduce the need for matching (low γ or high marginal productivity mitigating underprovision). In the mobile-capital setting, the framework integrates fiscal externalities with benefit spillovers, highlighting the role of the tax-base elasticity E: productive inputs can attract capital (lowering underprovision), but production spillovers to other regions attenuate this effect; the optimal grant must correct the net distortion. These insights are relevant for designing intergovernmental transfers for spending items like environmental quality, infrastructure, and education, where benefits span consumption and production channels and may spill across borders.

The paper contributes a unified theoretical analysis of optimal matching grants when local expenditures jointly provide public good benefits and function as public inputs, with both consumption and production spillovers. It derives optimal matching formulas for cases with immobile and perfectly mobile private capital, identifies comparative statics, and delineates when productive effects increase or decrease the optimal matching rate relative to the pure public-good case. Policy implications include that matching grant rates for expenditures with strong public-input characteristics and significant production spillovers (e.g., environmental policies) may optimally be lower or higher than for traditional public goods depending on γ versus β and on tax-base elasticities under capital mobility; in particular, when productive gains mitigate underprovision, lower matching may suffice, while large production spillovers call for higher matching to internalize external benefits. Future research should analyze richer strategic settings where jurisdictions play Nash games in taxes and/or expenditures with benefit spillovers, and explore empirical calibration to assess the magnitude of γ, β, and E in practice.

• The analysis is theoretical and relies on symmetric jurisdictions with representative immobile residents; no empirical calibration or data are provided. - The strategic interaction is modeled as a conventional Nash game where each jurisdiction treats others’ policies as given; more complex Nash games in taxes or expenditures with benefit spillovers are acknowledged but not analyzed due to tractability. - Matching grants are uniform and financed by lump-sum national taxation; local finance is via lump-sum taxes (immobile capital case) or a single capital tax (mobile capital case), which abstracts from richer tax instruments and heterogeneity. - Production and consumption spillovers are summarized by reduced-form parameters (β, γ), and the productive effect is captured through marginal productivity terms (f′(K), f_{kG}); potential non-linearities, heterogeneity across sectors, and dynamic considerations are not modeled. - The model assumes identical jurisdictions and does not consider political or administrative constraints beyond budget balance and uniformity.