Vertical Fiscal Externalities and Federal Tax-Transfers under Variable Factor Supplies

Abstract

1. Introduction

More often than not, in a federal economic system, for example, the USA, the EU or Canada, decentralized, that is, regional or national, fiscal policies may influence the fiscal actions of other regional/national, or even federal authorities. For example, a prevalent feature in federal economies is the co-sharing or the co-occupancy of the same tax-bases between the different levels of fiscal authority, for example, between different tiers of government, such as federal (high-level) and state (low-level).¹ This co-sharing of tax-bases potentially gives rise to a so-called ‘common pool problem’ whereby tax decisions by one level of government affect tax-payers decisions, which in turn affect the shared tax-bases. In this way, changes in state taxes can entail non-negligible effects on federal budgets and vice-versa. In the core of this common pool problem two issues of importance are the so-called ‘vertical fiscal externalities’ and ‘bottom-up’ or ‘top-down fiscal transfers’, or else negative or positive fiscal gap. Understanding the interaction of tax revenues and tax rates between federal and state governments is a significant issue in the design and implementation of federal tax systems.

‘Vertical fiscal externalities’ arise when state governments levy taxes discounting the effect on federal tax revenues and consequently the effect on other states in so far that the latter are entitled to a share of the federal tax revenues.² A pivotal conclusion of the relevant literature which considers models with a single factor in variable supply is that when tax-bases are shared among different tiers of government, and tax rates are specific, the emerging ‘vertical fiscal externalities’ are negative, implying inefficiently high non-cooperative specific taxes (e.g., Boadway and Keen 1996; Boadway et al. 1998; Keen 1998; Hoyt 2001; Keen and Kotsogiannis 2004; Köthenbürger 2004, Boadway and Tremblay 2012).³ With ad valorem taxes, however, the ‘vertical fiscal externality’ can be either positive or negative (e.g., Dahlby and Wilson 2003; Kotsogiannis and Martinez 2008; Karakosta 2010; Sas 2017).⁴

‘Bottom-up’ and ‘top-down transfers’ are the resource transfers from low to high-level governments, and vice-versa. The main conclusions of the relevant literature are that when the federal government pre-commits vis-a-vis the state governments, that is, centralized leadership, and tax bases are shared between them, then the federal government replicates, at all times, the unitary government’s optimal (second-best) tax setting in the presence of transfers (e.g., Boadway and Keen 1996; Boadway and Tremblay 2006; Kotsogiannis and Martinez 2008; Sas 2017).⁵ If, however, federal transfers are missing, then the federal government cannot replicate the unitary government’s second-best policy outcome (e.g., Keen and Kotsogiannis 2002).⁶ When tax bases do not perfectly overlap, then under centralized leadership the federal government cannot replicate the unitary government’s second-best policy, even in the presence of federal transfers (e.g., Koethenbuerger 2008).⁷

The objective of the paper is to re-examine the issues of the ‘vertical fiscal externality’ and of the ‘bottom-up/top-down transfers’, and their relationship in a model of a multistates, or jurisdictions, federal economy with federal-cum-state specific factor taxes, and where capital and labor are in variable supplies. Variable supply of capital is due to international mobility, across the states of the federation as well as between the federation and the rest of the world. The supply of labor is variable due to endogenous labor-leisure decision. Specific capital and labor taxes by the federal government finance the provision of a federal public consumption good, and specific capital and labor taxes by a state government finance the provision of a state public consumption good. Within this framework, we first derive the state’s simultaneous optimal setting of capital and labor taxes. We show that when labor and capital are substitutes in production, the optimal policy for a representative state calls for a capital and a labor tax. Then, we examine the impact of the capital and labor tax setting behavior by a state on the provision of the federal public consumption good, that is, ‘the vertical fiscal externality’. When capital and labor are substitutes in production, the sign of the ‘vertical fiscal externality’ can be reversed from negative to positive, implying inefficiently low instead of high noncooperative capital taxes. Intuitively, when state governments raise their own specific capital tax rates, capital leaves the federation and, on the one hand the capital tax base of the federal government contracts and as a result federal tax revenues fall. According to the relevant literature this result leads to inefficiently high noncooperative specific taxes. On the other hand, a rise in state capital taxes increases labor supply when capital and labor are substitutes in production. This expands the labor tax base of the federal government, and the total federal tax revenues can increase. Thus, when capital and labor are in variable supply and taxes are specific, the ‘vertical fiscal externality’, under plausible conditions, can be reversed from negative to positive.

We examine a unitary government’s optimal rule, that is, second-best optimum, for the provision of the local and federal public consumption goods, and we show that the federal tax-transfer policy can achieve this second-best optimum under centralized leadership. The optimal federal tax-transfer policy when capital and labor are substitutes in production requires a capital tax and, possibly, a ‘top-down transfer’. When capital and labor are complements in production the optimal federal policy is a capital subsidy, while transfers are ‘bottom-up’. Consequently, the relationship in production between capital and labor in variable supply is the driving force of our results.

2. The Model

Consider a federal economy with N symmetric states, populated with $Λ≥1$ identical and immobile across states individuals. Since households are assumed identical, following standard practice, we normalize $Λ=1$⁠, thus N also denotes the overall population of the federal economy. The federal economy is assumed small in world commodity and capital markets. Thus, in the federation, all commodity and factor markets are perfectly competitive, and commodity prices of the internationally traded goods and the rate of return to the internationally mobile capital are fixed and exogenous.

In a representative state, M freely tradable consumption goods are produced, and production technologies are assumed identical across all states. Several primary factors are used in the production of the traded goods, among which, capital (K), and labor (L), are in variable supply. The supply of capital is variable due to international mobility, while the supply of labor, an internationally and interstate immobile factor, is variable due to endogenous labor-leisure choice. All other factors are immobile and in fixed endowments.

The production side is represented by the Gross Domestic Product $(GDP)$ function. The GDP function is given by $R(p,K,L,Ω)=max{∑j=1Mpjxj:F(X,K,L,Ω)≤0}$ and denotes the maximum value of output production at world commodity prices p, given the country’s variable domestic supply of capital K and level of employment L, and the vector Ω of all other internationally immobile and in fixed supplies factors. $F(X,K,L,Ω)$ is the aggregate production possibilities set, where $X$ is the vector of goods produced. For simplicity, the vectors $(p,Ω)$ are omitted from the subsequent analysis because of our assumptions regarding fixed world commodity prices and fixed supplies of all other factors. Then, the GDP function reduces to R(K, L).⁸$K=K¯+k$ is the supply of capital in the representative state, $K¯$ is the state’s own capital endowment. If k < 0, the state is a net capital exporter to all other states of the federation and to the rest of the world, and if k > 0, the state is a net capital importer. The derivatives of the $R(.)$ function with respect to K and L, that is, $RK=∂R/∂K$ and $RL=∂R/∂L$⁠, denote the marginal revenue products of capital and labor. Strict concavity is assumed of the $R(.)$ function with respect to K and L, that is, $RKK(=∂RK/∂K)<0$ and $RLL(=∂RL/∂L)<0$⁠.⁹ Furthermore, $RKL(=∂RK/∂K)=RLK$⁠, can be either positive or negative, implying respectively, that capital and labor can be either complements or substitutes in production.¹⁰

A representative household derives utility from the consumption of the traded goods, leisure (l), and the two public consumption goods, all of which are considered normal in consumption.¹¹ A ‘state’ public consumption good (g) provided free of charge by the state authority to its residents, and a ‘federal’ public consumption good (G) provided free of charge by the federal authority to the residents of all states in the federation. Both these goods are internationally traded and their prices are given. The representative household’s preferences are depicted by the minimum expenditure function $E(g,G,L,u)=$$min{∑j=1j=MpjCj|U(C1,..CM,1−L,g,G)≥u}$⁠.¹² It denotes the minimum expenditure on the traded goods required to attain a given level of utility $(u),$ given the levels of leisure $(l=1−L)$⁠, and of consumption of the state and federal public goods, g and $G$⁠. The fixed commodity prices are tacitly omitted as arguments from the $E(.)$ function. The derivatives of the $E(.)$ function, E_g and E_G are negative implying that an increase in the consumption of the state and federal public consumption goods reduces the expenditure on private goods required to achieve a given level of utility u (e.g., Kotsogiannis et al. 2005; Raimondos-Møller and Woodland 2006). Thus, $−Eg>0$ and $−EG>0$ denote respectively, the so-called marginal willingness to pay for the consumption of the state and federal public goods, and $−Egg<0$ and $−EGG<0$⁠. The derivative E_u > 0, is the inverse of the marginal utility of income, and E_L captures the representative household’s reservation wage, denoted by $w˜$⁠, that is, $EL=w˜$ (e.g., Anderson and Neary 2016).¹³ The $E(.)$ function is strictly convex in L, that is, E_LL > 0, implying that an increase in labor supply, reduces leisure and thus it increases the reservation wage.

3. Optimal State Capital and Labor Taxes

4. Benchmark Case: The Unitary Government

That is, the unitary government is indifferent between spending a dollar for the provision of g or G when the marginal willingness to pay for the provision of a state’s public consumption good equals the federal marginal willingness to pay for the federal public consumption good. The optimality rule (14) determines the unitary optimal capital tax $(τ)$ so that the Marginal Cost of Public Funds (MCPF), i.e., $Kgτ,$ equals the marginal willingness to pay for the state public consumption good and the federal marginal willingness to pay for the provision of the federal public consumption good.²³ Thus, equation (14) determines the unitary second-best optimum.²⁴ We can state: 

5. Capital Taxes and Vertical Fiscal Externality

Next, we consider the ‘vertical fiscal externality’, which is the impact of the tax setting behavior of the representative state government on the provision of the federal public consumption good, i.e., $sign(Gt)$ given in equations (5). As reviewed, Boadway and Keen (1996) demonstrate that the sign of the ‘vertical fiscal externality’ is unambiguously negative when taxes are specific and only one factor of production is in variable supply. We revisit these results in the case of endogenous capital and labor supplies and specific factor taxes. For expositional clarity of the results, we consider the case of a federal economy with two identical states $(N=2)$⁠, for example, Home and Foreign. Thus, we examine the ‘vertical fiscal externality’ due to the tax setting behavior of one state, for example, say $t∗$ of Foreign, on the provision of the federal public consumption good, and on the welfare level of Home, that is, $du/dt∗$⁠.

Equations (15) and (16) indicate that in the present context of variable labor and capital factor supplies, even the change of a specific capital tax by one state entails an ambiguous impact on the provision of the federal public good, and on the welfare of the other state that is, the change of a state capital tax entails an ambiguous ‘vertical fiscal externality’. If $RL∗K∗∗$ is positive, then an increase in $t∗$ by Foreign entails an unambiguously negative ‘vertical fiscal externality’. This is to say that the non-cooperative equilibrium capital tax $t∗$ is inefficiently high relative to the cooperative one.²⁶ If, however, $RL∗K∗∗$ is negative, that is, K and L are substitutes in production, then the change of $t∗$ by Foreign leads to a positive ‘vertical fiscal externality’ if $(SRL∗K∗∗+TZL∗L∗∗)<0$⁠, i.e., $|SRL∗K∗∗|>TZL∗L∗∗.$ In this case, the non-cooperative equilibrium capital tax is inefficiently lower compared to the cooperative one. That is, if the increase in labor tax revenue due to capital outflow is larger than the decrease in capital tax revenue, then the increase in $t∗$ increases the federal tax revenue. Thus, the higher $t∗$ has a negative (ambiguous) effect on Home’s welfare when $RL∗K∗∗$ is positive (negative).²⁷ We state the following Proposition: 

5.1 Taxes on fixed factors income

5.2 Capital mobility only within the federation

The state labor markets equilibrium conditions, that is, equilibrium conditions (1) and the corresponding one for Foreign, and the federal capital market equilibrium condition (19) are used to determine the effects of changes in capital and labor taxes on the supplies of the two factors, that is, L, $L∗$ and $K(K∗)$⁠, in the two states. The relevant comparative statics are presented in Appendix by equations (A.7).

Equation (20) indicates that a change in Foreign’s capital tax $t∗$⁠, affects Home’s provision of the public consumption good g. Specifically, when capital and labor are substitutes in production, the emerging horizontal fiscal externality can be either positive or negative. That is, when $RLK<0,$ we get $gt∗≷0.$ When they are complements, the emerging horizontal fiscal externality is positive, that is, $RLK>0$ gives $gt∗>0$⁠. The horizontal fiscal externality remains even when the two states are symmetric.

Equation (21) indicates that unless capital and labor are of different relationship in production in the two states, for example, R_LK > 0 and $RL∗K∗∗<0$⁠, the resulting vertical fiscal externality is of ambiguous sign, that is, $Gt∗≷0$⁠. When the two states are symmetric, that is, $ZLL=ZL∗L∗∗$ and $RLK=RL∗K∗∗$⁠, and since by assumption $dK=−dK∗$⁠, then the resulting vertical fiscal externality is zero.³²

The previous analysis considers the two polar cases where (i) the federal economy is small in world capital markets, and capital is internationally mobile, (ii) capital is only mobile within the federation. We find that in the former case there is vertical but not horizontal fiscal externalities, while in the later, when states are symmetric, there is horizontal but not vertical fiscal externalities. When the federal economy is large in world capital markets, capital is internationally mobile, and a state increases its capital tax, then, part of its capital moves to the other states and to the rest of the world. In this case, we except both types of fiscal externalities to exist.³³ However, when states are symmetric the vertical fiscal externality is expected to be smaller than that in case (i), but still of ambiguous sign. The horizontal fiscal externality is expected to be smaller than that in case (ii), but still of ambiguous sign.

6. Federal Tax-Transfers with Centralized Leadership

We now address the issue of resource transfers between the different levels of government. Previous studies using models with one factor of production in variable supply, conclude that with specific taxes and under centralized leadership, resulting to negative vertical fiscal externality, the federal government sets a negative specific tax on the rate of return to the factor in variable supply, which can lead to bottom-up transfers (e.g., among others Boadway and Keen 1996). Subsequent studies demonstrate that with ad valorem taxation the above results can be reversed if the demand for the factor in variable supply is inelastic (e.g., Kotsogiannis and Martinez 2008; Karakosta 2010; Sas 2017). We revisit these results in the case of variable labor and capital supplies and specific factor taxes. We consider a two-stage game where the two levels of government pursue separately the setting of fiscal policy, for example, the choice of capital taxes $t$ and T, treating as exogenous the rates of federal and state labor taxes S and s. We assume that the federal government holds a first mover’s advantage vis-a-vis all states, that is, centralized leadership. In the first stage the federal government chooses the capital tax rate T and the level of the vertical transfer F in order to maximize the federal economy’s cumulative welfare level W = Nu. In the second stage a typical state government chooses its own specific capital tax t, in order to maximize its own welfare u, given T and F. As standard practice calls for, the subgame perfect equilibrium is solved backwards.

Equations (A.9)–(A.13) in the Appendix provide the details of this derivation.³⁴

Equation (26) indicates that when $Gt(1+tT)<0(>0)$⁠, then the marginal willingness to pay for the provision of the state public consumption good is lower (higher) than that for the federal public consumption good, that is, $−Eg<−NEG(−Eg>−NEG)$⁠.³⁵

Since t_F is negative, equation (27) indicates that the direction of the transfer between the state and the federal government depends on the sign of G_t. The sign of G_t, depends on the relationship between K and L in production, see discussion of equations (15) and (16). If G_t > 0, then, $−Eg>−NEG$⁠. That is, the increase in the federal labor income tax revenue due to the increase in the state capital tax is larger than the decrease in federal capital income tax revenue, leading to an increase in the federal public good, making its level too high. In this case, the transfer should go from the federal to the state government, that is, a ‘top-down transfer’. Intuitively, the transfer should go from the government providing the public good with lower marginal willingness to pay for its provision, in this case being the federal government, to that providing the public good with higher marginal willingness to pay for its provision, in this case being the state or regional government. The transfer will affect the state capital tax and the vertical externality. Since $GttF<0,$ the transfer reduces the vertical externality. If G_t < 0, then, $−Eg<−NEG$⁠. In this case the increase in the state capital tax reduces the provision of the federal public good, making its level too low. The transfer should go from the state to the federal government, that is, a ‘bottom-up transfer’.

7. Concluding Remarks

We reconsider the ‘vertical fiscal externality’ and the direction of federal-state transfers, and their relationship in a model of a multistates federal economy with federal-cum-state specific factor taxes and variable supplies of labor and capital in production. The supply of labor is variable due to endogenous labor-leisure decision and the supply of capital is variable due to its international mobility. We show that, first, when capital and labor are substitutes in production, then, (i) the optimal policy for a representative state calls for a capital and a labor tax, and (ii) the ‘vertical fiscal externality’ can be reversed from negative, implying inefficiently high noncooperative specific capital taxes, to positive, implying inefficiently low noncooperative specific capital taxes. Second, with co-sharing of tax bases and variable supplies of capital and labor, the federal government by precommiting to its policy achieves the unitary government’s second-best optimum. Specifically, with an exogenous labor tax, when capital and labor are complements in production, then, they are substitutes in production, the second-best optimum is achieved with a capital tax and possible ‘top-down transfers’.

In our model of a small open federation with international capital mobility, horizontal fiscal externalities do not exit. If, however, capital is only mobile within the federation, then both horizontal and vertical externalities exist. The sign of the vertical fiscal externality is generally ambiguous, while that of the horizontal fiscal externality is positive when capital and labor are complements in production, and ambiguous when they are substitutes. When states are symmetric, the horizontal fiscal externality remains while the vertical one seizes to exist.

Footnotes

Acknowledgements

We graciously acknowledge the constructive comments and suggestions of the Editor and an anonymous referee of the Journal, of participants in the 2018 Annual Meeting of Association of Southern European Theorists (ASSET), the 2017 Annual Conference of the European Trade Study Group (ETSG), and the 2016 Asian Meetings of the Econometrics Society (AMES). The authors remain responsible for any shortcomings of the paper.

Appendix

Effects of changes in $τ$ and $σ$ on $K$ and L

Comparative statics results for equations (5)

Using the results in (A.2) where $τ=t+T$⁠, we obtain the expressions in the text.

Comparative statics results for equation (17)

Taxes on fixed factors income

Differentiating the federal budget constraint with respect to $t∗$⁠, and since L and K are not affected by changes in $t∗$⁠, gives equation (18) in the text.

Capital mobility only within the Federation