Labor income taxes in an economic federation with proportional membership fees

It is useful to have a benchmark with which we can compare the results to be derived below. In line with much of the literature on fiscal federalism, the cooperative equilibrium will be used as the benchmark. Here a central planner determines the taxes in each state i, which means that the central planner solves the following maximization problemsubject towhere \( {\text{GNI}}_{i}^{ \circ } = {\text{GDP}}_{i}^{ \circ } = F\left[ {L_{i}^{ \circ } \left( {t_{i} } \right)} \right] \) holds when labor is immobile. Since the central planner can determine both the slope (\( t_{i} \)) and the intercept (\( T_{i} \)) of the labor income tax function in each member state, it is possible to implement any desired combination of work hours and private consumption for the consumer residing in state i. This implies that the central planner has a sufficient number of policy instruments at his/her disposal to implement the first best policy.¹² In the Appendix, it is shown that this policy can be summarized by the following equations:where \( \phi_{i}^{{\prime }} = \phi^{{\prime }} \left( {g_{i} } \right) \), \( \phi_{j}^{{\prime }} = \phi^{{\prime }} \left( {g_{j} } \right) \) and \( u_{c}^{i} = \partial u_{i} /\partial c_{i} \). The first equation in (4) implies complete redistribution in the sense that the same amount of the local public good will be provided in each member state, i.e., \( g_{i} = g_{j} \). The second equation in (4) shows that there is no incentive to implement a distortionary tax on labor income since the tax revenue in member state i can be raised via the lump-sum tax \( T_{i} \). As for the third equation in (4), it shows that the marginal rate of substitution between public and private spending (\( \phi_{i}^{{\prime }} /u_{c}^{i} \)) is set equal to the marginal cost of public funds (MCPF) which is one.

Let us now return to the case where the public sector is made up of separate federal and local governments. Beginning with the federal government, it is straightforward to show that the solution to its maximization problem, both when labor is immobile and mobile, implies that \( s_{i} \) and \( s_{j} \) are set so as to satisfy the first best rule \( \phi_{i}^{{\prime }} = \phi_{j}^{{\prime }} \). This equation, together with the federal government’s budget constraint, implicitly defines \( s_{i} \) and \( s_{j} \) as functions of the local governments’ tax instruments; \( s_{i} \left( {t_{i} ,T_{i} ,t_{j} ,T_{j} } \right) \) and \( s_{j} \left( {t_{i} ,T_{i} ,t_{j} ,T_{j} } \right) \). These functions will be referred to as federal reaction functions, and all interpretations below will be based on the premise that \( s_{i} \) and \( s_{j} \) are positive.

Let us now turn to the local government in member state \( i = 1, 2 \). It chooses \( t_{i} \) and \( T_{i} \) to maximize the welfare function \( U_{i} = u\left[ {c_{i}^{ \circ } \left( {t_{i} ,T_{i} } \right),h - L_{i}^{ \circ } \left( {t_{i} } \right)} \right] + \phi \left( {g_{i} } \right) \), subject to the budget constraint \( g_{i} = T_{i} + t_{i} w_{i}^{ \circ } \left( {t_{i} } \right)L_{i}^{ \circ } \left( {t_{i} } \right) - s_{i} F_{i} \left( {L_{i}^{ \circ } \left( {t_{i} } \right)} \right) \). In the Appendix, we solve this maximization problem both when government i acts as a Nash follower vis-à-vis the federal government (and vis-à-vis the other local government), and when government i exercises decentralized leadership vis-à-vis the federal government (but where government i treats the choices made by the other local government as exogenous). We summarize the results as follows;

To interpret part (i), we first recall that the optimal policy implemented under cooperation features \( t_{i} = 0 \). To explain why this result does not hold in the noncooperative Nash equilibrium, we conduct the following thought experiment. Assume that the local government initially has made an optimal choice of \( T_{i} \) conditional on that \( t_{i} \) is zero. Does the local government have an incentive to change \( t_{i} \) from this initial value? To address this question, we differentiate the local government’s objective function w.r.t. \( t_{i} \) and evaluate the resulting expression at \( t_{i} = 0 \). By using that the optimal choice of \( T_{i} \) satisfies \( \phi_{i}^{\prime } /u_{c}^{i} = 1 \), the welfare effect of an infinitesimal increase in \( t_{i} \) reduces to¹³where we recall that \( {\text{d}}L_{i}^{ \circ } / {\text{d}}t_{i} < 0 \). Equation (5) implies that if member state \( i \) is a net contributor of revenue to the federal government (\( s_{i} > 0 \)), then the local government has an incentive to implement a positive (marginal) tax on labor. The reason is that by increasing this tax from zero, employment is reduced and so is \( {\text{GDP}}_{i} = F\left[ {L_{i}^{ \circ } \left( {t_{i} } \right)} \right] \). The reduction in \( {\text{GDP}}_{i} \) implies that the fee paid to the federal government will be smaller (since the federal fee is proportional to GDP) which, on the margin, is welfare improving from the perspective of the local government. At the optimum, \( t_{i} \) is set equal to \( s_{i} \) because then the marginal distortionary effect of the tax equals the marginal benefit of reducing the fee paid to the federal government. We will refer to this as the proportional fee motive for taxing labor.

Let us now turn to part (ii) in Proposition 1. It shows that if the local government is able to exercise decentralized leadership, then it has no incentive to implement a distortionary tax on labor. To explain this result, we make the same thought experiment as in the preceding paragraph and assume that the government initially has made an optimal choice of \( T_{i} \) conditional on \( t_{i} = 0 \). A key difference here is that the decentralized leader recognizes the federal reaction function \( s_{i} \left( {t_{i} ,T_{i} ,t_{j} ,T_{j} } \right) \). In the Appendix, it is shown that \( \partial s_{i} /\partial T_{i} = 0.5 /Y_{i} \), and when we use this result in the decentralized leader’s first-order condition for \( T_{i} \), it follows that \( \phi_{i}^{{\prime }} /u_{c}^{i} = 2 \). This reflects that if the local government, ceteris paribus, increases its tax revenue via \( T_{i} \), then the federal government will respond by collecting half of that revenue so as to maintain complete redistribution between the member states. Therefore, when the local government exercises decentralized leadership, it recognizes that \( {\text{MCPF}}_{i} = 2 \). Does the local government in this situation have an incentive to let \( t_{i} \) deviate from zero? To address this question, we differentiate the local government’s objective function w.r.t. \( t_{i} \) and evaluate the resulting expression at \( t_{i} = 0 \), while we at the same time recognize that \( \phi_{i}^{{\prime }} /u_{c}^{i} = 2 \). This produces¹⁴Since the first term on the right-hand side (RHS) is the proportional fee motive to tax labor, it follows that two additional terms appear on the RHS of (6) compared with Eq. (5). The first additional term [the second term on the RHS of (6)] is positive and reflects that the local government has a higher marginal valuation of additional funds than the consumer (\( \phi_{i}^{{\prime }} > u_{c}^{i} \)). This implies that raising additional revenue via a tax on labor will be welfare improving. The second additional term reflects that the decentralized leader takes into account how the federal government will respond to an increase in the labor tax from the initial value \( t_{i} = 0 \). It can be shown that¹⁵The first term inside brackets reflects that if the local government increases its tax revenue via a higher tax on labor income, then the federal government will respond by collecting half of that additional tax revenue via a higher federal tax rate. The second term inside brackets reflects the federal government’s response to the proportional fee motive to tax labor at the local level. When \( s_{i} > 0 \), the federal government responds by increasing the federal tax rate so as to offset the reduction in federal revenue.¹⁶ Since the local government anticipates how the federal government will respond to an increase in \( t_{i} \), the proportional fee motive to tax labor vanishes under decentralized leadership. This can be seen if we substitute Eq. (7) into Eq. (6), which produces \( \partial U_{i} /\partial t_{i} |_{{t_{i} = 0}} = 0 \). Hence, the decentralized leader has no incentive to let the marginal tax rate deviate from zero when labor is immobile.

When labor is mobile, the consumer’s labor supply, \( l_{i}^{*} \), may differ from the amount of labor employed by the domestic firm, \( L_{i}^{*} \), while the local government, in turn, is restricted to tax labor income at the source. Hence, the local tax base for labor is \( w_{i}^{*} L_{i}^{*} \). Since the federal fee is proportional to \( {\text{GDP}}_{i}^{*} = F_{i} \left( {L_{i}^{*} } \right) \), government i’s maximization problem can be stated as follows:subject toGovernment i treats the policy instruments chosen by the other local government (\( t_{j} \) and \( T_{j} \)) as exogenous. If government i exercises decentralized leadership, then it takes effects via the federal reaction function \( s_{i} = s_{i} \left( {t_{i} ,T_{i} ,t_{j} ,T_{j} } \right) \) into account when choosing \( t_{i} \) and \( T_{i} \). For the analysis below, we introduce the short notation \( a_{i} = - 1 /\left[ {w_{i}^{*} \left( {\partial L_{i}^{*} /\partial t_{i} } \right)} \right] > 0 \) and observe that the after-tax wage, \( w_{j}^{n} \), and the tax base for labor in the other state, \( I_{j}^{*} \), are determined by:Equation (11) implies that the after-tax wage in the other state is decreasing in \( t_{i} \), that isIn the Appendix, we solve the local government’s problem and derive the following results;

As a point of reference, let us first consider what the outcome would be under cooperation, where a central planner makes all tax and expenditure decisions. This outcome is obtained by solving the same problem as in Sect. 3.1 with the exception that labor is mobile. It can be shown that the solution to this problem will be the same as that outlined in Sect. 3.1, i.e., the central planner implements the first best policy \( \phi_{i}^{{\prime }} /u_{c}^{i} = 1 \) and \( t_{i} = 0 \).

With this benchmark in mind, let us first interpret the tax policy implemented by the local government in the noncooperative Nash equilibrium. The first term on the right-hand side in Eq. (14) reflects a pecuniary motive to influence the after-tax wage in the other state. This type of motive is well understood from the literature on tax competition (see, e.g., Peralta and van Ypersele 2005; Itaya et al. 2008; Ogawa 2013). In the context of this model, it implies that if state i is a net exporter of labor (\( l_{i}^{*} > L_{i}^{*} \)), then an increase in the after-tax wage in the other state will have a positive effect on the factor income from abroad. This provides government i with an incentive to use tax policy to push up \( w_{j}^{n} \). An increase in \( w_{j}^{n} \) can be accomplished by subsidizing domestic labor via a reduction in \( t_{i} \). This reduces the export of labor to state j which contributes to push up \( w_{j}^{n} \). If state i instead is a net importer of labor (\( l_{i}^{*} < L_{i}^{*} \)), then the argument for a higher (marginal) tax rate is analogous. As for the second term on the RHS in Eq. (14), it shows that the proportional fee motive for taxing labor, which is discussed in Sect. 3.2, is present also when labor is mobile.

Turning to the tax policy implemented under decentralized leadership, we begin by recalling that the decentralized leader implements \( t_{i} = 0 \) in the closed economy. Part (ii) in Proposition 2 shows that this result does not hold when labor is mobile. Instead, labor mobility provides the decentralized leader with two distinct motives for implementing a nonzero tax on labor. The first is the pecuniary motive which was discussed in the previous paragraph, while the second is to use \( t_{i} \) to influence the size of the tax base for labor in the other state; \( I_{j}^{*} \left( {t_{i} ,t_{j} } \right) = w_{j}^{*} \left( {t_{i} ,t_{j} } \right)L_{j}^{*} \left( {t_{i} ,t_{j} } \right) \). The reason is that an increase in \( I_{j}^{*} \) leads to more tax revenue in state j (as long as \( t_{j} > 0 \)) which, ceteris paribus, provides the federal government with an incentive to redistribute resources from state j to state i via a lower federal tax rate \( s_{i} \). Since this effect works via the federal reaction function, it will only appear when the local government is able to exercise decentralized leadership. We will therefore refer to this as the decentralized leadership motive behind labor income taxation. If \( \partial I_{j}^{*} /\partial t_{i} > 0 \) (\( \partial I_{j}^{*} /\partial t_{i} < 0 \)), the decentralized leadership motive provides the local government with an incentive to set \( t_{i} \) higher (lower) than otherwise. The sign of \( \partial I_{j}^{*} /\partial t_{i} \) is determined by:where \( \varepsilon_{j}^{d} = \left( {{\text{d}}L_{j}^{d} / {\text{d}}w_{j} } \right)w_{j} /L_{j} < 0 \) is the elasticity of labor demand.¹⁷ Equation (16) implies:

Since the federal revenue mechanism in this paper resembles a formulaic transfer scheme, let us briefly relate the results derived above to the literature on incentive effects of formulaic transfer schemes. Smart (1998) analyzes how tax base equalization schemes distort the tax decisions of local governments. He shows that the formulaic transfer mechanism provides an upward pressure on the labor tax, which therefore becomes inefficiently high. Köthenbürger (2002), in turn, analyzes how a federal transfer mechanism affects the taxation of capital in a model with tax competition. Also, he concludes that the federal transfer mechanism has an upward pressure on the tax but since tax competition in itself leads to an inefficiently low taxation of capital, the upward pressure implied by the federal transfer mechanism tends to improve efficiency.

Whether the proportional fee motive addressed in this paper implies that the taxation of labor becomes more or less efficient is ambiguous. Recall that the most efficient tax policy (given that the government has access to a lump-sum instrument) involves setting \( t_{i} = 0 \). In the noncooperative Nash equilibrium, the proportional fee motive induces the local government to implement a higher tax on labor when \( s_{i} > 0 \). Whether this improves or reduces efficiency depends on whether the pecuniary motive [the first term in Eq. (14)] is negative or positive. Recall that if state i is a net exporter of labor, then the pecuniary motive provides government i with an incentive to implement a lower tax on labor. In this case, the proportional fee motive may lead to more efficiency since it provides government i with an incentive to implement a higher tax. If state i instead is a net exporter of labor (in which case the pecuniary motive provides government i with an incentive to implement a higher tax on labor), then the proportional fee motive exacerbates the inefficiency.

Also when the local government exercises decentralized leadership, the conclusion w.r.t. efficiency is unclear. If the two terms on the RHS of Eq. (15) have the same sign (which happens either if¹⁸\( l_{i}^{*} < L_{i}^{*} \) and \( |\varepsilon_{j}^{d} | > 1 \), or if \( l_{i}^{*} > L_{i}^{*} \) and \( |\varepsilon_{j}^{d} | < 1 \)), then the decentralized leadership motive leads to a more inefficient taxation of labor. The opposite argument applies if, instead, the two terms on the RHS of Eq. (15) have opposite signs (which happens either if \( l_{i}^{*} < L_{i}^{*} \) and \( |\varepsilon_{j}^{d}| < 1 \), or if \( l_{i}^{*} > L_{i}^{*} \) and \( |\varepsilon_{j}^{d}| > 1 \)).

Finally, recall that the analysis in this paper is based on the assumption that the federal spending is fixed at an exogenous level \( \overline{G} \). What would happen if we were to relax this assumption and instead let G be determined optimally by the federal government? For example, G could be a federal public good. Would the recognition that the size of the federal public good is endogenous influence the incentives underlying the policy implemented by a local government, which exercises decentralized leadership? The answer is no. The reason is that the decentralized leadership motive discussed above implies that government i in the present model already tries to max out the other states’ contribution to the federal budget. Making federal expenditure endogenous will not change this conclusion.

Let us now turn to the problem facing the local government when the federal fee is proportional to GNI. In this scenario, the local government’s problem is the same as that outlined in Sect. 4.1 with the exception that the local government’s budget constraint is given by \( g_{i} = T_{i} + t_{i} w_{i}^{*} L_{i}^{*} - s_{i} {\text{GNI}}_{i}^{*} \), where \( {\text{GNI}}_{i}^{*} = F_{i} \left( {L_{i}^{*} } \right) + {\text{FI}}_{i}^{*} \) and \( {\text{FI}}_{i}^{*} = w_{j}^{n} \left( {l_{i}^{*} - L_{i}^{*} } \right) \). The solution is presented in the Appendix, where the following results are derived:

Beginning with part (i), we see that the first two terms on the RHS of Eq. (21) are identical to, and can be interpreted in a similar way, as the terms that appear on the RHS of Eq. (14). The novelty is the appearance of the third term on the RHS of Eq. (21). It appears as a direct consequence of the fact that the federal fee is proportional to GNI instead of GDP. To explain why this term appears, let us conduct the same thought experiment as we used to interpret Proposition 1. Therefore, assume that the local government initially has made an optimal choice of \( T_{i} \) conditional on that \( t_{i} \) is zero. Then we ask whether the local government has an incentive to change \( t_{i} \) from zero? Differentiating the local government’s objective function w.r.t. \( t_{i} \), evaluating the resulting expression at \( t_{i} = 0 \) and using that the optimal choice of \( T_{i} \) satisfies \( \phi_{i}^{{\prime }} /u_{c}^{i} = 1 \) produce²¹whereSubstituting Eq. (24) into Eq. (23) producesThe first and second terms on the RHS of Eq. (25) reflect the pecuniary and the proportional fee motives for taxing labor. The novelty is the appearance of the third term on the RHS which reflects that the federal tax base is broader when it is based on \( {\text{GNI}}_{i}^{*} = {\text{GDP}}_{i}^{*} + {\text{FI}}_{i}^{*} \) instead of \( {\text{GDP}}_{i}^{*} \). This implies that the local government in member state \( i \) needs to take effects via \( {\text{FI}}_{i}^{*} \) into account (in addition to effects via \( {\text{GDP}}_{i}^{*} \)) when evaluating how \( t_{i} \) can be used to reduce the size of the fee paid to the federal level. From this perspective, the third term on the RHS of Eq. (25) reflects an extended proportional fee motive for taxing labor that arises when the federal fee is proportional to GNI. Since the payment to the federal level (all else equal) increases with \( {\text{FI}}_{i} \), government i has a motive to reduce \( {\text{FI}}_{i} \). Therefore, if \( \partial {\text{FI}}_{i} /\partial t_{i} > 0 \)\( \left( {\partial {\text{FI}}_{i} /\partial t_{i} < 0} \right) \), the local government has an incentive to set \( t_{i} \) lower (higher) than otherwise. This is captured by the third term on the RHS of Eq. (25), and this motive is also reflected by the third term on the RHS of the tax formula in Eq. (21). By using the comparative static properties of the functions that make up \( {\text{FI}}_{i} = w_{j}^{n} \left( {l_{i}^{*} - L_{i}^{*} } \right) \), it is possible to rewrite Eq. (21) to read (see “Appendix”)where we have used that the identity \( w_{i}^{n} = w_{j}^{n} \) implies that \( l_{i}^{*} \left( {t_{i} ,t_{j} } \right) = l_{i}^{*} \left[ {w_{j}^{n} \left( {t_{i} ,t_{j} } \right)} \right] \). Hence,Since \( a_{i} \) and \( \partial l_{i} /\partial w_{i}^{n} \) are both positive while \( \partial w_{j}^{n} /\partial t_{i} \) is negative, the second term on the RHS of (21′) is positive (including the minus sign). Since the first term on the RHS of (21′) reflects the pecuniary motive, the following result is readily available:

Turning to part (ii) in Proposition 3, we see that Eq. (22) is identical to Eq. (15). We therefore conclude that the incentive structure underlying the optimal (marginal) labor income tax implemented by a decentralized leader is independent of whether the federal government uses fees that are proportional to GDP or GNI, to collect its revenue.

The analysis above is based on the assumption that the federal government, in addition to financing the exogenous expenditure \( \overline{G} \), has an incentive to achieve redistribution between the two member states. Let us now relax the latter assumption and characterize the outcome when the federal government has no distributional motive and is only concerned with raising funds to finance the expenditure \( \overline{G} \). To do this, we first observe that when redistribution is no longer a concern for the federal government, then there is no motive for having a differentiated federal tax rate. Therefore, the federal government levies the same federal tax rate, s, on both state i and state j. If the federal fee is proportional to GDP, then the federal budget constraint becomes \( \overline{G} = s\left( {{\text{GDP}}_{i}^{*} + {\text{GDP}}_{j}^{*} } \right) \), but if the federal fee instead is proportional to GNI, then the federal budget constraint is given by \( \overline{G} = s\left( {{\text{GNI}}_{i}^{*} + {\text{GNI}}_{j}^{*} } \right) \). Since the factor incomes sum to zero over the two member states, it follows that \( {\text{FI}}_{i}^{*} = - {\text{FI}}_{j}^{*} \) and \( {\text{GDP}}_{i}^{*} + {\text{GDP}}_{j}^{*} = {\text{GNI}}_{i}^{*} + {\text{GNI}}_{j}^{*} \). This implies that the federal reaction function in either case is given by \( s = \overline{G} /\left( {{\text{GDP}}_{i}^{*} + {\text{GDP}}_{j}^{*} } \right) \). In the Appendix, we solve the decentralized leader’s problem when the federal fee is proportional to GDP and derive the following results:

The first term on the RHS reflects the pecuniary motive, while the second term reflects the proportional fee motive, for taxing labor. Recall from the analysis conducted earlier that when the federal government can implement state-specific federal taxes (\( s_{i} \)), then the proportional fee motive for taxing labor at the local level vanishes when the local government can exercise decentralized leadership. The reason was that the federal government responds to the proportional fee motive by increasing the state-specific federal tax rate \( s_{i} \) so as to balance out the negative impact on the federal tax base. However, when the federal government applies the same federal tax rate s on all member states, the federal response to something that happens in member state i will be imperfect. This means that a part of the proportional fee motive for taxing labor will remain for the decentralized leader in this scenario. This is captured by the term \( sY_{j}^{*} /\left( {Y_{i}^{*} + Y_{j}^{*} } \right) \) which appears on the RHS in Eq. (27).

As for the final term on the RHS of Eq. (27), it is analogous to the decentralized leadership motive that arises in Eqs. (15) and (22). To see this, recall that the decentralized leadership motive arises because government i has an incentive to increase the tax base in state j, since this induces the federal government to reduce the federal tax rate imposed on state i. Also when the federal budget constraint is given by \( \overline{G} = s\left( {{\text{GDP}}_{i}^{*} + {\text{GDP}}_{j}^{*} } \right) \), an increase in state j’s GDP will allow the federal government to reduce s. Since \( {\text{GDP}}_{j}^{*} = F_{j} \left[ {L_{j} \left( {t_{i} ,t_{j} } \right)} \right] \) and \( \partial L_{j}^{*} /\partial t_{i} > 0 \), it follows that an increase in \( t_{i} \) will have a positive effect on state j’s GDP. This provides government i with an incentive to set \( t_{i} \) higher than otherwise, and is captured by the final term on the RHS of Eq. (27) which is positive.