Unemployment and endogenous choice on tax instruments in a tax competition model: unit tax versus ad valorem tax

Abstract

This paper examines the endogenous choice of unit and ad valorem taxes in a model of tax competition with unemployment. Governments maximize objective functions that are a weighted sum of regional welfare and revenue. In the tax competition model, a high fixed wage rate generates unemployment and employment externalities. This effect can be either positive or negative because of freely mobile capital among regions. Without unemployment, revenue-maximizing governments choose unit taxes as their tax instrument to avoid revenue losses from intense tax competition under ad valorem taxes. However, with unemployment, positive employment externalities generate additional benefits for using ad valorem taxes to stimulate employment. Therefore, the present study shows that one region chooses an ad valorem tax, whereas the other chooses a unit tax, or that both governments use ad valorem taxes depending on employment externalities.

1 Introduction

Numerous studies have examined interregional tax competition following the pioneering studies of Zodrow and Mieszkowski (1986) and Wilson (1986).¹ One of the issues in the literature is to investigate which of the unit tax and the ad valorem tax is chosen as the government’s tax instrument (e.g., Aiura & Ogawa, 2013, 2019; Akai et al., 2011; Hoffmann & Runkel, 2016; Lockwood, 2004; Ogawa, 2016). Existing studies analyzed this issue by assuming a perfect labor market; therefore, unemployment does not exist. However, developed and developing countries have competed in their tax rates to encourage investment and create employment (OECD, 2017). By focusing on this aspect of tax competition, the present study addresses the issue of the government’s choice of tax instruments under unemployment.

Assuming constant returns to scale, a competitive market, and perfect labor markets, Lockwood (2004) examined the nature of the capital tax competition equilibrium with two different tax instruments by comparing equilibrium welfare. Under full employment, the author showed that unit taxes are superior to ad valorem taxes because ad valorem taxes can lead to harmful tax competition, resulting in underprovision of public goods. Using two-stage games with two symmetric countries, Akai et al. (2011) showed that all governments endogenously choose a unit tax regime. In contrast, Hoffmann and Runkel (2016) compared the welfare levels in each tax regime. They showed that the capital tax competition under ad valorem taxation might be less harmful than that under unit taxation with decreasing returns to scale and deductible capital costs. Furthermore, Ogawa (2016) discovered that asymmetric countries do not compete in the same tax instrument in a two-country two-stage game setting.

The issue of tax instrument selection has also been focused in the context of commodity tax competition models. Aiura and Ogawa (2013) demonstrated that governments endogenously choose an ad valorem tax; however, choosing a unit tax yields higher tax revenue for the government. Departing from a situation where one company in each country supplies homogeneous goods, Aiura and Ogawa (2019) compared the merits of tax methods in a model that includes firm entry and product diversity preferences. The results show that in such a commodity tax competition model, which tax improves welfare depends on the presence or absence of cross-border shopping and the government’s goal.

Respective studies have examined the effects of capital tax competition with unemployment (e.g., Eichner & Upmann, 2012; Exbrayat et al., 2012; Lee, 2021; Ogawa et al., 2006; Sato, 2009).² Although almost all assumed capital tax competition in the unit tax regime, unemployment can be depicted in various ways, including fixed wage, search friction, and labor union models.³ A key factor affecting the tax competition equilibrium is the employment externality caused by capital mobility and the technological relationship between capital and labor in production. If capital and labor supplement each other, then attracting capital improves employment for the own region, leading to the export of unemployment from one region to the other. In other words, the presence of unemployment derives employment externality.⁴ This new source of inefficiency generates new channels for the effects of different tax instruments. Therefore, the equilibrium outcomes differ from those found in the existing literature, which assumed full employment.

This paper examines the equilibrium properties under various tax regimes by endogenizing the government’s choice of tax instruments. Hence, we develop a capital tax competition model with unemployment and two tax instruments, along with a theoretical framework based on Akai et al. (2011). Regarding the labor market structure, our model differs from the existing literature in that the fixed wage is assumed in accordance with Ogawa et al. (2006).⁵ We investigate how the employment externality affects the equilibrium outcome using a two-stage game where each government chooses either a unit tax or an ad valorem tax on capital in the first stage and determines the tax rate in the second stage. Furthermore, we show the robustness of our results using numerical analysis assuming the asymmetries of the regions.

The critical factor in the government’s choice of tax instrument is an employment externality. If positive employment externalities are sufficiently small and the government’s preference for employment is sufficiently weak, each government chooses a unit tax as the tax instrument. For a small positive employment externality and preference for employment, the economy with unemployment is close to that under a perfect labor market or without preference for employment. Hence, the result is consistent with that of Lockwood (2004) and Akai et al. (2011). However, suppose that the positive employment externalities are large enough and the government’s preference for employment is strong enough. In such a case, ad valorem taxes are chosen as their tax instruments, similar to that of Hoffmann and Runkel (2016). Furthermore, surprisingly, if positive employment externalities and a desire to work are intermediate levels between the first two cases, one government chooses a unit tax, whereas the other opts for an ad valorem tax.⁶

Ad valorem taxation induces governments to lower their tax rates, compared to unit taxation (Lockwood, 2004).⁷ Without unemployment and the government’s preference for employment, this side effect causes severe tax competition. Therefore, both governments choose unit taxes as their tax instruments to avoid losses in such a problematic situation (Akai et al., 2011). However, if a positive externality exists and the regional government is interested in creating jobs, this unfavorable side effect could be transformed into a beneficial one that generates employment benefits. As a result, depending on the degree of employment externality and the government’s preference for employment, any combination of tax instruments can be strategically chosen. Extensive analyses confirm the robustness of these results.

The remainder of this paper is organized as follows: section 2 presents the basic framework of our theoretical analysis. Section 3 characterizes the tax competition equilibrium under the unit tax and ad valorem tax regimes. Section 4 presents extensions of our basic model by considering negative tax rates, negative employment externalities, and the asymmetricity of regions. Finally, Sect. 5 concludes the study.

2 The model

Consider a two-region economy where each region’s population is measured as $N_{i}$ ($i = 1,2$). Our model’s basic configuration is based on Ogawa et al. (2006). In region $i$, a continuum of identical competitive firms produces a homogeneous good using capital, labor, and land inputs ($K_{i}$, $L_{i}$, and $Z_{i}$, respectively). Let $Y_{i}$ be the output in region $i$. The production function in region $i$ is formulated as follows:

$$Y_{i} = F^{i} \left( {K_{i} ,L_{i} ,Z_{i} } \right).$$

We assume that ${F}^{i}$ is a constant-returns-to-scale. Moreover, the land input in each region is normalized to unity (i.e., ${Z}_{i}=1$).

By the assumptions that ${F}^{i}$ is a constant-returns-to-scale and ${Z}_{i}=1$, we have

$$Y_{i} = F^{i} \left( {\frac{{K_{i} }}{{Z_{i} }},\frac{{L_{i} }}{{Z_{i} }},1} \right)Z_{i} = F^{i} \left( {K_{i} ,L_{i} ,1} \right) \equiv f^{i} \left( {K_{i} ,L_{i} } \right)$$

The production function $f^{i}$ with fixed land input must be assumed to be concave with respect to $K_{i}$ and $L_{i}$ for well-behaved demand functions for capital and labor from firms’ profit maximization. Capital is freely mobile between the two regions, whereas labor and land are stuck to the original regions. Furthermore, the capital and land belong to the owners outside these two economies.

Each jurisdictional government taxes capital either via a unit tax $T_{i}$ or an ad valorem tax $t_{i}$ after the choice of the tax instruments. Profit maximization and perfect mobility of capital lead to

$$\begin{array}{*{20}c} {\left\{ {\begin{array}{*{20}c} {r = f_{K}^{i} - T_{i} ,} \\ {r = \left( {1 - t_{i} } \right)f_{K}^{i} ,} \\ \end{array} } \right.} \\ \end{array}$$

(1)

where $r$ is the posttax return on capital, and $f_{K}^{i}$ is the partial derivative with respect to $K_{i}$ (i.e., $f_{K}^{i} \equiv \partial f^{i} /\partial K_{i}$). We assume that the imperfect labor market and the imperfection are symbolized by the fixed wage.⁸ Let us denote the fixed wage rate in the region $i$ by $\overline{w}_{i}$. The labor demand satisfies the following:

$$\begin{array}{*{20}c} {\overline{w}_{i} = f_{L}^{i} ,} \\ \end{array}$$

(2)

$f_{L}^{i}$ is the partial derivative with respect to $L_{i}$ (i.e., $f_{L}^{i} \equiv \partial f^{i} /\partial L_{i}$). The capital market equilibrium condition is

$$K = K_{1} + K_{2}$$

(3)

In the labor market with a sufficiently high level of $\overline{w}_{i}$, we have $L_{i} < N_{i}$.

We assume that residents in each region have a preference for private consumption. The individual utility level is then determined solely by the resident’s income. By the presence of unemployment, the residents are categorized into two groups: the employed and the unemployed. Considering the absentee ownership of capital and land, the employed earn $\overline{w}_{i}$ times their labor supply, whereas the unemployed earn nothing. Therefore, if the welfare function is the utilitarian form and the marginal utility is assumed to be a constant, the regional welfare $W_{i}$ is defined over the regional income. Meanwhile, the Benthamite welfare function is the sum of the utility function of the employed $W_{i}^{e}$ and that of the unemployed $W_{i}^{u}$. Because $W_{i}^{e} = \lambda \overline{w}_{i} L_{i}$ and $W_{i}^{u} = \lambda \left( {N_{i} - L_{i} } \right) \times 0$, the regional welfare function becomes $W_{i} = \lambda \overline{w}_{i} L_{i}$:

$$W_{i} = \lambda \overline{w}_{i} L_{i} ,$$

where $\lambda$ is a positive constant ($\lambda > 0$). For analytical simplicity, $\lambda \overline{w}_{i} = 1$ is assumed without loss of generality.

Each jurisdictional government’s budget equation becomes

$$\begin{array}{*{20}c} {\left\{ {\begin{array}{*{20}c} {G_{i} = T_{i} K_{i} ,} \\ {G_{i} = t_{i} f_{K}^{i} K_{i} ,} \\ \end{array} } \right.} \\ \end{array}$$

(4)

where $G_{i}$ stands for region $i$ government’s tax revenue (or government spending which does not affect residents’ utility and firms’ productivity).

For the equilibrium analysis, the previous literature used the specified production function, satisfying the linear marginal productivity (e.g., Wildasin, 1991; Brueckner, 2004; Akai et al., 2011). In particular, we employ the following quadratic form of function:

$$\begin{array}{*{20}c} {f^{i} \left( {K_{i} ,L_{i} } \right) = \left[ {\delta_{i} + \left( {\alpha_{i} K_{i} + \beta_{i} L_{i} } \right) - \frac{{A_{i} K_{i}^{2} + B_{i} L_{i}^{2} }}{2} + \gamma_{i} K_{i} L_{i} } \right]\phi_{i} ,} \\ \end{array}$$

(5)

where $\delta_{i} \ge 0$, $\alpha_{i} > 0$, $\beta_{i} > 0$, $A_{i} > 0$, $B_{i} > 0$, and $\phi_{i} > 0$.⁹The specified production function (5) has linear marginal productivities (i.e., $f_{K}^{i} = \alpha_{i} - A_{i} K_{i} + \gamma_{i} L_{i}$ and $f_{L}^{i} = \beta_{i} - B_{i} L_{i} + \gamma_{i} K_{i}$). The marginal productivities of capital and labor are, respectively, decreasing in $K$ and $L$ (i.e., $f_{KK}^{i} = - A_{i} < 0$ and $f_{LL}^{i} = - B_{i} < 0$). The cross-derivative term could be either positive or negative depending on $\gamma_{i}$ (i.e., $f_{KL}^{i} = f_{LK}^{i} = \gamma_{i}$).

Using Eqs. (2, 5) yield the following:

$$\begin{array}{*{20}c} {L_{i} = c_{i} + \mu_{i} K_{i} ,} \\ \end{array}$$

(6)

where

$$c_{i} \equiv \frac{{\beta_{i} - \omega_{i} }}{{B_{i} }}, \omega_{i} \equiv \frac{{\overline{w}_{i} }}{{\phi_{i} }}{\text{, and }} \mu_{i} \equiv \frac{{\gamma_{i} }}{{B_{i} }}.$$

Equation (6) implies that the employment level increases with capital. We assume that $c_{i}$ is sufficiently small to ensure $L_{i} < N_{i}$. Equations (5, 6) provide

$$\begin{array}{*{20}c} {f_{K}^{i} = a_{i} - b_{i} K_{i} ,} \\ \end{array}$$

(7)

where

$$a_{i} \equiv \left[ {\alpha_{i} + \left( {\frac{{\beta_{i} - \omega_{i} }}{{B_{i} }}} \right)\gamma_{i} } \right]\phi_{i} {\text{ and }} b_{i} \equiv \left( {A_{i} - \frac{{\gamma_{i}^{2} }}{{B_{i} }}} \right)\phi_{i} .$$

Equation (7) has a functional form similar to that of Akai et al. (2011) if and only if $\gamma_{i} = 0$. With $\gamma_{i} \ne 0$, the intercept $a_{i}$ and the coefficient $b_{i}$ depend on $\gamma_{i}$. We focus on the case where the capital and labor are complementary to each other ($\gamma_{i} > 0$) until Sect. 4.

3 Competition in tax instruments and tax levels

This section explores a tax competition game involving the selection of a tax instrument. The game is noncooperative and has two stages. In the first stage of the game (stage 1), two jurisdictional governments simultaneously select a unit or an ad valorem tax as their tax instrument. The governments determine their tax levels in the second stage (stage 2) to maximize the values of their objective functions. We will assume that the two regions are symmetric throughout this section. Then, we will make the following assumption to ensure $Y_{i} > 0$:

Assumption 1

$a > bK.$

The previous literature makes the same assumption. Assumption 1 ensures that the marginal product of capital cannot be negative.

3.1 Determination of tax levels

We now consider tax level determination at stage 2 when (i) both regional governments opt for a unit tax (UU), (ii) both governments opt for an ad valorem tax (AA), or (iii) one jurisdictional government selects a unit tax, whereas the other chooses an ad valorem tax (UA or AU). $U$ and $A$ denote the unit and ad valorem tax options, respectively. The governments consider the following effects of their tax policy (see Appendix A for the derivation of the following equations):

$$UU: \frac{{\partial K_{i} }}{{\partial T_{i} }} = - \frac{1}{2b} < 0,\quad AA: \frac{{\partial K_{i} }}{{\partial t_{i} }} = - \frac{{a - bK_{i} }}{{\left( {2 - t_{i} - t_{j} } \right)b}} < 0,$$

$$AU\, or\, UA: \frac{{\partial K_{i} }}{{\partial T_{i} }} = - \frac{1}{{\left( {2 - t_{j} } \right)b}} < 0\, {\text{and}}\, \frac{{\partial K_{i} }}{{\partial t_{i} }} = - \frac{{a - bK_{i} }}{{\left( {2 - t_{i} } \right)b}} < 0,$$

where $a_{1} = a_{2} = a$ and $b_{1} = b_{2} = b$ ($i = 1,2$ and $i \ne j$).

Every region’s government is assumed to be a moderate Leviathan that is concerned with revenue $G_{i}$ and regional $W_{i}$ (Edwards & Keen, 1996; Wrede, 1998). Noteworthily, the welfare levels correspond to the employment levels (i.e., $W_{i} = L_{i}$). The objective function of region i government is specifically expressed as follows:¹⁰:

$$\begin{array}{*{20}c} {V_{i} = \theta_{i} W_{i} + \left( {1 - \theta_{i} } \right)G_{i} ,} \\ \end{array}$$

(8)

where the weight for the regional welfare (employment) level is $0 \le \theta_{i} \le 1$. If $\theta_{i} = 0$, the government’s objective function is identical to that of Akai et al. (2011). Hence, the governments behave like a pure Leviathan. In contrast, the government is purely benevolent if $\theta_{i} = 1$. Assuming symmetric regions, we focus on $\theta_{i} = \theta > 0$ ($i = 1,2$).

The first-order conditions for maximizing Eq. (8) with respect to $T_{i}$ and $t_{i}$ are

$$\begin{array}{*{20}c} {\frac{{\partial V_{i} }}{{\partial T_{i} }} = \theta \frac{{\partial L_{i} }}{{\partial T_{i} }} + \left( {1 - \theta } \right)\left( {K_{i} + T_{i} \frac{{\partial K_{i} }}{{\partial T_{i} }}} \right) = 0, } \\ \end{array}$$

(9a)

$$\begin{array}{*{20}c} {\frac{{\partial V_{i} }}{{\partial t_{i} }} = \theta \frac{{\partial L_{i} }}{{\partial t_{i} }} + \left( {1 - \theta } \right)\left[ {\left( {a - bK_{i} } \right)K_{i} + \left( {a - 2bK_{i} } \right)t_{i} \frac{{\partial K_{i} }}{{\partial t_{i} }}} \right] = 0,} \\ \end{array}$$

(9b)

where

$$\frac{{\partial L_{i} }}{{\partial T_{i} }} = \mu \frac{{\partial K_{i} }}{{\partial T_{i} }} < 0\, {\text{and}}\, \frac{{\partial L_{i} }}{{\partial t_{i} }} = \mu \frac{{\partial K_{i} }}{{\partial t_{i} }} < 0.$$

For Eq. (9a, 9b), increased taxes raise the government revenue while decreasing employment (regional welfare) level. Hence, if the governments have a stronger preference for the regional welfare (larger $\theta$), they are more willing to decrease the capital tax. Similarly, if an increased tax has a greater impact on regional employment (larger $\mu$), the governments are incentivized to lower the tax rate.

The effects of a higher tax on capital and labor inputs are shown in Eq. (9a, 9b), which are related to the externalities discovered in previous studies (e.g., Ogawa et al., 2006; Wildasin, 1989). We can verify these facts using the following equations:

$$\begin{array}{*{20}c} {\frac{{\partial V_{j} }}{{\partial T_{i} }} = \theta \frac{{\partial L_{j} }}{{\partial T_{i} }} + \left( {1 - \theta } \right)T_{j} \frac{{\partial K_{j} }}{{\partial T_{i} }} = \left( {1 - \theta } \right)\left( {\sigma + T_{j} } \right)\frac{{\partial K_{j} }}{{\partial T_{i} }}, } \\ \end{array}$$

(10a)

$$\begin{array}{*{20}c} {\frac{{\partial V_{j} }}{{\partial t_{i} }} = \theta \frac{{\partial L_{j} }}{{\partial t_{i} }} + \left( {1 - \theta } \right)\left( {a - 2bK_{i} } \right)t_{j} \frac{{\partial K_{j} }}{{\partial t_{i} }} = \left( {1 - \theta } \right)\left[ {\sigma + \left( {a - 2bK_{i} } \right)t_{j} } \right]\frac{{\partial K_{j} }}{{\partial t_{i} }},} \\ \end{array}$$

(10b)

where

$$\frac{{\partial L_{j} }}{{\partial T_{i} }} = \mu \frac{{\partial K_{j} }}{{\partial T_{i} }} = - \mu \frac{{\partial K_{i} }}{{\partial T_{i} }} > 0,\frac{{\partial L_{j} }}{{\partial t_{i} }} = \mu \frac{{\partial K_{j} }}{{\partial T_{i} }} = - \mu \frac{{\partial K_{i} }}{{\partial t_{i} }} > 0,\sigma \equiv \frac{\theta }{1 - \theta }\mu .$$

Regarding the intermediate parts of (10a) or (10b), the first and second terms represent employment and fiscal externalities, respectively. The effects on capital characterize both positive externalities. Larger $\mu$ derives the larger size of positive employment externality. Similarly, larger $\theta$ indicates a larger size of positive employment externality (relative to the fiscal externality). The degree of employment externality $\mu$ and the government’s preference for regional welfare (employment) $\theta$ will both have a quantitative impact on the payoff of policy choice and, as a result, the equilibrium strategy.

Because the employment externalities are linearly associated with the capital inflow effects of increased tax, the terms of the fiscal and employment externalities can be reduced using the coefficients of $\partial K_{j} /\partial T_{i}$ or $\partial K_{j} /\partial t_{i}$. The coefficient $\sigma$ denotes the degree of employment externality $\mu$ weighted by the government’s objective parameter $\theta$. Larger $\sigma$ leads to a higher degree of employment externalities or a higher level of government concern about regional welfare (employment); they completely magnify the effect of employment externalities. Therefore, we refer $\sigma$ as the key parameter for describing the equilibrium with employment externalities.

In terms of the equilibrium tax rates at stage 2, we obtain Lemma 1 (see Appendix B for the proof of Lemma 1):

Lemma 1

(i) If the governments choose a unit tax,

$$T_{1} = T_{2} = bK - \sigma .$$

(ii) If the governments choose an ad valorem tax,

$$t_{1} = t_{2} = \frac{bK - \sigma }{a}.$$

(iii) If region i government chooses a unit tax and the other opts for an ad valorem tax,

$$T_{i} = \frac{bK - \sigma }{2} + \frac{H}{4}\, {\text{and}}\, t_{j} = \frac{H}{{2\left( {a - bK} \right)}}, i,j = 1,2 \,{\text{and}}\, i \ne j,$$

where $H \equiv 6a - bK - \sigma - \sqrt {36a^{2} - 36abK + 25b^{2} K^{2} + 12a\sigma - 22bK\sigma + \sigma^{2} } .$

When $\sigma = 0$, the results from (i) through (iii) in Lemma 1 go back to those derived by Akai et al. (2011). There are two cases where $\sigma$ is zero. Given that $\mu$ represents the sign of the cross-derivatives of $K$ and $L$, $\mu = 0$ implies no employment externality. If this externality does not arise, $\sigma$ is zero. If $\theta = 0$, the same situation occurs. Without the government’s interest in regional welfare (employment), the employment level does not affect the government’s objective function in each case. Therefore, $\sigma$ is zero in this case.

Lemma 1 also indicates the possibility of a capital subsidy. Lump-sum taxes must be available to finance capital subsidies. However, negative government revenue can be considered because we assume a linear preference for employment and government revenue/expenditure. Furthermore, capital subsidies to stimulate employment can be observed in the real economy. Based on Lemma 1, the tax rates in cases (i) and (ii) are negative for $\sigma > bK$. This result implies that governments are motivated to use capital subsidies when positive externalities are sufficiently large, which is shown by the other literature on tax competition and unemployment (e.g., Ogawa et al., 2006; Tamai and Myles, 2022). If the lump-sum tax can only be used in limited circumstances to ensure nonnegative tax revenue, the unit or ad valorem tax rates could be negative. Furthermore, the output must be positive. To ensure the positive output, $\sigma < a$ is required.¹¹ Hence, we impose the following assumption:

Assumption 2

$a > \sigma.$

We need the following assumption to ensure positive capital demand.

Assumption 3

$a > 2bK.$

When $\sigma = 0$, $2a > 3bK$ is sufficient to ensure positive capital demand.¹² However, if $\sigma > 0$, higher productivity is needed because of the employment externality.

We now move on to the comparison between the equilibrium tax rates derived in Lemma 1. Because the tax rates must be converted to comparable forms, we use the effective tax rate $t_{i} = T_{i} /f_{K}^{i}$. This is the conversion of unit tax into the ad valorem rate and allows the best responses under different tax instruments to be converted into best responses under a unified notation. Following Akai et al. (2011), let $t_{i}^{mn}$ be the region $i$’s effective ad valorem tax rate in the Nash equilibrium when the region $i$ government chooses a tax instrument $m$ ($m = U,A$), and the other chooses a tax instrument $n$ ($n = U,A$). Lemma 1 and $t_{i} = T_{i} /f_{K}^{i}$ ($i = 1,2$) yield the following equations (see Appendix C for the derivation of the following equations):

$$\begin{array}{*{20}c} {t_{i}^{UU} = \frac{{2\left( {bK - \sigma } \right)}}{2a - bK}, } \\ \end{array}$$

(11a)

$$\begin{array}{*{20}c} {t_{i}^{AA} = \frac{bK - \sigma }{a}, } \\ \end{array}$$

(11b)

$$\begin{array}{*{20}c} {t_{i}^{UA} = \frac{{\left[ {2\left( {bK - \sigma } \right) + H} \right]\left[ {4\left( {a - bK} \right) - H} \right]}}{{4a\left[ {4\left( {a - bK} \right) - H} \right] - 2\left[ {2\left( {bK - \sigma } \right) + H} \right]\left( {a - bK} \right)}},} \\ \end{array}$$

(11c)

$$\begin{array}{*{20}c} {t_{i}^{AU} = \frac{H}{{2\left( {a - bK} \right)}}. } \\ \end{array}$$

(11d)

A comparison among Eq. (11a–11d) makes the following proposition (see Appendix D for the proof of Proposition 1):

Proposition 1.

There exists $\hat{\sigma } \in \left( {0,bK} \right)$, satisfying $t_{i}^{UA} = t_{i}^{AA}$. Then, $t_{i}^{UU} > t_{i}^{AU} > t_{i}^{UA} > t_{i}^{AA}$ holds if $\sigma < \hat{\sigma }$, whereas $t_{i}^{UU} > t_{i}^{AU} > t_{i}^{AA} \ge t_{i}^{UA}$ if $\hat{\sigma } \le \sigma < bK$. When $\sigma > bK$, the orders of tax rates become $t_{i}^{UU} < t_{i}^{AU} < t_{i}^{UA} < t_{i}^{AA}$.

The degree of employment externality affects the magnitude relationship of the tax rates under different scenarios. If the employment externality effect is not too strong, the lowest tax rate within the four scenarios is the tax rate when both governments choose an ad valorem tax. This is the case presented by Akai et al. (2011). However, if the employment externality effect is large enough, the smallest tax rate shifts from the tax rate when both governments choose an ad valorem tax to the tax rate when the region $i$’s government chooses a unit tax, and the other chooses an ad valorem tax. When governments prefer regional welfare (employment), the capital-attracting effects of unit and ad valorem tax cuts are more valued by the governments than pure Leviathans. In particular, if the employment externality effect is sufficiently large or the governments strongly prefer regional welfare, the capital-attracting effect of decreasing ad valorem tax relative to that of unit tax is strengthened. Therefore, the government sets the lowest rate of ad valorem tax for $\hat{\sigma } < \sigma$, facing the other unit tax as its tax instrument. If $\sigma > bK$, the tax rates are all negative. Hence, the magnitude relationship must be inversed in the case of $\sigma < bK$.

3.2 Determination of tax instruments

We consider the determination of tax instruments in stage 1. The region i government’s payoff is given as follows (see Appendix E for the proof of Lemma 2):

Lemma 2.

Let be $\nu \equiv \left( {1 - \theta } \right)^{ - 1} \theta c$. (i) If the governments choose a unit tax,

$$V_{i}^{UU} = \left( {1 - \theta } \right)\left( {\nu + \frac{{bK^{2} }}{2}} \right).$$

(ii) If the governments choose an ad valorem tax,

$$V_{i}^{AA} = \left( {1 - \theta } \right)\left[ {\nu + \frac{{\left( {2a - bK + \sigma } \right)bK^{2} }}{4a}} \right].$$

(iii) If region i government chooses a unit tax, and the other opts for an ad valorem tax,

$$V_{i}^{UA} = \left( {1 - \theta } \right)\left\{ {\nu + \left( {\frac{bK + \sigma }{2} + \frac{H}{4}} \right)\frac{{\left[ {2\left( {bK - \sigma } \right) + H} \right]\left( {a - bK} \right)}}{{2b\left[ {4\left( {a - bK} \right) - H} \right]}}} \right\},$$

$$V_{i}^{AU} = \left( {1 - \theta } \right)\left\{ {\nu + \left[ {\sigma + \frac{{\left( {a - b\Gamma } \right)H}}{{2\left( {a - bK} \right)}}} \right]\Gamma } \right\}, {\text{where}} \Gamma \equiv \frac{{\left( {a - bK} \right)\left( {6bK - H + 2\sigma } \right) - 2bKH}}{{2b\left[ {4\left( {a - bK} \right) - H} \right]}}.$$

Based on Lemma 1, the employment externality changes the magnitude relationship of the payoff chosen by the governments, depending on its size. The payoffs are shown in Table 1. In this regard, Lemma 2 yields the following result (see Appendix F for the proof of Proposition 2):

Table 1 Payoff table

Proposition 2.

There exist $\overline{\sigma }$ and $\tilde{\sigma }$, satisfying $V_{i}^{UA} \frac{ > }{ < }V_{i}^{AA} \Leftrightarrow \sigma \infty \overline{\sigma }$ and $V_{i}^{UU} \frac{ > }{ < }V_{i}^{AU} \Leftrightarrow \sigma \infty \tilde{\sigma }$, respectively. (i) If $0 \le \sigma < \min \left( {\overline{\sigma },\tilde{\sigma }} \right)$, each government chooses a unit tax in the Nash equilibrium. (ii) If $\min \left( {\overline{\sigma },\tilde{\sigma }} \right) < \sigma < \max \left( {\overline{\sigma },\tilde{\sigma }} \right)$, two possible cases exist: $\overline{\sigma } < \sigma < \tilde{\sigma }$ and $\tilde{\sigma } < \sigma < \overline{\sigma }$. When $\overline{\sigma } < \sigma < \tilde{\sigma }$, two Nash equilibria exist such that one government selects the same tax instrument as the other. When $\tilde{\sigma } < \sigma < \overline{\sigma }$, two Nash equilibria exist such that one government chooses a unit tax, and the other opts for an ad valorem tax. (iii) If $\sigma > \max \left( {\overline{\sigma },\tilde{\sigma }} \right)$, each government chooses an ad valorem tax in the Nash equilibrium.

To verify the results of Proposition 2, we provide numerical illustrations. Given the interaction of many parameters with each other and the vast number of possible combinations, focusing on all the cases is difficult. Hence, we fix the values of $a$, $b$, and $K$ as $a = 3$, $b = 1$, and $K = 1$, respectively. Figure 1 depicts three parametric regions, satisfying (i) $0 \le \sigma < \min \left( {\overline{\sigma },\tilde{\sigma }} \right)$, (ii) $\min \left( {\overline{\sigma },\tilde{\sigma }} \right) < \sigma < \max \left( {\overline{\sigma },\tilde{\sigma }} \right)$, and (iii) $\sigma > \max \left( {\overline{\sigma },\tilde{\sigma }} \right)$. Because $\sigma$ is defined over two parameters of $\mu$ and $\theta$, Fig. 1 shows that higher values of $\theta$ and $\mu$ tend to generate the result (ii) or (iii) in Proposition 2. In contrast, lower values lead to the result (i). Therefore, all cases of Proposition 2 hold under certain values of parameters.

Fig. 1

Region plot related to the critical values in Proposition 2

The intuition of Proposition 2 can be explained as follows: first, we focus on the result (i) where $0 \le \sigma < \min \left( {\overline{\sigma },\tilde{\sigma }} \right)$. In other words, the employment externality is small enough, or the government’s preference is weak enough, to support Akai et al.’s (2011) scenario. Taking $\sigma = 0$ as an example, the intuition for the result (i) is along with that of Lockwood (2004) and Akai et al. (2011). Assume that there is a slight increase in public goods spending for region 1. In this case, the increased unit tax will result in a capital outflow given by $\Delta$ from region 1. If region 2 adopts a unit tax, $\Delta$ units of capital are employed in region 2, and thus, region 2’s government revenue is increased by $T_{2} \Delta$. However, if region 2 uses ad valorem tax, it brings about a side effect that region 2 wishes to lower the tax rate because region 2’s tax revenue is given by $t_{2} f_{k}^{2} \Delta$ and $f_{k}^{2}$ decreases with $\Delta$. Therefore, in the first stage, both governments are willing to choose a unit tax rather than an ad valorem tax.

In the case of result (iii) where $\sigma > \max \left( {\overline{\sigma },\tilde{\sigma }} \right)$, the mechanism is opposite to the result in (i). Assume, as in (i), that the government of region 1 increases a unit tax to fund a small increase in public goods spending. If region 2 has an ad valorem tax, the region’s government is incentivized to lower the tax rate due to the previously mentioned side effect. A decrease in the tax rate to attract more capital leads to more employment. With a large $\sigma$, creating employment will cause large benefits relative to costs facing large elasticity of capital to the tax rate. This can be applied interactively to both regions. Hence, $V_{i}^{UU} < V_{i}^{AU}$ and $V_{i}^{UA} < V_{i}^{AA}$ hold: one region has no incentive to choose a unit tax if the other uses an ad valorem tax. Therefore, both governments will select an ad valorem tax as a tax instrument in the first stage.

We now turn to the result (ii) where $\min \left( {\overline{\sigma },\tilde{\sigma }} \right) < \sigma < \max \left( {\overline{\sigma },\tilde{\sigma }} \right)$. This case falls somewhere between (i) and (iii), and the payoffs have a complicated magnitude relationship for different values of $\sigma$. Thus, the symmetric regions can strategically choose different tax instruments depending on $\sigma$.

We consider the case where $\overline{\sigma } < \sigma < \tilde{\sigma }$. Suppose that region 2 chooses unit tax. This decision indicates to region 1 that region 2 has no intention of bringing intense tax competition. If region 1 chooses an ad valorem tax as the first stage tax instrument for increasing employment. Nonetheless, it could spark fierce tax competition. Region 1 does not receive enough benefits to cover the costs of intense tax competition because $\sigma$ is not sufficiently large (i.e., $V_{1}^{UU} > V_{1}^{AU}$ for $\sigma < \tilde{\sigma }$). Therefore, the best strategy for region 1 is to use the same unit tax as region 2. The same reasoning applies to region 2. Assume that region 2 imposes an ad valorem tax for $\overline{\sigma } < \sigma < \tilde{\sigma }$. This decision would indicate that region 2 intends to create intense tax competition for region 1. If region 1’s government chooses a unit tax to avoid intense tax competition, region 1 will see capital flow to region 2. As a result, region 1 does not add benefits from such a choice compared to when choosing a unit tax ($V_{1}^{UA} < V_{1}^{AA}$ for $\sigma > \overline{\sigma }$). Consequently, both regions choose an ad valorem tax in the first stage.

Finally, the case where $\tilde{\sigma } < \sigma < \overline{\sigma }$ can be explained as follows: suppose that region 2 selects a unit tax. If region 1 opts for an ad valorem tax, the tax rate must be reduced in order to maintain tax revenue, and the government expects to gain additional employment benefits. In contrast, when region 1 employs a unit tax rather than an ad valorem tax, there is no opportunity for additional benefits from job creation. Therefore, region 1 chooses an ad valorem tax in response to region 2’s choice of unit tax ($V_{1}^{UU} < V_{1}^{AU}$ for $\sigma > \tilde{\sigma }$). Consider the case where region 2 opts for an ad valorem tax. If region 1 imposes an ad valorem tax, region 2 must counterbalance tax cuts to prevent further capital flight. Meanwhile, region 1 is not required to deal with such an unfavorable effect by using a unit tax. Hence, region 1 opts for a unit tax, whereas region 2 opts for an ad valorem tax ($V_{1}^{UA} > V_{1}^{AA}$ for $\sigma < \overline{\sigma }$). These results show that the best strategy for one region is choosing the same tax instrument in response to that chosen by the other region.

The mechanisms of our main findings are explicated along with the existing literature (Akai et al., 2011; Hoffmann & Runkel, 2016; Lockwood, 2004; Ogawa, 2016). Although the analytical methods used in those studies differ (simple welfare comparisons versus endogenous decision processes with two-stage games), their intuitive mechanisms can be compared. Lockwood (2004) and Akai et al. (2011) showed that an ad valorem tax incentivizes governments to lower their tax rate because the rate of return on capital, that is, the marginal product of capital, decreases with capital inflow. As a decrease, when the rate of return falls, the government must lower the tax rate in order to maintain tax revenue. If the regional government has a strong interest in employment (regional welfare), this unfavorable side effect could be turned into a favorable one that generates employment benefits. Hence, if the government’s preference for employment is sufficiently weak, unit taxes are chosen by both governments, as shown by Lockwood (2004) and Akai et al. (2011).

However, the governments with sufficiently strong regional welfare (employment) preferences use ad valorem taxes, similar to Hoffmann and Runkel (2016). The factors influencing the choice of tax instruments differ between Hoffmann and Runkel (2016) and our paper. The goal of Hoffmann and Runkel (2016) is to combine tax instrument comparisons and the deductibility of capital costs from taxable income under full employment. According to Lockwood (2004), unit taxation outperforms ad valorem taxation in terms of welfare comparisons. Considering the economic rents generated by decreasing returns to scale and the deductible capital costs, only ad valorem taxation absorbs the rent generated in the private sector into the public sector. Therefore, ad valorem taxation resulting in a trade-off is superior to unit taxation if the deduction ratio exceeds a certain threshold value. The essential difference is that Hoffmann and Runkel (2016) focused on the welfare effects of ad valorem taxation through economic rents, whereas we focus on technological relationships in the manufacturing sector.

Furthermore, Ogawa (2016) points out that each government selects a different tax instrument in intermediate cases. Using an asymmetric two-region model, this previous study resulted in a single equilibrium. The most surprising aspect of this study is that, unlike Ogawa (2016), multiple equilibria occur in a symmetric two-region model. Extensive analyses ensure the robustness of our results.

4 Further analysis and discussion

We assumed in previous sections that positive employment externalities exist and that the two regions are symmetric. However, negative employment externalities may appear in the real world, and regional asymmetries exist. Hence, we discuss the possibility of capital subsidy (negative tax rate on capital) and examine the effects of negative employment externalities and the equilibrium properties in the economy with asymmetric regions.

Negative employment externalities. The degree of employment externality can be measured using the parameter $\mu$. A negative employment externality implies that $\mu$ is negative. Then, $\sigma < 0$ holds by its definition. In practice, the value of $\mu$ could be negative, resulting in a negative cross-derivative with respect to capital and labor. In the short term, certain types of production improvement may harm workers, such as the unskilled.¹³ Autor et al. (2003) developed a task model that precisely elucidates such a situation. The production function used in this study is related to the production technology developed by Autor et al. (2003). Therefore, $\sigma < 0$ should be included. If $\sigma$ is sufficiently small in the absolute value, the qualitative effects are the same as those shown in the previous sections.

Asymmetric regions. We consider the equilibrium tax instruments chosen by the governments in asymmetric regions. For the reason explained in Sect. 3, we set $a_{i}$, $b_{i}$, $c_{i}$, $\theta_{i}$, and $K$ as $a_{i} = 3$, $b_{i} = 1$, $c_{i} = 1$, $\theta_{i} = 0.5$, and $K = 1$, respectively. Three cases are considered hereafter.

First, we consider positive employment externalities in both regions. Focusing on two subcases, Table 2 shows the payoffs in cases of $\left( {\mu_{1} ,\mu_{2} } \right) = \left( {1.5,0.5} \right)$ and $\left( {\mu_{1} ,\mu_{2} } \right) = \left( {2,1.5} \right)$. Based on Table 2, regions 1 and 2, respectively, use an ad valorem tax and a unit tax if $\left( {\mu_{1} ,\mu_{2} } \right) = \left( {1.5,0.5} \right)$, whereas both governments choose an ad valorem tax if $\left( {\mu_{1} ,\mu_{2} } \right) = \left( {2,1.5} \right)$. These results are examples of Proposition 2 results (ii) and (iii) for asymmetric regions.

Table 2 Payoff table when $\mu_{i} > 0$

Now consider the case of positive employment externality in one region versus negative employment externality in the other. The payoffs are calculated as shown in Table 3 for two subcases where $\left( {\mu_{1} ,\mu_{2} } \right) = \left( {1.5, - 0.5} \right)$ and $\left( {\mu_{1} ,\mu_{2} } \right) = \left( {0.5, - 1.5} \right)$. The governments choose unit taxes as their tax instruments. This discovery corresponds to the result in (i) of Proposition 2.

Table 3 Payoff table when $\mu_{1} > 0$ and $\mu_{2} < 0$

Finally, Table 4 shows the payoffs when negative employment externalities are considered in both regions. For $\left( {\mu_{1} ,\mu_{2} } \right) = \left( { - 0.5, - 1.5} \right)$ and $\left( {\mu_{1} ,\mu_{2} } \right) = \left( { - 1.5, - 2} \right)$, the governments choose unit taxes. These numerical examples imply that, depending on the values of key parameters, all of the cases of Proposition 2 are realized.

Table 4 Payoff table when $\mu_{i} < 0$

Government’s objective function. The absentee ownership of capital and land simplifies the functional form of Eq. (8). However, if we remove such an assumption of absentee ownership, the earnings for employed and unemployed people invariably include capital and land income. Hence, when all residents equally share the capital and land, the government’s objective function should be written as

$$\begin{array}{*{20}c} {V_{i} = \theta \left[ {f^{i} \left( {K_{i} ,L_{i} } \right) + \left( {\frac{K}{2} - K_{i} } \right)r} \right] + \left( {1 - \theta } \right)G_{i} .} \\ \end{array}$$

(12)

The term $\left( {K/2 - K_{i} } \right)r$ on the RHS of Eq. (12) denotes the effect of terms of trade, which is vanished when the economy is in a symmetric equilibrium. If $\mu > 0$ and the regional welfare is monotonically increasing in $K_{i}$, the nature of the best response function based on (12) for each government is similar to that of absentee ownership of capital and land. Hence, both governments may choose ad valorem taxation or that the governments choose different tax instruments, as demonstrated in earlier sections of this paper. However, the presence of capital and land income has a quantitative impact on the levels of government payoffs. The actual equilibrium outcomes are more complicated than those shown in the preceding sections.

5 Conclusion

This paper examined the government’s choice of tax instruments in the context of unemployment. The government is expected to seek higher levels of welfare (employment) and tax revenue. If the positive employment externalities are small enough, governments choose unit taxes as their tax instruments, similar to Lockwood (2004) and Akai et al. (2011). By contrast, ad valorem taxes are chosen as their tax instruments if positive employment externalities are sufficiently large, which corresponds to the conclusion reached by Hoffmann and Runkel (2016). Furthermore, contrary to previous literature, multiple equilibria are found when positive employment externalities are intermediate between the former two cases. In that case, one government chooses a unit tax, whereas the other chooses an ad valorem tax, or one government selects the same tax instrument as the other. Therefore, this study demonstrates that, depending on the degree of employment externality and the government’s preference for seeking employment, all combinations are enabled.

Finally, we would like to discuss the research’s future direction. One possible extension would be able to investigate the relationship between leadership and competition in tax instruments and tax levels, as presented by Kempf and Rota-Graziosi (2010) and Ogawa (2013). This paper focused on tax instrument selection in a noncooperative game with governments adopting the Nash behavior. However, endogenizing leadership changes the equilibrium choices of tax instruments by affecting the payoffs. Another plausible extension would be to examine endogenizing the government’s objectives, as developed by Pal and Sharma (2013) and Kawachi et al. (2019). This study considered that the governments care about regional employment and their budgets. Based on our findings, employment externalities are expected to affect the payoff structure in strategic delegation games. Thus, by focusing on endogenizing the government’s objectives, the extended analysis would explain why some governments seek employment, whereas others act as Leviathans. This paper provides an analytical foundation for these potential extensions.