ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E22-2

Unemployment and Endogenous Choice on Tax
Instruments in a Tax Competition Model:
Unit Tax versus Ad Valorem Tax
byYuya Kikuchi and Toshiki Tamai

July 2022

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Unemployment and Endogenous Choice on Tax Instruments in a
Tax Competition Model: Unit Tax versus Ad Valorem Tax

Yuya Kikuchia and Toshiki Tamaib*aCollege of Business Administration and Information Science, Chubu University,
Matsumoto-cho 1200, Kasugai, Aichi 487-8501, Japanb

Graduate School of Economics, Nagoya University,
Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, Japan

Abstract
This paper examines the endogenous choice of the government’s tax instrument
between the unit and ad valorem taxes under tax competition with unemployment.
Governments seek to maximize their objective functions that are the weighted sum of
employment and revenue levels. Considering the tax competition model, a high fixed
wage rate generates not only unemployment but also employment externalities. Thiseffect can be either positive or negative because of the presence of capital freely
mobile among regions. Without unemployment, the revenue-maximizing
governments choose unit tax as their tax instruments to avoid revenue loss from
intense tax competition under ad valorem taxes. However, with unemployment,
positive employment externalities generate additional benefits to use ad valorem
taxes for stimulating employment. Therefore, the present study shows that one region
chooses an ad valorem tax and the other selects a unit tax, or both governments use
ad valorem taxes depending on employment externalities.
Keywords: Tax competition; Unit tax; Ad valorem tax; Unemployment
JEL Classifications: H20; H21; H77; J64*
Corresponding author. Tel: +81-52-789-2384. E-mail address: tamai@soec.nagoya-u.ac.jp. This work was supported by JSPS
KAKENHI (Grant Number: 20H01492). The authors are grateful to the seminar participants of Nagoya International Economics Study
Group Workshop at Hokkaido University for their constructive comments.

1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 recently focused issues in
the literature is to investigate which of the unit tax and the ad valorem tax is chosen as thegovernment’s tax instruments (e.g., Lockwood, 2004; Akai et al., 2011; Aiura and Ogawa, 2013;
Hoffmann and Runkel, 2016; Ogawa, 2016). Existing studies treated this issue by assuming aperfect labor market, and therefore, there is no unemployment. However, in reality, developed
and developing countries have competed in their tax rates to encourage investment and create
employment (OECD, 2017). The present study tackles the issue of the government’s choice of tax
instruments under unemployment by focusing on such an aspect of tax competition.
In an outstanding work, assuming constant returns to scale (CRS), competitive market, andperfect labor markets, Lockwood (2004) examined the nature of tax competition equilibrium withtwo different tax instruments by comparing the equilibrium outcomes in terms of welfare. Under
full employment, the author showed that unit taxes are superior to ad valorem taxes because if all
governments choose ad valorem taxes, a harmful tax competition can lead to underprovision of
public goods. Akai et al. (2011) showed that a unit tax regime is endogenously chosen by all
governments using two-stage games with two symmetric countries. By contrast, Aiura and Ogawa(2013) demonstrated that some countries might choose the ad valorem tax regime if countries’
asymmetries are sufficiently large. Departing from CRS production technology, Hoffmann andRunkel (2016) showed that the tax competition under an ad valorem taxation might be lessharmful than that under unit taxation with decreasing returns to scale. Furthermore, Ogawa (2016)
revealed that asymmetric countries do not compete in the same tax instrument.
Respective studies examined the effects of tax competition with unemployment (e.g., Ogawa
et al., 2006; Sato, 2009; Eichner and Upmann, 2012; Exbrayat et al., 2012). Although almost allof them assumed tax competition in the unit tax regime, unemployment can be illustrated inseveral ways: fixed wage, search friction, and labor union models.2 A key factor affecting the tax
competition equilibrium is the employment externality, which is caused by capital mobility andthe technological relationship between capital and labor in production. If capital and labor are a
supplement to each other, then attracting capital improves employment for own region, leading
to the export of unemployment from own region to the other. In other words, the presence of
unemployment derives employment externality.3 This additional source of inefficiency generates
1

See Zodrow (2010) for general review of the broad literature on tax competition.
See Sato (2009) for search friction model and Eichner and Upmann (2012) and Exbrayat et al. (2012) for labor union model.
Numerous studies found empirical evidence to support the effect of taxes on employment (e.g., Feld and Kirchgassner, 2003; Harden
and Hoyt, 2003; Bettendorf et al., 2009; Felix, 2009; Feldmann, 2011; Zirgulis and Šarapovas, 2017). Almost all of the literature foundthat employment is negatively associated with tax rates, implying positive employment externalities. On the contrary, Feldmann (2011)
showed empirical evidence that higher corporate taxes might lower the unemployment rate, leading to a negative employment
externality. We will discuss the possibility of negative employment externalities in Section 4.
2
32

new channels of the effects of different tax instruments. Therefore, the equilibrium outcomesdiffer from the existing literature, which assumed full employment.
This paper aims to examine the equilibrium properties under different tax regimes byendogenizing the government’s choice of tax instruments. Hence, we develop a tax competition
model with unemployment and two tax instruments, along with the theoretical framework basedon Akai et al. (2011). Our model differs from the existing literature with respect to the labormarket structure; the fixed wage is assumed in accordance with Ogawa et al. (2006). 4 We
investigate how the equilibrium outcome is affected by the employment externality using a twostage game where each government chooses either a unit tax or an ad valorem tax in the first stage
and determines the tax rate in the second stage. Furthermore, we examine the robustness of ourresults through numerical analyses assuming regions’ asymmetries.
The key factor in the government’s choice of tax instrument is employment externality. If
positive employment externalities are sufficiently small and the government’s preference for
employment is sufficiently weak, then each government chooses a unit tax as the tax instrument.
For a small degree of positive employment externality and preference for employment, theeconomy with unemployment is closed 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 positive employment externalities are sufficiently large, and thegovernment’s preference for employment is sufficiently strong. In such a case, ad valorem taxes
are chosen as their tax instruments, similar to that of Aiura and Ogawa (2013) and Hoffmann andRunkel (2016). Furthermore, surprisingly, if positive employment externalities and a taste for
employment are intermediate levels between the former two cases, one government chooses aunit tax, and the other opts for an ad valorem tax.
Using ad valorem tax induces governments to reduce their tax rate because the factor price ofcapital (i.e., the marginal product of capital) decreases with capital inflow. Thus, the government
facing capital price down must lower the tax rate to keep the tax revenue (Lockwood, 2004).5
Without unemployment and the government’s preference for employment, this side effect causes
severe tax-cutting competition. Therefore, both governments choose unit taxes as their tax
instruments to avoid their loss in such a difficult situation (Akai et al., 2011). However, if apositive externality exists and the regional government has an interest in creating employment,
this unfavorable side effect could be converted into a favorable one that generates employment
benefits. Therefore, all combinations of tax instruments can be strategically chosen based on the
degree of employment externality and the government’s preference for employment. The

4Several recent studies examined the nature of tax competition equilibrium with unemployment assuming fixed wage (e.g., Gillet
and Pauser, 2018; Kikuchi and Tamai, 2019; Tamai and Myles, 2022).
5
Focusing on a tariff war, Lockwood and Wong (2000) showed that the country switching from specific tariff to ad valorem tax has
an incentive to lower its tariff.

3

robustness of these results is verified through extensive analyses.
The remainder of this paper is organized as follows. Section 2 presents the basic framework of
our theoretical analyses. 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, negative employment externalities, and asymmetricity of regions. Finally, Section 5
concludes the study.

2. The model
Consider two-region economy where each region has the population measured as 𝑁! (𝑖 = 1,2).
In region 𝑖, a continuum of identical firms competitively produces a homogenous good using
capital, labor, and land inputs (𝐾! , 𝐿! , and 𝑍! , respectively). Let be 𝑌! as the output in region 𝑖.
The production function in region 𝑖 is formulated as follows:
𝑌! = 𝐹 ! (𝐾! , 𝐿! , 𝑍! ).
Assuming that 𝐹 ! is a homogenous of degree 1, the production function can be rewritten as
𝑌! = 𝐹 ! /𝐾! 𝐿!
, , 10 𝑍! = 𝑓 ! (𝑘! , 𝑙! )𝑍! ,𝑍! 𝑍!

where 𝑘! ≡ 𝐾! /𝑍! and 𝑙! ≡ 𝐿! /𝑍! . Capital is freely mobile between two regions, whereas laborand land are stuck to the original regions. Without loss of generality, the land input in each region
is normalized to unity (i.e., 𝑍! = 1).
Each jurisdictional government taxes on capital as a unit tax 𝑇! or an ad valorem tax 𝑡! after
the choice of the tax instruments. Profit maximization and perfect mobility of capital conditions
lead to8

𝑟 = 𝑓"! − 𝑇! ,𝑟 = (1 − 𝑡! )𝑓"! ,(1)

where 𝑟 is the post-tax return on capital that is common between two regions and 𝑓"! is thepartial derivative with respect to 𝐾! (i.e., 𝑓"! ≡ 𝜕𝑓 ! /𝜕𝐾! ). Following Ogawa et al. (2006), we
assume the imperfect labor market and the imperfection are symbolized as the fixed wage.6 Let
us denote the fixed wage rate in region 𝑖 by 𝑤=! . The labor demand satisfies the following:𝑤=! = 𝑓#! .

(2)

𝐾 = 𝐾$ + 𝐾% .

(3)

The capital market equilibrium condition is
In the labor market with a sufficiently high level of 𝑤=! , we have 𝐿! < 𝑁! .

6
Several assumptions about labor market imperfection have been made in previous studies. For instance, the efficiency wage model
presented by Yellen (1984) is the simplest way to justify our model.

4Each jurisdictional government’s budget equation becomes
𝐺! = 𝑇! 𝐾! ,A
𝐺! = 𝑡! 𝑓"! 𝐾! ,(4)

where 𝐺! stands for region 𝑖 government’s tax revenue (or government spending which does
not affect residents’ utility and firms’ productivity).
The previous literature used the specified production function for the equilibrium analysis (e.g.,
Wildasin, 1991; Brueckner, 2004; Akai et al., 2011). In particular, we use the following production
function:
𝑌! = D1 +𝛼! 𝐾! + 𝛽! 𝐿! 𝐴! 𝐾!% + 𝐵! 𝐿%! 𝛾! 𝐾! 𝐿!
−+
J 𝜙! 𝑍! ,
𝑍!
2𝑍!%
𝑍!%

(5)

which leads to
𝛼! 𝐴! 𝐾! 𝛾! 𝐿!
𝑓"! = D − % + % J 𝜙! = [𝛼! − 𝐴! 𝐾! + 𝛾! 𝐿! ]𝜙! ,𝑍!
𝑍!
𝑍!
𝑓#! = D𝛽! 𝐵! 𝐿! 𝛾! 𝐾!
− % + % J 𝜙! = [𝛽! − 𝐵! 𝐿! + 𝛾! 𝐾! ]𝜙! ,
𝑍!
𝑍!
𝑍!

𝑓&! = D1 +𝐴! 𝐾!% + 𝐵! 𝐿%! 𝛾! 𝐾! 𝐿!
𝐴! 𝐾!% + 𝐵! 𝐿%!
−
J
𝜙
=
D1
+
− 𝛾! 𝐾! 𝐿! J 𝜙! .
!
2
2𝑍!%
𝑍!%

Using Eqs. (2) and (5) yields the following:
𝐿! = P𝛽! + 𝛾! 𝐾! −𝑤=! 1 𝛽! + 𝛾! 𝐾! − 𝜔!
Q == 𝑐! + 𝜇! 𝐾! ,
𝜙! 𝐵!
𝐵!

(6)

where
𝜔! ≡

𝑤
=!
𝛽! − 𝜔!
𝛾!
, 𝑐! ≡, and 𝜇! ≡ .
𝜙!
𝐵!
𝐵!

Eq. (6) implies that the employment level increases with capital. We assume that 𝑐! is sufficiently
small to ensure 𝐿! < 1. Eqs. (5) and (6) provide
𝑓"! = D𝛼! − V𝐴! −𝛾!%
𝛽! − 𝜔!
W 𝐾! + /0 𝛾! J 𝜙! = 𝑎! − 𝑏! 𝐾! ,
𝐵!
𝐵!

(7)

where
𝛽! − 𝜔!
𝛾!%
𝑎! ≡ P𝛼! + /0 𝛾! Q 𝜙! and 𝑏! ≡ V𝐴! − W 𝜙! .
𝐵!
𝐵!
Eq. (7) has a functional form that is similar to that of Akai et al. (2011). Particularly, it is identicalto 𝛾! = 0. With 𝛾! ≠ 0, the intercept 𝑎! and the coefficient 𝑏! depend on 𝛾! . The employment
externality will quantitatively affect the payoff of policy choice and, therefore, the equilibrium
strategy. We focus on the case where positive employment externality 𝛾! > 0 until Section 4.

5

3. Competition in tax instruments and tax levels
In this section, we study a tax competition game with the choice of a tax instrument. The game isnoncooperative and has two stages. In the first stage of the game (stage 1), two jurisdictional
governments simultaneously choose their tax instruments out of a unit or an ad valorem tax. In
the second stage (stage 2), the governments determine their tax levels to maximize the values of
their objective functions. Throughout this section, we assume that the two regions are symmetric.
Then, we will make the following assumption to ensure 𝑌! > 0:
Assumption 1 𝑎 > 𝑏𝐾.
The same assumption is made in the previous literature. Assumption 1 ensures that the marginal
product of capital cannot be negative if the entire stock of capital is used in a region.

3.1. Determination of tax levels
We now consider the determination of tax levels at stage 2 when (i) both regional governmentschoose a unit tax (UU), (ii) both governments choose an ad valorem tax (AA), or (iii) one
jurisdictional government chooses a unit tax, whereas the other chooses an ad valorem tax (UAor AU). 𝑈 and 𝐴 denote the choice of unit tax and that of the ad valorem tax, respectively. The
governments consider the following effects of their tax policy (see Appendix A for the derivationof the following equations):
𝑈𝑈:

𝜕𝐾!
1
=−
< 0,
𝜕𝑇!
2𝑏

𝐴𝐴:

𝜕𝐾!
𝑎 − 𝑏𝐾!
=−< 0,
𝜕𝑡!
`2 − 𝑡! − 𝑡' a𝑏

𝐴𝑈 𝑜𝑟 𝑈𝐴:

𝜕𝐾!
1
𝜕𝐾!
𝑎 − 𝑏𝐾!
=−< 0 𝑎𝑛𝑑=−< 0,(2 − 𝑡! )𝑏𝜕𝑇!
𝜕𝑡!
`2 − 𝑡' a𝑏

where 𝑎$ = 𝑎% = 𝑎 and 𝑏$ = 𝑏% = 𝑏 (𝑖 = 1,2 and 𝑖 ≠ 𝑗).
Previous studies supposed that governments maximize their tax revenue like Leviathan. In
addition to the Leviathan objective, we consider that the governments care about their regional
employment levels. The region 𝑖 government’s objective function is formulated as follows:7

7

The absentee owenership of capital and land justifies this functional form. The Benthamite welfare function should be formulated
as 𝑊! = 𝑟𝜎𝐾 + 𝑤/ ! 𝐿! + 𝜋! + 𝑣(𝐺! ) = 𝐹 ! (𝐾! , 𝐿! , 𝑍! ) − (𝑟 + 𝑇! )𝐾! + 𝑟𝜎𝐾 + 𝑣(𝐺! ) . If 𝑣(𝐺! ) = (1 + 𝜂)𝐺! , we have 𝑊! =𝑓 ! (𝐾! , 𝐿! ) + (𝜎𝐾 − 𝐾! )𝑟 + 𝜂𝐺! . Using Eq. (8) is a more simple way to illustrate the effects of tax competition on equilibrium outcome(the same as capital, employment, and government revenue).

6𝑉! = 𝜃! 𝐿! + (1 − 𝜃! )𝐺! ,(8)

where the weight for the regional employment level is 0 ≤ 𝜃! ≤ 1. If 𝜃! = 0, the government’s
objective function is identical to that of Akai et al. (2011). Hence, the governments behave like apure Leviathan. By the assumption of symmetric regions, we focus on 𝜃! = 𝜃 > 0 (𝑖 = 1,2).
Regarding the equilibrium tax rates at stage 2, we obtain the following Lemma 1 (see Appendix
B for the proof of Lemma 1):
Lemma 1 (i) If the governments choose a unit tax,
𝑇$ = 𝑇% = 𝑏𝐾 − 𝜎, where 𝜎 ≡

𝜃
𝜇.
1−𝜃

(ii) If the governments choose an ad valorem tax,
𝑡$ = 𝑡% =𝑏𝐾 − 𝜎.
𝑎(iii) If region i government chooses a unit tax and the other opts for an ad valorem tax,
𝑇! =𝑏𝐾 − 𝜎 𝐻
𝐻+ and 𝑡' =, 𝑖, 𝑗 = 1,2 and 𝑖 ≠ 𝑗,2
42(𝑎 − 𝑏𝐾)

where 𝐻 ≡ 6𝑎 − 𝑏𝐾 − 𝜎 − √36𝑎% − 36𝑎𝑏𝐾 + 25𝑏% 𝐾 % + 12𝑎𝜎 − 22𝑏𝐾𝜎 + 𝜎 % .
The parameter, 𝜎, denotes a degree of employment externality weighted by the government’s
objective parameter. When 𝜎 = 0, the results from (i) through (iii) in Lemma 1 go back to thosederived by Lockwood (2004) and Akai et al. (2011). Considering that 𝜇 represents the sign ofcross-derivatives of 𝐾 and 𝐿, 𝜎 = 0 implies that no employment externality exists. If 𝜃 = 0,
the same situation occurs. Without the government’s interest in regional employment, theemployment externality does not arise. The employment level does not affect the government’s
objective function in each case. Therefore, this case is identical to the model presented by Akai
et al. (2011).
Lemma 1 also indicates the possibility of capital subsidy. In reality, positive lump-sum taxes
must be available for financing negative capital taxes. However, negative government revenue
can be considered because we assume the linear preference with respect to employment andgovernment revenue/expenditure. Furthermore, capital subsidies aiming to stimulate employment
can be observed in the real economy. Based on Lemma 1, the tax rates in cases (i) and (ii) arenegative for 𝜎 > 𝑏𝐾 . This result implies that the governments are motivated to use capitalsubsidies 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, 2021). If thelump-sum tax can be limitedly used for ensuring non-negative tax revenue, the unit or ad valorem
tax could be negative. Furthermore, the output must be positive. To ensure the positive output,

7𝜎 < 𝑎 is required.8 Hence, we impose the following assumption:
Assumption 2 𝑎 > 𝜎.
Furthermore, we need the following assumption to ensure positive capital demand.
Assumption 3 𝑎 > 2𝑏𝐾.
When 𝜎 = 0, 2𝑎 > 3𝑏𝐾 is sufficient to ensure positive capital demand.9 However, if 𝜎 > 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.
Hence, the tax rates must be converted to the comparable forms using 𝑡! 𝑓"! = 𝑇! . Following Akai
et al. (2011), let 𝑡!>? be the region 𝑖’s effective ad valorem tax rate in the Nash equilibrium when
the region 𝑖 government chooses tax instrument 𝑚 (𝑚 = 𝑈, 𝐴), and the other chooses tax
instrument 𝑛 (𝑛 = 𝑈, 𝐴). Lemma 1 and 𝑡! 𝑓"! = 𝑇! (𝑖 = 1,2) yield the following equations (seeAppendix C for the derivation of the following equations):2(𝑏𝐾 − 𝜎),2𝑎 − 𝑏𝐾(9a)

𝑏𝐾 − 𝜎,𝑎(9b)[2(𝑏𝐾 − 𝜎) + 𝐻][4(𝑎 − 𝑏𝐾) − 𝐻],
4𝑎[4(𝑎 − 𝑏𝐾 ) − 𝐻] − 2[2(𝑏𝐾 − 𝜎) + 𝐻](𝑎 − 𝑏𝐾 )

(9c)

𝐻
.
2(𝑎 − 𝑏𝐾 )(9d)𝑡!@@ =

𝑡!AA =
𝑡!@A =

𝑡!A@ =

A comparison among Eqs. (9a)–(9d) makes the following proposition (see Appendix D for the
proof of Proposition 1):
Proposition 1 There exists 𝜎s ∈ (0, 𝑏𝐾) , satisfying 𝑡!@A = 𝑡!AA . Then, 𝑡!@@ > 𝑡!A@ > 𝑡!@A >𝑡!AA holds if 𝜎 < 𝜎s , whereas 𝑡!@@ > 𝑡!A@ > 𝑡!AA ≥ 𝑡!@A if 𝜎s ≤ 𝜎 < 𝑏𝐾 . When 𝜎 > 𝑏𝐾 , theorders of tax rates become 𝑡!@@ < 𝑡!A@ < 𝑡!@A < 𝑡!AA .
The degree of employment externality affects the magnitude relation of the tax rates under
different scenarios. If the employment externality effect is not too strong, the lowest tax rate
within four scenarios is the tax rate when both governments choose an ad valorem tax. This8𝑏𝐾 > 𝜎 requires that the employment externality is sufficiently small to ensure positive values of 𝐺! . This condition andAssumption 1 lead to 𝑎 > 𝜎.
9
See Appendix C. For instance, Eq. (C1) is positive if Assumption 3 holds.

8

corresponds to the case shown by Akai et al. (2011). However, if the employment externality
effect is sufficiently large, the smallest tax rate changes from the tax rate when both governmentschoose an ad valorem tax to that when the region 𝑖’s government chooses a unit tax and the otherchooses an ad valorem tax. When the governments have preferences for employment, the capitalattracting effects of cuts of unit and ad valorem taxes are more evaluated by the governments thanpure Leviathans. In particular, if the employment externality effect is sufficiently large or thegovernments have a strong preference for employment, the capital-attracting effect of decreasing
ad valorem tax relative to that of unit tax is strengthened. Therefore, the government sets thelowest rate of ad valorem tax for 𝜎s < 𝜎, facing the other unit tax as its tax instrument. If 𝜎 >𝑏𝐾, the tax rates are all negative. Hence, the magnitude relationship must be inversed in the caseof 𝜎 < 𝑏𝐾.
The region i government’s payoff is given as follows (see Appendix E for the proof of Lemma
2):
Lemma 2. Let be 𝜆 ≡ (1 − 𝜃)E$ 𝜃𝑐. (i) If the governments choose a unit tax,
𝑉!@@ = (1 − 𝜃) V𝜆 +

𝑏𝐾 %
W.
2

(ii) If the governments choose an ad valorem tax,
𝑉!AA = (1 − 𝜃) D𝜆 +(2𝑎 − 𝑏𝐾 + 𝜎)𝑏𝐾 %J.
4𝑎(iii) If region i government chooses a unit tax and the other opts for an ad valorem tax,
𝑉!@A = (1 − 𝜃) 8𝜆 + /𝑏𝐾 + 𝜎 𝐻 [2(𝑏𝐾 − 𝜎) + 𝐻](𝑎 − 𝑏𝐾)
+ 0
w,
2
4
2𝑏[4(𝑎 − 𝑏𝐾 ) − 𝐻]

𝑉!A@ = (1 − 𝜃) 8𝜆 + D𝜎 +(𝑎 − 𝑏Γ)𝐻(𝑎 − 𝑏𝐾)(6𝑏𝐾 − 𝐻 + 2𝜎) − 2𝑏𝐾𝐻
J Γw , where Γ ≡.
2(𝑎 − 𝑏𝐾)2𝑏[4(𝑎 − 𝑏𝐾) − 𝐻]

Based on the result of Lemma 1, the employment externality changes the magnitude relationof the payoff chosen by the governments, depending on its size. Table 1 represents the payoffs.
Regarding this point, Lemma 2 derives the following result (see Appendix F for the proof of
Proposition 2):
Proposition 2. There exist 𝜎y and 𝜎z , satisfying 𝑉!@A ⋛ 𝑉!AA ⇔ 𝜎 ⋚ 𝜎y and 𝑉!@@ ⋛ 𝑉!A@ ⇔𝜎 ⋚ 𝜎z, respectively. (i) If 0 ≤ 𝜎 < min(𝜎y, 𝜎z), each government chooses a unit tax in the Nashequilibrium. (ii) If min(𝜎y, 𝜎z) < 𝜎 < max(𝜎y, 𝜎z), two possible cases exist: 𝜎y < 𝜎 < 𝜎z and 𝜎z <𝜎 < 𝜎y. When, 𝜎y < 𝜎 < 𝜎z, two Nash equilibria exist such that one government selects the sametax instrument as the other. When 𝜎z < 𝜎 < 𝜎y, two Nash equilibria exist such that one government

9

chooses a unit tax and the other opts for an ad valorem tax. (iii) If 𝜎 > max(𝜎y, 𝜎z), eachgovernment chooses an ad valorem tax in the Nash equilibrium.
The intuition of Proposition 2 can be explained as follows. First, we focus on the result (i)where 0 ≤ 𝜎 < min(𝜎y, 𝜎z). In other words, the employment externality is sufficiently small, or
the government’s preference is weak enough to hold Akai et al.’s (2011) scenario. Taking 𝜎 = 0
as an example, the intuition for the result (i) is along with that of Lockwood (2004) and Akai etal. (2011). Suppose that a small increase in public goods expenditure occurs in region 1. In thiscase, financed by increased unit tax, it will cause a capital outflow given by ∆ from region 1. Ifregion 2 adopts a unit tax, ∆ unit of capital is employed in region 2, and thus, region 2’s
government revenue is increased by 𝑇% ∆. 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 isgiven by 𝑡% 𝑓F% ∆ and 𝑓F% decreases with ∆. Therefore, in the first stage, both governments arewilling to choose a unit tax rather than an ad valorem tax.
In the case of the result in (iii) where 𝜎 > max(𝜎y, 𝜎z), the mechanism is contrary to the resultin (i). As same as the case in (i), suppose that region 1’s government increases a unit tax to financea small increase in public goods expenditure. If region 2 uses an ad valorem tax, it means that
region 2’s government has an incentive to reduce the tax rate because of the side effect mentioned
above. A decrease in the tax rate to attract more capital leads to a more increase in employment.
With large 𝜎 , 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!@@ < 𝑉!A@ and 𝑉!@A < 𝑉!AA hold; one region has no incentive to choose a unit tax if the otheruses an ad valorem tax. Therefore, both governments will select an ad valorem tax as the tax
instrument in the first stage.
We now turn to the result (ii) where min(𝜎y, 𝜎z) < 𝜎 < max(𝜎y, 𝜎z). This case is an intermediate
between (i) and (iii), and the payoffs exhibit complicated magnitude relation for different valuesof 𝜎. Thus, the regions may strategically choose different tax instruments depending on 𝜎, even
if the regions are symmetric.
We consider the case where 𝜎y < 𝜎 < 𝜎z. Suppose that region 2 chooses unit tax. This choice isthe signal to region 1 that region 2 has no intention to bring intense tax competition. If region 1chooses an ad valorem tax as the tax instrument in the first stage for creating more employment.
Nevertheless, it may cause fierce tax-cutting competition. Region 1 does not obtain benefits
enough to cover the costs of intense tax competition because 𝜎 is not sufficiently large (i.e.,
𝑉$@@ > 𝑉$A@ for 𝜎 < 𝜎z). Therefore, the best strategy for region 1 is to choose a unit tax the sameas region 2. The same logic can be applied to region 2. Next, suppose that region 2 uses an advalorem tax for 𝜎y < 𝜎 < 𝜎z. This choice would be the signal that region 2 intends to cause an

10intense tax competition for region 1. If region 1’s government chooses unit tax to avoid intense
tax competition, region 1 faces capital outflow to region 2. As a result, region 1 does not add
benefits from such a choice compared when choosing a unit tax (𝑉$@A < 𝑉$AA for 𝜎 > 𝜎y ).
Consequently, both regions choose an ad valorem tax in the first stage.
Finally, the case where 𝜎z < 𝜎 < 𝜎y can be explained as follows. Suppose that region 2 selects
unit tax. If region 1, in turn, chooses an ad valorem tax, lowering the tax rate is required to keep
the tax revenue, and then, the government expects to obtain additional employment benefits. By
contrast, when region 1 uses a unit tax instead of an ad valorem tax, there is no opportunity for
extra benefits from creating employment. Therefore, region 1 chooses an ad valorem tax inresponse to region 2’s choice of unit tax (𝑉$@@ < 𝑉$A@ for 𝜎 > 𝜎z). We now consider the case that
region 2 chooses an ad valorem tax. If region 1 uses an ad valorem tax, region 1 must countervail
tax-cutting by region 2 to prevent additional capital flight. On the contrary, by using a unit tax,
region 1 has no necessity to cope with such an unfavorable effect. Hence, region 1 selects a unit
tax when region 2 uses an ad valorem tax (𝑉$@A > 𝑉$AA for 𝜎 < 𝜎y). These results show that thebest strategy for one region is choosing the same tax instrument in response to that chosen by theother region.
To close the analysis in this section, we provide numerical examples of critical values. Given
that many parameters interact with each other and tremendous numbers of their 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 illustrates the contour plot related to 𝜎y with respect to 𝜇 and𝜃, whereas Figure 2 shows the contour plot related to 𝜎z with respect to 𝜇 and 𝜃. Given that thelevel line of 0 is isoquant of the critical value, the area below (above) the 0-level line means 𝜎 <𝜎y (𝜎 > 𝜎y) in Figure 1 or 𝜎 < 𝜎z (𝜎 > 𝜎z) in Figure 2. Both figures imply that higher values of 𝜃and 𝜇 tend to generate the result (ii) or (iii) in Proposition 2.

4. Further analyses
In the previous sections, we assume that positive employment externalities exist and two regions
are symmetric. However, in the real world, negative employment externalities may appear, andthere are asymmetries in regions. Hence, we discuss the possibility of capital subsidy (negative
tax rate on capital) and examine the effects of negative employment externalities and theequilibrium properties in the economy with asymmetric regions.
Negative employment externalities. A degree of employment externality can be measured usingparameter 𝜇. A negative employment externality implies that 𝜇 is negative. Then, 𝜎 < 0 holds
by its definition. In reality, the value of 𝜇 is plausible to be negative, leading to a negative cross11derivative with respect to capital and labor. A certain kind of improvement in production may
damage workers, such as the unskilled, in the short term.10 The task model developed by Autoret al. (2003) precisely elucidates such a situation. The production function used in this study isrelated to Autor et al.’s (2003) production technology. Therefore, 𝜎 < 0 should be included. If
𝜎 is sufficiently small in the absolute value, the qualitative effects are the same as those shownin the previous sections.
Asymmetric regions. We consider the equilibrium tax instruments chosen by the governments
in asymmetric regions. For the reason explained in Section 3, we set 𝑎! , 𝑏! , 𝑐! , 𝜃! , and 𝐾 as
𝑎! = 3, 𝑏! = 1, 𝑐! = 1, 𝜃! = 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 (𝜇$ , 𝜇% ) = (1.5,0.5) and (𝜇$ , 𝜇% ) = (2,1.5). Based on
Table 2, regions 1 and 2, respectively, use an ad valorem tax and a unit tax if (𝜇$ , 𝜇% ) = (1.5,0.5),
whereas both governments choose an ad valorem tax if (𝜇$ , 𝜇% ) = (2,1.5). These results areexamples of the results (ii) and (iii) in Proposition 2 for asymmetric regions.
We now move on to the case of positive employment externality in one region and negative
employment externality in the other. The payoffs are calculated as shown in Table 3 for two
subcases where (𝜇$ , 𝜇% ) = (1.5, −0.5) and (𝜇$ , 𝜇% ) = (0.5, −1.5) . In this case, thegovernments select unit taxes as their tax instruments. This finding corresponds to the result in (i)of Proposition 2.
Finally, considering negative employment externalities in both regions, the payoffs are given
in Table 4. For (𝜇$ , 𝜇% ) = (−0.5, −1.5) and (𝜇$ , 𝜇% ) = (−1.5, −2), the governments choose
unit taxes. These numerical examples imply that all the cases of Proposition 2 are realized
depending on the values of key parameters.

5. Conclusion
This paper examined the government’s choice of tax instruments under unemployment. The
government is assumed to seek larger employment and tax revenue. If positive employment
externalities are sufficiently small, the governments choose unit taxes as their tax instruments. By
contrast, if positive employment externalities are sufficiently large, ad valorem taxes are chosen
as their tax instruments. Furthermore, suppose positive employment externalities are intermediate
between the former two cases. In that case, one government chooses a unit tax, and the other opts
for an ad valorem tax, or one government selects the same tax instrument that is chosen by the10
Frey and Osborne (2017) estimated that approximately 47% of the total US employment is in the high-risk category that jobs could
be automated relatively soon.

12other. Therefore, this study demonstrates that all combinations are enabled based on the degreeof employment externality and the government’s preference for seeking employment.
The mechanisms of our main findings are explicated along with the existing literature.
Lockwood (2004) showed that an ad valorem tax gives governments an incentive to lower theirtax rate because the rate of return on capital, that is, the marginal product of capital, decreases
with capital inflow. Thus, the government facing a decrease in the rate of return must reduce thetax rate to keep the tax revenue. With positive employment externalities, this unfavorable side
effect could be turned into a favorable one that generates employment benefits if the regional
government has a strong interest in creating employment. Hence, if the government’s preference
for employment is sufficiently weak, both governments choose unit taxes, as shown by Lockwood(2004) and Akai et al. (2011). However, the governments with sufficiently strong preferences for
employment use ad valorem taxes, similar to Aiura and Ogawa (2013) and Hoffmann and Runkel(2016), who allowed asymmetricity of regions. Most surprisingly, multiple equilibria can occur
in intermediate cases. The robustness of our results is ensured by extensive analyses.
Finally, we would like to mention the future direction of this research. One possible extension
is 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 focusedon choices of tax instruments in a noncooperative game with governments taking the Nashbehavior. However, endogenizing leadership changes equilibrium choices of tax instruments byaffecting the payoffs. Another plausible extension is to examine endogenizing the government’s
objectives as developed by Pal and Sharma (2013) and Kawachi et al. (2019). In this study, we
considered that the governments care about regional employment and their budgets. Based on ourfindings, employment externalities are expected to affect the payoff structure in strategicdelegation games. Hence, focusing on endogenizing the government’s objectives, the extended
analysis would elucidate why some governments aim to seek employment, whereas others act as
Leviathans. This paper provides an analytical basis for these future extensions.

13Appendix
A. Effects of tax policy on capital
Unit tax. Eqs. (1), (3), and (7) yield the following system:
𝑎$ − 𝑏$ 𝐾$ − 𝑇$ = 𝑎% − 𝑏% 𝐾% − 𝑇% ,𝐾$ + 𝐾% = 𝐾.
Solving the above system with respect to 𝐾! and applying 𝑎$ = 𝑎% = 𝑎 and 𝑏$ = 𝑏% = 𝑏 to
the solution, we obtain
𝐾! =𝑏𝐾 − 𝑇! + 𝑇'
.
2𝑏

(A1)

The first-order and second-order partial derivatives with respect to 𝑇$ are
𝜕𝐾!
1
𝜕 % 𝐾!
=−< 0 and
= 0.
𝜕𝑇!
2𝑏
𝜕𝑇!%

(A2)

Ad valorem tax. Using a similar way to derive the effect of a change in unit tax on capital, Eqs.
(1), (3), and (7) give𝐾! =`1 − 𝑡' a𝑏𝐾 + `𝑡' − 𝑡! a𝑎`2 − 𝑡! − 𝑡' a𝑏

.

(A3)

The first-order and second-order partial derivatives with respect to 𝑡$ are derived as
𝜕𝐾! −`2 − 𝑡! − 𝑡' a𝑎 + „`1 − 𝑡' a𝑏𝐾 + `𝑡' − 𝑡! a𝑎…
𝑎 − 𝑏𝐾!
=
=−
< 0,
%
𝜕𝑡!
`2 − 𝑡! − 𝑡' a𝑏`2 − 𝑡 − 𝑡 a 𝑏
!

(A4)

'

𝜕𝐾!
𝜕 % 𝐾! 𝑏`2 − 𝑡! − 𝑡' a 𝜕𝑡! − (𝑎 − 𝑏𝐾! )2(𝑎 − 𝑏𝐾! )
2
𝜕𝐾!
=−
% =
%
% = 2 − 𝑡 − 𝑡 𝜕𝑡 < 0.
𝜕𝑡!
!
'
!
`2 − 𝑡! − 𝑡' a 𝑏`2 − 𝑡! − 𝑡' a 𝑏
Unit vs. Ad Valorem. Without loss of generality, we suppose that region 1 chooses a unit tax
and region 2 opts for an ad valorem tax. Eqs. (1), (3), and (7) yield 𝑎$ − 𝑏$ 𝐾$ − 𝑇$ =(1 − 𝑡% )(𝑎% − 𝑏% 𝐾% ). Solving this with respect to 𝐾! , we obtain
𝐾$ =(1 − 𝑡% )𝑏𝐾 + 𝑎𝑡% − 𝑇$𝑏𝐾 − 𝑎𝑡% + 𝑇$
and 𝐾% =.
(2 − 𝑡% )𝑏(2 − 𝑡% )𝑏(A5)Hence, the partial derivatives are𝜕𝐾$1
𝜕 % 𝐾$=−𝑎𝑛𝑑= 0,(2 − 𝑡% )𝑏
𝜕𝑇$
𝜕𝑇$%

(A6)

𝜕𝐾%
𝑎 − 𝑏𝐾%
𝜕 % 𝐾%
𝑎 − 𝑏𝐾%
1 𝜕𝐾%=−< 0 𝑎𝑛𝑑
=−
=
< 0.
%
%
(2 − 𝑡% )𝑏(2 − 𝑡% ) 𝑏 2 − 𝑡% 𝜕𝑡%
𝜕𝑡%
𝜕𝑇%

(A7)

14

B. Proof of Lemma 1
Unit vs. Unit. In the second stage, the unit tax level must satisfy
𝜕𝑉!
𝜕𝐿!
𝜕𝐺!
𝜕𝐾!
𝜕𝐾!
=𝜃
+ (1 − 𝜃)= 𝜃𝜇+ (1 − 𝜃) /𝐾! + 𝑇!
0 = 0,
𝜕𝑇!
𝜕𝑇!
𝜕𝑇!
𝜕𝑇!
𝜕𝑇!

(B1)

𝜕 % 𝑉!
𝜕 % 𝐾!
𝜕𝐾!
𝜕 % 𝑉!
𝜕𝐾!
(1
=
𝜃𝜇
+
−
𝜃)
V2
+
𝑇
W = 2(1 − 𝜃)
< 0.
!
%
%
%
𝜕𝑇!
𝜕𝑇!
𝜕𝑇!
𝜕𝑇!
𝜕𝑇!
Solving Eq. (B1) yields
𝐾! = −𝑇!

𝜕𝐾!
𝜕𝐾!
−𝜎
.
𝜕𝑇!
𝜕𝑇!

(B2)

Inserting Eqs. (A1) and (A2) into Eq. (B2) and applying 𝑇$ = 𝑇% , we obtain
𝑇$ = 𝑇% = 𝑏𝐾 − 𝜎 and 𝐾$ = 𝐾% =

𝐾
.
2

(B3)

Ad Valorem vs. Ad Valorem. The optimality conditions in the second stage are
𝜕𝑉!
𝜕𝐿!
𝜕𝐺!
𝜕𝐾!
𝜕𝐾!
=𝜃
+ (1 − 𝜃)= 𝜃𝜇+ (1 − 𝜃) P(𝑎 − 𝑏𝐾! )𝐾! + (𝑎 − 2𝑏𝐾! )𝑡!
Q = 0, (B4)
𝜕𝑡!
𝜕𝑡!
𝜕𝑡!
𝜕𝑡!
𝜕𝑡!
𝜕 % 𝑉!
2(𝑎 − 𝑏𝐾! )𝑡!
2
𝜕𝐾!
)(𝑎 − 2𝑏𝐾! )𝑡!
J< 0.
% = D2(𝑎 − 2𝑏𝐾! + 2 − 𝑡 − 𝑡 +2 − 𝑡! − 𝑡' 𝜕𝑡!
𝜕𝑡!
!
'
Eq. (B4) leads to(𝑎 − 𝑏𝐾! )𝐾! = −(𝑎 − 2𝑏𝐾! )𝑡!

𝑑𝐾!
𝑑𝐾!
−𝜎
.
𝑑𝑡!
𝑑𝑡!

(B5)

With the symmetric region condition, Eqs. (A3), (A4), and (B5) provide
𝑡$ = 𝑡% =𝑏𝐾 − 𝜎𝐾
and 𝐾$ = 𝐾% = .
𝑎
2

(B6)

Unit vs. Ad Valorem. The first-order conditions are given by Eqs. (B1) and (B4). The secondorder conditions are𝜕 % 𝑉$
𝜕 % 𝐾$
𝜕𝐾$
𝜕 % 𝐾$
𝜕𝐾$
(1
=
𝜃𝜇
+
−
𝜃)
V2
+
𝑇
W = 2(1 − 𝜃)
< 0,
!
%
%
%
𝜕𝑇$
𝜕𝑇$
𝜕𝑇$
𝜕𝑇$
𝜕𝑇$
𝜕 % 𝑉%
𝜕 % 𝐾%
𝜕𝐾%
𝜕𝐾% %
𝜕 % 𝐾%
(1
)
(𝑎
)𝑡
=
𝜃𝜇
+
−
𝜃)
D2(𝑎
−
2𝑏𝐾
−
2𝑏𝑡
+
−
2𝑏𝐾
J < 0,
/
0
%
%
% %
𝜕𝑡%
𝜕𝑡%
𝜕𝑡%%
𝜕𝑇%%
𝜕𝑡%%
where
𝜕 % 𝐾%
𝜕𝐾%
𝜕𝐾% %
𝜕 % 𝐾%)(𝑎)𝑡𝜎+ 2(𝑎 − 2𝑏𝐾%
− 2𝑏𝑡% /0 + − 2𝑏𝐾% %
𝜕𝑡%
𝜕𝑡%
𝜕𝑇%%
𝜕𝑡%%
=

[𝜎 + (4 + 𝑡% )𝑎 − 8𝑏𝐾% ] 𝜕𝐾%.
2 − 𝑡%
𝜕𝑡%

Eqs. (A5), (A6), (A7), (B1), and (B4) provide

15

𝜕𝐾$
𝜕𝐾$
𝑇$ + 𝜎
−𝜎
=
,
(2
𝜕𝑇$
𝜕𝑇$
− 𝑡% )𝑏(𝑎 − 2𝑏𝐾% )𝑡% + 𝜎
𝜕𝐾𝜕𝐾(𝑎 − 𝑏𝐾% )𝐾% = −(𝑎 − 2𝑏𝐾% )𝑡% % − 𝜎 % ⇔ 𝐾% =.
(2 − 𝑡% )𝑏𝜕𝑡%𝜕𝑡%
𝐾$ = −𝑇$

(B7)
(B8)

Solving Eqs. (B7) and (B8) with respect to 𝑇$ and 𝑡% gives
𝑇$ =𝑏𝐾 − 𝜎 𝐻
𝐻+ and 𝑡% =,2
42(𝑎 − 𝑏𝐾)

(B9)

where
𝐻 ≡ 6𝑎 − 𝑏𝐾 − 𝜎 − ‡36𝑎% − 36𝑎𝑏𝐾 + 25𝑏% 𝐾 % + 12𝑎𝜎 − 22𝑏𝐾𝜎 + 𝜎 % .
Using Eqs. (B7), (B8), and (B9), we arrive at
𝐾$ =(𝑎 − 𝑏𝐾)(6𝑏𝐾 − 𝐻 + 2𝜎) − 2𝑏𝐾𝐻
[2(𝑏𝐾 − 𝜎) + 𝐻](𝑎 − 𝑏𝐾)
and 𝐾% =.
2𝑏[4(𝑎 − 𝑏𝐾) − 𝐻]
2𝑏[4(𝑎 − 𝑏𝐾 ) − 𝐻]

(B10)

C. Derivation of equations from (9a) to (9d)Unit vs. Unit. Using Eqs. (1), (7), and (B3), we obtain𝑟=

2𝑎 − 3𝑏𝐾 + 2𝜎
.
2

(C1)

By the definition of 𝑡!>? , Eq. (C1) leads to𝑡!@@ =𝑏𝐾 − 𝜎 2(𝑏𝐾 − 𝜎)=.
𝑏𝐾
2𝑎 − 𝑏𝐾𝑎−2

Ad Valorem vs. Ad Valorem. Eqs. (1), (7), and (B6) provide𝑟=(2𝑎 − 𝑏𝐾)(𝑎 − 𝑏𝐾 + 𝜎)
.
2𝑎

(C2)

In a similar way to deriving Eq. (9a), we arrive at𝑡!AA =𝑏𝐾 − 𝜎.
𝑎

Unit vs. Ad Valorem. Eqs. (1), (7), (B9), and (B10) engender𝑟 = (1 − 𝑡% )(𝑎 − 𝑏𝐾% ) =[2(𝑎 − 𝑏𝐾) − 𝐻]{2[4𝑎 − (3𝑏𝐾 + 𝜎)] − 𝐻}.
4[4(𝑎 − 𝑏𝐾) − 𝐻]

(C3)

Using the definition of 𝑡!>? , we obtain
𝑏𝐾 − 𝜎 𝐻[2(𝑏𝐾 − 𝜎) + 𝐻][4(𝑎 − 𝑏𝐾) − 𝐻]
𝐻
2 +4=
@A
𝑡! =
, 𝑡!A@ =.
𝑎 − 𝑏𝐾!
4𝑎[4(𝑎 − 𝑏𝐾) − 𝐻] − 2[2(𝑏𝐾 − 𝜎) + 𝐻](𝑎 − 𝑏𝐾)
2(𝑎 − 𝑏𝐾)16

D. Proof of Proposition 1𝑡!@@ vs. 𝑡!AA : Eqs. (9a) and (9b) lead to𝑡!@@ − 𝑡!AA =2(𝑏𝐾 − 𝜎) 𝑏𝐾 − 𝜎 (𝑏𝐾 − 𝜎)𝑏𝐾−=
⋛ 0 ⇔ 𝑏𝐾 ⋛ 𝜎.
(2𝑎 − 𝑏𝐾)𝑎2𝑎 − 𝑏𝐾𝑎(D1)

𝑡!A@ vs. 𝑡!AA : Using Eqs. (9b) and (9d), we obtain𝑡!A@ − 𝑡!AA =𝐻
𝑏𝐾 − 𝜎 𝑎𝐻 − 2(𝑏𝐾 − 𝜎)(𝑎 − 𝑏𝐾)−=.
2(𝑎 − 𝑏𝐾)𝑎2𝑎(𝑎 − 𝑏𝐾 )The numerator can be decomposed to𝑎𝐻 − 2(𝑏𝐾 − 𝜎)(𝑎 − 𝑏𝐾) = 𝑀$ − 𝑀% ,where
𝑀$ = 6𝑎% − 3𝑎𝑏𝐾 + 2𝑏% 𝐾 % + 𝑎𝜎 − 2𝑏𝐾𝜎 > 0,
𝑀% = 𝑎‡36𝑎% − 36𝑎𝑏𝐾 + 25𝑏% 𝐾 % + 12𝑎𝜎 − 22𝑏𝐾𝜎 + 𝜎 % > 0.
We have𝑀$% − 𝑀%% = 4𝑏𝐾(𝑎 − 𝑏𝐾)(𝑏𝐾 − 𝜎)(2𝑎 − 𝑏𝐾 + 𝜎) ⋛ 0 ⇔ 𝑏𝐾 ⋛ 𝜎.
Therefore, we obtain𝑡!A@ ⋛ 𝑡!AA ⇔ 𝑏𝐾 ⋛ 𝜎.

(D2)

𝑡!@A vs. 𝑡!AA : Eqs. (9b) and (9c) yield
𝑡!@A − 𝑡!AA =[2(𝑏𝐾 − 𝜎) + 𝐻][4(𝑎 − 𝑏𝐾) − 𝐻]𝑏𝐾 − 𝜎−.
4𝑎[4(𝑎 − 𝑏𝐾) − 𝐻] − 2[2(𝑏𝐾 − 𝜎) + 𝐻](𝑎 − 𝑏𝐾)𝑎

The above equation can be transformed to𝑎(𝑂$ − 𝑂% )`𝑡!@A − 𝑡!AA a = (𝑂G − 𝑂H ),where
𝑂$ = (3𝑎 − 𝑏𝐾)‡36𝑎% − 36𝑎𝑏𝐾 + 25𝑏% 𝐾 % + 12𝑎𝜎 − 22𝑏𝐾𝜎 + 𝜎 % > 0,
𝑂% = 10𝑎% + 𝑎𝑏𝐾 − 𝑏% 𝐾 % − 5𝑎𝜎 + 3𝜎𝑏𝐾 > 0,
𝑂G = [4𝑎% − (𝑎 − 𝑏𝐾)(𝑏𝐾 − 𝜎)]‡36𝑎% − 36𝑎𝑏𝐾 + 25𝑏% 𝐾 % + 12𝑎𝜎 − 22𝑏𝐾𝜎 + 𝜎 % > 0,
𝑂H = 24𝑎G − 18𝑎% 𝑏𝐾 + 13𝑎𝑏% 𝐾 % + 𝑏G 𝐾 G + 𝜎[10𝑎% − 𝑏𝐾(4𝑏𝐾 − 3𝜎) − 𝑎(10𝑏𝐾 + 3𝜎)]> 0.
Taking the difference of the squared terms of 𝑂! provides
𝑂$% − 𝑂%% = 8(𝑎 − 𝑏𝐾)Ψ > 0,
𝑂G% − 𝑂H% = 8(𝑎 − 𝑏𝐾)(𝑏𝐾 − 𝜎)Φ ⋛ 0 ⇔ (𝑏𝐾 − 𝜎)Φ ⋛ 0,where
Ψ ≡ (2𝑎 − 𝑏𝐾)[14𝑎(𝑎 − 𝑏𝐾) + 3𝑏% 𝐾 % ] + 𝜎[14𝑎(𝑎 − 𝑏𝐾) + (2𝑏𝐾 + 𝜎)𝑏𝐾 + 2𝑎(6𝑎 − 𝜎)]
> 0,

17

Φ ≡ 𝑏𝐾(2𝑎 − 𝑏𝐾)(4𝑎% − 5𝑎𝑏𝐾 + 3𝑏% 𝐾 % )− [(24𝑎% − 𝜎 % )(𝑎 − 𝑏𝐾) + 5𝑏% 𝐾 % (5𝑎 − 𝑏𝐾)+ (10𝑎% − 13𝑎𝑏𝐾 + 𝑏% 𝐾 % )𝜎]𝜎.
The sign of Φ is crucial to determine the sign of 𝑂G% − 𝑂H% . We can easily verify
Φ = 𝑏𝐾(2𝑎 − 𝑏𝐾)(4𝑎% − 5𝑎𝑏𝐾 + 3𝑏% 𝐾 % ) > 0 if 𝜎 = 0.
If 𝜎 = 0, 𝑂G% − 𝑂H% > 0. Therefore, 𝑡!@A > 𝑡!AA holds if 𝜎 = 0.
We now consider 0 < 𝜎 < 𝑏𝐾. Then, we have sgn(𝑂G% − 𝑂H% ) = sgnΦ. The derivative of Φwith respect to 𝜎 gives𝑑Φ= −{3(8𝑎% − 𝜎 % )(𝑎 − 𝑏𝐾) + 𝑏% 𝐾 % [5(5𝑎 − 𝑏𝐾) + 2𝜎] + 2𝑎𝜎(10𝑎 − 13𝑏𝐾)} < 0.
𝑑𝜎Furthermore, we have Φ = −16𝑎G 𝑏𝐾 if 𝜎 = 𝑏𝐾. These properties show that a critical point 𝜎sexists such that Φ = 0 and 𝜎s ∈ (0, 𝑏𝐾) for 𝜎 ∈ [0, 𝑏𝐾]. Therefore, we obtain
𝑂G% − 𝑂H% > 0 for 𝜎 ∈ [0, 𝜎s)
𝑂G% − 𝑂H% ≤ 0 for 𝜎 ∈ [𝜎s, 𝑏𝐾]. Finally, we arrive at𝑡!@A ≥ 𝑡!AA for 𝜎 < 𝜎s while 𝑡!@A ≤ 𝑡!AA for 𝜎 ≥ 𝜎s

(D3)

𝑡!@A vs. 𝑡!A@ : Eqs. (9c) and (9d) derive𝑡!@A − 𝑡!A@ =[2(𝑏𝐾 − 𝜎) + 𝐻][4(𝑎 − 𝑏𝐾) − 𝐻]𝐻−.
4𝑎[4(𝑎 − 𝑏𝐾) − 𝐻] − 2[2(𝑏𝐾 − 𝜎) + 𝐻](𝑎 − 𝑏𝐾) 2(𝑎 − 𝑏𝐾 )This equation can be rewritten as(𝑎 − 𝑏𝐾)(𝑂$ − 𝑂% )`𝑡!@A − 𝑡!A@ a = 2(𝑁$ − 𝑁% ),where
𝑁$ = (30𝑎G − 21𝑎% 𝑏𝐾 + 15𝑎𝑏% 𝐾 % + 𝑏G 𝐾 G − 𝑎% 𝜎 − 6𝑎𝑏𝐾𝜎 − 3𝑏% 𝐾 % 𝜎 + 𝑎𝜎 % ) > 0,
𝑁% = [5𝑎% + 𝑏% 𝐾 % − 𝑎(𝑏𝐾 + 𝜎)]‡36𝑎% − 36𝑎𝑏𝐾 + 25𝑏% 𝐾 % + 12𝑎𝜎 − 22𝑏𝐾𝜎 + 𝜎 % > 0.
As we have 𝑂$ > 𝑂% , sgn`𝑡!@A − 𝑡!A@ a = sgn(𝑁$ − 𝑁% ) holds. Calculation shows
𝑁$% − 𝑁%% = −4(𝑎 − 𝑏𝐾)% (𝑏𝐾 − 𝜎){(5𝑎 − 6𝑏𝐾)(2𝑎 − 𝑏𝐾)𝑏𝐾+ [30𝑎(𝑎 − 𝜎) + 2𝑏𝐾(𝑎 + 𝑏𝐾)]𝜎}.
Therefore, we obtain𝑡!@A ⋚ 𝑡!A@ ⇔ 𝑁$% − 𝑁%% ⋚ 0 ⇔ 𝑏𝐾 ⋛ 𝜎.
𝑡!@@ vs. 𝑡!A@ : Eqs. (9a) and (9d) lead to𝑡!@@ − 𝑡!A@ ==2(𝑏𝐾 − 𝜎)𝐻4(𝑏𝐾 − 𝜎)(𝑎 − 𝑏𝐾) − (2𝑎 − 𝑏𝐾)𝐻−=
2𝑎 − 𝑏𝐾2(𝑎 − 𝑏𝐾 )2(2𝑎 − 𝑏𝐾 )(𝑎 − 𝑏𝐾)4(𝑏𝐾 − 𝜎)(𝑎 − 𝑏𝐾) − (2𝑎 − 𝑏𝐾)𝐻.
2(2𝑎 − 𝑏𝐾)(𝑎 − 𝑏𝐾 )Using this equation, we obtain

18

(D4)

2(2𝑎 − 𝑏𝐾)(𝑎 − 𝑏𝐾)`𝑡!@@ − 𝑡!A@ a = 𝑄$ − 𝑄% ,where
𝑄$ = (2𝑎 − 𝑏𝐾)‡36𝑎% − 36𝑎𝑏𝐾 + 25𝑏% 𝐾 % + 12𝑎𝜎 − 22𝑏𝐾𝜎 + 𝜎 % > 0,
𝑄% = 12𝑎% − 12𝑎𝑏𝐾 + 5𝑏% 𝐾 % + 2𝑎𝜎 − 3𝑏𝐾𝜎 > 0.
After some calculations, we have
𝑄$% − 𝑄%% = 8𝑏𝐾(𝑎 − 𝑏𝐾)(2𝑎 − 𝜎)(𝑏𝐾 − 𝜎) ⋛ 0 ⇔ 𝑏𝐾 ⋛ 𝜎.
Therefore, we arrive at𝑡!@@ ⋛ 𝑡!A@ ⇔ 𝑏𝐾 ⋛ 𝜎.

(D5)Comparison between tax rates. Eqs. (D1)–(D5) derive the following results for 𝜎 < 𝑏𝐾:𝑡!@@ > 𝑡!A@ > 𝑡!@A > 𝑡!AA for 𝜎 < 𝜎s,𝑡!@@ > 𝑡!A@ > 𝑡!AA ≥ 𝑡!@A for 𝜎 ≥ 𝜎s.
However, these orders are inverted in the case of 𝜎 > 𝑏𝐾 because the tax rates are negative.
Furthermore, if 𝜎 > 𝑏𝐾, 𝜎 > 𝜎s holds. Consequently, we obtain𝑡!@@ < 𝑡!A@ < 𝑡!@A < 𝑡!AA for 𝜎 > 𝑏𝐾.

E. Proof of Lemma 2
Unit vs. Unit. Inserting Eqs. (4), (6), (9a), and (B3) into Eq. (8) yields
𝜎𝐾 (𝑏𝐾 − 𝜎)𝐾𝑏𝐾 %𝑉!@@ = 𝜃𝑐 + (1 − 𝜃) D+J = (1 − 𝜃) V𝜆 +W.
222
Ad Valorem vs. Ad Valorem. Using (4), (6), (8), (9b), and (B6), we obtain
𝑉!AA = 𝜃𝑐 + (1 − 𝜃) P

𝜎𝐾
𝑏𝐾 𝑏𝐾 − 𝜎 𝐾+ /𝑎 − 0 /
0 Q
2
2
𝑎
2

= (1 − 𝜃) D𝜆 +(2𝑎 − 𝑏𝐾)(𝑏𝐾 − 𝜎)𝐾 + 2𝑎𝜎𝐾J
4𝑎

= (1 − 𝜃) D𝜆 +(2𝑎 − 𝑏𝐾 + 𝜎)𝑏𝐾 %J.
4𝑎Unit vs. Ad Valorem. Eqs. (4), (6), (8), (9c), and (B10) lead to
𝑉!@A = 𝜃𝑐 + (1 − 𝜃)(𝜎𝐾! + 𝑇! 𝐾! )= (1 − 𝜃) 8𝜆 + /𝑏𝐾 + 𝜎 𝐻 [2(𝑏𝐾 − 𝜎) + 𝐻](𝑎 − 𝑏𝐾)
+ 0
w.
2
4
2𝑏[4(𝑎 − 𝑏𝐾 ) − 𝐻]

Similarly, Eqs. (4), (6), (8), (9d), and (B10) provide
𝑉!A@ = 𝜃𝑐 + (1 − 𝜃)[𝜎𝐾! + 𝑡! (𝑎 − 𝑏𝐾! )𝐾! ] = (1 − 𝜃) 8𝜆 + D𝜎 +19(𝑎 − 𝑏Γ)𝐻J Γw.
2(𝑎 − 𝑏𝐾)

F. Proof of Proposition 2
𝑉!@A vs. 𝑉!AA : Lemma 2 gives𝑏𝐾 + 𝜎 𝐻 [2(𝑏𝐾 − 𝜎) + 𝐻](𝑎 − 𝑏𝐾) (2𝑎 − 𝑏𝐾 + 𝜎)𝑏𝐾 %sgn`𝑉!@A − 𝑉!AA a = sgn 8/
+ 0
−
w.
2
4
2𝑏[4(𝑎 − 𝑏𝐾) − 𝐻]
4𝑎
The terms if the curly brackets on the right-hand side of the above equation become𝑏𝐾 + 𝜎 𝐻 [2(𝑏𝐾 − 𝜎) + 𝐻](𝑎 − 𝑏𝐾) (2𝑎 − 𝑏𝐾 + 𝜎)𝑏𝐾 %
+ 0
−
/
2
4
2𝑏[4(𝑎 − 𝑏𝐾 ) − 𝐻]4𝑎=[2(𝑏𝐾 − 𝜎) + 𝐻][2(𝑏𝐾 + 𝜎) + 𝐻](𝑎 − 𝑏𝐾) (2𝑎 − 𝑏𝐾 + 𝜎)𝑏𝐾 %−8𝑏[4(𝑎 − 𝑏𝐾) − 𝐻]
4𝑎=

𝐽$ − 𝐽%,4𝑎𝑏[4(𝑎 − 𝑏𝐾) − 𝐻]

where
𝐽$ = 24𝑎G (𝑎 − 2𝑏𝐾) + 𝑎% (6𝑎% − 𝜎 % ) + 9𝑎𝑏% 𝐾 % (𝑎 − 𝑏𝐾) + (20𝑎% − 3𝑏% 𝐾 % )𝑏% 𝐾 %+ 6𝑎% (𝑎% − 2𝑏𝐾𝜎) + 12𝑎𝑏% 𝐾 % 𝜎 + 4𝑏G 𝐾 G 𝜎 + 𝑏𝐾𝜎 % (𝑎 − 𝑏𝐾) > 0,
𝐽% = [(𝑎 − 𝑏𝐾)(5𝑎% + 𝑏% 𝐾 % ) + 𝑎% (𝑎 − 𝜎) + 𝑏𝐾𝜎(𝑎 + 𝑏𝐾)]× ‡36𝑎% − 36𝑎𝑏𝐾 + 25𝑏% 𝐾 % + 12𝑎𝜎 − 22𝑏𝐾𝜎 + 𝜎 % > 0.
Hence, we havesgn`𝑉!@A − 𝑉!AA a = sgn(𝐽$ − 𝐽% ),(F1)Using 𝐽$ and 𝐽% makes
𝐽$% − 𝐽%% = 4𝑏𝐾(𝑎 − 𝑏𝐾)(2𝑎 − 𝑏𝐾 + 𝜎)Ω,where
Ω ≡ 24𝑎G 𝑏% 𝐾 % − 42𝑎% 𝑏G 𝐾 G + 23𝑎𝑏H 𝐾 H − 4𝑏I 𝐾 I− [72𝑎H − 60𝑎G 𝑏𝐾 − 12𝑎% 𝑏% 𝐾 % + 17𝑎𝑏G 𝐾 G − 8𝑏H 𝐾 H − 12𝑎G 𝜎 + 10𝑎% 𝑏𝐾𝜎+ 7𝑎𝑏% 𝐾 % 𝜎 + 4𝑏G 𝐾 G 𝜎 − 𝑎𝑏𝐾𝜎 % ]𝜎.
Regarding Ω, we have
Ω = 𝑏% 𝐾 % (2𝑎 − 𝑏𝐾)(12𝑎% − 15𝑎𝑏𝐾 + 4𝑏% 𝐾 % ) > 0 for 𝜎 = 0,
Ω = −32𝑎% 𝑏% 𝐾 % (𝑎 − 𝑏𝐾)(2𝑎 − 𝑏𝐾 + 𝜎)(9𝑎% − 12𝑎𝑏𝐾 + 5𝑏% 𝐾 % ) < 0 for 𝜎 = 𝑏𝐾.
Furthermore, Ω is monotonically decreasing in 𝜎 because of𝑑Ω= 12𝑎% [3𝑎(𝑎 − 2𝑏𝐾) + (𝑎 − 𝑏𝐾)𝑏𝐾 + 𝑎(3𝑎 − 2𝜎)] + [17𝑎 − 8(1 − 𝜎)]𝑏G 𝐾 G
𝑑𝜎+ (20𝑎 + 14𝑏𝐾 − 3𝜎)𝑎𝑏𝐾𝜎 < 0.
Hence, there exists 𝜎y ∈ (0, 𝑏𝐾) , which is Ω = 0 ( 𝐽$% − 𝐽%% = 0 ⇔ 𝐽$ = 𝐽% ). Using this, we
obtain20

𝐽$ ⋛ 𝐽% ⇔ Ω ⋛ 0 ⇔ 𝜎 ⋚ 𝜎y.

(F2)

𝑉!@A ⋛ 𝑉!AA ⇔ 𝜎 ⋚ 𝜎y.

(F3)Eqs. (F1) and (F2) derive

𝑉!@@ vs. 𝑉!A@ : Using Lemma 2, we havesgn`𝑉!@@ − 𝑉!A@ a = sgn 8(𝑎 − 𝑏Γ)𝐻𝑏𝐾 %− D𝜎 +J Γw.
22(𝑎 − 𝑏𝐾)

The terms in the curly brackets on the right-hand side of the above equation are(𝑎 − 𝑏Γ)𝐻𝑏𝐾 %− D𝜎 +JΓ2
2(𝑎 − 𝑏𝐾)=

𝐼$ − 𝐼%„2𝑎 + 3𝑏𝐾 − 𝜎 − √36𝑎% − 36𝑎𝑏𝐾 + 25𝑏% 𝐾 % + 12𝑎𝜎 − 22𝑏𝐾𝜎 + 𝜎 % …

%,

where
%

𝐼$ = (𝑎 − 𝑏𝐾) ›2𝑎 + 3𝑏𝐾 − 𝜎 − ‡36𝑎% − 36𝑎𝑏𝐾 + 25𝑏% 𝐾 % + 12𝑎𝜎 − 22𝑏𝐾𝜎 + 𝜎 % œ 𝑏% 𝐾 %> 0,
𝐼% = (𝑎 − 𝑏𝐾) ›6𝑎% − 𝑎𝑏𝐾 + 5𝑏% 𝐾 % − (3𝑎 − 𝑏𝐾)𝜎− (𝑎 + 𝑏𝐾)‡36𝑎% − 36𝑎𝑏𝐾 + 25𝑏% 𝐾 % + 12𝑎𝜎 − 22𝑏𝐾𝜎 + 𝜎 % œ× ›6𝑎% − 𝑎𝑏𝐾 + 5𝑏% 𝐾 % + (7𝑎 − 4𝑏𝐾 − 𝜎)𝜎− (𝑎 + 𝑏𝐾 + 𝜎)‡36𝑎% − 36𝑎𝑏𝐾 + 25𝑏% 𝐾 % + 12𝑎𝜎 − 22𝑏𝐾𝜎 + 𝜎 % œ.
Therefore, we obtainsgn`𝑉!@@ − 𝑉!A@ a = sgn(𝐼$ − 𝐼% ).

(F4)

𝐼$ − 𝐼% = 2(𝑎 − 𝑏𝐾)(𝑃$ − 𝑃% ),(F5)

In Eq. (F4), we havewhere
𝑃$ = (6𝑎G + 5𝑎% 𝑏𝐾 + 2𝑎𝑏% 𝐾 % + 2𝑏G 𝐾 G + 5𝑎% 𝜎 + 2𝑏% 𝐾 % 𝜎 − 2𝑎𝜎 % )× ‡36𝑎% − 36𝑎𝑏𝐾 + 25𝑏% 𝐾 % + 12𝑎𝜎 − 22𝑏𝐾𝜎 + 𝜎 % > 0,
𝑃% = 36𝑎H + 12𝑎G 𝑏𝐾 + 5𝑎% 𝑏% 𝐾 % + 14𝑎𝑏G 𝐾 G + 8𝑏H 𝐾 H + 36𝑎G 𝜎 − 10𝑎% 𝑏𝐾𝜎 − 14𝑎𝑏% 𝐾 % 𝜎+ 8𝑏G 𝐾 G 𝜎 − 7𝑎% 𝜎 % + 6𝑎𝑏𝐾𝜎 % − 16𝑏% 𝐾 % 𝜎 % + 2𝑎𝜎 G > 0.
To verify the sign of Eq. (F5), we consider the difference between squared terms of 𝑃$ . For
each end of the domain of 𝜎, their values are
𝑃$% − 𝑃%% = 12𝑏G 𝐾 G (𝑎 − 𝑏𝐾)% (2𝑎 − 𝑏𝐾)% (4𝑎 + 3𝑏𝐾) > 0 for 𝜎 = 0,21

𝑃$% − 𝑃%% = −64𝑏% 𝐾 % (𝑎 − 𝑏𝐾)% (9𝑎H + 12𝑎G 𝑏𝐾 − 4𝑎% 𝑏% 𝐾 % + 4𝑎𝑏G 𝐾 G − 𝑏H 𝐾 H ) < 0 for 𝜎= 𝑏𝐾.
Furthermore, 𝑃$% − 𝑃%% is decreasing in 𝜎 because𝑑(𝑃$% − 𝑃%% )= −8(𝑎 − 𝑏𝐾)% {𝑏𝐾(2𝑎 − 𝑏𝐾)(36𝑎G − 35𝑎𝑏% 𝐾 % − 2𝑏G 𝐾 G )𝑑𝜎+ 10𝑏𝐾(24𝑎G − 18𝑎% 𝑏𝐾 − 10𝑎𝑏% 𝐾 % − 3𝑏G 𝐾 G )𝜎
+ 6(18𝑎G + 43𝑎% 𝑏𝐾 + 12𝑎𝑏% 𝐾 % − 11𝑏G 𝐾 G )𝜎 %+ [(144𝑎 − 35𝜎)𝑎 + 88𝑎𝑏𝐾 + 126𝑏% 𝐾 % ]𝜎 G } < 0.
These properties of 𝑃$% − 𝑃%% show that the critical point 𝜎z exists such that 𝑃$% − 𝑃%% = 0 and
0 ≤ 𝜎z < 𝑏𝐾. Given that 𝜎 < 𝜎z ⇒ 𝑃$% − 𝑃%% > 0 and 𝜎 > 𝜎z ⇒ 𝑃$% − 𝑃%% < 0, Eqs. (F4) and (F5)lead to𝑉!@@ ⋛ 𝑉!A@ ⇔ 𝐼$ ⋛ 𝐼% ⇔ 𝑃$ ⋛ 𝑃% ⇔ 𝑃$% − 𝑃%% ⋛ 0 ⇔ 𝜎 ⋚ 𝜎z.

(F6)

We now consider three cases in turn: 0 ≤ 𝜎 < min(𝜎y, 𝜎z), min(𝜎y, 𝜎z) < 𝜎 < max(𝜎y, 𝜎z), andmax(𝜎y, 𝜎z) < 𝜎 . If 0 ≤ 𝜎 < min(𝜎y, 𝜎z), we derive 𝑉!@@ > 𝑉!A@ and 𝑉!@A > 𝑉!AA from Eqs.
(F3) and (F6). Choosing a unit tax as a policy instrument is a dominant strategy. Hence, bothgovernments choose a unit tax. If min(𝜎y, 𝜎z) < 𝜎 < max(𝜎y, 𝜎z), we obtain 𝑉!@@ > 𝑉!A@ and
𝑉!@A < 𝑉!AA for 𝜎y < 𝜎 < 𝜎z, and 𝑉!@@ < 𝑉!A@ and 𝑉!@A > 𝑉!AA for 𝜎z < 𝜎 < 𝜎y. For the casewhere 𝜎y < 𝜎 < 𝜎z (𝜎z < 𝜎 < 𝜎y), one government obtains more benefits by choosing the sametax instrument (the tax instrument other than that) chosen by the other government. Therefore,
one government chooses a unit tax, and the other opts for an ad valorem tax. If max(𝜎y, 𝜎z) < 𝜎,then 𝑉!@@ < 𝑉!A@ and 𝑉!@A < 𝑉!AA hold. Considering that choosing an ad valorem tax is adominant strategy, both governments use ad valorem taxes as tax instruments. Finally, theseresults hold for 𝜎 > 𝑏𝐾 by the monotonicity and continuity of functions used for the proof in
the case of 𝜎 < 𝑏𝐾 if equilibrium values exist.

22

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Tables
Table 1. Payoff table
Region 2

Region 1

Unit
Ad Valorem

Unit

Ad Valorem

𝑉$@@ , 𝑉%@@
𝑉$A@ , 𝑉%@A

𝑉$@A , 𝑉%A@
𝑉$AA , 𝑉%AA

Table 2. Payoff table when 𝜇! > 0
𝜇$ = 1.5, 𝜇% = 0.5
Region 1

Unit

Ad Valorem

Unit

0.9444, 0.6111

0.9400, 0.6069

Ad Valorem

0.9495, 0.6123

0.9453, 0.6078

𝜇$ = 2, 𝜇% = 1.5
Region 1

Region 2

Region 2Unit

Ad Valorem

Unit

0.8403, 0.6736

0.8525, 0.6930

Ad Valorem

0.8661, 0.6817

0.8760, 0.6992

Table 3. Payoff table when 𝜇$ > 0 and 𝜇% < 0
𝜇$ = 1.5, 𝜇% = −0.5
Region 1

Unit

Ad Valorem

Unit

1.1944, 0.5278

1.1596, 0.5165

Ad Valorem

1.1910, 0.5274

1.1594, 0.5165

𝜇$ = 0.5, 𝜇% = −1.5
Region 1

Region 2

Region 2Unit

Ad Valorem

Unit

1.1944, 0.5278

1.1067, 0.5063

Ad Valorem

1.1683, 0.5246

1.0995, 0.5060

25

Table 4. Payoff table when 𝜇! < 0
𝜇$ = −0.5, 𝜇% = −1.5
Region 1

Unit

Ad Valorem

Unit

0.9444, 0.6111

0.8673, 0.5588

Ad Valorem

0.8849, 0.5914

0.8393, 0.5535

𝜇$ = −1.5, 𝜇% = −2
Region 1

Region 2

Region 2Unit

Ad Valorem

Unit

0.8403, 0.6736

0.7497, 0.5927

Ad Valorem

0.7500, 0.6250

0.7081, 0.5781

26

Figures=
Figure 1. Contour plot related to 𝝈

27

Figure 2. Contour plot related to 𝝈28