The combined use of specific and ad valorem taxation as a policy response to the welfare losses caused by international oligopoly is explored. With Nash competition between countries, taxation is inferior to quantity control. In contrast, when countries cooperate production control and taxation lead to identical outcomes. If a single country regulates the oligopoly, taxation can strictly dominate production control.oligopoly

Gareth Myles

English

Research Papers in Economics

Taxation and international oligopoly 
Gareth Myles
University of Exeter and Institute for Fiscal Studies
Abstract
The combined use of specific and ad valorem taxation as a policy response to the welfare
losses caused by international oligopoly is explored. With Nash competition between
countries, taxation is inferior to quantity control. In contrast, when countries cooperate
production control and taxation lead to identical outcomes. If a single country regulates the
oligopoly, taxation can strictly dominate production control.
I would like to thank Myrna Wooders and the anonymous referee for helpful comments.
Citation: Myles, Gareth, (2001) "Taxation and international oligopoly." Economics Bulletin, Vol. 8, No. 4 pp. 1−9
Submitted: May 24, 2001.  Accepted: September 4, 2001.
URL: http://www.economicsbulletin.com/2001/volume8/EB−01H00001A.pdf1 Introduction
This note analyzes the use of ad valorem and speci¯c taxation to control interna-
tional oligopoly. The outcome with taxation is contrasted to direct government
control of production under cooperation between countries, Nash competition
and single-government intervention.
It is a common perception that direct production control is the optimal
second-best policy. With a single tax instrument this is correct: taxation is in-
ferior to production control both with collusion between governments and when
governments act independently. Using the two tax instruments, this remains
the case with Nash competition. In contrast, if governments collude, then both
production control and taxation achieve the same outcome. Furthermore, when
a single country regulates the duopoly, taxation may achieve an outcome strictly
better than that of direct production control.
2T h e F r a m e w o r k
Consider a two-country world in which a traded good is produced by a duopolis-
tic industry. Each country plays host to one of the ¯rms. The preferences of
the consumers in each country are represented by the aggregate social welfare
function
U = U (Xj) ¡ `j;U 0 > 0;U 00 < 0;j=1 ;2; (1)
where Xj is consumption of the duopoly's output in country j and `j is labor
supply. The wage rate in both countries is ¯xed at unity. With price qj for
the consumption good, the budget constraint for the consumer in country j is
qjXj = `j+¼j; where ¼j is pro¯t income. Maximizing (1) subject to the budget
constraint generates an inverse demand function qj = Ã(Xj), with Ã(Xj) ´
U0 (Xj). The indirect utility function can be derived as
U = U
¡
Ã
¡1 (qj)
¢
¡ qjÃ
¡1 (qj)+¼j ´ V (qj)+¼j: (2)
The ¯rms produce with costs determined by the cost function
C = C
¡
x1
i + x2
i
¢
;C(0) ¸ 0;C0 > 0;C00 ¸ 0;i=1 ;2; (3)
where x
j
i is the supply of ¯rm i to country j. Firm 1 is located in country 1.
With speci¯c tax tj and ad valorem tax ¿j, the pro¯t level of ¯rm i is
¼i =
2 X
j=1
h
[1 ¡ ¿j]Ã
³
x
j
1 + x
j
2
´
¡ tj
i
x
j
i ¡ C
¡
x1
i + x2
i
¢
: (4)
When ¯rms choose output levels, they act as Cournot-Nash competitors. Lump-
sum taxes cannot be used, so each ¯rm must at least break-even.
The following two assumptions are imposed:
1Assumption 1. The pro¯t of ¯rm i is a strictly concave function of
¡
x1
i;x 2
i
¢
and, when ¿j = tj =0 ;j=1 ;2; there exists e x ´
¡
e x1
1;e x2
1;e x1
2;e x2
2
¢
such that
¼i (e x) > 0;i=1 ;2.
Assumption 2. The inverse demand function satis¯es limXj!1 Ã(Xj)=0
and limXj!0 Ã(Xj) · K<1.
3 Cooperative Control
With cooperative control, the governments choose their policies to maximize
the sum of country welfare levels.
3.1 Quantity Control
De¯ne x by Ã(x)=C0 (x);x is the output consistent with marginal cost pricing.
Proposition 1 (i) If ¼i (x) < 0; the optimal outcome under government control
of production occurs at the highest output level consistent with zero pro¯t for the
¯rms. (ii) If ¼i (x) ¸ 0 the optimal outcome has marginal cost pricing.
Proof. The situation is symmetric for the two countries, so the maximiza-
tion of joint welfare is equivalent to the maximization of individual welfare.
Hence the solution
©
b x1
1;b x2
1;b x1
2;b x2
2
ª
to the optimization
max
2 X
j=1
h
U
³
x
j
1 + x
j
2
´
¡ C
¡
x1
j + x2
j
¢i
s.t. ¼i ¸ 0;i =1 ;2; (5)
where C
¡
x1
j + x2
j
¢
is labor demand in country j; is the same as the solution
fb x1;b x2g to the simpler optimization
maxU (xj) ¡ C (xj)s . t .¼i ¸ 0;i =1 ;2; (6)
in the sense that b xj = b x1
i + b x2
i = b x
j
1 + b x
j
2. The Kuhn-Tucker conditions for the
optimization (6) are
U0 ¡ C0 + ¸
£
Ã + Ã
0xj ¡ C0¤
=0 ; (7)
and
Ãxj ¡ C ¸ 0;¸¸ 0;¸[Ãxj ¡ C]=0 : (8)
Since U0 ´ Ã, there are two potential solutions. The ¯rst has Ã>C 0 and
Ãxj ¡ C =0 . I nt h i sc a s e ,Ã + Ã
0xj ¡ C0 < 0, so the optimum is given by
the greater of the two solutions to ¼i (xj) = 0. The second potential solution
involves Ã = C0 so marginal cost pricing ensues with Ãxj ¡ C ¸ 0.
Denote the optimal output in the marginal cost pricing outcome, which is
also the consumption level in both countries, by x¤ and the optimal output
level when ¼i (x) < 0b yx¤¤. Note that x¤ is ¯rst-best and (in the absence of
lump-sum transfers) x¤¤ is second-best optimal.
23.2 Taxation
Let qj ´ qj (t1;t 2;¿1;¿2) be the equilibrium price in country j as a function of
the tax rates determined via optimization by the ¯rms. The cooperative tax
optimum is the solution to
max
ft1;t2;¿1;¿2g
2 X
j=1
V (qj)+¼j; (9)
subject to
¼i =
2 X
j=1
[[1 ¡ ¿j]Ã ¡ tj]x
j
i¡C ¸ 0;i =1 ;2; [¿jqj + tj]
h
x
j
1 + x
j
2
i
=0 ;j =1 ;2:
(10)
The ¯rst constraint in (10) is pro¯tability for the ¯rms and the second govern-
ment budget balance.
Theorem 2 (i) If ¼i (x) ¸ 0; an ad valorem tax ¿ =1and a speci¯c tax
t = ¡C0 lead to marginal cost pricing and attain the ¯rst-best outcome x¤.( i i )
If ¼i (x) < 0, the optimal combination of ad valorem and speci¯c taxes achieves
the second-best outcome x¤¤.
Proof. The proof is a straightforward adaptation of Proposition 1 in Myles
(1996).
The explanation for this result is that the ad valorem tax reduces the per-
ceived market power of the ¯rms and the speci¯c tax acts as a ¯xed price-per-
unit. These ensure that the ¯rms expand output and the welfare loss due to
the imperfect competition is eliminated. Hence coordination by countries can
control international oligopolies by the use of domestic tax instruments alone;
trade policies are unnecessary.
4 Nash Equilibrium
The next two results show that quantity control is strictly superior to taxation
with Nash competition between governments.
Given U0
³
x
j
1 + x
j
2
´
= qj;j=1 ;2 it follows that
`j = U0
³
x
j
1 + x
j
2
´h
x
j
1 + x
j
2
i
¡ ¼j: (11)
With quantity control, utility in country j c a nt h e nb ew r i t t e na s
U = U
³
x
j
1 + x
j
2
´
+
h
x
j
0
j Ã
³
x
j
0
1 + x
j
0
2
´
¡ x
j
j0Ã
³
x
j
1 + x
j
2
´i
¡ C
¡
x1
j + x2
j
¢
;j6= j0:
(12)
The central term in (12) represents the trade surplus (or de¯cit) of country j in
its trade with country j0.
3The optimization problem for country j is to maximize (12) through choice
of x
j
j and x
j0
j ; subject to the conditions
x
j
jÃ
³
x
j
1 + x
j
2
´
+ x
j0
j Ã
³
x
j0
1 + x
j0
2
´
¡ C
¡
x1
j + x2
j
¢
¸ 0;x
j0
j ¸ 0: (13)
Theorem 3 The Nash equilibrium in quantity setting is identical to the coop-
erative outcome.
Proof. The symmetric equilibrium between countries is described by
Ã ¡ xoÃ
0 ¡ C0 + ¸
£
Ã + xhÃ
0 ¡ C0¤
+ ¹h =0 ; (14)
Ã + xoÃ
0 ¡ C0 + ¸
£
Ã + xoÃ
0 ¡ C0¤
+ ¹o =0 ; (15)
where xh ´ x
j
j;x o ´ x
j0
j ;j6= j0 and ¹h;¹ o are the multipliers on the correspond-
ing non-negativity constraints: Conditions (14) and (15) have two solutions: (i)
¸ =0 ;x o =0 ;Ã = C0; and (ii)
£
xh + xo¤
Ã ¡ C =0 ;x o > 0;x h > 0: These
correspond to the allocations x¤ and x¤¤ respectively.
It is interesting to note that this theorem shows that even though choice is
non-cooperative, the ¯rst-best is still sustained. There is no need for explicit
cooperation between the governments to coordinate outputs.
With taxation, imposing budget balance gives the objective function in coun-
try j
U
³
x
j
1 + x
j
2
´
¡Ã
³
x
j
1 + x
j
2
´
x
j
j0+
h
[1 ¡ ¿j0]Ã
³
x
j0
1 + x
j0
2
´
¡ tj0
i
x
j0
j ¡C
¡
x1
j + x2
j
¢
:
(16)
Theorem 4 With Nash competition, taxation is inferior to production control.
Proof. Assume marginal cost pricing is the solution under production con-
trol so the pro¯t constraint is not binding. Using symmetry and budget balance
the ¯rst-order condition arising from the optimization of (16) is then
@xh
@¿
£
Ã ¡ 2xhÃ
0 ¡ C0¤
+
@xo
@¿
£
Ã +2[ 1¡ ¿]xhÃ
0 ¡ C0¤
=0 : (17)
Since @xh
@¿ has the opposite sign to @xo
@¿ (using the comparative statics of pro¯t
maximization), Ã = C0 can never be a solution to (17). Since marginal cost
pricing is ¯rst-best, the tax outcome must be inferior.
This conclusion is very much in the °avour of a standard strategic trade
policy result. Even though there are two tax instruments available, Nash equi-
librium in taxation is still ine±cient.
5 Single-Country Control
This section determines the outcome when country 2 chooses not to intervene
in the market. In this case the only policy intervention arises from the actions
of country 1.
45.1 Quantity Control
Assume country 1 acts alone in controlling the output of the ¯rm located within
its borders and that it holds Nash conjectures about the behavior of ¯rm 2.
Country 1 therefore seeks to maximize (12) and ¯rm 2 has objective
max
fx1
2;x2
2g
x1
2Ã
¡
x1
1 + x1
2
¢
+ x2
2Ã
¡
x2
1 + x2
2
¢
¡ C
¡
x1
2 + x2
2
¢
: (18)
The necessary conditions are given by
Ã
¡
x1
1 + x1
2
¢
¡ x1
2Ã
0 ¡
x1
1 + x1
2
¢
¡ C0 ¡
x1
1 + x2
1
¢
=0 ; (19)
where the substitution U0 ¡
x1
1 + x1
2
¢
´ Ã
0 ¡
x1
1 + x1
2
¢
has been used,
Ã
¡
x2
1 + x2
2
¢
+ x2
1Ã
0 ¡
x2
1 + x2
2
¢
¡ C0 ¡
x1
1 + x2
1
¢
=0 ; (20)
and
Ã
³
x
j
1 + x
j
2
´
+ x
j
2Ã
0
³
x
j
1 + x
j
2
´
¡ C0 ¡
x1
2 + x2
2
¢
=0 ;j=1 ;2: (21)
The equilibrium is the simultaneous solution to (19)-(21) and is denoted
b x ´
¡
b x1
1;b x2
1;b x1
2;b x2
2
¢
. Assumption 3 is imposed.
Assumption 3. The pro¯t of ¯rm 1 is positive at the equilibrium b x,t h a ti s
¼1 (b x) > 0:
Proposition 5 (i) If C00 =0then b x1
2 =0 , b x2
1 = b x2
2, the good is sold at marginal
cost in country 1, and country 1 runs a trade surplus. (ii) If C00 > 0 then
b x2
1 < b x2
2;b x1
1 > b x1
2, total production is greater in country 1, and consumption is
greater in country 1.
Proof. (i) Denote the common value of constant marginal cost by c.i t
follows from (19) and (21) that
Ã
¡
b x1
1 + b x1
2
¢
¡ b x1
2Ã
0 ¡
b x1
1 + b x1
2
¢
= c = Ã
¡
b x1
1 + b x1
2
¢
+ b x1
2Ã
0 ¡
b x1
1 + b x1
2
¢
; (22)
so b x1
2 = 0 and Ã
¡
b x1
1 + b x1
2
¢
= c: Hence from (20) and (21), b x2
1 = b x2
2.
(ii) From (19) and (21) it follows that C0 ¡
b x1
1 + b x2
1
¢
>C 0 ¡
b x1
2 + b x2
2
¢
so b x1
1 +
b x2
1 > b x1
2 + b x2
2. The relative values of marginal costs imply from (20) and (21)
that b x2
1Ã
0 ¡
b x2
1 + b x2
2
¢
> b x2
2Ã
0 ¡
b x2
1 + b x2
2
¢
so b x2
1 < b x2
2. Combining these inequalities,
it follows that b x1
1 > b x1
2. Finally, equating (19) and (20) via marginal cost gives
Ã
¡
b x1
1 + b x1
2
¢
= Ã
¡
b x2
1 + b x2
2
¢
+ b x2
1Ã
0 ¡
b x2
1 + b x2
2
¢
+ b x1
2Ã
0 ¡
b x1
1 + b x1
2
¢
or Ã
¡
b x1
1 + b x1
2
¢
<
Ã
¡
b x2
1 + b x2
2
¢
which implies b x1
1 + b x1
2 > b x2
1 + b x2
2.
5.2 Taxation
In the presence of taxation, ¯rm i maximizes pro¯t
¼i =[ 1¡ ¿]Ã
¡
x1
1 + x1
2
¢
x1
i +Ã
¡
x2
1 + x2
2
¢
x2
i ¡tx1
i ¡C
¡
x1
i + x2
i
¢
;i =1 ;2: (23)
5The optimal choice of output in country 1 is determined by
[1 ¡ ¿]Ã
¡
x1
1 + x1
2
¢
+[ 1¡ ¿]Ã
0 ¡
x1
1 + x1
2
¢
x1
i ¡ t ¡ C0 ¡
x1
i + x2
i
¢
=0 ;i =1 ;2;
(24)
which can be simpli¯ed using government budget balance to give
Ã
¡
x1
1 + x1
2
¢
+[ 1¡ ¿]Ã
0 ¡
x1
1 + x1
2
¢
x1
i ¡ C0 ¡
x1
i + x2
i
¢
=0 ;i =1 ;2: (25)
The optimal choice in country 2 follows from
Ã
¡
x2
1 + x2
2
¢
+ Ã
0 ¡
x2
1 + x2
2
¢
x2
i ¡ C0 ¡
x1
i + x2
i
¢
=0 ;i =1 ;2: (26)
The solution of (25) and (26) is denoted by x(¿) ´
¡
x1
1 (¿);x 2
1 (¿);x 1
2 (¿);x 2
2 (¿)
¢
.
Since an increase in ¿ reduces perceived market power, output will increase
as ¿ increases, reaching a feasible maximum at ¿ = 1 with the speci¯c tax, t,
equal to the negative of marginal cost. The optimal solution is therefore ¿ =1 ;
with output e x ´ x(1); and t = ¡C
¡
x1
i (1) + x2
i (1)
¢
. The question now is
whether this solution provides a greater welfare level than the direct control of
production.
5.2.1 Case 1: C00 =0
Theorem 6 With single country regulation, direct control of production and
the use of optimal ad valorem and speci¯c taxes lead to the same level of welfare
when marginal cost is constant.
Proof. When C00 =0 ; (26) shows that b x2
1 = b x2
2 = e x2
1 = e x2
2 so both equilibria
result in marginal cost pricing in country 1. Since b x1
2 = 0 this implies b x1
1 = e x1
1+
e x1
2. Under quantity control, country 1 is supplied entirely by ¯rm 1. Output, and
labor use are therefore higher for ¯rm 1 than with taxation. Exports of country
1 are the same in both cases, so the trade surplus of country 1 must be greater
with quantity control. Welfare di®erences between the two are determined by
the sum of the labor supply and trade balance e®ects. The value of imports
into country 1 with taxation is e x1
2c. The labor usage with quantity control is
b x1
1c and with taxation e x1
1c. However, since b x1
1 = e x1
1 + e x1
2, these e®ects exactly
cancel.
5.2.2 Case2: C00 > 0
In this case the same factors are at work: the labor use of country 1 is higher
with production control but this is o®set by a greater trade surplus. The nature
of costs then suggests that the production-control equilibrium is likely to provide
a lower level of welfare since the higher level of production in country 1 must
be generated by a technology with increasing marginal cost. To show that this
reasoning can be substantiated, the equilibria are now explicitly calculated for
a parameterized example.
Example. Assume that the utility function is quadratic with
U = a
h
x
j
1 + x
j
2
i
¡
b
2
h
x
j
1 + x
j
2
i2
¡ `j;j =1 ;2; (27)
6τ
1 0.8 0.6 0.4 0.2 0
7600
7400
7200
7000
6800
6600
6400
6200
6000
Figure 1: Welfare as a Function of ¿
and that the cost function is also quadratic
C = F + d
£
x1
i + x2
i
¤
+ e
£
x1
i + x2
i
¤2
;i =1 ;2: (28)
With parameter values a =2 0 0 ,b =1 ,d =1 0 ,e = 1, the equilibrium in the
quantity control case is given by b x1
1 =5 7 ,b x2
1 =9 :5, b x1
2 =1 4 :25, b x2
2 =3 8w i t h
q1 =1 2 8 :75 and marginal cost for ¯rm 1 is 143. As already noted, the trade-
balance e®ect drives price below marginal cost. The welfare level is 6238.5¡F.
The equilibrium with taxation is x1
1 (¿)=x1
2 (¿)= 570
21¡5¿ and x2
1 (¿)=x2
2 (¿)=
190 ¿¡3
5¿¡21. Figure 1 plots the welfare level of country 1 as a function of ¿.
Figure 1 shows the level of welfare attained by production control can be
exceeded by the use of taxation, a consequence of taxation reducing the use of
labor in country 1 relative to production control. For F =0 ; this occurs for ad
valorem tax rates in excess of 0.22.
Theorem 7 The use of taxation can generate a higher level of welfare than
direct production control in single country regulation.
6 Conclusions
When countries cooperate, the paper showed that the combined use of speci¯c
and ad valorem taxation is at least as e®ective as direct production control. The
use of the taxes succeeds in eliminating the welfare losses in the international
economy due to the existence of imperfect competition. Furthermore, no other
policy intervention (such as tari®s etc.) are necessary to control the oligopoly. It
is interesting to note how the structure of these taxes (with the optimal speci¯c
tax being negative) di®ers substantially from the policy of cigarette taxation in
the European Union (where a positive speci¯c tax is employed).
7With competition between countries, the Nash equilibrium in quantity-setting
is ¯rst-best so no coordination is required. In contrast, competition in tax-
setting leads to an inferior outcome. If a single country acts in isolation, it can
do at least as well by employing taxation than it can through controlling the
production of its \home\ ¯rm. An example demonstrated that taxation can
sometimes be strictly preferable.
7R e f e r e n c e s
Myles, G.D. (1996) "Imperfect competition and the optimal combination of ad
valorem and speci¯c taxation", International Tax and Public Finance, 3,2 9-
44.
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Taxation and international oligopoly

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