Competition Policy at the Intensive andExtensive Margins in General Equilibrium

Kenji Fujiwara
Keita Kamei
Research Group of Economics and Management
No. 2016-E02
2016.11

Discussion Paper Series

Department of Law, Economics and Public Policy
Faculty of Literature and Social Sciences
Yamagata University
Yamagata, Japan

Competition Policy at the Intensive andExtensive Margins in General Equilibrium
Kenji Fujiwara∗
School of Economics, Kwansei Gakuin University
Keita Kamei
Faculty of Literature and Social Sciences, Yamagata University

Abstract
This paper examines welfare eﬀects of competition policies in a general equilibrium model in which perfectly competitive and oligopolistic
industries coexist and compete for a common factor of production. We
first show that increasing the number of oligopolistic firms raises welfare if the oligopolists’ production technology exhibits non-increasing
returns to scale. Then, we address another competition policy modeled by an increase in the portion of perfectly competitive industries,
finding that this policy improves welfare if decreasing returns of theoligopolists’ technology are strong enough. These results suggest thatthe degree of returns to scale plays a key role for welfare-enhancing
competition policy.
Keywords: Competition policy, General oligopolistic equilibrium (GOLE),
Welfare, Returns to scale.
JEL Classifications: L4, L13.

∗

Corresponding author: School of Economics, Kwansei Gakuin University. Uegahara
1-1-155, Nishinomiya, Hyogo, 662-8501, Japan. Tel: +81-798-54-7066. Fax: +81-798-510944. E-mail: kenjifujiwara@kwansei.ac.jp.

11Introduction

Economic impacts of competition policy have been one of the most important
subjects, and received considerable attention in economics. If competition
policy is simply defined as an increase in the number of imperfectly competitive firms, it raises welfare because ‘when the number of firms becomesvery large, the market price tends to the competitive price.’ (Tirole, 1988,
p. 220) Besides this classical view on competition policy, recent evidence
suggests that competition policy leads to economic growth by inducing firms
to innovate and reduce costs.1 To sum up, the existing literature allows us toclaim that competition policy has a positive eﬀect on economic growth andwelfare.2
This paper theoretically reexamines the eﬀects of competition policy in a
general equilibrium context. While the previous theoretical literature gives
useful insights into competition policy, its scope is largely restricted to apartial equilibrium model. The partial equilibrium approach is undoubtedly
convenient, but it is diﬃcult to straightforwardly apply its implications toa situation in which the presence of oligopolistic industries is significant. In
order to evaluate the economy-wide impacts of competition policy, a generalequilibrium model is inevitably called for. Nevertheless, it has been recognized that integrating oligopoly and general equilibrium theories leads to anumber of theoretical problems, e.g. non-existence of equilibrium. To ourknowledge, Neary (2003, 2016) first overcomes such diﬃculties by inventing
a general oligopolistic equilibrium (GOLE) model.3 By assuming a contin1

World Bank (2012) provides a concise but useful review of empirical assessments on
the impacts of competition policies.
2
We should note that there is a literature that finds negative welfare eﬀects of increased
competition. For instance, Lahiri and Ono (1988) demonstrate in an asymmetric oligopoly
model that increasing the number of ineﬃcient firms can reduce welfare. Mankiw and
Whinston (1986) and Suzumura and Kiyono (1987), in addition, show that increasing thenumber of firms is welfare-reducing in the presence of economies of scale.
3
Although Neary (2016) is published in 2016, its first working paper version was released
in 2002. Colacicco (2015) is a comprehensive survey on the GOLE model and some of its

2uum of industries and modeling ‘firms as having market power in their own
industry but not in the economy as a whole,’ (Neary, 2016, p. 670) he oﬀers
a theoretically consistent model that has a variety of applications. Neary(2003), among others, applies his model to competition policy, showing thatan increase in the oligopolistic firms raise welfare when there is a technological diﬀerence across industries.4 Kamei (2014), on the contrary, finds thatthe same competition policy reduces welfare in the presence of increasing
returns to scale.
Our purpose of this paper is to supplement the above strand of literatureby extending the GOLE model of the previous studies. While Neary (2003)
and Kamei (2014) commonly suppose that all industries are oligopolized, we
split the whole industry into a set of perfectly competitive industries and aset of oligopolistic industries. This extension gives us two definitions of competition policy. The first is an increase in the number of oligopolistic firms,
and the second is an increase in the share of perfectly competitive industries.
We call in this paper the former policy a competition policy at the intensive
margin and the latter a competition policy at the extensive margin, following Kreickemeier and Meland (2013). Then, we obtain the following resultson the welfare eﬀect of these competition policies. First, the competition
policy at the intensive margin raises welfare if the production technologyof oligopolistic firms exhibits non-increasing returns to scale. Second, it is
impossible to analytically show that the competition policy at the extensive
margin improves welfare, but our numerical analysis finds that this policy
is beneficial if the oligopolists’ production technology is suﬃciently decreasing returns to scale. Accordingly, these results commonly suggest that bothkinds of competition policies enhance welfare if oligopolistic firms’ production
technology is suﬃciently decreasing returns.
applications.
4
If all industries are identical, which Neary (2003) calls a ‘featureless economy,’ thecompetition policy has no eﬀect. Chen and Liu (2014) also obtain a similar result.

3This paper is organized as follows. Section 2 presents a model. Section 3
addresses the impacts of the two competition policies. Section 4 concludes.

2

Model

Suppose a continuum of goods in a unit interval [0, 1], and a representative
consumer maximizes utility∫ 1

ln x(z)dz,(1)0

under the budget constraint∫ 10

p(z)x(z)dz ≤ I,

where x(z) is consumption of Good z, p(z) is its price, and I is nominal
income. Thus, letting λ be a Lagrangean multiplier, the first-order condition
for utility maximization is 1/x(z) = λp(z).
All goods are produced from labor only. We then assume perfect competition with a unitary input coeﬃcient in industries [0, ze], and Cournot
competition among n ≥ 2 firms in industries [ze, 1]. And, denoting by w andy(z) the wage rate and per-firm output in industry z, the cost function ofindustry z is assumed to be w[y(z)]1/θ , θ > 0, which is derived from theproduction function y(z) = [l(z)]θ , where l(z) is labor input.5 As mentioned
in Introduction, we define competition policy in two ways. The first is anincrease in n (competition policy at the intensive margin) and the second isan increase in ze (competition policy at the extensive margin).
Following Neary (2003, 2016), we normalize marginal utility of income
λ to unity for the following reason. In the present model, λ is an endogenous variable that is determined in general equilibrium, and depends on thestrategic variable of oligopolistic firms, i.e. output in the present model of5

From this formulation, θ > 1 (resp. θ < 1) implies increasing (resp. decreasing)returns to scale, and θ = 1 corresponds to constant returns to scale.

4

Cournot oligopoly. Hence, truly rational firms would choose output by fully
taking account of this eﬀect, ‘which opens a Pandora’s Box of technical difficulties.’ (Neary, 2016, p. 670) But, such diﬃculties can be resolved byassuming that oligopolists are large in their product market but small in thewhole economy because this assumption implies that oligopolists take λ as
given in choosing output. Thus, the perceived inverse demand function eachfirm faces becomes
1,
i=1 yi (z)p(z) = ∑n(2)

and the profit of each firm is defined by p(z)yi (z) − w[yi (z)]1/θ . Under thisdefinition of profit, the first-order condition for profit maximization is1n−1w= [y(z)] θ −1 ,2n y(z)
θin the symmetric equilibrium where all firms produce the same amount. From
this equation, total output in industry z becomes[(n − 1)θ
ny(z) = n
n2 w

]θ

.

(3)

It follows from the assumption of perfect competition and unitary input
coeﬃcient that p(z) = w and Y (z) = 1/w in industry z in industries [0, ze],
where Y (z) is industry-wide output. Making use of this result and (3), thelabor market-clearing condition is
L =

∫ e
z

∫ 1

Y (z)dz +

0

e
z

1

n[y(z)] θ dz = ze

1(n − 1)θ
+ (1 − ze),wnw

where the first term in the right-hand side is aggregate labor demand in theperfectly competitive industries, and the second term is the counterpart in
the oligopolistic industries. Solving this equation for w gives the equilibrium
wage rate:

zen + (1 − ze) (n − 1)θ
.
(4)nLThis completes the description of the model. Once the equilibrium wage ratew=

is determined as above, all of the other endogenous variables are explicitly
derived.
5

3

Eﬀects of Competition Policies

Drawing on the model developed in the last section, this section examines
the eﬀects of competition policies. At this stage, we make the following
restriction on parameters θ and n in order to ensure the non-negative profits
of oligopolistic firms.6
Assumption. 0 < θ < n/(n − 1).
Given this assumption, the eﬀect of two competition policies on the equilibrium wage is identified as follows.
Lemma 1. Both the competition policy at the intensive margin and at theextensive margin raises the equilibrium wage.
Proof. By diﬀerentiating (4) with respect to n and ze, we have∂w
(1 − ze) θ
=
> 0,
∂n
n2 L

∂w
n − (n − 1)θ
=
> 0,
e
∂z
nL

which leads to the result. ||It is natural that the competition policy at the intensive margin raises
the wage rate because it increases aggregate labor demand in the oligopolistic industries and aggregate economy. The positive eﬀect of the competition
policy at the extensive margin is, in contrast, a little more complicated thanthe competition policy at the intensive margin. When ze increases, labor demand in perfectly competitive industries increases by 1/w units, but labordemand in oligopolistic industries decreases by (n − 1)θ/(nw) units.7 Since
Substituting (3) into the definition of profit, the maximized profit is given by [n − (n −1)θ]/n2 .
7This is checked by looking at the labor market-clearing condition in the previous
section.
66

we have assumed θ < n/(n − 1) for guaranteeing the non-negative profitof oligopolists, an increase in labor demand in perfectly competitive industries outweighs a decrease in labor demand in oligopolistic industries, which
eventually results in a higher wage rate.
Note here that the profit maximization condition in the competitive industries p(z) = w and the above lemma jointly tell that both types of competition policies raise the price of competitive goods.
Let us next examine the eﬀect of the competition policies on the priceof oligopolized goods. Substituting (4) into (3), and relating the resulting
expression to (2), the price of oligopolized goods depends on primitive parameters as follows.
{p(z) = n−1(n − 1)θLn [zen + (1 − ze) (n − 1)θ]

}−θ

.

(5)

Therefore, diﬀerentiating (5) with respect to n and ze, the eﬀect of two competition policies is respectively obtained as
dp(z)
θL∆
=
dn
n3 [zen + (1 − ze) (n − 1)θ]2{[ ((n − 1)θLn [zen + (1 − ze) (n − 1)θ]
)

}−1−θ

(6)

]

∆ ≡ (1 − ze) (n − 1)2 θ2 + ze 2n2 − 4n + 1 − (n − 1)2 θ − zen(n − 1)(n − 1)[n − (n − 1)θ]θ2 L
dp(z)
=
dze
n2 [zen + (1 − ze) (n − 1)θ]2{(n − 1)θLn [zen + (1 − ze) (n − 1)θ]

}−1−θ

. (7)

On the one hand, according to Eq. (7), the competition policy at theextensive margin unambiguously raises the price of oligopolized goods. On
the other hand, Eq. (6) suggests that the eﬀect of the competition policy
at the intensive margin is generally unclear because the first term in ∆ isnegative and the second term can be both positive and negative. However,Eq. (6) allows us to infer that an increase in n reduces the product price if
non-increasing returns to scale (θ ≤ 1) prevail in the oligopolistic industries.
7But, it is possible that the same policy can raise the product price if the production technology of oligopolistic industries is subject to suﬃciently strong
economies of scale. The reason is as follows. When the number of oligopolistic firms increases, each firm reduces output and the goods price tends to fall
as a first-order eﬀect. However, as proved in Lemma 1, this policy involves
a rise in the wage rate by increasing aggregate labor demand in the wholeoligopolistic industries, which puts upward pressure on the product price asa second-order eﬀect. If the degree of increasing returns is suﬃciently large,
this second-order eﬀect dominates the first-order eﬀect, and thereby raises
the price of oligopolized goods.
Having identified the price eﬀect of competition policies, we now address
their welfare eﬀect. The welfare eﬀect of increasing the number of oligopolistic firms is summarized as follows.
Proposition 1. The competition policy at the intensive margin improves
welfare if and only if[]Γ ≡ (1 − ze) (n − 1)2 θ2 + zen(2n − 3) − (n − 1)2 θ − zen(n − 1) < 0.

(8)

Proof. In order to compute the impact of an increase in n on welfare, let
us define welfare in the present model. Substituting the demand function
x(z) = 1/p(z) into the direct utility function (1), welfare W is obtained as
W =

[
∫ e
z
0

]

[

]

∫ 1
1
1
dz +
dz = −ze ln p1 − (1 − ze) ln p2 ,
p(z)
p(z)
e
z

(9)

where p1 is the price of perfectly competitive goods given by (4), and p2 is
the price of oligopolized goods given by (5).
Taking into account Eqs. (4) and (5), diﬀerentiating (9) with respect ton yields
dW
dn

= −ze

dp1 /dn
dp2 /dn− (1 − ze)
p1
p2
8

= −

(1 − ze) Γ,n(n − 1) [zen + (1 − ze) (n − 1)θ]where Γ is defined by (8). Consequently, we see that dW/dn > 0 if and only
if Γ < 0. ||(Figure 1 around here)
While condition (8) contains three parameters θ, ze and n and somewhat
complicated, it is arguably most helpful to look at it as a condition on θ.
Figure 1 illustrates condition (8) as a condition on θ. The left-hand sideof (8) is a quadratic function of θ, and is negative if θ = 0 and θ = 1.
This implies that the competition policy above necessarily improves welfare
if the production technology exhibits non-increasing returns to scale (θ ≤ 1).
However, exactly the same policy can be welfare-reducing if θ is larger thane 8 That is, the competition policy at the intensive margin is welfare-reducingθ.

if the production technology of oligopolistic firms is under suﬃciently strong
economies of scale.
These findings are intuitively interpreted as follows. As noted earlier,when n increases, the equilibrium wage and the price of competitive goods
rise, which has a negative eﬀect on welfare. That is, the competition policy
for oligopolistic industries serves as an ‘anti-competitive’ policy for perfectlycompetitive industries. However, if the oligopolistic firms’ production technology is non-increasing returns (θ ≤ 1), the competition policy above leads
to a decrease in the price of oligopolized goods, and positively aﬀects welfare. As a result, the overall eﬀect is determined by the interaction of theseconflicting eﬀects. The above proposition states that the competition policy
at the intensive margin improves welfare if θ ≤ 1 since the positive eﬀect
8e

θ is a solution to the quadratic equation that Γ = 0, and derived as√(n − 1)2 − zen(2n − 3) + De,θ=
2 (1 − ze) (n − 1)2[]2
where D ≡ zen(2n − 3) − (n − 1)2 + 4ez (1 − ze) n(n − 1)3 > 0.

9

dominates the negative eﬀect under θ ≤ 1. That is, unless θ is large enough,the pro-competitive eﬀect on the price of oligopolized goods dominates theanti-competitive eﬀect on the price of competitive goods.
However, the same no longer holds for the case with θ > 1. In particular,
if the degree of increasing returns is suﬃciently high, the competition policy
ends up raising the price of all goods, thereby reducing welfare.
Having addressed the competition policy at the intensive margin, we turnto the eﬀect on an increase in ze. Its eﬀect on welfare is formally stated in:
Proposition 2. The welfare eﬀect of the competition policy at the extensive
margin is generally ambiguous. However, if the labor endowment is suﬃciently large, the welfare eﬀect is positive under θ → 0 and negative underθ → n/(n − 1).
Proof. Diﬀerentiating (9) with respect to ze, a small increase in ze aﬀects
welfare as follows.
dW
dze

(

p2
= ln
p1
[


)

− ze

dp2 /dze
dp1 /dze− (1 − ze)
p1
p2

n
= ln
 (n − 1)θ

]θ [L
zen + (1 − ze) (n − 1)θ

]1−θ 
−[n − (n − 1)θ] [ze + (1 − ze) θ].
zen + (1 − ze) (n − 1)θ

While it is analytically impossible to determine the sign of the right-handside, its limits are obtained as follows.
(

)

dW
L
−1>0
when θ → 0 :
→ ln
zen
dze
[( ) 1 ]ndWn n−1when θ →
:
→ ln
< 0.
n−1
dze
L
These inequalities allow us to establish the proposition. ||The ambiguous result above is not surprising by invoking that total eﬀect
is decomposed into the following two sub-eﬀects. On the one hand, when the10

range of perfectly competitive industries expands and θ is close to zero, theconsumer can consume more of cheaper goods than before and hence benefits
from this policy. But, note that this eﬀect serves to worsen the consumer
utility if θ is suﬃciently large. On the other hand, as shown in Lemma 1 andEq. (7), this policy has a detrimental eﬀect by raising the price of all goods.
All we can say is that the present competition policy reduces welfare if θ isclose to n/(n − 1) because both of the two sub-eﬀects above have negative
impacts.
(Figure 2 around here)
However, we can obtain a little more via a numerical example in Figure 2
that shows the relationship between the welfare eﬀects of competition policy
at extensive margin and returns to scale in cases of z̃ = 0.8, z̃ = 0.5 andz̃ = 0.2, respectively.9 It illustrates that the present type of competition policy raises welfare if θ is small enough. The intuition behind this observation is
explained as follows. If θ is suﬃciently small, namely, the degree of decreasing returns is suﬃciently strong, both marginal cost and price of oligopolized
goods also becomes very high. Therefore, by removing a portion of suchineﬃcient oligopoly industries, the consumer can consume more of low-pricegoods (perfectly competitive goods) than before, and welfare improves. Conversely, as θ is larger, the negative welfare eﬀect through the increase in wage
rate and goods prices becomes dominant, and thereby reduces welfare.

4

Concluding Remarks

Applying a GOLE model, we have explored the welfare eﬀects of two competition policies. The first is a competition policy at the intensive margin,
which is modeled by an exogenous increase in the number of oligopolistic
firms. Whether this policy improves welfare highly depends on the degree of
9

The parameters are set as n = 4 and L = 10.

11returns to scale of the oligopolistic firms. More concretely, if the production
technology is non-increasing returns, the policy raises welfare, but the opposite holds if the production technology exhibits suﬃciently strong economies
of scale. The second policy is a competition at the extensive margin, i.e. an
expansion of the perfectly competitive industries. We have shown througha numerical calculation that this policy enhances welfare if the oligopolists’production technology is subject to suﬃciently strong decreasing returns. To
summarize, the production technology of the oligopolistic industries plays a
key role in the welfare eﬀect of two competition policies.
This paper has hopefully contributed to literature in the sense that theabove results are not obtained in the previous studies, but they admittedly
rest on a number of restrictive assumptions. First, we have specified relevant
functional forms such as a utility function and a production function so as tosimplify analysis. Second, our results are purely theoretical, and empiricalre-evaluation is required to qualify the welfare eﬀects of competition policies.
These are left as future research agenda.

References[1] Chen, P. and D. Liu (2014), ‘Trade Liberalization and Competition Levels,’ B.E. Journal of Theoretical Economics, 15(1), 25-47.
[2] Colacicco, R. (2015), ‘Ten Years of General Oligopolistic Equilibrium: A
Survey,’ Journal of Economic Surveys, 29(5), 965-992.
[3] Kamei, K. (2014), ‘Pro-Competitive Eﬀect, Division of Labor, and Firm
Productivity,’ Economics Letters, 124(1), 132-135.
[4] Kreickemeier, U. and F. Meland (2013), ‘Non-traded Goods, Globalization and Union Influence,’ Economica, 80(320), 774-792.

12

[5] Lahiri, S. and Y. Ono (1988), ‘Helping Minor Firms Reduces Welfare,’Economic Journal, 98(393), 1199-1202.
[6] Mankiw, N. G. and M. D. Whinston (1986), ‘Free Entry and Social Eﬃciency,’ RAND Journal of Economics, 17, 48-58.
[7] Neary, P. (2003), ‘The Road Less Travelled: Oligopoly and Competition
Policy in General Equilibrium,’ in Arnott, R., B. Greenwald, R. Kanbur
and B. Nalebuﬀ (eds.), Economics for an Imperfect World: Essays in
Honor of Joseph E. Stiglitz, Cambridge: MIT Press, 485-500.
[8] Neary, P. (2016), ‘International Trade in General Oligopolistic Equilibrium,’ Review of International Economics, 24(4), 669-698.
[9] Suzumura, K. and K. Kiyono (1987), ‘Entry Barriers and Economic Welfare,’ Review of Economic Studies, 54, 157-167.
[10] Tirole, J. (1988), The Theory of Industrial Organization, Cambridge:
MIT Press.
[11] World Bank (2012), ‘Viewpoint 331.’

13

Γ

dW
dn

<0

1
O

θe

θ

n
n−1

dW
dn

Figure 1: Condition (8)

14

>0

a
] 
a
] 

a
G:G]

a
] 

ǉ

Figure 2: Welfare eﬀects of competition policy at the extensive margins andreturns to scale15