International Journal of Industrial Organization
25 (2007) 173 – 185
www.elsevier.com/locate/econbase
Welfare reductions from small cost reductions in
differentiated oligopoly☆
X. Henry Wang, Jingang Zhao⁎
University of Missouri-Columbia, USA
University of Saskatchewan, Canada
Received 27 September 2004; received in revised form 23 January 2006; accepted 26 February 2006
Available online 2 June 2006
Abstract
This paper compares the welfare effects of cost reductions in differentiated Bertrand and Cournot
oligopolies. Using a linear demand system, the paper provides the critical levels of output shares below
which a small cost reduction by a Bertrand or Cournot firm reduces welfare. It then shows that the critical
value is larger under Bertrand competition than Cournot competition.
© 2006 Elsevier B.V. All rights reserved.
JEL classification: C72; D43; L13
Keywords: Bertrand Equilibrium; Cournot Equilibrium; Cost reduction; Welfare effects
1. Introduction
This paper studies the welfare effects of a single firm's cost reduction in Bertrand and Cournot
oligopolies with differentiated goods. It continues the previous literature on the welfare effects of
technological changes (or cost reductions) and shocks. Lahiri and Ono (1988) and Zhao (2001)
provided conditions on market share for cost reductions to reduce welfare. Recently, Février and
Linnemer (2004) provided conditions on variance or covariance to determine the welfare effects
of shocks. Their analysis unified earlier studies on the effects of shocks by Bergstrom and Varian
☆
We would like to thank Professor Simon Anderson and two anonymous referees for valuable comments and
suggestions.
⁎ Corresponding author. Department of Economics, University of Saskatchewan, 9 Campus Drive, Saskatoon, SK,
Canada S7N 5A5. Tel.: +1 306 966 5217; fax: +1 306 966 5232.
E-mail address: [email protected] (J. Zhao).
0167-7187/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.ijindorg.2006.02.003
174
X.H. Wang, J. Zhao / Int. J. Ind. Organ. 25 (2007) 173–185
(1985), Salant and Shaffer (1999), Long and Soubeyran (2001), and Linnemer (2003). All of
these papers focused on Cournot oligopolies with homogeneous goods.1
In contrast, the present paper derives analogous results for Bertrand and Cournot oligopolies
with differentiated goods. It shows that in both models, a small cost reduction by one firm is
welfare-decreasing if and only if its output share falls below a critical level determined by cost and
demand parameters. It also shows that the critical value is larger under Bertrand competition than
Cournot competition.
The rest of the paper is organized as follows. Sections 2 and 3 study the Bertrand model and the
Cournot model with differentiated goods, respectively, and examine the effects of cost reductions
in each model. Section 4 compares these effects; Section 5 concludes the paper. The Appendix
provides proofs.
2. The Bertrand model
Consider the following Bertrand–Shubik demand system originated in Shubik (1980):
qi ð pÞ ¼ qi ð p1 ; N ; pn Þ ¼ V −pi −gð pi −p̄ Þ;
ð1Þ
Pn
where V N 0 is the common quantity intercept, pi is i's price, p̄ ¼
i¼1 pi =n is the industry
average price, γ ≥ 0 is the substitutability parameter, and the price space is given by
P ¼ fpaRnþ : V −pi þ gðpi −p̄ÞN0; i ¼ 1; N ; ng:
Firm i's cost and profit functions are given by Ci(qi) = ciqi and πi( p) = (pi − ci)qi( p), where
ci N 0 is its constant marginal (average) cost. For convenience, we label firms such that
c1 ≤ c2 ≤ ··· ≤ cn, so firm 1 is the most efficient firm and firm n is the least efficient firm.
The Bertrand equilibrium is a price vector pB = ( p1B, …, pnB) such that each piB is firm i's best
B
B
B
response to others' choices p−i
= ( p1B, …, pi−1
, pi+1
, …, pnB). Precisely, each piB is the solution of
B
B
Max {πi( pi, p−i)| pi ≥ 0}, and p solves the following n first-order conditions:
Api ð pÞ
¼ 0; i ¼ 1; ⋯; n; or;
Api
2
2d
6 −g
Bp ¼ 6
4 ⋮
−g
−g : : :
2d : : :
⋮
⋱
−g : : :
30 1 0 1
d1
−g
p1
B p2 C B d2 C
−g 7
7B C ¼ B C; where
⋮ 5@ ⋮ A @ ⋮ A
2d
pn
dn
d ¼ n þ ðn−1Þg; and di ¼ nV þ dci for each i:
1
ð2Þ
ð3Þ
ð4Þ
The exceptions are Anderson, de Palma, and Kreider (2001) and Bernstein and Federgruen (2004), which study
respectively the profit effects of taxation and cost reduction in a Bertrand model.
X.H. Wang, J. Zhao / Int. J. Ind. Organ. 25 (2007) 173–185
175
Solving Eq. (3) for p leads to the following Bertrand equilibrium values:2
pBi ¼
n
nðV −c̄Þ
nð1 þ gÞc̄ þ ðn þ ðn−1ÞgÞci
1X
þ
; where c̄ ¼
cj ;
2n þ ðn−1Þg
n j¼1
2n þ ð2n−1Þg
ð5Þ
qBi ¼
X
n þ ðn−1Þg B
nðn þ ðn−1ÞgÞ
ðpi −ci Þ; QB ¼
ðV −c̄Þ; and
qBi ¼
n
2n þ ðn−1Þg
ð6Þ
pBi ¼
n
ðqB Þ2 :
n þ ðn−1Þg i
ð7Þ
We assume throughout that all firms produce positive outputs at the above Bertrand
equilibrium, which is equivalent to the following assumption A1:3
A1.
nV ð2n þ ð2n−1ÞgÞ þ gðn þ ðn−1ÞgÞ
n2 ðg
P
þ 1Þðg þ 2Þ−2ngðg þ 1Þ þ
j p n cj
2
g
Ncn :
We will omit discussions about the average price, because p̄ = (∑ pi) / n = V − Q̄ = V − (∑qi) / n
follows directly from the industry output. Note that the markup ( piB − ci) is given by
pBi −ci ¼
nðV −c̄ Þ
nð1 þ gÞðc̄ −ci Þ
þ
;
2n þ ðn−1Þg 2n þ ð2n−1Þg
so more efficient firms (i.e., firms with smaller ci) have larger markups and outputs. Note
that Eq. (6) implies the markup/output ratios are the same across all firms (i.e., ( piB − ci) / qiB =
n / (n + (n − 1)γ) for all i).
An inspection of Eqs. (5)–(7) leads directly to the following effects of a small reduction in firm
i's marginal cost on prices, outputs and profits at the Bertrand equilibrium: (i) it decreases firm i's
price, and increases its output and profits; (ii) it decreases all other firms' prices, outputs and
2
The inverse matrices for solving Eqs. (3) and (14), and for deriving Eq. (13) from Eq. (1), can all be obtained by
applying Lemma 2 in Zhao and Howe (2004). Alternatively, as suggested by a referee, the inverse in Eq. (3) can be
obtained by a known formula for B = {I − βH} / α, where α = 1 / (2n + (2n − 1)γ), β = γα b 1 / n, I is the identity matrix, and H
nP
o
n
o
P
k¼l
k
k k−1
¼ a I þ k¼l
is the matrix of ones. Specifically, B−1 ¼ a
H ¼ afI þ ½b=ð1−nbÞHg. Multik¼0 ½bH
k¼0 ½b n
plying B− 1 into the right-hand side of Eq.(3) gives Eq. (5). Eqs. (6) and (7) follow directly by utilizing Eq. (5) and the
demand equations in Eq. (1).
3
For each k and (V, c−k) ∈ Rn++, A1 determines a feasible range (cL, cU) for our comparative statics
analysis on ck. It is not difficult to show thatP the upper bound cU = cU(k) and lower bound cL = cL(k)
8
þ
P
< ½a3 cn −ðna1 V þga2 jpk;n cj Þ if kbn
na1 V þga2
c
jpk j
ga2
; and cL ¼ ½a c −ðna V þ ga Pn−2 c Þþ
where a 1 = 2n + (2n − 1)γ, a 2 = (n + (n − 1)γ,
are cU ¼
a3
2
: 3 n−1 1
j¼1 j
if k ¼ n;
ga2
and a3 = n2 (γ + 1)(γ + 2) − 2nγ(γ + 1) + γ2, and t+ = 0 if t ≤ 0, = t if t N 0.
176
X.H. Wang, J. Zhao / Int. J. Ind. Organ. 25 (2007) 173–185
profits; and (iii) it increases the industry output.4 Note that the own effect on firm i's output
dominates the combined output effects on all other firms, so that industry output increases after a
cost reduction by firm i. Industry output increases because it depends on average cost (see Eq. (6))
and because i's cost reduction reduces average cost.
The standard method used to compute consumer surplus is to assume a representative
consumer with a quasi-linear utility function whose utility maximization, subject to a budget
constraint, generates the demand system.5 It is straightforward to verify that the utility function
yielding the demand system in Eq. (1) is given by u(q) = a′q − q′Aq / 2, where a′ = (V,···,V) is an
1 × n vector, q denotes the n × 1 quantity vector, and
2
g
1þ
6 gn
1 6
6
A¼
6 n
1 þg6 ⋮
4
g
n
g
:::
n
g :::
1þ
n
⋮
g
n
⋱
:::
3
g
n
g
n
⋮
1þ
g
n
7
7
7
7:
7
5
ð8Þ
The consumer surplus is given by CS = u(q) − p′q = q′Aq / 2, which is obtained by applying the
first-order conditions of utility maximization.
PStraightforward calculations show that, at the
Bertrand equilibrium, industry profit PB ¼ nj¼1 pBj , total consumer surplus CSB, and total
welfare W B, respectively, are given by
PB ¼
n
X
n
ðqB Þ2 ;
n þ ðn−1Þg j¼1 j
CSB ¼
n
X
1
g
ðQB Þ2 ; and
ðqBj Þ2 þ
2ð1 þ gÞ j¼1
2nð1 þ gÞ
W B ¼ PB þ CSB ¼
n
X
3n þ ð3n−1Þg
g
ðQ BÞ2 ;
ðqB Þ2 þ
2ð1 þ gÞðn þ ðn−1ÞgÞ j¼1 j
2nð1 þ gÞ
ð9Þ
ð10Þ
ð11Þ
where qjB and Q B are given by Eq. (6).
We now study how small changes in costs affect total profits, as well as consumer surplus and
total welfare under price competition. Proposition 1 below characterizes such effects in terms of
the magnitudes of marginal costs.
Proposition 1. Under Bertrand competition, a small reduction in firm i's marginal cost
(i) reduces total profits ⇔ ci is above a critical level ciB⁎;
(ii) always increases consumer surplus; and
(iii) reduces total welfare ⇔ ci is above a critical level ciB⁎⁎, where the critical levels ciB⁎ and
ciB⁎⁎ are given in Eqs. (24) and (26) in the Appendix.
4
Closed form expressions of these effects are given in part 3 of Appendix B, available from the authors or at IJIO's
supplementary material website (http://www.mgmt.purdue.edu/centers/ijio/eo/eo.htm).
5
Spence (1976) provides an alternative formulation that yields the same measurement.
X.H. Wang, J. Zhao / Int. J. Ind. Organ. 25 (2007) 173–185
177
For convenience, we call all firms with marginal costs below the industry average level (i.e.,
c̄ − ci N 0) efficient firms, and all firms with marginal costs above the industry average level (i.e.,
c̄ − ci b 0) inefficient firms. It is straightforward to show that ciB⁎⁎ N ciB⁎. Hence, if a cost
reduction increases total profits, it also will increase total welfare. From the Proof of Proposition
1, a small cost reduction by each efficient firm will raise total welfare. If cost reduction by an
inefficient firm lowers total welfare, cost reduction by any less efficient firm also will lower
total welfare.
By parts (i–ii) of Proposition 1, following a small reduction in firm i's marginal cost, consumer
surplus will always increase, although total profits will decrease if firm i is a high-cost firm. Total
profits fall because the output of a high-cost firm increases at the expense of the outputs of more
efficient firms. By part (iii) of Proposition 1, total welfare will decrease if firm i's marginal cost is
above the critical level ciB⁎⁎. That is, if firm i is sufficiently inefficient the decrease in total profits
will outweigh the increase in consumer surplus, resulting in a decrease in total welfare.6
Corollary 1 below shows that the effects of small cost reductions on performances also can be
characterized using output shares. By Eq. (6), a firm's equilibrium output share is
tiB ¼
qBi
1 ð1 þ gÞð2n þ ðn−1ÞgÞ c̄ −ci
d
¼ þ
:
nð2n þ ð2n−1ÞgÞ
QB n
V −c̄
ð12Þ
Therefore, under Bertrand competition, firms' output shares have the reverse order of their
marginal costs: the most efficient firm has the largest share, and the least efficient firm has the
smallest share. It follows from Eq. (12) that at the Bertrand equilibrium all efficient firms have
output shares greater than 1 / n, and all inefficient firms have output shares less than 1 / n.
Corollary 1. Under Bertrand competition, a small reduction in firm i's marginal cost
(i) reduces total profits ⇔ its output share is below a critical level tΠB;
B
(ii) reduces total welfare ⇔ its output share is below a critical level tW
, where the critical levels
B
B
tΠ and tW are given in Eqs. (23) and (25) in the Appendix.
Our results in Proposition 1 can be extended to other models of price competition in a
straightforward manner. Here we briefly consider the logit model.7 Firm i's profit is πi( p) = ( pi − ci)
si − Ki, i = 1, …, n, where pi, ci and Ki are i's price, marginal cost, and sunk entry cost, respectively, and
si is its market share or logit demand. As shown in Anderson and de Palma (2001), the Bertrand
equilibrium is determined by ( pi − ci) = μ / (1 − si), for all i, where μ N 0 is a demand parameter. Let pj⁎
6
The perverse welfare effect of cost reductions cannot arise in a homogenous Bertrand model, in which only the most
efficient firm (firm 1) has a positive output in equilibrium. The equilibrium price is either firm 1's monopoly price (when
it is below firm 2's marginal cost c2) or equal to c2. In the former case, small reductions in cj ( j ≠ 1) have no effect, and a
small reduction in c1 increases both consumer surplus and firm 1's profit (and therefore total welfare). In the latter case, a
small reduction in c1 will not affect consumer surplus but will increase firm 1's profit (and therefore total welfare); small
reductions in cj ( j ≥ 3) have no effect; and a small reduction in c2 will raise consumer surplus. Firm 1's profit will be
lower but total welfare will be higher.
7
Anderson et al. (1992) provided extensive coverage of both the logit model discussed here and the Hotelling model.
Part 1 of Appendix B provides detailed derivations for the logit model. Note that the Bertrand–Shubik demand system (1)
does not include the Hotelling model as a special case. However, one can easily show that similar results as in Proposition
1 hold for the Hotelling model. Part 2 of Appendix B derives such results.
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X.H. Wang, J. Zhao / Int. J. Ind. Organ. 25 (2007) 173–185
and sj⁎ be j's equilibrium price and market share, respectively, then the effect of a small reduction in ci
on welfare (W⁎ = Π⁎ + CS⁎) is
⁎
⁎
⁎
⁎
Xn
ðsj Þ2
AW ⁎
ð1−si Þ3
si ð1−si Þ2
¼
þ
⁎
⁎
⁎
⁎
⁎
⁎ −1;
j¼1
Aci
1−si þ ðsi Þ2 ð1−aÞ½1−si þ ðsi Þ2 1−sj þ ðsj Þ2
"
#
⁎
⁎
⁎
ðsj Þ2
AW ⁎
ð1−si Þ3
si Xn
⁎
N0⇔
1−si þ
and
⁎
⁎
⁎
⁎ N1;
j¼1
Aci
ð1−aÞ
1−si þ ðsi Þ2
1−sj þ ðsj Þ2
P
⁎
⁎
⁎
where a ¼ nj¼1 ½ðsj Þ2 =ð1−sj þ ðsj Þ2 Þ satisfies 0 b α b 1. The above inequality implicitly defines a
critical value for i's market share, below which its small cost reduction decreases welfare.
3. The Cournot model
The inverse demand functions corresponding to the Bertrand–Shubik model (1) are:
pi ðqÞ ¼ pi ðq1 ; N ; qn Þ ¼ V −
1
g
ðqi þ gq̄ Þ ¼ V −qi þ
ðqi −q̄Þ;
1þg
1þg
ð13Þ
P
where q̄ð ni¼1 qi Þ=n is the average quantity, V and γ are the same as in Eq. (1), and the quantity
space is {q ∈ R+n:V − qi + γ(qi − q̄ ) / (1 + γ) N 0, i = 1, …, n}. The first-order conditions for the firms'
profit maximization can be arranged as
2 g
2 1þ
6
n
6
g
6
Cq ¼ 6
n
6
⋮
6
4
g
n
g
:::
n g
:::
2 1þ
n
⋮
g
n
⋱
:::
g
n
g
n
3
0 1 0
1
7 q1
ð1 þ gÞðV −c1 Þ
7
7B q2 C B ð1 þ gÞðV −c2 Þ C
7B C ¼ B
C:
7@ ⋮ A @
A
⋮
⋮
7
ð1 þ gÞðV −cn Þ
g 5 qn
2 1þ
n
ð14Þ
Solving Eq. (14) for q leads to the following Cournot equilibrium values:8
pCi ¼
ðn þ gÞV
ngðn þ gÞc̄
nci
þ
þ
;
2n þ ðn þ 1Þg ð2n þ gÞð2n þ ðn þ 1ÞgÞ 2n þ g
ð15Þ
qCi ¼
X
nð1 þ gÞ C
n2 ð1 þ gÞ
ðpi −ci Þ; QC ¼
ðV −c̄Þ; and
qCi ¼
nþg
2n þ ðn þ 1Þg
ð16Þ
pCi ¼
nþg
ðqC Þ2 :
nð1 þ gÞ i
ð17Þ
8
The inverse of matrix C can be obtained the same way as B − 1 (see footnote 2). Specifically, C − 1 = (n / (2n + γ))
{I − [γ / (2n + (n + 1)γ)]H}. Multiplying C − 1 into the right hand side of Eq. (14) gives Eq. (15). Eqs. (16) and (17) follow
directly by utilizing Eq. (15) and the inverse demand Eq. (13).
X.H. Wang, J. Zhao / Int. J. Ind. Organ. 25 (2007) 173–185
179
Similar to the Bertrand model, we make the following assumption A2, which guarantees a
positive output for each firm at the above Cournot equilibrium:9
A2.
ð2n þ gÞV þ g
P
j p n cj
nðg þ 2Þ
Ncn :
It is useful to note that A2 follows from A1, so positive outputs under Bertrand competition
imply positive outputs under Cournot competition.
By Eq. (16), (pic − ci) / qic = (n + γ) / (n(1 + γ)), so all firms' markup/output ratios are the same. It
is useful to note that the markup (pic − ci) is given by
pCi −ci ¼
ðn þ gÞðV −c̄ Þ ðn þ gÞðc̄ −ci Þ
þ
;
2n þ ðn þ 1Þg
2n þ g
so more efficient firms (i.e., with smaller ci) will have larger markups and outputs.
Differentiating Eqs. (15)–(17) shows immediately that the effects of a small reduction in firm
i's marginal cost on prices, outputs and profits at the Cournot equilibrium are similar to those at
the Bertrand equilibrium: (i) it decreases firm i's price, and increases its output and profits; (ii) it
decreases all other firms' prices, outputs and profits; and (iii) it increases the industry output.10
We next study the effect of small changes in costs on total profits as well as consumer surplus
and total welfare in theP
Cournot model. Direct calculations show that, at the Cournot equilibrium,
industry profits PC ¼ nj¼1 pCj , consumer surplus (CSC) and total welfare (W C), respectively, are
given by
PC ¼
n
nþg X
ðqC Þ2 ;
nð1 þ gÞ j¼1 j
CSC ¼
n
X
1
g
ðQ C Þ2 ;
ðqC Þ2 þ
2ð1 þ gÞ j¼1 j
2nð1 þ gÞ
W C ¼ PC þ CSC ¼
n
3n þ 2g X
g
ðQ C Þ2 ;
ðqCj Þ2 þ
2nð1 þ gÞ j¼1
2nð1 þ gÞ
ð18Þ
ð19Þ
ð20Þ
where qjC and QC are given by Eq. (16).
Proposition 2 characterizes the effects of small cost reductions under Cournot competition.
9
Similar to A1 in the Bertrand model, A2 determines a feasible range (ccL, ccU) for each ck in the Cournot model, where
and ccL = ccL(k) are its upper and lower bounds as given below:
8
P
½nðg þ 2Þcn −ðð2n þ gÞV þ g j p k;n cj Þþ
>
>
>
if kbn
<
P
ð2n þ gÞV þ g j p k cj
g
c
Pn−2
ccU ¼
; and cL ¼
þ
nðg þ 2Þ
>
½nðg þ 2Þcn−1 −ðð2n þ gÞV þ g j¼1 cj Þ
>
>
:
if k ¼ n:
g
ccU = ccU(k)
10
See part 3 of Appendix B for closed form expressions of these effects.
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X.H. Wang, J. Zhao / Int. J. Ind. Organ. 25 (2007) 173–185
Proposition 2. Under Cournot competition, a small reduction in firm i's marginal cost
(i) reduces total profits ⇔ ci is above a critical level ciC⁎;
(ii) always increases consumer surplus; and
(iii) reduces total welfare ⇔ ci is above a critical level ciC⁎⁎, where the critical levels ciC⁎ and
ciC⁎⁎ are given respectively by Eqs. (28) and (30) in the Appendix.
It is straightforward to show that ciC⁎⁎ N ciC⁎. Hence, if a cost reduction increases total profits it
also will increase total welfare. What is more interesting is the negative welfare effect: a firm's
cost reduction decreases total welfare if its marginal cost is above the critical level ciC⁎⁎ in Eq.
(30). The explanations for this counter-intuitive result are similar to those for the welfare effects
under Bertrand competition discussed in Section 2.
The next corollary characterizes the welfare effects of a Cournot firm's small cost reductions
using its output shares. From Eq. (16), firm i's equilibrium output share is
tiC ¼
qCi
1 ð2n þ ðn þ 1ÞgÞ ðc̄ −ci Þ
d
:
¼ þ
nð2n þ gÞ
ðV −c̄ Þ
QC n
ð21Þ
It follows immediately from Eq. (21) that output shares in the Cournot model are ranked
similarly as in the Bertrand model: the most efficient firm has the largest share, and the least
efficient firm has the smallest share. Moreover, all efficient firms have output shares greater than
1 / n and all inefficient firms have output shares less than 1 / n.
Corollary 2. Under Cournot competition, a small reduction in firm i's marginal cost
C
(i) reduces total profits ⇔ its output share is below a critical level tΠ
;
c
(ii) reduces total welfare ⇔ its output share is below a critical level tw, where the critical levels
C
C
tΠ
and tw
are given in Eqs. (27) and (29) in the Appendix.
4. Bertrand vs. Cournot
By comparing our results in Sections 3 and 4, we obtain the following three differences
between Bertrand and Cournot competition that are relevant to our comparative statics analysis:
(I) total output under Cournot competition is lower than that under Bertrand competition; (II) a
larger market share is required at Bertrand equilibrium than at Cournot equilibrium for a cost
reduction to have a positive welfare effect; and (III) the distribution of output shares is wider
under Bertrand competition than under Cournot competition.
By Eqs. (6) and (16), QC − QB = − n(n − 1)γ2(V − c̄ ) / {2n + (n − 1)γ)(2n + (n + 1)γ)} b 0, so
Claim I holds. By Eqs. (25) and (29),
B
C
tW
−tW
¼
g2 ½3n3 þ ð3n2 −1Þng þ ðn2 þ 2n−3Þg2 N0;
ð2n þ ðn−1ÞgÞð2n þ ð2n þ 1ÞgÞð3n þ ð3n−1ÞgÞð3n þ 2gÞ
ð22Þ
so Claim II holds. By Eqs. (12) and (21), the difference between firm i's output shares in Bertrand
and Cournot equilibria is
tiB −tiC ¼
ðn−1Þg3
ðc̄ −ci Þ
d
:
ð2n þ ð2n−1ÞgÞð2n þ gÞ V −c̄
X.H. Wang, J. Zhao / Int. J. Ind. Organ. 25 (2007) 173–185
181
This difference is positive (i.e., tiB N tiC) for all efficient firms and negative (i.e., tiB b tiC) for all
inefficient firms. Hence, all efficient firms have larger output shares under Bertrand competition
than under Cournot competition, and all inefficient firms have smaller output shares under
Bertrand competition than under Cournot competition. Claim III also follows from an
examination of the output shares within each model. By Eqs. (12) and (21),
tiB −tjB ¼
ð1 þ gÞð2n þ ðn−1ÞgÞ ðcj −ci Þ
d
; and
nð2n þ ð2n−1ÞgÞ
ðV −c̄ Þ
tiC −tjC ¼
2n þ ðn þ 1Þg ðcj −ci Þ
d
:
nð2n þ gÞ
ðV −c̄ Þ
It follows from (1 + γ)(2n + (n − 1)γ) / (2n + (2n − 1)γ) N (2n + (n + 1)γ) / (2n + γ) that tiB − tjB N
for all i b j (recall the assumption that c1 ≤ c2 ≤ … ≤ cn). Hence, the difference in output
shares between any two firms is larger under Bertrand competition than under Cournot
competition.
The above three differences lead to the next proposition.
tiC − tjC
Proposition 3. For each inefficient firm i (i.e., ci N c̄ ), a small reduction in ci will decrease
welfare under Bertrand competition if it decreases welfare under Cournot competition.
Example 1 below shows a case in which small cost reductions increase (decrease) welfare in
Cournot (Bertrand) competition. Note that earlier works comparing Bertrand and Cournot
equilibria showed that prices at Bertrand equilibria are intrinsically lower than at Cournot
equilibria,11 and that total welfare at Bertrand equilibria are higher than at Cournot equilibria in
our model.12
Example 1. Consider n = 3 with V = 12, (c1, c2, c3) = (1, 4.5, 9.7) and γ = 0.8. The Bertrand and
Cournot equilibria and their performances are:
ðpB1 ; pB2 ; pB3 Þ ¼ ð5:933; 7:543; 9:935Þ; ðpC1 ; pC2 ; pC3 Þ ¼ ð6:136; 7:680; 9:975Þ;
ðpB1 ; pB2 ; pB3 Þ ¼ ð37:311; 14:197; 0:085Þ; ðpC1 ; pC2 ; pC3 Þ ¼ ð37:490; 14:374; 0:107Þ;
QB ¼ 12:590; QC ¼ 12:209; CSB ¼ 33:715; CSC ¼ 31:556; W B ¼ 85:307; W C ¼ 83:527;
APB =Ac3 ¼ 0:415N0; APC =Ac3 ¼ 0:375N0; AW B =Ac3 ¼ 0:055N0; AW C =Ac3 ¼ −0:015:
11
See Singh and Vives (1984), Hackner (2000), and Amir and Jin (2001).
Hsu and Wang (2005) compared welfare at Bertrand and Cournot equilibria in a differentiated goods oligopoly.
Although their focus is on the Hackner (2000) model with quality differentiation and identical costs, their results can be
generalized in a straightforward manner to show that in the present model with unequal costs both consumer surplus and
total welfare are higher under Bertrand competition than under Cournot competition.
12
182
X.H. Wang, J. Zhao / Int. J. Ind. Organ. 25 (2007) 173–185
So a small cost reduction by firm 3 decreases total welfare under Bertrand competition but
increases total welfare under Cournot competition. It reduces total profits in both cases.
5. Conclusion and extension
This paper has provided closed-form expressions for the critical levels of output shares below
which a small cost reduction by a Bertrand or Cournot firm reduces welfare in linear oligopolies
with differentiated goods. It has shown that the critical value is larger under Bertrand competition
than under Cournot competition.
In addition to extending our analysis to Bertrand oligopolies with non-linear demand or nonlinear costs or both, we suggest three other closely related future studies. First, it would be useful
to evaluate the welfare effects of shocks under Bertrand competition, which are extensions of
recent works under Cournot competition, such as Salant and Shaffer (1999), Long and Soubeyran
(2001), Linnemer (2003), and Février and Linnemer (2004), that studied such effects under
Cournot competition. Second, it would be useful to extend the profit effects of taxation polices in
a symmetric Bertrand model seen in Anderson, de Palma, and Kreider (2001) to the welfare
effects of taxation polices in symmetric or asymmetric Bertrand models. Third, it would be useful
to extend the welfare effects of the trade policies seen in Lahiri and Ono (1997), Leahy and
Montagna (2001), and Neary (2002) to our differentiated Bertrand or Cournot model.
Appendix A
Proof of Proposition 1 and Corollary 1. (i) By Eqs. (5) and (6), differentiating Eq. (9) gives
(
)
B
B
n
n
X
X
Aq
Ap
APB
2n
j
j
¼
qB
¼2
qBj
−qB
n þ ðn−1Þg j¼1 j Aci
Aci
Aci i
j¼1
2nð1 þ gÞ
gðn þ ðn−1ÞgÞ
B
B
Q −qi
¼
2n þ ð2n−1Þg nð1 þ gÞð2n þ ðn−1ÞgÞ
2nð1 þ gÞQB
gðn þ ðn−1ÞgÞ
B
−t ;
¼
2n þ ð2n−1Þg nð1 þ gÞð2n þ ðn−1ÞgÞ i
ð22Þ
which leads to
APB
gðn þ ðn−1ÞgÞ
B
N 0ftiB b tP
¼
nð1 þ gÞð2n þ ðn−1ÞgÞ
Aci
fci −c̄ N
ð2n þ ð2n−1ÞgÞ2 ðV −c̄ Þ
ð1 þ gÞ2 ð2n þ ðn−1ÞgÞ2
;
ð23Þ
which is equivalent to ci N ciB⁎, where
⁎
cBi
nð2n þ ð2n−1ÞgÞ2 V þ gðn þ ðn−1ÞgÞððn−1Þg2 þ ð5n−2Þg þ 4nÞ
¼
P
jpi
ðn−1Þð1 þ gÞ2 ð2n þ ðn−1ÞgÞ2 þ ð2n þ ð2n−1ÞgÞ2
cj
:
ð24Þ
X.H. Wang, J. Zhao / Int. J. Ind. Organ. 25 (2007) 173–185
183
(ii) Differentiating Eq. (10) with respect to ci gives
n
AqBj
ACSB
1 X
g
AQB
¼
qBj
þ
QB
1 þ g j¼1
Aci
Aci nð1 þ gÞ
Aci
gðn þ ðn−1ÞgÞ
ðn þ ðn−1ÞgÞ B
QB −
q b0:
¼−
ð2n þ ðn−1ÞgÞð2n þ ð2n−1ÞgÞ
2n þ ð2n−1Þg i
(iii) Finally, utilizing the above results leads to
AW B APB ACSB
gðn þ ðn−1ÞgÞ
ð3n þ ð3n−1ÞgÞ B
QB −
q
¼
þ
¼
ð2n þ ðn−1ÞgÞð2n þ ð2n−1ÞgÞ
2n þ ð2n−1Þg i
Aci
Aci
Aci
ð3n þ ð3n−1ÞgÞQB
gðn þ ðn−1ÞgÞ
B
−t ;
¼
ð2n þ ðn−1ÞgÞð3n þ ð3n−1ÞgÞ i
2n þ ð2n−1Þg
which leads to
APB
gðn þ ðn−1ÞgÞ
B
N 0ftiB b tW
¼
ð2n þ ðn−1ÞgÞð3n þ ð3n−1ÞgÞ
Aci
fci −c̄ N
ð2n þ ð2n−1ÞgÞ2 ð3n þ ðn−1ÞgÞðV −c̄ Þ
ð1 þ gÞð2n þ ðn−1ÞgÞ2 ð3n þ ð3n−1ÞgÞ
;
ð25Þ
which is equivalent to ci N ciB⁎⁎, where ciB⁎⁎ is equal to
nð2n þ ð2n−1ÞgÞð3n þ ðn−1ÞgÞV þ gðn þ ðn−1ÞgÞðð3n2 −4n þ 1Þg2 þ 11n2 g−6ng þ 8n2 Þ
P
jpi
ðn−1Þð1 þ gÞð2n þ ðn−1ÞgÞ2 ð3n þ ð3n−1ÞgÞ þ ð2n þ ð2n−1ÞgÞ2 ð3n þ ðn−1ÞgÞ
cj
:
□
ð26Þ
Proof of Proposition 2 and Corollary 2. (i) By Eqs. (15) and (16), differentiating Eq. (18) gives
n
AqCj
APC 2ðn þ gÞ X
2ðn þ gÞ
g
C
C
C
Q −qi
¼
q
¼
nð1 þ gÞ j¼1 j Aci
2n þ g 2n þ ðn þ 1Þg
Aci
2ðn þ gÞQC
g
C
−t ;
¼
2n þ ðn þ 1Þg i
2n þ g
which leads to
APC
g
C
N 0ftiC b tP
¼
2n þ ðn þ 1Þg
Aci
fci −c̄ N
ð2n þ gÞ2 ðV −c̄ Þ
ð2n þ ðn−1ÞgÞ2
;
ð27Þ
which is equivalent to ci N ciC⁎, where
C⁎
ci
nð2n þ gÞ2 V þ ðnð2n þ ðn þ 1ÞgÞ2 −ð2n þ gÞ2 Þ
¼
P
jpi
ðn−1Þð2n þ ðn þ 1ÞgÞ2 þ ð2n þ gÞ2
cj
:
ð28Þ
184
X.H. Wang, J. Zhao / Int. J. Ind. Organ. 25 (2007) 173–185
(ii) Differentiating Eq. (19) with respect to ci gives
n
AqCj
ACSC
1 X
g
AQC
QC
¼
qCj
þ
1 þ g j¼1
Aci
Aci nð1 þ gÞ
Aci
gðn þ gÞ
n
QC −
qC b0:
¼−
ð2n þ gÞð2n þ ðn þ 1ÞgÞ
2n þ g i
(iii) Finally, utilizing the above results leads to
AW C APC ACSC
gðn þ gÞ
3n þ 2g C
QC −
q
¼
þ
¼
ð2n þ gÞð2n þ ðn þ 1ÞgÞ
2n þ g i
Aci
Aci
Aci
ð3n þ 2gÞQC
gðn þ gÞ
−tiC ;
¼
ð2n þ ðn þ 1ÞgÞð3n þ 2gÞ
2n þ g
which leads to
AW C
gðn þ gÞ
C
N 0ftiC b tW
¼
Aci
ð2n þ ðn þ 1ÞgÞð3n þ 2gÞ
fci −c̄ N
ð2n þ gÞ2 ð3n þ ðn þ 2ÞgÞ
ð2n þ ðn þ 1ÞgÞ2 ð3n þ gÞ
ð29Þ
ðV −c̄Þ;
which is equivalent to ci N ciC⁎⁎, where ciC⁎⁎ is equal to
nð2n þ gÞ2 ð3n þ ðn þ 2ÞgÞV þ gð8n3 −4n2 þ 3n3 g þ 6n2 g−4ng þ n2 g2 þ ng2 −g2 Þ
P
jpi
ðn−1Þð2n þ ðn þ 1ÞgÞ2 ð3n þ gÞ þ ð2n þ gÞ2 ð3n þ ðn þ 2ÞgÞ
cj
:
□
ð30Þ
Proof of Proposition 3. Using the fact that tiB b tiC for inefficient firms, by Eq. (22), Corollary 1
C
C
B
and Corollary 2, ∂W C / ∂ci N 0 ⇒ tiC b tW
⇒ tiB b tiC b tW
b tW
⇒ ∂W B / ∂ci N 0. □
Appendix B. Supplemetary data
Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.ijindorg.2006.02.003.
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