The Complete Welfare Effects of Cost Reductions by Oligopoly Firms

The Complete Welfare Effects of Cost Reductions by Oligopoly Firms

Donald J. Smythe and Jingang Zhao*
California Western School of Law
Iowa State University

April 2002




Abstract: This paper completely characterizes the welfare effects of cost reductions and technological
spill-overs in a general Cournot model. It shows that a cost reduction by an individual firm (a
technological spill-over within a set of firms) increases welfare if and only if the firm's market share (the
weighted sum of the members’ shares) is greater than a weighted sum of the other firms' (the non-
members’) market shares. Such characterizations based on market shares can be directly applied to
simulation studies and related empirical analyses.
JEL Codes: D43, L13

Keywords: Cost reduction, oligopoly, technological spillover, welfare


* Address all correspondence to Jingang Zhao, Department of Economics, 260 Heady Hall, Iowa State
University, Ames, Iowa 50011-1070, jingang@iastate.edu, Fax: (515) 294-0221, Tel: (515) 294-5245.

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The Complete Welfare Effects of Cost Reductions by Oligopoly Firms

Abstract: This paper completely characterizes the welfare effects of cost reductions and
technological spill-overs in a general Cournot model. It shows that a cost reduction by an
individual firm (a technological spill-over within a set of firms) increases welfare if and only if
the firm's market share (the weighted sum of the members’ shares) is greater than a weighted
sum of the other firms' (the non-members’) market shares. Such characterizations based on
market shares can be directly applied to simulation studies and related empirical analyses.

1. Introduction

Technological advancements have long been considered the main driving force
behind economic growth and development. Recent research has shown, however, that it is a
mistake to presume cost reductions will always increase social welfare. Lahiri and Ono
(1988), for instance, show that cost reductions in a Cournot model with general demand and
linear costs will actually decrease welfare if the cost-reducing firm is sufficiently small, and
Zhao (2001) shows that cost reductions in a linear Cournot model will decrease welfare if
and only if the cost-reducing firm’s market share is smaller than 1/(2n+2).

These results have limited applications either because of the linearity assumptions or
because they do not precisely characterize the conditions under which cost reductions will be
welfare-decreasing. In this paper, we provide a precise characterization of the conditions
under which cost reductions will be welfare-decreasing in a model with general demand and
costs. Our results indicate that a firm’s cost reduction (a cost-reducing technological spill-
over within a group of firms) will increase social welfare if and only if the firm's market
share (the weighted sum of the group members’ market shares) is greater than a threshold
determined by a weighted sum of the other firms’ (the non-members’) market shares.

As it turns out, our results not only replicate those of the previous studies as special

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cases, they also provide a number of new insights. In particular, they precisely characterize
conditions under which an industry-wide technological spill-over will decrease welfare.
Moreover, since our characterizations use only market shares and the first and second order
derivatives of the demand and cost functions, they should have direct applications in
simulation studies and other empirical analyses.

The paper is organized as follows: Section 2 describes the problem, Section 3
analyzes cost reductions by an individual firm, and Section 4 analyzes technological spill-
overs within a group of firms. Section 5 concludes, and the Appendix provides the proofs.
2. Description of the problem

A Cournot oligopoly with n firms and a homogeneous good is defined by an inverse
demand function, p(X) = p(?xj), and n cost functions Ci(xi), i ? N = {1, 2, ..., n }. This is
equivalent to an n-person game in normal form:
(1)
? = {N, Zi, ?i},
where Z i= [0, ?) is the production set for each firm i? N, and
?

i(x) = p(X) xi – Ci(xi), x = (x1, ..., xn) ? Z = ?i?NZi,
is firm i’s profit function. A Cournot equilibrium is a vector of outputs satisfying the best
response property (i.e., each xi is i's best response to x-i = (x1, ..., xi-1, xi+1, ..., xn)), and is
defined by the solution to the following first order conditions:
(2) ??

i/?xi = p(X) + xip’(X) - C'i(xi) = 0, i = 1, …, n.
For each firm i, define the following two terms:
(3)
?i(xi, X) = p’(X) + xi p’’(X), and ?i(xi, X) = p’(X) - C''i(xi).
The existence of a Cournot equilibrium is guaranteed by the condition that ?i(xi, X)

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? 0 for all xi ? X and all i (see Novshek, 1985). The requirements for a unique equilibrium
are beyond the scope of this paper; some recent results are surveyed in Zhang and Zhang
(1996). Let social welfare (i.e., the sum of consumer and producer surplus) be given by
(4)

W = CS + ??
X
i = õ
ó P(t)dt – ?Cj(xj);
0
and let firm i’s cost function be given by
(5)

Ci(xi) = cixi + di(xi), ci > 0,
where di(xi) includes the constant and second or higher order terms in its Taylor series. It is
useful to note that C''i = d''i, all i. We make the following standard assumptions called A1:
A1: The following conditions hold: (i) p’< 0, C'j > 0, p’’ and C''j are continuous, all j;
(ii) ?j(xj, X) < 0, and ?j(xj, X) < 0, for all xj ? X, all j; (iii) a unique Cournot equilibrium
exists; and (iv) all firms have positive market shares at the Cournot equilibrium.1
Without loss of generality, we assume MCi(q) = ci+d'i(q) ? ci+1+d ' i+1(q) = MCi+1(q) for
any q > 0 and for i = 1, …, (n-1). Let si = xi/X denote firm i’s market share. Then, (2) and
A1 lead to the following relation:
(6)
s1 ? ... ? sn-1 ? sn > 0.
Let a decrease in ck (i.e., a decrease in the constant part of MCk) denote firm k’s cost
reduction. Such a cost reduction is welfare-increasing if ?W/?ck < 0. By differentiating the
n equations in (2), one obtains (see the appendix for a proof) the following results:
?X
1
?xk 1+?
?xj
-?
(7)

i?k?i
j
? =
=
=
ck ?k? < 0; ?ck
?k? < 0; and for j ? k, ?ck ?k? > 0;

1
“?i<0” in part (iii) can be weakened to “?i ? 0,” which is equivalent to (P’+XP’’) ? 0 (see Shapiro,
1989). We assume “< 0” for simplicity. Parts (iii) and (iv) together imply ?2?i/?x2i = (2P’ + xiP’’- C’’(xi)) <
0. These assumptions are implicit in all related studies, such as Mankiw and Whinston (1986).

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where ?k is given by (3), and ?i and ? are given by
(8)
?i = ?i/?i > 0, and ? = 1+??j > 0.
By (7), a cost reduction by firm k increases both the firm's own and total industry
outputs, and decreases the individual outputs of all other firms.
3. The welfare effects of an individual firm’s cost reduction

Define the term ?k by
(9)

?k = ?k?/p’ = (p’?X/?ck)-1 > 0,
which is the inverse of the product of p’ and ?X/?ck. Proposition 1 below provides a general
characterization of the sign of ?W/?ck.

Proposition 1: Let si be i’s market share, let ?i and ?k be given by (8) and (9), and
assume A1 holds. Then the sign of ?W/?ck is determined as below:
?W
?
(10)
i?k?isi
? < 0 ? s
c
k >

k
1+?k+?i?k?i
(11)

? ?k-1?
?
i=1 i(si-sk) < (1+?k)sk + ? n
i=k+1 i(sk-si).

To summarize, a cost reduction by firm k increases welfare if and only if its market
share is larger than the critical value given by (10). If the firm’s market share is smaller than
the critical value, its cost-reduction will decrease social welfare. This is because the
increases in consumer surplus and the cost-reducing firm’s profits are out-weighed by the
decreases in the other firms’ profits.
Corollaries 1 and 2 below summarize four special cases.

Corollary 1: Assume A1 holds in an oligopoly defined by (1). (i) ?W/?c1 < 0; (ii)
if (1) is symmetric, so that Ci(q) ? Cj(q) for all i?j, then ?W/?ck < 0 for all k; (iii) if (1) is

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linear, so that p = a-X and Ci(xi) = cixi, all i, then
(12)
?W/?ck < 0 ? sk > 1/(2n+2).
Parts (i) and (ii) follow directly from (11). They show that a cost-reduction by the
most efficient firm or by any firm in a symmetric Cournot oligopoly always increases
welfare. Part (iii) pertains to linear Cournot oligopolies. By di(xi) = 0 and p = a-X, one has
?i = ?i = p’ = -1, ?i = 1, and ? = ?k = (n+1). Hence, (10) becomes sk > (1- sk)/(2n+1). This
leads to (12), which is the characterization reported in Zhao (2001).
Corollary 2 below further characterizes the results for cases in which firms’ cost
functions have identical and constant second order derivatives, so that Ci(xi) = cixi+x2id/2 or
C''i = d''i = d, all i. Let ?, E and H be defined as
(13)
? = (p’-d)/p’,
(14)
E = d lnX /d lnp’ = p’/(Xp’') , and
(15)
H = ?s2i = ?(x2i/X2),
where E and H are the price elasticity of demand and the Herfindahl index.

Corollary 2: Given an oligopoly defined by (1), assume that C''i = d''i = d, all i, and
that A1 holds, and let ?, E and H be given by (13-15). Then, (10) becomes
?W
E+H
(16)

? < 0 ? s
.
c
k >
k
(1+?)[1+(n+?)E]

4. The welfare effects of a technological spill-over

Define a technological spill-over within a set of firms T ? ? (2? |?|=k<n) as a
simultaneous and equal reduction in c


i for all i ?T. Let ci = ci(?) = c
–i+?, all i?T, and ci = c–i,
all i?T. Then the welfare effects of a spill-over within T are represented by ?W/??.

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Proposition 2 below characterizes the output effects.

Proposition 2: Assume A1 holds in an oligopoly defined by (1). Let ? ? ? denote
the set of firms that benefits from a technological spill-over (the "members"), and let ?i, ?i
and ? be given by (3) and (8). Then the output effects of the spill-over are as follows:
?X 1
1
??i?Txi
?X
(17)
? =
= (1+?
?
? ?i?T? < 0;
i?T?i)
i
??
?? < 0;
?xi 1
?X
?xi
?X
(18)
? =
for i?T; and
= -?
?
? - ?i
i
i
??
??
?? > 0 for i?T.

By (17) and (18), non-members’ outputs always decrease, and both the sum of the
members' outputs and total industry output always increase. Define ?0 and ?i as follows:
?
1
?
1
(19)
?0 =
1 = ? > 0, ?
> 0,
X
k =
1 =
?X
p’?i?T? p’
?k?i?T
?k
i
??
?i
??
where ?i and ? are given by (3) and (8). Using these formulae and differentiating (4) with
respect to ?, we obtain the following characterization of the sign of ?W/??:

Proposition 3: Assume A1 holds in an oligopoly defined by (1). Let ? ? ? denote
the set of firms that benefit from a technological spill-over (the "members"), and let ?i and
?i be given by (8) and (19). Then
(20)

?W/?? < 0 ? ?i?T(?0+?i-?i)si > ?i?T?isi.

In other words, a technological spill-over increases welfare if and only if the
weighted sum of the members’ market shares is greater than the weighted sum of the non-
members’ market shares as defined in (20).
In a linear Cournot oligopoly, one has: ?0 = ?i = (n+1)/k, ?i = 1, and ? = ?k = (n+1).
Hence, the right side of (20) becomes ?i?T(2n+2)si/k >1, or

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(21)

?i?Tsi > k/(2n+2),
which is the same as Proposition 6 in Zhao (2001).
For cases in which Ci(xi) = cixi+x2id/2, one has
(1+?)p
1
p
?
’
’'X(E+H)
0 + ?i = k(p’-d) (n+?+E), and ??isi = (p’-d) .
Since (20) is equivalent to ?i?T(?0+?i)si > ??isi, these expressions imply
k(E+H)
(22)
?W/?? < 0 ? ?i?Tsi >
.
(1+?)[1+(n+?)E]

Finally, Corollary 3 below characterizes the conditions under which an industry wide
spill-over decreases social welfare.

Corollary 3: Assume A1 holds in an oligopoly defined by (1). Let c i= ci(?) = c–i+?,
all i?N; and let ?i and ?i be given by (8) and (19). Then
(23)
?W/?? > 0 ? ?(?0+?i-?i)si < 0
p’ ??x2
(24)

? X <
i
2 ?? .
In such counter-intuitive situations, the increase in consumer surplus is more than
offset by a decrease in total industry profits.2
5. Concluding remarks

We have completely characterized the welfare effects of cost reductions and

2
One obtains the complete characterization of conditions in which an industry-wide spill-over
decreases total industry profits using ??/?? = pX’+ Xp’X’ - ??Ci(xi)/?? = -X + p’ ?(X-xi)?xi/?? and
Proposition 2. In symmetric oligopolies with linear costs and constant elasticity of demand, Kimmel (1992)
has shown that an industry-wide spill-over decreases total profits when ? > -1, where ? is the (constant)
elasticity of demand.

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technological spill-overs in a Cournot oligopoly with general demand and costs. Our results
indicate that a small reduction in a firm's marginal costs (a technological spill-over within a
set of firms) decreases welfare if and only if the firm's market share (the weighted sum of the
members’ market shares) is less than a weighted sum of the other firms’ (the non-members’)
market shares.
Our results have a number of implications for public policy. To cite a few examples,
they imply that foreign aid programs which help less developed nations adopt better
technologies may be (globally) welfare-decreasing, firms’ attempts to raise their rivals’ costs
(see Salop and Scheffman, 1983) may be welfare-increasing, and patents may increase social
surplus even in the short-run. Moreover, since the characterizations use only market shares
and the first and second order derivatives of demand and cost functions, they should have
applications in simulation studies and other empirical analyses.

Appendix
Proof of (7): This is a special case of Proposition 2 with k = 1.

Q.E.D
Proof of Proposition 1: Differentiating (4) with respect to ck, one gets
?W
?X ??Ci(xi)
?xi ?Ck(xk)
?Ci(xi)
?xi
? = p
-
= p
-
= –x
.
c
?
- ?i?k
k + ?(p-C'
k
?ck
?ck
?ck
?ck
?ck
i)?ck
Using (2), (7) and (8), the above expression becomes:
?W
?xi
?xk
?xi
? = –x
- p

c
k - p’ ?xi
= –xk - p’ xk
’ ?i?kxi
k
?ck
?ck
?ck
1+?
-?
Xp’
= –x
i?k?i
j
k - p’ xk
?k? -p’ ?i?kxi?k? = ?k?{–sk(1+?k+?i?k?i) + ?i?k?isi},
By Xp’/(?k?) > 0, (10) holds. (11) follows from (10).



Q.E.D
Proof of Corollary 1: The discussions following the corollary form a proof.
Q.E.D

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Proof of Corollary 2: By (3), (8-9), and (13-15), and by C''i = d''i = d, one has:
(P1)
?i = p’ – d, ??i = (np’+Xp’’)/(p’–d), and
? = 1+??i = (p’–d+np’+Xp’’)/(p’–d);
p’+p’'?x
Xp’'
p’
x2
Xp’'(E+H)
(P2)
??
isi
i
isi =
(p
+?
’–d) = (p’–d) {Xp’'
X2} = (p’–d) ; and
(p’–d)
(p’–d+np’+Xp’’)
(P3)
1+?k +??i = (1+??i)(1+ p )
’
= (1+?)
(p’–d)

(1+?)p’
d
Xp’'
(1+?)p’
1
= (p
] =
’–d) [1- p’ + n + p’
(p’–d) [n+?+ E].
It is straightforward to see that (10) is equivalent to sk > ??isi /(1+?k+??i). By (P2) and
(P3), the above expression becomes sk > (E+H) /{(1+?)[1+(n+?)E]}.

Q.E.D
Proof of Proposition 2: Consider first the case in which Ci(xi) = (c–i+?)xi+di(xi), i = 1, …,
k; and Ci(xi) = c–ixi+di(xi), i = (k+1), …, n; i.e., T = {1, …, k}. (2) now becomes
(P4)
p + xip’ = C'i = (c–i+?)+d'i, i = 1, …, k; and


p + xip’ = C'i = c–i+d'i, i = (k+1), …, n.
Differentiating the above n equations with respect to ?, we have
?X
?xi
?xi
(P5)
(p’ + xip’’) ? + p = 1+ C''
?
’ ??
i ?? , i = 1, …, k; and
?X
?xi
?xi
(p’ + xip’’) ? + p = C'' , i = (k+1), …, n.
?
’ ??
i ??
Using (3) and matrix form, (P5) becomes:
é ?1 .. 0 0 .. 0 ?1 ù
x '
1
1
æ .. ö
ê .. .. .. .. .. .. .. ú çæ ..
ç ÷
x '
÷
ö
(P6)
ë 0 .. ?
1
k
0
..
0
?k û ç k
ç 0 ÷
x '
k+1 ÷

é 0 .. 0 ?
=
.
k+1
..
0
?k+1 ù
ê
ç
ç .. ÷
..
÷
..
..
..
..
..
..
.. ú
ê 0 .. 0 0 .. ?
x '
ç 0 ÷
n
?n ú
n
ë
è
ç X
è 0 ø
’
ø
÷
-1 .. -1
-1
.. -1
1 û

10

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