Vol. 73 (2001), No. 1, pp. 81-93
Journal of Economics
Zeitschrift
f~ir
National6konomie
9 Springer-Verlag 2001 - Printed in Austria
Collusive Outcomes in Price Competition
Krishnendu Ghosh Dastidar
Received March 23, 1999; revised version received June 6, 2000
In this paper we provide a sufficient condition for collusive outcomes in a single-shot
game of simultaneous price choice in a homogeneous product market with symmetric
firms and strictly convex costs. We also prove the counterintuitive result : if the second
derivative of the cost function is nonincreasing in output, it is easier to sustain collusion
when the number of firms increases.
Keywords: price competition, collusion, convex cost.
JEL classification: L13.
1 Introduction
In this paper we consider a single-shot game of simultaneous price choice
in a h o m o g e n e o u s product oligopoly, with symmetric firms and strictly
convex cost functions. In this context we derive two main results. First,
we provide a sufficient condition for collusive outcomes in such a game.
Next, we prove the counterintuitive result: if the second derivative of the
cost function is nonincreasing in output, it is easier to sustain collusion
when the number o f firms increases. We also provide a specific numerical
example to illustrate this point.
There is substantial literature in Industrial Organization theory explaining collusive outcomes in oligopolistic interactions (see Tirole, 1988, and
Vives, 1999, for a detailed discussion of such issues). However, these results
are mainly obtained within a repeated-game framework. For example, in
infinitely repeated games we have the folk theorem (see Friedman, 1971).
There it is shown that if the discount factor is large enough then certain
"trigger strategies" can lead to collusive outcomes in a subgame-perfect
82
K.G. Dastidar
equilibrium. In the case of finitely repeated games we have the results
derived by Benoit and Krishna (1985), or by Kreps et al. (1982).
Till date the theoretical literature does not have any result showing that
collusive outcomes in oligopoly (in price or quantity competition without
conjectural variations) can be sustained as a Nash equilibrium in a singleshot game. 1 For example, if firms are quantity choosers then in a singleperiod game, the joint-profit-maximizing output cannot be sustained as a
Cournot-Nash equilibrium (the first-order conditions for Cournot equilibrium make this obvious). Similarly, collusive outcomes cannot be sustained
as a Nash equilibrium in a single-shot game of price competition in a differentiated product market.
However, the situation alters dramatically if we consider price competition in a homogeneous product market, with strictly convex costs. Dastidar
(1995) shows that for firms with symmetric and convex costs the resulting
Nash equilibria are necessarily nonunique. A whole range of prices can
be sustained as Bertrand equilibria under such situations and it is possible
for the joint-profit-maximizing price to fall within this range of Bertrand
equilibrium prices. This was noted independently by Dastidar (1993), and
Stahler (1997). The present paper derives conditions under which this result
holds. To be more specific, it provides a sufficient condition for collusive
outcomes in price competition, in a homogeneous product market, when
firms are symmetric and have strictly convex costs. Intuitively, the idea is
to arrange things so that the joint-profit-maximizing price lies within the
Nash equilibrium interval.
Conventional wisdom suggests that when the number of firms increases
in an oligopoly, the outcome tends to be more competitive. For example,
in a standard Cournot oligopoly with convex costs, if the number of firms
increases, then the resulting equilibrium approaches the competitive one.
However, we show that in price competition, under certain conditions, it
may actually be easier to sustain collusive outcomes when the number of
firms increases. This is somewhat counterintuitive and goes against conventional wisdom. We will provide a specific numerical example where collusion is not possible with two firms but is possible with three or more firms.
1 It may be noted that collusion can be supported with appropriate conjectural variation or supply functions. In both cases, like ours, the collusive outcome is one of the
multiple equilibria that exist. Similarly, competition in contracts for customers involving facilitating practices, such as most-favored-consumer or meet-or-release clauses,
may induce collusive pricing (see Salop, 1986, and Vives, 1999).
Collusive Outcomes in Price Competition
83
The plan of the paper is as follows. We first provide the model of our exercise. Next, we derive our main results and provide some possible intuitions
behind them. The proofs of the propositions are given in the appendix.
Lastly, we provide some concluding remarks.
We now introduce the model of our exercise.
2 The Model
Consider a one-shot game of simultaneous price choice, in a h o m o g e n e o u s
product oligopoly where there are n firms. We assume the following:
1. The demand F(P) is twice continuously differentiable and 3 positive
numbers pmax and Qmax such that F ( P max) = 0, f ( 0 ) = Qmax, and
FI(P) < O, V P.
2. The revenue function P F ( P ) is concave.
3. Cost functions Ci (Qi) are twice continuously differentiable and strictly
convex (that is C~I(Qi) > 0 for all Qi > 0). For simplicity we assume
that Ci(O) = 0. Also assume that C~(Q) > 0 and C~(0) < pmax gi.
Since we consider symmetric firms only, we denote all costs by C (Qi).
4. We assume that in price competition a firm always supplies the demand
it faces. 2
5. We assume that the firm which quotes the lowest price gets all the demand
and, given Assumption 4, it must serve it. The higher price quoting firm
gets nothing and sells zero. H o w e v e r when firms quote the same price
they share the demand equally.
M o r e formally let Pi be the price quoted by the i-th firm.
Let Si (P1, P2 . . . . . Pi . . . . . Pn) be the amount sold by the i-th firm when
it quotes Pi and the other firms quotes Pj (j ~ i). Under Assumptions 4
2 This assumption can be rationalized by assuming that when a finn sets a price Pi
this represents a commitment with customers to supply the forthcoming demand. This
is plausible when there are large costs of turning customers away (see Dixon, 1990).
This may be the case in regulated industries (for example, in the supply of electricity
or telephone) or the result of consumer-protection laws. For example, it is typical of
"common carrier" regulation to require firms to meet all demand at the set prices. If
the supply of a product is exhausted, the customer may take a "rain check" (a coupon
to purchase the good at the posted price at a later date); see Vires (1990, 1999) in this
respect.
84
K. G. Dastidar
and 5 we have the following:
Si (P1, P2 . . . . . Pi . . . . . Pn) =
0
if Pi > Pj for some j,
]--F(Pi)if Pi <_ Pj V j and P~ = Pi,
----
m
where k = 1, 2 . . . . . m and m < n,
F(Pi)
if Pi < Pj 'r
Define rr(P) = P F ( P ) - C ( F ( P ) ) and 7~(P, m) = ( 1 / m ) P F ( P ) C ( ( 1 / m ) F ( P ) ) , where 2 < m _< n. Clearly zr(P) is the profit going to a
firm when it supplies the market alone and ~ ( P , m) is the profit when it
shares the market equally with m firms.
We will consider a game of simultaneous price choices in such a set
up. Let El(P1, P 2 , . . . , Pi . . . . . Pn) be the payoff to the i-th firm. In our
framework clearly we have the following:
Ei (Pi, P2 . . . . . Pi . . . . . Pn) =
0
if Pi > Pj for some j,
~ ( P , m)
if Pi < Pj, V j and Pk = Pi,
where k = 1, 2 . . . . . m and m _< n,
[7r(P)
if Pi < Pj, Vj.
Define/3 (m) s.t. :~ (/3 (m), m) = 0. Also define/5 (m) s.t. ~ (/5 (m), m) =
x(/5(m)). Dastidar (1995) shows that for each m, /3(m) and/5(m) exist
and they are unique in [0, pmax). He also shows the following:
/3(n) < /5(n) ,
(la)
~ ( P , n) > 0, VP > / 3 ( n ) ,
(lb)
z?(P, n) _> 7r(P) > zr(P - s), VP E [/3(n),/5(n)] ,
(lc)
Dastidar (1995) shows that in a game of pure price competition, all
firms quoting the same price P E [/3(n), /5(n)] is a pure-strategy Bertrand
equilibrium. The logic of this is very simple. Consider the case of firm i.
When all other firms quote any price, P 6 [/5 (n),/5 (n))], the best response
for the i-th firm is to quote the same price. By quoting the same price it gets
~ ( P , n). We know that ~ ( P , n) > 0, VP 6 [/3(n),/5(n)] [see Eq. (lb)].
If it undercuts (i.e., charges P - e) it gets ~r(P - e). We also know that
Collusive Outcomes in Price Competition
85
7c(P - s) < 7v(P) < ~ ( P , n), VP 6 [/3(n),/3(n)] [from Eq. (lc)]. By
charging more it gets zero. Hence it is optimal for it to quote the same price.
Therefore, we get that all firms quoting the same price P E [/3(n),/5(n)]
is a pure-strategy Bertrand equilibrium. Since/3 (n) < /5 (n) [see Eq. (la)]
such equilibria are necessarily nonunique.
The intuition behind this result is the following. A unilateral price reduction from a price charged by all firms means that the price-cutting firm
has to supply the entire demand (given our Assumption 4). Since costs
are strictly convex this leads to a disproportionate increase in cost. In the
equilibrium range of prices a firm refrains from price reduction because
the increase in additional revenue (because of larger sales) is less than the
increase in costs. This is the reason for the existence of multiple equilibria and also why prices above marginal costs can be sustained in a Nash
equilibrium which is not possible under constant marginal costs due to the
well-known Bertrand paradox. We now introduce the following which will
be extensively used in our analysis. Let P* (n) = arg max,,_>0 ~ (P, n) and
pm = argmaxe_>0 :r(p, n). Also let Qm = F(pm).
Clearly P m is the monopoly price, Q m the monopoly quantity, and P * (n)
the joint-profit-maximizing price. That is, P* (n) is the price which maximizes industry profits when all firms quote the same price and share the
market equally. It is similar to the solution for a monopolist who employs n
plants. It may also be noted that our assumptions imply that pm and P* (n)
are unique.
Note that P* (n) may lie between/3 (n) and/5 (n). To illustrate our point,
we provide the following example in a homogeneous-product duopoly.
2.1 Example 1
Consider a single-shot game of price competition satisfying Assumptions
1 to 5. Let the demand be given by F ( P ) = 10 - P and costs by C ( Q i ) =
3Q 2 (where i = 1, 2). Routine calculations show that P* = 8, /3 = 6,
and/5 = 8.1818. From our discussions above, we know that both firms
quoting the same price in the range [6, 8.1818] is a pure-strategy Bertrand
equilibrium. Since P* (the joint-profit-maximizing price) lies between/3
and/5, our example shows that it is possible to sustain collusive outcomes
in a single-shot game of price competition. Figure 1 below graphically
portrays our example.
We now provide the general sufficient condition for P* (n) to lie between
K. G. Dastidar
86
~
p*
=8
t5
=8.18
pmax
p
= 10
Fig. 1
fi(n) and/5(n). We show that a lower bound on CII(Qi) serves as a sufficient condition. Intuitively it means that if the cost function is sufficiently
convex then P* will always lie in that range. We also show that if C'I(Qi)
is nonincreasing in Qi, it is easier to sustain collusion when the number of
firms rises.
3 The Results
In this section, we formally state our main results. Define:
k = arg maxo<e<pn, ax
f1~p ~ .
Proposition 1: If C11(Qi) > (2n/(n - 1 ) ) ( - 1 / U ( k ) )
[(1/n)Qm, Qmax] then P*(n) c [/3(n),/5(n)].
for all Qi
Proof." Given in the appendix.
Corollary 1: If F(P) is concave and if C11(Qi) > - 4 / F 1 ( 0 ) for all Qi
[(1/n)Qm, Qmax] then P*(n) ~ [/3(n),/5(n)] for any n.
Proof." Concavity of F(P) implies that k = 0. Since 2n/(n - 1) is de-
Collusive Outcomes in Price Competition
87
creasing in n and since n > 2 our claim follows immediately from Proposition 1.
[]
Comment: The above proposition and its corollary shows that if costs
are sufficiently convex then it is possible to sustain collusion in a oneshot game of price competition. A possible intuition for Proposition 1 is
as follows. Firstly, note that for our result to hold /5 (n) should be high
enough so that P*(n) falls below it. Now/5(n) is the price which equates
7r(P) with ~ ( P , n). At any price below/5(n), re(P) is less than 7?(P, n).
At any price higher than /5(n) (but less than pmax) it is the other way
round. That is/5(n) is the lowest price at which undercutting does not pay.
Note that if costs are very convex [i.e., C"(.) is sufficiently high] benefits
to undercutting (at lower prices and hence higher possible demand) are
relatively lower as costs increase disproportionately when a firm has to
supply additional demand. This implies that if costs are more convex then
/5(n) will be higher. We now argue that greater convexity of costs tends
to push up P*(n) though the effect of greater convexity on P*(n) is less
pronounced. The reason is as follows. There are two opposing effects of
more convexity on P* (n). Note that P* (n) solves :U (P, n) = 0. If costs are
more convex then marginal costs tend to be higher. This effect increases
P*(n) [see the first-order condition of maximization of J?(P, n) in the
appendix]. Also note that ~ ' ( P , n) is negatively sloping at the point where
z?t(P, n) = 0. From Eq. (3c) in the appendix it is clear that 77'(P, n) is
steeper if C"(.) is higher. This effect tends to decrease P* (n). These two
effects imply that the effect of more convexity on P* (n) is less pronounced
than that on/5 (n). This means if costs are very convex then/5 (n) will tend
to be higher than P*(n).
It may also be noted that we do not need any other conditions like
appropriate conjectural variations and/or supply functions. We have shown
that simple Bertrand conjectures together with sufficient convexity of costs
can support collusive outcomes.
We now state our second main result.
Proposition 2: If C"(Qi) is nonincreasing in Qi then it is easier to sustain
collusion when the number of firms rises.
Proof." Given in the appendix.
88
K.G. Dastidar
Comment: A possible intuition behind Proposition 2 is the following. 3 If
the number of firms rises both/5(n) and P*(n) tend to fall. We now argue
that/5 (n) falls less than P* (n) if the condition for Proposition 2 is satisfied.
Note that P*(n) (joint-profit-maximizing price) is the same price which a
multiplant monopolist, with n firms, will set. An increase in the number
of firms implies that the collusive price will fall since the monopolist who
increases the number of employed plants will face lower marginal costs
at each of his plants. It may be noted that /5(n) is the lowest price at
which undercutting does not pay (see the comment after Proposition 1).
The increase in n has two opposing effects on/5(n). On the one hand, due
to lower marginal costs at each plant there are more incentives for price
reduction. On the other hand, increasing the number of firms implies a
demand effect such that price reduction is penalized stronger since price
reduction implies more individual demand (under our assumption that a
firm must supply all demand it encounters) and thereby higher marginal
costs. A possible interpretation of Proposition 2 could then be that an
increase in the number of firms makes the demand effect more important
if the second derivative of the cost function does not increase with output.
If the demand effect is more important, then the effect of an increase in n
on /5(n) is less pronounced than that on P*(n). This means that with a
rising n both decrease but/5(n) tends to decrease less than P*(n). We give
below an example to illustrate Proposition 2.
3.1 Example2
Consider the following homogeneous-product oligopoly. Demand is given
by F(P) = 10 - P. Costs are C(Qi) = (3/2)Q/2. Routine calculations
lead to the following:
Case 1 (when n = 2): /3(2) = 4.28,/5(2) = 6.92, and P*(2) = 7.14;
Case 2 (when n = 3): /3(3) = 3.33,/5(3) = 6.66, and P*(3) = 6.66;
Case 3 (when n = 4): /3(4) = 2.72,/5(4) = 6.52, and P*(4) = 6.36.
In the above example when the number of firms is two, then the collusive
outcome cannot be sustained as a Nash equilibrium in a single-shot game
as the collusive price P* lies outside the Bertrand-equilibrium range. However, when n is three or higher collusive outcomes can be sustained.
3 I mn indebted to one of the referees for this intuition.
Collusive Outcomes in Price Competition
89
4 Conclusion
In this paper we derived a sufficient condition for the existence of collusive
outcomes in a single-shot game of price competition in a homogeneousproduct market. We also proved the counterintuitive result that under certain plausible conditions it becomes easier to sustain collusion in a one-shot
game of price competition when the number of firms increases.
Appendix
Proof of Proposition 1
For notational simplification we will denote/3(n) by/3,/3(n) by/3, and
P*(n) by P*.
We know that/3 solves zr(P) = 77(P, n) for P c [0, pmax). That is
/3 F(/3) - C(F(/3)) = 1_/3 F(/3) - C(1F(/3))
l't
/3--
n
(2)
Since C(.) is strictly convex,
--
>
n
(']lF(fi),.
F(P)1
n
/
",H
/
This implies that
/3 > C'(, n1 F ( / 5 ) ) .
(3a)
Now we have,
P* solves ~ ' ( P , n) = 0.
Note that
since
F'(P)<O,
P*> C'(1F(p*)].
k/t
]
(3b)
K.G. Dastidar
90
Now we have
~1'(P,n) =
/*" 1
\
Ii
~nF(P))F
(P)
lfpF"(P)+2FI(P)-C
I(FI(p))2CtI(1F(p))]
.
(3c)
12
Note that
tion 2.
pF'I(P) + 2 F ' ( P )
C"(.)
Since CI(-) and
< 0, since
> 0,
PF(P)
is concave by Assump-
ff11(p, n) < 0 if
F"(P)
> 0.
(3d)
From (3c) we have
fr'I(p,n) = l [FII(P)(p - c l ( 1 F ( p ) ) ) + 2FI(P)
I (F'(p))2C'I(1F(p)) ] .
(3e)
n
The above implies that
,(1
)
<OandP >C nF(P) .
~'"(P,n) <0ifF"(P)
(3f)
Therefore (3d) and (3f) together imply that
~ " ( P , n ) < 0 if
P - CI(1F(p))>_O.
Note that the function P - CI((1/n)F(P))
0, the function has a negative value. From
C'((1/n)F(P)) and P* > C((1/n)F(P*)).
A such that A - C((1/n)F(A)) = 0. Using
we have
for all P > A,
(3g)
is increasing in P. At P =
(3a) and (3b) we get /3 >
Therefore there exists a P =
(3g) and the above argument
~ " ( P , n) < 0.
Equation (3h) implies that for all prices greater than or equal to
(3h)
A, ~d(P, n)
Collusive Outcomes in Price Competition
91
is negatively sloping. From (3a) and (3b) we have that t5, p . > A. Therefore,
p* < /5 ,',
> ~1(/5, n) _< 0
t/1
-
\
/
-
',, # F ' ( # ) + F(P) - C ~ n F ( P ) ) F (P) < 0
,~
> F'(/5))[/5
,',
- C(1F(/5))]
',, /5 - C ' ( F ( [ ' ) )
F(/5)
>_
F'(P)
_< -F(/~)
[since U(.) < 0].
Therefore, from (2) we have the following:
n
n
',,
n
1
C(F(/5)
- C
F(f')
- C'(F(/5))
1
-
C(F
1
- C
F
n
f (/5)
> - - -
F
> - - F/(fi)
C'(F
(4)
F'(~)
From Taylor's theorem we know that 3 x E [ (1/ n ) F ( /5), (1/ n ) F ( [') +
((n - 1 ) / n ) F ( f i ) ] such that the following is true:
C
F(/5) +
f(/5)
n
=C(
1F(f'))+C(1F(P))E
n-
F(/5)]
1 tl
rn - 1
]2
+ ~C (x)[ n g(/5)
1C,(x ) n - 1F
2
n
Combining Eq. (5) with (4) we get:
(5)
92
K.G. Dastidar
/5>p.
1,,
> ~C (x)[
.: FC'(x)>_
n
F(/5)
n---1
] ->
---/7I(/5) '
,
wherex 6
wherex~
F(/5), F(/5)
F(/5),F
.
]
(6)
Note that from the definition o f k we get that - ( 2 n / ( n - 1))(1/F'(k)) is
the m a x i m u m value of the function - (2n / (n - 1)) ( 1 / F ~( P ) ) . We know that
/5 < p m and so F(/5) > F ( P m) = Qm. We a l s o h a v e V P , Qmax > F(P).
Therefore
..
x ~
f(/5), f(/5)
.
Hence if Ctt(Qi) > - ( 2 n / ( n - 1))(1/F'(k)) for all Qi ~ [(1/n)Q m,
Qmax] then from (6) we have P* _</5. Hence the proposition follows. []
Proof of Proposition 2
From Proposition 1 we know that with given demand and cost functions,
the sufficient condition for collusion to be sustained is
C'(Qi) > - -n-1
for all Qi 6
En Qm,
1
Qmax .
The above condition is more likely to hold when the number of firms
increases. The reasoning is as follows. The right-hand side of the condition decreases with n. Hence the condition is more likely to be fulfilled when n increases. However, since ( l / n ) Q m falls with n, the range
[ ( l / n ) Qm, Qmax] for which the condition should hold increases with n.
Note that, the condition must hold for lower Qi if it holds for higher ones,
since C" (Qi ) is nonincreasing in Qi.
[]
Acknowledgements
I am indebted to Anjan Muldaerji and Xavier Vives for guidance and encouragement.
Comments by Amit Bhaduri, Satish Jain, Prabhat Patnaik, Kunal Sengupta and two
anonymous referees of this journal were extremely helpful. The paper was first written
when the author was a "visiting scholar" at the department of Economics, Harvard
University. Fellowship Grant from the Ford Foundation (fellowship no. 15976021,
Grant No. 890-0264-1) is gratefully acknowledged. An earlier version was circulated
as Dastidar (1999). The usual disclaimer applies.
Collusive Outcomes in Price Competition
93
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Address of author: Krishnendu Ghosh Dastidar, Centre for Economic Studies and
Planning, School of Social Sciences, Jawaharlal Nehru University, New Delhi 110067,
India (e-mail: [email protected])