Download Product differentiation, kinked demand and collusion

Product differentiation, kinked demand and collusion
Antonio D’Agata
Department of Political and Social Science
University of Catania
Via Vittorio Emanuele, 8
95131 Catania (Italy)
Tel. +39 095 70305209
[email protected]
Abstract. We first show that in homogeneous product Cournot oligopolies with kinked demand the
market configurations supported by collusion depend upon how kinky the demand curve is. Specifically,
the kinkier the demand curve, the closer the collusive market configuration set is to the monopolistic one.
Then, we extend the analysis to the (linear version of) 1979 Salop’s model and show that three different
kinds of kinked market demand curves can emerge, and, therefore, three different collusive market
configuration sets can be supported. As an application of the previous results, we show that collusion may
confirm the Principle of Minimum Product Differentiation even in one-shot market games.
Keywords: Product differentiation, kinked demand curve, collusion, Cournot oligopoly.
JEL Classification numbers: D43, L13.
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1. Introduction
Collusion in homogeneous markets with price- or quantity-setting firms is well known to be not
stable and scholars have spent great effort in trying to uncover facilitating practices. The kinked
demand approach by Sweezy (1939) and Hall and Hitch (1939) has greatly contributed to
understand the stability of collusion, and still attracts great attention (see Maskin and Tirole
(1988), Bashkar (1988), Sen (2004), Lu and Wright (2010), Garrod (2012)). This approach has
been criticized for not providing a theory of price determination and for its peculiar behavioral
non-Nash-like hypothesis on competitors (see Stigler (1947), Primeaux and Bomball (1974),
Bashkar, Machine and Reid (1991)). Salop (1979) addresses these criticisms by showing that,
within a differentiated product model, kinks in demand may emerge endogenously and with
Nash-like assumptions on competitors. Salop does not deal with collusion and successive works
have dealt only with collusion in differentiated markets and repeated interaction (Friedman and
Thisse (1992), Jehiel (1992)). Thus, two issues are still open: 1) whether kinked demand in
homogeneous markets and with Nash-like firms support collusive market configurations, and 2)
to what extent collusive market configurations can be supported by the endogenously determined
kinked demand curves emerging in differentiated product models. Section 2 deals with the first
issue by showing how and under which conditions kinked demand can support oligopolistic
equilibria in homogeneous product markets with Nash-like firms. Section 3 shows that in a linear
differentiated product model three different kinds of kinked demand can emerge and analyses the
implications of this result for the sustainable collusive market configurations. Homogeneity of
products is problematic in product differentiation models, thus Section 4 deals with the
implications of the previous analysis for the Minimum Product Differentiation Principle. Section
5 provides some final remarks.
2. Kinked demand and collusion with Nash-like firms
Consider the market of a homogeneous product x with the linear function p = A – bq, A,b > 0.
Production is carried out at zero cost and firms are quantity-setting and follow the Cournot-Nash
conjecture. Denote by p(1) and q(1) the monopolistic price/quantity configuration and for n =
2,3,4,…., a n-Cournot (symmetric market) equilibrium is a price/quantity couple (p(n), q(n)) such
that p(n) = A – bq(n) and q(n)/n is the profit maximizing level of production for each firm, given the
aggregate production level (n-1)q(n)/n of all other firms. In general, for n = 1,2,…..:
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q(n) =
n
A
A
⋅ , p(n) =
( n + 1) b
( n + 1)
(1)
Collusive firms in a n-firm oligopoly seek to ensure the lowest possible production level in
{
}
interval [q(1), q(n)]. Set Cx (n) = ( p, q) p = A − bq, q (1) ≤ q ≤ q ( n ) is the collusive interval for the nfirm oligopoly in market x with demand function p = A – bq. For (p*, q*)∈Cx(n),
{
}
C x (n; p*, q*) = ( p, q ) ( p, q) ∈ Cx (n), q* ≤ q ≤ q ( n ) is the competitive portion of Cx(n) with respect to
market configuration (p*, q*). The next result follows immediately from (1), while Fact 2 follows
immediately from Fact 1:
Fact 1. Every (p*, q*) ∈ Cx (n) is a n-Cournot equilibrium for a market with demand function p
= A(q*,n) – b(q*,n)q, where A( q*,n ) = (n + 1)( A − bq*) and b ( q*, n ) =
n( A − bq*)
.
q*
Fact 2. If (p*, q*), (p**, q**) ∈ Cx (n) with q* ≤ q**, then b(q*,n) ≥ b(q**,n) ≥ b, where the last
inequality holds as an equality only if q** = q(n).
The following result, whose proof is provided in the Appendix, highlights the role of kinks in
supporting collusive behavior in homogeneous markets with Nash-like firms:
Fact 3. If (p*, q*) ∈ Cx (n) , then (p*, q*) is a n-Cournot equilibrium for the market with demand
function:
 A − bq for 0 ≤ q ≤ q *
p=
 A '− b ' q for q ≥ q *
(2)
with A' – A = (b' – b)q*, A' ≥ A(q*,n) and b' ≥ b(q*,n), where A(q*,n) and b(q*,n) are defined in Fact 1.
Example. Figure 1 illustrates Fact 2 for two price/quantity configurations in Cx (2) :
(a) the monopolistic configuration p (1) =
A
A
and q (1) =
(point b) is supported as a duopolistic
2
2b
equilibrium by the market demand curve Abd defined by function:
 A − bq for 0 ≤ q ≤ q (1)

p = 3
(1)
 2 A − 2bq for q ≥ q ,
(3)
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(b) configuration pˆ =
2
3 A
A and qˆ = ⋅ (point e) is supported as duopolistic equilibrium by the
5
5 b
market demand curve Aef defined by function:
 A − bq for 0 ≤ q ≤ qˆ

p = 6
4
 5 A − 3 bq for q ≥ qˆ ,
(4)
Fact 3 ensures that these configurations are supported as duopolistic equilibria also for demand
curves with steeper competitive regions (for the concepts of competitive and monopoly regions of
kinked demand curves, see Salop (1979 p. 143)).
FIGURE 1 AROUND HERE
Next result is an immediate consequence of Facts 2 and 3.
Fact 4. Let (p*, q*) ∈ Cx (n) be a n-Cournot equilibrium for the market with a given demand
function defined by (2), then (p**,q**)∈ Cx (n, p*, q*) is a n-Cournot equilibrium for the market
with demand function:
 A − bq for 0 ≤ q ≤ q **
p=
 A ''− b ' q for q ≥ q **
where A′′= A + (b′ – b)q** and b' is the slope of the specific demand function given by (2).
Referring to Figure 1, Fact 4 means that, for example, since demand curve Abd supports (p(1),
q(1)) as duopolistic equilibrium, any kinked demand curve obtained by curves D and D'
intersecting along segment bg will also support a duopolistic equilibrium at the kink.
Facts 2 and 3 can be reinterpreted as follows. Consider, for example, a duopoly facing an initial
demand function p = A – bq. The monopolistic configuration (p(1), q(1)) is not a stable collusive
equilibrium for the Cournotian duopoly, but it can become so if at this configuration an
“appropriate” kink, as defined by (3) and illustrated by curve Abd in Figure 1, is somewhat
generated. Similar argument for configuration ( pˆ , qˆ ) with reference to kinked demand (4) and
illustrated by curve Aed. Now the issue is how to generate kinks. Next section shows that in a
differentiated product model three kinds of kinks can be generated on the demand curve market
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of an homogeneous product by the existence of goods or boundaries “close enough” to the
original one in the product space.
3.
Product differentiation, kinked demand and collusion
Consider linear version of 1979 Salop’s model, with consumers uniformly distributed along the
segment L = [ x, x ] , each buying only one unit of good. Set L defines also the product space, so
each point in L describes also a product. Consumer c ∈ L has a reservation price for product x∈L
equal to R(c, x) = max [0, A – 2b|c – x| ].1 Consumer c buys product x at price px only if R(c,x) - px
≥ 0. If products x1 and x2 in L are sold at prices p x1 and p x2 , then consumer c buys product x1 if
R(c, x1 ) - p x1 ≥ R(c, x2 ) – p x2 .
Our aim is to provide a complete list of kinked demand curves and, consequently, of potentially
collusive configurations which are generated by the existence of differentiated products or
boundaries “close enough” to the relevant one. Four possible cases can occur:2 (a) isolated
market; (b) market with unilateral boundary constraint; (c) market with unilateral competition;
(d) market with bilateral competition; (d) market with unilateral competition and boundary
constraint.3
Figure 2 illustrates cases (b)-(e), case (a) being obvious. Market of product x1 exhibits a
boundary constraint, market of product x2 ( x3 ) shows unilateral (bilateral) competition, finally,
market of product x4 exhibits unilateral competition with boundary constraint. We show that in
these markets demand curves show different kinks in terms of the slopes of the competitive
regions. Then, on the basis of the results in the previous section, we show that different kinks in
demand curves have different impact on the possibility for collusion. Since all claims are
obtained by standard calculations and as direct implications of Facts 1-4, the proofs of the results
are omitted.
FIGURE 2 AROUND HERE
1
Parameters in R(c,x) are chosen in order to allow a direct comparison with the results in the previous section.
A fifth case, that is markets with bilateral boundary constraints, is not considered as empirically implausible and
analytically obvious once taken into account case (b) below.
3
In Salop’s circular city model only cases (a), (c) and (d) can occur.
2
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(a)
Isolated markets. The inverse demand function of the isolated market of product x is:
p = A − bq .
Next market types exhibit interdependencies with other products, so their market demand curves
may have kinks with monopoly as well as competitive regions (see Salop (1979)).4
(b)
Market with boundary constraint. Product x1 ’s inverse demand curve is:
 A − bq1 for 0 ≤ q1 ≤ 2 H1
p1 = 
 A + 2bH1 − 2bq1 for q1 ≥ 2 H1 ,
(5)
where H1 is the distance of product x1 from the constraining boundary.
(c)
Market with unilateral competition. Product x2 ’s inverse demand curve is:
 A − bq2 for 0 ≤ q2 ≤ q2
p2 =  2b( x3 − x2 ) + 2 A + p3 4b
− q2 for q2 ≥ q2

3
3
where q2 = 2( x3 − x2 −
(6)
A − p3
) is the demand level at which the market of x2 overlaps the market
2b
of product x3 (see Salop (1979, Figure 4)).
(d)
Market with bilateral competition. Under symmetry of markets x2 and x4 with respect to
market x3 (that is: x4 − x3 −
A − p4
A − p2 5
= x4 − x3 −
), the inverse demand curve of x3 is:
2b
2b
 A − bq3 for 0 ≤ q3 ≤ q3

p3 =  2b( x4 − x2 ) + ( p4 + p2 )
− 2bq3 for q3 ≥ q3

2
(7)
A − p2
) is the demand level at which the market of product 3 overlaps
2b
products x2 and x4 markets.
where q3 = 2( x4 − x2 −
(e)
Market with unilateral competition and boundary constraint. Under symmetry of markets
A − p3
x3 and x4 with respect to market x3 and to the boundary (that is: H 4 = x4 − x3 −
, where H4
2b
is the distance of product x4 from the constraining boundary),6 the inverse demand of x4 is:
4
We do not consider the supercompetitive region (Salop (1979, p. 143)).
Without symmetry, market of product x3 overlaps markets x2 and x4 at different prices, so demand curve exhibits
two kinks. The kink associated to the low level of production is the one in case (c), while the kink associated to the
highest production level is the one given by (7). By contrast, symmetry ensures that market of product x3 overlaps
markets x2 and x4 at the same price, so demand curve exhibits only one kink of the kind indicated by (7). From the
point of view of kink types and opportunities of collusion, symmetry does not yield any loss of information.
5
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 A − bq4 for 0 ≤ q4 ≤ q4
p4 = 
 2b(2 H 4 + x4 − x3 ) + p3 − 4bq4 for q4 ≥ q4
where q4 = 2 H 4 (= 2( x4 − x3 −
(8)
A − p3
) is the demand level at which the market of product 4
2b
overlaps the product 3 market and the boundary.
Once noticed that, from Fact 1, b( q
(1)
,n )
= nb , by Facts 1, 3 and 4 and from (1),(5)-(8) it follows:
Proposition: The following assertions hold true:
(1) in isolated markets (product x), only the n-Cournot equilibrium configuration ( p ( n ) , q ( n ) )
defined by (1) can be supported as a n-Cournot equilibrium;
(2) in markets with boundary constraint or with bilateral competition (products x1 and x3 ), any
configuration in set C xi (n,
2
n A
A,
) (i = 1, 3) can be supported as a n-Cournot
n+2 n+2 b
equilibrium;
(3) in markets with unilateral competition (product
C x2 ( n ,
x2 ), any configuration in set
4A
3nA
,
) can be supported as a n-Cournot equilibrium;
3n + 4 (3n + 4)b
(4) in markets with unilateral competition and boundary constraints (product x4 ), any
configuration in set C x4 ( n,
4A
nA
,
) can be supported as a n-Cournot equilibrium.
n + 4 ( n + 4)b
To be more specific, for example, in the isolated duopolistic market x, only the duopolistic
configuration p1(2) =
A (2) 2 A
, q1 = ⋅ can be supported as an 2-Cournot equilibrium, and so on. In
3
3 b
markets with boundary constraints or with bilateral competition any price/quantity configuration
in set C xi (2, p (1) , q (1) ) , i = 1,3, can be supported as a duopolistic equilibrium. In markets with
unilateral competition any price/quantity configuration (p*, q*)∈ Cx2 (2, pˆ , qˆ ) can be supported as
a duopolistic equilibrium, where pˆ =
2
3 A
A and qˆ = ⋅ as introduced in Example (b). Finally, in
5
5 b
6
The reason for assuming symmetry is, mutatis mutandis, similar to the one provided for the previous case (d) (See
footnote 5). Without symmetry, in this case the kink associated at the low level of production may be either of type
(b) or of type (c).
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markets with unilateral competition and boundary constraints any price/quantity configuration in
set C x4 ( n, p (1) , q (1) ) can be supported as a n-Cournot equilibrium, with n ≤ 4.
4. The Minimum Product Differentiation Principle in one-shot games
We show, by means of an example, that the results in Section 3 imply that oligopolists may
choose the same product even in one-shot games, unlike the extant literature (See Section 1).
Example. Consider the model in Section 3 with L =
3A
3A
, where x = 0 and x =
. Firms first
2b
2b
choose the location, then the level of production (Friedman and Thisse (1993)). There are six
firms, from a to f, each firm producing only one product. We focus on two scenarios defining
symmetric equilibria. First, consider the case in which three products, x1 , x2 , x3 , are produced.
Good x1 (resp. good x2 , resp. good x3 ), is produced by firms a and b (resp. firms c and d, resp.
firms e and f). Suppose also that, xi =
(2i − 1) A
, i = 1,2,3, and that duopolies choose the
4b
price/quantity configuration at the kink, that is qi =
and equal to π =
A
A
and pi = . Firms’ profits are identical
2b
2
A2
. Only markets with bilateral competition and markets with unilateral
8b
competition exist, so from Proposition (2) and (4) in the previous section, the levels of production
are optimal, given the location of plants. Consider now the case characterized by the
monopolistic production of six products, xi = 1,2,3,4,5,6, located as follows: xi =
(2i − 1) A
, where
8b
good x1 is produced by firm a, good x2 by firm b, and so on. The common monopolistic price is
again p =
A
A
and the associated level of production for each product is q = , hence the total
2
4b
2
profits for each firm are once again equal to π ' =
A
. It is immediate to check that also this
8b
scenario defines an equilibrium in quantities. Therefore, the former collusive scenario is a
symmetric equilibrium in both quantities and locations.
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5.
Conclusions
Section 2 deals only with market configurations which can potentially be supported by collusion.
Section 3 shows that actual market configurations depend upon the specific position of the kink
in the competitive portions of the collusive sets Cxi (n) , if any. This, in turn, depends upon the
existence of boundaries or products “close enough” in the product space (see (5)-(8)). The
example provided in Section 4 shows that it may be optimal for firms to produce the same
product. Our analysis can deal with the case of multiproduct firm (see, for example, Brander and
Eaton (1984)). This extension could suggest that location can be used strategically for supporting
collusion.
APPENDIX
Lemma. Consider a market with demand function:
 A − bq for 0 ≤ q ≤ q *
p=
 A '− b ' q for q ≥ q *
where A – A' = (b – b')q* and q* ≥ 0. Then (p*, q*) is an n-Cournot equilibrium if
n
A'
n
A
⋅ ≤ q* ≤
⋅ .
( n + 1) b '
( n + 1) b
Proof. The assertion follows from the fact that inequalities
n
A'
n
A
⋅ ≤ q* ≤
⋅ imply that
(n + 1) b '
( n + 1) b
∂π i (q * / n)
∂π i '(q * / n)
= A − (n + 1)bqi ≥ 0 and
= A '− (n + 1)b ' qi ≤ 0 , where
∂qi
∂qi
π ( qi ) = ( A − b ( n + 1) qi ) qi and π '( qi ) = ( A '− b '( n + 1) qi ) qi .
n A
⋅ . By the fact that
Proof of Fact 3. Let (p*, q*) ∈ Cx(n), then q (1) ≤ q* ≤ q ( n ) =
n +1 b
q* =
n
A( q*, n )
n
A( q*, n )
n A
⋅ ( q*,n ) it follows that
⋅ ( q*, n ) = q* ≤
⋅ . By Lemma, the claim is valid for
(n + 1) b
( n + 1) b
n +1 b
A' = A( q*, n ) and b ' = b ( q *,n ) . For A' ≥ A( q*, n ) and b ' ≥ b ( q*, n ) with A' – A = (b' – b)q* the inequalities
A ' A − bq *
n
A'
n A
⋅ ≤ q* ≤
⋅ a fortiori have to be satisfied, because
=
+ q *.
(n + 1) b '
n +1 b
b'
b'
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- 10 -
D'
A
(1)
p
D
C(2)
b
e
p(2)
g
q(1)
q(2) d
f
c
Figure 1. Demand curves supporting collusive market configurations
- 11 -
A
Figure 2. Types of markets
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