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 Ema...
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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.


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,…..: -2-

q(n) =

n A A ⋅ , p(n) = ( n + 1) b ( n + 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 *


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 ,



(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 ,


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


of an homogeneous product by the existence of goods or boundaries “close enough” to the original one in the product space.


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.



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



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 ,


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 −


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


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 −


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:


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


 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 −


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


,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


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).


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.




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.


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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b e




q(2) d



Figure 1. Demand curves supporting collusive market configurations

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Figure 2. Types of markets

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