Abstract
In this chapter we argue, using a simple model, that Cournot’s founding contribution to oligopoly theory should not be reduced to quantity competition but viewed instead as introducing a particular competitive conduct of the producers, between the two extreme conducts that will be singled out much later by Bertrand’s critique: collusion and pure price competition. We then analyse how producers can coordinate their conduct through “facilitating practices” of which the best-price guarantee is an example, and so implement Cournot’s regime of competition. As a second factor shaping the intensity of competition, we let producers differentiate their products either to attract different types of consumers or to respond to their taste for variety. When products are neither too-close substitutes nor too-close complements, the collusive regime may become enforceable, which was excluded in Cournot’s case of non-cooperative producers of a homogeneous good. The last factor we examine as a source of producers’ market power is concentration, looking at the way it varies with the number of firms in a symmetric context, and then how it varies with the dispersion of market shares.
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Notes
- 1.
Cournot adopts a dual approach in his complementary monopoly model (Sonnenschein, 1968), with which he studies producers’ concurrence (Cournot, 1838, chap.IX), by contrast with producers’ competition (ibid., chap.VII). Two producers sell goods that can only be used if combined, say one unit of each to get one unit of a composite good, so that there is no point in distinguishing x i and x j. Instead of using the inverse demand function \(D^{-1}\left ( x_{i}+x_{j}\right ) \), one can now use the demand function \(D\left ( p_{i}+p_{j}\right ) \) for the quantity of the composite good demanded at the total price p i + p j where p i and p j are the prices to be paid to the two producers (marginal costs being again assumed nil). By duality, the Cournot solution in quantities for the homogeneous duopoly can be translated into a Cournot solution in prices for the complementary monopoly.
- 2.
We read “by properly adjusting his price” in the English translation of Cournot’s Researches (by N. Bacon 1897), but this translation is incorrect and has confused prominent readers who did not have access to the original French text (see Magnan de Bornier, 1992).
- 3.
The minimum of the two prices in the market size constraint may be viewed as a particular “pricing scheme” that does not allow upward price manipulability.
- 4.
To define an interval in the real line, we use parentheses \(\left ( .\right ) \) or brackets \(\left [ .\right ] \) to indicate whether the endpoints are excluded or included, respectively.
- 5.
For example, Kalai and Satterthwaite (1994) and Doyle (1988) get the implementation of the collusive price (supposing that firms use price strategies only, ignoring their rivals’ market shares). By introducing discount possibilities below list prices in a second stage, though, Holt and Scheffman (1987) get the Cournot price as the maximal implementable price. In a specific model, d’Aspremont and Dos Santos Ferreira (2005) obtain by combining the “most-favoured-customer” and the “meet-or-release” clauses, the Cournot solution as the subgame perfect equilibrium of a two-stage duopoly game. An experimental approach to the effects of facilitating practices has been initiated by Grether and Plott (1984).
- 6.
In the whole of this chapter, we keep Cournot’s (provisional) assumption of zero production costs. An equivalent assumption would involve identical linear production costs cx, allowing to use the transformations \(p-c= \widetilde {p}\) and \(\widetilde {D}\left ( \widetilde {p}\right ) =D\left ( \widetilde {p}+c\right ) \). Firm i’s profit is then \(\widetilde {p}_{i}x_{i}\) if \(x_{i}\leq \widetilde {D}\left ( \widetilde {p}_{i}\right ) /2\) and \( \widetilde {p}_{i}\widetilde {D}\left ( \widetilde {p}_{i}\right ) /2-c\left ( x_{i}-\widetilde {D}\left ( \widetilde {p}_{i}\right ) /2\right ) \), with a cost proportional to the excess of output over demand.
- 7.
In Anderson et al. (1992, chap. 3) conditions are given to show how a representative consumer’s utility function and the corresponding demand function can be derived from an underlying population of consumers making discrete choices (and conversely). The special case where this representative utility is CES is also considered.
- 8.
Vectors are denoted in bold.
- 9.
We use the simplifying notations:
$$\displaystyle \begin{aligned} \partial _{i}F\left( \mathbf{x}\right) =\frac{\partial F\left( \mathbf{x} \right) }{\partial x_{i}}\text{ and }\epsilon _{i}F\left( \mathbf{x}\right) = \frac{\partial F\left( \mathbf{x}\right) }{\partial x_{i}}\frac{x_{i}}{ F\left( \mathbf{x}\right) }\text{.} \end{aligned}$$ - 10.
A second limit case is the Leontief case of perfect complementarity (s = 0), where the quantity and price aggregators are respectively given by \(X\left ( \mathbf {x}\right ) =\min \left ( \mathbf {x}\right ) \) and \(P\left ( \mathbf {p} \right ) =\sum _{i}p_{i}\). A third limit case is the Cobb-Douglas case (s = 1 ), where the quantity and price aggregators are respectively given by \( X\left ( \mathbf {x}\right ) =\prod \nolimits _{i}x_{i}\) and \(P\left ( \mathbf {p} \right ) =\prod \nolimits _{i}p_{i}\).
- 11.
The curves are computed for n = 2, s = 10 and linear demand \(D\left ( P\right ) =2-P\). The collusive solution is given by the monopoly price and quantity P m = X m = 1 and, for each duopolist, \(p^{\text{m} }=0.5^{1/\left ( 1-s\right ) }\simeq 1.08\) and \(x^{\text{m}}=0.5^{s/\left ( s-1\right ) }\simeq 0.46\).
- 12.
The relative position of the two frontiers would be inverted in the case of complementary goods. The two frontiers are reminiscent of the two demand curves considered by Chamberlin: the curve DD’, showing “the falling off in sales which would attend an increase in price, provided other prices did not also increase,” and the curve dd’, showing “the demand for the product of any one seller at various prices on the assumption that his competitors’ prices are always identical with his,” the latter “evidently [] much less elastic than the former” (Chamberlin, 1933, p. 90). Chamberlin’s curves are explicitly the branches of the kinked demand curve of Hall and Hitch (1939, pp. 23–24 and 29n) and, implicitly, of Sweezy (1939, p. 569).
- 13.
In the complementary monopoly regime, where producers are concurring, “an association of monopolists, working for their own interest, in this instance will also work for the interest of consumers, which is exactly the opposite of what happens with competing producers” (Cournot 1838 [1897], p. 103). See also Ellet (1839, pp. 79–80) for a similar observation in a spatial model of complementary monopoly.
- 14.
Equilibrium prices are always lower at price equilibria than at quantity equilibria, independently of goods being substitutable or complementary. This is consistent with Vives (1985).
- 15.
See also d’Aspremont et al. (2000b).
- 16.
A similar example has been developed for the linear-quadratic utility case by d’Aspremont and Motta (2000b).
- 17.
The condition is \(\left ( n-1\right ) /2\leq a/\sqrt {\phi }-1\leq n\). The least integer satisfying this condition is \( \underline {n}=\left \lceil a/ \sqrt {\phi }-1\right \rceil \), and the greatest integer \(\overline {n} =\left \lfloor 2a/\sqrt {\phi }-1\right \rfloor \). Hence, \(\overline {n}- \underline {n}\leq \left ( 2a/\sqrt {\phi }-1\right ) -\left ( a/\sqrt {\phi } -1\right ) =a/\sqrt {\phi }\leq \underline {n}+1\), or \(\overline {n}= \underline {n }+k\), with \(k=0,1,\ldots , \underline {n}+1\).
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d’Aspremont, C., Dos Santos Ferreira, R. (2021). Modelling the Intensity of Competition. In: The Economics of Competition, Collusion and In-between. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-63602-9_1
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