Abstract
In this chapter, we subdivide the oligopolistic sector into several separable groups and adapt our concept of oligopolistic equilibrium by simplifying accordingly both the market share and the market size constraints. In the following we introduce another path to further simplify firms’ conjectures: each firm anticipates the income to be spent in its group as if all groups were independent. Then, we further exploit the subdivision into groups by restricting our analysis to the limit case where perfect substitutability holds within each group. Assuming Cournot competition within each group leads to the concept of Cournotian monopolistic competition equilibrium. Through an example and a simple existence proposition, we show that this concept is less demanding than the Cournot-Walras equilibrium concept. In the last section, we present an empirical application to an industry divided into two groups, a dominant group and a competitive fringe, and, finally, examine in a theoretical example the limit of collusion.
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Notes
- 1.
One can refer here to the empirical study of Hottman et al. (2016) who observe (i) “that the typical sector is characterized by a few large firms with substantial markets shares and a competitive fringe of firms with trivial markets shares, and (ii) that “appeal” (quality or taste) explains in large part the success of the firms in the dominant group. We should add that the conduct, in particular a collusive conduct, of the dominant firms reinforces this success.
- 2.
As Dixit and Stiglitz (1977), we prefer to use the more general term “group” than the more specific term “industry.” As well emphasised by Neary (2004, p. 161), “previous writers had debated the appropriate definition of an “industry,” or, in Chamberlin’s preferred term, a “group.” Typically, definitions were given in terms of cross-elasticities of demand, sometimes of both direct and inverse demand functions. […] DS cut through all this fog: instead of restricting the demand functions by imposing arbitrary limits on inter- and intra-industry substitutability, they made a single restriction on the utility function, which implies that (in symmetric equilibria) all products within an industry should have the same degree of substitutability with other goods.”
- 3.
We denote \({\mathbf {x}}^{k}\equiv \left ( x_{i}^{k}\right ) _{i=1}^{N_{k}}\in \mathbb {R}_{+}^{N_{k}}\) and \(X^{k}:{\mathbf {x}}^{k}\mapsto X^{k}\left ( \mathbf { x}^{k}\right ) \in \mathbb {R}_{+}\).
- 4.
\(s_{i}^{k}\) is the elasticity (in absolute value) of \(x_{i}^{k}/X_{k}=\) \(x_{i}^{k}/H_{i}^{k} \left ( {\mathbf {p}}^{k}\mathbf {,}X_{k}\right ) \) with respect to \( p_{i}^{k}/P_{k}=p_{i}^{k}/\partial _{X}e^{k}\left ( {\mathbf {p}}^{k}\mathbf {,} X_{k}\right ) \), where P k denotes the shadow price \(\partial _{X}e^{k}\left ( {\mathbf {p}}^{k}\mathbf {,}X_{k}\right ) \) of the composite good k (see d’Aspremont and Dos Santos Ferreira, 2016, Appendix).
- 5.
Notice that the elasticities of the two frontiers in the space \( x_{i}^{k}\times p_{i}^{k}\) (with quantity and price in the horizontal and vertical axes, respectively, according to the Marshallian tradition) are here taken as the two corresponding expressions for \(-\left ( dp_{i}^{k}/dx_{i}^{k}\right ) \left ( x_{i}^{k}/p_{i}^{k}\right ) \). In d’Aspremont and Dos Santos Ferreira (2020, equations (8) and (9)) we have adopted the opposite convention, using \(-\left ( dx_{i}^{k}/dp_{i}^{k}\right ) \left ( p_{i}^{k}/x_{i}^{k}\right ) \) for the elasticities of the two frontiers.
- 6.
So, \(\partial _{i}^{-}P^{k}\left ( p,...,p\right ) =1\) and \(\partial _{i}^{+}P^{k}\left ( p,...,p\right ) =0\) in case (i) and \(\partial _{i}^{-}X^{k}\left ( x,...,x\right ) =1\) and \(\partial _{i}^{+}X^{k}\left ( x,...,x\right ) =0\) in case (ii). As a consequence, \(\sigma _{i}^{k}\equiv -\epsilon _{k}\widehat {D}^{k}\left ( P^{k}\left ( p,...,p\right ) ,\mathbf {P} ^{-k}\left ( {\mathbf {p}}^{-k}\right ) ,Y\right ) /\alpha _{i}^{k}\) for a downward price deviation in case (i) and \(\sigma _{i}^{k}\equiv -\epsilon _{k}\widehat {D}^{k}\left ( P^{k}\left ( p,...,p\right ) ,{\mathbf {P}}^{-k}\left ( {\mathbf {p}}^{-k}\right ) ,Y\right ) \epsilon _{i}P^{k}\left ( p,...,p\right ) \) for a downward quantity deviation in case (ii).
- 7.
The granularity hypothesis concerning the macroeconomic effects of the behaviour of large firms, was introduced by Gabaix (2011) to explain aggregate fluctuations when all productivity shocks are firm-level idiosyncratic shocks. It was then applied to a multi-sector model of international trade by di Giovanni and Levchenko (2012).
- 8.
Gravity equations have been introduced by Tinbergen (1962) for analysing bilateral trade flows.
- 9.
More generally, by the consumer’s first-order condition,
$$\displaystyle \begin{aligned} \frac{p_{i}^{k\ast }x_{i}^{k\ast }}{Y^{k\ast }}=\alpha _{i}^{k\ast }\epsilon _{X}e^{k}\left( {\mathbf{p}}^{k\ast },X_{k}\left( {\mathbf{x}}^{k\ast }\right) \right) \text{,} \end{aligned}$$where \(\epsilon _{X}e^{k}\left ( {\mathbf {p}}^{k\ast },X_{k}\left ( \mathbf {x} ^{k\ast }\right ) \right ) \) is not necessarily equal to one.
- 10.
- 11.
For references see Roberts and Sonnenschein (1977), Novshek and Sonnenschein (1978), Mas-Colell (1982), Gary-Bobo (1989), Codognato and Gabszewicz (1993). A recent paper by Azar and Vives (2020) also aims at building a tractable general equilibrium model of oligopoly with large firms, as we do, but using Cournot-Walras equilibrium and allowing for ownership diversification.
- 12.
The pricing scheme would now be a continuous increasing function Ψ from \(\mathbb {R}_{+}^{N}\) to \(\mathbb {R}_{+}^{K}\), associating with each vector of price signals ψ a single market price vector Ψ(ψ). As mentioned in Chap. 2, the essential property to get the Cournot solution is the manipulability of the market prices by each individual producer.
- 13.
We also make this assumption in d’Aspremont et al. (2010).
- 14.
In Costa and Dixon (2011), aggregate demand is supposed to depend also on government public expenditure. Their purpose is to examine the effects of fiscal policy.
- 15.
\(\lambda =\left ( a\mu _{1}^{P}-bY\right ) /\mu _{2}^{P}\), with \(\mu _{1}^{P}\) and \(\mu _{2}^{P}\) the first and second moments of the price distribution.
- 16.
- 17.
See Colacicco (2015) for more discussion and a survey of various applications of the GOLE approach to topics in international trade.
- 18.
- 19.
For multistage demand systems see Hausman et al. (1994).
- 20.
Folgers’ competitive toughness parameter is estimated in a tight range between 0.12 and 0.14. Maxwell House’s toughness parameter are in an equally tight range of 0.02–0.04.
- 21.
If Y∕P < bP −σ, all the income is spent in the oligopolistic sector, because the marginal utility of the composite good X deflated by its price P is larger than the corresponding marginal utility of the numeraire good, namely 1.
- 22.
This markup is not necessarily close to zero, because the goods produced by competitive firms may be sufficiently differentiated among themselves to keep each producer so to say in its own dedicated niche.
- 23.
This amounts to choose the unit of the goods produced by the collusive firms so as to obtain the marginal cost
$$\displaystyle \begin{aligned} c^{1}=\frac{2^{\left( 1-s\right) /\left( 1-s^{1}\right) }\left( \sigma -1\right) +P_{2}^{1-s}\left( s-1\right) }{2^{\left( 1-s\right) /\left( 1-s^{1}\right) }\sigma +P_{2}^{1-s}s}\text{,} \end{aligned}$$and to choose the unit of the numeraire good so as to obtain the parameter value
$$\displaystyle \begin{aligned} b=2^{\left( s^{1}-s\right) /\left( s^{1}-1\right) }\left( 2^{\left( 1-s\right) /\left( 1-s^{1}\right) }+P_{2}^{1-s}\right) ^{\left( \sigma -s\right) /\left( 1-s\right) }\text{.} \end{aligned}$$
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d’Aspremont, C., Dos Ferreira, R.S. (2021). Competition within and Between Groups of Firms. In: The Economics of Competition, Collusion and In-between. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-63602-9_3
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