On the social (sub)optimality of divisionalization under product differentiation

Abstract

We revisit the interplay between differentiation and divisionalization in a duopoly version of Ziss (Econ Lett 59:133–138, 1998). We model divisionalization as a discrete problem to prove that (i) firms may choose not to become multidivisional; and (ii) there may arise asymmetric outcomes in mixed strategies, due to the existence of multiple symmetric equilibria. If industry-wide divisionalization is the unique equilibrium, it can be socially efficient provided goods are almost perfect substitutes. Even small degrees of differentiation may suffice to make industry-wide divisionalization socially desirable because of the prevalence of consumers’ taste for variety over the replication of fixed costs.

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Notes

  1. 1.

    Of course, this has a lot to do with strategic delegation (Vickers 1985; Fershtman and Judd 1987; Sklivas 1987). The relationship between divisionalization and delegation is explicitly considered in Ziss (1999), Bárcena-Ruiz and Espinosa (1999) and González-Maestre (2000). For a reconstruction of the debate, see Lambertini (2017).

  2. 2.

    However, going multidivisional may facilitate implicit collusion (Dargaud and Jacques 2015).

  3. 3.

    Corchón and González-Maestre (2000) extend the analysis to the case of concave demand functions, to prove that either perfect competition or a natural oligopoly may obtain at equilibrium as the cost of divisionalization shrinks. See also González-Maestre (2001), where an analogous result emerges from a discrete choice model of spatial differentiation.

  4. 4.

    The details of calculations related to the welfare analysis with d divisions are omitted for the sake of brevity, as the relevant expressions of the critical thresholds of F pertaining to the welfare assessment keep the properties and signs illustrated in Sect. 4.

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Acknowledgements

We would like to thank Giacomo Corneo, Flavio Delbono and two anonymous referees for precious comments and suggestions. The usual disclaimer applies.

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Correspondence to Luca Lambertini.

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Appendix

Appendix

In order to characterise the Nash equilibrium in mixed strategies relevant for region II, we may assume that firm 1 (resp., 2) attaches probability \( \mathfrak {p}\in \left[ 0,1\right] \) (resp., \(\mathfrak {q}\in \left[ 0,1 \right] \)) to strategy S. Firm 1 must choose \(\mathfrak {p}\) so as to make firm 2 indifferent between S and M, and the problem of firm 2 in choosing \(\mathfrak {q}\) is analogous. The relevant expected profits for the two firms, calculated along columns or rows, are the following:

$$\begin{aligned} E\pi _{2}\left( S\right)= & {} \mathfrak {p}\pi _{SS}^{{ CN}}+\left( 1-\mathfrak {p} \right) \pi _{SM}^{{ CN}} \nonumber \\ E\pi _{2}\left( M\right)= & {} \mathfrak {p}\pi _{MS}^{{ CN}}+\left( 1-\mathfrak {p} \right) \pi _{{ MM}}^{{ CN}} \end{aligned}$$
(A1)
$$\begin{aligned} E\pi _{1}\left( S\right)= & {} \mathfrak {q}\pi _{SS}^{{ CN}}+\left( 1-\mathfrak {q}\right) \pi _{SM}^{{ CN}} \nonumber \\ E\pi _{1}\left( M\right)= & {} \mathfrak {q}\pi _{MS}^{{ CN}}+\left( 1-\mathfrak {q} \right) \pi _{{ MM}}^{{ CN}} \end{aligned}$$
(A2)

Solving the system \(E\pi _{i}\left( S\right) =E\pi _{i}\left( M\right) ,\)\( i=1,2,\) we obtain

$$\begin{aligned} \mathfrak {p}^{*}=\mathfrak {q}^{*}=\frac{\pi _{{ MM}}^{{ CN}}-\pi _{SM}^{{ CN}} }{\pi _{{ MM}}^{{ CN}}-\pi _{SM}^{{ CN}}+\pi _{SS}^{{ CN}}-\pi _{MS}^{{ CN}}} \end{aligned}$$
(A3)

which is positive and lower than one in region II.

The probabilities of observing each of the four possible outcomes are

$$\begin{aligned} P\left( S,S\right)= & {} \mathfrak {p}^{*}\mathfrak {q}^{*}=\left( \mathfrak {p}^{*}\right) ^{2}=\left( \mathfrak {q}^{*}\right) ^{2} \nonumber \\ P\left( M,M\right)= & {} \left( 1-\mathfrak {p}^{*}\right) \left( 1-\mathfrak {q }^{*}\right) =\left( 1-\mathfrak {p}^{*}\right) ^{2}=\left( 1-\mathfrak {q}^{*}\right) ^{2} \nonumber \\ P\left( M,S\right)= & {} P\left( S,M\right) =\left( 1-\mathfrak {p}^{*}\right) \mathfrak {q}^{*}=\left( 1-\mathfrak {q}^{*}\right) ^{2}\mathfrak {p} ^{*} \end{aligned}$$
(A4)

The last step consists in assessing the probability for firms to play one of the two Nash equilibria in pure strategies, \(P\left( \mathcal {N}\right) =P\left( S,S\right) +P\left( M,M\right) \) against the probability of making a mistake, \(P\left( \mathcal {M}\right) =P\left( M,S\right) +P\left( S,M\right) \), with

$$\begin{aligned} P\left( \mathcal {N}\right) -P\left( \mathcal {M}\right) =\frac{\left( \pi _{{ MM}}^{{ CN}}-\pi _{SM}^{{ CN}}-\pi _{SS}^{{ CN}}+\pi _{MS}^{{ CN}}\right) ^{2}}{\left( \pi _{{ MM}}^{{ CN}}-\pi _{SM}^{{ CN}}+\pi _{SS}^{{ CN}}-\pi _{MS}^{{ CN}}\right) ^{2}}>0. \end{aligned}$$
(A5)

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Lambertini, L., Pignataro, G. On the social (sub)optimality of divisionalization under product differentiation. J Econ 128, 225–238 (2019). https://doi.org/10.1007/s00712-019-00669-5

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Keywords

  • Divisionalization
  • Product differentiation
  • Duopoly

JEL Classification

  • L13
  • L22
  • L41