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

- 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.
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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## 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:

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

which is positive and lower than one in region *II*.

The probabilities of observing each of the four possible outcomes are

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

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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