Multiproduct firms and horizontal product differentiation

I am having difficulty modelling the effect of horizontal product differentiation between multiproduct firms. I've been looking at models specifically by Hotelling and d'Aspremont. Hotelling concluded that in duopolistic competition, an equilibrium was found through firms differentiating minimally, (i.e. designing products of similar attributes) whilst d’Aspremont shows the opposite with the equilibria resting at maximal product differentiation.

However, both of these principles do not take into account multiproduct firms, where firms could produce different varieties of the same product (take shoes for example) where some are highly differentiated and others may be similar to a competitor. Is there a model that explains this situation?

• Could you specify the kind of multi-product firms you have in mind? N>2 products over a single line or k lines of products where only two firms compete, with scale economies? – Yann Dec 5 '16 at 22:21

Here is a link to Schmalansee's classic paper. Use SSRI to look forward in time from this paper. Useful search terms in Econlit or Google Scholar would be "crowding the product space" or "product space congestion."

In the strict classical Hotelling model, it is impossible to have an equilibrium with 3 agents over the segment (deviation is always profitable). Some modification have been done to keep the transportation costs+inelastic demand with $n$ firms (with multiple segments and a common point, like a flake).

By the way, d'Aspremont et al. modified the cost function to reach "their" equilibrium.

• It is not true that equilibrium is always symmetric. There is a characterisation of the classic equilibria (linear cost, as in Hotelling's paper) for $n$ players in Eaton (1975) and for $n > 5$ asymmetric equilibria exist. – Giskard Dec 13 '16 at 14:07
• Accurate! I'm glad my answer raised a bit of attention to this unanswered question, nevertheless I would be happy to delete it (and in this case your link would disappear). Do you think you could write an answer? – Yann Dec 13 '16 at 16:52
• No need to delete it. Perhaps edit out the always symmetric part. I don't know what the exact question is. I think your comment under the question is important. It is a shame it was not answered. – Giskard Dec 13 '16 at 21:18