# Existence of market sharing equilibrium in hotelling model

I'm reading a paper on Competition in Two-Sided Markets. The model is a Hotelling-type model, with consumers on some interval, choosing their preferred firm based on price and 'distance' to the different firms (see below). In it, there's an assumption made that guarantees that there's no single monopoly that serves all consumers, which amounts to the condition:

$$4t_1t_2>(\alpha_1\alpha_2)^2$$

But I'm not sure how this is derived. Annoyingly, the paper says: "it turns out that the necessary and sufficient condition for a market sharing equilibrium is $$4t_1t_2>(\alpha_1\alpha_2)^2$$", without providing the derivation. I'd appreciate a hint on how to derive this, or an actual derivation of this condition.

These screenshots are from page 8 from Competition in Two-Sided Markets.

• I suspect that showing this involves showing that the consumer on one side of the spectrum, has more utility from purchasing from the firm located at their side of the spectrum, rather than paying $t$. – Пафну́тий Dec 7 '19 at 18:34
• Update: I've actually managed to derive the condition. The condition comes from the Hessian matrix having a negative determinant. I'd appreciate hearing why a negative Hessian matrix is necessary and sufficient for a market-sharing equilibrium. Happy to give the bounty for a satisfactory explanation of this :) – user526463 Dec 10 '19 at 1:33