# 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 $t$. 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 :) Dec 10 '19 at 1:33

Market sharing simply means that both platforms have some market share. So the condition can probably be obtained by finding the closed-form solution for market shares (not only implicitly as in (7)) and imposing the condition that all market shares are positive. This is akin to each platform having an interior solution.

I am not sure which Hessian you are referring to, but my guess is that you mean the hessian of a firm's optimization problem (choosing both of their prices) taking the prices of the other platform as given. In that case, if the Hessian is negative definite, regardless of the other firm's prices, an interior solution exists.