Quantity competition in Hotellings linear city

We augment the Shapley–Shubik (1977) market game to include a spatial dimension à la Hotelling (1929). Taking firms’ locations as given, we study and characterise, through several propositions, lemmata, and a theorem, the main equilibrium predictions of this new model. When both firms locate in the centre and there is no product differentiation at all, we derive a counterexample in which both firms charge a price that is greater than marginal cost. Intriguingly, we show that even when both firms are in the same location, it is possible for the Law of One Price (LOOP) to fail, i.e., the exact same good sells at different prices across two platforms that are a priori identical. We derive similar (equal- and unequal-price) counterexamples in the context where the firms locate at the extreme ends of the city. Now, it is well known that in the traditional Hotelling model, a pure-strategy Nash equilibrium (PSNE) fails to exist when the two firms are closely spaced and near the centre of the city. In our main result, we allow the firms to be arbitrarily close to each other, and propose two counterexamples in which a PSNE exists. In one, the LOOP holds, while in the other, it fails.