Equilibrium existence in the Hotelling model with convex production costs

We revisit the spatial duopoly model à la Hotelling (1929), to show that, provided the parameter scaling marginal cost is sufficiently high, quadratic production costs guarantee equilibrium existence in presence of linear transportation costs and a uniform distribution, with minimum product differentiation and no undercutting. We also discuss the conditions under which partial coverage arises. Finally, we extend the duopoly game to generalise the condition for the existence of the pure-strategy equilibrium to a class of convex production cost functions.