We consider an unexplored potential of the Hotelling price competition framework. We relax the standard assumption of full market coverage, which requires the marginal transportation cost to be low relative to the consumer maximum willingness to pay and implies that the indifferent consumer gets strictly positive utility at equilibrium. We identify a parametric interval where the marginal transportation cost is relatively high and duopolists act as follows: they have an incentive to set relatively low prices, so as to fully cover the market, but not as low as to be able to steal consumers from the competitor, with the effect that the indifferent consumer gets zero equilibrium utility. We refer to this scenario as Monopolistic Duopoly, in that the cross-price elasticity of firm demands is positive when the competitor reduces its price, as in the standard Hotelling game, but zero when the competitor's price rises, as in a monopolistic setting.