Optimal Product Differentiation in a Circular Model

Since circular model was introduced in Salop (1979), it has been the workhorse for analyzing spatial competition among differentiated firms. A common assumption in this literature is that firms are evenly spaced on the circle, even when entry is allowed. We characterize conditions for even spacing to be an equilibrium, using a two-stage (location-then-price) circular model with general transport cost function. Under duopoly competition, we characterize a mild sufficient condition -- the first derivative of transport cost is concave (together with an assumption governing the transport cost difference to the two firms). If one only considers pure strategy equilibrium in prices, this sufficient condition is weakened to the first derivative of transport cost being -1-concave. These conditions ensure that firms' profits are concave in their prices when firms are evenly spaced and that even spacing maximizes profits. Under oligopoly competition (N>= 2 firms), we characterize a necessary condition for even spacing to be an equilibrium. This necessary condition requires a firm's profit to be concave in location at the symmetric location. It involves the third derivative of transport cost function, so having convex transport cost in general is neither necessary nor sufficient to determine equilibrium location choice. Our results have implications for studies employing circular models, especially in terms of welfare analysis which depends on firms' location choices