Imitation by price and quantity setting firms in a differentiated market

Abstract

t r a c tWe study the evolution of imitation behaviour in a differentiated market where firms are located equidistantly on a (Salop) circle. Firms choose price and quantity simultaneously, leaving open the possibility for non-market-clearing outcomes. The strategy of the most successful firm is imitated. Behaviour in the stochastically stable outcome depends on the level of market differentiation and corresponds exactly with the Nash equilibrium outcome of the underlying stage game. For high level of differentiation, firms end up at the monopoly outcome. For intermediate level of differentiation, they gravitate to a "mutually non-aggressive" outcome where price is higher than the monopoly price. For low level of differentiation, firms price at a mark-up above the marginal cost. Marketclearing always results endogenously.

Imitation and price competition in a differentiated market

“…A consumer maximises utility and purchases only if the net utility from doing so is at least 0, in which case, the firm providing the highest (non-negative) net utility is chosen. 5 We follow Apesteguia and Selten (2005) and Khan and Peeters (2015) in describing the 4 Even though this assumption is guided by our motivation to study a differentiated market, it is noteworthy that Hehenkamp and Wambach (2010) show that the equidistant locations of firms on a Salop circle is the predicted long-run outcome in a two-stage evolutionary model where firms choose location by imitation in the first stage, and choose the Nash equilibrium price corresponding to the first stage location profile in the second stage. 5 When the maximum net utility that a consumer receives on purchase is exactly equal to 0, we assume the good is purchased.…”

“…The following proposition presents the unique symmetric pure strategy Nash equilibrium for varying levels of the market differentiation parameter (the proof of which can be found in Khan and Peeters, 2015). (ii) Suppose τ ∈ [ 2 3 n (β − c), n (β − c) ].…”

“…First, if it takes a (series of) single experimentation(s) to reach a set of absorbing states from any other absorbing state and more than one experimentation to exit this set, then the stochastically stable set is contained in this set of 9 For more details, the reader is referred to Young (1993) or Kandori et al (1993). 10 Finiteness of the strategy space is a technical requirement for stochastic stability. Alós-Ferrer (2006) shows that in general, Nash equilibrium or other rest points properties of a continuous strategy space may not carry over to the discrete strategy case.…”

“…This nuance is not captured in the differentiated market structure used in the earlier papers, but is conveniently encapsulated by the Salop circle -a firm is more affected by, and in turn affects to a greater extent, the immediately neighbouring firms. Our paper is most related to Khan and Peeters (2015), which analyses the long-run equilibrium under imitation in such a spatially differentiated market with firms setting both price and quantity, and finds that the stochastically stable state coincides uniquely with the Nash equilibrium. We postpone a comparative discussion of the current paper and Khan and Peeters (2015) to the concluding section.…”

“…Our paper is most related to Khan and Peeters (2015), which analyses the long-run equilibrium under imitation in such a spatially differentiated market with firms setting both price and quantity, and finds that the stochastically stable state coincides uniquely with the Nash equilibrium. We postpone a comparative discussion of the current paper and Khan and Peeters (2015) to the concluding section.…”