Modeling the intensity of competition

Within the context of a symmetric duopoly with linear demand and costs, we construct a parameterized family of price-setting games, where the parameter $\gamma\in[0,2]$ measures the degree or intensity of competition; $\gamma = 0$ corresponds to collusion, a particular value of $\gamma$ between 0 and 1 corresponds to the Cournot outcome, $\gamma=1$ corresponds to the Bertrand outcome and, in general, as $\gamma$ increases the intensity of competition increases. All the games within the parameterized family share the same strategic properties. We also construct a parameterized family of quantity-setting games, where the parameter $\beta\in[0,2]$ measures the intensity of competition; $\beta = 0$ corresponds to collusion, $\beta=1$ corresponds to the Cournot outcome and a particular value of $\beta$ between 1 and 2 corresponds to the Bertrand outcome. As $\beta$ increases, the intensity of competition increases. As an example of the potential usefulness of this approach, we show that, contrary to the view first put forward by Schumpeter (but later challenged by Arrow), the incentive to introduce a cost-reducing innovation is an increasing function of the intensity of competition (that is, an increasing function of $\gamma$ in the price-setting case and of $\beta$ in the quantity-setting case).