When Stackelberg and Cournot equilibria coincide

We take a new look at the comparison between the Stackelberg equilibrium and the Cournot equilibrium. We show that, when the elasticity of the inverse market demand equals the curvature of the inverse market demand weighted by the Lerner Index, a generic Stackelberg leader sets the same quantity and earns the same profit as a generic Stackelberg follower. When the curvature of the inverse market demand equals the total number of firms in the industry, a coincidence among the quantities produced by a first mover, a second mover, and a generic firm facing Cournot competition occurs.