Three firms produce an identical product. The market demand is as follows: q(p) = Qo if p ≤ 2, q(p) = 0 if p > 2. Production costs are zero for all firms. Each firm has a capacity constraint of 5 units. In each period firms set their prices for that period simultaneously. This is repeated infinitely many times. The common discount factor of firms is δ < 1. Let Qo = 15. What is the lowest δ for which there exists a (Nash) equilibrium of the infinitely repeated game where the firms play the “grim strategy”?
Grim strategy: In the first period, set p = pM, where pM is the monopoly price. In each subsequent period, set p = pM if all firms have set p = pM in all past periods, otherwise set p = 0.