Find the unique nash equilibrium of the first-stage game

Question

Augmented Competition:

Consider two firms playing a two-stage game with discount factor δ. In the first stage they play a Cournot quantity-setting game in which each firm has costs c_i(q_i) = 10q_i for i ∈ {1, 2} and the demand is given by p(q) = 100 - q, where q = q₁ + q₂. In the second stage, after the results of the Cournot game are observed, the firms play the following standard setting game:

a. Find the unique Nash equilibrium of the first-stage game and the two pure-strategy Nash equilibria of the second-stage game.

b. As far as the two firms are considered, what are the symmetric Paretooptimal outcomes of each stage-game?

c. For which values of δ can the Pareto-optimal outcomes be supported as a subgame-perfect equilibrium?

d. Assume that δ = 0.5. What is the "best" symmetric subgame-perfect equilibrium that the players can support?

e. What happens to the best symmetric subgame-perfect equilibrium that the players can support as δ drops toward zero?