8.6 Investment in the Future: Consider two firms that play a Cournot competition game with demand $p = 100 - q$ and costs for each firm given by $c_i(q_i) = 10q_i$. Imagine that before the two firms play the Cournot game firm
8.5 Exercises • 171
1 can invest in cost reduction. If it invests the costs of firm 1 will drop to $c_1(q_1) = 5q_1$. The cost of investment is $F > 0$. Firm 2 does not have this investment opportunity.
a. Find the value $F^*$ for which the unique subgame-perfect equilibrium involves firm 1 investing.
b. Assume that $F > F^*$. Find a Nash equilibrium of the game that is not subgame perfect.