Limit Capacity: Two firms, A and B, will engage in Cournot competition for widgets. Before they do so, the firms can invest into their production capacities. A small plant costs $50,000 and has a capacity of producing up to 100 widgets. A large plant costs $175,000 and has unlimited capacity. The timing is

Investment stage:

oFirst, firm A invests either small or large.

oThen, firm B invests small, large, or not at all.

oNote: Firm A has to invest either small or large; only firm B can stay out.

Production stage: Firms simultaneously produce q_A, q_B widgets (if they invested, and up to their capacity)

Marginal costs of production are 0 and the inverse demand function is given by p(Q) = 900 - Q, where Q = q_A + q_B is the total number of produced widgets.

(a)Assume for the moment that firm B does not exist, or equivalently that it does not invest. What is the optimal strategy for firm A?

(b)Returning to the original problem, sketch the game tree and solve for subgame-perfect equilibrium with generalized backward induction! (Hint: To do so, start by solving the six production subgames, i.e. solving for the equilibrium q_A, q_B after both firm invested large, both rms invested small, etc.) Why does firm A choose to build a different size plant than in part (a)?

(c)If firm B could commit to a plant size (small, large, or none) in advance, and A would have to respond optimally to this action, which plant size would B choose? (I.e. First B invests small, large or not at all, then A invests small or large, and then they play Cournot.)

B does not commit to a large investment because it always leads to a negative profit.

(d)Now consider a variation of the game, where the firms simultaneously decide to invest into no plant, a small plant or a large plant (A can also invest into no plant now), before engaging in Cournot competition: What are the subgame-perfect equilibria of this game? (Hint: The investment stage is now a simultaneous 3 x 3 game, in which the firms payoffs are given by the payoffs calculated in part (b).)