In the faraway town of Saint James two firms, Bilge and Chem, compete in the soft-drink market (Coke and Pepsi aren’t in this market yet). They sell identical products, and since their good is a liquid, they can easily choose to produce fractions of units. Since they are the only two firms in this market, the price of the good (in dollars), P, is determined by P = (30 - QB - QC), where QB is the quantity produced by Bilge and QC is the quantity produced by Chem (each measured in liters). At this time both firms are considering whether to invest in new bottling equipment that will lower their variable costs.
(i) If firm j decides not to invest, its cost will be
where j stands for either B (Bilge) or C (Chem).
(ii) If a firm decides to invest, its cost will be
where j stands for either B (Bilge) or C (Chem). This new cost function reflects the fixed cost of the new machines (20) as well as the lower variable costs.
The two firms make their investment choices simultaneously, but the payoffs in this investment game will depend on the subsequent duopoly games that arise. The game is thus really a two-stage game: decide to invest, and then play a duopoly game.
(a) Suppose both firms decide to invest. Write the profit functions in terms of QB and QC for the two firms. Use these to find the Nash equilibrium of the quantity-setting game. What are the equilibrium quantities and profits for both firms? What is the market price?
(b) Now suppose both firms decide not to invest. What are the equilibrium quantities and profits for both firms? What is the market price?
(c) Now suppose that Bilge decides to invest, and Chem decides not to invest. What are the equilibrium quantities and profits for both firms? What is the market price?
(d) Write out the two-by-two game table of the investment game between the two firms. Each firm has two strategies: Investment and No Investment. The payoffs are simply the profits found in parts (a), (b), and (c).
(e) What is the subgame-perfect equilibrium of the overall two-stage game?