Two fuel brands, Shell and Esso, are considering opening a gas station at the same highway intersection. There is a one-time $50 irreversible cost from opening a station. Once active, stations compete by choosing prices simultaneously. Both stations have a gas per-liter cost of $4; there are no ongoing fixed costs. The firms anticipate that aggregate fuel demand at the intersection will be Q = 100-10P, and consumers perceive gasoline as a homogeneous good, i.e., consumers shop for the lowest price across any active firm. 1. (10 points) Each firm makes a Yes/No decision about whether to enter the market simultane- ously. For each possible combination of entry decisions, describe the equilibrium price and profits (including start-up costs) for each firm. The question describes a simultaneous entry decision game, followed by Bertrand competition between firms with homogeneous goods. We know that for the symmetric case, both firms choose a price at the marginal in the Bertrand equilibrium. If only one firm enters, then it will choose the monopolist solution: maxp(100-10P) (P-4) p=7 = 30 x 3-50=40. Hence, the matrix form of the entry games is: Shell Esso E NE E -50,-50 40,0 NE 0,40 0,0 . 5 points for the correct solution of the Bertrand situation. 5 points for the correct final answer 2. (5 points) Find all Nash equilibria of the entry game between the two firms. The Nash equilibria are (NE,E), (E,NE), and a strategy profile with mixed strategies. 4 points for the two pure strategy Nash 1 point for the mentioning mixed strategy Nash 3. (5 points) What would the equilibrium profits be if one firm had a marginal cost of 87 while the other had a marginal cost of 84? Let's assume that Shell has a marginal cost equal to 84. In the Bertrand equilibrium of the asymmetric cost case, firms set prices at the highest marginal cost, and the firm with the lowest marginal cost serves the whole market. Shell's profit if both firms enter is now: (100-10x7) (7-4)-50-40. The monopolist profit of Esso is now 22.5. The new payoff matrix is: Shell Esso E NE 40.-50 40.0 NE 0,22.5 0.0 Now, Enter is a dominant strategy for Shell. The only Nash equilibrium is (E,NE). 2.5 points for the new Bertrand profits 2.5 points for the new Nash equilibrium