Mathematical Oligopoly Problem. Suppose a duopoly is operating using the following information. The firms are denoted Firm a and Firm b.
Demand: P = $1,250 – Q
Q = Qa + Qb, (this is stating that the market quantity is the sum of the two firm’s quantities)
TRa (Total Revenue for Firm a) = $1,250Qa – Qa2 – Qa*Qb
MRa (Marginal Revenue for Firm a) = $1,250 – 2Qa – Qb MCa (Marginal Cost for Firm a) = MCb (Marginal Cost for Firm b) = $50
Fixed Costs = $0, for both firms.
a. Use the Cournot solution method to find the output-reaction equations for both firms
b. Use the Cournot solution method to find Qa*, Qb*, Q*, P*
c. Suppose now that Firm a is the Stackelberg leader. Suppose further that when Firm a has prior knowledge about Firm b’s reaction curve, they now have an MRa = $650 – $1Qa. Use the Stackelberg solution method to find Qa*, Qb*, Q*, P*.
d. Assume the products sold by these firms are identical. Use the Bertrand model for identical p6roducts to solve for P* and Q*.
e. Compare/contrast the findings from parts 16b, 16c, and 16d. Explain any differences.