# Sequential price setting SPNE

Firms A and B are in an oligopoly. They both face the linear market demand curve $$X=A-\alpha P$$, where $$X$$ is total market demand, and $$P$$ is price. Assume constant marginal costs $$C_{A}$$ and $$C_{B}$$, where $$C_{B}.

What will be the SPNE outcome if firm A sets price first? Is the following correct?

So long as firm A sets $$P>C_{B}$$, then it will get 0 profits (as firm B will profitably undercut and take all market demand). Therefore, it is indifferent between any $$P>C_{B}$$. If, however, A sets $$P \leq C_{B}$$, then firm B will not be able to profitably undercut, and so market demand will be split between the two firms, at some $$P \leq C_{B}$$. But then firm A will be making negative profits, and so can profitably deivate to setting some P>$$C_{B}$$.

Is this analysis correct? Or can firm A set some $$p\leq C_{B}$$ in an SPNE?

Your logic is correct. You need one more step. Using the same logic you can show that, in any SPNE, firm $$A$$ must set a price greater than the price that maximizes $$B$$'s profits. Firm $$B$$ then chooses such price.