Random duopoly:-

Consider a market that is shared by two firms A and B. The demand curve is p = e - q, where q is the total quantity produced and e is a constant that is either e₁ or e₂. The marginal cost of both firms is m.

(a) Suppose that B goes first and selects qB. Then after observing qB, A follows and selects qA. Both firms know the demand curve. What are qA and qB?

(b) Now suppose that the value of e is known to A, but B knows only that e has the possible values e1 and e2 with probabilities p₁ and p₂ = 1 - p₁, respectively. Define e = p₁e₁ + p₂e₂. Again B must select qB first and A selects qA after observing qB. Assuming A and B each act to maximize the expected value of profit, find qB and qA.