Consider two firms that act as Cournot competitors and face the inverse demand f
ID: 1253626 • Letter: C
Question
Consider two firms that act as Cournot competitors and face the inverse demand function p(.), where p'(Y1 + Y2) < 0. The total cost function for firm i is given by Ci(.), where Ci'(Yi) > 0 i = 1,2.a) Do such duopolists produce a Pareto efficient level of output? Show your work and clearly explain why or why not.
b)Would a lump sum subsidy placed on one or both duopolists work to entice the firms to produce a Pareto efficient level of output" Show your work and clearly explain why or why not.
Explanation / Answer
A) No. Pareto efficiency implies price equals marginal cost. P = Ci' Each firm will profit maximize. Call V profit. Vi = P(Qi+Qj)*Qi - Ci Take the derivative and set equal to zero. P'*Qi + P - Ci' = 0 P - Ci' = -P'*Qi > 0 Therefore, P > Ci' B) No, a lump sum subsidy would not affect anything because it would not affect the profit maximizing price and input. Let's call the subsidy S. Vi = P(Qi+Qj)*Qi - Ci + S Take the derivative and set equal to zero. P'*Qi + P - Ci' = 0 P - Ci' = -P'*Qi > 0 Still, P > Ci' A per-unit subsidy would, actually work, though. Let's call the per-unit subsidy U(Qi). Vi = P(Qi+Qj)*Qi - Ci + U(Qi) Take the derivative and set equal to zero. P'*Qi + P - Ci' + U'= 0 P - Ci' = -P'*Qi - U' > 0 P = Ci' iff U' = -P'*Qi Each firm will increase output to the efficient level in order to obtain more of the the per-unit subsidy. But you would have to get the per-unit subsidy from somewhere.