Bonus question Competition à la Cournot with homogenous goods and asymmetric incomplete information. Consider a duopoly that competes à la Cournot (choosing quantities simultane- ously), facing inverse demand function p(Q) = a – Q where Q = q1 + q2, where the cost function for firm 1 is c1(41) = cqı with c > 0 which is common knowledge for both firms. The cost function for firm 2 is not common knowledge and can be c2(92) = CH92 with probability 0 € (0,1) or c2(92) = CL92 with probability 1 – 0, where cL < CH. The notation cL means low cost and cH means high cost. Firm 2 can be a new entrant to the industry, or could have just invented a new technology. Firm 2 knows its marginal cost which means it has private information since it know if it is cz or CH. Firm 1 on the other hand does not know the marginal cost of firm 2 (we don't either) and therefore it has less information than firm 2 and therefore constitutes asymmetric information. Firm 2 may want to choose a different (presumably lower) quantity if its marginal cost is high than if it is low. Firm 1, for its part, should anticipate that firm 2 may tailor its quantity to ist cost in this way. Find the best response of firm 2 to any quantity q1 that firm 1 chooses when it has marginal a. cost cí and cL. (Hint: maximize profits for firm 2 separately when marginal cost is cH and when it is cL)
Bonus question Competition à la Cournot with homogenous goods and asymmetric incomplete information. Consider a duopoly that competes à la Cournot (choosing quantities simultane- ously), facing inverse demand function p(Q) = a – Q where Q = q1 + q2, where the cost function for firm 1 is c1(41) = cqı with c > 0 which is common knowledge for both firms. The cost function for firm 2 is not common knowledge and can be c2(92) = CH92 with probability 0 € (0,1) or c2(92) = CL92 with probability 1 – 0, where cL < CH. The notation cL means low cost and cH means high cost. Firm 2 can be a new entrant to the industry, or could have just invented a new technology. Firm 2 knows its marginal cost which means it has private information since it know if it is cz or CH. Firm 1 on the other hand does not know the marginal cost of firm 2 (we don't either) and therefore it has less information than firm 2 and therefore constitutes asymmetric information. Firm 2 may want to choose a different (presumably lower) quantity if its marginal cost is high than if it is low. Firm 1, for its part, should anticipate that firm 2 may tailor its quantity to ist cost in this way. Find the best response of firm 2 to any quantity q1 that firm 1 chooses when it has marginal a. cost cí and cL. (Hint: maximize profits for firm 2 separately when marginal cost is cH and when it is cL)
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:Bonus question
Competition à la Cournot with homogenous goods and asymmetric
incomplete information. Consider a duopoly that competes à la Cournot (choosing quantities simultane-
ously), facing inverse demand function p(Q) = a – Q where Q = q1 + 92, where the cost function for firm 1
is c1 (91) = cqı with c > 0 which is common knowledge for both firms. The cost function for firm 2 is not
common knowledge and can be c2(42) = Cí42 with probability 0 € (0, 1) or c2(92) = CL92 with probability
1- 0, where CL < CH- The notation Cz means low cost and CH means high cost. Firm 2 can be a new
entrant to the industry, or could have just invented a new technology. Firm 2 knows its marginal cost which
means it has private information since it know if it is Cz or CH. Firm 1 on the other hand does not know
the marginal cost of firm 2 (we don't either) and therefore it has less information than firm 2 and therefore
constitutes asymmetric information. Firm 2 may want to choose a different (presumably lower) quantity if
its marginal cost is high than if it is low. Firm 1, for its part, should anticipate that firm 2 may tailor its
quantity to ist cost in this way.
Find the best response of firm 2 to any quantity q1 that firm 1 chooses when it has marginal
cost cH and cL- (Hint: maximize profits for firm 2 separately when marginal cost is CH and when it is cL)
а.
b.
Find the optimal response of firm 1 to any quantity q2 that firm 2 chooses under the information
it has. (Hint: maximize expected profit for firm 1 according to the belief that q2 (q1; CH) arises with proba-
bility 0 and q2 (q1; Cz) with probability (1 – 0) subject to the best response functions of firm 2 i.e. q2 (91; CH)
and q2 (41; Cz) found in part a). Find the Bayesian Nash equilibrium quantities qt, q (CH), q (CL) and total
supply Q, and Që,-
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