Cournot duopoly model of incomplete information A homogeneous product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q, and 2, respectively They choose their quantities simultaneously The market price: P(O=a-o, where a is a constant number and 2=q+q2 Firm 1's cost function: C((=cq1 All the above are common knowledge
Cournot duopoly model of incomplete information ◼ A homogeneous product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q1 and q2 , respectively. ◼ They choose their quantities simultaneously. ◼ The market price: P(Q)=a-Q, where a is a constant number and Q=q1+q2 . ◼ Firm 1’s cost function: C1 (q1 )=cq1 . ◼ All the above are common knowledge 11
Cournot duopoly model of incomplete information contd a Firm 2s marginal cost depends on some factor(e.g. technology) that only firm 2 knows Its marginal cost can be HIGH: cost function C2(@2 =cH2 LOW: cost function: C2(a2)=cL2 Before production, firm 2 can observe the factor and know exactly which level of marginal cost is in However, firm 1 cannot know exactly firm 2s cost. Equivalently, it is uncertain about firm 2's payoff firm 1 believes that firm 2 s cost function is C2(92)=cH2 with probability 0, and C2(2)=cL2 with probability 1-0 All the above are common knowledge The harsanyi transformation a Independent types
Cournot duopoly model of incomplete information cont’d ◼ Firm 2’s marginal cost depends on some factor (e.g. technology) that only firm 2 knows. Its marginal cost can be ➢ HIGH: cost function: C2 (q2 )=cHq2 . ➢ LOW: cost function: C2 (q2 )=cLq2 . ◼ Before production, firm 2 can observe the factor and know exactly which level of marginal cost is in. ◼ However, firm 1 cannot know exactly firm 2’s cost. Equivalently, it is uncertain about firm 2’s payoff. ◼ Firm 1 believes that firm 2’s cost function is ➢ C2 (q2 )=cHq2 with probability , and ➢ C2 (q2 )=cLq2 with probability 1–. ◼ All the above are common knowledge ◼ The Harsanyi Transformation ◼ Independent types 12
Cournot duopoly model of incomplete information contd A solution for the Cournot duopoly model of incomplete information Firm 2 knows exactly its marginal cost is high or low If its marginal cost is high, i. e C2(@)=CH2, then, for any given q1, it will solve Max 2la-(g1+92)-cHl FOC: a-g1 0 a 2 a2(cH )is firm 2's best response to q,, if its marginal cost is high
Cournot duopoly model of incomplete information cont’d A solution for the Cournot duopoly model of incomplete information Firm 2 knows exactly its marginal cost is high or low. • If its marginal cost is high, i.e. 2 2 2 C (q ) = cH q , then, for any given 1 q , it will solve . . 0 [ ( ) ] 2 2 1 2 − + − st q Max q a q q cH • FOC: ( ) 2 1 2 0 ( ) 1 2 H 2 H 1 H a − q − q − c = q c = a − q − c • ( ) 2 H q c is firm 2's best response to 1 q , if its marginal cost is high. 13
Cournot duopoly model of incomplete information contd Firm 2 knows exactly its marginal cost is high or low If its marginal cost is low, i.e. C2( q2)=CLq2, then,for any given q,, it will solve Max 2la-(g1+92)-C,] st q,≥0 ·FoC:a-q1-2q2-C=0→q2(cn)=(a-q1-c1) g2(c,) is firm 2's best response to g,, if its marginal cost is low
Cournot duopoly model of incomplete information cont’d Firm 2 knows exactly its marginal cost is high or low. • If its marginal cost is low, i.e. 2 2 2 C (q ) = cLq , then, for any given 1 q , it will solve . . 0 [ ( ) ] 2 2 1 2 − + − st q Max q a q q cL • FOC: ( ) 2 1 2 0 ( ) 1 2 L 2 L 1 L a − q − q − c = q c = a − q − c • ( ) 2 L q c is firm 2's best response to 1 q , if its marginal cost is low. 14
Cournot duopoly model of incomplete information contd Firm 1 knows exactly its cost function C(qu=cqu Firm 1 does not know exactly firm 2's marginal cost is high or low But it believes that firm 2s cost function is C2(g2)=cHg2 with probability 0, and C2(92)=C,a2 with probability 1-6 Equivalently, it knows that the probability that firm 2'S quanti ity is q2(cH)is 8, and the probability that firm 2's quantity is g2(c,)is 1-8. So it solves Max 8xqila-(q1+2(CH))-cI L S.q1≥0
Cournot duopoly model of incomplete information cont’d • Firm 1 knows exactly its cost function 1 1 1 C (q ) = cq . • Firm 1 does not know exactly firm 2's marginal cost is high or low. • But it believes that firm 2's cost function is 2 2 2 C (q ) = cH q with probability , and 2 2 2 C (q ) = cLq with probability 1− • Equivalently, it knows that the probability that firm 2's quantity is ( ) 2 H q c is , and the probability that firm 2's quantity is ( ) 2 L q c is 1− . So it solves . . 0 (1 ) [ ( ( )) ] [ ( ( )) ] 1 1 1 2 1 1 2 + − − + − − + − st q q a q q c c Max q a q q c c L H 15