Cournot duopoly model of incomplete information contd Firm 1's problem Max 8xqla-( q1+q2(cH))-c +(1-)×qh[a-(q+q2(c)-c st ≥0 ●FOC: a-2q-q2(c)-c]+(1-a-2q-q2(cn)-c]=0 Hence Bla-a2 cH)-c]+(1-bla-q2C)-c q1 g1 is firm 1's best response to the belief that firm 2 chooses g2(ch)with probability 8, and g2(cr)with probability 1-0 16
Cournot duopoly model of incomplete information cont’d • Firm 1's problem: . . 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 • FOC: [a −2q1 − q2 (cH ) −c]+ (1−)[a −2q1 − q2 (cL ) −c]= 0 Hence, 2 [ ( ) ] (1 )[ ( ) ] 2 2 1 a q c c a q c c q − H − + − − L − = • 1 q is firm 1's best response to the belief that firm 2 chooses ( ) 2 H q c with probability , and ( ) 2 L q c with probability 1− 16
Cournot duopoly model of incomplete information contd ● Now we have q2 ch=(a-g1-CH) q2(C1)=(a-qh-C1) a-q2(CH)-c]+(1-a-q2(cL)-c] We have three equations and three unknowns. Solving these gives us
Cournot duopoly model of incomplete information cont’d • Now we have ( ) 2 1 ( ) 2 H 1 H q c = a − q − c ( ) 2 1 ( ) 2 L 1 L q c = a − q − c 2 [ ( ) ] (1 )[ ( ) ] 2 2 1 a q c c a q c c q − H − + − − L − = • We have three equations and three unknowns. Solving these gives us 17
Cournot duopoly model of incomplete information contd a2(cu)=(a-2cH+c+- q2(c1)=:(a-2cL+c) a-2c+ber+(1-6) q1 ·Fim1 chooses g1 Firm 2 chooses g2(cH)if its marginal cost is high, or g2(cr) if its marginal cost IS loW This can be written as(qu, (q2(cH), q2(cr) One is the best response to the other A Nash equilibrium solution called Bayesian Nash equilibrium
Cournot duopoly model of incomplete information cont’d ( ) 6 1 ( 2 ) 3 1 ( ) * 2 H H H L q c a c c c − c − = − + + ( ) 6 ( 2 ) 3 1 ( ) * 2 L L H L q c = a − c + c + c − c 3 * 2 (1 ) 1 H L a c c c q − + + − = • Firm 1 chooses * 1q • Firm 2 chooses ( ) * 2 H q c if its marginal cost is high, or ( ) * 2 L q c if its marginal cost is low. • This can be written as ( * 1q , ( ( ) * 2 H q c , ( ) * 2 L q c )) • One is the best response to the other • A Nash equilibrium solution called Bayesian Nash equilibrium. 18
Cournot duopoly model of incomplete information(version one) cont'd 2(cn)=2(a-2cn+c)+(CH-c) q2(c1)=(a-2c1+c)+(cH-c1) a-2c+BcH+(1-0)c q This can be written as (g1, (@2(cH),2(cL) Firm 1 chooses g1 which is its best response to firm 2's 2(cH)A2(CL)) (and the probability) If firm 2' s marginal cost is HIGH then firm 2 choosesq2(cH) which is its best response to firm 1s q1 If firm 2's marginal cost is LOW then firm 2 chooses q2(CL)which is its best response to firm 1's q1 A Nash equilibrium solution called Bayesian Nash equilibrium 19
Cournot duopoly model of incomplete information (version one) cont’d ( ) 6 1 ( 2 ) 3 1 ( ) * 2 H H H L q c a c c c − c − = − + + ( ) 6 ( 2 ) 3 1 ( ) * 2 L L H L q c = a − c + c + c − c 3 * 2 (1 ) 1 H L a c c c q − + + − = • This can be written as ( * q1 , ( ( ) * 2 H q c , ( ) * 2 L q c )) • Firm 1 chooses * q1 which is its best response to firm 2's ( ( ) * 2 H q c , ( ) * 2 L q c ) (and the probability). • If firm 2's marginal cost is HIGH then firm 2 chooses ( ) * 2 H q c which is its best response to firm 1's * 1q . • If firm 2's marginal cost is LOW then firm 2 chooses ( ) * 2 L q c which is its best response to firm 1's * 1q . • A Nash equilibrium solution called Bayesian Nash equilibrium. 19
Cournot duopoly model of incomplete information(version two) 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 All the above are common knowledge
Cournot duopoly model of incomplete information (version two) ◼ 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 . ◼ All the above are common knowledge 20