3.3. Stackelberg Model Matilde Machado Slides available from: http://www.eco.uc3m.es/OI-I-MEI/ � 3.3. Stackelberg Model 2-period model � Same assumptions as the Cournot Model except that � firms decide sequentially. In the first period the leader chooses its quantity. This � decision is irreversible and cannot be changed in the second period. The leader might emerge in a market because of historical precedence, size, reputation, innovation, information, and so forth. In the second period, the follower chooses its quantity � after observing the quantity chosen by the leader (the quantity chosen by the follower must, therefore, be along its reaction function). ������������������������ ��������������� � ����������������� �
3.3. Stackelberg Model Important Questions: 1. Is there any advantage in being the first to choose? 2. How does the Stackelberg equilibrium compare with the Cournot? ������������������������ ��������������� � ����������������� 3.3. Stackelberg Model Let’s assume a linear demand P(Q)=a-bQ Mc 1 =Mc 2 =c In sequential games we first solve the problem in the second period and afterwards the problem in the 1st period. 2nd period (firm 2 chooses q 2 given what firm 1 has chosen in the 1st period q 1 ): ( ) ( ) Π = + − = − + − Max P q q c q a b q q c q 2 ( ) ( ) 1 2 2 1 2 2 q 2 given ������������������������ ��������������� � ����������������� �
3.3. Stackelberg Model ( ) ( ) Π = + − = − + − Max P q q c q a b q q c q 2 ( ) ( ) 1 2 2 1 2 2 q 2 ∂Π = 2 ⇔ a − bq − bq − c = FOC: 0 2 0 ∂ q 2 1 2 − − a bq c ⇔ q = R q = = * 1 ( ) Cournot's reaction function 2 2 1 b 2 ������������������������ ��������������� � ����������������� 3.3. Stackelberg Model In the 1st period (firm 1 chooses q1 knowing that firm 2 will react to it in the 2nd period according to its reaction function q 2 =R 2 (q 1 )): ( ) ( ) Max Π = P q + q − c q = a − b q + R q − c q 1 ( ) ( ( )) 1 2 1 1 2 1 1 q 1 ∂Π 1 = ⇔ − − − ′ − = a bq bR q bq R q c FOC: 0 2 ( ) ( ) 0 ∂ q 1 2 1 1 2 1 1 − − a bq c 1 ⇔ − − + − = a bq b bq c 2 1 0 b 1 1 2 2 − a c 1 1 ⇔ − bq + bq + bq = 2 0 1 1 1 2 2 2 − − − a c a c a c 3 ⇔ − = ⇔ = = N > N = bq q q q * 0 1 1 b 1 1 b 2 2 2 3 ������������������������ ��������������� � ����������������� �
3.3. Stackelberg Model q * Given we solve for q 2 1 − − − − a c a c a c a c 1 1 3 = − = − = = N < N q q q q * * b b b b 2 1 2 2 2 2 2 2 2 4 4 q > q * * Therefore 1 2 ( ) ( ) − − − − a c a c a c a c 3 2 + = + = > = N q q Q * * 1 2 b b b b 2 4 4 3 a>c ( ) − a c + a c 3 3 = − = − = > p a bQ a b c * * b 4 4 a + c 2 < = N p p * But 3 ������������������������ ��������������� � ����������������� 3.3. Stackelberg Model The equilibrium profits of both firms: ( ) ( ) + − − − a − c 2 a − c 2 a c a c a c a c ( ) 3 Π = p − c q = − c = = > Π N = 1* * * 1 b b b 1 b 4 2 4 2 8 9 ( ) ( ) a − c 2 a − c 2 a + c a − c a − c a − c ( ) 3 Π = − = − = = < Π N = p c q c 2* * * b b b b 2 2 4 4 4 4 16 9 Note: The profit of firm 1 must be at least as large as in Cournot because firm 1 could have always obtain the Cournot profits by N , to which firm 2 would have choosing the Cournot quantity q 1 replied with its Cournot quantity q 2 N =R 2 (q 1 N ) since firm 2’s reaction curve in Stackelberg is the same as in Cournot. ������������������������ ��������������� � ����������������� �
3.3. Stackelberg Model Conclusion: > q q * * a) (the leader produces more) 1 2 p > c * b) (There will be a DWL) Π > Π 1* 2* c) (the leader has higher profits, there is an advantage of being the first to choose) Q > Q N ⇒ p < p N * * d) The leader has a higher profit for two reasons: 1) the leader knows that by increasing q1 the follower will reduce q2 (strategic substitutes). 2) the decision is irreversible (otherwise the leader would undo its choice and we would end up in Cournot again) The sequential game (Stackelberg) leads to a more competitive equilibrium than the simultaneous move game (Cournot). ������������������������ ��������������� ����������������� 3.3. Stackelberg Model Graphically: The isoprofit curves for firm 1 are derived as: ( ) Π = − + − q q a b q q c q 1 ( , ) ( ) 1 2 1 2 1 therefore: ( ) π = − + − = − − − a b q q c q aq bq bq q cq 2 ( ) 1 2 1 1 1 1 2 1 ( ) ⇔ = − − − π bq q a c q bq 2 1 2 1 1 ( ) a − c π ⇔ q = − q − 2 b 1 q 1 ∂ q π ∂ q π 2 2 = − + = − < 2 2 1 ; 0 ∂ q q ∂ q q 2 2 3 1 1 1 1 ������������������������ ��������������� �! ����������������� �
3.3. Stackelberg Model Graphically(cont): Given q2, firm 1 chooses its best q2 response i.e. the isoprofit curve that corresponds to the maximum profit given q2 π ’< π M =(1/b)((a-c)/2)^2 Isoprofit = π M =1 single point q M q’ q’’ q1 ������������������������ ��������������� �� ����������������� 3.3. Stackelberg Model Graphically(cont): The reaction function intercepts the isoprofit curves where the slope becomes zero (i.e. horizontal) R q = Π q q ⇔ Π R q q = 1 1 ( ) arg max ( , ) ( ( ), ) 0 1 2 1 2 1 1 2 2 q 1 Moreover we know that: Π dq 1 Π = π ⇒ Π + Π = ⇔ = − Π q q dq dq 1 1 1 2 1 ( , ) 0 1 2 1 1 2 2 dq 1 1 2 dq = = q R q 2 therefore at the best response ( ) the derivative is zero : 0 1 1 2 dq = 1 q R q ( ) 1 1 2 ������������������������ ��������������� �� ����������������� �
3.3. Stackelberg Model Graphically(cont): the optimum of the leader (firm 1) is in a tangency point (S) of the isoprofit curve with the reaction curve of the follower (firm 2). (C) would be q2 the Cournot equilibrium, where the reaction curves cross and where dq 2 /dq 1 =0 q 1 S >q 1 N q M q 2 S <q 2 N C S q M q1 ������������������������ ��������������� �� ����������������� 3.3. Stackelberg Model Graphically(cont): q2 S +q 2 S >q 1 N +q 2 N q 1 q M C S -1 q M q1 ������������������������ ��������������� �� ����������������� �
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