1. Frank Cowell: Microeconomics Exercise 10.7 MICROECONOMICS Principles and Analysis Frank Cowell March 2007

2. Frank Cowell: Microeconomics Ex 10.7(1): Question purpose: examine equilibrium concepts in a very simple duopoly purpose: examine equilibrium concepts in a very simple duopoly method: determine best-response behaviour in a model where each firm takes other outputs as given method: determine best-response behaviour in a model where each firm takes other outputs as given

3. Frank Cowell: Microeconomics Ex 10.7(1): iso-profit curve By definition, profits of firm 2 are By definition, profits of firm 2 are   2 = pq 2  [C 0 + cq 2 ]  where q 2 is the output of firm 2  C 0, c are parameters of the cost function Price depends on total output in the industry Price depends on total output in the industry  p = p(q 1 + q 2 )  =  0   [q 1 + q 2 ] So profits of firm 2 as a function of (q 1, q 2 ) are So profits of firm 2 as a function of (q 1, q 2 ) are   2 =  0 q 2   [q 1 + q 2 ]q 2  [C 0 + cq 2 ] The iso-profit contour is found by The iso-profit contour is found by  setting  2 as a constant  plotting q 1 as a function of q 2

4. Frank Cowell: Microeconomics Ex 10.7(1): firm 2’s iso-profit contours q1q1 q2q2 profit   Output space for the two firms   Contour for a given value of    Contour map    0 q 2   [q 1 + q 2 ]q 2  [C 0 + cq 2 ] = const   As q 1 falls for given q 2 price rises and firm 2’s profits rise

5. Frank Cowell: Microeconomics Ex 10.7(2): Question method: Use the result from part 1 Use the result from part 1 Use Cournot assumption to get firm 2’s best response to firm 1’s output (2’s reaction function) Use Cournot assumption to get firm 2’s best response to firm 1’s output (2’s reaction function) By symmetry find the reaction function for firm 1 By symmetry find the reaction function for firm 1 Nash Equilibrium where both these functions are satisfied Nash Equilibrium where both these functions are satisfied

6. Frank Cowell: Microeconomics Ex 10.7(2): reaction functions and CNE Firm 2 profits for given value  q of firm 1’s output: Firm 2 profits for given value  q of firm 1’s output:   2 =  0 q 2   [  q 1 + q 2 ]q 2  [C 0 + cq 2 ] Max this with respect to q 2 Max this with respect to q 2 Differentiate to find FOC for a maximum: Differentiate to find FOC for a maximum:   0   [  q 1 + 2q 2 ]  c =  Solve for firm 2’s output: Solve for firm 2’s output:  q 2 = ½[  0  c]/   ½  q 1  this is firm 2’s reaction function  2 By symmetry, firm 1’s reaction function  1 is By symmetry, firm 1’s reaction function  1 is  q 1 = ½[  0  c]/   ½  q 2 Substitute back into  2 to find Cournot-Nash solution Substitute back into  2 to find Cournot-Nash solution  q 1 = q 2 = q C = ⅓[  0  c]/ 

7. Frank Cowell: Microeconomics Ex 10.7(2): firm 2’s reaction function   Output space as before   Isoprofit map for firm 2   For given q 1 find q 2 to max 2’s profits   Firm 2’s reaction function    2 (q 1 ) gives firm 2’s best output response to a given output q 1 of firm 1 q1q1 q2q2 2(∙)2(∙)   Repeat for other given values of q 1 profit   Plot locus of these points   Cournot assumption:   Each firm takes other’s output as given

8. Frank Cowell: Microeconomics Ex 10.7(2): Cournot-Nash   Firm 2’s contours and reaction function   Firm 1’s contours   CN equilibrium at intersection    1 (q 2 ) gives firm 1’s best output response to a given output q 2 of firm 2 q1q1 q2q2 2(∙)2(∙) 1(∙)1(∙) qCqC   Using the Cournot assumption…   …each firm is making best response to other exactly at q C   Firm 1’s reaction function

9. Frank Cowell: Microeconomics Ex 10.7(3): Question method: Use reaction functions from part 2 Use reaction functions from part 2 Find optimal output if one firm is a monopolist Find optimal output if one firm is a monopolist Joint profit max is any output pair that sums to this monopolist output Joint profit max is any output pair that sums to this monopolist output

10. Frank Cowell: Microeconomics Ex 10.7(3): joint profits Total output is q = q 1 + q 2 Total output is q = q 1 + q 2 The sum of the firms’ profits can be written as: The sum of the firms’ profits can be written as:   1  2 =  0 q 1   [q 1 + q 2 ]q 1  [C 0 + cq 1 ] +  0 q 2   [q 1 + q 2 ]q 2  [C 0 + cq 2 ] =  0 q   q] 2  [2C 0 + cq] =  0 q   q] 2  [2C 0 + cq] Maximise this with respect to q Maximise this with respect to q  differentiate to find FOC for a maximum:   0  2  q  c =  Solve for joint-profit maximising output: Solve for joint-profit maximising output:  q = ½[  0  c]/  However, breakdown into ( q 1, q 2 ) components is undefined However, breakdown into ( q 1, q 2 ) components is undefined

11. Frank Cowell: Microeconomics Ex 10.7(3): Joint-profit max   Reaction functions of the two firms   Cournot-Nash equilibrium   q 1 + q 2 = q M   Firm 1’s profit-max output if a monopolist q1q1 q2q2 2(∙)2(∙) 1(∙)1(∙) qCqC (0,q M ) (q M,0) qJqJ   Firm 2’s profit-max output if a monopolist   Output combinations that max joint profit   Symmetric joint profit maximisation   q J = ½ q M

12. Frank Cowell: Microeconomics Ex 10.7(4): Question method: Use firm 2’s reaction function from part 2 (the “follower”) Use firm 2’s reaction function from part 2 (the “follower”) Use this to determine opportunity set for firm 1 (the “leader”) Use this to determine opportunity set for firm 1 (the “leader”)

13. Frank Cowell: Microeconomics Ex 10.7(4): reaction functions and CNE Firm 2’s reaction function  2 : Firm 2’s reaction function  2 :  q 2 = ½[  0  c]/   ½q 1 Firm 1 uses this reaction in its calculation of profit: Firm 1 uses this reaction in its calculation of profit:   1 =  0 q 1   [q 1 +  2 (q 1 )]q 1  [C 0 + cq 1 ] =  0 q 1   [q 1 + [½[  0  c]/   ½q 1 ] ]q 1  [C 0 + cq 1 ] = ½[  0  c   q 1 ] q 1  C 0 Max this with respect to q 1 Max this with respect to q 1 Differentiate to find FOC for a maximum: Differentiate to find FOC for a maximum:  ½[  0  c ]   q 1 = 0 So, using firm 2’s reaction function again, Stackelberg outputs are So, using firm 2’s reaction function again, Stackelberg outputs are  q S 1 = ½[  0  c]/  (leader)  q S 2 = ¼[  0  c]/  (follower)

14. Frank Cowell: Microeconomics Ex 10.7(4): Stackelberg   Firm 2’s reaction function   Firm 1’s opportunity set   Firm 1’s profit-max using this set q1q1 q2q2 2(∙)2(∙) qCqC (q M,0) qSqS profit

15. Frank Cowell: Microeconomics Ex 10.7(5): Question method: compute profit compute profit plot in a diagram with (  1,  2 ) on axes plot in a diagram with (  1,  2 ) on axes

16. Frank Cowell: Microeconomics C0C0  M  { Ex 10.7(5): Possible payoffs 0  C  C   J  J   S  S  1 2   Profit space for the two firms   Monopoly profits (only one firm present)   Attainable profits for two firms   J = [  0  c] 2 /[8  ]  C 0   M = [  0  c] 2 /[4  ]  C 0   S 2 = [  0  c] 2 /[16  ]  C 0   S 1 = [  0  c] 2 /[8  ]  C 0   C = [  0  c] 2 /[9  ]  C 0   Symmetric joint profit maximisation   Cournot profits   Stackelberg profits 11 22 °  M    J = [  0  c] 2 /[4  ]  2C 0   max profits all to firm 1 (but with two firms present)

17. Frank Cowell: Microeconomics Ex 10.7: Points to remember Cournot best response embodied in  functions Cournot best response embodied in  functions Cooperative solution found by treating firm as a monopolist Cooperative solution found by treating firm as a monopolist Leader-Follower solution found by putting follower’s reaction into leader’s maximisation problem Leader-Follower solution found by putting follower’s reaction into leader’s maximisation problem