Frank Cowell Microeconomics March 2007 Exercise 10 7

Frank Cowell: Microeconomics March 2007 Exercise 10. 7 MICROECONOMICS Principles and Analysis Frank Cowell

Ex 10. 7(1): Question Frank Cowell: Microeconomics n n 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

Ex 10. 7(1): iso-profit curve Frank Cowell: Microeconomics n By definition, profits of firm 2 are u u u n Price depends on total output in the industry u u n p = p(q 1 + q 2 ) = b 0 b[q 1 + q 2] So profits of firm 2 as a function of (q 1, q 2) are u n P 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 P 2 = b 0 q 2 b[q 1 + q 2]q 2 [C 0 + cq 2] The iso-profit contour is found by u u setting P 2 as a constant plotting q 1 as a function of q 2

Ex 10. 7(1): firm 2’s iso-profit contours Frank Cowell: Microeconomics §Output space for the two firms §Contour for a given value of P §Contour map q 2 § b 0 q 2 b[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 prof it q 1

Ex 10. 7(2): Question Frank Cowell: Microeconomics method: n Use the result from part 1 n Use Cournot assumption to get firm 2’s best response to firm 1’s output (2’s reaction function) n By symmetry find the reaction function for firm 1 n Nash Equilibrium where both these functions are satisfied

Ex 10. 7(2): reaction functions and CNE Frank Cowell: Microeconomics n Firm 2 profits for given value`q of firm 1’s output: u n n Max this with respect to q 2 Differentiate to find FOC for a maximum: u n u q 2 = ½[b 0 c]/b ½`q 1 this is firm 2’s reaction function c 2 By symmetry, firm 1’s reaction function c 1 is u n b 0 b[`q 1 + 2 q 2] c = 0 Solve for firm 2’s output: u n P 2 = b 0 q 2 b[`q 1 + q 2 ]q 2 [C 0 + cq 2 ] q 1 = ½[ b 0 c]/ b ½ `q 2 Substitute back into c 2 to find Cournot-Nash solution u q 1 = q 2 = q. C = ⅓[b 0 c]/b

Ex 10. 7(2): firm 2’s reaction function Frank Cowell: Microeconomics §Output space as before §Isoprofit map for firm 2 §For given q 1 find q 2 to max 2’s profits §Repeat for other given values of q 1 §Plot locus of these points q 2 § Cournot assumption: § Each firm takes other’s output as prof it • • • given § Firm 2’s reaction function § c 2(q 1) gives firm 2’s best output c 2(∙) response to a given output q 1 of firm 1 q 1

Ex 10. 7(2): Cournot-Nash Frank Cowell: Microeconomics §Firm 2’s contours and reaction function §Firm 1’s contours §Firm 1’s reaction function §CN equilibrium at intersection q 2 c 1(∙) § c 1(q 2) gives firm 1’s best output response to a given output q 2 of firm 2 § Using the Cournot assumption… § …each firm is making best response q. C • to other exactly at q. C c 2(∙) q 1

Ex 10. 7(3): Question Frank Cowell: Microeconomics method: n Use reaction functions from part 2 n Find optimal output if one firm is a monopolist n Joint profit max is any output pair that sums to this monopolist output

Ex 10. 7(3): joint profits Frank Cowell: Microeconomics n n Total output is q = q 1 + q 2 The sum of the firms’ profits can be written as: u n Maximise this with respect to q u u n differentiate to find FOC for a maximum: b 0 2 bq c = 0 Solve for joint-profit maximising output: u n P 1 + P 2 = b 0 q 1 b[q 1 + q 2]q 1 [C 0 + cq 1] + b 0 q 2 b[q 1 + q 2]q 2 [C 0 + cq 2] = b 0 q b[q]2 [2 C 0 + cq] q = ½[b 0 c]/b However, breakdown into (q 1 , q 2) components is undefined

Ex 10. 7(3): Joint-profit max Frank Cowell: Microeconomics §Reaction functions of the two firms §Cournot-Nash equilibrium §Firm 1’s profit-max output if a monopolist §Firm 2’s profit-max output if a monopolist q 2 §Output combinations that max joint profit §Symmetric joint profit maximisation c 1(∙) (0, q. M) § q 1 + q 2 = q. M • q. C • § q. J = ½ q. M • q. J c 2(∙) • (q. M, 0) q 1

Ex 10. 7(4): Question Frank Cowell: Microeconomics method: n Use firm 2’s reaction function from part 2 (the “follower”) n Use this to determine opportunity set for firm 1 (the “leader”)

Ex 10. 7(4): reaction functions and CNE Frank Cowell: Microeconomics n Firm 2’s reaction function c 2: u n Firm 1 uses this reaction in its calculation of profit: u n n P 1 = b 0 q 1 b[q 1 + c 2(q 1)]q 1 [C 0 + cq 1] = b 0 q 1 b[q 1 + [½[b 0 c]/b ½q 1 ] ]q 1 [C 0 + cq 1] = ½[b 0 c bq 1] q 1 C 0 Max this with respect to q 1 Differentiate to find FOC for a maximum: u n q 2 = ½[b 0 c]/b ½q 1 ½[b 0 c ] bq 1 = 0 So, using firm 2’s reaction function again, Stackelberg outputs are u u q. S 1 = ½[b 0 c]/b (leader) q. S 2 = ¼[b 0 c]/b (follower)

Ex 10. 7(4): Stackelberg Frank Cowell: Microeconomics §Firm 2’s reaction function §Firm 1’s opportunity set §Firm 1’s profit-max using this set q 2 q. C • q. S of pr • it c 2(∙) • (q. M, 0) q 1

Ex 10. 7(5): Question Frank Cowell: Microeconomics method: n compute profit n plot in a diagram with (P 1 , P 2) on axes

Ex 10. 7(5): Possible payoffs Frank Cowell: Microeconomics P 2 §Profit space for the two firms §Attainable profits for two firms §Symmetric joint profit maximisation • (0, P ) M §max profits all to firm 1 (but with two firms present) §Monopoly profits (only one firm present) §Cournot profits §Stackelberg profits • § PJ = [b 0 c]2 /[8 b] C 0 § 2 PJ = [b 0 c]2 /[4 b] 2 C 0 (PJ, PJ) (PC, PC) • § PM = [b 0 c]2 /[4 b] C 0 § PC = [b 0 c]2 /[9 b] C 0 (PS 1, PS 2) § PS 1 = [b 0 c]2 /[8 b] C 0 { C 0 0 ° (P • , 0) M P 1 § PS 2 = [b 0 c]2 /[16 b] C 0

Ex 10. 7: Points to remember Frank Cowell: Microeconomics n n n Cournot best response embodied in c functions Cooperative solution found by treating firm as a monopolist Leader-Follower solution found by putting follower’s reaction into leader’s maximisation problem