MICROECONOMICS Principles and Analysis Frank Cowell Exercise 10.7 MICROECONOMICS Principles and Analysis Frank Cowell March 2007

Ex 10.7(1): Question 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 By definition, profits of firm 2 are P2 = pq2  [C0 + cq2] where q2 is the output of firm 2 C0, c are parameters of the cost function Price depends on total output in the industry p = p(q1 + q2) = b0  b[q1 + q2] So profits of firm 2 as a function of (q1, q2) are P2 = b0q2  b[q1 + q2]q2  [C0 + cq2] The iso-profit contour is found by setting P2 as a constant plotting q1 as a function of q2

Ex 10.7(1): firm 2’s iso-profit contours Output space for the two firms Contour for a given value of P Contour map q2 b0q2  b[q1 + q2]q2  [C0 + cq2] = const As q1 falls for given q2 price rises and firm 2’s profits rise profit q1

Ex 10.7(2): Question method: 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) By symmetry find the reaction function for firm 1 Nash Equilibrium where both these functions are satisfied

Ex 10.7(2): reaction functions and CNE Firm 2 profits for given value`q of firm 1’s output: P2 = b0q2  b[`q1 + q2]q2  [C0 + cq2] Max this with respect to q2 Differentiate to find FOC for a maximum: b0  b[`q1 + 2q2]  c = 0 Solve for firm 2’s output: q2 = ½[b0  c]/b  ½`q1 this is firm 2’s reaction function c2 By symmetry, firm 1’s reaction function c1 is q1 = ½[b0  c]/b  ½`q2 Substitute back into c2 to find Cournot-Nash solution q1 = q2 = qC = ⅓[b0  c]/b

Ex 10.7(2): firm 2’s reaction function Output space as before Isoprofit map for firm 2 For given q1 find q2 to max 2’s profits Repeat for other given values of q1 q1 q2 Plot locus of these points Cournot assumption: Each firm takes other’s output as given profit Firm 2’s reaction function c2(q1) gives firm 2’s best output response to a given output q1 of firm 1 • • • c2(∙) • • •

Ex 10.7(2): Cournot-Nash Firm 2’s contours and reaction function Firm 1’s contours Firm 1’s reaction function CN equilibrium at intersection q2 c1(∙) c1(q2) gives firm 1’s best output response to a given output q2 of firm 2 Using the Cournot assumption… …each firm is making best response to other exactly at qC qC • c2(∙) q1

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

Ex 10.7(3): joint profits Total output is q = q1 + q2 The sum of the firms’ profits can be written as: P1 + P2 = b0q1  b[q1 + q2]q1  [C0 + cq1] + b0q2  b[q1 + q2]q2  [C0 + cq2] = b0q  b[q]2  [2C0 + cq] Maximise this with respect to q differentiate to find FOC for a maximum: b0  2bq  c = 0 Solve for joint-profit maximising output: q = ½[b0  c]/b However, breakdown into (q1 , q2) components is undefined

Ex 10.7(3): Joint-profit max 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 q2 Output combinations that max joint profit Symmetric joint profit maximisation c1(∙) q1 + q2 = qM (0,qM) • qC • qJ = ½ qM qJ • c2(∙) • q1 (qM,0)

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

Ex 10.7(4): reaction functions and CNE Firm 2’s reaction function c2: q2 = ½[b0  c]/b  ½q1 Firm 1 uses this reaction in its calculation of profit: P1 = b0q1  b[q1 + c2(q1)]q1  [C0 + cq1] = b0q1  b[q1 + [½[b0  c]/b  ½q1 ] ]q1  [C0 + cq1] = ½[b0  c  bq1] q1  C0 Max this with respect to q1 Differentiate to find FOC for a maximum: ½[b0  c ]  bq1 = 0 So, using firm 2’s reaction function again, Stackelberg outputs are qS1 = ½[b0  c]/b (leader) qS2 = ¼[b0  c]/b (follower)

Ex 10.7(4): Stackelberg • • • profit Firm 2’s reaction function Firm 1’s opportunity set Firm 1’s profit-max using this set q2 qC • qS • profit c2(∙) • q1 (qM,0)

Ex 10.7(5): Question method: compute profit plot in a diagram with (P1 , P2) on axes

Ex 10.7(5): Possible payoffs Profit space for the two firms • (0, PM) Attainable profits for two firms Symmetric joint profit maximisation max profits all to firm 1 (but with two firms present) Monopoly profits (only one firm present) Cournot profits Stackelberg profits PJ = [b0  c]2 /[8b]  C0 (PJ,PJ) • 2PJ = [b0  c]2 /[4b]  2C0 (PC,PC) PM = [b0  c]2 /[4b]  C0 PC = [b0  c]2 /[9b]  C0 • (PS,PS) 1 2 PS1 = [b0  c]2 /[8b]  C0 C0 { ° • PS2 = [b0  c]2 /[16b]  C0 (PM,0) P1

Ex 10.7: Points to remember 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