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Frank Cowell: Microeconomics March 2007 Exercise 10.7 MICROECONOMICS Principles and Analysis Frank Cowell Ex 10.7(1): Question Frank Cowell: Microeconomics 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 By definition, profits of firm 2 are 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 = pq2 [C0 + cq2] where q2 is the output of firm 2 C0, c are parameters of the cost function 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 Frank Cowell: Microeconomics 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 q1 Ex 10.7(2): Question Frank Cowell: Microeconomics 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 Frank Cowell: Microeconomics Firm 2 profits for given value`q of firm 1’s output: Max this with respect to q2 Differentiate to find FOC for a maximum: q2 = ½[b0 c]/b ½`q1 this is firm 2’s reaction function c2 By symmetry, firm 1’s reaction function c1 is b0 b[`q1 + 2q2] c = 0 Solve for firm 2’s output: P2 = b0q2 b[`q1 + q2]q2 [C0 + cq2] 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 Frank Cowell: Microeconomics 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 Plot locus of these points q2 Cournot assumption: Each firm takes other’s output as given • • • • • • Firm 2’s reaction function c2(q1) gives firm 2’s best output c2(∙) response to a given output q1 of firm 1 q1 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 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 qC • to other exactly at qC c2(∙) q1 Ex 10.7(3): Question Frank Cowell: Microeconomics 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 Frank Cowell: Microeconomics Total output is q = q1 + q2 The sum of the firms’ profits can be written as: Maximise this with respect to q differentiate to find FOC for a maximum: b0 2bq c = 0 Solve for joint-profit maximising output: P1 + P2 = b0q1 b[q1 + q2]q1 [C0 + cq1] + b0q2 b[q1 + q2]q2 [C0 + cq2] = b0q b[q]2 [2C0 + cq] q = ½[b0 c]/b However, breakdown into (q1 , q2) 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 q2 Output combinations that max joint profit Symmetric joint profit maximisation c1(∙) (0,qM) q1 + q2 = qM • qC • qJ = ½ qM • qJ c2(∙) • (qM,0) q1 Ex 10.7(4): Question Frank Cowell: Microeconomics 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 Frank Cowell: Microeconomics Firm 2’s reaction function c2: 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: q2 = ½[b0 c]/b ½q1 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 Frank Cowell: Microeconomics Firm 2’s reaction function Firm 1’s opportunity set Firm 1’s profit-max using this set q2 qC • qS • c2(∙) • (qM,0) q1 Ex 10.7(5): Question Frank Cowell: Microeconomics method: compute profit plot in a diagram with (P1 , P2) on axes Ex 10.7(5): Possible payoffs Frank Cowell: Microeconomics P2 Profit space for the two firms • (0, P ) Attainable profits for two firms Symmetric joint profit maximisation M 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 2PJ = [b0 c]2 /[4b] 2C0 (PJ,PJ) (PC,PC) • PM = [b0 c]2 /[4b] C0 PC = [b0 c]2 /[9b] C0 (PS1,PS2) PS1 = [b0 c]2 /[8b] C0 { C0 0 ° (P• ,0) M P1 PS2 = [b0 c]2 /[16b] C0 Ex 10.7: Points to remember Frank Cowell: Microeconomics 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