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Exercise 10.7

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.

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Exercise 10.7

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

  2. 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

  3. 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

  4. Ex 10.7(1): firm 2’siso-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

  5. 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

  6. 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

  7. q2 q1 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 • 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(∙) • • •

  8. 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

  9. 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

  10. 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

  11. 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)

  12. 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”)

  13. 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)

  14. Ex 10.7(4): Stackelberg • 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)

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

  16. Ex 10.7(5): Possible payoffs P2 • 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 0 (PM,0) P1

  17. 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

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