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Oligopoly (Game Theory)

Oligopoly (Game Theory). Oligopoly: Assumptions. Many buyers Very small number of major sellers (actions and reactions are important) Homogeneous product (usually, but not necessarily) Perfect knowledge (usually, but not necessarily) Restricted entry (usually, but not necessarily).

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Oligopoly (Game Theory)

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  1. Oligopoly (Game Theory)

  2. Oligopoly: Assumptions • Many buyers • Very small number of major sellers (actions and reactions are important) • Homogeneous product (usually, but not necessarily) • Perfect knowledge (usually, but not necessarily) • Restricted entry (usually, but not necessarily)

  3. Oligopoly Models • “Kinked”Demand Curve • Cournot (1838) • Bertrand (1883) • Nash (1950s): Game Theory

  4. “Kinked” Demand Curve P Elastic p* Inelastic D or d Q or q Q* or q*

  5. “Kinked” Demand Curve P Where do p* and q* come from? Elastic p* Inelastic D or d Q* or q* Q or q

  6. Cournot Competition • Assume two firms with no entry allowed and homogeneous product • Firms compete in quantities (q1, q2) • q1 = F(q2) and q2 = G(q1) • Linear (inverse) demand, P=a–bQ where Q = q1+q2 • Assume constant marginal costs,i.e. TCi =cqi for i=1,2 • Aim: Find q1and q2and hence p, i.e. find the equilibrium.

  7. Cournot Competition Firm 1 (w.o.l.o.g.) Profit = TR -TC 1 = P.q1-c.q1 [P =a-bQ and Q =q1+q2, hence P =a-b(q1+q2) P =a-bq1-bq2]

  8. Cournot Competition 1= Pq1-cq1 1= (a-bq1-bq2)q1-cq1 1= aq1-bq12-bq1q2-cq1

  9. Cournot Competition 1 = aq1-bq12-bq1q2-cq1 To find the profit maximising level of q1 for firm 1, differentiate profit with respect to q1and set equal to zero.

  10. Cournot Competition Firm 1’s “Reaction” curve

  11. Cournot Competition Next graph with q1 on the horizontal axis and q2 on the vertical axis Do the same steps to find q2 Note: We have two equations and two unknowns so we can solve for q1 and q2

  12. Cournot Competition q2 COURNOT EQUILIBRIUM q1

  13. Cournot Competition Step 1: Rewrite q1 Step 3: Factor out 1/2 Step 2: Cancel b Step 4: Sub. in for q2

  14. Cournot Competition Step 5: Multiply across by 2 to get rid of the fraction Step 6: Simplify

  15. Cournot Competition Step 7: Multiply across by 2 to get rid of the fraction Step 8: Simplify

  16. Cournot Competition Step 9: Rearrange and bring q1 over to LHS. Step 10: Simplify Step 11: Simplify

  17. Cournot Competition Step 12: Repeat above for q2 Step 13: Solve for price (go back to demand curve) Step 14: Sub. in for q1 and q2

  18. Cournot Competition Step 14: Simplify

  19. Cournot Competition

  20. Cournot Competition: Summary

  21. Cournot v. Bertrand Cournot Nash (q1, q2): Firms compete in quantities, i.e. Firm 1 chooses the best q1 given q2 and Firm 2 chooses the best q2 given q1 Bertrand Nash (p1, p2): Firms compete in prices, i.e. Firm 1 chooses the best p1 given p2 and Firm 2 chooses the best p2 given p1 Nash Equilibrium (s1, s2): Player 1 chooses the best s1 given s2 and Player 2 chooses the best s2 given s1

  22. Bertrand Competition: Bertrand Paradox Assume two firms (as before), a linear demand curve, constant marginal costs and a homogenous product. Bertrand equilibrium: p1=p2=c (This implies zero excess profits and is referred to as the Bertand Paradox)

  23. Perfect Competition v. Monopoly v. Cournot Oligopoly Given Perfect Competition Monopoly

  24. Perfect Competition v. Monopoly v. Cournot Oligopoly

  25. Perfect Competition v Monopoly v Cournot Oligopoly Qm < Qco <QPC Pm>Pco>Ppc

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