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Oligopoly (Game Theory)
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Oligopoly: Assumptions u Many buyers u Very small number of major sellers (actions and reactions are important) u Homogeneous product (usually, but not necessarily) u Perfect knowledge (usually, but not necessarily) u Restricted entry (usually, but not necessarily)
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Oligopoly Models 1. “Kinked” Demand Curve 2. Cournot (1838) 3. Bertrand (1883) 4. Nash (1950s): Game Theory
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“Kinked” Demand Curve P Q or q Elastic Inelastic p*p* Q* or q* D or d
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“Kinked” Demand Curve P Q or q Elastic Inelastic p*p* Q* or q* Where do p* and q* come from? D or d
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Cournot Competition u Assume two firms with no entry allowed and homogeneous product u Firms compete in quantities (q 1, q 2 ) u q 1 = F(q 2 ) and q 2 = G(q 1 ) u Linear (inverse) demand, P = a – bQ where Q = q 1 + q 2 u Assume constant marginal costs, i.e. TC i = cq i for i = 1,2 u Aim: Find q 1 and q 2 and hence p, i.e. find the equilibrium.
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Cournot Competition Firm 1 (w.o.l.o.g.) Profit = TR - TC 1 = P.q 1 - c.q 1 [P= a - bQ and Q = q 1 + q 2, hence P= a - b(q 1 + q 2 ) P= a - bq 1 - bq 2 ]
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Cournot Competition 1 =Pq 1 -cq 1 1 =( a - bq 1 - bq 2 )q 1 - cq 1 1 = a q 1 - bq 1 2 - bq 1 q 2 - cq 1
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Cournot Competition 1 = a q 1 -bq 1 2 -bq 1 q 2 -cq 1 To find the profit maximising level of q 1 for firm 1, differentiate profit with respect to q 1 and set equal to zero.
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Cournot Competition Firm 1’s “Reaction” curve
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Cournot Competition Do the same steps to find q 2 Next graph with q 1 on the horizontal axis and q 2 on the vertical axis Note: We have two equations and two unknowns so we can solve for q 1 and q 2
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Cournot Competition q2q2 q1q1 COURNOT EQUILIBRIUM
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Cournot Competition Step 1: Rewrite q 1 Step 2: Cancel b Step 3: Factor out 1/2 Step 4: Sub. in for q 2
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Cournot Competition Step 5: Multiply across by 2 to get rid of the fraction Step 6: Simplify
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Cournot Competition Step 7: Multiply across by 2 to get rid of the fraction Step 8: Simplify
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Cournot Competition Step 9: Rearrange and bring q 1 over to LHS. Step 10: Simplify Step 11: Simplify
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Cournot Competition Step 12: Repeat above for q 2 Step 13: Solve for price (go back to demand curve) Step 14: Sub. in for q 1 and q 2
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Cournot Competition Step 14: Simplify
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Cournot Competition
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Cournot Competition: Summary
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Cournot v. Bertrand Cournot Nash (q 1, q 2 ): Firms compete in quantities, i.e. Firm 1 chooses the best q 1 given q 2 and Firm 2 chooses the best q 2 given q 1 Bertrand Nash (p 1, p 2 ): Firms compete in prices, i.e.Firm 1 chooses the best p 1 given p 2 and Firm 2 chooses the best p 2 given p 1 Nash Equilibrium (s 1, s 2 ): Player 1 chooses the best s 1 given s 2 and Player 2 chooses the best s 2 given s 1
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Bertrand Competition: Bertrand Paradox Assume two firms (as before), a linear demand curve, constant marginal costs and a homogenous product. Bertrand equilibrium: p 1 = p 2 = c (This implies zero excess profits and is referred to as the Bertand Paradox)
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Perfect Competition v. Monopoly v. Cournot Oligopoly Perfect Competition Given Monopoly
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Perfect Competition v. Monopoly v. Cournot Oligopoly
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Perfect Competition v Monopoly v Cournot Oligopoly Q m < Q co < Q PC P m > P co > P pc
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