Oligopoly (Game Theory)

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)

Oligopoly Models 1. “Kinked” Demand Curve 2. Cournot (1838) 3. Bertrand (1883) 4. Nash (1950s): Game Theory

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

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

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.

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 ]

Cournot Competition  1 =Pq 1 -cq 1  1 =( a - bq 1 - bq 2 )q 1 - cq 1  1 = a q 1 - bq bq 1 q 2 - cq 1

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.

Cournot Competition Firm 1’s “Reaction” curve

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

Cournot Competition q2q2 q1q1 COURNOT EQUILIBRIUM

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

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

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

Cournot Competition Step 9: Rearrange and bring q 1 over to LHS. Step 10: Simplify Step 11: Simplify

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

Cournot Competition Step 14: Simplify

Cournot Competition

Cournot Competition: Summary

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

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)

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

Perfect Competition v. Monopoly v. Cournot Oligopoly

Perfect Competition v Monopoly v Cournot Oligopoly Q m < Q co < Q PC P m > P co > P pc