1. Models for Interactions of Small Numbers of Firms

2.  Small number of competitors  Recognize their interdependence ◦ And react to the actions of rivals  Protected by Barrier(s) to Entry ◦ Natural  Economies of Scale  Patents/Access to Technology  Cost of establishing reputation/name recognition ◦ Strategic  Limit pricing  Excess capacity  Aggressive entry response: reputation for “toughness”

3.  Cournot (Augustin Cournot, 1838) ◦ Firms choose quantities assuming rivals do not react  Stackleberg (Heinrich von Stackleberg, 1934) ◦ Leader moves first, anticipates follower’s action  Bertrand (Joseph Bertrand, 1883) ◦ Firms choose prices assuming rivals do not react ◦ Homogeneous products ◦ Differentiated products  Nash Equilibrium ◦ Each firm chooses its best response to its rival ◦ No firm can improve its profit by altering its choice

4.  Two firms (duopoly)  Linear demand: P = 30 – Q  Constant Marginal Cost: MC = 0  Consider firm 1’s choice of Q ◦ R 1 = P 1 Q 1 = Q 1 [30 – (Q 1 + Q 2 )] ◦ R 1 = 30Q 1 – Q 1 2 - Q 1 Q 2 ◦ Set MR 1 = 0 = 30 – 2Q 1 - Q 2 ◦ Solve for Q 1 : 2Q 1 = 30 – Q 2 ◦ => Q 1 = 15 – ½ Q 2 Firm 1’s Reaction Curve ◦ Firm 2’s Reaction Curve: Q 2 = 15 – ½ Q 1  Substitute and Solve: Q 1 = 15 – ½(15 – ½ Q 1 ) ◦ Q 1 = Q 2 = 10, P = 30 – 20 = 10 ◦ π i = $10(10) = $100, π T = $200

5.  Compare Cournot Q, P, and π ◦ Q T = 20, P = $10, π T = $200  To Perfect Competition ◦ P = MC = 0, Q T = 30 – P = 30, π T = $0  And to Collusion: Firms split monopoly Q ◦ R M = P(Q) = Q(30 – Q) = 30Q – Q 2 ◦ MR M = 30 – 2Q = 0 => Q = 15, Q 1 = Q 2 = 7.5 ◦ P = 30 – 15 = $15, π T = $15(15) = $225  Long Run: Entry ◦ Q i = [1/(n+1)]Q C, Q T = [n/(n + 1)]Q C, n = No. of firms ◦ (P – MC)/P = -1/(nE p ) for MC > 0  Graphs

6.  Firm 1: Leader, goes first  Firm 2: Follower, goes 2 nd  Start with Firm 2’s Cournot reaction curve ◦ Q 2 = 15 – ½Q 1  Leader’s Revenue, Marginal Revenue, and Q ◦ R 1 = P 1 Q 1 = 30Q 1 – Q 1 2 - Q 1 Q 2 ◦ R 1 = 30Q 1 – Q 1 2 – Q 1 (15 – ½Q 1 ) = 15Q 1 – ½Q 1 2 ◦ MR 1 = 15 – Q 1 = 0 => Q 1 = 15  Find Firm 2’s quantity: Q 2 = 15 – ½Q 1 = 7.5  P = 30 – (Q1 + Q2) = 30 -22.5 = 7.5  π = (P-MC)(Q1 + Q2) = $7.5(22.5) = $168.75

7.  Firms choose prices assuming rival does not react  Homogeneous good: perfect substitutes ◦ Firm with lowest price takes entire market ◦ Firms choosing the same price split market Q ◦ Linear demand, constant MC ◦ Nash Equilibrium: P = MC with only 2 firms  Graph

8.  Firms produce imperfect substitutes  Firms simultaneously choose prices  Q 1 = 12 – 2P 1 + P 2, Q 2 = 12 – 2P 2 + P 1  Fixed Cost = 20, Variable Cost = 0  π 1 = P 1 Q 1 – 20 = P 1 (12 – 2P 1 + P 2 ) -20 ◦ = 12P 1 – 2P 1 2 + P 1 P 2 - 20  ∆π 1 /∆P 1 = 12 – 4P 1 + P 2 = 0  Firm 1’s Reaction Curve: P 1 = 3 + ¼P 2  Firm 2’s Reaction Curve: P 2 = 3 + ¼P 1

9.  Firm 1’s Reaction Curve: P 1 = 3 + ¼P 2  Firm 2’s Reaction Curve: P 2 = 3 + ¼P 1  Substitute for P 2 in firm 1’s reaction curve and solve for P 1  P 1 = 3 + ¼(3 + ¼P 1 ) = 3¾ + (1/16)P 1  P 1 = P 2 = $4.00  Q 1 = Q 2 = 12 – 2(4) + 4 = 8  π 1 = π 2 = P(Q) – 20 = 4(8) – 20 = $12.00

10.  P 1 = P 2 = $4.00, Q = 16, π 1 = π 2 = $12  Compare to Collusion ◦ P 1 = P 2, Q 1 + Q 2 = 12 - 2P + P +12 - 2P + P ◦ Q = 24 – 2P, π = P(Q) – 20 = 24P – 2P 2 – 20 ◦ ∆π/∆P = 24 – 4P = 0 => P 1 = P 2 = $6.00 ◦ Q = 24 – 2(6) = 12, Q 1 = Q 2 = 6 ◦ π 1 = π 2 = 6(6) – 20 = $16  Compare to Perfect Competition ◦ P 1 = P 2 = MC = 0 ◦ Q 1 = Q 2 = 12 ◦ π 1 = π 2 = 0  Graph

11.  Suppose firm 1 chooses first ◦ π 1 = P 1 Q 1 – 20 = P 1 (12 – 2P 1 + P 2 ) - 20 ◦ Substitute: P 2 = 3 + ¼P 1 ◦ π 1 = 12P 1 – 2P 1 2 + 3P 1 + ¼P 1 2 - 20 ◦ ∆π 1 /∆P 1 = 15 – 3.5(P 1 ) = 0  P 1 = $4.29, P 2 = 3 + ¼ ($4.29) = $4.07  Q 1 = 12 – 2(4.29)+ 4.07 = 7.49  Q 2 = 12 – 2(4.07) + 4.29 = 8.15  π 1 = 4.29(7.49) – 20 = $12.13  π 2 = 4.07(8.15) – 20 = $13.17

12. Firm 2 Cheat Charge $4 Collude Charge $6 Firm 1 Cheat Charge $4 $12, $12$20, $4 Collude Charge $6 $4, $20$16, $16

13. Prisoner B ConfessDon’t Confess Prisoner A Confess -5, -5-1, -10 Don’t Confess -10, -1-2, -2