Basic Oligopoly Models

1
Chapter 9

• Basic Oligopoly Models

2
Several Interacting Firms

• Oligopoly refers to markets in which firms
  interact strategically.
• Each firms decision depends upon what it
  believes other firms will do.
• More than one firm, but not many.
• Interdependence of decisions makes managers
  problem complex.
• Number of possible beliefs about other firms
  behavior leads to many models of oligopoly.

3
Assumptions

• All models assume
• 1. Few firms who try to predict reactions of
  other firms.
• 2. Barriers to entry even in the long run.
• except for contestable markets, which assumes
  free entry and exit.

4
Sweezy Model (Kinked Demand)

• Firms choose prices.
• Each firm believes a price reduction will be
  matched but a price increase will not be matched.
• Demand is more elastic to price increase than to
  price decrease.

5
Kink in Demand

• At current price P, demand is more elastic for a
  price increase than for a price decrease.

P
price is matched
price not matched
P
D
Q
6
Equilibrium P and Q

• At current Q, there is a gap in MR.
• At lower Q, MR gt MC. At higher Q, MR lt MC.
• Therefore, Q is optimal.

7
Predictions

• MC may shift and leave the optimal Q and P
  unchanged.
• Model explains reluctance of firms to change
  prices, even as MC changes.
• Therefore, price can be sticky.

8
Cournot (Quantity Competition)

• Firms choose quantities, market price depends on
  total quantity supplied.
• Firms assume other firms quantities remain the
  same.
• Each firm has a reaction function that determines
  the best quantity for any given level of other
  firms quantities.

9
A Duopoly Example

• The equilibrium choices are given by the
  quantities that satisfy both reaction functions.
• Firm 1s problem choose Q1 to
• max ?(Q1,Q2) P(Q1,Q2)?Q1 ? TC1(Q1)
• where
• P(Q1,Q2) 10 ? (Q1Q2)
• Set MR1(Q1,Q2) MC1(Q1)

10
Deriving MR

• Firm 1s demand can be written
  P (10 ? Q2) ? Q1
• Firm 2's quantity is assumed constant and so is
  part of the intercept. The slope of demand is
  ?1.
• MR1 (10 ? Q2) ? 2Q1
• Same intercept, twice the slope.

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Firm 1s Reaction Function

• If MC1 0, then set
• MR1 (10 ? Q2) ? 2Q1 0 MC
• and solve for Q1 as a function of Q2
• Q1(Q2) 5 ? (1/2)Q2
• This is firm 1s reaction function.
• It specifies what Q1 should be for any Q2.

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Firm 2s Reaction Function

• Firm 2s problem is similar and reaction function
  is
• Q2(Q1) 5 ? (1/2)Q1
• This is due to symmetry of the example.
• Solve these two equations for the two unknowns,
  Q1 and Q2.

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Solve for Q2

• Q2(Q1) 5 ? (1/2)Q1
• 5 ? (1/2) 5 ? (1/2)Q2
• 2.5 (1/4)Q2
• Q2 (4/3)(2.5) 10/3.

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Solve for Q1

• Similarly
• Q1(Q2) 5 ? (1/2)Q2 5 ? (1/2)(10/3)
• 5 ? 10/6 20/6 10/3.

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Price and Profits

• P(Q1,Q2) 10 ? Q 10 ? 20/3 10/3.
• Profits for firm 1 are
• ?1(Q1,Q2) P?Q1 ? TC1
• (10/3)(10/3) ? 0 100/9 11.11
• Total profits of both firms are
• ?1 ?2 200/9 22.22.

16
Collusion

• MRM 10 ? 2QM 0 MC gt QM 5
• If the firms had colluded on quantity, they would
  choose to each produce half of the monopoly
  quantity QM Q1 Q2.
• or QM 5 and P 10 ? 5 5.
• Total profits are
• ?M PM?QM ? 0 5?5 25
• Each firm gets ?i (1/2) 25 12.5 gt 11.11.

17
Excel Worksheet (click)
18
Better Off Colluding

• Each firm makes a higher profit if the two firms
  collude and act like a monopolist.
• This is one explanation for the desire of firms
  to horizontally merge.
• Anti-trust regulation prohibits collusion because
  it produces less surplus..
• We will return to this later.

19
Stackelberg (Price Leader)

• One firm (follower) chooses a quantity taking the
  other firms quantity as given.
• Price leader chooses a quantity that considers
  the supply of follower.
• The follower behaves like a Cournot firm.
• The leader is more sophisticated.

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Residual Demand
Sfollower

• At P1, follower supply equals demand.
• At P2, no follower supply.
• At intermediate prices, leader faces residual D.

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• Leaders reaction function takes into account the
  followers reaction function.
• Using the residual demand, the leader determines
  its MR and sets it equal to MC.

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Algebraic Example

• Market demand P 50 ? (QLQF)
• Costs TCL 5 2QL
• TCF 5 2QF
• Followers reaction function set
• MRF MCF
• MRF (50 ? QL) ? 2QF 2 MCF
• Solve for QF 24 ? (1/2)QL

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The Leaders QL

• Substitute the follower's reaction function into
  market demand function to get the residual
  demand
• P 50 ? 24 ? (1/2)QL ? QL
• 26 ? (1/2) QL
• Set MRL 26 ? QL 2 MCL
• to get QL 24.

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The Followers QF

• Substitute QL into followers reaction function
• QF 24 ? (1/2)24 12.
• Substitute Q QL QF 36 into market demand
• P 50 ? 36 14.

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Profits

• ?L(QL) P?QL ? 5 ? 2QL
• 14?24 ? 5 ? 2?24 283.
• ?F(QF) P?QF ? 5 ? 2QF
• 14?12 ? 5 ? 2?12 139.

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Bertrand (Price Competition)

• Firms choose their own prices, taking the other
  prices as given.
• Firms sell perfect substitutes.
• Buyers choose the firm with the lowest price.
• Firm with the lowest price captures the market.

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Firm 1s Demand

• QD if P1 lt P2
• Q1 (1/2)QD if P1 P2
• 0 if P1 gt P2
• Given P2, firm 1 chooses P1 lt P2.
• Given P1, firm 2 chooses P2 lt P1.
• Outcome is P MC.

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Equilibrium

• Equilibrium price equals marginal cost.

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Comparing the ModelsAn Example

• demand P 1,000 ? Q1 ? Q2
• costs TCi 4Qi i 1,2
• Competitive market Q Q1 Q2
• Ppc MC 4, Qpc 996, ?i 0
• Monopoly MRM 1,000 ? 2Q 4 MC
• QM 498, PM 502
• ?M (PM ? 4)QM (502 ? 4)498 248,004

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Competition vs. Monopoly

• In general, QM (1/2) Qp
• Total surplus is also half of competition

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Cournot Reaction Functions

• Q1 498 ? (1/2)Q2
• Q2 498 ? (1/2)Q1
• Q1 Q2 332 (1/3)Qpc
• Pc 1,000 ? 664 336, ?i 110,224.
• In general, if there are N Cournot firms
• Qi Qpc/(N1)

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• Cournot produces more than Monopoly but less than
  Competition.
• Surplus is also more than Monopoly but less than
  Competition.

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Collusion and Bertrand

• Cournot collusion is the same as monopoly
• Qi (1/2)QM 249.
• Pc PM 502, ?i (1/2)?M 124,002.
• Bertrand is the same as perfect competition PB
  PPC MC, QB Qpc, ?i 0

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Entry in Oligopoly

• Contestable Markets Free entry and exit
• If ? gt 0, entry occurs, P falls until P AC.
• If ? lt 0, exit occurs, P increases until P AC.
• LR equilibrium implies ? 0.
• Same as long-run perfect competition, even if
  there are few firms.