Chapter6: Oligopoly

1
Chapter6 Oligopoly

• This section studies oligopolistic market with
  game theory.
• In this lecture, I start with two firm Cournot,
  Bertrand and Stackelberg models. Then I extend
  them to n-firms settings. Finally, I will look
  into repeated games

2
Basic setup

• Game theoretic analysis
• There are two or more firms
• Each firm maximizes its profit
• One firms action can influence profit of the
  other firms

3
Basic setup

• Assumptions
• Price taking consumers
• Homogenous products (relaxed in chapter7)
• No entry (relaxed in chapter7)
• Presence of market power
• Firms choose price or quantity

4
Two firms models

• Cournot duopoly
• Market demand Q 1000 P
• There are two firms with symmetric cost
  structure F0, MC280
• Residual demand of firm1 q1 Q - q2
• Each firm chooses its output. Market price is
  given by the demand restriction
• Firm1 equates MR derived from such residual
  demand to MC
• The solution concept is the Nash equilibrium
  mutual best response

5
Two firms models
Figure4-6

P
D(P)
520
Residual demand
MC
280
MR
240
760
1000
Q
6
Two firms models

• When we solve it mathematically, typically we use
  inverse demand function P 1000 Q instead of
  residual demand
• Firm1s problem
• Maxq1 Pq1 280q1
• ?Maxq1 (1000 q1- q2)q1 280q1
• Firm1 solves this problem for any given level of
  q2.
• First order condition
• (1000 q1 q2) q1 280

7
Two firms models

• Intuition is embedded on the partial
  differentiation procedure q2 is given fixed
• Solve the first order condition with respect to
  q1
• q1 360 q2/2 (Reaction function)
• Equilibrium quantities are given by solving a
  system of equation consists of reaction functions
  of all firms
• In this case, two firms are fully symmetric. In
  such case, it is often quicker to impose q1q2
• q1 360 q1/2 ? q1 q2 240, P 520

8
Two firms models
Figure6-2

p2
129600
Profit possibility frontier
57600
Cournot
129600
57600
p1
9
Two firms models

• Cournot vs Monopoly (or Cartel)
• Monopoly in this setting
• MaxQ (1000 - Q)Q 280
• Foc 1000 2Q 280
• ? Q 360 (lower than Cournot), P 640 (higher)
• Joint profit 129600 The two firms would split
  it, often equally
• In the case of Cournot duopoly, profit of each
  firm is 57600

10
Two firms models
Figure6-2

q2
R1
240
R2
q1
240
11
Two firms models

• Bertrand Model
• Demand and cost structures are the same as before
• Q 1000 P, C(q) 280q
• Here, each firm chooses its p, instead of q
• Solution concept is again the Nash equilibrium

12
Two firms models
Figure6-4

P
D(P)
P2
Residual demand
MC
Q
13
Two firms models

• Note that residual demand is not differentiable
• Let p1 p2 MC 280. When p1 p2, suppose
  demand is split half by half
• If a firm deviates and set price upwards Its
  demand will be zero
• If a firm deviates and set price downwards Its
  profit will be negative
• Therefore, p1p2280 is the equilibrium prices
• If p2p1gtMC, each firm has an incentive to cut
  its price
• If p2p1ltMC, each firm has an incentive to raise
  its price
• Therefore, such prices cannot be equilibrium ones
• In fact, p1p2280 is the unique set of
  equilibrium prices The market outcome coincides
  with the perfect competition model

14
Two firms models

• Stackelberg model
• Again, demand and cost structures are the same as
  before
• Q 1000 P, C(q) 280q
• Now, firm1 sets its quantity before firm2 does
• It is crucially important to understand what
  corresponds to the first mover. It has to be able
  to make an irreversible decision. Making such
  decision is called commitment
• In practice, it is a substantial matter how to
  make such commitment. Such issue will be
  discussed in chapter13

15
Two firms models

q2
R1
Following its reaction curve, firm2 produces at
this level
R2
Speculating such behavior of firm2, firm1 has an
incentive to reverse its claim and produce at
this level
Suppose firm1 claims to produce at this level at
the first place
q1
16
Two firms models

• The solution concept is subgame perfection
• Such game can be solved by backward induction
• Start from behavior of firm2, given any q1 chosen
• It follows its reaction function solved before
  q2 360 q1/2 R2(q1)
• Firm1 speculates such behavior and makes its
  irreversible choice of its output
• Maxq1 1000 (q1 R2(q1))q1 280q1
• (640 q1/2) 280q1

17
Two firms models

• First order condition gives a solution to q1
  directly q1 360
• Substituting it to R2 gives q2 180
• From demand, market price is obtained P 460
• Stackelberg vs Cournot
• Stackelberg leader produces more than the Cournot
  level (360 vs 240), while the follower produces
  less (180)
• Stackelberg leader earns more profit than the
  Cournot level (64800vs57600) while the follower
  earns less (32400)
• Market price is lower in the case of Stackelberg
  (460vs520), with higher total unit of output
  (540vs480)

18
Multi-firm models

• N-firm Cournot model
• Settings
• There are N number of firms, which is exogenously
  given
• Market demand P(Q)
• Cost C(q), symmetric across firms
• Each firm i maximizes its profit
• Maxqi P(q1 q2 qN)qi C(qi)
• First order condition
• P(q1 q2 qN)qi qiP(q1 q2 qN)
  C(qi)
• Note that at this process, we are assuming ?qj/ ?
  qi 0 ?i?j

19
Multi-firm models

• To illustrate the property of this market, it is
  useful to derive Lerner Index
• The first order condition above gives
• P MC/P - qi/P?P/?qi
• Use qi Q/N (symmetry) and ?P/?qi dP/dQ (chain
  rule) to get P MC/P -Q/NPdP/dQ
  1/N1/e
• N1 case coincides with monopoly. As N increases,
  LI declines. As N?8, LI converges towards zero
  (perfect competition)

20
Multi-firm models

• To solve the model, further restriction is put on
  the functional forms
• Demand P a - bQ
• Cost C(qi) mqi
• Solving PQ
• FOC can be written as a b(q1 q2 qN) -
  bqi m
• Impose symmetry q2 q3 qN N 1q
• FOC is now written as a 2bqi b(N - 1)q m
• Solve it with respect to qi
• qi (a m)/2b (N 1)/2q (reaction
  function)
• Be careful with the manipulation around qi

21
Multi-firm models

• Impose symmetry again qi q
• q a m/(N 1)b
• Q Nq (a m)N/(N 1)b
• P a bQ a Nm/N 1
• Comparison to monopoly and perfect competition
  results
• Competition results P m, Q (a m)/b
• Monopoly results P (a m)/2, Q (a m)/2b
• As N1, Cournot result coincides that of
  monopoly. As N?8, the result converges to that of
  perfect competition

22
Repeated games

• Multi-period model
• Fixed number of players interact repeatedly over
  time
• Each player knows history of action of the other
  players, and take such history into account to
  make its decision
• Such repeated game is called super-game as a
  whole
• Solution concept is again Subgame-Perfection

23
Repeated games

• An example of two firms model
• Recall the two firm Cournot example discussed
  before
• P 1000 Q, C(q) 280m
• Cournot outcome q 240, p 57600 for both
  firms
• Cartel outcome q 180, p 64800
• For further analysis, we need to obtain
  switchoutcome
• One firm produces 240 and another do 180
• P 580, p1 72000, p2 54000
• Intuition Each firm faces more elastic demand
  curve than as collective cartel. Thus, starting
  at the cartel setting, raising q / lowering price
  would lead to higher profit.

24
Repeated games

• Restrict our attention to the case where each
  firm produces either 240 (Cournot) or 180
  (Cartel) outcomes
• Payoff matrix
• Observe that q 240 is a dominant strategy

25
Repeated games

• Information structure
• Assume that at the time a firm makes decision on
  its output, it does not know output of another
  firm
• Also, assume that direct communication for
  quantity fixing (at q 180) is illegal
• Under the repetition setting, firms may be able
  to achieve the cartel outcome by punishing

26
Repeated games

• Strategy of a firm
• Suppose firm1 takes the following strategy at
  time t
• Produce q1180 if q2 has been set 180 until
  previous period (equilibrium path)
• Produce q2240 otherwise (punishment)
• Given such strategy, firm2 may stay at q1180, or
  alternatively, raise its output to q2240
  (cheat), take higher profit at a period, with
  lower profit onwards

27
Repeated games

• Is such strategy subgame perfect? Illustration
  by a two periods example
• Scenario
• Firm1 produces q1180 at period1
• Firm1 produces q1180 at period2 if firm2
  produces q2180 at period1 / Otherwise, it
  produces q1240
• Following backward induction, start from looking
  at perod2 it is always a dominant strategy to
  choose q2240. q1q2240 is a Nash equilibrium
  outcome.
• Which means that the strategy of firm1 above is
  not subgame perfect.
• Speculating such outcome at period2, both firms
  choose q240 also at period1

28
Repeated games

• Such intuition can be extended to T periods case
• At period T (which is the last subgame),
  q1q2240 is Nash equilibrium outcome
• Given such outcome, at period T-1, q1q2240 is
  subgame perfect
• Given such outcome, at period T-2, q1q2240 is
  subgame perfect
• Speculating such outcome, at period1, q1q2240
  is subgame perfect

29
Repeated games

• However, such strategy can be subgame perfect in
  the infinitely repeated game, or if the last
  period is unknown
• At any given period, cartel outcome is chosen if
• Current profit if cheats (switch) Current
  cartel profit
• lt Value of future stream of cartel profit
• Value of future stream of Cournot
  profit

30
Repeated games

• Some economic applications
• If the market demand is supposed to be growing
  over time, cartel tends to sustain, since LHS of
  the inequality above tends to be greater
• If cheating is difficult to detect, cartel is
  difficult to sustain
• Think about the case where actions of the
  opposite player will be revealed after three
  periods Then, RHS of the inequality above will
  include net profit during the three periods
• In another case, suppose there are many players
  and only the market price can be observed. In
  such case, small cheating of the firm with
  smaller market share would be difficult to
  detect, particularly when market demand
  fluctuates over time
• So far, we have assumed that entry / exit is
  disallowed. If allowed, cartel would attract
  potential entrants.