Applied Microeconomics

1
Applied Microeconomics

• Oligopoly

2
Outline

• The Bertrand Paradox
• Differentiated price competition
• Quantity competition
• Capacity competition
• Repeated Games

3
Readings

• Kreps Chapter 22
• Perloff Chapter 13 and 14
• Tirole Chapters 5 and 6
• Zandt Chapters 11 and 12.A
• Case The GE/Honeywell Merger

4
Oligopoly

• Characteristics
• Small number of firms
• Product differentiation may or may not exist
• Barriers to entry
• Strategic interaction
• Examples cars, steel, aluminum, computers

5
Price Competition

• Consider a duopoly market where two identical
  firms, i1,2, produce a homogeneous good at
  constant marginal cost clt1
• Suppose that both firms set prices in the
  interval c,1 simultaneously
• Firm i has demand function Di(pi,p-i)
• 1-pi for piltp-i
• (1-pi)/2 for pip-i
• 0 for pigtp-i

6
Strategic-Form Game

• This gives the strategic-form game with
• Players i1,2
• Strategy sets Sic,1 for i1,2
• Payoffs ui(pi,p-i)Di(pi,p-i)(pi-c) for i1,2
• To find the NE in this game, we cannot use the
  best-reply correspondence technique, but need to
  reason deductively
• In order to do this it is useful to plot the
  profit that a single firm would earn in this
  market as a function of price

7
Monopoly Profits for a Firm with c0.5
8
Nash Equilibrium

• If one firm sets a price higher than the monopoly
  price pm(1c)/2, the other firm can gain by
  charging pm and earn the monopoly profit
  (1-pm)(pm-c)
• If one firm sets a price p?(c,pm, then the
  other firm can gain by charging a price p?(c,p)
  and earn (1-p)(p-c)
• If one firm sets price pc, the other firms
  best reply is to play c or any higher price
• The unique NE in pure strategies is for both
  firms to set price equal to marginal cost and
  earn zero profits!

9
The Bertrand Paradox

• Hence, although there are only two firms in the
  market, firms behave as in a perfectly
  competitive market
• This is know as the Bertrand Paradox
• Solutions
• Product differentiation
• Quantity competition
• Capacity constraints
• Repeated interaction

10
Product Differentiation

• Change the setting so that the products are
  differentiated with demand functions
  Di(pi,p-i)1-pibp-i for 1-pibp-igt0 and 0
  otherwise, where 0ltblt1
• This gives the game with
• Players i1,2
• Strategy sets Sic,8) for i1,2
• Payoffs ui(pi,p-i)Di(pi,p-i)(pi-c) for i1,2

11
Best-Reply Correspondences

• In this setting, we can use the best-reply
  correspondences to find the NE
• Given that firm 2 is playing p2, the best-reply
  of firm 1 is given by the solution of
  Maxp1c(1-p1bp2 )(p1-c)
• This give the first-order condition-p1c1-p1bp2
  0
• Hence, B1(p2)(1cbp2)/2 and by symmetry
  B2(p1)(1cbp1)/2

12
Best-Reply Correspondences
p2
B1(p2)
B2(p1)
Nash Equilibrium
(1c)/(2-b)
(1c)/2
p1
(1c)/2
(1c)/(2-b)
13
Nash Equilibrium

• Note that each firm wants to charge a higher
  price the higher is the price of the competitor
  this means that prices are strategic complements
• In a NE both firms must be playing a best-reply
• p1(1cbp2)/2
• p2(1cbp1)/2
• Solving this system gives us the unique
  NEp1p2(1c)/(2-b)
• Hence, product differentiation gives us
  equilibrium prices above marginal cost

14
Quantity Competition

• Actually, quantity competition is more natural
  when products are completely homogeneous
• We have already seen Game Theory I that
  quantity (Cournot) competition gives us
  equilibrium prices above marginal cost

15
Capacity Constraints

• Another way to solve the Bertrand paradox is to
  assume that firms compete in two stages
• In a first stage, both firms simultaneously
  decide how much production capacity xi to build
  at a cost of c0 per unit
• In the second stage, both firms compete in prices
  given that they can only produce up to their
  capacity level xi at a unit cost c
• Under mild assumptions, this model gives us the
  same equilibrium prices as in the quantity
  competition model!

16
Repeated Interaction

• If the price competition game is repeated, firms
  can achieve a higher price by creating a cartel
  of colluding firms and threatening to punish any
  deviating firm in future periods by lowering the
  price
• Collusion can either be explicit (illegal in most
  countries), or implicit

17
Example U.S. Turbine Generator Industry

• In the 1950s three firms were active in the
  industry GE, Westinghouse, and Allis-Chalmers
  were making huge profits
• In the 1960s levels of profits were so small that
  Allis-Chalmers was driven out
• In the 1970s GE and Westinghouse were once again
  making enormous profits
• Changes in profitability were due to changes in
  rivalry in the 1950s the firms found a clever
  and illegal way of coordinating prices

18
Example Turbine Generator Industry

• Suppliers would ask the firms to submit bids for
  a new turbine generator
• At the time of the solicitation of bids, the
  firms would consult a lunar calendar and
  coordinate
• Days 1-17 GE got the contract
• Days 18-25 Westinghouse
• Days 26-28 Allis-Chalmers
• The time periods were more or less proportional
  to the market shares of the firms

19
Collusion

• Examples of successful cartels OPEC,
  International Bauxite Assoc., Mercurio Europeo
• Examples of unsuccessful cartels copper, tin,
  coffee, tea, cocoa
• What determines the success of a cartel?
• To answer this question, we need to know
  something about repeated games

20
Finitely Repeated Games

• Suppose the above strategic-form game is repeated
  T times and that the players observe the outcome
  after each play
• In each period t1,2,,T both players
  simultaneously choose whether to fink or
  cooperate
• This gives a history of play at time t of the
  form ht(s0,s1,,st-1) (where s0 is empty)
• Example ht(s0,(Fink,Fink)1,,(Fink,Fink)t-1)

21
Finitely Repeated Games

• A strategy xi in Xi for player i prescribes a
  choice (either fink or cooperate) for any
  possible history ht
• The payoffs to each player is the sum of the
  per-period payoffs V(xi,x-i)ui(si1,s-i1)
  ui(siT,s-iT)
• A Nash equilibrium is a strategy profile x such
  that V(xi,x-i)V(xi,x-i) for all i1,2 and all
  xi in Xi
• A subgame-perfect equilibrium prescribes a NE for
  any subgame

22
Finitely Repeated Games

• We can find the unique subgame-perfect
  equilibrium working backwards
• In period T, for any history hT both players must
  play the unique stage-game NE (Fink, Fink)
• In period T-1, for any history hT-1 both players
  must play the unique stage-game NE (Fink, Fink)
• And so on until period 1, where both players must
  play the unique stage-game NE (Fink, Fink)
• Result If a stage game with a unique NE is
  repeated a finite number of times, the unique
  subgame-perfect equilibrium is for all players to
  always play the NE!

23
Infinitely Repeated Games

• Suppose instead that the stage-game is repeated
  an infinite number of times
• The only other thing that is different from the
  previous set-up is that payoffs now are the
  discounted sum of stage-game payoffsV(xi,x-i)ui
  (si1,s-i1)aui(si2,s-i2)a2ui(si3,s-i3) where
  0ltalt1
• The discount factor can either be interpreted as
  incorporating the time-value of money, a1/(1r),
  or as the probability that the game continues in
  the next period

24
Infinitely Repeated Games

• Consider the following grim trigger strategy
• Start playing cooperate
• Continue playing cooperate as long as nothing
  else has been played
• If something else is played, thereafter always
  play fink
• To check whether it is a NE for both players to
  use this strategy, it is enough to check that no
  player has incentives to deviate in any single
  period t

25
Infinitely Repeated Games

• If my opponent plays the above strategy and I
  always cooperate, I get a payoff 1aa2
• If I instead fink in period t, I get a payoff of
  1aa2at-12at00
• To compare these payoffs, the following formula
  for summing infinite sequences with 0ltalt1 is
  convenient atat1at/(1-a)
• Hence, cooperation can be sustained in a NE (in
  fact a SPE) given that 2atat/(1-a) or 1/2a

26
Infinitely Repeated Games

• For general strategic-form games, for each player
  i define the collusive payoff UiC, the highest
  deviation payoff UiD, and the punishment payoff
  UiP which we assume to be a NE of the stage game
• A trigger strategy for each players can be
  sustained as a SPE if UiDaUiP/(1-a)UiC/(1-a)
  for all i
• Rewriting this expression gives(UiD-UiC)/(UiD-Ui
  P)a

27
Infinitely Repeated Game

• Let UiP be player is payoff in a NE of the stage
  game
• The Folk Theorem Take any outcome of the stage
  game that gives each player i a payoff that
  exceeds UiP. Then, if the discount factor is
  close enough to 1, there is a SPE of the repeated
  game that gives this outcome round after round
• Problem embarrassment of riches

28
Collusion

• If we assume that the price competition game is
  repeated infinitely, we can in the same manner
  sustain collusion at a price pgtc
• Three conditions for successful collusion follow
  from (UiD-UiC)/(UiD-UiP)a
• The future must be sufficiently important
• The gains from deviating cannot be too large
• The losses from being punished must be
  sufficiently large

29
Collusion

• Other factors important for the success of
  collusion
• Potential entrants must be kept out
• Compliance must be observable
• The collusive agreement must be sufficiently
  simple
• Small number of firms

30
Conclusion

• The Bertrand paradox is that price competition
  among few firms with a homogeneous good gives
  prices equal to marginal cost
• It can be solved using product differentiation,
  capacity constraints, quantity competition, and
  repeated interaction
• According to the Folk theorem any outcome with
  payoffs exceeding that of a NE can be sustained
  as a SPE for sufficiently large discount factor
• Collusion is more likely to be successful for
  large discount factor, small gains from
  deviation, severe punishment for deviating,
  observable compliance, small number of firms,
  simple agreement, and entry barriers