Transcript MICROECONOMIC THEORY

Chapter 15
Imperfect Competition
Nicholson and Snyder, Copyright ©2008 by Thomson South-Western. All rights reserved.
Short-Run Decisions: Pricing & Output
• When there are only a few firms in a
market, predicting output and price can
be difficult
– how aggressively do firms compete?
– how much information do firms have about
rivals?
– how often do firms interact?
Short-Run Decisions: Pricing & Output
• Bertrand model
– two identical firms choosing prices
simultaneously for identical products
• end up with situation similar to perfect
competition
• Cartel model
– firms act as a group
• end up with the monopoly outcome
Short-Run Decisions: Pricing & Output
In the Bertrand model, output would be Q*
and price would be P*
Price
Under the cartel model, output would be
Q** and price would rise to P**
P**
MC=AC
P*
D
MR
Q**
Q*
Quantity
Short-Run Decisions: Pricing & Output
• Cournot model
– firms set quantities rather than prices
• end up with a result between the Bertrand and
the cartel models
Short-Run Decisions: Pricing & Output
It is important to know where the industry ends
up because total welfare depends on price
and quantity
Price
Under Bertrand, there is no DWL
P**
MC=AC
P*
The cartel model
implies a DWL
D
MR
Q**
Q*
Quantity
Bertrand Model
• Two identical firms producing identical
products at a constant MC = c
• Firms choose prices p1 and p2
simultaneously
– single period of competition
• All sales go to the firm with the lowest
price
– sales are split evenly if p1 = p2
Nash Equilibrium of the Bertrand Model
• The only pure-strategy Nash equilibrium
is p1* = p2* = c
– both firms are playing a best response to
each other
• neither firm has an incentive to deviate to some
other strategy
Nash Equilibrium of the Bertrand Model
• If p1 and p2 > c, a firm could gain by
undercutting the price of the other and
capturing all the market
• If p1 and p2 < c, profit would be negative
Nash Equilibrium of the Bertrand Model
• The same result will arise for any
number of firms n  2
• The Nash equilibrium of the n-firm
Bertrand game is p1* = p2* = … = pn*= c
Bertrand Paradox
• The Nash equilibrium of the Bertrand
model is identical to the perfectly
competitive outcome
• It is paradoxical that competition
between as few as two firms would be
so tough
Cournot Model
• Each firm chooses its output qi of an
identical product simultaneously
• Total industry output Q = q1 + q2 +…+ qn
determines the market price P(Q)
– P(Q) is the inverse demand curve
corresponding to the market demand curve
Cournot Model
• Each firm recognizes that its own
decisions about qi affect price
– P/qi  0
• However, each firm believes that its
decisions do not affect those of any other
firm
– qj /qi = 0 for all j i
Cournot Model
• The FOC for profit maximization are
i
 P Q   P ' Q q i  C 'i( q i )  0
q i
• The firm maximizes profit where MRi =
MCi
Cournot Model
• Price exceeds marginal cost by
P ' Q q i
Cournot Model
• Price will exceed marginal cost, but
industry profits will be lower than in the
cartel model
– social welfare is greater in the Cournot
model than in the cartel situation
Cartel Model
• In the cartel model, each firm chooses qi
for each firm so as to maximize total
industry profits
n
n
n
j 1
j 1
j 1
    j  P Q  q j  C (j q j )
Cartel Model
• The FOC for a maximum gives

q i
 n
j

 j 1
n

  P Q   P ' Q q j  C 'i( q i )  0

j 1

• This is the same result as Cournot,
except that price exceeds marginal cost
by
n
P ' Q q j  P ' Q Q
j 1
Natural Springs Duopoly
• Assume that there are two owners of
natural springs
– firm’s cost of pumping and bottling qi liters
is Ci(qi) = cqi
– each firm has to decide how much water
to supply to the market
• The inverse demand function is
P(Q) = a – Q
Natural Springs Duopoly
• In the Bertrand game the two firms set
price equal to marginal cost
P* = c
total output = Q* = a – c
*i = 0
total profit for all firms = * = 0
Natural Springs Duopoly
• The solution for the Cournot model is
similar
1 = P(Q)q1 – cq1 = (a – q1 – q2 – c)q1
2 = P(Q)q2 – cq2 = (a – q1 – q2 – c)q2
q1 
q2 
a q2 c
2
a  q1  c
2
Natural Springs Duopoly
• The Nash equilibrium will be
q1* = q2* = (a – c)/3
total output = Q* = (2/3)(a – c)
P* = (a + 2c)/3
1* = 2* = (1/9)(a – c)2
total profit for all firms = * = (2/9)(a – c)2
Natural Springs Duopoly
• The objective function for a perfect cartel
involves joint profits
1 + 2 = (a – q1 – q2 – c)q1 + (a – q1 – q2 – c)q2
• The FOCs for a maximum are


1  2  
1  2   a  2q 1  2q 2  c  0
q 1
q 2
Natural Springs Duopoly
• These FOCs do not pin down the market
shares for the firms in a perfect cartel
total output = Q* = (1/2)(a – c)
P* = (1/2)(a + c)
total cartel profit = * = (1/4)(a – c)2
Cournot Best-Response Diagrams
• We can also show each firm’s bestresponse function graphically
– the intersection of these best-response
functions is the Nash equilibrium
Cournot Best-Response Diagrams
The intersection of the firms’ bestresponse functions is the Nash
equilibrium
q2
a-c
BR1(q2)
The Nash equilibrium is where q1* = q2*
= (a – c)/3
a c
2
a c
3
BR2(q1)
a c
a c
3
2
a-c
q1
Cournot Best-Response Diagrams
A change in a firm’s marginal cost will
shift its best-response function
q2
BR1(q2)
If firm 1’s marginal cost rises, its bestresponse-function will shift in and there
will be a new Nash Equilibrium
BR2(q1)
q1
Varying the Number of Cournot Firms
• The Cournot model can represent the
whole range of outcomes by varying the
number of firms
– n =   perfect competition
– n = 1  perfect cartel / monopoly
total output = Q* = (1/2)(a – c)
P* = (1/2)(a + c)
total cartel profit = * = (1/4)(a – c)2
Varying the Number of Cournot Firms
• In equilibrium, identical firms will produce
the same share of output qi = Q/n
• The difference between price and
marginal cost becomes P’(Q)Q/n
– this wedge term gets smaller as the number
of firms gets larger
Prices or Quantities?
• Moving from price competition to quantity
competition changes the outcome
dramatically
– an advantage of the Cournot model is the
realistic implication that the increases in the
number of firms makes the market more
competitive
• but real-world firms tend to set prices rather than
quantities
Capacity Constraints
• Firms must have unlimited capacity for
the Bertrand model to generate the
Bertrand paradox
– more realistically, firms may not have an
unlimited ability to meet all demand
Capacity Constraints
• Consider a two-stage game
– firms build capacity in the first stage
– firms choose prices p1 and p2 in the second
stage
– sales of firms cannot exceed the capacity
chosen in the first stage
Capacity Constraints
• If the cost of building capacity is
sufficiently high, the equilibrium of this
game is the same as the Nash
equilibrium of the Cournot model
– firms choose the price at which quantity
demanded equals total capacity
Product Differentiation
• The possibility of product differentiation
introduces some uncertainty into what
we mean by the market for a good
Product Differentiation
• The law of one price may not hold
– demanders may now have preferences
about which suppliers to purchase the
product from
– there are now many closely related, but not
identical, products to choose from
Product Differentiation
• We will take the market to be a group of
closely related products that are more
substitutable among each other than with
goods outside the group
– measure substitutability by the cross-price
elasticity
Product Differentiation
• We will assume that there are n firms
competing in a particular market
– each product has its own attributes, ai
• The product’s attributes affect its demand
qi(pi, P-i, ai, A-i)
– where P-i is a list of all other firms’ prices and
A-i is a list of the attributes of other firms’
products
Product Differentiation
• Firm i’s total cost is
Ci(qi, ai)
and profit is
i = piqi – Ci(qi, ai)
Product Differentiation
• The FOCs for a maximum are
i
q i C i q i
 q i  pi


0
p i
p i q i p i
i
q i C i q i
 pi


0
a i
a i q i a i
Product Differentiation
• At the profit-maximizing level of output,
marginal revenue is equal to marginal
cost
• Additional differentiation activities should
be pursued up to the point at which the
additional revenues they generate are
equal to their marginal costs
Hotelling’s Beach
• Suppose we are examining the case of
ice cream stands located on a beach
– assume that demanders are located
uniformly along the beach
• one at each unit of beach
– ice cream cones are costless to produce but
carrying them back to one’s place on the
beach results in a cost of td 2
• t = temperature
• d = distance
Spatial Differentiation
L
Ice cream stands are located at points A
and B along a linear beach of length L



a
x
b
Suppose that a person is standing at point x
Spatial Differentiation
• A person located at point x will be
indifferent between stands A and B if
pa + t(x – a)2 = pb + t(b – x)2
where pa and pb are the prices charged
by each stand
Spatial Differentiation
• Solving for x we get
b  a pB  p A
x

2
2t( b  a )
• If the two stands charge an equal price,
the indifferent consumer is located
midway between a and b
Spatial Differentiation
• The Nash equilibrium prices are
t
p A  b  a 2L  a  b 
*
3
t
p B  b  a 4L  a  b 
*
3
Spatial Differentiation
• Profits for the two firms are
*
A 
*
B 
t
18
b  a 2L  a  b 2
t
18
b  a 4L  a  b 2
Tacit Collusion
• Tacit collusion is not the same as an
explicit cartel
– can only be enforced through punishments
internal to the market
Tacit Collusion
• Repeating the stage game T times does
not change the outcome
– the only subgame perfect equilibrium is to
repeat the stage-game Nash equilibrium in
each of the T periods
Tacit Collusion
• If the stage game is repeated infinitely,
the folk theorem applies
– any feasible and individually rational payoff
can be sustained each period as long as
the discount factor () is close enough to 1
Tacit Collusion
• Suppose two firms in a duopoly agree to
tacitly collude to sustain the monopoly
price by using a grim trigger strategy
• Successful tacit collusion provides the
profit stream
V
collude
M
M
  M  1 
2 M



 ...  


2
2
2
 2  1   
Tacit Collusion
• If a firm deviates, it will earn all of the
monopoly profit for itself in the current
period
– the deviation will trigger the grim strategy of
marginal cost pricing for all future periods
– the stream of profits from deviating is Vdeviate
= M
Tacit Collusion
• For deviation not to be profitable, it must
be that Vcollude  Vdeviate
  M  1 


  M
 2  1   

1
2
Tacit Collusion
• Suppose only 2 firms produce a medical
device that is produced at constant
average and marginal cost of $10
• The demand for the device is
Q = 5,000 – 100P
Tacit Collusion
• If the Bertrand game is played in a
single period, each firm will charge $10
and a total of 4,000 devices will be sold
• At the monopoly price, each firm would
earn a profit of $20,000
Tacit Collusion
• Collusion at the monopoly price is
sustainable if
 1 
20,000
  40,000
 1  

1
2
Tacit Collusion
• Now, suppose there are n firms
– monopoly profit is $40,000, but each firm’s
share is 40,000/n
• n firms can successfully collude on the
monopoly price if
40,000  1 

  40,000
n  1  
  1
1
n
Investment, Entry, and Exit
• Even when making long-run decisions,
an oligopolist must consider how rivals
will respond
• Crucial to these decisions is how easy it
is to reverse a decision once it has been
made
Investment, Entry, and Exit
• Absent strategic considerations, a firm
would value flexibility and reversibility
• But commitment has value as well
– firm can gain first-mover advantage
Sunk Costs and Commitment
• Sunk costs are expenditures on
irreversible investments
– these allow the firm to produce in the
market but have no residual value if the
firm leaves the market
– could include expenditures on unique types
of equipment or job-specific training of
workers
First-Mover Advantage in the Stackelberg Model
• This model is similar to the duopoly
version of the Cournot model except
firms move sequentially
– firm 1 (the leader) chooses q1 first
– firm 2 (the follower) chooses q2 after
seeing q1
First-Mover Advantage in the Stackelberg Model
• We can solve the model by backward
induction
– begin with output of the follower (q2)
• this results in a best-response function for Firm
2 [BR2(q1)]
– substitute BR2(q1) into Firm 1’s profit
function
1 = P(q1 + BR2(q1))q1 – C1(q1)
First-Mover Advantage in the Stackelberg Model
• The FOC is
1
 P Q   P ' Q q 1  P ' Q BR '2 q 1 q 1  C '1( q 1 )  0
q 1
S
– this is the same FOC as in the Cournot
model except for the addition of the
strategic effect of Firm 1’s output on Firm 2
(S)
First-Mover Advantage in the Stackelberg Model
• The strategic effect will lead Firm 1 to
produce more than it would have in a
Cournot model
– this leads Firm 2 to lower output
– if Firm 2 lowers output, the market price
will rise, increasing Firm 1’s revenue from
existing sales
First-Mover Advantage in the Stackelberg Model
• The strategic effect would not occur if
– the leader’s output was unobservable to
the follower
– the leader could reverse its output choice
in secret
• The leader must be able to commit or
else firms are back in the Cournot game
Stackelberg Springs
• Recall the natural springs duopoly
discussed earlier
– this time we will assume they choose
output levels sequentially
– Firm 1 is assumed to be the leader
– Firm 2 is assumed to be the follower
Stackelberg Springs
• Solving for Firm 2’s output, we get its
best-response function
q2 
a  q1  c
2
• Substituting Firm 2’s best-response
function into Firm 1’s profit function,
1


 a  q1  c 
1  a  q 1  
  c q 1  a  q 1  c q 1
2
2




Stackelberg Springs
• Taking the FOC,
 1
 a  2q 1  c   0
q 1 2
• This means that
q 
*
1
a  c 
2
1
2


  a c
8
*
1
q 
*
2
a  c 
4
1
a  c 2
 
16
*
2
Contrast with Price Leadership
• In the Stackelberg game, the leader
uses a “top dog” strategy
– aggressively overproduces to force the
follower to scale back production
– the leader earns more (than it would in the
Cournot game), while the follower earns
less
Contrast with Price Leadership
• The leader could follow a “puppy dog”
strategy
– increases its price, producing less output
than in a simultaneous-move game
– acts less aggressively, leading its rival to
compete less aggressively
Contrast with Price Leadership
• The crucial difference between these
two games is that the slopes of the
best-response functions differ
– “top dog” strategy leads to a downwardsloping best-response function for Firm 2
– “puppy dog” strategy leads to an upwardsloping best-response function for Firm 2
Strategic Entry Deterrence
• In some cases, first-mover advantages
may be large enough to deter all entry
by rivals
– however, it may not always be in the firm’s
best interest to create that large a capacity
Deterring Entry of a Natural Spring
• We will now add an entry stage to the
Stackelberg Natural Springs example
– Firm 2 must decide whether to enter the
market after seeing Firm 1’s output level
– entry for Firm 2 requires a sunk cost, K2
• Firm 1 incurred sunk cost before the start of the
game
– we will assume a = 120 and c = 0
Deterring Entry of a Natural Spring
• We start by calculating Firm 1’s profit if it
accommodates entry
– this was done in earlier example
q1acc = (a – c)/2 = 60
1acc = (a – c)2/8 = 1,800
Deterring Entry of a Natural Spring
• Next, we compute Firm 1’s profit if it
deters entry
– Firm 1 needs to produce and amount high
enough that Firm 2 cannot earn enough
profit to cover sunk cost
Deterring Entry of a Natural Spring
• Firm 2’s best-response function is
q2 = (120 – q1)/2
• Substituting into Firm 2’s profit function
gives us
 120  q
2  
2

det
1
2

  K 2

Deterring Entry of a Natural Spring
• Setting Firm 2’s profit to zero yields
q1det  120  2 K 2
1det  2 K 2( 120  2 K 2 )
Deterring Entry of a Natural Spring
• The final step is to compare 1acc with
1det
• The level of K2 at which the firm would be
indifferent is K2 = 77
– if K2 < 77, entry is cheap and Firm 1 would
have to increase its output to 102 to deter
entry
Signaling
• The ability to signal is another firstmover advantage
– if a second mover has incomplete
information about the market, it may try to
watch the first-mover to learn about market
conditions
– the first mover may distort its actions to
manipulate what the second mover learns
Entry-Deterrence Model
• Consider a game where two firms
choose a price for their differentiated
products
– Firm 1 is a first mover
– Firm 2 is a second mover
Entry-Deterrence Model
• Firm 1 has private information about its
marginal costs
– High costs with a probability of Pr(H)
– Low costs with a probability of Pr(L) = 1 –
Pr(H)
• In period 1, Firm 1 serves the market
alone
– at the end of the period, Firm 2 observes p1
and considers entry
Entry-Deterrence Model
• If Firm 2 enters, it faces a sunk cost of
K2 and learns the true nature of Firm 1’s
costs
• The firms then behave as duopolists in
the second period
– choosing prices for differentiated products
Entry-Deterrence Model
• If Firm 2 does not enter, it obtains a
payoff of zero
– Firm 1 serves the market alone
• Assume there is no discounting
between periods
Entry-Deterrence Model
• Let Dit = duopoly profit for firm i if Firm 1
is of type t (low-cost, high-cost)
• Assume that D2L < K2 < D2H
– Firm 2 earns more than its entry cost only if
Firm 1 is high-cost
Entry-Deterrence Model
• If Firm 1 is low cost, it has only one
relevant action
– setting the monopoly price (p1L)
• If Firm 1 is high cost, it has two possible
actions
– set the monopoly price associated with its
type (p1H)
– choose the same price as the low-cost type
(p1L)
Entry-Deterrence Model
• Let M1t = Firm 1’s monopoly profit if it is
of type t
• Let R = the loss in Firm 1’s profit if it is
high-cost, but chooses p1L
Entry-Deterrence Model
Separating Equilibrium
• In a separating equilibrium, the different
types of the first-mover must choose
different actions
• There is only one possibility for Firm 1
– the low-cost type chooses p1L
– the high-cost type chooses p1H
Separating Equilibrium
• Firm 2 sees Firm 1’s actions
– stays out is Firm 1 charges p1L
– enters if Firm 1 charges p1H
• Would a high-cost Firm 1 prefer to
charge a price of p1L?
– only if
R < M1H – D1H
Pooling Equilibrium
• If R < M1H – D1H, the high type would
like to pool with the low type if pooling
deters entry
– pooling deters entry if Firm 2’s prior belief
that Firm 1 is the high type is low enough
that Firm 2’s expected payoff from entering
is less than zero
Predatory Pricing
• The incomplete-information model of
entry deterrence may explain why a firm
would engage in predatory pricing
– charging an artificially low price to prevent
potential rivals from entering or to force
existing rivals to exit
Barriers to Entry
• In order for a market to be oligopolistic,
there must be barriers to entry
– sunk cost to enter
– government intervention (patents, licensing)
– search costs faced by consumers
– product differentiation (brand loyalty)
– entry deterrence by existing firms
Long-Run Equilibrium
• Suppose there are a large number of
symmetric firms that are potential
entrants into a market
– make the decision simultaneously
• Entry requires a sunk cost, K
• Let n = number of firms that decide to
enter
Long-Run Equilibrium
• Let g(n) = profit earned by a firm (not
including sunk cost)
– we would expect g’(n) < 0
Long-Run Equilibrium
• The sub-game perfect equilibrium
number of firms (n*) will satisfy two
conditions
– they earn enough to cover their entry costs
• g(n*)  K
– an additional firm cannot cover its entry
cost
• g(n*+1) < K
Long-Run Equilibrium
• Is the long-run equilibrium efficient?
• A benevolent social planner would
choose n to maximize
CS(n) + ng(n) – nK
– CS(n) is equilibrium consumer surplus
– ng(n) is equilibrium gross profits
– nK is total expenditure on sunk entry costs
Long-Run Equilibrium
• The long-run equilibrium number of
firms (n*) may be greater or less than
the social optimum (n**) depending on
two effects
– the appropriability effect
– the business-stealing effect
Long-Run Equilibrium
• The appropriability effect
– the social planner takes account of
increased consumer surplus from lower
prices
– firms do not
• This implies that n** > n*
Long-Run Equilibrium
• The business-stealing effect
– entry causes the profits of existing firms to
fall
– the marginal firm does not consider the
drop in other firms’ profits when making its
entry decision (the social planner would)
• This implies that n* > n**
Feedback Effect
• The feedback effect is that the more
profitable a market is for a given number
of firms, the more firms will enter the
market, making the market more
competitive and less profitable than it
would be if the number of firms was fixed
Monopoly on Innovation
• The dissipation effect
– competition dissipates some of the profit
from innovation and thus reduces the
incentives to innovate
• The replacement effect
– firms gain less in incremental profit and
thus have less incentive to innovate if the
new product replaces an existing product
Competition for Innovation
• New firms are not always more
innovative than existing ones
– the dissipation effect may counteract the
replacement effect
• Dominant firms apply for “sleeping
patents” to prevent entry
– patents that are never implemented
Important Points to Note:
• One of the most basic oligopoly
models, the Bertrand model, involves
two identical firms that set prices
simultaneously
– the equilibrium resulted in the Bertrand
paradox
• even though the oligopoly is as concentrated
as possible, the two firms act as perfect
competitors
Important Points to Note:
• The Bertrand paradox is not the
inevitable outcome in an oligopoly but
can be escaped by changing
assumptions
– allowing for quantity competition,
differentiated products, search costs,
capacity constraints, or repeated play
leading to collusion
Important Points to Note:
• As in the Prisoners’ Dilemma, firms
could profit by coordinating on a less
competitive outcome
– this outcome will be unstable unless
firms can explicitly collude by forming a
legal cartel or tacitly collude in a
repeated game
Important Points to Note:
• For tacit collusion to sustain supercompetitive profits, firms must be
patient enough that the loss from a
price war in future periods to punish
undercutting exceeds the benefit from
undercutting in the current period
Important Points to Note:
• Whereas a nonstrategic monopolist
prefers flexibility to respond to
changing market conditions, a
strategic oligopolist may prefer to
commit to a single choice
– the firm can commit to the choice if it
involves a sunk cost that cannot be
recovered if the choice is later reversed
Important Points to Note:
• A first mover can gain an advantage
by committing to a different action
from what it would choose in the Nash
equilibrium of a simultaneous game
– to deter entry, the first mover should
commit to reducing the entrant’s profits
– if it does not deter entry, the first mover
should commit to a strategy that leads its
rival to compete less aggressively
Important Points to Note:
• Holding the number of firms in an
oligopoly constant in the short run, an
introduction of a factor that softens
competition will raise firms’ profit
– an offsetting effect in the long run is that
entry will now be more attractive
• reducing oligopoly profit
Important Points to Note:
• Innovation may be even more
important than low prices for total
welfare in the long run
– determining which oligopoly structure is
the most innovative is difficult because
offsetting effects are involved
• dissipation
• replacement