Download Document

Chapter 5
Business Strategy
in Oligopoly Markets
EC 170 Industrial Organization
Introduction
• In the majority of markets firms interact with few
competitors
• In determining strategy each firm has to consider rival’s
reactions
– strategic interaction in prices, outputs, advertising …
• This kind of interaction is analyzed using game theory
– assumes that “players” are rational
• Distinguish cooperative and noncooperative games
– focus on noncooperative games
• Also consider timing
– simultaneous versus sequential games
EC 170 Industrial Organization
Oligopoly Theory
• No single theory
– employ game theoretic tools that are appropriate
– outcome depends upon information available
• Need a concept of equilibrium
– players (firms?) choose strategies, one for each player
– combination of strategies determines outcome
– outcome determines pay-offs (profits?)
• Equilibrium first formalized by Nash: No firm wants to
change its current strategy given that no other firm
changes its current strategy
EC 170 Industrial Organization
Nash Equilibrium
• Equilibrium need not be “nice”
– firms might do better by coordinating but such coordination may
not be possible (or legal)
• Some strategies can be eliminated on occasions
– they are never good strategies no matter what the rivals do
• These are dominated strategies
– they are never employed and so can be eliminated
– elimination of a dominated strategy may result in another being
dominated: it also can be eliminated
• One strategy might always be chosen no matter what the
rivals do: dominant strategy
EC 170 Industrial Organization
An Example
• Two airlines
• Prices set: compete in departure times
• 70% of consumers prefer evening departure, 30% prefer
morning departure
• If the airlines choose the same departure times they share
the market equally
• Pay-offs to the airlines are determined by market shares
• Represent the pay-offs in a pay-off matrix
EC 170 Industrial Organization
The example (cont.) What is the
The Pay-Off Matrixequilibrium for this
game?
The left-hand
American
number is the
pay-off to
Morning
Evening
Delta
Morning
(15, 15)
(30, 70)
(70, 30)
The right-hand
number is the
(35, 35)
pay-off
to
American
Delta
Evening
EC 170 Industrial Organization
If AmericanThe example
If American
(cont.)
chooses a morning
The morning departure
chooses
an
evening
The morning departure
Pay-Off Matrix
departure, DeltaThe
departure, Delta is also a dominated
is a dominated
will choose
strategy for American and
will so
still choose
strategy for Delta and
evening
American
can be eliminated. evening again can be eliminated
The Nash Equilibrium must
therefore be one in which
Morning
Evening
both airlines choose
an evening departure
Morning
(15, 15)
(30, 70)
Delta
Evening
(70, 30)
EC 170 Industrial Organization
(35, 35)
The example (cont.)
• Now suppose that Delta has a frequent flier program
• When both airline choose the same departure times Delta
gets 60% of the travelers
• This changes the pay-off matrix
EC 170 Industrial Organization
If Delta
The
example (cont.)
chooses
morning
TheaPay-Off
Matrix
However, a morning departure
departure, American
But ifAmerican
Delta will
is
haschoose
nostill a dominated strategy for Delta
chooses dominated
an eveningstrategy
(Evening isAmerican
still a dominant strategy.
evening
departure,
American American
knows
willand
choose
this
so
Morning
Evening
morning
chooses
a morning
departure
Morning
(18, 12)
(30, 70)
Delta
Evening
(70,
(70,30)
30)
EC 170 Industrial Organization
(42, 28)
Nash Equilibrium Again
• What if there are no dominated or dominant strategies?
• The Nash equilibrium concept can still help us in
eliminating at least some outcomes
• Change the airline game to a pricing game:
– 60 potential passengers with a reservation price of $500
– 120 additional passengers with a reservation price of $220
– price discrimination is not possible (perhaps for regulatory reasons
or because the airlines don’t know the passenger types)
– costs are $200 per passenger no matter when the plane leaves
– the airlines must choose between a price of $500 and a price of
$220
– if equal prices are charged the passengers are evenly shared
– Otherwise the low-price airline gets all the passengers
• The pay-off matrix is now:
EC 170 Industrial Organization
The example (cont.)
If Delta prices high
The
Pay-Off low
Matrix
If both price highand
American
then both get 30then American gets
passengers.
If Delta prices
Profit
low
all 180 passengers.American
and
perAmerican
passengerhigh
isProfit per passenger If both price low
they each get 90
then$300
Delta gets
is $20
passengers.
PH = $500
PL = $220
all 180 passengers.
Profit per passenger
Profit per passenger
is $20
is $20
P = $500 ($9000,$9000)
($0, $3600)
H
Delta
PL = $220
($3600, $0)
EC 170 Industrial Organization
($1800, $1800)
Nash
Equilibrium
There
is no simple (cont.)
way to choose between
(PH, PH) is a Nash
these
equilibria.
even so, the Nash concept
equilibrium.
Matrix
(PHThe
, PL)Pay-Off
cannotBut
be
(PL, PL) is
aequilibria
Nash
has
eliminated
half
of
the
outcomes
as
If both are pricing
a Nash equilibrium.
equilibrium.
and familiarity
highCustom
then neither
wants
If American prices
If both are pricing
might
lead low
boththen
to Delta shouldAmerican
to change
(PL, PHprice
) cannot
low then neither wants
highbealso price low
a Nash equilibrium.
to change
PH = $500
PL = $220
If American prices
There are two Nash
high then Delta should
equilibria to this version
also pricePhigh
($9000,$9000)
($0, $3600)
of the$9000)
game
H = $500 ($9000,
“Regret” might
Delta
cause both to
price low
PL = $220
($3600, $0)
($1800, $1800)
EC 170 Industrial Organization
Nash Equilibrium
(cont.)
There
is no simple
(PH, PH) is a Nash
waybe
to choose between
equilibrium.
The
Pay-Off
Matrix
There
are two(PNash
,
P
)
cannot
H
L
(Pat
a Nash
L, P
L) iswe
these
equilibria,
but
least
have
If
both
are
pricing
equilibria to thisa version
Nash equilibrium.
equilibrium.
eliminated
half
of
the
outcomes as
high thenofneither
wants
the game
If American prices
American
If both are pricing
possible
equilibria
to change low then Delta would
(PL, PH) cannot be
low then neither wants
want to price low, too.
a Nash equilibrium.
to change
PH = $500
PL = $220
If American prices
high then Delta should
also pricePhigh
($9000,$9000)
$9000)
H = $500 ($9000,
($0, $3600)
Delta
PL = $220
($3600, $0)
EC 170 Industrial Organization
($1800, $1800)
The only sensible
choice
for
Nash
Equilibrium
(cont.)
consideration
of thethat Delta
Suppose
Delta is PH knowingSometimes,
that American
The
Pay-Off
Matrix
of
moves
can help
set its price first
will follow
with
PHifand
each
will
Delta
can see
that
it timing
sets
find the equilibrium This means that
earn
$9000.
theus
Nash
a high
price,
then So,
American
equilibria
is (PH, PH)
PH, PL cannot be
will
do bestnow
by also
American
an equilibrium
pricing high. Delta
outcome
earns $9000
This means
PH = $500
PL = $220
Delta canthat
alsoPsee
that
L,PH
if it sets a low
price,beAmerican
cannot
an
will
do$500
bestequilibrium
by
pricing$3,000)
low.
PH =
($9000,$9000)
($0, $3600)
($3,000,
Delta will then earn $1800
Delta
PL = $220
($3600, $0)
EC 170 Industrial Organization
($1800, $1800)
Oligopoly Models
• There are three dominant oligopoly models
– Cournot
– Bertrand
– Stackelberg
• They are distinguished by
– the decision variable that firms choose
– the timing of the underlying game
• But each embodies the Nash equilibrium concept
EC 170 Industrial Organization
The Cournot Model
• Start with a duopoly
• Two firms making an identical product (Cournot supposed
this was spring water)
• Demand for this product is
P = A - BQ = A - B(q1 + q2)
where q1 is output of firm 1 and q2 is output of firm 2
• Marginal cost for each firm is constant at c per unit
• To get the demand curve for one of the firms we treat the
output of the other firm as constant
• So for firm 2, demand is P = (A - Bq1) - Bq2
EC 170 Industrial Organization
The Cournot model (cont.)
$
P = (A - Bq1) - Bq2
The profit-maximizing
A - Bq1
choice of output by firm
2 depends upon the
A - Bq’1
output of firm 1
Marginal revenue for
Solve this
firm 2 is
c
output
MR2 = (A - Bq1)for
- 2Bq
q22
MR2 = MC
A - Bq1 - 2Bq2 = c
If the output of
firm 1 is increased
the demand curve
for firm 2 moves
to the left
Demand
MC
MR2
q*2
 q*2 = (A - c)/2B - q1/2
EC 170 Industrial Organization
Quantity
The Cournot model (cont.)
q*2 = (A - c)/2B - q1/2
This is the best response function for firm 2
It gives firm 2’s profit-maximizing choice of output for any
choice of output by firm 1
There is also a best response function for firm 1
By exactly the same argument it can be written:
q*1 = (A - c)/2B - q2/2
Cournot-Nash equilibrium requires that both firms be on
their best response functions.
EC 170 Industrial Organization
Cournot-Nash Equilibrium
q2
(A-c)/B
(A-c)/2B
qC2
If firm 2 produces
The best response
The Cournot-Nash
(A-c)/B then firm
function for firm 1 is
equilibrium is at
1 will choose to
q*1 = (A-c)/2B - q2/2
Point
C atfunction
the intersection
Firm 1’s best
response
produce no output
ofIfthe
best
response
firm
2 produces
functions
nothing
then firm The best response
1 will produce thefunction for firm 2 is
C
monopoly output q*2 = (A-c)/2B - q1/2
Firm
2’s best response function
(A-c)/2B
qC1 (A-c)/2B
(A-c)/B
EC 170 Industrial Organization
q1
Cournot-Nash Equilibrium
q*1 = (A - c)/2B - q*2/2
q2
q*2 = (A - c)/2B - q*1/2
(A-c)/B
 q*2 = (A - c)/2B - (A - c)/4B
+ q*2/4
Firm 1’s best response function
 3q*2/4 = (A - c)/4B
 q*2 = (A - c)/3B
(A-c)/2B
(A-c)/3B
C
 q*1 = (A - c)/3B
(A-c)/2B
Firm 2’s best response function
q1
(A-c)/B
(A-c)/3B
EC 170 Industrial Organization
Cournot-Nash Equilibrium (cont.)
•
•
•
•
•
•
•
•
In equilibrium each firm produces qC1 = qC2 = (A - c)/3B
Total output is, therefore, Q* = 2(A - c)/3B
Recall that demand is P = A - BQ
So the equilibrium price is P* = A - 2(A - c)/3 = (A + 2c)/3
Profit of firm 1 is (P* - c)qC1 = (A - c)2/9
Profit of firm 2 is the same
A monopolist would produce QM = (A - c)/2B
Competition between the firms causes their total output
to exceed the monopoly output. Price is therefore lower
than the monopoly price
• But output is less than the competitive output (A - c)/B
where price equals marginal cost and P exceeds MC
EC 170 Industrial Organization
Numerical Example of Cournot Duopoly
•
•
•
•
•
•
•
•
•
Demand: P = 100 - 2Q = 100 - 2(q1 + q2); A = 100; B = 2
Unit cost: c = 10
Equilibrium total output: Q = 2(A – c)/3B = 30;
Individual Firm output: q1 = q2 = 15
Equilibrium price is P* = (A + 2c)/3 = $40
Profit of firm 1 is (P* - c)qC1 = (A - c)2/9B = $450
Competition: Q* = (A – c)/B = 45; P = c = $10
Monopoly: QM = (A - c)/2B = 22.5; P = $55
Total output exceeds the monopoly output, but is less
than the competitive output
• Price exceeds marginal cost but is less than the
monopoly price
EC 170 Industrial Organization
Cournot-Nash Equilibrium (cont.)
• What if there are more than two firms?
• Much the same approach.
• Say that there are N identical firms producing identical
products
• Total output Q = q1 + q2 + … This
+ qN denotes output
• Demand is P = A - BQ = A - B(q
+ q2 +firm
… +other
qN)
of 1every
than
1
• Consider firm 1. It’s demand curve
canfirm
be written:
P = A - B(q2 + … + qN) - Bq1
• Use a simplifying notation: Q-1 = q2 + q3 + … + qN
• So demand for firm 1 is P = (A - BQ-1) - Bq1
EC 170 Industrial Organization
The Cournot model (cont.)If the output of
the other firms
is increased
the demand curve
for firm 1 moves
to the left
$
P = (A - BQ-1) - Bq1
The profit-maximizing
choice of output by firm A - BQ-1
1 depends upon the
output of the other firms A - BQ’
-1
Marginal revenue for
Solve this
firm 1 is
c
for output
MR1 = (A - BQ-1) - 2Bq
q1 1
Demand
MC
MR1
q*1
MR1 = MC
A - BQ-1 - 2Bq1 = c  q*1 = (A - c)/2B - Q-1/2
EC 170 Industrial Organization
Quantity
Cournot-Nash Equilibrium (cont.)
q*1 = (A - c)/2B - Q-1/2
How do we solve this
As the
number
of
for
q*
?
1
The firms
are
identical.
As the
number
firms increases
output of
 q*1 = (A - c)/2B - (N - 1)q*1So
/2 in equilibrium they
firms
of each
firmincreases
falls
have identical
 (1 + (N - 1)/2)q*1 = (A - c)/2Bwillaggregate
output
As
the
number
ofof
outputs
As
increases
the
number
 q*1(N + 1)/2 = (A - c)/2B
firms
firmsincreases
increasesprice
profit
 q*1 = (A - c)/(N + 1)B
tends
marginal
cost
of to
each
firm falls
 Q*-1 = (N - 1)q*1
 Q* = N(A - c)/(N + 1)B
 P* = A - BQ* = (A + Nc)/(N + 1)
Profit of firm 1 is P*1 = (P* - c)q*1 = (A - c)2/(N + 1)2B
EC 170 Industrial Organization
Cournot-Nash equilibrium (cont.)
• What if the firms do not have identical costs?
• Once again, much the same analysis can be used
• Assume that marginal costs of firm 1 are c1 and of firm 2
Solve this
are c2.
• Demand is P = A - BQ = A - B(q1 + q2for
) output
A symmetric result
q1
• We have marginal revenue for
firm
1
as
before
holds for output of
• MR1 = (A - Bq2) - 2Bq1
firm 2
• Equate to marginal cost: (A - Bq2) - 2Bq1 = c1
 q*1 = (A - c1)/2B - q2/2
 q*2 = (A - c2)/2B - q1/2
EC 170 Industrial Organization
Cournot-Nash Equilibrium
q2
(A-c1)/B
R1
q*1 = (A - c1)/2B - q*2/2
The equilibrium
As the marginal
output cost
of firm
2 q*22 = (A - c2)/2B - q*1/2
of firm
What
happens
increases
and
falls its
bestof
response

q*2 =to
(Athis
- c2)/2B - (A - c1)/4B
firmcurve
1 equilibrium
fallsshifts to when + q* /4
2
costs
change?
the
right
 3q*2/4 = (A - 2c2 + c1)/4B
 q*2 = (A - 2c2 + c1)/3B
(A-c2)/2B
R2
C
 q*1 = (A - 2c1 + c2)/3B
(A-c1)/2B
(A-c2)/B
q1
EC 170 Industrial Organization
Cournot-Nash Equilibrium (cont.)
• In equilibrium the firms produce
qC1 = (A - 2c1 + c2)/3B; qC2 = (A - 2c2 + c1)/3B
• Total output is, therefore, Q* = (2A - c1 - c2)/3B
• Recall that demand is P = A - BQ
• So price is P* = A - (2A - c1 - c2)/3 = (A + c1 +c2)/3
• Profit of firm 1 is (P* - c1)qC1 = (A - 2c1 + c2)2/9B
• Profit of firm 2 is (P* - c2)qC2 = (A - 2c2 + c1)2/9B
• Equilibrium output is less than the competitive level
• Output is produced inefficiently: the low-cost firm should
produce all the output
EC 170 Industrial Organization
A Numerical Example with Different Costs
• Let demand be given by: P = 100 – 2Q; A = 100, B =2
• Let c1 = 5 and c2 = 15
• Total output is, Q* = (2A - c1 - c2)/3B = (200 – 5 – 15)/6 =
30
• qC1 = (A - 2c1 + c2)/3B = (100 – 10 + 15)/6 = 17.5
• qC2 = (A - 2c2 + c1)/3B = (100 – 30 + 5)/3B = 12.5
• Price is P* = (A + c1 +c2)/3 = (100 + 5 + 15)/3 = 40
• Profit of firm 1 is (A - 2c1 + c2)2/9B =(100 – 10 +5)2/18 =
$612.5
• Profit of firm 2 is (A - 2c2 + c1)2/9B = $312.5
• Producers would be better off and consumers no worse off
if firm 2’s 12.5 units were instead produced by firm 1
EC 170 Industrial Organization
Concentration and Profitability
•
•
•
•
•
Assume that we have N firms with different marginal costs
We can use the N-firm analysis with a simple change
Recall that demand for firm 1 is P = (A - BQ-1) - Bq1
But then demand for firm i is P = (A - BQ-i) - Bqi
Equate this to marginal cost ci
But Q*-i + q*i = Q*
A - BQ-i - 2Bqi = ci
and A - BQ* = P*
This can be reorganized to give the equilibrium condition:
A - B(Q*-i + q*i) - Bq*i - ci = 0
 P* - Bq*i - ci = 0  P* - ci = Bq*i
EC 170 Industrial Organization
Concentration and profitability (cont.)
P* - ci = Bq*i
The price-cost margin
Divide by P* and multiply the right-hand
side
by is
Q*/Q*
for each
firm
determined by its own
P* - ci
BQ* q*i
market share and overall
=
P*
P* Q*
market demand elasticity
The verage price-cost margin
But BQ*/P* = 1/ and q*is
= si
i/Q*
determined
by industry
so: P* - ci = si concentration as measured by
 the Herfindahl-Hirschman Index
P*
Extending this we have
P* - c
H
=
P*

EC 170 Industrial Organization
Price Competition: Bertrand
• In the Cournot model price is set by some market clearing
mechanism
• Firms seem relatively passive
Check that with
• An alternative approach is to assume thatthis
firms
compete
demand
andin
prices: this is the approach taken by Bertrand
these costs the
• Leads to dramatically different results monopoly price is
• Take a simple example
$30 and quantity
40 units
– two firms producing an identical product (spring is
water?)
– firms choose the prices at which they sell their water
– each firm has constant marginal cost of $10
– market demand is Q = 100 - 2P
EC 170 Industrial Organization
Bertrand competition (cont.)
• We need the derived demand for each firm
– demand conditional upon the price charged by the other firm
• Take firm 2. Assume that firm 1 has set a price of $25
– if firm 2 sets a price greater than $25 she will sell nothing
– if firm 2 sets a price less than $25 she gets the whole market
– if firm 2 sets a price of exactly $25 consumers are indifferent
between the two firms
– the market is shared, presumably 50:50
• So we have the derived demand for firm 2
– q2 = 0
– q2 = 100 - 2p2
– q2 = 0.5(100 - 50) = 25
if p2 > p1 = $25
if p2 < p1 = $25
if p2 = p1 = $25
EC 170 Industrial Organization
Bertrand competition (cont.)
• More generally:
Demand is not
continuous. There
is a jump at p2 = p1
p2
– Suppose firm 1 sets price p1
• Demand to firm 2 is:
q2 = 0 if p2 > p1
p1
q2 = 100 - 2p2 if p2 < p1
q2 = 50 - p1 if p2 = p1
• The discontinuity in
demand carries over to
profit
100 - 2p1
50 - p1
EC 170 Industrial Organization
100 q2
Bertrand competition (cont.)
Firm 2’s profit is:
p2(p1,, p2) = 0
if p2 > p1
p2(p1,, p2) = (p2 - 10)(100 - 2p2)
if p2 < p1
p2(p1,, p2) = (p2 - 10)(50 - p2)
if p2 = p1
Clearly this depends on p1.
For whatever
reason!
Suppose first that firm 1 sets a “very high” price:
greater than the monopoly price of $30
EC 170 Industrial Organization
What price
Atshould
p2 = (cont.)
pfirm
Bertrand competition
1= 2
So, if p1 falls to $30, $30, firm 2
If
p1 = $30, then
set?
firm
2
should
just
With p1 > $30, Firm 2’s profitgets
looks
like
this:
half
of
2 will only earn a
undercut p1 a bit andthe firm
monopoly
What
if
firm
1
Firm
2’s
Profit
positive
profit
by cutting its
The
monopoly
get almost all the
profit
prices
at
$30?
price
to $30
or less
price
of $30
monopoly profit
p2 < p1
p2 = p 1
p 2 > p1
$10
$30
EC 170 Industrial Organization
p1
Firm 2’s Price
Bertrand competition (cont.)
Now suppose that firm 1 sets a price less than $30
Firm 2’s profit looks like this:As long as p1 > c = $10,
Firm
2 should
What
price aim
Of2’scourse,
Firm
Profit firm
just to undercut
should
firm 2
1 will then
set firm
now?1
undercut firm 2
p < p1
so on
Then firm 2and
should
also2 price
at $10. Cutting price below cost
gainsWhat
the whole
but loses
if firmmarket
1
money
every customer p2 = p1
priceson
at $10?
p 2 > p1
$10
p1 $30
EC 170 Industrial Organization
Firm 2’s Price
Bertrand competition (cont.)
• We now have Firm 2’s best response to any price set by
firm 1:
– p*2 = $30
if p1 > $30
– p*2 = p1 - “something small” if $10 < p1 < $30
– p*2 = $10
if p1 < $10
• We have a symmetric best response for firm 1
– p*1 = $30
if p2 > $30
– p*1 = p2 - “something small” if $10 < p2 < $30
– p*1 = $10
if p2 < $10
EC 170 Industrial Organization
The
best response
Bertrand
competition (cont.)
The best response
function
for
These best response
functions
look like this
function for
firm 1
p2
firm 2
R
1
R2
$30
The equilibrium
The Bertrand
isequilibrium
with both has
firms
bothpricing
firms charging
at
marginal
$10
cost
$10
$10
$30
p1
EC 170 Industrial Organization
Bertrand Equilibrium: modifications
• The Bertrand model makes clear that competition in prices is very
different from competition in quantities
• Since many firms seem to set prices (and not quantities) this is a challenge
to the Cournot approach
• But the Bertrand model has problems too
– for the p = marginal-cost equilibrium to arise, both firms need enough
capacity to fill all demand at price = MC
– but when both firms set p = c they each get only half the market
– So, at the p = mc equilibrium, there is huge excess capacity
• This calls attention to the choice of capacity
– Note: choosing capacity is a lot like choosing output which brings us
back to the Cournot model
• The intensity of price competition when products are identical that the
Bertrand model reveals also gives a motivation for Product differentiation
EC 170 Industrial Organization
An Example of Product Differentiation
Coke and Pepsi are nearly identical but not quite. As a
result, the lowest priced product does not win the entire
market.
QC = 63.42 - 3.98PC + 2.25PP
MCC = $4.96
QP = 49.52 - 5.48PP + 1.40PC
MCP = $3.96
There are at least two methods for solving this for PC and PP
EC 170 Industrial Organization
Bertrand and Product Differentiation
Method 1: Calculus
Profit of Coke: pC = (PC - 4.96)(63.42 - 3.98PC + 2.25PP)
Profit of Pepsi: pP = (PP - 3.96)(49.52 - 5.48PP + 1.40PC)
Differentiate with respect to PC and PP respectively
Method 2: MR = MC
Reorganize the demand functions
PC = (15.93 + 0.57PP) - 0.25QC
PP = (9.04 + 0.26PC) - 0.18QP
Calculate marginal revenue, equate to marginal cost, solve
for QC and QP and substitute in the demand functions
EC 170 Industrial Organization
Bertrand competition and product differentiation
Both methods give the best response functions:
The Bertrand
PP
PC = 10.44 + 0.2826PP
Note
that these
equilibrium
is
PP = 6.49 + 0.1277PC
are
at upward
their
sloping
intersection
These can be solved
for the equilibrium
$8.11
prices as indicated
RC
RP
B
$6.49
$10.44
EC 170 Industrial Organization
PC
$12.72
Bertrand Competition and the Spatial Model
• An alternative approach is to use the spatial model from
Chapter 4
–
–
–
–
a Main Street over which consumers are distributed
supplied by two shops located at opposite ends of the street
but now the shops are competitors
each consumer buys exactly one unit of the good provided that its
full price is less than V
– a consumer buys from the shop offering the lower full price
– consumers incur transport costs of t per unit distance in travelling
to a shop
• What prices will the two shops charge?
EC 170 Industrial Organization
Xm
Bertrand
and
the
spatial
model
What
if
shop
1 raises
marks the
Assume that shop 1 sets
price?
location ofprice
the p and shopits
2
sets
1
Price
marginal buyer— price p
2
one who is
indifferent between
p’1buying either firm’s
good
Price
p2
p1
x’m
Shop 1
xm
All consumers to xthe
Shop 2
m moves to the
left of xm buy left:
fromsome consumers
And all consumers
shop 1 switch to shop
to 2the right buy from
shop 2
EC 170 Industrial Organization
is the fraction
Bertrand and This
the spatial
model
of consumers who
p1 +
= p2 + t(1 buy from firm 1
m
2tx = p2 - p1 + t
m
How
is
x
xm(p1, p2) = (p2 - p1 + t)/2t
determined?
txm
xm)
There are n consumers in total
So demand to firm 1 is D1 = N(p2 - p1 + t)/2t
Price
Price
p2
p1
xm
Shop 1
Shop 2
EC 170 Industrial Organization
Bertrand equilibrium
Profit to firm 1 is p1 = (p1 - c)D1 = N(p1 - c)(p2 - p1 + t)/2t
p1 = N(p2p1 - p12 + tp1 + cp1 - cp2 -ct)/2t Solve this
Thistoispthe best
for p1
Differentiate with respect
1
response function
N
for +firm
1 =0
(p2 - 2p
t + c)
p1/ p1 =
1
2t
p*1 = (p2 + t + c)/2 This is the best response
function for
firma 2
What about firm 2? By symmetry,
it has
similar best response function.
p*2 = (p1 + t + c)/2
EC 170 Industrial Organization
Bertrand and Demand
p2
p*1 = (p2 + t + c)/2
R1
p*2 = (p1 + t + c)/2
2p*2 = p1 + t + c
R2
= p2/2 + 3(t + c)/2
 p*2 = t + c
c+t
(c + t)/2
 p*1 = t + c
(c + t)/2
EC 170 Industrial Organization
c+t
p1
Stackelberg
• Interpret in terms of Cournot
• Firms choose outputs sequentially
– leader sets output first, and visibly
– follower then sets output
• The firm moving first has a leadership advantage
– can anticipate the follower’s actions
– can therefore manipulate the follower
• For this to work the leader must be able to commit to its
choice of output
• Strategic commitment has value
EC 170 Industrial Organization
Stackelberg Equilibrium: an example
• Assume that there are two firms
with
identical
products
Both
firms
have constant
marginal
costs ofbe:
$10,
• As in our earlier Cournot example,
let demand
– P = 100 - 2Q = 100 - 2(q1 + q2) i.e., c = 10 for both firms
• Total cost for for each firm is:
– C(q1) = 10q1; C(q2) = 10q2
• Firm 1 is the market leader and chooses q1
• In doing so it can anticipate firm 2’s actions
• So consider firm 2. Demand is:
– P = (100 - 2q1) - 2q2
• Marginal revenue therefore is:
– MR2 = (100 - 2q1) - 4q2
EC 170 Industrial Organization
Equate
This ismarginal
firm 2’s revenue
Stackelberg
equilibrium
(cont.)
Solve
this
equation
withresponse
marginal cost
best
for
output
function
MR2 = (100 - 2q1) - 4q2
But
firm q12knows
q2
what
q2 is going
MR = (100 - 2q1) - 4q2 = 10 = c
Firm 1 knows that
to be
this is how firm 2
q*2 = 22.5 - q1/2 From earlier example (slide
22) we know
will reactSotofirm
firm11’s
can
Demand for firm 1 is:that 22.5 is the monopoly output.
This
is an
output
choice
anticipate firm 2’s
P = (100 - 2q2) - 2q1 important result. The Stackelberg
leader
22.5
reaction
the same output as a monopolist would.
P = (100 - 2q*2) -chooses
2q1
But
2 is not excluded
P = (100 - (45
- q1)) -marginal
2q1 firm revenue
Equate
S from the market
11.25
 P = 55 - q1
with
cost
Solvemarginal
this equation
R
Marginal revenue for firm 1 is:
for output q1
MR1 = 55 - 2q1
55 - 2q1 = 10  q*1 = 22.5  q*2 = 11.25
EC 170 Industrial Organization
2
22.5
45
q1
Stackelberg equilibrium (cont.)
Aggregate output is 33.75
Leadership benefits
q2
Firm
1’s best
response
So the equilibrium price is $32.50
the leader
firm
1 but
Leadership
benefits
45
function
“like”
Firm 1’s profit is (32.50 - 10)22.5
R1
harms
theisfollower
consumers
but
firm
Compare
firm2’s
2 this with
 p1 = $506.25
reduces
aggregate
the
Cournot
Firm 2’s profit is (32.50 - 10)11.25
profits
equilibrium
22.5
 p2 = $253.125
C
We know (see slide 22) that the
15
S
Cournot equilibrium is:
11.25
qC1 = qC2 = 15
R2
q1
The Cournot price is $40
45
15 22.5
Profit to each firm is $450
EC 170 Industrial Organization
Stackelberg and Commitment
• It is crucial that the leader can commit to its output choice
– without such commitment firm 2 should ignore any stated intent by
firm 1 to produce 45 units
– the only equilibrium would be the Cournot equilibrium
• So how to commit?
– prior reputation
– investment in additional capacity
– place the stated output on the market
• Finally, the timing of decisions matters
EC 170 Industrial Organization