ec 170 industrial organization chapter 5 business strategy in oligopoly markets
Post on 21-Jan-2016
222 Views
Preview:
TRANSCRIPT
EC 170 Industrial Organization
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.)The Pay-Off Matrix
American
Delta
Morning
Morning
Evening
Evening
(15, 15)
The left-handnumber is the
pay-off toDelta
The right-handnumber is the
pay-off toAmerican
(30, 70)
(70, 30) (35, 35)
What is theequilibrium for this
game?
EC 170 Industrial Organization
The example (cont.)The Pay-Off Matrix
American
Delta
Morning
Morning
Evening
Evening
(15, 15)
If Americanchooses a morning
departure, Deltawill choose
evening
(30, 70)
(70, 30) (35, 35)
If Americanchooses an evening
departure, Deltawill still choose
evening
The morning departureis a dominated
strategy for Delta and so can be eliminated.
The Nash Equilibrium must therefore be one in which
both airlines choose an evening departure
(35, 35)
The morning departureis also a dominated
strategy for American and again can be eliminated
EC 170 Industrial Organization
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
The example (cont.)The Pay-Off Matrix
American
Delta
Morning
Morning
Evening
Evening
(18, 12) (30, 70)
(70, 30) (42, 28)
However, a morning departureis still a dominated strategy for Delta (Evening is still a dominant strategy.
If Deltachooses a morning
departure, Americanwill choose
eveningBut if Delta
chooses an eveningdeparture, American
will choosemorning
American has no dominated strategy
American knowsthis and so
chooses a morningdeparture
(70, 30)
EC 170 Industrial Organization
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.)The Pay-Off Matrix
American
Delta
PH = $500
($9000,$9000) ($0, $3600)
($3600, $0) ($1800, $1800)
PH = $500
PL = $220
PL = $220
If both price highthen both get 30
passengers. Profitper passenger is
$300
If Delta prices highand American lowthen American getsall 180 passengers.
Profit per passengeris $20
If Delta prices lowand American high
then Delta getsall 180 passengers.
Profit per passengeris $20
If both price lowthey each get 90
passengers.Profit per passenger
is $20
EC 170 Industrial Organization
Nash Equilibrium (cont.)The Pay-Off Matrix
American
Delta
PH = $500
($9000,$9000) ($0, $3600)
($3600, $0) ($1800, $1800)
PH = $500
PL = $220
PL = $220
(PH, PL) cannot bea Nash equilibrium.If American prices
low then Delta shouldalso price low
($0, $3600)
(PL, PH) cannot bea Nash equilibrium.If American prices
high then Delta shouldalso price high
($3600, $0)
(PH, PH) is a Nashequilibrium.
If both are pricinghigh then neither wants
to change
($9000, $9000)
(PL, PL) is a Nashequilibrium.
If both are pricinglow then neither wants
to change
($1800, $1800)
There are two Nashequilibria to this version
of the game
There is no simple way to choose betweenthese equilibria. But even so, the Nash concept has eliminated half of the outcomes as equilibria
Custom and familiaritymight lead both to
price high
“Regret” mightcause both to
price low
EC 170 Industrial Organization
Nash Equilibrium (cont.)The Pay-Off Matrix
American
Delta
PH = $500
($9000,$9000) ($0, $3600)
($3600, $0) ($1800, $1800)
PH = $500
PL = $220
PL = $220
(PH, PL) cannot bea Nash equilibrium.If American prices
low then Delta wouldwant to price low, too.
($0, $3600)
(PL, PH) cannot bea Nash equilibrium.If American prices
high then Delta shouldalso price high
($3600, $0)
(PH, PH) is a Nashequilibrium.
If both are pricinghigh then neither wants
to change
($9000, $9000)
(PL, PL) is a Nashequilibrium.
If both are pricinglow then neither wants
to change
($1800, $1800)
There are two Nashequilibria to this version
of the game
There is no simpleway to choose between
these equilibria, but at least we have eliminated half of the outcomes as
possible equilibria
EC 170 Industrial Organization
Nash Equilibrium (cont.)The Pay-Off Matrix
American
Delta
PH = $500
($9000,$9000) ($0, $3600)
($3600, $0) ($1800, $1800)
PH = $500
PL = $220
PL = $220
($0, $3600)
($3600, $0)
($3,000, $3,000)
($1800, $1800)
Delta can see that if it sets a high price, then American
will do best by also pricing high. Delta
earns $9000
Suppose that Deltacan set its price first
Delta can also see that if it sets a low price, American
will do best by pricing low. Delta will then earn $1800
The only sensible choice for Delta is PH knowing that American will follow with PH and each will
earn $9000. So, the Nash equilibria now is (PH, PH)
($1800, $1800)
Sometimes, consideration of the timing of moves can help us find the equilibrium This means that
PH, PL cannot be an equilibrium
outcomeThis means that PL,PH
cannot be an equilibrium
EC 170 Industrial Organization
Oligopoly Models
• There are three dominant oligopoly models– Cournot
– Bertrand
– Stackelberg
• They are distinguished by– the decision variable that firms choose
– the timing of the underlying game
• But each embodies the Nash equilibrium concept
EC 170 Industrial Organization
The Cournot Model
• Start with a duopoly
• Two firms making an identical product (Cournot supposed this was spring water)
• Demand for this product is
P = A - BQ = A - B(q1 + q2)
where q1 is output of firm 1 and q2 is output of firm 2
• Marginal cost for each firm is constant at c per unit
• To get the demand curve for one of the firms we treat the output of the other firm as constant
• So for firm 2, demand is P = (A - Bq1) - Bq2
EC 170 Industrial Organization
The Cournot model (cont.)
P = (A - Bq1) - Bq2$
Quantity
A - Bq1
If the output offirm 1 is increasedthe demand curvefor firm 2 moves
to the left
A - Bq’1
The profit-maximizing choice of output by firm 2 depends upon the output of firm 1
DemandMarginal revenue for firm 2 is
MR2 = (A - Bq1) - 2Bq2MR2
MR2 = MC
A - Bq1 - 2Bq2 = c
Solve thisfor output
q2
q*2 = (A - c)/2B - q1/2
c MC
q*2
EC 170 Industrial Organization
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 Equilibriumq2
q1
The best response function for firm 1 isq*1 = (A-c)/2B - q2/2
The best response function for firm 1 isq*1 = (A-c)/2B - q2/2
(A-c)/B
(A-c)/2B
Firm 1’s best response function
The best response function for firm 2 isq*2 = (A-c)/2B - q1/2
The best response function for firm 2 isq*2 = (A-c)/2B - q1/2
(A-c)/2B
(A-c)/B
If firm 2 producesnothing then firm1 will produce themonopoly output
(A-c)/2B
If firm 2 produces(A-c)/B then firm1 will choose to
produce no output
Firm 2’s best response function
The Cournot-Nashequilibrium is at
Point C at the intersectionof the best response
functions
C
qC1
qC2
EC 170 Industrial Organization
Cournot-Nash Equilibrium
q2
q1
(A-c)/B
(A-c)/2B
Firm 1’s best response function
(A-c)/2B
(A-c)/B
Firm 2’s best response function
C
q*1 = (A - c)/2B - q*2/2
q*2 = (A - c)/2B - q*1/2
q*2 = (A - c)/2B - (A - c)/4B + q*2/4
3q*2/4 = (A - c)/4B
q*2 = (A - c)/3B(A-c)/3B
q*1 = (A - c)/3B
(A-c)/3B
EC 170 Industrial Organization
Cournot-Nash Equilibrium (cont.)
• In equilibrium each firm produces qC1 = qC
2 = (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)qC
1 = (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)qC
1 = (A - c)2/9B = $450• Competition: Q* = (A – c)/B = 45; P = c = $10• Monopoly: QM = (A - c)/2B = 22.5; P = $55• Total output exceeds the monopoly output, but is less
than the competitive output• Price exceeds marginal cost but is less than the
monopoly price
EC 170 Industrial Organization
Cournot-Nash Equilibrium (cont.)
• What if there are more than two firms?
• Much the same approach.
• Say that there are N identical firms producing identical products
• Total output Q = q1 + q2 + … + qN
• Demand is P = A - BQ = A - B(q1 + q2 + … + qN)
• Consider firm 1. It’s demand curve can be written:P = A - B(q2 + … + qN) - Bq1
• Use a simplifying notation: Q-1 = q2 + q3 + … + qN
This denotes outputof every firm other
than firm 1
• So demand for firm 1 is P = (A - BQ-1) - Bq1
EC 170 Industrial Organization
The Cournot model (cont.)
P = (A - BQ-1) - Bq1$
Quantity
A - BQ-1
If the output ofthe other firms
is increasedthe demand curvefor firm 1 moves
to the leftA - BQ’-1
The profit-maximizing choice of output by firm 1 depends upon the output of the other firms
DemandMarginal revenue for firm 1 is
MR1 = (A - BQ-1) - 2Bq1MR1
MR1 = MC
A - BQ-1 - 2Bq1 = c
Solve thisfor output
q1
q*1 = (A - c)/2B - Q-1/2
c MC
q*1
EC 170 Industrial Organization
Cournot-Nash Equilibrium (cont.)
q*1 = (A - c)/2B - Q-1/2
How do we solve thisfor q*1?The firms are identical.
So in equilibrium theywill have identical
outputs
Q*-1 = (N - 1)q*1
q*1 = (A - c)/2B - (N - 1)q*1/2
(1 + (N - 1)/2)q*1 = (A - c)/2B
q*1(N + 1)/2 = (A - c)/2B
q*1 = (A - c)/(N + 1)B
Q* = N(A - c)/(N + 1)B
P* = A - BQ* = (A + Nc)/(N + 1)
As the number offirms increases output
of each firm falls As the number of
firms increasesaggregate output
increases As the number offirms increases price
tends to marginal cost
Profit of firm 1 is P*1 = (P* - c)q*1 = (A - c)2/(N + 1)2B
As the number offirms increases profit
of each firm falls
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 are c2.
• Demand is P = A - BQ = A - B(q1 + q2)
• We have marginal revenue for firm 1 as before
• MR1 = (A - Bq2) - 2Bq1
• Equate to marginal cost: (A - Bq2) - 2Bq1 = c1
Solve thisfor output
q1
q*1 = (A - c1)/2B - q2/2
A symmetric resultholds for output of
firm 2
q*2 = (A - c2)/2B - q1/2
EC 170 Industrial Organization
Cournot-Nash Equilibrium
q2
q1
(A-c1)/B
(A-c1)/2B
R1
(A-c2)/2B
(A-c2)/B
R2C
q*1 = (A - c1)/2B - q*2/2
q*2 = (A - c2)/2B - q*1/2
q*2 = (A - c2)/2B - (A - c1)/4B + q*2/4
3q*2/4 = (A - 2c2 + c1)/4B
q*2 = (A - 2c2 + c1)/3B
q*1 = (A - 2c1 + c2)/3B
What happens to thisequilibrium when
costs change?
As the marginalcost of firm 2
falls its best responsecurve shifts to
the right
The equilibriumoutput of firm 2increases and of
firm 1 falls
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• qC
1 = (A - 2c1 + c2)/3B = (100 – 10 + 15)/6 = 17.5• qC
2 = (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
A - BQ-i - 2Bqi = ci
This can be reorganized to give the equilibrium condition:
A - B(Q*-i + q*i) - Bq*i - ci = 0
But Q*-i + q*i = Q*and A - BQ* = P*
P* - Bq*i - ci = 0 P* - ci = Bq*i
EC 170 Industrial Organization
Concentration and profitability (cont.)
P* - ci = Bq*i
Divide by P* and multiply the right-hand side by Q*/Q*
P* - ci
P*=
BQ*
P*
q*i
Q*
But BQ*/P* = 1/ and q*i/Q* = si
so: P* - ci
P*=
si
The price-cost marginfor each firm is
determined by its ownmarket share and overallmarket demand elasticity
Extending this we have
P* - cP*
= H
The verage price-cost margin is determined by industry
concentration as measured by the Herfindahl-Hirschman Index
EC 170 Industrial Organization
Price Competition: Bertrand
• In the Cournot model price is set by some market clearing mechanism
• Firms seem relatively passive
• An alternative approach is to assume that firms compete in prices: this is the approach taken by Bertrand
• Leads to dramatically different results
• Take a simple example– two firms producing an identical product (spring 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
Check that withthis demand andthese costs the
monopoly price is$30 and quantity
is 40 units
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 if p2 > p1 = $25
– q2 = 100 - 2p2 if p2 < p1 = $25
– q2 = 0.5(100 - 50) = 25 if p2 = p1 = $25
EC 170 Industrial Organization
Bertrand competition (cont.)
• More generally:– Suppose firm 1 sets price p1
• Demand to firm 2 is:
p2
q2
q2 = 0 if p2 > p1p1
q2 = 100 - 2p2 if p2 < p1
100100 - 2p1
q2 = 50 - p1 if p2 = p1
50 - p1
Demand is notcontinuous. Thereis a jump at p2 = p1
• The discontinuity in demand carries over to profit
EC 170 Industrial Organization
Bertrand competition (cont.)
Firm 2’s profit is:
2(p1,, p2) = 0 if p2 > p1
2(p1,, p2) = (p2 - 10)(100 - 2p2) if p2 < p1
2(p1,, p2) = (p2 - 10)(50 - p2) if p2 = p1
Clearly this depends on p1.
Suppose first that firm 1 sets a “very high” price: greater than the monopoly price of $30
For whateverreason!
EC 170 Industrial Organization
Bertrand competition (cont.)
With p1 > $30, Firm 2’s profit looks like this:
Firm 2’s Price
Firm 2’s Profit
$10 $30 p1
p2 < p1
p2 = p1
p2 > p1
What priceshould firm 2
set?
The monopolyprice of $30
What if firm 1prices at $30?
So, if p1 falls to $30, firm 2 should just
undercut p1 a bit and get almost all the monopoly profit
If p1 = $30, then firm 2 will only earn a
positive profit by cutting its price to $30 or less
At p2 = p1 = $30, firm 2 gets half of
the monopoly
profit
EC 170 Industrial Organization
Bertrand competition (cont.)Now suppose that firm 1 sets a price less than $30
Firm 2’s Price
Firm 2’s Profit
$10 $30p1
p2 < p1
p2 = p1
p2 > p1
Firm 2’s profit looks like this:
What priceshould firm 2
set now?
As long as p1 > c = $10, Firm 2 should aim
just to undercutfirm 1
What if firm 1prices at $10?
Then firm 2 should also price at $10. Cutting price below costgains the whole market but loses
money on every customer
Of course, firm 1 will then
undercut firm 2 and so on
EC 170 Industrial Organization
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
Bertrand competition (cont.)These best response functions look like this
p2
p1$10
$10
R1
R2
The best responsefunction for
firm 1The best response
function forfirm 2
The equilibriumis with both
firms pricing at$10
The Bertrandequilibrium has
both firms chargingmarginal cost
$30
$30
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
QC = 63.42 - 3.98PC + 2.25PP
QP = 49.52 - 5.48PP + 1.40PC
MCC = $4.96
MCP = $3.96
There are at least two methods for solving this for PC and PP
Coke and Pepsi are nearly identical but not quite. As a result, the lowest priced product does not win the entire market.
EC 170 Industrial Organization
Bertrand and Product Differentiation
Method 1: CalculusProfit of Coke: C = (PC - 4.96)(63.42 - 3.98PC + 2.25PP)
Profit of Pepsi: P = (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 differentiationBoth methods give the best response functions:PC = 10.44 + 0.2826PP
PP = 6.49 + 0.1277PC
PC
PP
RC
$10.44
RP
Note that theseare upward
sloping
The Bertrandequilibrium is
at theirintersection
B
$12.72
$8.11
$6.49
These can be solved for the equilibrium prices as indicated
EC 170 Industrial Organization
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
Bertrand and the spatial model
Shop 1 Shop 2
Assume that shop 1 setsprice p1 and shop 2 sets
price p2
Price Price
p1
p2
xm
All consumers to theleft of xm buy from
shop 1And all consumers
to the right buy fromshop 2
What if shop 1 raisesits price?
p’1
x’m
xm moves to theleft: some consumers
switch to shop 2
Xm marks the location of the
marginal buyer—one who is
indifferent between buying either firm’s
good
EC 170 Industrial Organization
Bertrand and the spatial model
Shop 1 Shop 2
Price Price
p1
p2
xm
How is xm
determined?
p1 + txm = p2 + t(1 - xm)
2txm = p2 - p1 + t
xm(p1, p2) = (p2 - p1 + t)/2t
This is the fractionof consumers whobuy from firm 1
So demand to firm 1 is D1 = N(p2 - p1 + t)/2t
There are n consumers in total
EC 170 Industrial Organization
Bertrand equilibrium
Profit to firm 1 is 1 = (p1 - c)D1 = N(p1 - c)(p2 - p1 + t)/2t
1 = N(p2p1 - p12 + tp1 + cp1 - cp2 -ct)/2t
Differentiate with respect to p1
1/ p1 =N
2t(p2 - 2p1 + t + c) = 0
Solve thisfor p1
p*1 = (p2 + t + c)/2
What about firm 2? By symmetry, it has a similar best response function.
This is the bestresponse function
for firm 1
p*2 = (p1 + t + c)/2
This is the best response function for firm 2
EC 170 Industrial Organization
Bertrand and Demand
p*1 = (p2 + t + c)/2 p2
p1
R1
p*2 = (p1 + t + c)/2
R2
(c + t)/2
(c + t)/2
2p*2 = p1 + t + c
= p2/2 + 3(t + c)/2
p*2 = t + c
c + t
p*1 = t + c
c + t
EC 170 Industrial Organization
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
• As in our earlier Cournot example, let demand be:– P = 100 - 2Q = 100 - 2(q1 + q2)
• Total cost for for each firm is:– C(q1) = 10q1; C(q2) = 10q2
Both firms have constantmarginal costs of $10,
i.e., c = 10 for both firms
• 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
Stackelberg equilibrium (cont.)
MR2 = (100 - 2q1) - 4q2
MR = (100 - 2q1) - 4q2 = 10 = c
Equate marginal revenuewith marginal costSolve this equation
for output q2
q*2 = 22.5 - q1/2
q2
q1
R2
22.5
45
This is firm 2’sbest response
function
Firm 1 knows thatthis is how firm 2
will react to firm 1’soutput choice
Firm 1 knows thatthis is how firm 2
will react to firm 1’soutput choice So firm 1 can
anticipate firm 2’sreaction
So firm 1 can anticipate firm 2’s
reaction
Demand for firm 1 is:
P = (100 - 2q2) - 2q1
But firm 1 knowswhat q2 is going
to be
P = (100 - 2q*2) - 2q1
P = (100 - (45 - q1)) - 2q1
P = 55 - q1
Marginal revenue for firm 1 is:
MR1 = 55 - 2q1
Equate marginal revenuewith marginal cost
55 - 2q1 = 10
Solve this equationfor output q1
q*1 = 22.522.5
q*2 = 11.25
11.25
From earlier example (slide 22) we know that 22.5 is the monopoly output. This is an
important result. The Stackelberg leader chooses the same output as a monopolist would.
But firm 2 is not excluded from the marketS
EC 170 Industrial Organization
Firm 1’s best responsefunction is “like”
firm 2’s
Stackelberg equilibrium (cont.)
Aggregate output is 33.75
So the equilibrium price is $32.50 q2
q1
R2
22.5
45
Compare this withthe Cournotequilibrium
Compare this withthe Cournotequilibrium
22.5
11.25
Firm 1’s profit is (32.50 - 10)22.5
1 = $506.25
Firm 2’s profit is (32.50 - 10)11.25
2 = $253.125
45R1
SCWe know (see slide 22) that the
Cournot equilibrium is:
qC1 = qC
2 = 15
15
15
The Cournot price is $40
Profit to each firm is $450
Leadership benefitsthe leader firm 1 butharms the follower
firm 2
Leadership benefitsconsumers but
reduces aggregateprofits
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
top related