Price Theory
Lecture 10: Price Competition Topics for today’s lecture . . .
1. Bertrand competition
2. Collusion
3. Hotelling competition
4. Product dierentiation Bertrand competition Denition: Bertrand competition
A form of oligopoly competition in which the rms in a market simultaneously select the
prices they will charge.
The Bertrand model of competition tends to work well for industries in which rms can easily
adjust the quantities they produce in response to consumer demand.The market for widgets
00
6 MC
30
D
300
P
0 Q
Alpha Foundry and Beta Forges engage in Bertrand competition in the market for widgets; a homogeneous good.
Demand in the market is, Q = 300 10P:
Note: This is the same demand curve as we used with Cournot competition in the
previous lecture.
Both rms have a constant marginal cost of MCA = MCB = $6.Residual demand in Bertrand competition 00
6 MC
30
D
300
P
0 Q
PB
Given that widgets are a homogeneous good, consumers purchase exclusively from the rm with the lowest price.
Consumers split their purchases between rms if rms choose the same price.
For any price PB chosen by Beta, demand for Alpha’s widgets will be,
QA =
8><
>:
0 PA > PB
150 5PA PA = PB
300 10PA PA < PB
:Nash equilibrium with discontinuous demand
00
6 MC
30
D
300
P
0 Q
PB
In a Nash equilibrium, each rm’s price is a best-response to the price of its rival.
The discontinuity in the residual demands means that we cannot construct best-response functions for rms engaged
in Bertrand competition.
Instead, we will employ a process of elimination, excluding cases in which a rm can increase its prot by unilaterally
altering its price.Step 1: Neither rm will choose a price below marginal cost
00
PSA < 0
PA
6 MC
30
D
300
P
0 Q
PB
Suppose that Alpha chooses a price that lies below marginal cost.
Alpha’s producer surplus is negative as it incurs a loss on each sale.
If, instead, Alpha set its price equal to marginal cost, its producer surplus would be zero.
It follows that pricing at marginal cost cannot be a best-response.Step 2: Both rms choose the same price
00
PSA
6 MC
30
D
300
P
0 Q
PB
Suppose that Alpha chooses a price that is greater than PB.
Alpha makes no sales and receives a producer surplus of zero.
If, instead, Alpha matches Beta’s price, it sells to half the market and receives a
positive surplus.
It follows that selecting a higher price than the rival rm cannot be a best-response.Step 3: Firms do not choose a price above marginal cost 00
PSA gain
loss
6 MC
30
D
300
P
0 Q
PB
Suppose that Alpha and Beta choose the
same price, in excess of marginal cost.
Alpha sells to half the market and
receive a positive surplus.
If, instead, Alpha chooses a price that is
just below PB, it captures the entire
market.
The surplus lost from existing sales is
arbitrarily small, relative to the gain
from more than doubling sales.Nash equilibrium prices
00
CS
120 240
6 MC
30
D
300
P
0 Q
In a Nash equilibrium, both rms set their
price equal to marginal cost.
Each rm sells 120 widgets, half
market demand at the price $6.
The producer surplus of each rm is
zero.
The outcome is ecient; there is no
deadweight loss.
Neither rm can gain by unilaterally
altering its price. (You should check this.)Quiz 1
Suppose that natural gas is a homogeneous good, and that demand for gas is given by the
function,
Q = 75 2P:
The marginal cost of producing gas is $7.50. If two rms (A and B) engage in Bertrand
competition, in the Nash equilibrium,
(a) QA
= 15 and QB
= 15.
(b) QA
= 30 and QB
= 30.
(c) QA= 37:5 and QB
= 37:5.
(d) QA
= 60 and QB
= 60.Quiz 2
Suppose that natural gas is a homogeneous good, and that demand for gas is given by the
function,
Q = 75 2P:
The marginal cost of producing gas is $7.50. If two rms (A and B) engage in Bertrand
competition, which of the following statements is false?
(a) The Nash equilibrium prices are PA
= $7:50 and PB
= $7:50.
(b) The Nash equilibrium consumer surplus is larger than under monopoly.
(c) In a Nash equilibrium the deadweight loss is smaller than under monopoly, but larger
than under perfect competition.
(d) The entry of a third rm would not alter the eciency of the market.The Bertrand paradox
Market outcomes depend critically on whether it is prices that are determined by rms’
choices of quantities, or quantities that are determined by rms’ choices of prices.
In Cournot competition, small changes in strategy lead to small changes in price and producer
surplus.
Firms are accommodating, willing to give one another enough room to make a prot.
In Bertrand competition, a small change in price could be the dierence between selling
nothing and capturing the entire market.
Firms are cut-throat, unwilling to yield anything to their rivals.CollusiFraming oligopoly as a game
We have examined both Cournot competition and Bertrand competition as games.
In both games we assume that rms choose their strategies simultaneously.
Both games have nite-horizons, in that they are played once only.
In many markets rms interact repeatedly, over an indenite time horizon.
Repeated interaction over an indenite time-horizon creates the possibility of cooperation
between rms.
Firms have an incentive to cooperate as competition reduces the total producer surplus in
a market.Denition: Collusion
An agreement between rms to coordinate prices, quantities, or some other aspect of market
behaviour.
By colluding, the rms in a market can mimic the behaviour of a monopolist, increasing
industry prots while creating a (larger) deadweight loss.Collusion in the market for widgets
00
CS
DWL
240
PSA PSB
18
60 120
6 MC
30
D
300
PThe incentive to cheat
00
PSA
18
60 120
6 MC
30
D
300
PThe punishment for cheating
00
gain
loss
720
collude
1440
0 t
Suppose that the rms utilise theQuiz 3
Which of the following factors would facilitate collusion in a market?
(a) The rms in a market heavily discount future prots, relative to prots earned today.
(b) It is dicult to distinguish between
uctuations in consumer demand, and the eects
of another rm cheating.
(c) The additional prots gained by cheating are large, relative to the future prots lost if
the collusive agreement collapses.
(d) A rm can rapidly change its price if a rival is observed to be cheating.
grim-trigger strategy.
Collude by choosing the price $18 so
long as the other rm does the same.
If either rm cheats, choose the price
$6 (marginal cost) thereafter, thereby
reducing both rms’ surpluses to $0.
This strategy will sustain collusion so long
as both rms place suciently high value
on future prots.
0 Q
We have already established that the
prices PA = PB = $18 do not constitute a
mutual best-response.
If Alpha shaves its price to slightly less
than $18, Alpha captures the entire
market.
Alpha’s producer surplus becomes
(arbitrarily close to) $1440.
0 Q
Suppose that Alpha and Beta are engaged
in an innite-horizon repeated game.
The two rms agree to select the prices
PA = PB = $18 (the monopoly price)
each time they interact.
Each rm receives a producer surplus
of $720.
Both consumer surplus and
deadweight loss are $720.
Note: The welfare cost of the collusive
agreement is the same as for monopoly.onExercise: Collusion with N rms
Suppose that natural gas is a homogeneous good, and that demand for gas is given by the
function Q = 75 2P. The marginal cost of producing gas is $7.50, and rms compete by
selecting prices.
1. Find the monopoly price, quantity, and producer surplus, for this market. (Hint: The rst
step is to derive inverse demand.)
2. If N rms collude by selecting the monopoly price, what is each rm’s producer surplus?
3. If a rm cheats on the agreement by shaving its price, what producer surplus would it
receive?
4. Suppose that the colluding rms employ the grim-trigger strategy to punish cheating.
How does the number of colluding rms aect their ability to cooperate?Exercise solutions
1. The rst step is to rearrange the demand function to derive inverse demand,
P = 37:5 0:5Q:
Marginal revenue can then be found using the double the slope rule,
MR = 37:5 Q:
Setting marginal revenue equal to marginal cost,
37:5 Q = 7:5 or Q = 30:
The monopoly price is P = 37:5 0:5 30 = $22:5, and the corresponding producer
surplus is PS = 30 (22:5 7:5) = $450.Exercise solutions
2. If there are N rms in the market, with all rms selecting the monopoly price, then each
rm will sell a quantity Qrm = 30=N. Therefore, each rm’s producer surplus is,
PSrm =
30
N
(22:5 7:5) =
450
N
:
3. If a rm cheats by shaving its price, it will sell (close to) the monopoly quantity at (close
to) the monopoly price. Therefore, its producer surplus would be (close to) $450.
4. As the number of rms in the market increases, the surplus a rm earns from colluding
decreases, while the payo to cheating remains the same. It follows that sustaining a
collusive agreement becomes more dicult.Denition: Dierentiated products
Products that dier in one or more characteristic that is signicant to consumers.
When products are dierentiated some consumers may be willing to pay a premium for their
preferred product.Linear city
Imagine a city in which 300 consumers live on Main Street, which is one kilometre long.
The people of Linear City dislike travelling to visit a store.
A consumer in Linear City suers $10 of disutility for each kilometre they travel.
There are only two stores in Linear City, Alice’s Groceries and Brett’s Bargains, located at
either end of Main Street.
The groceries sold by the two stores are identical.
The marginal cost of groceries is $15.
The stores compete by selecting prices.Consumers in linear city
1km
x 1 x
Alice Consumer Brett
Suppose that the people of Linear City
are evenly distributed along Main Street.
Each consumer demands one unit of
groceries, regardless of price.
Consumers choose a store based on price
and location.
The eective price of Alice’s groceries
to a consumer at point x is PA + 10x.
The eective price of Brett’s
groceries is PB + 10(1 x).Quiz 4
1km
Alice John Brett
Paul George
If John buys his groceries from Alice, we
can conclude that,
(a) both Paul and George will also buy
from Alice.
(b) Paul will buy from Alice, and
George will buy from Brett.
(c) Paul will buy from Alice, and
George’s choice is uncertain.
(d) George will buy from Brett, and
Paul’s choice is uncertain.Market share in Hotelling competition
The indierent consumer is the consumer for whom the cost of purchasing from Alice is
equal to the cost of purchasing from Brett.
All consumers to the left of the indierent consumer prefer Alice, while those to the right
prefer Brett.
If the indierent consumer is located at a point x then,
PA + 10x = PB + 10(1 x) or x =
PB PA + 10
20
:
Therefore, demand for Alice’s groceries is the number of consumers in the market (300),
times Alice’s market share x. (Brett’s demand is 300(1 x).)Advanced: Best-response functions in Hotelling competition
Alice’s Groceries prot function can be written,
A = 300x(PA MCA) = 300
PB PA + 10
20
(PA 15):
For any given price PB selected by Brett’s Bargains, the price that maximises Alice’s prots
solves the rst-order condition,
@A
@PA
= 300
PB 2PA + 25
20
= 0:
Solving for PA gives us Alice’s best response function,
PA =
PB + 25
2
:
Note: You will not be assessed on the methods employed in this slide.Alice’s best-response function
00
15
12.5 20
PA =
PB + 25
2
PB
0 PA
We can plot Alice’s best-response function in PA-PB space.
Alice’s location is a source of market power:
If Brett prices at marginal cost, Alice’s best-response is $20|a price in excess of marginal cost.
A fraction 0.25 of consumers will buy from Alice at this price, to avoid the cost of travelling to Brett’s.The Nash equilibrium 00 25 12.5 25 PA = PB + 25 2 12.5 PB = PA + 25 2 PB 0 PA
Brett’s best-response function is, PB = PA + 252:
Substituting for PB into Alice’s best-response function, PA = 12:5+12 PA + 252= 18:75+PA4:
Collecting like terms 0:75PA = 18:75 or PA = $25.
Substituting for PA into Brett’s best-response function PB = $25.Exercise: Equilibrium market shares and prots
There are 300 consumers living in linear city. Each consumer demands a single unit of groceries, and the indierent consumer is located at, x = PB PA + 10 20 :
The marginal cost of each rm is MC = $15, and the equilibrium prices are PA = PB = $25.
1. Calculate the equilibrium market share of each store.
2. What is each store’s equilibrium prot?
3. Calculate each store’s Lerner index of market power.Exercise solutions
1. To nd the equilibrium market shares, substitute the prices into the equation for the indifferent consumer, x = PB PA + 10 20 = 25 25 + 10 20 = 1 2:
Therefore, each store sells to half the market (150 consumers).
2. Alice’s equilibrium prot is, A = 150 (PA MC) = 150 (25 15) = $1500:
Brett’s equilibrium prot is, B = 150 (PB MC) = 150 (25 15) = $1500:Exercise solutions
3. Alice’s Lerner index is, LA = PA MC PA = 25 15 25 = 0:4:
Brett’s Lerner index is, LB = PB MC PB = 25 15 25 = 0:4:Deriving market power from location
strongly prefer Alice strongly prefer Brett
Alice Brett
The Hotelling model illustrates that differences in location may be sufficient to negate the Bertrand paradox.
Each store enjoys market power due to the reluctance of nearby consumers to travel to the more distant retailer.
While stores do compete for consumers closer to the centre, this competition is not sufficient to eliminate all prots.Other interpretations of spatial product differentiation
The Linear City example treats the concept of distance literally.
An alternative interpretation is that the spatial dimension represents an attribute of a product.
Motorcars might be classed as either family cars, located at the left end of the line, or
sports cars located at the right end.
Music could be divided into classical at the left end, and popular music at the right end.
The location of a consumer on the line then represent her/his preference over the attribute in
question, while the travel cost represents the intensity of that preference.Exercise: The intensity of consumer preferences
Two rms, Alpha and Beta, are engaged in Hotelling competition; Alpha at the left end of a
line of unit length, and Beta at the right end. The marginal cost of production is MC.
There are 1000 consumers uniformly distributed along the line. Each consumer demands a
single unit of output and faces a travel cost of T per unit traveled.
1. Find the location of the indierent consumer as a function of PA, PB, MC and T.
2. If the best-response functions of the two rms are,
PA =
PB + MC + T
2
and PB =
PA + MC + T
2
;
nd the Nash equilibrium prices.
3. What is the relationship between the travel cost T, and the equilibrium prices?Exercise solutions
3. The equilibrium prices are PA
= PB
= MC + T. Each rm’s markup over marginal cost is
exactly equal to the travel cost T. In Hotelling competition, each rm’s market power is
increasing in the intensity of consumer preferences.
Note: As the intensity of preferences T approaches zero, the equilibrium prices approach
marginal cost. At T = 0 the products of the two rms are perfect substitutes and the
Bertrand Paradox applies.Key concepts from today’s lecture
You can use these concepts (as search terms) to conduct further research into the topics
covered in today’s lecture:
Bertrand competition
Discontinuity in demand
Price shaving
Bertrand paradox
Collusion
Grim-trigger strategy
Temporary punishment
Hotelling competition
Diifferentiated products
Indierent consumer
Spatial product differentiation
Distribution of consumers Further reading & exercises
The further readings provide additional context to the lecture material, and reinforce core concepts. All readings and exercises can be found in Microeconomics 5th edition, by Besanko and Braeutigam.
Chapter 13, section 13.2.
Where the readings and lecture materials dier, the lecture materials take precedence.
