1. Lecture 10 The Bertrand Model
ECON 4100: Industrial Organization
Lecture 10
The Bertrand Model

2. Oligopoly Models There are three dominant oligopoly models Cournot
Bertrand
Stackelberg
Now we will consider the Bertrand Model

3. 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: => Bertrand
This leads to dramatically different results
Take a simple example
two firms producing an identical product
firms choose the prices at which they sell their water
each firm has constant marginal cost of $10
market demand is Q = P
Check that with
this demand and
these costs the
monopoly price is
$30 and quantity
is 40 units

4. 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 = p2 if p2 < p1 = $25
q2 = 0.5( ) = 25 if p2 = p1 = $25

5. Bertrand competition (cont.)
Demand is not
continuous. There
is a jump at p2 = p1
More generally:
Suppose Firm 1 sets price p1 p2
Demand to Firm 2 is:
q2 = 0 if p2 > p1 p1
q2 = p2 if p2 < p1
q2 = 50 - p1 if p2 = p1
The discontinuity in demand carries over to profit p1
100 q2
50 - p1

6. Bertrand competition (cont.)
Firm 2’s profit is:
p2(p1,, p2) = 0 if p2 > p1
p2(p1,, p2) = (p2 - 10)( p2) if p2 < p1
For whatever
reason!
p2(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

7. Bertrand competition (cont.)
If p1 = $30, then Firm 2 will only earn a positive profit by cutting its price to $30 or less
What price
should Firm 2
set?
At p2 = p1 = $30, Firm 2 gets half of the monopoly profit
Bertrand competition (cont.)
So, if p1 falls to $30, Firm 2 should just
undercut p1 a bit and get almost all the monopoly profit
With p1 > $30, Firm 2’s profit looks like this:
What if Firm 1
prices at $30?
The monopoly
price of $30
Firm 2’s Profit
p2 < p1
p2 = p1
p2 > p1 p1
$10
$30
Firm 2’s Price

8. Bertrand competition (cont.)
Now suppose that Firm 1 sets a price less than $30
As long as p1 > c = $10, Firm 2 should aim
just to undercut
Firm 1
Firm 2’s profit looks like this:
What price
should Firm 2
set now?
Of course, Firm 1 will then undercut Firm 2 and so on
Firm 2’s Profit
p2 < p1
Then Firm 2 should also price at $10. Cutting price below cost gains the whole market but loses money on every customer
What if Firm 1
prices at $10?
p2 = p1
p2 > p1 p1
$10
$30
Firm 2’s Price

9. Bertrand competition (cont.)
We now have Firm 2’s best response to any price set by Firm 1:
p*2 = $ if p1 > $30
p*2 = p1 - “something small” if $10 < p1 < $30
p*2 = $ if p1 < $10
We have a symmetric best response for Firm 1
p*1 = $ if p2 > $30
p*1 = p2 - “something small” if $10 < p2 < $30
p*1 = $ if p2 < $10

10. Bertrand competition (cont.)
The best response
function for
Firm 1
The best response
function for
Firm 2
These best response functions look like this p2 R1
The Bertrand
equilibrium has
both firms charging
marginal cost R2
$30
The equilibrium
is with both
firms pricing at
$10
$10 p1
$10
$30

11. Bertrand Equilibrium: modifications
Bertrand: 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 = marginal cost 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

12. 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 = PC PP
MCC = $4.96
QP = PP PC
MCP = $3.96
There are at least two methods for solving this for PC and PP

13. Bertrand and Product Differentiation
Method 1: Calculus
Profit of Coke: pC = (PC )( PC PP)
Profit of Pepsi: pP = (PP )( PP PC)
Differentiate with respect to PC and PP respectively
Method 2: MR = MC
Reorganize the demand functions
PC = ( PP) QC
PP = ( PC) QP
Calculate marginal revenue, equate to marginal cost, solve for QC and QP and substitute in the demand functions

14. Bertrand competition and product differentiation
Both methods give the best response functions:
The Bertrand
equilibrium is
at their
intersection
PC = PP PP
Note that these
are upward
sloping RC
PP = PC
These can be solved for the equilibrium prices as indicated RP
$8.11 B
$6.49 PC
$10.44
$12.72

15. 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?

16. Bertrand and the spatial model
Xm marks the location of the marginal buyer—one who is indifferent between buying either firm’s good
Bertrand and the spatial model
What if shop 1 raises
its price?
Assume that shop 1 sets
price p1 and shop 2 sets
price p2
Price
Price
p’1 p2 p1 xm
x’m
Shop 1
All consumers to the
left of xm buy from
shop 1
xm moves to the
left: some consumers
switch to shop 2
Shop 2
And all consumers
to the right buy from
shop 2

17. Bertrand and the spatial model
This is the fraction
of consumers who
buy from Firm 1
Bertrand and the spatial model
p1 + txm = p2 + t(1 - xm)
How is xm
determined?
=> 2txm = p2 - p1 + t
=> xm(p1, p2) = (p2 - p1 + t)/2t
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

18. This is the best response function for Firm 2
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
for p1
This is the best
response function
for Firm 1
Differentiate with respect to p1 N
p1/ p1 =
(p2
- 2p1
+ t + c)
= 0 2t
p*1 = (p2 + t + c)/2
This is the best response function for Firm 2
What about Firm 2? By symmetry, it has a similar best response function.
p*2 = (p1 + t + c)/2

19. Bertrand and Demand p2 p*1 = (p2 + t + c)/2 p*2 = (p1 + t + c)/2
c + t
=> p*2 = t + c
(c + t)/2
=> p*1 = t + c p1
(c + t)/2
c + t

20. Next: Stackelberg 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