•
In a wide variety of markets firms compete in prices
– Internet access
–
Restaurants
–
Consultants
– Financial services
• With monopoly setting price or quantity first makes no difference
• In oligopoly it matters a great deal
– nature of price competition is much more aggressive the quantity competition

• In the Cournot model price is set by some market clearing mechanism
•
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 products
– each firm has constant marginal cost of c
– inverse demand is P = A – B.Q
– direct demand is Q = a – b.P with a = A/B and b= 1/B

•
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 p
1
– if firm 2 sets a price greater than p
1 she will sell nothing
– if firm 2 sets a price less than p
1 she gets the whole market
– if firm 2 sets a price of exactly p
1 consumers are indifferent between the two firms: the market is shared, presumably 50:50
• So we have the derived demand for firm 2
– q
2
– q
2
– q
2
= 0
= (a – bp
2
)/2
= a – bp
2 if p
2
> p
1 if p
2
= p
1 if p
2
< p
1

p
2
•
This can be illustrated as follows:
•
Demand is discontinuous
•
The discontinuity in demand carries over to profit p
1
There is a jump at p
2
= p
1
a - bp
1
(a - bp
1
)/2 a q
2

Firm 2’s profit is: p
2
(p
1,
, p
2
) = 0 if p
2
> p
1 p
2
(p
1,
, p
2
) = (p
2
- c)(a - bp
2
) if p
2
< p
1 p
2
(p
1,
, p
2
) = (p
2
- c)(a - bp
2
)/2 if p
2
= p
1
For whatever reason!
Clearly this depends on p
1
.
Suppose first that firm 1 sets a “very high” price: greater than the monopoly price of p M = (a +c)/2b

What price
With p
1 Firm 2 will only earn a set?
> (a + c)/2b, Firm 2’s profit looks like this:
Firm 2’s Profit
At p2 = p1 price to (a + c)/2b or less undercut p
1
The monopoly firm 2 gets half of the
So firm 2 should just monopoly profit price a bit and get almost all the p
2
< p
What if firm 1
1 monopoly profit prices at (a + c)/2b?
c (a+c)/2b p
1 p
2
= p
1 p
2
> p
1
Firm 2’s Price

Now suppose that firm 1 sets a price less than (a + c)/2b
Firm 2’s profit looks like this:
Firm 2’s Profit
Of course, firm 1
What price will then undercut firm 2 and so on p < p
Then firm 2 should also price at c. Cutting price below cost gains the whole market but loses
What if firm 1 prices at c?
p
2
= p
1
1
> c, c p
1
(a+c)/2b p
2
> p
1
Firm 2’s Price

• We now have Firm 2’s best response to any price set by firm 1:
– p*
2
– p*
2
– p*
2
= (a + c)/2b
= p
= c
1
- “something small” if p
1
> (a + c)/2b if c < p
1
< (a + c)/2b if p
1
< c
• We have a symmetric best response for firm 1
– p*
1
– p*
1
– p*
1
= (a + c)/2b
= p
= c
2
- “something small” if p
2
> (a + c)/2b if c < p
2
< (a + c)/2b if p
2
< c

function for
The best response
These best response functions look like this function for p
2 firm 2
R
1
R
2
(a + c)/2b
The Bertrand c firms pricing at marginal cost c c (a + c)/2b p
1

• 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 extreme version of the difference seems somewhat forced
• Two extensions can be considered
– impact of capacity constraints
– product differentiation

• For the p = c equilibrium to arise, both firms need enough capacity to fill all demand at p = c
•
But when p = c they each get only half the market
• So, at the p = c equilibrium, there is huge excess capacity
•
So capacity constraints may affect the equilibrium
• Consider an example
– daily demand for skiing on Mount Norman Q = 6,000 – 60P
– Q is number of lift tickets and P is price of a lift ticket
– two resorts: Pepall with daily capacity 1,000 and Richards with daily capacity 1,400, both fixed
– marginal cost of lift services for both is $10

•
Is a price P = c = $10 an equilibrium?
– total demand is then 5,400, well in excess of capacity
•
Suppose both resorts set P = $10: both then have demand of 2,700
•
Consider Pepall:
– raising price loses some demand
– but where can they go? Richards is already above capacity
– so some skiers will not switch from Pepall at the higher price
– but then Pepall is pricing above MC and making profit on the skiers who remain
– so P = $10 cannot be an equilibrium

•
Assume that at any price where demand at a resort is greater than capacity there is efficient rationing
– serves skiers with the highest willingness to pay
•
Then can derive residual demand
•
Assume P = $60
– total demand = 2,400 = total capacity
– so Pepall gets 1,000 skiers
– residual demand to Richards with efficient rationing is Q =
5000 – 60P or P = 83.33 – Q/60 in inverse form
– marginal revenue is then MR = 83.33 – Q/30

•
Residual demand and MR:
•
Suppose that Richards sets
$83.33
P = $60. Does it want to
Price change?
– since MR > MC Richards does not want to raise price and lose skiers
$60
$ 36.66
MR
Demand
– since Q
R
= 1,400 Richards is $10 at capacity and does not
1,400 want to reduce price
•
Same logic applies to Pepall so P = $60 is a Nash equilibrium for this game.
MC
Quantity

•
Logic is quite general
– firms are unlikely to choose sufficient capacity to serve the whole market when price equals marginal cost
• since they get only a fraction in equilibrium
– so capacity of each firm is less than needed to serve the whole market
– but then there is no incentive to cut price to marginal cost
•
So the efficiency property of Bertrand equilibrium breaks down when firms are capacity constrained

•
Original analysis also assumes that firms offer homogeneous products
•
Creates incentives for firms to differentiate their products
– to generate consumer loyalty
– do not lose all demand when they price above their rivals
• keep the “most loyal”

Coke and Pepsi are similar but not identical. As a result, the lower priced product does not win the entire market.
Econometric estimation gives:
Q
C
= 63.42 - 3.98P
C
+ 2.25P
P
MC
C
= $4.96
Q
P
= 49.52 - 5.48P
P
+ 1.40P
C
MC
P
= $3.96
There are at least two methods for solving for P
C and P
P

Method 1: Calculus
Profit of Coke: p
C
= (P
C
Profit of Pepsi: p
P
= (P
P
- 4.96)(63.42 - 3.98P
- 3.96)(49.52 - 5.48P
P
C
+ 2.25P
P
+ 1.40P
C
)
)
Differentiate with respect to P
C and P
P respectively
Method 2: MR = MC
Reorganize the demand functions
P
C
= (15.93 + 0.57P
P
) - 0.25Q
C
P
P
= (9.04 + 0.26P
C
) - 0.18Q
P
Calculate marginal revenue, equate to marginal cost, solve for Q
C and Q
P and substitute in the demand functions

Both methods give the best response functions:
P
C
= 10.44 + 0.2826P
P
P
P
P
P
= 6.49 + 0.1277P
C
These can be solved for the equilibrium prices as indicated
$8.11
The equilibrium prices
$6.49
are each greater than marginal cost intersection
B
$10.44
$12.72
R
C
R
P
P
C

•
An alternative approach: spatial model of Hotelling
– 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
•
Recall the broader interpretation
•
What prices will the two shops charge?

p’
1
Price
marginal buyer—one who price p
1 buying either firm’s good price p
2 p
2 p
1
Shop 1 x’ x m
All consumers to the left of x m m x m
And all consumers moves to the buy from shop 1 left: some consumers shop 2 switch to shop 2

p
1
p
1
+ tx m = p
2
+ t(1 - x m )

2tx m = p
2
- p
1
+ t
 x m (p
1
, p
2
) = (p
2
- p
1
+ t)/2t
How is x m
There are N consumers in total
So demand to firm 1 is D 1 = N(p
2
Price
- p
1 of consumers who buy from firm 1
Price p
2 x m
Shop 1 Shop 2

Profit to firm 1 is p
1
= N(p
2 p
1
- p
1
2 p
1
= (p
+ tp
1
1
- c)D 1 = N(p - c)(p
This is the best
2
+ cp
1
- cp
Differentiate with respect to p
1
- p
Solve this for firm 1 for p
1
1
+ t)/2t
N
p
1
/
 p
1
= (p
2
- 2p
1
+ t + c) = 0
2t p*
1
= (p
2
+ t + c)/2
This is the best response similar best response function.
p*
2
= (p
1
+ t + c)/2

p*
1
= (p
2
p
2
+ t + c)/2 p*
2
= (p
1
+ t + c)/2
2p*
2
= p
1
+ t + c
= p
2
/2 + 3(t + c)/2
 p*
2
= t + c
 p*
1
= t + c c + t
(c + t)/2
Profit per unit to each firm is t
(c + t)/2
Aggregate profit to each firm is Nt/2 c + t
R
1
R
2 p
1

•
Two final points on this analysis
• t is a measure of transport costs
– it is also a measure of the value consumers place on getting their most preferred variety
– when t is large competition is softened
• and profit is increased
– when t is small competition is tougher
• and profit is decreased
•
Locations have been taken as fixed
– suppose product design can be set by the firms
• balance “business stealing” temptation to be close
• against “competition softening” desire to be separate

q
2 •
Best response functions are very different with Cournot and Bertrand
– they have opposite slopes
– reflects very different forms of competition
– firms react differently e.g. to an increase in costs p
2
Firm 1
Firm 2
Firm 1
Cournot q
1
Firm 2 Bertrand p
1

q
2
– suppose firm 2’s costs increase
– this causes Firm 2’s Cournot best response function to fall
• at any output for firm 1 firm 2 now wants to produce less
– firm 1’s output increases and firm 2’s falls
Firm 1 passive response by firm 1 p
2 – Firm 2’s Bertrand best response function rises
• at any price for firm 1 firm 2 now wants to raise its price
– firm 1’s price increases as does firm 2’s aggressive response by firm 1
Cournot
Firm 2
Firm 1
Firm 2 q
1 p
1
Bertrand

•
When best response functions are upward sloping (e.g.
Bertrand) we have strategic complements
– passive action induces passive response
•
When best response functions are downward sloping
(e.g. Cournot) we have strategic substitutes
– passive actions induces aggressive response
• Difficult to determine strategic choice variable: price or quantity
– output in advance of sale – probably quantity
– production schedules easily changed and intense competition for customers – probably price

•
As noted earlier, products can be differentiated horizontally or vertically
•
In many respects, which type of differentiation prevails reflects underlying consumer preferences
•
Are the meaningful differences between consumers about what makes for quality and not about what quality is worth (Horizontal Differentiation); Or
•
Are the meaningful differences between consumers not about what constitutes good quality but about how much extra quality should be valued (Vertical
Differentiation)

•
Consider the study of the retail gasoline market in southern California by Hastings (2004)
•
Gasoline is heavily branded. Established brands like Chevron and Exxon-Mobil have contain special, trademarked additives that are not found in discount brands, e.g. RaceTrak.
• In June 1997, the established brand Arco gained control of 260 stations in Southern
California formerly operated by the discount independent, Thrifty
•
By September of 1997, the acquired stations were converted to Arco stations. What effect did this have on branded gasoline prices?

•
If consumers regard Thrifty as substantially different in quality from the additive brands, then losing the Thrifty stations would not hurt competition much while the entry of 260 established
Arco stations would mean a real increase in competition for branded gasoline and those prices should fall.
•
If consumers do not see any real quality differences worth paying for but simply valued the Thrifty stations for providing a low-cost alternative, then establish brand prices should rise after the acquisition.
•
So, behavior of gasoline prices before and after the acquisition tells us something about preferences.

•
Tracking differences in price behavior over time is tricky though
•
Hastings (2004) proceeds by looking at gas stations that competed with Thrifty’s before the acquisition
(were within 1 mile of a Thrifty) and ones that do not. She asks if there is any difference in the response of the prices at these two types of stations to the conversion of the Thrifty stations
•
Presumably, prices for both types were different after the acquisition than they were before it. The question is, is there a difference between the two groups in these before-and-after differences? For this reason, this approach is called a difference-indifferences model.

Hastings observes prices for each station in Feb, June,
Sept. and December of 1997, i.e., before and after the conversion. She runs a regression explaining station i’s price in each of the four time periods, t p it
= Constant +
 i
+

1
X it
+

2
Z it
+

3
T i
+ e it
 i is an intercept term different for each that controls for differences between each station unrelated to time
X it is 1 if station i competes with a Thrifty at time t and 0 otherwise.
Z it is 1 if station i competes with a station directly owned by a major brand but 0 if it is a franchise.
T i is a sequence of time dummies equal reflecting each of the four periods. This variable controls for the pure effect of time on the prices at all stations.

The issue is the value of the estimated coefficient

1
Ignore the contractual variable Z it for the moment and consider two stations: firm 1that competed with a
Thrifty before the conversion and firm 2 that did not.
In the pre-conversion periods, X it is positive for firm
1 but zero for firm 2. Over time, each firm will change its price because of common factors that affect them over time. However, firm 1 will also change is price because for the final two observations, X it is zero.
Before After Difference
Firm 1: α i
+ β
1
Firm 2: α j
α i
+ time effects - β
1
+ time effects
α j
+ time effects time effects

Thus, the estimated coefficient

1 captures the difference in movement over time between firm 1 and firm 2.
Hastings (2004) estimates
 to be about -0.05. That is,
1 firms that competed with a Thrifty saw their prices rise by about 5 cents more over time than did other firms
Before the conversion, prices at stations that competed against Thrifty’s were about 2 to 3 cents below those that did not. After the removal of the Thrifty’s, however, they had prices about 2 to 3 cents higher than those that did not.
Conversion of the Thrifty’s to Arco stations did not intensify competition among the big brands. Instead, it removed a lost cost alternative.