Intro
The Bertrand Model
The Cournot Model
Product Differentiation
Imperfect Competition
Lecture 4, ECON 4240 Spring 2017
summary of Snyder et al. (2015)
University of Oslo
07.Feb.2017
1/19
summary of Snyder et al. (2015)
Imperfect Competition
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
Outline
In this lecture and in the rest of the course we consider
situations in which the assumptions necessary for the welfare
theorems to hold are violated, so the market equilibria are
inefficient
Now: few firms: oligopoly
Next: externalities and public goods
Second half of the course: asymmetric information
2/19
summary of Snyder et al. (2015)
Imperfect Competition
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
Oligopoly - Short Run Decisions
Definition
An oligopoly is a market with relatively few firms, but more than
one.
Focus here: "short-term" decisions of firms in oligopolistic
markets: price and quantity ("long-run" decisions = entry,
investment, research and development, advertising are the
topic of ECON 4820)
Benchmark: Bertrand model of oligopolistic competition
We show that the Bertrand model does a poor job at
describing oligopolies
We twist the Bertrand model to obtain models that are simple
(enough) but somewhat realistic
What is this for? Theory of oligopolistic markets used in
anti-trust cases
summary of Snyder et al. (2015)
Imperfect Competition
3/19
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
The Bertrand Model
2 identical firms, called 1 and 2: identical costs, identical
products (everything works identically with n firms, but the
results are more striking with 2 firms)
Marginal cost is constant (call it c) - hence equal to the
average cost
Many identical consumers
Firms choose their prices (p1 and p2 ) simultaneously
All sales go to the firm with the lowest price (and are split
evenly if p1 = p2 )
4/19
summary of Snyder et al. (2015)
Imperfect Competition
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
Nash Equilibrium of the Bertrand Game
The only (pure-strategy) Nash equilibrium of the Bertrand
game is p1∗ = p2∗ = c
Nash equilibrium (in this context): p1 = p1∗ is the best
response of firm 1 to p2 = p2∗ . At the same time, p2 = p2∗ is
the best response of firm 2 to p1 = p1∗
To see that this is a Nash equilibrium:
Note that firm 1 has no profitable deviation: for p2 = c,
setting p1 = c ensures profits equal to 0, while for p1 > c firm
1 does not sell any unit, hence profits=0, while for p1 < c firm
1 sells to all consumers, but it incurs a loss for every unit sold
Same argument holds for firm 2
5/19
summary of Snyder et al. (2015)
Imperfect Competition
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
Nash Equilibrium of the Bertrand Game
To see that this is THE ONLY Nash equilibrium: suppose in
equilibrium p2 ≥ p1 (note this case is as general as it can be)
If p1 > c, then firm 2 would be better off choosing p1 − ε for
some small enough ε rather than p2 ≥ p1
If p1 < c then firm 1 is incurring losses as on each unit sold it
makes negative profits of p1 − c < 0, but then firm 1 would be
better off charging a price strictly larger than p2 (so as to sell
nothing)
If p1 = c and p2 > p1 , then firm 1 would be better off
increasing the price a little
So we are left with p1 = p2 = c
6/19
summary of Snyder et al. (2015)
Imperfect Competition
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
Bertrand Paradox
Paradox: the Bertrand model suggests that in a market with
only 2 firms we should expect firms to act as if they were in
perfect competition (that is, setting a price equal to their
marginal cost)
Possible "fixes"
Introduce capacity constraints for firms (one way to do that is
to assume that firms choose quantity instead of prices, as in
the Cournot model)
Assume firms sell differentiated products
Assume firms interact with each other multiple times and can
tacitly collude over price (we will skip this)
Assume firms choose their prices sequentially (we will skip this)
7/19
summary of Snyder et al. (2015)
Imperfect Competition
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
The Cournot Model - 2 firms
Same as Bertrand, except that firms choose (simultaneously)
quantities q1 and q2 , and the price is the one at which
quantity demanded is equal to Q := q1 + q2
Let P(Q) be the inverse of the demand curve D(P) (demand
curve is assumed downward sloping, thus invertible)
Profits: πi = (P(Q) − c)qi ; i = 1, 2
∗ −i is
Nash equilibrium: each qi maximizes πi for given q−i
"the other firm"
first-order-condition (FOC): ∂∂ πqii = P(Q) + P 0 (Q)qi = c
2
second-order-condition (SOC): ∂∂ 2 πqi = 2P 0 (Q) + P 00 (Q)qi < 0
i
assume here that the (SOC) holds
8/19
summary of Snyder et al. (2015)
Imperfect Competition
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
The Cournot Model - 2 firms
P(Q) + P 0 (Q)qi = marginal revenue; c= marginal cost
As P 0 (Q)qi < 0, we have P(Q) > c, (unlike Bertrand)
Example: say P(Q) = a − bQ
so FOC: a − bQ − bqi = c
Focus on symmetric equilibrium (in this case, no reason for the
equilibrium to be unique): a − b2qi − bqi = c → qi = a−c
3b and
(a−c)
2c
a
so P(Q) = a − b2 3b = 3 + 3 > c (to see that the last
inequality holds, note that if c > a no market equilibrium price
can ensure non-negative profits to the firms)
9/19
summary of Snyder et al. (2015)
Imperfect Competition
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
The Cournot Model - n firms
Assume there are n firms,
We focus on symmetric equilibrium: qi =
Equilibrium: first-order-condition:
Q
n
for all i = 1, .., n
P(Q) + P 0 (Q) Qn
=c
The "wedge" between price and marginal cost is P 0 (Q) Qn
which is decreasing in n: for very large n, Cournot gives the
same prediction as perfect competition, for n = 1 Cournot is
equal to the monopoly case
Unlike Bertrand, Cournot can be used to study the effect of
the number of firms on the degree of competition
10/19
summary of Snyder et al. (2015)
Imperfect Competition
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
Price or Quantities
Difference between Bertrand and Cournot:
In Bertrand, starting from equal prices, a small reduction in
price allows a firm to steal the entire market: competition is
very intense
In Cournot, starting from equal quantities, a small reduction or
increase in quantity of one firm has only a marginal effect on
price and market shares: competition is not-so-intense
Cournot model seems unrealistic: in reality firms set prices.
BUT: think of Cournot model as a model in which first firms
choose their capacity, then set their prices in order to sell all
units that they can produce
Lesson to be learned: models of oligopoly with capacity
constraints do not suffer from the Bertrand paradox
11/19
summary of Snyder et al. (2015)
Imperfect Competition
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
Product Differentiation
Another way around the Bertrand paradox: differentiated
products
Vertical differentiation: (every consumer likes product 1 better
than 2, e.g. Ipad with 128 GB vs Ipad with 32 GB)
Horizontal differentiation: (some consumers prefer product 1
to 2, others have opposite preferences: e.g. Ipad Air vs Ipad
Mini)
12/19
summary of Snyder et al. (2015)
Imperfect Competition
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
Product Differentiation
Problem with Bertrand model: starting from the equilibrium, if
a firm increases the price even a tiny bit, it immediately loses
all its customers
With differentiated products it is not the case that a marginal
change in price shifts a large amount of consumers from a
product to another one, competition is not-so tough and in
equilibrium P > c
Issue: where do we put the boundaries of a market? Are Ipads
and Samsung tablets differentiated products in the same
market? Or is Apple a monopolist in the market for Ipads and
Samsung tablets are just substitutes?
13/19
summary of Snyder et al. (2015)
Imperfect Competition
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
Hotelling’s Beach
Example of horizontal differentiation: Hotelling’s Beach
Two identical ice-cream kiosks (A and B), selling 1 identical
product and located at a and b > a on a beach of length 1
(the two ends are 0 and L)
Ice-cream sellers’ costs are 0
For consumers, walking on the beach has cost td 2 where d is
the distance to the kiosk
Consumers are evenly spread on the beach and for each unit of
distance there is consumer
14/19
summary of Snyder et al. (2015)
Imperfect Competition
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
Hotelling’s Beach
Let prices be low enough that all consumers buy 1 ice-cream,
only choice is where to buy it..
Let x be the location of the consumer indifferent between
going to a or to b, so:
pb −pa
pa + t(x − a)2 = pb + t(x − b)2 → x(pa , pb ) = b+a
2 + 2t(b−a)
Every consumer located on y < x will go to a, every consumer
located on y > x will go to b
15/19
summary of Snyder et al. (2015)
Imperfect Competition
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
Hotelling’s Beach
Profits of kiosk A: πA (pa ) = x(pa , pb )pa
FOC:
b+a
2
pb −2pa
+ 2t(b−a)
= 0 → pa =
(b 2 −a2 )t+pb
2
Profits of kiosk B: πB (pb ) = (1 − x(pa , pb ))pb
2pb −pa
FOC: 1 − b+a
2 − 2t(b−a) = 0 → pb =
2t(b−a)−(b 2 −a2 )t+pa
2
So: pa = 3t (b − a)(b + a + 2), pb = 3t (b − a)(4 − (b + a))
Even if kiosks select prices at the same time (as in Bertrand)
here prices are above marginal costs
Here if a kiosk increases the price a bit it makes more profits
on each unit sold and loses only the marginal customers (the
customers located around x switch to the other kiosk)
16/19
summary of Snyder et al. (2015)
Imperfect Competition
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
Hotelling’s Beach
πa =
πb =
t
2
18 (b − a)(2 + a + b)
t
2
18 (b − a)(4 − a − b)
Note t measures a cost associated with buying an ice-cream
Nevertheless here profits of both kiosks increase with t
To see why this is the case, note that for t = 0 we are back to
Bertrand and we know in that case pa = pb = 0, so πa = πb = 0
dπb
a
Note also that dπ
dt > 0 and dt > 0 holds because we assume
prices and t are so low that all consumers buy an ice-cream.
For t large enough, this is not true anymore at the prices we
found
In fact for t = +∞, πa = πb = 0..
17/19
summary of Snyder et al. (2015)
Imperfect Competition
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
Consumer Search
Third way out the Bertrand paradox: consumers need to spend
time and/or money to learn about prices of different firms
These search models suffer from a problem that is the opposite
of the Bertrand paradox: even with small costs to search for
new price offers, the basic search models suggest the only
equilibrium has all firms charging the monopoly price
Note that this is an equilibrium: if a consumer expects to find
the monopoly price everywhere, she buys from the first
store/firm available
18/19
summary of Snyder et al. (2015)
Imperfect Competition
Intro
The Bertrand Model
The Cournot Model
Product Differentiation
Consumer Search
This equilibrium is unique: suppose prices are different from
the monopoly price, than any firm setting the lowest price of
all would have an incentive to increase the price a bit
This is called the Diamond paradox
One "solution" to the paradox: assuming that consumers have
differentiated search costs
19/19
summary of Snyder et al. (2015)
Imperfect Competition