ECO 5341 Price Competition with Differentiated
Products
Saltuk Ozerturk (SMU)
Saltuk Ozerturk (SMU)
Price Competition
Price Competition
Price Competition with Differentiated Products
Suppose two firms produce differentiated products at a unit
cost c = 0.
The firms are competing by simultaneously setting prices
Firms’ products are viewed as imperfect substitutes by
consumers.
Saltuk Ozerturk (SMU)
Price Competition
Price Competition
Suppose firm i sets price pi ∈ [0, ∞) when the rival firm j sets
a price pj ∈ [0, ∞). Then the demand qi for Firm i’s product
is given by
qi (pi , pj ) = 10 − αpi + pj
which implies
q1 (p1 , p2 ) = 10 − αp1 + p2
q2 (p1 , p2 ) = 10 − αp2 + p1
We assume that α > 1 so that own-price effect is larger than
the cross-price effect.
Saltuk Ozerturk (SMU)
Price Competition
Price Competition
Strategic (Normal) Form of the game:
Players: Two Firms N = {1, 2}
Strategies: Firm i ∈ N chooses price pi ∈ [0, +∞) .
Payoffs of the firms:
π1 (p1 , p2 ) = p1 q1 (p1 , p2 ) = p1 (10 − αp1 + p2 )
π2 (p1 , p2 ) = p2 q2 (p1 , p2 ) = p1 (10 − αp2 + p1 )
Both firms want to maximize profits.
Saltuk Ozerturk (SMU)
Price Competition
Price Competition
Deriving the Best Response Function of Firm 1
Given any p2 by Firm 2, Firm 1 chooses p1 to maximize
π1 (p1 , p2 )
=
p1 (10 − αp1 + p2 )
⇒ π1 (p1 , p2 ) = 10p1 − αp12 + p2 p1
First order derivative with respect to p1 yields the first order
condition
10 − 2αp1 + p2
=
0
⇒ p1∗ (p2 ) =
5
1
+
p2
α 2α
That is, Firm 1 sets a higher price as the rival firm’s price p2
increases and lower price as α increases.
Saltuk Ozerturk (SMU)
Price Competition
Price Competition
Deriving the Best Response Function of Firm 2
Given any p1 by Firm 1, Firm 2 chooses p2 to maximize
π2 (p1 , p2 )
=
p2 (10 − αp2 + p1 )
⇒ π2 (p1 , p2 ) = 10p2 − αp22 + p1 p2
First order derivative with respect to p2 yields the first order
condition
10 − 2αp2 + p1
=
0
⇒ p2∗ (p1 ) =
5
1
+
p1
α 2α
That is, Firm 2 sets a higher price as the rival firm’s price p1
increases and lower price as α increases.
Saltuk Ozerturk (SMU)
Price Competition
Price Competition
The Nash equilibrium pair (p1∗ , p2∗ ) solves the system of
equations described by the best responses.
p1∗ (p2∗ ) =
1 ∗
5
+
p
α 2α 2
p2∗ (p1∗ ) =
5
1 ∗
+
p
α 2α 1
Saltuk Ozerturk (SMU)
Price Competition
Price Competition
The Nash equilibrium pair (p1∗ , p2∗ ) solves the system of
equations described by the best responses.
p1∗ (p2∗ ) =
5
1 ∗
+
p
α 2α 2
p1∗
1
5
= +
α 2α
p2∗ (p1∗ ) =
5
1 ∗
+
p
α 2α 1
5
1 ∗
+
p
α 2α 1
5
1
5
⇒
= + 2+
p1∗
α 2α
4α2
2
4α − 1
10α + 5
⇒
p1∗ =
4α2
2α2
(2α − 1)(2α + 1) ∗ 5(2α + 1)
⇒
p1 =
4α2
2α2
10
⇒ p1∗ = p2∗ =
2α − 1
p1∗
Saltuk Ozerturk (SMU)
Price Competition
Price Competition
Note that the Nash Equilibrium Price pair
p1∗ = p2∗ =
10
2α − 1
is drastically different than the unique Nash Equilibrium Price
pair of the Bertrand duopoly with identical products. With
identical products, the NE was p1∗ = p2∗ = c which would
imply here that
p1∗ = p2∗ = 0
since we assumed that c = 0.
Note that with
10
2α − 1
as α approaches to infinity we again have p1∗ = p2∗ = 0.Why?
p1∗ = p2∗ =
Saltuk Ozerturk (SMU)
Price Competition