ECO 5341 Bertrand Price Competition with
Homogenous Products
Saltuk Ozerturk (SMU)
January 2016
Saltuk Ozerturk (SMU)
Bertrand Price Competition
Bertrand Price Competition
Bertrand Price Competition
Consider a market with two firms, Firm 1 and Firm 2. Both
firms produce homogenous (identical) products at a unit cost
c where 0 < c < 1.
Two firms are competing by simultaneously setting prices of
an identical product to place on the market.
Firms’ products are viewed identically to consumers — all
consumers buy from the firm with a lower price.
When the firms charge the same price, the firms split the
market and each firm captures exactly half of the market
demand.
Saltuk Ozerturk (SMU)
Bertrand Price Competition
Bertrand 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

 (1 − pi ) , if pi < pj ;
(1−pi )
qi (pi , pj ) =
, if pi = pj ;
 2
0, if pi > pj .
Saltuk Ozerturk (SMU)
Bertrand Price Competition
Bertrand 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:

(p − c) (1 − p1 ) , if p1 < p2 ;

 1
(p1 − c) (1 − p1 )
π1 (p1 , p2 ) =
, if p1 = p2 ;

2

0, if p1 > p2 .

(p − c) (1 − p2 ) , if p2 < p1 ;

 2
(p2 − c) (1 − p2 )
π2 (p1 , p2 ) =
, if p1 = p2 ;

2

0, if p2 > p1 .
Saltuk Ozerturk (SMU)
Bertrand Price Competition
Bertrand Price Competition
Is there any Nash Equilibria with p1 6= p2 , that is, an
equilibrium in which one firm sets a higher price than the
other firm?
Suppose, without loss of generality, there is a NE in which
firm 1 sets p1 = a and p2 = b where
c < p2 = b < p1 = a < 1
That is, Firm 1 sets a higher price than Firm 2. For this to be
a NE, each firm must be best responding to each other. In
this proposed equilibrium we have
π1 (p1 = a, p2 = b) = 0
since p2 < p1 .
Saltuk Ozerturk (SMU)
Bertrand Price Competition
Bertrand Price Competition
Is there any Nash Equilibria with p1 6= p2 (continued)
But note that, instead of setting p1 = a and receiving 0, Firm
1 can set p1 = b and get
π1 (p1 = b, p2 = b) =
(b − c) (1 − b)
>0
2
which is better than 0. Hence we cannot have a NE in which
firm 1 (or firm 2) sets a higher price than its rival.
Saltuk Ozerturk (SMU)
Bertrand Price Competition
Bertrand Price Competition
Is there any Nash Equilibria with p1 = p2 > c, that is, an
equilibrium in which the two firms both set a price strictly
hugher than their unit marginal cost c?
Suppose, there is a NE in which firms set p1 = a and p2 = a
where
c < p1 = p2 = a < 1
For this to be a NE, each firm must be best responding to each
other. In this proposed equilibrium we have
π1 (p1 = a, p2 = a) =
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(a − c) (1 − a)
>0
2
Bertrand Price Competition
Bertrand Price Competition
But note that, instead of setting p1 = a and receiving
(a − c) (1 − a)
, Firm 1 can set p1 = a − ε where ε > 0 is an
2
arbitrarily small number (like cutting its price by a cent). In
this case, firm 1 would get the whole market and receive
π1 (p1 = a − ε, p2 = a) = (a − ε − c) (1 − a + ε) > 0
(a − c) (1 − a)
.
2
Hence we cannot have a NE in which two firms both set a
price strictly higher than their unit marginal cost c
which is better than
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Bertrand Price Competition
Bertrand Price Competition
Is p1 = p2 = c a Nash Equilibrium?
If both firms set a price equal to their marginal cost, they
share the market but receive a zero profit.
π1 (p1 = c, p2 = c) =
π2 (p1 = c, p2 = c) =
(p1 − c) (1 − p1 )
=0
2
(p2 − c) (1 − p1 )
=0
2
Is there a profitable devitation for any of the two firms. The
answer is no. If a firm deviates and sets a lower price than c,
this firm will sell each unit a loss and make negative profits. If
the same firm deviates and sets a higher price than c, it will
not be able to sell anything and continue to receive zero.
Therefore p1 = p2 = c is the unique Nash Equilibrium.
Saltuk Ozerturk (SMU)
Bertrand Price Competition