In: Economics
Two Cournot firms produce slightly different products. Product prices depend on both firms' outputs and are determined by the following equations P1 = 70 - 2Q1 - Q2, P2 = 100 - Q1- 2Q2. Both Firm 1 and Firm 2 have constant marginal cost of $10 and zero fixed cost. Firm 1 chooses Q1 and Firm 2 chooses Q2.
Firm 1's inverse demand equation is
Firm 2's inverse demand equation is
Marginal Cost of firm 1 and firm 2 is $10. Fixed costs are 0 for
both firms this imply total cost is
for firm 1 and
for firm 2.
Let
represents profit of firm 1.
Putting
in
taking first partial derivative of
with respect to
:
Setting
equal to 0 and solving for
imply:
So, firm 1's best response as a function of firm 2's output is
Let
represents profit of firm 2.
Putting
in
taking first partial derivative of
with respect to
:
Setting
equal to 0 and solving for
imply:
So, firm 2's best response as a function of firm 1's output is
Putting best response of firm 2 i.e
in
:
Put
in
When
and
then:
so,
When
and
then:
So,
Nash equilibirum : Firm 1 will sell 10 units ()
at price 30 (
)
and Firm 2 will sell 20 units (
)
at price 50 (
).
Firm 1's profit is represented by
. Put
and
in
:
In equilibrium, profit of firm 1 is $200.
Firm 2's profit is represented by
. Put
and
in
:
In equilibrium, profit of firm 2 is $800.