Best Response Functions & Nash Equilibrium in a Cournot Duopoly

• Mar 28th 2011, 01:26 PM
leesperare
Best Response Functions & Nash Equilibrium in a Cournot Duopoly
Hello! I'm trying to figure out how to find the best response functions and a Nash equilibrium when given a utility (profit) function that has a quadratic in the denominator.

For example: Consider the following version of a Cournot duopoly. Each firm produces \$\displaystyle q_i\$ ≥ 0. There is a constant production cost, \$\displaystyle c\$. The resulting profit (utility) functions are:

\$\displaystyle u_1(q_1,q_2)=q_1/(1+(q_1+q_2)^2) - c\$

\$\displaystyle u_2(q_1,q_2)=q_2/(1+(q_1+q_2)^2) - c\$

1) Find the best response function for each firm.
2) Find a Nash equilibrium or show that there is none.

Thanks!
• Mar 29th 2011, 01:50 AM
SpringFan25
taking firm1 as an example
Differentiate u() with respect to q1, treating q2 as a constant. You'll need to use the quotient rule.

you'll probably end up with a nasty looking fraction with a quadratic numerator and a messy denominator.

The "trick" in this type of problem is normally that you can ignore the denominator, you want the first derivative=0 which means the numerator must be zero. Find the value of q1 that makes the numerator zero.

If you need further help post your entire solution up to the point you get stuck.