# Find the Nash Equilibrium for a Cournot Game

Consider a Cournot game with $2$ firms. Firm $i$ has constanct marginal cost $C_i$, where $C_1 \lt C_2$. Inverse demand is linear: $p(q)=A-q$ (where $A \gt 2C_2 - C_1$). Find the Nash Equilibrium.

-
I have tried to make your question readable using LaTeX, but you should check it says what you want. If this is homework, you should tag it as such and perhaps show what you have tried –  Henry May 4 '12 at 15:46

Firstly you have to calculate the profit function (assuming no fixed cost). $$\pi(q_i,q_j)=p(q_i+q_j)\cdot q_i - c_i \cdot q_i=(A - q_i-q_j - c_i)\cdot q_i$$In order to find the Nash equilibrium, both functions $\pi(q_i,q_j)$ and $\pi(q_j,q_i)$ must be maximized. Taking the partial derivatives: $$\frac{\partial \pi(q_i,q_j)}{\partial q_i}=0 \ , i=1,2$$ you obtain the Nash equilibrium: $$q_1^{*}=\frac{A+C_2-2C_1}{3}$$ $$q_2^{*}=\frac{A+C_1-2C_2}{3}$$