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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.

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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: $$q_1^{*}=\frac{A+C_2-2C_1}{3}$$ $$q_2^{*}=\frac{A+C_1-2C_2}{3}$$

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