Example
Zach and Yann compete in the market for coffee. They face a demand `P = 1400 - 2Q`, and both have the same marginal cost equal to 200.
Zach's best response is `Q_Z = 300 - \frac{Q_Y}{2}`. Yann's best response is `Q_Y = 300 - \frac{Q_Z}{2}`.
$$ \begin{align*} Q_Z &= 300 - \frac{Q_Y}{2} \\ Q_Z &= 300 - \frac{300 - \frac{Q_Z}{2}}{2} \\ Q_Z &= 300 - \frac{300}{2} + \frac{Q_Z}{4} \\ Q_Z - \frac{Q_Z}{4} &= 300 - 150 \\ \frac{3}{4} Q_Z &= 150 \\ Q_Z &= 200 \end{align*} $$Finally, plug `Q_Z = 200` into Yann's best response
$$ \begin{align*} Q_Y &= 300 - \frac{Q_Z}{2} \\ Q_Y &= 300 - \frac{200}{2} \\ Q_Y &= 300 - 100 \\ Q_Y &= 200 \end{align*} $$The Cournot equilibrium is (200, 200).
Question
The inverse demand on the market for coffee is `P = 3696 - 1 ( Q_Z + Q_Y )`.
Zach faces marginal costs equal to `198`.
Yann faces marginal costs equal to `330`.
What is the Cournot Equilibrium?
In equilibrium, `Q_Z = 1210.0` and `Q_Y = 1078.0`.
Zach's revenue:
$$ \begin{align*} R \left( Q_Z \right) &= P Q_Z \\ &= \left( 3696 - 1 ( Q_Z + Q_Y ) \right) Q_Z \\ &= \left( 3696 - 1 Q_Z - 1 Q_Y \right) Q_Z \\ &= 3696 Q_Z - 1 Q_Z^2 - 1 Q_Y Q_Z \end{align*} $$
Zach's marginal revenue:
$$ \begin{align*} MR (Q_Z) \begin{align*} MR ( Q_Z ) &= 3696 - 1 Q_Y - 2 \times 1 Q_Z \\ &= 3696 - 1 Q_Y - 2 Q_Z \end{align*} \end{align*} $$Zach's marginal cost:
$$ \begin{align*} MC (Q_Z) = 198 \end{align*} $$Zach maximizes profit when `MR \left( Q_Z \right) = MC \left( Q_Z \right)`
\begin{align*} MC ( Q_Z ) &= MR ( Q_Z ) \\ 198 &= 3696 - 1 Q_Y - 2 Q_Z \\ 2 Q_Z &= 3696 - 198 - 1 Q_Y \\ Q_Z &= \frac{3696 - 198}{2} - \frac{Q_Y}{2} \\ Q_Z &= 1749.0 - \frac{Q_Y}{2} \end{align*}Yann's revenue:
$$ \begin{align*} R \left( Q_Y \right) &= P Q_Y \\ &= \left( 3696 - 1 ( Q_Z + Q_Y ) \right) Q_Y \\ &= \left( 3696 - 1 Q_Y - 1 Q_Z \right) Q_Y \\ &= 3696 Q_Y - 1 Q_Y^2 - 1 Q_Z Q_Y \end{align*} $$ Yann's marginal revenue: $$ \begin{align*} MR (Q_Y) \begin{align*} MR ( Q_Y ) &= 3696 - 1 Q_Z - 2 \times 1 Q_Y \\ &= 3696 - 1 Q_Z - 2 Q_Y \end{align*} \end{align*} $$ Yann's marginal cost: $$ \begin{align*} MC (Q_Y) = 330 \end{align*} $$ Yann's best response solves \begin{align*} MC ( Q_Y ) &= MR ( Q_Y ) \\ 330 &= 3696 - 1 Q_Y - 2 Q_Z \\ 2 Q_Y &= 3696 - 330 - 1 Q_Z \\ Q_Y &= \frac{3696 - 330}{2} - \frac{Q_Z}{2} \\ Q_Y &= 1683.0 - \frac{Q_Z}{2} \end{align*}
Plug `Q_Y = 1683.0 - \frac{Q_Z}{2}` into Zach's best response:
$$ $$ \begin{align*} Q_Z &= 1749.0 - \frac{Q_Y}{2} \\ Q_Z &= 1749.0 - \frac{ 1683.0 - \frac{Q_Z}{2} }{ 2 } \\ Q_Z &= 1749.0 - \frac{ 1683.0 }{ 2 } + \frac{Q_Z}{4} \\ Q_Z - \frac{Q_Z}{4} &= 1749.0 - \frac{ 1683.0 }{ 2 } \\ \frac{3}{4} Q_Z &= 907.5 \\ Q_Z &= \frac{4}{3} 907.5 \\ Q_Z &= 1210.0 \end{align*} $$ $$So Zach's quantity is `Q_Z = 1210.0`. Plug that into Yann's best response function:
$$ $$ \begin{align*} Q_{ Y } &= 1683.0 - \frac{Q_Z}{2} \\ &= 1683.0 - \frac{ 1210.0 }{2} \\ &= 1683.0 - 605.0 \\ &= 1078.0 \end{align*} $$ $$