Example
Zach and Yann are competitors in the coffee market. Consumers cannot tell their coffee beans apart.
They face a demand characterized by `P = 1000 - 2Q`.
Zach's Best Response
Zach maximizes profit `\pi_Z ( Q_Z ) = P Q_Z - c Q_Z`.
$$ \begin{align*} MR_Z &= MC_Z \\ 1000 - 4Q_Z - 2Q_Y &= 200 \\ 800 - 2Q_Y &= 4Q_Z \\ Q_Z &= 200 - \frac{Q_Y}{2} \end{align*} $$Zach's best response is `Q_Z = 200 - \frac{Q_Y}{2}`.
Yann's Best Response
Yann maximizes profit `\pi_Y ( Q_Y ) = P Q_Y - c Q_Y`.
$$ \begin{align*} MR_Y &= MC_Y \\ 1000 - 4Q_Y - 2Q_Z &= 200 \\ 800 - 2Q_Z &= 4Q_Y \\ Q_Y &= 200 - \frac{Q_Z}{2} \end{align*} $$Yann's best response is `Q_Y = 200 - \frac{Q_Z}{2}`.
Question
The demand is `P = 249 - 1 ( Q_Z + Q_Y )`.
The marginal cost for Zach is `83`.
What is Zach's best response if `Q_Y = 76`?
Zach's best response is `Q_Z = 45.0`.
$$
\begin{align*}
R \left( Q_Z \right) &= P Q_Z \\
&= \left( 249 - 1 ( Q_Z + Q_Y ) \right) Q_Z \\
&= \left( 249 - 1 Q_Z - 1 Q_Y \right) Q_Z \\
&= 249 Q_Z - 1 Q_Z^2 - 1 Q_Y Q_Z
\end{align*}
$$
Zach's marginal costs being `MC ( Q_Z ) = 83`, the quantity `Q_Z` that maximizes Zach's profit solves
$$ \begin{align*} MC ( Q_Z ) &= MR ( Q_Z ) \\ 83 &= 249 - 1 Q_Y - 2 Q_Z \\ 2 Q_Z &= 249 - 83 - 1 Q_Y \\ Q_Z &= \frac{249 - 83}{2} - \frac{Q_Y}{2} \\ Q_Z &= 83.0 - \frac{Q_Y}{2} \end{align*} $$Plug Yann's quantity `Q_Y = 76` into Zach's best response function:
$$ $$ \begin{align*} Q_{ Z } &= 83.0 - \frac{Q_Y}{2} \\ &= 83.0 - \frac{ 76 }{2} \\ &= 83.0 - 38.0 \\ &= 45.0 \end{align*} $$ $$