Best Response Functions

A best response is the profit-maximizing quantity for a firm given the quantity the other firm has chosen.

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 = 116 - 4 ( Q_Z + Q_Y )`.

The marginal cost for Zach is `4`.

What is Zach's best response if `Q_Y = 12`?

Zach's best response is `Q_Z = 8.0`.

$$ \begin{align*} R \left( Q_Z \right) &= P Q_Z \\ &= \left( 116 - 4 ( Q_Z + Q_Y ) \right) Q_Z \\ &= \left( 116 - 4 Q_Z - 4 Q_Y \right) Q_Z \\ &= 116 Q_Z - 4 Q_Z^2 - 4 Q_Y Q_Z \end{align*} $$

Zach's marginal costs being `MC ( Q_Z ) = 4`, the quantity `Q_Z` that maximizes Zach's profit solves

$$ \begin{align*} MC ( Q_Z ) &= MR ( Q_Z ) \\ 4 &= 116 - 4 Q_Y - 8 Q_Z \\ 8 Q_Z &= 116 - 4 - 4 Q_Y \\ Q_Z &= \frac{116 - 4}{8} - \frac{Q_Y}{2} \\ Q_Z &= 14.0 - \frac{Q_Y}{2} \end{align*} $$

Plug Yann's quantity `Q_Y = 12` into Zach's best response function:

$$ $$ \begin{align*} Q_{ Z } &= 14.0 - \frac{Q_Y}{2} \\ &= 14.0 - \frac{ 12 }{2} \\ &= 14.0 - 6.0 \\ &= 8.0 \end{align*} $$ $$