# 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 = 1741 - 9 ( Q_Z + Q_Y ).

The marginal cost for Zach is 13.

What is Zach's best response if Q_Y = 178?

Zach's best response is Q_Z = 7.0.

\begin{align*} R \left( Q_Z \right) &= P Q_Z \\ &= \left( 1741 - 9 ( Q_Z + Q_Y ) \right) Q_Z \\ &= \left( 1741 - 9 Q_Z - 9 Q_Y \right) Q_Z \\ &= 1741 Q_Z - 9 Q_Z^2 - 9 Q_Y Q_Z \end{align*}

Zach's marginal costs being MC ( Q_Z ) = 13, the quantity Q_Z that maximizes Zach's profit solves

\begin{align*} MC ( Q_Z ) &= MR ( Q_Z ) \\ 13 &= 1741 - 9 Q_Y - 18 Q_Z \\ 18 Q_Z &= 1741 - 13 - 9 Q_Y \\ Q_Z &= \frac{1741 - 13}{18} - \frac{Q_Y}{2} \\ Q_Z &= 96.0 - \frac{Q_Y}{2} \end{align*}

Plug Yann's quantity Q_Y = 178 into Zach's best response function:

 \begin{align*} Q_{ Z } &= 96.0 - \frac{Q_Y}{2} \\ &= 96.0 - \frac{ 178 }{2} \\ &= 96.0 - 89.0 \\ &= 7.0 \end{align*}