# Best Response Functions

A best response is a firm's profit-maximizing quantity 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 = 1400 - 2Q.

## Zach's Best Response

Zach's marginal cost is 200. He maximizes profit when the marginal cost is equal to the marginal revenue: \begin{align*} MC_Z &= MR_Z \\ 200 &= 1400 - 4Q_Z - 2Q_Y \\ 4Q_Z &= 1200 - 2Q_Y \\ Q_Z &= 300 - \frac{Q_Y}{2} \end{align*}

Zach's best response is Q_Z = 300 - \frac{Q_Y}{2}.

## Yann's Best Response

Yann's marginal cost is 200. He maximizes profit when the marginal cost is equal to the marginal revenue: \begin{align*} MC_Y &= MR_Y\\ 200 &= 1400 - 4Q_Y - 2Q_Z \\ 4Q_Y &= 1200 - 2Q_Z \\ Q_Y &= 300 - \frac{Q_Z}{2} \end{align*}

Yann's best response is Q_Y = 300 - \frac{Q_Z}{2}.

### Question

The demand is P = 428 - 3 ( Q_Z + Q_Y ).

The marginal cost for Zach is 98.

What is Zach's best response if Q_Y = 84?

Zach's best response is Q_Z = 13.0.

\begin{align*} R \left( Q_Z \right) &= P Q_Z \\ &= \left( 428 - 3 ( Q_Z + Q_Y ) \right) Q_Z \\ &= \left( 428 - 3 Q_Z - 3 Q_Y \right) Q_Z \\ &= 428 Q_Z - 3 Q_Z^2 - 3 Q_Y Q_Z \end{align*}

\begin{align*} MR ( Q_Z ) &= 428 - 3 Q_Y - 2 \times 3 Q_Z \\ &= 428 - 3 Q_Y - 6 Q_Z \end{align*}

Zach's marginal costs being MC ( Q_Z ) = 98, the quantity Q_Z that maximizes Zach's profit solves \begin{align*} MC ( Q_Z ) &= MR ( Q_Z ) \\ 98 &= 428 - 3 Q_Y - 6 Q_Z \\ 6 Q_Z &= 428 - 98 - 3 Q_Y \\ Q_Z &= \frac{428 - 98}{6} - \frac{Q_Y}{2} \\ Q_Z &= 55.0 - \frac{Q_Y}{2} \end{align*}

Plug Yann's quantity Q_Y = 84 into Zach's best response function:  \begin{align*} Q_{ Z } &= 55.0 - \frac{Q_Y}{2} \\ &= 55.0 - \frac{ 84 }{2} \\ &= 55.0 - 42.0 \\ &= 13.0 \end{align*}