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 = 604 - 6 ( Q_Z + Q_Y )`.

The marginal cost for Zach is `76`.

What is Zach's best response if `Q_Y = 42`?

Zach's best response is `Q_Z = 23.0`.

$$ \begin{align*} R \left( Q_Z \right) &= P Q_Z \\ &= \left( 604 - 6 ( Q_Z + Q_Y ) \right) Q_Z \\ &= \left( 604 - 6 Q_Z - 6 Q_Y \right) Q_Z \\ &= 604 Q_Z - 6 Q_Z^2 - 6 Q_Y Q_Z \end{align*} $$

$$ \begin{align*} MR ( Q_Z ) &= 604 - 6 Q_Y - 2 \times 6 Q_Z \\ &= 604 - 6 Q_Y - 12 Q_Z \end{align*} $$

Zach's marginal costs being `MC ( Q_Z ) = 76`, the quantity `Q_Z` that maximizes Zach's profit solves $$ \begin{align*} MC ( Q_Z ) &= MR ( Q_Z ) \\ 76 &= 604 - 6 Q_Y - 12 Q_Z \\ 12 Q_Z &= 604 - 76 - 6 Q_Y \\ Q_Z &= \frac{604 - 76}{12} - \frac{Q_Y}{2} \\ Q_Z &= 44.0 - \frac{Q_Y}{2} \end{align*} $$

Plug Yann's quantity `Q_Y = 42` into Zach's best response function: $$ $$ \begin{align*} Q_{ Z } &= 44.0 - \frac{Q_Y}{2} \\ &= 44.0 - \frac{ 42 }{2} \\ &= 44.0 - 21.0 \\ &= 23.0 \end{align*} $$ $$