# Cournot Equilibrium

The market is in equilibrium when firms set quantities that are the best responses to each other.

## Example

Zach and Yann compete in the market for coffee. They face a demand P = 1400 - 2Q, and both have the same marginal cost equal to 200.

Zach's best response is Q_Z = 300 - \frac{Q_Y}{2}. Yann's best response is Q_Y = 300 - \frac{Q_Z}{2}. \begin{align*} Q_Z &= 300 - \frac{Q_Y}{2} \\ Q_Z &= 300 - \frac{300 - \frac{Q_Z}{2}}{2} \\ Q_Z &= 300 - \frac{300}{2} + \frac{Q_Z}{4} \\ Q_Z - \frac{Q_Z}{4} &= 300 - 150 \\ \frac{3}{4} Q_Z &= 150 \\ Q_Z &= 200 \end{align*}

Finally, plug Q_Z = 200 into Yann's best response \begin{align*} Q_Y &= 300 - \frac{Q_Z}{2} \\ Q_Y &= 300 - \frac{200}{2} \\ Q_Y &= 300 - 100 \\ Q_Y &= 200 \end{align*}

The Cournot equilibrium is (200, 200).

### Question

The inverse demand on the market for coffee is P = 13080 - 4 ( Q_Z + Q_Y ).

Zach faces marginal costs equal to 2064.

Yann faces marginal costs equal to 552.

What is the Cournot Equilibrium?

In equilibrium, Q_Z = 792.0 and Q_Y = 1170.0.

Zach's revenue: \begin{align*} R \left( Q_Z \right) &= P Q_Z \\ &= \left( 13080 - 4 ( Q_Z + Q_Y ) \right) Q_Z \\ &= \left( 13080 - 4 Q_Z - 4 Q_Y \right) Q_Z \\ &= 13080 Q_Z - 4 Q_Z^2 - 4 Q_Y Q_Z \end{align*}

Zach's marginal revenue: \begin{align*} MR ( Q_Z ) &= 13080 - 4 Q_Y - 2 \times 4 Q_Z \\ &= 13080 - 4 Q_Y - 8 Q_Z \end{align*}

Zach's marginal cost: \begin{align*} MC (Q_Z) = 2064 \end{align*}

Zach maximizes profit when MR \left( Q_Z \right) = MC \left( Q_Z \right)

\begin{align*} MC ( Q_Z ) &= MR ( Q_Z ) \\ 2064 &= 13080 - 4 Q_Y - 8 Q_Z \\ 8 Q_Z &= 13080 - 2064 - 4 Q_Y \\ Q_Z &= \frac{13080 - 2064}{8} - \frac{Q_Y}{2} \\ Q_Z &= 1377.0 - \frac{Q_Y}{2} \end{align*}

Yann's revenue: \begin{align*} R \left( Q_Y \right) &= P Q_Y \\ &= \left( 13080 - 4 ( Q_Z + Q_Y ) \right) Q_Y \\ &= \left( 13080 - 4 Q_Y - 4 Q_Z \right) Q_Y \\ &= 13080 Q_Y - 4 Q_Y^2 - 4 Q_Z Q_Y \end{align*} Yann's marginal revenue: \begin{align*} MR ( Q_Y ) &= 13080 - 4 Q_Z - 2 \times 4 Q_Y \\ &= 13080 - 4 Q_Z - 8 Q_Y \end{align*} Yann's marginal cost: \begin{align*} MC (Q_Y) = 552 \end{align*} Yann's best response solves \begin{align*} MC ( Q_Y ) &= MR ( Q_Y ) \\ 552 &= 13080 - 4 Q_Y - 8 Q_Z \\ 8 Q_Y &= 13080 - 552 - 4 Q_Z \\ Q_Y &= \frac{13080 - 552}{8} - \frac{Q_Z}{2} \\ Q_Y &= 1566.0 - \frac{Q_Z}{2} \end{align*}

Plug Q_Y = 1566.0 - \frac{Q_Z}{2} into Zach's best response:  \begin{align*} Q_Z &= 1377.0 - \frac{Q_Y}{2} \\ Q_Z &= 1377.0 - \frac{ 1566.0 - \frac{Q_Z}{2} }{ 2 } \\ Q_Z &= 1377.0 - \frac{ 1566.0 }{ 2 } + \frac{Q_Z}{4} \\ Q_Z - \frac{Q_Z}{4} &= 1377.0 - \frac{ 1566.0 }{ 2 } \\ \frac{3}{4} Q_Z &= 594.0 \\ Q_Z &= \frac{4}{3} 594.0 \\ Q_Z &= 792.0 \end{align*}  So Zach's quantity is Q_Z = 792.0. Plug that into Yann's best response function:  \begin{align*} Q_{ Y } &= 1566.0 - \frac{Q_Z}{2} \\ &= 1566.0 - \frac{ 792.0 }{2} \\ &= 1566.0 - 396.0 \\ &= 1170.0 \end{align*}