# Cournot Equilibrium

The market is in equilibrium when firms set quantitys that are best responses to one another.

## 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 = 73800 - 10 ( Q_Z + Q_Y ).

Zach faces marginal costs equal to 5820.

Yann faces marginal costs equal to 1560.

What is the Cournot Equilibrium?

In equilibrium, Q_Z = 2124.0 and Q_Y = 2550.0.

Zach's revenue:

\begin{align*} R \left( Q_Z \right) &= P Q_Z \\ &= \left( 73800 - 10 ( Q_Z + Q_Y ) \right) Q_Z \\ &= \left( 73800 - 10 Q_Z - 10 Q_Y \right) Q_Z \\ &= 73800 Q_Z - 10 Q_Z^2 - 10 Q_Y Q_Z \end{align*}

Zach's marginal revenue:

\begin{align*} MR (Q_Z) \begin{align*} MR ( Q_Z ) &= 73800 - 10 Q_Y - 2 \times 10 Q_Z \\ &= 73800 - 10 Q_Y - 20 Q_Z \end{align*} \end{align*}

Zach's marginal cost:

\begin{align*} MC (Q_Z) = 5820 \end{align*}

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

\begin{align*} MC ( Q_Z ) &= MR ( Q_Z ) \\ 5820 &= 73800 - 10 Q_Y - 20 Q_Z \\ 20 Q_Z &= 73800 - 5820 - 10 Q_Y \\ Q_Z &= \frac{73800 - 5820}{20} - \frac{Q_Y}{2} \\ Q_Z &= 3399.0 - \frac{Q_Y}{2} \end{align*}

Yann's revenue:

\begin{align*} R \left( Q_Y \right) &= P Q_Y \\ &= \left( 73800 - 10 ( Q_Z + Q_Y ) \right) Q_Y \\ &= \left( 73800 - 10 Q_Y - 10 Q_Z \right) Q_Y \\ &= 73800 Q_Y - 10 Q_Y^2 - 10 Q_Z Q_Y \end{align*} Yann's marginal revenue: \begin{align*} MR (Q_Y) \begin{align*} MR ( Q_Y ) &= 73800 - 10 Q_Z - 2 \times 10 Q_Y \\ &= 73800 - 10 Q_Z - 20 Q_Y \end{align*} \end{align*} Yann's marginal cost: \begin{align*} MC (Q_Y) = 1560 \end{align*} Yann's best response solves \begin{align*} MC ( Q_Y ) &= MR ( Q_Y ) \\ 1560 &= 73800 - 10 Q_Y - 20 Q_Z \\ 20 Q_Y &= 73800 - 1560 - 10 Q_Z \\ Q_Y &= \frac{73800 - 1560}{20} - \frac{Q_Z}{2} \\ Q_Y &= 3612.0 - \frac{Q_Z}{2} \end{align*}

Plug Q_Y = 3612.0 - \frac{Q_Z}{2} into Zach's best response:

 \begin{align*} Q_Z &= 3399.0 - \frac{Q_Y}{2} \\ Q_Z &= 3399.0 - \frac{ 3612.0 - \frac{Q_Z}{2} }{ 2 } \\ Q_Z &= 3399.0 - \frac{ 3612.0 }{ 2 } + \frac{Q_Z}{4} \\ Q_Z - \frac{Q_Z}{4} &= 3399.0 - \frac{ 3612.0 }{ 2 } \\ \frac{3}{4} Q_Z &= 1593.0 \\ Q_Z &= \frac{4}{3} 1593.0 \\ Q_Z &= 2124.0 \end{align*} 

So Zach's quantity is Q_Z = 2124.0. Plug that into Yann's best response function:

 \begin{align*} Q_{ Y } &= 3612.0 - \frac{Q_Z}{2} \\ &= 3612.0 - \frac{ 2124.0 }{2} \\ &= 3612.0 - 1062.0 \\ &= 2550.0 \end{align*}