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 = 21300 - 5 ( Q_Z + Q_Y )`.

Zach faces marginal costs equal to `1920`.

Yann faces marginal costs equal to `210`.

What is the Cournot Equilibrium?

In equilibrium, `Q_Z = 1178.0` and `Q_Y = 1520.0`.

Zach's revenue:

$$ \begin{align*} R \left( Q_Z \right) &= P Q_Z \\ &= \left( 21300 - 5 ( Q_Z + Q_Y ) \right) Q_Z \\ &= \left( 21300 - 5 Q_Z - 5 Q_Y \right) Q_Z \\ &= 21300 Q_Z - 5 Q_Z^2 - 5 Q_Y Q_Z \end{align*} $$

Zach's marginal revenue:

$$ \begin{align*} MR (Q_Z) \begin{align*} MR ( Q_Z ) &= 21300 - 5 Q_Y - 2 \times 5 Q_Z \\ &= 21300 - 5 Q_Y - 10 Q_Z \end{align*} \end{align*} $$

Zach's marginal cost:

$$ \begin{align*} MC (Q_Z) = 1920 \end{align*} $$

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

\begin{align*} MC ( Q_Z ) &= MR ( Q_Z ) \\ 1920 &= 21300 - 5 Q_Y - 10 Q_Z \\ 10 Q_Z &= 21300 - 1920 - 5 Q_Y \\ Q_Z &= \frac{21300 - 1920}{10} - \frac{Q_Y}{2} \\ Q_Z &= 1938.0 - \frac{Q_Y}{2} \end{align*}

Yann's revenue:

$$ \begin{align*} R \left( Q_Y \right) &= P Q_Y \\ &= \left( 21300 - 5 ( Q_Z + Q_Y ) \right) Q_Y \\ &= \left( 21300 - 5 Q_Y - 5 Q_Z \right) Q_Y \\ &= 21300 Q_Y - 5 Q_Y^2 - 5 Q_Z Q_Y \end{align*} $$ Yann's marginal revenue: $$ \begin{align*} MR (Q_Y) \begin{align*} MR ( Q_Y ) &= 21300 - 5 Q_Z - 2 \times 5 Q_Y \\ &= 21300 - 5 Q_Z - 10 Q_Y \end{align*} \end{align*} $$ Yann's marginal cost: $$ \begin{align*} MC (Q_Y) = 210 \end{align*} $$ Yann's best response solves \begin{align*} MC ( Q_Y ) &= MR ( Q_Y ) \\ 210 &= 21300 - 5 Q_Y - 10 Q_Z \\ 10 Q_Y &= 21300 - 210 - 5 Q_Z \\ Q_Y &= \frac{21300 - 210}{10} - \frac{Q_Z}{2} \\ Q_Y &= 2109.0 - \frac{Q_Z}{2} \end{align*}

Plug `Q_Y = 2109.0 - \frac{Q_Z}{2}` into Zach's best response:

$$ $$ \begin{align*} Q_Z &= 1938.0 - \frac{Q_Y}{2} \\ Q_Z &= 1938.0 - \frac{ 2109.0 - \frac{Q_Z}{2} }{ 2 } \\ Q_Z &= 1938.0 - \frac{ 2109.0 }{ 2 } + \frac{Q_Z}{4} \\ Q_Z - \frac{Q_Z}{4} &= 1938.0 - \frac{ 2109.0 }{ 2 } \\ \frac{3}{4} Q_Z &= 883.5 \\ Q_Z &= \frac{4}{3} 883.5 \\ Q_Z &= 1178.0 \end{align*} $$ $$

So Zach's quantity is `Q_Z = 1178.0`. Plug that into Yann's best response function:

$$ $$ \begin{align*} Q_{ Y } &= 2109.0 - \frac{Q_Z}{2} \\ &= 2109.0 - \frac{ 1178.0 }{2} \\ &= 2109.0 - 589.0 \\ &= 1520.0 \end{align*} $$ $$