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 = 27264 - 8 ( Q_Z + Q_Y )`.

Zach faces marginal costs equal to `2256`.

Yann faces marginal costs equal to `1152`.

What is the Cournot Equilibrium?

In equilibrium, `Q_Z = 996.0` and `Q_Y = 1134.0`.

Zach's revenue: $$ \begin{align*} R \left( Q_Z \right) &= P Q_Z \\ &= \left( 27264 - 8 ( Q_Z + Q_Y ) \right) Q_Z \\ &= \left( 27264 - 8 Q_Z - 8 Q_Y \right) Q_Z \\ &= 27264 Q_Z - 8 Q_Z^2 - 8 Q_Y Q_Z \end{align*} $$

Zach's marginal revenue: $$ \begin{align*} MR ( Q_Z ) &= 27264 - 8 Q_Y - 2 \times 8 Q_Z \\ &= 27264 - 8 Q_Y - 16 Q_Z \end{align*} $$

Zach's marginal cost: $$ \begin{align*} MC (Q_Z) = 2256 \end{align*} $$

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

\begin{align*} MC ( Q_Z ) &= MR ( Q_Z ) \\ 2256 &= 27264 - 8 Q_Y - 16 Q_Z \\ 16 Q_Z &= 27264 - 2256 - 8 Q_Y \\ Q_Z &= \frac{27264 - 2256}{16} - \frac{Q_Y}{2} \\ Q_Z &= 1563.0 - \frac{Q_Y}{2} \end{align*}

Yann's revenue: $$ \begin{align*} R \left( Q_Y \right) &= P Q_Y \\ &= \left( 27264 - 8 ( Q_Z + Q_Y ) \right) Q_Y \\ &= \left( 27264 - 8 Q_Y - 8 Q_Z \right) Q_Y \\ &= 27264 Q_Y - 8 Q_Y^2 - 8 Q_Z Q_Y \end{align*} $$ Yann's marginal revenue: $$ \begin{align*} MR ( Q_Y ) &= 27264 - 8 Q_Z - 2 \times 8 Q_Y \\ &= 27264 - 8 Q_Z - 16 Q_Y \end{align*} $$ Yann's marginal cost: $$ \begin{align*} MC (Q_Y) = 1152 \end{align*} $$ Yann's best response solves \begin{align*} MC ( Q_Y ) &= MR ( Q_Y ) \\ 1152 &= 27264 - 8 Q_Y - 16 Q_Z \\ 16 Q_Y &= 27264 - 1152 - 8 Q_Z \\ Q_Y &= \frac{27264 - 1152}{16} - \frac{Q_Z}{2} \\ Q_Y &= 1632.0 - \frac{Q_Z}{2} \end{align*}

Plug `Q_Y = 1632.0 - \frac{Q_Z}{2}` into Zach's best response: $$ $$ \begin{align*} Q_Z &= 1563.0 - \frac{Q_Y}{2} \\ Q_Z &= 1563.0 - \frac{ 1632.0 - \frac{Q_Z}{2} }{ 2 } \\ Q_Z &= 1563.0 - \frac{ 1632.0 }{ 2 } + \frac{Q_Z}{4} \\ Q_Z - \frac{Q_Z}{4} &= 1563.0 - \frac{ 1632.0 }{ 2 } \\ \frac{3}{4} Q_Z &= 747.0 \\ Q_Z &= \frac{4}{3} 747.0 \\ Q_Z &= 996.0 \end{align*} $$ $$ So Zach's quantity is `Q_Z = 996.0`. Plug that into Yann's best response function: $$ $$ \begin{align*} Q_{ Y } &= 1632.0 - \frac{Q_Z}{2} \\ &= 1632.0 - \frac{ 996.0 }{2} \\ &= 1632.0 - 498.0 \\ &= 1134.0 \end{align*} $$ $$