Example
Zach and Yann compete in the market for coffee beans. The price follows $$P = 1000 - 2 Q = 1400 - 2 \left( Q_Z + Q_Y \right)$$ where `Q_Z` denotes Zach's quantity, and `Q_Y` denotes Yann's quantity.
Zach's profit
Zach's revenue function is \begin{align*} R_Z &= PQ_Z \\ &= \left( 1400 - 2 \left( Q_Z + Q_Y \right) \right) Q_Z \\ &= 1400 Q_Z - 2Q_Z^2 - 2 Q_Y Q_Z \end{align*}
Zach's cost function is $$C_Z = 200 Q_Z$$
So Zach's profit function is \begin{align*} \pi_Z &= R_Z - C_Z \\ &= 1400 Q_Z - 2Q_Z^2 - 2 Q_Y Q_Z - 200 Q_Z \\ &= 1200 Q_Z - 2Q_Z^2 - 2 Q_Y Q_Z \end{align*}
Yann's profit
Yann's revenue function is \begin{align*} R_Y &= PQ_Y \\ &= \left( 1400 - 2 \left( Q_Z + Q_Y \right) \right) Q_Y \\ &= 1400 Q_Y - 2Q_Z Q_Y - 2 Q_Y^2 \end{align*}
Yann's cost function is $$C_Y = 200 Q_Y$$
So Yann's profit is \begin{align*} \pi_Y &= R_Y - C_Y \\ &= 1400 Q_Y - 2Q_Z Q_Y - 2 Q_Y^2 - 200Q_Y \\ &= 1200 Q_Y - 2Q_Z Q_Y - 2 Q_Y^2 \end{align*}
Question
The price for coffee follows `P = 3 - 67 Q`.
Zach's marginal cost is equal to 3, and Yann's marginal cost is 3.
In function of `Q_Z` and `Q_Y`, what is Zach's revenue? What is his cost? What is his profit?
In function of `Q_Z` and `Q_Y`, what is Yann's revenue? What is his cost? What is his profit?
Zach's revenue: $$ \begin{align*} R \left( Q_Z \right) &= P Q_Z \\ &= \left( 3 - 67 ( Q_Z + Q_Y ) \right) Q_Z \\ &= \left( 3 - 67 Q_Z - 67 Q_Y \right) Q_Z \\ &= 3 Q_Z - 67 Q_Z^2 - 67 Q_Y Q_Z \end{align*} $$
Zach's cost: $$ 3 Q_Z $$
Zach's profit: $$ \pi \left( Q_Z \right) = 3 Q_Z - 67 Q_Z Q_Y - 67 Q_Z^2 - 3 Q_Z $$
Yann's revenue: $$ \begin{align*} R \left( Q_Y \right) &= P Q_Y \\ &= \left( 3 - 67 ( Q_Z + Q_Y ) \right) Q_Y \\ &= \left( 3 - 67 Q_Y - 67 Q_Z \right) Q_Y \\ &= 3 Q_Y - 67 Q_Y^2 - 67 Q_Z Q_Y \end{align*} $$
Yann's cost: $$ 3 Q_Y $$
Yann's profit: $$ \pi \left( Q_Y \right) = 3 Q_Y - 67 Q_Y Q_Z - 67 Q_Y^2 - 3 Q_Y $$