# Profit Functions

A duopolist's profit is their revenue minus their cost. $$\pi = R - C$$

## 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 = 51 - 79 Q.

Zach's marginal cost is equal to 49, and Yann's marginal cost is 36.

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( 51 - 79 ( Q_Z + Q_Y ) \right) Q_Z \\ &= \left( 51 - 79 Q_Z - 79 Q_Y \right) Q_Z \\ &= 51 Q_Z - 79 Q_Z^2 - 79 Q_Y Q_Z \end{align*}

Zach's cost: $$49 Q_Z$$

Zach's profit: $$\pi \left( Q_Z \right) = 51 Q_Z - 79 Q_Z Q_Y - 79 Q_Z^2 - 49 Q_Z$$

Yann's revenue: \begin{align*} R \left( Q_Y \right) &= P Q_Y \\ &= \left( 51 - 79 ( Q_Z + Q_Y ) \right) Q_Y \\ &= \left( 51 - 79 Q_Y - 79 Q_Z \right) Q_Y \\ &= 51 Q_Y - 79 Q_Y^2 - 79 Q_Z Q_Y \end{align*}

Yann's cost: $$36 Q_Y$$

Yann's profit: $$\pi \left( Q_Y \right) = 51 Q_Y - 79 Q_Y Q_Z - 79 Q_Y^2 - 36 Q_Y$$