**Introduction:**In this section we apply the first-order condition for an extremum to derive the marginal output rule. In economic theory this rule constitutes, in combination with the production rule, the decision model for a producer who aims to maximize his profits.

**Model:**The variable in this model is $y$, the output of the production process.

Furthermore, three functions are part of this model:

- $R(y)$: the revenue function,
- $C(y)$: the cost function,
- $\pi(y)$: the profit function,

We assume $C(0)=0$.

**Maginal output rule:**The output quantity $y>0$ that maximizes the profit $\pi(y)=R(y)-C(y)$ satisfies the equation

\[

MR(y) = MC(y).\]

**Production rule:**If $y>0$ is the output quantity that maximizes profit, the producer will produce if

\[

AR(y) \geq AC(y),

\]

with $AR(y)=\dfrac{R(y)}{y}$ the average revenue function and $AC(y)=\dfrac{C(y)}{y}$ the average cost function.