Marginal output rule and production rule

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,

such that $\pi(y)=R(y)-C(y)$.

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.