Supply function producer

Introduction:  The Marginal output rule and production rule are introduced in a setting where the price is fixed. Using this approach we find a supply function, which gives for each price the output quantity that maximizes profit.

Model:

• The variable in this model is $y$, the output of the production process.
• The parameter in this model is $p$, the price.

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

Supply function: The supply function of a producer with profit function $\pi (y) = py- C(y)$ is given by
$$y(p) =\left \{ \begin{array}{lll} MC^{-1} (p) & \mbox{if} & p \geq \min AC(y),\\[1mm] 0               & \mbox{if} & p < \min AC(y). \end{array} \right .$$