First Order Conditions for Maximization Problem

Question

Could someone explain why the second maximization problem follows from the first? And how did we get the FOC in that form?

For $max_{p' \in R^L} py(p')$ are we holding p fixed?

Thank you!

Answer 1

It looks like we're maximizing a firm's revenue subject to being in some constraint set $Y$. That is our problem is

$$ max_{y \in Y} R(p, y)$$

where $R(p,y) = p \cdot y$ is the revenue. Then we can write

$$y(p) = \arg \max_{y \in Y} R(p,y)$$

That is, give me a price $p$ and I will tell you the $y$ that maximizes it. We can view this as a function $y(p): P \to Y$. That is it maps from the set of prices to the set of outputs $Y$.

We can equivalently write this problem as

$$\max_{p' \in \mathbb{R}^N} R(p, y(p'))$$

Now the problem says give me a price $p$ (hold it fixed) and I can find a $y$ indirectly by finding $p'$ that will maximize revenue.

The first order condition for this is \begin{align} \dfrac{\partial}{\partial p'} p \cdot y(p') &= 0\\ [D_p y(p)] p &= 0 \end{align}