# Separating and supporting hyperplane theorems

I am having trouble understanding the separating and supporting hyperplane theorems. I've read what I can online but am just not able to develop any intuition. Can someone please give a basic outline of these theorems using economic context? And any graphical interpretations would be extremely helpful.

• I think while you are perfectly right to want to develop intuition in this area, the question is very broad. For an example on the use of the separation theorem you probably want to read about the first and second welfare theorems. I recommend the book by Mas-Collel, Whinston, Green. – Giskard Oct 25 '15 at 18:23

I will sketch a solution for your question on the separating hyperplane theorem together with one economic application to producer theory (for more details please look at MWG).

Theorem: Suppose $B \subset \mathbb{R}^N$ is a convex and closed and that $x \notin B$. Then there is a $p \in \mathbb{R}^N$ with $p \neq 0$ and a value $c \in \mathbb{R}$ such that $px>c$ and $py < c$ for every $y \in B$

Application (Producer theory): Suppose the production set $Y$ is closed and convex. Then every efficient production $y \in Y$ is a profit maximizing production for some nonzero price vector $p \geq 0$.

Sketch of proof: Start with an efficient $y$ and define $P_y$ as the "better than" set with respect to $y$. This set is clearly convex (pick $y', y'' \in P_y$ and their convex combination is clearly "in it"). Next realize that because $y$ is efficient $P_y \cup Y = \emptyset$. We know have all the ingredients to apply our theorem above, and in particular $\exists p \neq 0$ such that $p y' \geq py'''$ $\forall$ $y' \in P_y$ and $\forall$ $y''' \in Y$. Because $y'$ can be chosen arbitrarily close to $y$ (remember $y' \in P_y$) we can straightforwardly show that $y$ is profit maximizing for $p$

Other "famous applications" that you may want to look at are:

1) The "Second Fundamental theorem of Welfare Economics

2) The saddle point approach to dynamic programming developed by Karlin and Uzawa.