Here is a statement of Farkas Lemma from the Wikipedia. Let $A$ be an $m \times n$ matrix and $b$ an $m$-dimensional vector. Then, exactly one of the following two statements is true:
- There exists an $x \in \mathbb{R}^n$ such that $Ax = b$ and $x \geq 0$.
- There exists a $y \in \mathbb{R}^m$ such that $A^T y \geq 0$ and $b^T y < 0$.
This result has a simple geometric interpretation: either $b$ lies in the cone formed by the columns of $A$ or it is possible to find a vector $y$ such that $y$ forms an acute angle with all columns of $A$ and an obtuse angle with $b$.
I was wondering if there is a way to make the result intuitive purely at the algebraic level of solving linear equations and inequalities?
The lemma is important in finance and the geometric intuition is not much help there since the vectors are payoffs and prices of financial assets which have no natural geometric meaning. Gale's Theory of Linear Economic Models has a purely algebraic proof, but that is an opaque induction argument.
EDIT: A little more about my application. We have $m$ assets and $n$ possible future states of nature. $A_{ij}$ is the payoff by asset $i$ in state $j$. $b_i$ is the price of asset $i$. $x_j$ is the value of one dollar in state $j$. $y_i$ is the amount of asset $i$ in a portfolio.
So Farkas' lemma tells us that either (1) there is a way of assigning a non-negative price to a dollar in each state in a way such that the price of each asset is just the sum total of the value of its payoffs, or (2) there is a portfolio whose price is negative, so you get paid for holding it, but whose payoffs are non-negative, which means that you do not have to pay anything back. I want to understand, in terms which make economic sense, why (1) and (2) should be mutually exclusive. Acute and obtuse angles are no help here.