Hawkins–Simon condition

The Hawkins–Simon condition refers to a result in mathematical economics, attributed to David Hawkins and Herbert A. Simon,[1] that guarantees the existence of a non-negative output vector that solves the equilibrium relation in the input–output model where demand equals supply. More precisely, it states a condition for ${\displaystyle [\mathbf {I} -\mathbf {A} ]}$ under which the input–output system

${\displaystyle [\mathbf {I} -\mathbf {A} ]\cdot \mathbf {x} =\mathbf {d} }$

has a solution ${\displaystyle \mathbf {\hat {x}} \geq 0}$ for any ${\displaystyle \mathbf {d} \geq 0}$. Here ${\displaystyle \mathbf {I} }$ is the identity matrix and ${\displaystyle \mathbf {A} }$ is called the input–output matrix or Leontief matrix after Wassily Leontief, who empirically estimated it in the 1940s.[2] Together, they describe a system in which

${\displaystyle \sum _{j=1}^{n}a_{ij}x_{j}+d_{i}=x_{i}\quad i=1,2,\ldots ,n}$

where ${\displaystyle a_{ij}}$ is the amount of the ith good used to produce one unit of the jth good, ${\displaystyle x_{j}}$ is the amount of the jth good produced, and ${\displaystyle d_{i}}$ is the amount of final demand for good i. Rearranged and written in vector notation, this gives the first equation.

Define ${\displaystyle [\mathbf {I} -\mathbf {A} ]=\mathbf {B} }$, where ${\displaystyle \mathbf {B} =\left[b_{ij}\right]}$ is an ${\displaystyle n\times n}$ matrix with ${\displaystyle b_{ij}\leq 0,i\neq j}$.[3] Then the Hawkins–Simon theorem states that the following two conditions are equivalent

(i) There exists an ${\displaystyle \mathbf {x} \geq 0}$ such that ${\displaystyle \mathbf {B} \cdot \mathbf {x} >0}$.

(ii) All the successive leading principal minors of ${\displaystyle \mathbf {B} }$ are positive, that is

${\displaystyle b_{11}>0,{\begin{vmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{vmatrix}}>0,\ldots ,{\begin{vmatrix}b_{11}&b_{12}&\dots &b_{1n}\\b_{21}&b_{22}&\dots &b_{2n}\\\vdots &\vdots &\ddots &\vdots \\b_{n1}&b_{n2}&\dots &b_{nn}\end{vmatrix}}>0}$

For a proof, see Morishima (1964),[4] Nikaido (1968),[3] or Murata (1977).[5] Condition (ii) is known as Hawkins–Simon condition. This theorem was independently discovered by David Kotelyanskiĭ,[6] as it is referred to by Felix Gantmacher as Kotelyanskiĭ lemma.[7]