The Hawkins–Simon condition refers to a result in mathematical economics, attributed to David Hawkins and Herbert A. Simon, 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 [ I − A ] {\displaystyle [\mathbf {I} -\mathbf {A} ]} under which the input–output system
[ I − A ] ⋅ x = d {\displaystyle [\mathbf {I} -\mathbf {A} ]\cdot \mathbf {x} =\mathbf {d} }
has a solution x ^ ≥ 0 {\displaystyle \mathbf {\hat {x}} \geq 0} for any d ≥ 0 {\displaystyle \mathbf {d} \geq 0} . Here I {\displaystyle \mathbf {I} } is the identity matrix and A {\displaystyle \mathbf {A} } is called the input–output matrix or Leontief matrix after Wassily Leontief, who empirically estimated it in the 1940s. Together, they describe a system in which
∑ j = 1 n a i j x j + d i = x i i = 1 , 2 , … , n {\displaystyle \sum _{j=1}^{n}a_{ij}x_{j}+d_{i}=x_{i}\quad i=1,2,\ldots ,n}
where a i j {\displaystyle a_{ij}} is the amount of the ith good used to produce one unit of the jth good, x j {\displaystyle x_{j}} is the amount of the jth good produced, and d i {\displaystyle d_{i}} is the amount of final demand for good i. Rearranged and written in vector notation, this gives the first equation. Define [ I − A ] = B {\displaystyle [\mathbf {I} -\mathbf {A} ]=\mathbf {B} } , where B = [ b i j ] {\displaystyle \mathbf {B} =\left[b_{ij}\right]} is an n × n {\displaystyle n\times n} matrix with b i j ≤ 0 , i ≠ j {\displaystyle b_{ij}\leq 0,i\neq j} . Then the Hawkins–Simon theorem states that the following two conditions are equivalent
(i) There exists an x ≥ 0 {\displaystyle \mathbf {x} \geq 0} such that B ⋅ x > 0 {\displaystyle \mathbf {B} \cdot \mathbf {x} >0} . (ii) All the successive leading principal minors of B {\displaystyle \mathbf {B} } are positive, that is
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