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]

See also

• Diagonally dominant matrix
• Perron–Frobenius theorem
• Sylvester's criterion

References

1. Hawkins, David; Simon, Herbert A. (1949). "Some Conditions of Macroeconomic Stability". Econometrica. 17 (3/4): 245–248. JSTOR 1905526.
2. Leontief, Wassily (1986). Input-Output Economics (2nd ed.). New York: Oxford University Press. ISBN 0-19-503525-9.
3. Nikaido, Hukukane (1968). Convex Structures and Economic Theory. Academic Press. pp. 90–92.
4. Morishima, Michio (1964). Equilibrium, Stability, and Growth: A Multi-sectoral Analysis. London: Oxford University Press. pp. 15–17.
5. Murata, Yasuo (1977). Mathematics for Stability and Optimization of Economic Systems. New York: Academic Press. pp. 52–53.
6. Kotelyanskiĭ, D. M. (1952). "О некоторых свойствах матриц с положительными элементами" [On Some Properties of Matrices with Positive Elements] (PDF). Mat. Sb. N.S. 31 (3): 497–506.
7. Gantmacher, Felix (1959). The Theory of Matrices. Vol. 2. New York: Chelsea. pp. 71–73.

Further reading

• McKenzie, Lionel (1960). "Matrices with Dominant Diagonals and Economic Theory". In Arrow, Kenneth J.; Karlin, Samuel; Suppes, Patrick (eds.). Mathematical Methods in the Social Sciences. Stanford University Press. pp. 47–62. OCLC 25792438.
• Takayama, Akira (1985). "Frobenius Theorems, Dominant Diagonal Matrices, and Applications". Mathematical Economics (Second ed.). New York: Cambridge University Press. pp. 359–409.