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 $[\mathbf {I} -\mathbf {A} ]$ under which the input–output system

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

has a solution $\mathbf {\hat {x}} \geq 0$ for any $\mathbf {d} \geq 0$ . Here $\mathbf {I}$ is the identity matrix and $\mathbf {A}$ 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

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

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

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

(i) There exists an

\mathbf {x} \geq 0

such that

\mathbf {B} \cdot \mathbf {x} >0

.

(ii) All the successive leading principal minors of

\mathbf {B}

are positive, that is

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]

References

^ Hawkins, David; Simon, Herbert A. (1949). "Some Conditions of Macroeconomic Stability". Econometrica. 17 (3/4): 245–248. doi:10.2307/1905526. JSTOR 1905526.
^ Leontief, Wassily (1986). Input-Output Economics (2nd ed.). New York: Oxford University Press. ISBN 0-19-503525-9.
^ ^a ^b Nikaido, Hukukane (1968). Convex Structures and Economic Theory. Academic Press. pp. 90–92. ISBN 978-1-4832-6668-8.
^ Morishima, Michio (1964). Equilibrium, Stability, and Growth: A Multi-sectoral Analysis. London: Oxford University Press. pp. 15–17. ISBN 978-0-19-828145-0.
^ Murata, Yasuo (1977). Mathematics for Stability and Optimization of Economic Systems. New York: Academic Press. pp. 52–53. ISBN 978-1-4832-7129-3.
^ 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.
^ Gantmacher, Felix (1959). The Theory of Matrices. Vol. 2. New York: Chelsea. pp. 71–73. ISBN 978-0-8218-1393-5.

Hawkins–Simon condition

See also

References

Further reading