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 [math]\displaystyle{ [\mathbf{I} - \mathbf{A}] }[/math] under which the input–output system
- [math]\displaystyle{ [\mathbf{I} - \mathbf{A}] \cdot \mathbf{x} = \mathbf{d} }[/math]
has a solution [math]\displaystyle{ \mathbf{\hat{x}} \geq 0 }[/math] for any [math]\displaystyle{ \mathbf{d} \geq 0 }[/math]. Here [math]\displaystyle{ \mathbf{I} }[/math] is the identity matrix and [math]\displaystyle{ \mathbf{A} }[/math] 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
- [math]\displaystyle{ \sum_{j=1}^{n} a_{ij} x_{j} + d_{i} = x_{i} \quad i = 1, 2, \ldots, n }[/math]
where [math]\displaystyle{ a_{ij} }[/math] is the amount of the ith good used to produce one unit of the jth good, [math]\displaystyle{ x_{j} }[/math] is the amount of the jth good produced, and [math]\displaystyle{ d_{i} }[/math] is the amount of final demand for good i. Rearranged and written in vector notation, this gives the first equation.
Define [math]\displaystyle{ [\mathbf{I} - \mathbf{A}] = \mathbf{B} }[/math], where [math]\displaystyle{ \mathbf{B} = \left[ b_{ij} \right] }[/math] is an [math]\displaystyle{ n \times n }[/math] matrix with [math]\displaystyle{ b_{ij} \leq 0, i \neq j }[/math].[3] Then the Hawkins–Simon theorem states that the following two conditions are equivalent
- (i) There exists an [math]\displaystyle{ \mathbf{x} \geq 0 }[/math] such that [math]\displaystyle{ \mathbf{B} \cdot \mathbf{x} \gt 0 }[/math].
- (ii) All the successive leading principal minors of [math]\displaystyle{ \mathbf{B} }[/math] are positive, that is
- [math]\displaystyle{ b_{11} \gt 0, \begin{vmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{vmatrix} \gt 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} \gt 0 }[/math]
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
References
- ↑ Hawkins, David; Simon, Herbert A. (1949). "Some Conditions of Macroeconomic Stability". Econometrica 17 (3/4): 245–248.
- ↑ Leontief, Wassily (1986). Input-Output Economics (2nd ed.). New York: Oxford University Press. ISBN 0-19-503525-9. https://archive.org/details/inputoutputecono0000leon.
- ↑ 3.0 3.1 Nikaido, Hukukane (1968). Convex Structures and Economic Theory. Academic Press. pp. 90–92. https://books.google.com/books?id=NMVgDAAAQBAJ&pg=PA90.
- ↑ Morishima, Michio (1964). Equilibrium, Stability, and Growth: A Multi-sectoral Analysis. London: Oxford University Press. pp. 15–17. https://books.google.com/books?id=m-cjAAAAMAAJ&pg=PA15.
- ↑ Murata, Yasuo (1977). Mathematics for Stability and Optimization of Economic Systems. New York: Academic Press. pp. 52–53. https://books.google.com/books?id=5DOjBQAAQBAJ&pg=PA52.
- ↑ Kotelyanskiĭ, D. M. (1952). "О некоторых свойствах матриц с положительными элементами". Mat. Sb.. N.S. 31 (3): 497–506. http://www.mathnet.ru/links/75fc598ee896f36e1c2671f38c1e778d/sm5545.pdf.
- ↑ Gantmacher, Felix (1959). The Theory of Matrices. 2. New York: Chelsea. pp. 71–73. https://books.google.com/books?id=cyX32q8ZP5cC&pg=PA71.
Further reading
- McKenzie, Lionel (1960). "Matrices with Dominant Diagonals and Economic Theory". in Arrow, Kenneth J.; Karlin, Samuel; Suppes, Patrick. 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. https://books.google.com/books?id=j6PLOBFotPQC&pg=PA359.
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