Hawkins–Simon condition explained

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

[ I - A ] under which the input–output system[ I - A ] ⋅ x = d has a solution \hat{x } \geq 0 for any d \geq 0 . Here I is the identity matrix and 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 ^{n} _{j =1} a _{ij} x _{j} + d _{i} = x _{i} 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

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

(i) There exists an

x \geq 0 such that B ⋅ x > 0 .

(ii) All the successive leading principal minors of

B are positive, that isb _{11} > 0 , \begin{vmatrix} b _{11} & b _{12} \ b _{21} & b _{22} \end{vmatrix} > 0 , \ldots, \begin{vmatrix} b _{11} & b _{12} & ... & b _{1n} \ b _{21} & b _{22} & ... & b _{2n} \ \vdots & \vdots & \ddots & \vdots \ b _{n1} & b _{n2} & ... & b _{nn} \end{vmatrix} > 0

For a proof, see Morishima (1964),^[4] Nikaido (1968), 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

Further reading

• Book: McKenzie, Lionel . Lionel McKenzie . Matrices with Dominant Diagonals and Economic Theory . Mathematical Methods in the Social Sciences . Kenneth J. . Arrow . Kenneth Arrow . Samuel . Karlin . Samuel Karlin . Patrick . Suppes . Patrick Suppes . Stanford University Press . 1960 . 47–62 . 25792438 .
• Book: Takayama, Akira . Frobenius Theorems, Dominant Diagonal Matrices, and Applications . Mathematical Economics . New York . Cambridge University Press . Second . 1985 . 359–409 . https://books.google.com/books?id=j6PLOBFotPQC&pg=PA359 .

Notes and References

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