Hawkins–Simon condition

Result in mathematical economics on existence of a non-negative equilibrium output vector

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 ] {\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.[2] 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} .[3] 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
b 11 > 0 , | b 11 b 12 b 21 b 22 | > 0 , … , | b 11 b 12 … b 1 n b 21 b 22 … b 2 n ⋮ ⋮ ⋱ ⋮ b n 1 b n 2 … b n n | > 0 {\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

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. ^ a b 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.