Appendix

 Characteristic roots and vectors Since, in this paper, price systems are expressed as characteristic vectors of non-negative matrices, it is useful to state the following theorems, which have been proven in the theory of matrices: Definition: The characteristic root l0 of a square matrix A is called a maximum characteristic root if, for all the remaining characteristic roots, li of matrix A,  |li|  £  |l0|   holds true. Theorem (1) A non-negative matrix A ³0 always has a non-negative maximum characteristic root l0, and a non-negative characteristic vector X0. For any arbitrarily chosen l > l0,  (lI - A)-1 ³ 0;   and(lI - A)-1 /dl£ 0. Note: If  l0 is a maximum characteristic root of matrix A, it is also the maximum characteristic root of matrix A’. Theorem (2) Let A ³ 0 be irreducible. Then there exists a positive maximum characteristic root l0 and accompanying it a positive characteristic vector X0. No other positive characteristic vector of matrix A that is linearly independent of X0 exists. For any arbitrarily chosen l > l0,  (lI - A)-1 > 0;   and(lI - A)-1 /dl< 0. Theorem (3) Given two non-negative matrices A and A* with maximum characteristic roots l0 and l*0, A

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