The Second Eigenvalue of the Leontief Matrix

Andras Brody

Economic Systems Research, 1997, vol. 9, issue 3, 253-258

Abstract: According to Frobenius, a positive matrix possesses a unique positive eigenvector which belongs to a positive eigenvalue. This eigenvalue is of the largest absolute magnitude and the matrix admits no other positive eigenvector. If an arbitrary positive vector is repeatedly premultiplied by such a matrix, then the result tends towards this positive eigenvector. It is the second largest eigenvalue that determines the speed of convergence. The estimate of the second eigenvalue of a purely random flow coefficient matrix shows that its expected absolute magnitude declines monotonically with the size of the matrix. Hence, the larger the system is the faster is the convergence. A prescribed exactness of the eigenvector (of equilibrium prices or quantities) will be reached after a few—perhaps just a couple of—iterations in a large system.

Keywords: Equilibrium; convergence stability; market (search for similar items in EconPapers)
Date: 1997
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DOI: 10.1080/09535319700000018

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