Leontief Meets Markov: Sectoral Vulnerabilities Through Circular Connectivity

Abstract

Economists have been aware of the mapping between an Input-Output (I-O, hereinafter) table and the adjacency matrix of a weighted digraph for several decades (Solow, Econometrica 20(1):29–46, 1952). An I-O table may be interpreted as a network in which edges measure money flows to purchase inputs that go into production, whilst vertices represent economic industries. However, only recently the language and concepts of complex networks (Newman 2010) have been more intensively applied to the study of interindustry relations (McNerney et al. Physica A Stat Mech Appl, 392(24):6427–6441, 2013). The aim of this paper is to study sectoral vulnerabilities in I-O networks, by connecting the formal structure of a closed I-O model (Leontief, Rev Econ Stat, 19(3):109–132, 1937) to the constituent elements of an ergodic, regular Markov chain (Kemeny and Snell 1976) and its chance process specification as a random walk on a graph. We provide an economic interpretation to a local, sector-specific vulnerability index based on mean first passage times, computed by means of the Moore-Penrose inverse of the asymmetric graph Laplacian (Boley et al. Linear Algebra Appl, 435(2):224–242, 2011). Traversing from the most central to the most peripheral sector of the economy in 60 countries between 2005 and 2015, we uncover cross-country salient roles for certain industries, pervasive features of structural change and (dis)similarities between national economies, in terms of their sectoral vulnerabilities.

The ternary water cycle of ‘nature–society–trade’ in the inland arid areas

“…The Leontief inverse matrix is the core of the input–output technique, which reveals the complex economic linkages between the various sectors of the national economy (Wirkierman et al, 2021), expressed by the formula: $$$$ L={\left(I-{A}^R\right)}^{-1} $$$$ where I is the unit matrix and A R is the direct consumption factor matrix.…”

“…The Leontief inverse matrix is the core of the input-output technique, which reveals the complex economic linkages between the various sectors of the national economy (Wirkierman et al, 2021), expressed by the formula:…”