Embodied Resource Flows and Product Flows: Combining the Absorbing Markov Chain with the Input-Output Model

We develop the absorbing Markov chain (AMC) for describing in detail the network of paths through an industrial system taken by an embodied resource from extraction through intermediate products and finally consumer products. We refer to this as a resource-specific network. This work builds on a recent literature in industrial ecology that uses an AMC to quantify the number of times a resource passes through a recycling sector before ending up in a landfill. Our objective is to incorporate into that analysis an input-output (IO) table so that the resource paths explicitly take account of the interdependence of sectors through their reliance on intermediate products. This feature makes it possible to track multiple resources simultaneously and consistently and to represent both resources and products in mixed units. Hypothetical scenarios about technological changes and changes in consumer demand are analyzed using an IO model, and model solutions generate the AMC database. A numerical example is provided. AMC analysis describes the resource-specific networks using matrices that are derived not from the Leontief inverse but from a generalized variant of the Ghosh inverse matrix. The Leontief inverse and especially the Ghosh inverse (although often not identified as such) have been used extensively to analyze ecological systems, and this paper extends these approaches for use in studying material cycles in industrial systems. Constructing the AMC formalizes the resource-specific network analysis and generalizes the content and interpretation of the Ghosh matrix. Path-based analyses derived from AMC theory are discussed in relation to the set of techniques called Structural Path Analysis (SPA). The paper concludes by identifying the three most critical enhancements to the IO model needed for analyzing material cycles: the simultaneous incorporation of waste-processing sectors, stock and flow relationships, and international trade. The idea is to implement an AMC after each model extension. The modeling framework is intended for analyses such as: tracking a resource extracted in one region to landfills in other regions, evaluating ways to intensify secondary recovery at key junctures in-between. There are other ways, of course, to approach such an analysis, but the combination of an extended IO model and an AMC, representing both resources and products in mixed units, provides a comprehensive, systematic and standardized approach that includes many features that are valued in industrial ecology and builds directly on a number of active research programs.