Chapter 5 explores variations to the commodity-by-industry input–output framework introduced in Chapter 4, expanding the basic input–output framework to include distinguishing between commodities and industries, i.e., the supply of specific commodities in the economy and the use of those commodities by collections of businesses defined as industries. The chapter introduces the fundamental commodity-by-industry accounting relationships and how they relate to the basic input–output framework. Alternative assumptions are defined for handling the common accounting issue of secondary production, and economic interpretations of those alternative assumptions are presented. The formulations of commodity-driven and industry-driven models are also presented along with illustrations of variants on combining alternative assumptions for secondary production. Finally, the chapter illustrates the problem encountered with commodity-by-industry models, such as non-square commodity–industry systems, mixed technology options, or the interpretation of negative elements.

Chapter 2 introduces Leontief’s conceptual input–output framework and explains how to develop the fundamental mathematical relationships from the interindustry transactions table. The key assumptions associated with the basic Leontief model and implications of those assumptions are recounted and the economic interpretation of the basic framework is explored. The basic framework is illustrated with a highly aggregated model of the US economy. In addition, the “price model” formulation of the input–output framework is introduced to explore the role of prices in input–output models. Appendices to this chapter include a fundamental set of mathematical conditions for input–output models, known as the Hawkins–Simon conditions.