The formulation of the indirect transactions between the sectors of an economic system is a long-standing open problem. There have been numerous attempts to define and mathematically formulate this concept in various  scientific fields in literature . The current indirect effects formulations for an economic system represent the sum of the subsequent gross outputs of goods and services at each step of a production chain to supply the final demands, generally after the first gross outputs from each sector in the system. These existing formulations cannot quantify the indirect interactions  between any two sectors . The indirect transactions between two sectors of an economic system and the corresponding indirect requirements  are introduced in the present article , based on the system decomposition theory . This novel concept of the indirect transactions is also compared with some existing indirect effect formulations, and the theoretical advancements brought by the proposed methodology are discussed. It is shown theoretically and through illustrative examples that the proposed indirect transactions better describe and quantify the indirect interactions than the current indirect effects formulations. The indirect requirements matrices for the US economy using aggregated input-output tables for multiple years are also presented and briefly analyzed.
The indirect transactions between sectors of an economic system has been a long-standing open problem. There have been numerous attempts to define and mathematically formulate this concept in various other scientific fields in literature as well. The existing indirect effects formulations , however, cannot quantify the indirect interactions and transactions between any two sectors of an economic system. Therefore, although the direct and total requirement matrices are formulated and used for economic system analysis, the indirect requirements matrices has never been formulated before. Based on the system decomposition theory, the indirect transactions between two sectors of an economic system and the corresponding indirect requirements matrices are introduced in the present article for the first time . This novel concept of the indirect transactions is also compared with some existing indirect effect formulations, and the theoretical advancements brought by the proposed methodology are discussed. It is shown theoretically and through illustrative examples that the proposed indirect transactions accurately describe and quantify the indirect interactions and transactions, unlike the current indirect effects formulations. The indirect requirements matrices for the US economy using aggregated input-output tables for multiple years are also presented and briefly analyzed.
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