  The indirect transactions between sectors of an economic system has been a
long-standing open problem. There have been numerous attempts to conceptually
define and mathematically formulate this notion in various other scientific
fields in literature as well. The existing direct and indirect effects
formulations, however, can neither determine the direct and indirect
transactions separately nor quantify these transactions between two individual
sectors of interest in a multisectoral economic system. The novel concepts of
the direct, indirect and transfer (total) transactions between any two sectors
are introduced, and the corresponding requirements matrices and coefficients
are systematically formulated relative to both final demands and gross outputs
based on the system decomposition theory in the present manuscript. It is
demonstrated theoretically and through illustrative examples that the proposed
requirements matrices accurately define and correctly quantify the
corresponding direct, indirect, and total interactions and relationships. The
proposed requirements matrices for the US economy using aggregated input-output
tables for multiple years are then presented and briefly analyzed.
