Income distributions in multi-sector analysis; Miyazawa's fundamental equation of income formation revisited

In standard type 1 input-output models, households' activities are part of the exogenous final demand. This means that their scale and composition are exogenously determined. That is, if some other final demand categories change (say public investment or exports) this does not influence the behaviour of the household categories. In type 2 input-output models households' activities are explained endogenously to capture the possibility of mutual interaction between household categories and productive sectors. In this area, Miyazawa (1976) proposed a novel way of modeling the endogenization of households' activities. In modeling terms, Miyazawa's proposition resulted in the so-called 'fundamental equation of income formation', core of which is an extended input coefficients matrix. This extended coefficients matrix produced several new types of multiplier matrices and explains industrial gross output and households' income in terms of non-household final demand in great detail. The model is traditionally solved by inverting the new extended coefficients matrix, which often generates highly com-plex outcomes in terms of convoluted multiplier matrices. Consequently, the link be-tween final demand impulses, gross outputs and income formation is not straightfor-ward, working sometimes in different directions. Regarding this aspect, as we shall show, there is a second way to solve Miyazawa's fundamental equation, which is much more transparent. This second way shows that Miyazawa type endogenization means that gross output and (remaining) final demand are directly linked via a new type of coefficients matrix. This matrix is the sum of the traditional matrix of intermediate input coefficients and a number of matrices of rank 1, each one corresponding to an en-dogenized households category. The existence of this matrix makes several new appli-cations possible including the study of shifts over time in the distribution of income and (un)employment between the households categories involved. In an appendix we briefly focus on the link between the use of matrices of rank 1 in, respectively, Leontief, Sraffa and Miyazawa input-output economics