1. The argument developed in the previous chapter was based on the tacit assumption that the production of goods by means of goods and labour is independent of the feeding of labour. It was described in terms of the matrices of input and output coefficients and the vector of labour-input coefficients, all of which were considered as technologically given constants. As a matter of fact, however, it is difficult to obtain precise figures of these coefficients, especially the figures of the labour-input coefficients, without knowing how and at what level people are fed. The productivity of labour depends not only on technology in the narrowest sense but also on the workers' state of health, their living and working conditions and so on, as well as on domestic troubles between them and their wives, which will often occur when they are paid poor wages. We must further-more remember the historical fact that in a slave economy, with the wages fixed at a subsistence level, the productivity of labour was low, so that it was replaced by a more productive system, the capitalist economy. It is not surprising to see that outputs of goods might increase, even though the allocation of available goods among industries and families became unfavourable for the former; in fact, the positive indirect effect on outputs of a transfer of goods from industries to families causing an improvement in the welfare of the workers might be so strong as to overcome the negative direct effect on outputs of the decrease in industrial inputs. Thus the production of goods and the feeding of men should be treated as an inseparable process.
We now convert our notation into an entirely new one. Let xi(t) be the amount of good i available at the beginning of period t, and m(t) the number of people living in the economy at the same point of time. xi(t) is allocated among industries and families for the production of goods and for the production (or feeding) of men, respectively; the former part is denoted by xiI(t), the latter part by xiF(t). Obviously, they add up to xi(t),
unless some of good i is exposed to the wind and rain without being used by any of the industries and families. There are n kinds of goods. x(t) denotes the n-dimensional vector (x1(t), x2(t), . . . , xn(t)); similarly, xI(t) and xF(t) represent (x1I(t), x2I(t), . . . , xnI(t)) and (x1F(t), x2F(t), . . . , xnF(t)), respectively.