Let us assume that there are two lines of production, (1) and (2), each of which produces both producer goods and consumer goods, and that aij = the coefficients of inputs necessary for the production of these goods, p1and p2 = their unit prices; w = the wage rate (the quantities of labor being assigned by the coefficients a01 and a02); and r = the rate of profit. We then have:
To this system corresponds the following system of values:
Let it be remembered that since the two products (1) and (2) are not destined by nature, one for use as equipment and the other for consumption, this system does not describe an equilibrium of supply and demand for each department. The conditions for that equilibrium, which are assumed to be achieved, are external to the model.
We define two parameters of improvement in productivity, π1 and π2, specific to each of the branches (1) and (2). Let us assume, for simplicity, that it is the same, π, in both cases. Let us go on to assume that the system of values for Phase 1 is as follows:
from which we get:
Assuming that the same quantity of direct labor becomes capable of setting to work twice as much equipment and raw material and, for simplicity, in the same proportions aij so as to provide twice the quantity of end products (that is, if π = 0.5), we have for Phase 2:
from which we get:
The table below will then show the evolution of the system of values obtained with the same global quantity of labor, left unchanged.
The results, meaning the increase in the net product (from 1.00 to 2.00) are independent of distribution (no assumptions having been made regarding wages or the rate of profit).
| Phase 1 | Phase 2 | |
| Production | 1.0v1 + 1.0v2 = 2.45 | 2.0v′1 + 2.0v′2 = 4.92 |
| – Productive consumption | 0.7v1 + 0.5v2 = 1.45 | 1.4v′1 + 1.0v′2 = 2.92 |
| = Net Product | 0.3v1 + 0.5v2 = 1.00 | 0.6v′1 + 1.0v′2 = 2.00 |
Rising productivity can be expressed by falling prices while nominal incomes remain unchanged or by nominal incomes’ increases with unchanged unit prices. Here prices are doubled:
If, however, we examine the evolution of a system expressed in prices, we have to introduce an assumption regarding the way income is distributed.
The previous system, expressed in price terms, namely:
completed by an assumption regarding wages, e.g., that:
can be reduced to a system of “production of commodities by means of commodities only” which here is as follows:
the solutions of which are:
For the next phase the system becomes:
The results (relative prices and rate of profit) will depend on the way that wages evolve. If we assume an unchanged real wage, that is, if
the reduced system becomes:
the solutions of which are p′1/p′2 = 0.98, from which we get the comparative table, established in price terms, given below:
| Phase 1 | Phase 2 | |
| Production | 1.0p1 + 1.0p2 = 2.08 | 2.0p′1 + 2.0p′2 = 4.04 |
| – Productive consumption | 0.7p1 + 0.5p2 = 1.24 | 1.4p′1 + 1.0p′2 = 2.42 |
| = Net Product | 0.3p1 + 0.5p2 = 0.84 | 0.6p′1 + 1.0p′2 = 1.62 |
| of which, wages | 0.2p1 + 0.2p2 = 0.42 | 0.2p′1 + 0.2p′2 = 0.40 |
| and profits | 0.1p1 + 0.3p2 = 0.42 | 0.4p′1 + .8p′2 = 1.22 |
It will be noted that comparison between the two phases is obscured by the fact that the solution of the system gives relative prices, p1/p2 and p′1/p′2, which differ according to the evolution of wages. We do know, from our assumption, that the system of Phase 2 will enable us to obtain, with the same total quantity of labor, twice as much physical product (use-values) from (1) and (2). But if we assume p1 = p′1 = 1, we have p2 unequal to p′2, since p1/p2 and p′1/p′2 both depend on the way distribution takes place. Here p2 = 1.08 and p′2 = 1.02.
The net product, which is the measurement of the growth in value that is independent of distribution (in my model, this net product increases in value terms from 1.00 to 2.00), here increases from 0.84 to 1.62 (a growth rate of 93 percent) when we analyze the evolution of