The Positive Theory of Capital
APPENDIX TO PAGE 327 [Book VI, Chapter V]
Amount of Subsistence Fund Necessary Before Entering on a Production Period of Given Length
If one year be the period of the production process and the stage period also be one year, so that no new goods, finished and ready for consumption, are turned out under a year's time, then, obviously, before beginning such a process, there must be on hand a fund of subsistence containing sufficient to cover the entire wants of the workers for one year, and that in a finished state. If we call the Subsistence Fund S, and the year's Want Y, then, in this case, S = Y.
If two years be the production period, and the stage period, as before, be annual, it is necessary that, at the beginning of the production period, there should be on hand one year's supply finished, and a second year's supply half finished. In each year the finished year's supply is consumed by the workers, while the half-finished is finished by the workers of the second stage—thus securing the subsistence for the next year—and a fresh year's supply is put in hands by the workers of the first stage, and, in turn, half finished. Here, therefore, if we call the half-finished year's supply a half year's supply, S = 1½ Y.
Similarly for a three years' production process, with annual stages, we require one year's want entirely covered, another 2/3 covered, and another 1/3 covered: or, one year's supply finished, another 2/3 finished, another 1/3 finished. In each year, then, the finished year's supply is consumed, the 2/3 finished is finished by the workers of the third stage, the 1/3 finished becomes 2/3 finished by the workers of the second stage, and a further year's supply is newly created by the workers of the first stage, and is finished to the extent of 1/3—whereby, at the end of the year, the status quo is restored, and continuous provision is guaranteed. S, therefore, here = 1Y × 2/3 Y × 1/3 Y = 2Y.
Similarly, if the stage is still one year, then in a four years' process S=(1 + ¾ + ½ + ¼)Y = 2½Y:
in a five years' process S=(1 + 4/5 + 3/5 + 2/5 + 1/5)Y = 3Y:
If we look closely into these figures we shall easily discover the law that underlies them: Every production period requires a fund of subsistence containing sufficient to cover half a year more than half the production period.
Suppose we continue our inquiry under the assumption of a different stage period, say, half a year. Here it is quite the same whether the stage period occurs under the division of labour or not, the only thing essential being that, every half-year, finished consumption goods are turned out from the total process. To enter upon a one year's process, with half-yearly stages, what we require is a finished supply for one half-year—during which no fresh consumption goods are turned out—and half-finished supply for the second half-year. During each six months, then, the finished supply is consumed; the half-finished is finished by the workers of the second stage; and a new six months' supply is begun and half finished by the workers of the first stage, whereby the status quo is restored. S here = ½Y + ½ × ½Y = ½Y + ¼Y = ¾Y.
Similarly in a two years' production process, with half-yearly stages, we require ½Y + ½ × ¾Y + ½ × ½Y + ½ × ¼Y = (½ + 3/8 + ¼ + 1/8)Y = 1¼Y, while in a three years' period we require ½ + ½ × 5/6 + ½ × 4/6 + ½ × 3/6 + ½ × 2/6 + ½ × 1/6 = ½ + 5/12 + 4/12 + 3/12 + 2/12 + 1/12 = 1¾Y.
Here, again, the underlying law is plain: If the stage period be six months the fund necessary contains subsistence for three months longer than half the production period.
If we were to carry out our inquiry still further we should find, similarly, that, where the stage is three months, the fund must contain six weeks' more subsistence, where it is one month, must contain two weeks' more subsistence, than half the production period. And thus we arrive at the general formula of p. 327, that the fund of means of subsistence most be sufficient for half the production period plus half the usual stage period.
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