Reducing cost of production to dated labour was implicit in the English classical economists but was revived first by Dmitriev and then by the Austrians and Wicksell in their approach to ‘originary’ factors of production. The approach then fell into disuse until revived by Sraffa.
It is not often noticed however that Sraffa used a very different mathematical method than Dmitriev and this led him false conclusions.
Why you can reduce to dated quantities of labour is easiest presented in the ‘labour commanded form’ used by Sraffa. There are other arguments from a labour embodied position but the labour commanded form is easiest to grasp.
Profits=investment+capitalist consumption – the Kalecki profits formula
That investment is resolvable to wages and capital costs and rent, capital costs are resolvable to wages and rent. Rent is an extraction from the surplus so rearranging to the RHS.
We can resolve capitalist consumption in the same manner.
(note as an aside as wages =1-r rent=capitalist consumption).
Dmitriev used a brilliant device to counter the criticism of Ricardo by Marx (later revived by Walras) that Ricardo’s system was unresolved in having insufficient terms to be able to resolve wages and prices – in other words it was under-determined. Dmitriev reduced all costs to labour and discounted all costs at the profits (interest rate). Dmitriev essentially mathematised the Annuity approach to fixed capital that Ricardo’s successors such as Mill, Torrens and McCulloch used to argue for a ‘pure’ labour theory of value.
Sraffa adopts a similar method but he did not discount labour costs, only ‘constant’ capital. As a result he came to the spurious view that costs were only reducable to labour costs at zero profits and except at zero profits there would also be a capital residue. This has also led to nonsense such as Bose’s theorum.
I think he avoided doing this because of his objectivism, the approach was clearly inspired by Dmitriev, so the revised equation must have been deliberate. It implied that costs=labour costs+cost of waiting – the classical view from Senior. This to Sraffa would have been unobservable and unobjective.
However we can see from our exploration above that profits are resolvable to wages + capitalist consumption, and substituting into the formula, as per Dmitriev, interest has an objective meaning in terms of the labour costs of producing capitalist consumption goods. A different twist on exploitation theory.
I am greatful to Ian Wright for the last argument though I hope this is a simpler means at arriving at it in terms of accounting identities.
Perhaps if Sraffa has pursued this course he would have realised the ‘accountants method’ of dealing with fixed capital had real meaning and the cost of maintaining fixed capital was independently calculable in terms of labour costs – joint production then becoming a trivial problem to resolve.