Engels outlines some of this in his Supplement,
“Had Marx an opportunity to go over the third volume once more, he would doubtless have extended this passage considerably. As it stands, it gives only a sketchy outline of what is to be said on the point in question. Let us, therefore, examine it somewhat closer.” (p 896)
Engels sets out clearly the period during this operation of the Law of Value, by which commodities exchange at their values is then specified clearly by Engels.
“In a word: the Marxian law of value holds generally, as far as economic laws are valid at all, for the whole period of simple commodity production — that is, up to the time when the latter suffers a modification through the appearance of the capitalist form of production. Up to that time, prices gravitate towards the values fixed according to the Marxian law and oscillate around those values, so that the more fully simple commodity production develops, the more the average prices over long periods uninterrupted by external violent disturbances coincide with values within a negligible margin. Thus, the Marxian law of value has general economic validity for a period lasting from the beginning of exchange, which transforms products into commodities, down to the 15th century of the present era. But the exchange of commodities dates from a time before all written history — which in Egypt goes back to at least 2500 B.C., and perhaps 5000 B.C., and in Babylon to 4000 B.C., perhaps to 6000 B.C.; thus, the law of value has prevailed during a period of from five to seven thousand years.” (p 899 - 900)
The commencement of capitalist production, then, from around the fifteenth century, though only on a minor scale, begins to transform these exchange values into prices of production, through a long historical process. Capital invades those areas first where a low organic composition of capital means that a high rate of profit prevails, typically agriculture. The more capital invades such an area, the more supply is increased, and consequently prices are forced down below exchange values, and profits along with it, providing an incentive for capital to turn its attention to some other area of production with a low organic composition of capital. But, now capital has not only brought about prices of production in agriculture that differ from exchange values, it has had a ripple effect.
Every producer that consumes the products of agriculture now buys those products not at their exchange value, but at their new lower price of production. These lower price inputs, now reduce the exchange value of their own commodity, even if it is still being produced by a direct producer, not capitalistically. As Marx puts it,
“The foregoing statements have at any rate modified the original assumption concerning the determination of the cost-price of commodities. We had originally assumed that the cost-price of a commodity equalled the value of the commodities consumed in its production. But for the buyer the price of production of a specific commodity is its cost-price, and may thus pass as cost-price into the prices of other commodities. Since the price of production may differ from the value of a commodity, it follows that the cost-price of a commodity containing this price of production of another commodity may also stand above or below that portion of its total value derived from the value of the means of production consumed by it. It is necessary to remember this modified significance of the cost-price, and to bear in mind that there is always the possibility of an error if the cost-price of a commodity in any particular sphere is identified with the value of the means of production consumed by it. Our present analysis does not necessitate a closer examination of this point.”
Increasingly then, it is impossible to identify exchange values, because, even non-capitalistically produced commodities' cost prices, are comprised, themselves, of prices of production, of the commodities used in their production. Marx and Engels not only were conscious that transformed output prices were at the same time input prices, for other commodities, it was a central aspect of their theory of how exchange values are historically superseded by prices of production! Marx and Engels knew that input prices had to be transformed simultaneously with output prices, and Marx says so above, but, as with so much of Marx's explication, he deals with the core idea first, before moving on to its further elaboration. Unfortunately, he did not live long enough to provide that further elaboration.
Although, Schmidt had been led astray by his desire to provide this mathematical solution, he had clearly learned invaluable lessons from the first two volumes of Capital, Engels says.
“His is the honour of independently finding the correct explanation developed by Marx in the third part of the third volume for the hitherto inexplicable sinking tendency of the rate of profit, and, similarly, of explaining the derivation of commercial profit out of industrial surplus-value, and of making a great number of observations concerning interest and ground-rent, in which he anticipates ideas developed by Marx in the fourth and fifth parts of the third volume.” (p 13)
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