The Transformation of Labour Values into Prices of Production
As we saw, in Capital Marx adopted the labour theory of value. However, just like Ricardo, he too realised that such a theory was inconsistent with the assumption of a uniform rate of profits expressing in analytic terms a central feature of the capitalistic mode of production, namely competition. Marx tackled the problem, in Book 3 of Capital, through the so- called transformation of labour values into prices of production.11 Marx’s idea was to show that this transformation did not modify the substance of the results reached on the basis of the labour theory of value, in particular insofar as the thesis of exploitation was concerned.
Let us recall that Marx called v the variable capital (the value of labour power employed in the productive process), c constant capital (the value of means of production employed in the productive process), s the surplus value, corresponding to surplus labour, or in other words the labour employed in excess of the requirements to reconstitute the means of subsistence. The rate of exploitation is equal to the ratio between surplus labour s and necessary labour v. If we assume that competition in the labour market brings out uniform working conditions in the different sectors of the economy, and in particular an equal length of working day, and that the subsistence wage is the same for all workers, then the rate of exploitation corresponds to the ratio between surplus value and variable capital, s/v, and is the same for each individual worker, for each sector and for the economic system as a whole.
However, the condition of a uniform rate of exploitation in all sectors of the economy is inconsistent with the assumption of a uniform rate of profits. Let us indicate the different sectors with 1,2,..., n. The condition of equal rates of exploitation is expressed by:
The assumption of uniform rate of profits (computed for each sector as the ratio between profits and value of capital advanced, which includes both constant and variable capital, or wages) is expressed by:12
Let us divide both numerator and denominator of the different terms of this series of equalities respectively by v1, v2, ..., vn. We get:
At the denominator we thus have the ratio between constant and variable capital, c/v, which Marx called the organic composition of capital, plus 1. At the numerator we have the rates of exploitation of the different sectors, by assumption all equal. As a consequence, the series of equalities (3) - which we have just deduced from the assumption of uniform profit rates - hold if, and only if, the denominators, too, are all equal. Uniformity of profit rates hence requires that the organic compositions of capital in the different sectors also be all equal:
Here we disregard the complications that might arise from the presence of fixed capital goods.
However, there is no reason for this to happen. In fact, each sector adopts a technology specific to it. Thus, the assumption of a uniform rate of profits contradicts the assumption that the quantities of labour contained are a correct measure of the exchange values of commodities produced and of means of production.
Marx recognised this difficulty and proposed transformation of the magnitudes expressed in terms of labour values that do not comply with the condition of a uniform rate of profits into magnitudes expressed in terms of prices of production. In order to do this, he added to the production costs of each sector (given by the sum of constant and variable capital) the profits for that sector, computed by applying the average rate of profit calculated for the system as a whole, expressed by s/(c + v), to the capital advanced for the sector. Let us consider a two-sector economy; we then have
where A and B represent the quantities of product obtained in the first and second sector respectively, expressed in terms of labour values (that is, A = c1 + v1 + s1 and B = c2 + v2 + s2), while pj and p2 represent the
prices of production of the two commodities and constitute the two unknown variables determined by the two equations, the rate of profits being known (since, let us recall, r = (s1 + s2)/(v1 + v2 + c1 + c2)).
However, the solution cannot be considered satisfactory: costs and advanced capital are expressed in terms of labour contained, while it is obvious that capitalists compute their profit rate as ratio of profits and capital advanced measured in terms of prices, not of labour values.1 
-  Marx (cf. 1867-94, vol. 3, pp. 261-72) recognised the existence of this difficulty but putit aside, considering it as practically irrelevant when referring to aggregate magnitudesrepresenting the economic system as a whole. In sum, Marx imposed a double constraint:(i) equality between total surplus value created in the economy and total value of profitsand (ii) equality between the total value of the product of the various sectors in terms oflabour contained and its value in terms of prices of production. However, the twoconstraints are simultaneously satisfied only in very rare circumstances. Objections to Marx’s solution were raised on many sides, in particular by Bohm-Bawerk (1896). Ladislaus von Bortkiewicz (1868-1931) tried to formulate a correctedversion of Marx’s proposal (Bortkiewicz, 1906-7, 1907) by adopting as unit of measurement for each of the two commodities a and b the quantity of that commoditycorresponding to a unit oflabour contained. In this way the prices ofproduction p1 andp2 can be interpreted as those multiplicative coefficients that allow us to move on frommagnitudes measured in terms of labour contained to corresponding magnitudes