There are many different interpretations and solutions to the transformation problem (how are values “transformed” into prices). They range from ignoring it, making it more qualitative, reworking the math or definitions of value, etc. I’m not versed enough in all different interpretations, but one that I feel that I “get” and could explain is that offered by Ian Wright. Though his politics are too Trot for me, his correction of the transformation problem makes sense to me and I’ve wondered about doing a write up on it. It is under the framework of Input Output analysis with natural prices (equilibrium prices that act as an “gravitational attractor” for long term prices, this is also in the sphere of work done by economists such as Sraffa and Pasinetti), and he effectively takes Anwar Shaikh’s critique of Sraffa’s students (the Neo Ricardians) and applies this critique to their own model to show how the labor values of a good gives you natural prices but it requires care to ensure one is using the appropriate commensurate measure of labor in a commodity.

Effectively his response, and that of Shaikh from what I can tell, is that the profit of capitalists appears as if it comes out of nowhere and can’t be explained by the standard definition of labor value in Sraffa’s economic model. But the profit that capitalists make (after spending on means of production) get spent back into the economy (in the sphere of circulation) on real use values (say for their class consumption) which have a labor content due to being products of labor. If you “complete that path” of capitalist consumption goods back to their source (labor) and add that labor back into the labor content of a commodity, then boom - labor values march natural prices in Sraffa’s own model.

In other other terms: the standard view of a labor value is the amount of labor needed to create one unit of net output (the quantity of goods available for consumption after reinvesting some portion of them as materials for production). So this is labor that’s directly required for the production of the unit net good plus labor for the materials required for the unit net good. But, capitalists consume part of this net product and laborers only consume part of it. If the prices of the unit net good matched this measure of labor then prices would be low enough that workers could theoretically buy back with their money-wage their entire net product - leaving none for the capitalists. Some “markup” appears to exist and classical Sraffian model couldn’t explain this markup in terms of labor.

But if you measure a “non-standard” labor value (a la Wright) as the amount of labor needed to produce the commodities that only the working class consumes then you get a labor value that also explains this mysterious “markup”.

A non-standard labor value counts the amount of labor that is directly and indirectly required to produce a good due to technical conditions, but also the amount of (surplus) labor that is socially/institutionally required by workers to create their real wage (i.e. their consumed product)

I don’t claim to understand all of it, or be the best at explaining it. So my apologies for the difficulties. But it is an explanation that meshes with me, and my math-brain seems to get it (the original papers require knowledge of Linear Algebra). It also "cleans up Sraffa’s and Pasinetti’s economic work and brings that back into alignment with Marx.

Wright also has a dynamic version of this description which better gets at what is usually meant when people say a labor theory of value (the feedback loop between prices, values, and social division of labor)