the transformation problem is the problem of finding a general rule by which to transform the "values" of commodities (based on their socially necessary labour content, according to his labour theory of value) into the "competitive prices" of the marketplace.Clearly, if Society is not homogenous because whatever it supervenes on is not homogenous, then 'socially necessary labour' is going to differ by time and place. If competitive prices are informationally efficient signals- for example if standard assumptions obtain- then they themselves solve the transformation problem. We know from Axtell's work that it would be silly to search for an independent algorithmic solution- i.e. something purely 'Supply based'- to verify this result because that solution would be in a much higher complexity class to the usual price equation featuring Supply and Demand. Now, the work of Kantorovich showed that provided the Central Planner knew the 'right' Social Welfare Function and had sufficient computing power, then they could compute 'shadow prices' just as good as actual open market price signals. Lange et al, took a different 'Market Socialism' approach where Enterprises would behave entrepreneurially unless 'divorce between ownership and control' really were a scandal- in which case Capitalism too has the same problem but in a more acute form because mechanism design is constrained by the legal framework whereas Commissars can just get shoot malefactors pour encourager les autres.
In other words, Marxists and Market fundamentalists start level. What is sauce for the goose is sauce for the gander. Obviously, there is one big difference. So long as the Soviet style Command Economy was paradigmatic, Marxists were on the defensive. Sure they could do 'catch up growth'. But could they do 'endogenous growth'? It may be that under Xi, the answer will turn out to be 'sure! The C.P can control competitive enterprises and thus have all the information aggregating benefits of the Market while eliminating rent seeking behaviour when it comes to Regulatory action w.r.t sources of Market Failure.'
In other words, the Marxists could have the last laugh here.
The problem is that 'socially necessary labour' would turn out to include that of the masseuse and golf caddie and cocktail waiter and call girl and everybody else a given entrepreneur considers useful for his continued functioning. In other words, 'socially necessary' is a movable feast. It might feature lots of Gulags and Secret Policemen or else Hedge fund bosses and Drug kingpins salting away billions. The notion would have no normative force. Thus solving the Marxian transformation problem would transform the subject into shit.