I don't know if I would use the word 'disproven', but Marx' labor theory of value is wrong in the extent that it fails to explain the empirical reality in a capitalist society. Most significantly perhaps, that the value added by human labour has no necessary link with the price at which goods are sold, giving rise to the infamous "transformation problem" of converting the value as determined by Marx' theory into the price by which it is sold. He tries to solve this in subsequent volumes of Capital, but the lack of success in this endavour is probably why he didn't publish them.
There is a reason why neo-classical economics and specifically the marshallian cross completely trashed the LTV. And there is a reason why nobody really bothers with it today and uses the marginal theory of value instead.
There are also numerous plain logical issues, contradictions and circularities that are explained in both inadequate and unessecarily complicated ways, like for example the abstract and inquantifiable concept of socially necessary labour time. Not even to mention the lack of reasoning for this concept. You can easily find articles online sketching out the most important of these.
I'm not personally anti-Marxist and Smith had a labour theory of value as well, but Marx rests a lot of really heavy exploitation arguments on top of his value theory, so I think that a lot of the critique is warranted.
Well the LTV doesn't explain everything, certainly. But it remains a useful contribution in the context of Volume 1 (which, lest we forget, includes the caveat throughout that commodities exchange at their values).
Why it is useful, is that it places the worker at the centre of production (and therefore value).
People always criticise Capital either because of the caveats Marx introduces or because they fail to remember those caveats. The whole point of them is to demonstrate that capitalism, as a system, is fundamentally unstable on its own terms.
I'm not an expert Marxist or anything but I think the reason for socially necessary labour time is fairly clear as well. If you're talking about production in general it makes sense to have a marker for how long it generally takes a worker in society to produce something. You can then make comparisons with other societies, or examine how this variable will change with developments in technology or modifications to the length of the working day.
My feeling is that not publishing his material about the transformation problem was probably more to do with him not being able to get his notes in order before he died, but I take your point. Volume 3 is a bit of bastard if I'm honest and it doesn't all stack up. The point is not to laud Marx as a saint or dismiss him because his work is incomplete. The point is to build on what is useful and get rid of the rest.
