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Most criticisms of the Labor Theory of Value I have come across go something like this quote from Carl Menger:

There is no necessary and direct connection between the value of a good and whether, or in what quantities, labor and other goods of higher order were applied to its production. A non-economic good (a quantity of timber in a virgin forest, for example) does not attain value for men since large quantities of labor or other economic goods were not applied to its production. Whether a diamond was found accidentally or was obtained from a diamond pit with the employment of a thousand days of labor is completely irrelevant for its value.

This is perfectly obvious and Marx even mentions this:

A commodity, such as iron, corn, or a diamond, is therefore, so far as it is a material thing, a use value, something useful. This property of a commodity is independent of the amount of labour required to appropriate its useful qualities.

But putting aside the natural resource example for a moment, in the long term in a competitive market isn't it the case that if there is a situation where someone is able to sell something for more than the cost of the labor that went into it, someone else should be able to step in and sell that thing for a lower price? Eventually prices will go down until they are stopped by another constraint. That constraint being the cost of all the labor required to produce the product.

In situations where this doesn't happen isn't it an indication of some anti-competitive force? Monopoly of a natural resource, monopoly of a technology through a patent, externalities that result in a small number of large companies, cooperation between large companies to set prices, etc...

So is it fair to say that in the long term in competitive markets the labor theory of value holds?

If the theory is "at any point in time a commodity is worth as much as the cost of labor that went into it" then it is obviously wrong and I can't imagine any classical economist would have believed it. But if the theory is "over time the cost of a good will tend towards the cost of the amount of labor that went into it assuming the market is competitive" then it still seems plausible.

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    $\begingroup$ Adam Smith: "The value of any commodity ... is equal to the quantity of labour which it enables him to purchase or command. Labour, therefore, is the real measure of the exchangeable value of all commodities. "The real price of every thing, what every thing really costs to the man who wants to acquire it, is the toil and trouble of acquiring it. What every thing is really worth to the man who has acquired it, and who wants to dispose of it or exchange it for something else, is the toil and trouble which it can save to himself, and $\endgroup$ – user18 Dec 23 '20 at 2:33
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    $\begingroup$ which it can impose upon other people. What is bought with money or with goods is purchased by labor as much as what we acquire by the toil of our own body. That money or those goods indeed save us this toil. They contain the value of a certain quantity of labor which we exchange for what is supposed at the time to contain the value of an equal quantity. Labor was the first price, the original purchase-money that was paid for all things. It was not by gold or by silver, but by labor, that all the wealth of the world was originally purchased; and its value to those $\endgroup$ – user18 Dec 23 '20 at 2:33
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    $\begingroup$ who possess it and who want to exchange it for some new productions, is precisely equal to the quantity of labor which it can enable them to purchase or command." $\endgroup$ – user18 Dec 23 '20 at 2:33
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    $\begingroup$ Smith would add many qualifications (sometimes somewhat contradictory). But Smith, Ricardo, Marx, and other classical economists all believed in some version of the LTV. They were certainly confused (as are you) and this confusion was not fully cleared up until the 1870s. $\endgroup$ – user18 Dec 23 '20 at 2:35
  • $\begingroup$ What happened 1870? Léon Walras and his pioneering 1874 work Elements of Pure Economics $\endgroup$ – Jesper Hybel Dec 23 '20 at 2:40
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No it does not for several reasons. Labor theory of value (LTV) central tenant is that there is some intrinsic component to value determined objectively by labor (see Brue and Grant History of Economic Thought 7ed). This implies that there exist some unit of labor in which every value could be measured. In addition, it means that the value ($V$) is some sort of function labor $L$ i.e. $V=f(L)$. As stated by Adam Smith in Wealth of Nations who first developed the LTV (and as mentioned in the comments as well):

"The value of any commodity ... is equal to the quantity of labour which it enables him to purchase or command. Labour, therefore, is the real measure of the exchangeable value of all commodities.

Hence, at the core of labor theory of value is proposition that all value can be somehow objectively measured by labor, and that it should depend on labor.

Well there are several problems with that proposition:

  1. What about things that require great labor inputs but are worthless (which is situation that can and empirically does arise even in competitive markets as preferences and tastes change and something that was once desired might be trash today)?

The LTV actually has no good answer for that. Adam Smith tried to answer this objection by distinguishing between value in use and value in exchange. As Smith further writes in his Wealth of Nations:

The word value, it is to be observed, has two different meanings, and sometimes expresses the utility of some particular object, and sometimes the power of purchasing other goods which the possession of that object conveys. The one may be called "value in use"; the other, "value in exchange." The things which have the greatest value in use have frequently little or no value in exchange; and, on the contrary, those which have the greatest value in exchange have frequently little or no value in use. Nothing is more useful than water: but it will purchase scarce anything; scarce anything can be had in exchange for it. A diamond, on the contrary, has scarce any value in use; but a very great quantity of other goods may frequently be had in exchange for it.

This is basically the argument that Marx repeats (although he had different name for it), as mentioned above (indeed I am always surprised that Marx is often quoted regarding LTV given he did not do virtually any original work on the LTV and just took it from Ricardo and Smith).

The problem with this is that it is already quite scientifically undesirable as it is essentially a cop-out. Labor is the source of value, well, except in cases where it isn't because the object is not useful. So already LTV from the get go admits it is not a theory that can explain all value, which is scientifically undesirable as we want to generalize as much as possible. Now, if we would not have alternative theories of value this could be forgiven but given we do have them and they can explain more this in itself is argument for ditching LTV (e.g. we can safely ditch Newtons theory of gravity in favor of relativity as the latter is more general, but even though theory of relativity can't explain everything we cannot ditch because there is no other competing theory that can).

With regards to your main question, already this point 1 proves that in competitive markets LTV does not need to hold as it is completely valid outcome under perfect competition that value of produced good will be 0 (or indeed even negative like with oil futures).

  1. How to measure value in labor (the problem of transformation)?

One of the central tenets of labor theory of value as mentioned above is that labor must be some sort of real measure of value of all commodities. However, due to the problem mentioned in point 1 it is also necessary to argue that value of labor will not correspond exactly to the price.

Hence, this gives raise to the question how to actually establish some correspondence between prices and labor which is also known as the problem of transformation (see Samuelson; 1971). This was in fact a problem that Ricardo devoted considerable amount of time to solve, although he was ultimately unsuccessful. Samuelson argues there is no general solution to the problem, and although there are some scholars claiming to solve it there are no generally accepted solutions.

In regards to your question, assuming competitive market does not provide solution to the above problem and without the above being possible it is very hard to sustain claim that value can be somehow objectively measured by labor.

  1. Empirically value seems to be orthogonal on labor inputs and this hold regardless of how competitive industry is.

There simply does not seem to be any relationship between value and labor inputs. Relative prices fluctuate widely even when labor inputs stay same. This can occur both in theoretical models of perfect competition (e.g. consider two perfectly sectors two goods model with some idiosyncratic demand shocks), and frequently occurs in reality.

Now even if there would be a discrepancy between price and the value of labor that went into producing the product, if there is any objective value determined by labor then you would at the very least expect relative prices to depend on relative labor inputs but they do not and there is no indication that they start doing so in competitive industries. Hence $V \perp L$.

This also connects to the previous point, but for LTV to hold it is simply not enough to just show that something that was produced with labor is happens to have value - the value of it should depend on labor input and ultimately be measurable. What even more it should be the measuring stick of the value. But this is clearly inconsistent with observation that relative prices change and fluctuate widely even when labor inputs do not.

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  • $\begingroup$ Does point 3 mean there is no direct relation or also no indirect relation? Because to me it sounds logical that the amount of labor input is part of the supply and thus indirectly contributes to the value. Like if there is a demand for something and it takes 1 day to make something more people will try to make it then if it takes 1 year to make it. Thus labor input will influence the supply and thus the price. Maybe I'm missing the definition of value, but I thought it would be price in modern economic theories. $\endgroup$ – Hakaishin Dec 23 '20 at 10:16
  • $\begingroup$ @Hakaishin 1. As the point 3 is not enough to show that labor in some cases influences price. Under LTV there labor must be able to serve as some sort of unit of value and so for example something that takes 2h to make should be twice as valuable than something that takes 1h to make - we do not observe that empirically. 2. Yes value should be closely related to price (although does not need to be exactly equivalent in all cases) - this is already failure of LTV and why modern economics uses subjective theory of value instead - LTV always needs some hot fixes because price does not have any $\endgroup$ – 1muflon1 Dec 23 '20 at 10:29
  • $\begingroup$ discernible relationship with labor inputs. For example, labor inputs of firms are relatively stable - companies usually do not fire employees at a whim on hourly basis. Yet prices can fluctuate even hourly at commodity markets for example. Even in general markets again usually when people are fired hired it is long process yet stores change prices sometimes daily. So the question is why they do that if labor component of the goods is same. LTV does not have any good answer for that save for saying that they fluctuate due to the value of usefulness- which is already half abandoning the LTV $\endgroup$ – 1muflon1 Dec 23 '20 at 10:35
  • $\begingroup$ In addition to prices fluctuating relative prices fluctuate as well. The last nail to the coffin is that it seems it is even impossible to solve the transformation problem generally (hence labor cannot serve as measurement of value). Now subjective theory of value that states that value depends on people’s subjective judgments and preferences (marginal utility) can explain everything what LTV does and even things that LTV cannot, hence there really is no use for LTV anymore. $\endgroup$ – 1muflon1 Dec 23 '20 at 10:41

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