The problems with the Labour Theory of Value

The labour theory of value ranges from unfalsifiable, through contradictory to empirically false, depending on how much mental gymnastics one wants to make.

If you want to know why theories of value in general are silly and what economists use today, click here.

1. Falsifiability

Why is falsifiability important?

The output of a company is a function \( y = P(w, c, l) \) where \( P \) is production, \( w \) is work (labour), \( c \) is capital (machines) and \( l \) is land. Mind that \( l \) is land and not labour. The claim of the labour theory of value is that \( P \) only depends on \( w \) (labour). If \( M(w) \) is a function that takes labour hours as input and outputs market prices then \( M(P(w, c, l)) - M(w) \) is the “surplus value” (profit).

Marxist authors usually write that production \( P = c + v + s \), where \( c \) is constant capital (machines), \( v \) is variable capital (recurrent expenses and wages) and \( s \) is the surplus value (profit, which marxists claim to be hours of work the capitalist steals from workers).

What experiments could we perform to falsify the LTV?

Just add more land or capital to a company and watch how the output increases. The Marxist reply is that capital and land are only made useful through labour and thus you would need to account for all the past labor that went into that capital and land. The function would become: \( P = (w, C(w), L(w)) \) where \( C(w) \) and \( L(w) \) are work embedded in the capital and land.

In principle this may see like a logical solution, but how do you test your new functions \( C(w) \) and \( L(w) \)? The answer is that you don’t, you would have to go back in time since forever with extremely precise data to be able to do this. Making your hypothesis harder to test is seldom a good strategy.

The gemstone problem: I found a very beautiful gemstone in a river, it was smoothed by the water current and had amazing rare colors. I sold it for a pretty penny. Most people would have to get a normal stone and grind and polish until they could get the same price I did. The Marxist reply is that what matters is not the direct labour input but the average socially necessary labour input. Now we get \( P = (S(w), C(w), L(w)) \) where \( S(w)\) is the “average socially necessary” labour. Ok how do we measure \( S(w) \)? We don’t. It’s never even properly defined.

Diamond and water problem: Obviously anyone lost in a desert would pay a much higher price \( M \) for an water bottle than for a diamond, even if the diamond takes much more labour to produce. The Marxist reply ranges from Marx’s “use-value” but no further inquire, passing through “labour is only as an anchor to price” (more on this later), to a “don’t think about it”:

Use-value is an expression of a certain relation between the consumer and the object consumed. Political economy, on the other hand, is a social science of the relations between people. It follows that ‘use-value as such’ lies outside the sphere of investigation of political economy. Marx excluded use value (or, as it now would be called, ‘utility’) from the field of investigation of political economy on the ground that it does not directly embody a social relation.

– The Theory of Capitalist Development - Paul M. Sweezy - Monthly Review - 1970 - Chapter 2, page 26

Ok so we have this production function \( y = P \) of 3 variables, we can’t study the output or isolate any of the variables because they are too complex and we can’t indirectly study \( P \) because \( M \) only matters sometimes or “don’t think about it”. We can’t study the function, we can’t test it. The LTV is effectively a dogma.

2. The organic composition of capital should converge across all industries

Marx gives a different name to our function \(P\): \(lv = c + v + s\)

Where \(lv\) is the labour value embedded in the commodity, \(c\) is the value of constant capital (the means of production: machines, buildings, etc), \(v\) is the variable capital (money used to pay salaries), \(s\) is the surplus value (i.e. the profit).

These variables are used to define three new variables:

\( \pi \) (pi) is the rate of profit (surplus value divided by the sum of constant and variable capital):

\[ \pi = \frac{s}{c + v} \hspace{3em} (1) \]

\( \epsilon \) (epsilon) is the rate of exploitation (surplus value divided by the variable capital)

\[ \epsilon = \frac{s}{v} \hspace{3em} (2) \]

\( \omega \) (omega) is the organic composition of capital (constant capital divided by variable capital)

\[ \omega = \frac{c}{v} \hspace{3em} (3) \]

Now dividing the numerator and denominator of \( \pi \) by \( v \) and substituting \( \frac{s}{v}\) for \( \epsilon \) and \( \frac{c}{v} \) for \( \omega \) we get:

\[ \pi = \frac{ \frac{s}{v} }{ \frac{c}{v} + 1 } = \frac{ \epsilon }{ \omega + 1 } \hspace{3em} (4) \]

But he rate of profit \( \pi \) tends to equalize among industries and so does the rate of exploitation \( \epsilon \), specially in freer economies [1][2]. But note that if \( \pi \) and \( \epsilon \) equalize among all industries so should \( \omega \):

\[ \pi = \frac{\epsilon}{\omega + 1} \hspace{3em} (5) \]

Yet it’s trivial to find perfectly functional industries with vastly different \( \omega \) (different ratios of constant and variable capital). Example: a jiu jitsu gym which needs basically only the mat and lights as constant capital but the possibly expensive salaries of instructors as variable capital and a bitcoin mining farm which needs hundreds of thousands of dollars in constant capital but only a few sys admins and security guards as variable capital.

The standard marxist response is to say that Marx talked about “the tendencies” and that capitalism is distorting the truth and thus we should look at the “aggregates” across an economy. And in fact marxists have done so and arrived at the conclusion that equation (5) is “accurate to a large degree”.

The issues with “aggregates” are:

1. If we accept the use of “aggregates” then “accurate to a large degree” still doesn’t solve the problem since the LTV predicts that all “value” should come from labor and not that “value” is correlated to labour. The amount to which “aggregates” are not accurate could very well be caused by the not-surplus-value profit and then we could say that the LTV was falsified no matter how “accurate to a large degree” our data is. “Aggregates” are an ad-hoc solution to a bad hypothesis and not an attempt at falsifying it;

2. Using averages to smooth out “distortions” is completely arbitary and implicitly supposes that the LTV is correct and that there are “distortions” to be smoothed out. One could also use \( min \{ \epsilon_{n} \} \) or \( max \{ \omega_{n} \} \) on the set of companies in each sector;

3. Using specific sectors is completely arbitray, one could define a “sector” however they wanted and in fact marxists disconsider the financial sector on the basis of it being “unproductive”;

4. Using averages for the entire economy takes (5) from a testable hypothesis to a tautology. See:

Look at equations (1) through (3) again. If we have two different sets of \( s \), \( v \) and \( c \) we will also have two different sets of \( \pi \), \( \epsilon \) and \( \omega \) and thus we can compare any combination of those, including (5), to make a prediction. If we only have one set of “aggregates” and substitute it on equation (5) then it will always be true \(1=1\) because we obtained equation (5) exactly by combining \(s\), \(v\) and \(c\).

Let’s see an example: pick any numbers for \(s\), \(v\) and \(c\) and pretend they are the average values of a hypothetical economy. I’m going to use \(s=10\), \(c=5\) and \(v=8\) but I encourage you to try with as many numbers as you like.

from (1)

\[ \pi = \frac{s}{c+v} = \frac{10}{5 + 8} = \frac{10}{13} \hspace{3em} (6) \]

from (2)

\[ \epsilon = \frac{s}{v} = \frac{10}{8} = \frac{5}{4} \hspace{3em} (7) \]

from (3)

\[ \omega = \frac{c}{v} = \frac{5}{8} \hspace{3em} (8) \]

Now substituting on (5)

\[ \pi = \frac{\epsilon}{\omega + 1} = \frac{ \frac{5}{4} }{ \frac{5}{8} + 1 } = \frac{ \frac{5}{4} }{ \frac{5}{8} + \frac{8}{8} } = \frac{ \frac{5}{4} }{ \frac{13}{8} } = \frac{5}{ \frac{13}{2} } = \frac{10}{13} \hspace{3em} (9) \]

Yep, (9) is the same as (6).

It seems Marx realized this problem at some point but decided not to think about it:

It appears therefore that here the theory of value is irreconcilable with the actual movement of things, irreconcilable with the actual phenomena of production, and that, on this account, the attempt to understand the latter must be given up.

– Das Kapital Vol 3, “The Theory of Capitalist Development - Paul M. Sweezy” Page 111 and Bohm

To this day some Marxists still believe Bortkiewicz’s “solution”, which if anything only makes the problem worse.

3. Tendency of the rate of profit to fall

This is an extract from Das Kapital Volume III, Chapter 13.

Assuming a given wage and working-day, a variable capital, for instance of 100, represents a certain number of employed labourers.

Variable capital here are the wages payed to workers. This term is sometimes taken to mean slightly different things.

Suppose £100 are the wages of 100 labourers for, say, one week. If these labourers perform equal amounts of necessary and surplus-labour, if they work daily as many hours for themselves, i.e., for the reproduction of their wage, as they do for the capitalist, i.e., for the production of surplus-value, then the value of their total product = £200, and the surplus-value they produce would amount to £100. The rate of surplus-value, s/v, would = 100%. But, as we have seen, this rate of surplus-value would nonetheless express itself in very different rates of profit, depending on the different volumes of constant capital c and consequently of the total capital C, because the rate of profit = s/C. The rate of surplus-value is 100%:

This is how the same rate of surplus-value would express itself under the same degree of labour exploitation in a falling rate of profit, because the material growth of the constant capital implies also a growth - albeit not in the same proportion - in its value, and consequently in that of the total capital.

If it is further assumed that this gradual change in the composition of capital is not confined only to individual spheres of production, but that it occurs more or less in all, or at least in the key spheres of production, so that it involves changes in the average organic composition of the total capital of a certain society, then the gradual growth of constant capital in relation to variable capital must necessarily lead to a gradual fall of the general rate of profit, so long as the rate of surplus-value, or the intensity of exploitation of labour by capital, remain the same.

(Emphasis mine)

Yet there is absolutely no evidence for a falling rate of profit. Mind that living standards have been increasing basically no stop since Marx.

The most important conclusion of Marx’s theory of capitalism is that the rate of profit would tend to decline over time as a result of technological change. Marx called his law of the tendency of the rate of profit to fall “in every respect the most important law of modern political economy” (G. 748). In a letter to Engels, Marx claimed that this law was one of his most important achievements over classical economics (SC. 194).1

– Marx’s Theory of the Falling Rate of Profit - Fred Moseley