## Wednesday, April 01, 2015

### Modeling the labor theory of value

In The Demise of Marx’s Labour Theory of Value and the ‘New Interpretation’ [pdf] Ernesto Screpanti uses a simple one-commodity model to show the theoretical incoherence in the Labor Theory of Value (LTV). (I don't think the model Screpanti's own invention; I've seen the model, and variants of it, for many years.) You can look at the article for the details of the model and the definitions.

The model states that value (exchange-value) is equal to the labor used in current production, plus a fraction \$0<a<1\$ of value from previous production. Price is equal to the cost of labor plus a fraction of the price of previous production, with the latter receiving a return on investment, \$0<r<1\$.

 (1) \$v=l+av\$ (2) \$p=l+(1+r)ap\$ therefore (3) \$v=l/{1-a}\$ (4) \$p=l/{1-(1+r)a}\$

A little algebra shows that value = price if and only if the rate of return on capital is zero.

There are some problems with this model, and not because it's simplified. First of all, it assumes that value and price are constant over time, i.e. that production is not growing. Otherwise, we would have to say \$v_n=l+av_{n-1}\$ and \$p_n=l+(1-r)ap_{n-1}\$, and have to worry about changing values over time. If production is not growing, then it is perhaps legitimate to say that the rate of profit really must be zero. However, this objection doesn't matter, because it is not a model of Marx's LTV. Absolutely central to Marx's theory is that capitalists do not pay workers for all their labor; capitalists only pay the cost of labor power, the amount of Socially Necessary Abstract Labor Time (SNLT) to produce the goods the workers must consume to continue working and reproduce the working class. Thus, a more accurate model is that value represents the total labor, whereas price represents the cost of labor power plus the cost of capital. Furthermore, capitalists want a return on their total investment, which includes both capital used and advance wages. Therefore, a more accurate model would be:

 (1) \$v=l_p+l_s+av\$ (2) \$p=(1+r)(l_p+ap)\$ therefore (3) \$v={l_p+l_s}/{1-a}\$ (4) \$p=l_p/{1/{1-r}-a\$

with \$l_p\$ the cost of labor power (in SNLT) and \$l_s\$ being the surplus labor. Therefore total labor \$l_t=l_p+l_s\$, and the rate of exploitation \$l_e=l_s/l_t\$

Because we're introducing the extra term, we can set value = price and just get a relation between the rate of exploitation and the rate of profit.

\$l_e=l_s/{l_p+l_s}=1-{1/{1+r}-a}/{1-a}=r/{(1+r)(1-a)}\$
\$r=1/{1-(1-a)l_e}-1\$

Both of these equations are well-behaved in the relevant ranges (\$0<l_e,a<1;r>0\$). They also show that holding the exploitation rate constant, as capital requirements increase (as \$a\$ gets larger), the rate of profit must decrease.

1. I have a few questions.

1. What is price, what is value? If I understand correctly, price is the amount that a firm needs to pay to produce a unit, and value is the amount they can exchange a unit for.

2. You set price = value, but I was confused because your next post does not use the same condition. What's the meaning of setting price = value?

3. Why is "a" in Eq (1) the same as "a" in Eq (2)?

I had some other questions but perhaps I should make sure I understand these first.

1. 1. Price is money price; value is exchange value, which is the total amount of embodied labor. Yes, your interpretation is correct.

2. The LTV asserts that (theoretically, ideally) prices just represent value. With the appropriate choice of units, they should be identical. The linked paper asserts (incorrectly in my opinion) that this identity cannot hold for positive rates of profit (i.e. for r > 0).

I do actually assume that price equals value in the next post, but only implicitly. if v1 = p1 and v2 = p2, then v1/v2 = p1/p2.

3. a is a dimensionless physical parameter: it is the proportion of previous production that is reinvested in current production, i.e. "capital". It should be the same regardless of the method of representation.

2. Thank you for the responses. My next questions are about the second model.

1. In order for employment to be profitable for both sides, wages must be somewhere between \$l_t\$ and \$l_p\$. Is this correct?

2. This model appears to assume the actual wages will be \$l_p\$. Is this assumption meant as hypothesis, or merely a simplification?

(I think there's a typo in the last paragraph. The relevant range is that a < 1, not r < 1.)

1. To clarify question 2, I know you said that "Absolutely central to Marx's theory is that capitalists do not pay workers for all their labor; capitalists only pay the cost of labor power". However, it seems extreme to say that workers do not extract any profit at all, only subsistence wages.

Although now that I think about it, \$l_p\$ represents the subsistence wages of the *marginal* worker. So maybe I'm answering my own question here.

2. 1. Yes.

2. We are choosing our units so that the cost of labor power is one unit of money.

2a. Well, the lower bound is the marginal subsistence. However, Marx notes that the actual cost of labor power is politically determined within the bounds of subsistence and zero-exploitation.

(Yes, that's a typo. Fixed. Thanks. \$r\$ can be \$>1\$.)

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