**1.0 Introduction**

A simple labor theory of value holds in two special cases.

- The rate of profits in the system of prices of production is zero.
- The vector of direct labor coefficients is an eigenvector of the Leontief input-output matrix corresponding to the maximum eigenvalue.

I do not know if I've worked through this alone before. A more rigorous approach would prove the uniqueness of the solution.

**2.0 The Setting**

Suppose a capitalist economy is observed at a given point in time. *n* commodities are being produced, each by a separate industry.
Suppose the technique in use can be characterized by a row vector **a**_{0} and a *n* x *n* square matrix **A**.

The *j*th element of **a**_{0} is the amount of labor directly employed in the *j*th industry in producing one unit of a
commodity output from that industry. "We suppose labour to be uniform in quality or, what amounts to the same thing, we assume
any differences in quality to have previously been reduced to equivalent differences in quantity
so that each unit of labour receives the same wage…" - Piero Sraffa (1960).
I guess the idea is that relative wages are more or less stable.

The *j*th column of **A** is the goods used up in producing one unit of a commodity output.
For example, suppose iron is produced by the first industry and steel is produced by the second industry. *a*_{1,2} is then
the kilotons of iron needed to produce a kiloton of steel. Assume that every good enters directly or indirectly into
the production of each commodity. Iron enters indirectly into the production of tractors if steel enters directly into the tractor
industry. Assume a surplus product, also known as a net output, exists.

**2.1 Quantity Flows**

Let **y** be the column vector of net outputs and **q** the column vector of gross outputs, both in physical terms.
In Leontief's work, **y** is taken as given. Gross outputs and net outputs are related as:

y=q-Aq

Or:

q= (I-A)^{-1}y

The labor force needed to produce this net product is:

L=a_{0}q=a_{0}(I-A)^{-1}y

One might as well take units in which labor is measured to be such that this labor force is unity. Employment is such that the net output is produced, the capital goods in producing the net output are reproduced, the capital goods used in producing those capital goods are reproduced, and so on.

**2.2 Labor Values**

Let **e**_{j} be the *j*th column of the identity matrix. The labor force needed to produce this net output is:

v_{j}=a_{0}(I-A)^{-1}e_{j}

That is, the (direct and indirect) labor needed to produce a net output of one unit of the *j*th commodity is *v*_{j}.
The row vector of labor values is:

v=a_{0}(I-A)^{-1}

(I could put an aside here about geometric series and an infinite sum of labor time, assuming the current technology was used forever in the past.)

The employment needed to produce a given net output is the sum of the labor values of the individual commodities
in net output, **v** **y**. One can think of this post as showing one way of decomposing the observed net
output and employed workers. With this way of thinking, no assumptions have been made about returns to scale.

Labor values support one way of doing accounting in models like this. One could ask about how much employment would have decreased or increased if final demand had been decreased or increased by some specified quantities of specified commodities.

**2.3 Prices of Production**

Take **y** as numeraire. At any time, market prices are such that different industries are making different rates of profits.
Under competitive conditions, without barriers to entry in the various industries, a kind of leveling process is going on.

One can imagine a vector of prices such that this leveling process is already completed with the observed technique and wage. Let **p** be
that row vector of prices of production, with all industries obtaining the same rate of profits:

pA(1 +r) +wa_{0}=p

where *r* is the rate of profits and *w* the wage.
That is, **p** is a price vector consistent with the observed technique and wage. Since **y** is numeraire, one has:

py= 1

The point is to show that prices of production are labor values in special cases.

**3.0 The First Special Case: No Profits**

Assume that the rate of profits is zero. The claim is that prices of production are labor values.

First, consider the equation for the numeraire:

vy=a_{0}(I-A)^{-1}y=a_{0}q

By assumption, the amount of labor employed is one unit. So using labor values for prices satisfies the equation for the numeraire. Furthermore, if the rate of profits is zero, the wage is unity. (One might do a bit of algebra here.)

I want to show:

vA+a_{0}=v

But this is true if and only if:

a_{0}=v(I-A)

Or:

a_{0}(I-A)^{-1}=v

But this is the definition of labor values. So if the rate of profits is zero, prices of production are labor values.

**4.0 The Second Special Case: Equal Organic Compositions Of Capital**

Suppose that:

a_{0}A= λa_{0}

where λ is the eigenvalue with the maximum modulus. By the Perron-Frobenius theorem, this eigenvalue is positive and less than unity. All of the elements of the vector of direct labor coefficients are positive.

Under this special case, the solution to the price equations is:

p=v

and:

r=R(1 -w)

where:

R= (1 - λ)/λ

Suppose:

v= (1/(1 - λ))a_{0}

By the definition of labor values:

v(I-A) =a_{0}

Or:

(1/(1 - λ)) (a_{0}-a_{0}A) =a_{0}

Using the special case assumptions, one has:

(1/(1 - λ)) (a_{0}- λa_{0}) =a_{0}

Thus, in this special case, labor values are directly proportional to direct labor coefficients.

I want to show:

vA(1 + ((1 - λ)/λ)(1 -w)) +wa_{0}=v

Or:

(1/(1 - λ))a_{0}A((1/λ) - ((1 - λ)/λ)w) +wa_{0}= (1/(1 - λ))a_{0}

Or:

(λ/(1 - λ))a_{0}((1/λ) - ((1 - λ)/λ)w) +wa_{0}= (1/(1 - λ))a_{0}

But the left-hand side is simply (1/(1 - λ)) **a**. So labor values are prices of production
in the special case. Furthermore, prices of production do not vary with distribution in this case.

**5.0 Conclusion**

Suppose the organic composition of capital does not vary among industries. That is, the vector
of direct labor coefficients is the specified eigenvector of the Leontief input-output matrix.
So prices of production associated with the observed technique and net output are labor values.
How does capital obtain profits in this special case? This is Marx's question in the first volume
of *Capital*.

Objections to the lack of realism of this special case and to the conditions needed to define prices of production are not on point. If you have a theory explaining returns to capital, it should apply in this special case. The question, I gather, is more salient if you think there is something fair about commodities being priced at labor values.

The answer cannot be entrepreneurship, since the returns to entrepreneurship are a non-equilibrium phenomenon. For half a century, economists have known that the answer cannot be supply and demand of capital. For that answer, one must have a unit in which capital can be measured independently of prices. I suppose one can create a self-consistent model with intertemporal utility maximization by households, including households whose income is entirely from returns to ownership. But the mechanics of how such a model works disagree with traditional notions of substitution and scarcity.

A valid answer, it seems to me, must invoke some concept of power. This answer need not be exactly Marx's. The Post Keynesian theory of growth, in which large corporations set the rate of growth, might be part of an answer applicable in some times and places.

## 3 comments:

I think there is a third. The so called Average Industry System where all industries are proportional and the matrix is of rank 1.

Thanks. I have not worked through that example.

I think it could be interesting to study that system. For example how prices behave with different numeraires, how capital-labor and capital-output ratios behave and how differential rates of profits influences the formers.

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