**1.0 Introduction**
I have been reading Robin Hahnel. Hahnel argues even more strongly than Steedman did that labor values are redundant.
And he argues for the importance of the fundamental Sraffian theorem. I think this may be Hahnel's coinage. Anyways'
this is my working my way through some of what I think he is saying.

Hahnel has some interesting things to say, not discussed here, about analyzing environmental concerns in a Sraffian framework.
I ignore the chapter in Hahnel (2017) on the moral critique of capitalism. Following Eatwell (2019) and others,
I hold that mainstream economists do not have a theory of value and distribution, anyways.

**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**.
Let the column vector **d** denote the quantities of each commodity paid to the workers for a unit of labor.

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).

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. That is:

0 < λ_{PF}(**A**) < 1

where λ_{PF}(**A**) is the dominant eigenvalue of the matrix **A**. The dominant eigenvalue is
also known as the Perron-Frobenius root.

**3.0 Prices of Production**
Suppose the wage purchases the specified bundle of commodities. And also assume the wage is advanced. One can
define the input-output matrix with wage goods included:

**A**^{+} = **A** + **d** **a**_{0}

I think that Sraffa treats the input-output matrix as **A**^{+} in chapter 1 of his book.

The system of equations for prices of production is:

**p** **A**^{+} (1 + *r*) = **p**

where **p** is a row vector, and *r* is the rate of profits.
One can show that, given a surplus product, not including wage goods, a solution exists.

**Fundamental Sraffian Theorem:** The rate of profits, *r*, in the system of prices of production is positive if and only if:

0 < λ_{PF}(**A**^{+}) < 1

In fact, the rate of profits is:

*r* = 1/λ_{PF}(**A**^{+}) - 1

Under these assumptions, the price of each produced commodity is positive with the above rate of profits.
And this economically meaningful solution is unique, up to the specification of a numeraire.

**4.0 Increased Surplus Product**
Suppose one or more of the elements of **A**^{+} decrease. Then 1 - λ_{PF}(**A**^{+}),
which is strictly positive, increases. The surplus product that capitalists capture is increased
by decreased components of the real wage and by decreased commodity inputs into production.

Suppose that the real wage is given and that an innovation results in a new technique, **B**, being available. This technique
might have increased coefficients and decreases in other coefficients, as compared to **A**. It might even have a new column
or delete a column for an industry that is not used to directly produce a wage good. This new technique is adopted at the given wage if and only if:

1 - λ_{PF}(**B**^{+}) > 1 - λ_{PF}(**A**^{+})

Suppose further that:

1 - λ_{PF}(**B**) < 1 - λ_{PF}(**A**)

Then the maximum rate of profits, at a wage of zero, decreases. Suppose no reswitching exists.
I think this is what is meant
by Capital-Using, Labor-Saving technical change. This is also known as Marx-biased technical change.
Marx's law of the tendency of the rate of profits to fall, presented in volue 3 of *Capital*, is not justified
by this analysis.

**5.0 Quantity Flows**
This framework can also be used to examine the rate of growth.
Suppose employment, at an instant of time, is unity:

*L* = **a**_{0} **q** = 1

where **q** isthe column vector of gross outputs.
In this formulation, employment increases at the rate of growth.

Let consumption out of the surplus product be in the composition of
the column vector **e**, and let *c* be the level of such consumption.
It is most coherent to take this consumption as not made by the workers:

We could hardly imagine that, when the workers had a surplus to spend on beef. their physical need
for wheat was unchanged. -- Robinson (1961)

So prices of production associated with this treatment of qunatity flows are as above.

Let the column vector **j** represent investment goods. These are part of the surplus product.
Then the column vector **q** of gross outputs satisfies the following equation:

**q** = **A**^{+} **q** + *c* **e** + **j**

The above is extremely general. I now consider a steady-state rate of growth.
Assume constant returns to scale in every industry.
The vector of investment goods is in the same proportion as existing capital goods:

**j** = *g* **A**^{+} **q**

Here I present a derivation, since I typed this out to check myself.
Substituting the specification of investment goods and after some
algebraic manipulations, one has:

*c* **e** = [**I** - (1 + *g*)**A**^{+}] **q**

Assuming the rate of growth is less than the maximum, one has:

*c* [**I** - (1 + *g*)**A**^{+}]^{-1}**e** = **q**

Premultiplying by the row vector of labor coefficients, one has:

*c* **a**_{0} [**I** - (1 + *g*)**A**^{+}]^{-1}**e** = **a**_{0} **q** = 1

The solution of the system of equations for quantity flows is:

*c* = 1/{**a**_{0} [**I** - (1 + *g*)**A**^{+}]^{-1}**e**}

The maximum rate of growth is:

*g*_{max} = 1/λ_{PF}(**A**^{+}) - 1

The level of consumption out of the surplus product is lower, the higher the rate of growth and vice versa.
One can also consider the impact on the rate of growth of changes in the elements of the matrix **A**^{+}.
I believe one can prove the following:

**Theorem:** The steady state rate of growth, *g* is higher if:

- Consumption out of the surplus product, where the surplus product does not include wages, is lower.
- Necessary wages are lower.
- The dominant eigenvalue, λ
_{PF}(**A**), of the input-output matrix is lower.

The theorem highlights dilemmas in development economics. One does not want to obtain a higher rate of
growth by lowering wages for those who are already pressed. It does not help for foreign aid to end up
in luxury consumption either. In chosing the technique out of a range of possibilities, one would like
the one that maximizes the rate of growth. Unless the rate of growth equals the rate of profits, that is,
consumption out of the surplus product does not occur, the cost-minimizing technique is unlikely to
be efficient in this sense.

**6.0 Conclusion**
The theory of value and distribution has a family resemblance to modern formulations of classical and Marxian
political economy. Labor values are not discussed. It is focused on prices of production. Yet, with its consideration
of dynamic changes in dominant eigenvalues, it seems to be consistent with
an analysis
of the formal and real subsumption of labor to capital. The formulation in this post can easily be generalized
in various ways, Hahnel emphasizes inputs from nature and mentions the theory's consistency with homogeneous labor
inputs. The analysis of growth should include technical change. I am interested in fixed capital. Some issues arise
with general joint production, but the model is open in any case.

**References**
- John Eatwell. 2019. 'Cost of production' and the theory of the rate of profit.
*Contributions to Political Economy*.
- Robin Hahnel. 2017.
*Radical Political Economy: Sraffa Versus Marx*. Routledge.
- Joan Robinson. 1961. Prelude to a critique of economic theory.
*Oxford Economic Papers*, New Series. 13 (1): 53-58.