**1.0 Introduction**
I recently stumbled across McKiernan (2017), an Austrian response to Sraffa's book.
This is a weird working paper. He works through Sraffa, with apparently no
knowledge of all the textbooks
explaining the book.
McKiernan correctly notes that Sraffa provides little context about
his points. And the mathematics is not always explicit. Naturally, McKiernan
makes mistakes. If he ever revisits this, I think he would want to break
it up into several papers. (Fabra (1991) is the strangest published response
to Sraffians of which I know.)

I only want to focus on one point, Sraffian subsytems. But before doing that, let
me at least point out something insightful in McKiernan's paper.
He points out one of what I later call a fluke case:

However, Sraffa makes no note of cases in which a
scarcity of land is offset by using a more expensive method of production,
and that employment meets present desire for the product exactly
when applied to
all available land, so that only one method is used. In this case, *two*
variables, *p*_{corn} and ρ, correspond to *one* equation. (p. 22)

I think I could also construct a fluke case with intensive rent, which is closer to
his point.

**2.0 A Criticism Of Subsystems**
McKieran mistakenly asserts:

But consider the pricing of a commodity that were not produced in surplus,
as could happen in these models with a purely infrastructural commodity. A
sub-system for this commodity would have a net production of 0, but these
models have all presumed a need for active renewal, so there would be an expenditure
of labor. If returns to scale were co-incident, so that the sub-system
might be embodied, that sub-system standing alone would produce a wage of 0,
but wages are presumed to be shared across sub-systems (else Sraffa’s argument
falls apart). Hence, it appears that prices of commodities of this infrastructural
sort must be whatever one makes of division by 0. (p. 8)

(If McKieran ever revisits this, I wish he would give a more explicit and formal
definition of *co-incident returns*. I think I get his point, but I do
not think I have ever seen this notion in the economics literature.)

Anyways, if the net output of a commodity is zero, the decomposition of the given
quantity flows into Sraffian subsytems, with nothing left over, will
result in no labor being directed towards producing a net output of that
commodity. So one rather gets a quotient of zero divided by zero, not a division
of a positive quantity of zero. Nevertheless, one can still find a Sraffian
subsystem for producing that commodity.

**3.0 A Decomposition of One Set of Quantity Flows**
I take an example from my FAQ on the Labor Theory of Value.
Consider an economy with the observed quantity flows shown in Table 1.
In this little model economy, wheat, iron, and labor are used to produce
a net output of 500 quarters of wheat and 8 tons iron. From the postulated
observations, one cannot tell whether iron is somehow consumed or whether
this economy is undergoing steady growth.
Given, say, the real wage, one can calculate prices of production.
But many questions are not adressed here.

**Table 1: Given Quantity Flows**
**Inputs** | | **Output** |

74 qr. wheat & 37 t. iron & 592 worker. | -> | 592 qr. wheat |

18 qr. wheat & 3 t. iron & 48 workers | -> | 48 t. iron |

I find Leontief coefficients of production useful. Table 2 results from dividing the first of Table 1 by 592 qr. wheat
and the second row by 48 t. iron.

**Table 2: Leontief Coefficients**
**Inputs** | | **Output** |

1/8 qr. & 1/16 t. & 1 workers | -> | 1 qr. wheat |

3/8 qr. & 1/16 t. & 1 workers | -> | 1 t. iron |

Suppose these coefficients of production are used to calculate inputs when gross outputs of are approximately
588.2 quarters wheat and 39.2 tons iron. Table 3 results. The net output of the wheat subsytem is 500 quarters
wheat, produced from an input of 32000/51 workers. That is 64/51 ≈ 1.25 workers are embodied in each quarter
of wheat.

**Table 3: Wheat Subsystem**
**Inputs** | | **Output** |

1250/17 ≈ 73.5 qr. & 625/17 ≈ 36.8 t. & 10000/17 ≈ 588.2 workers | | 10000/17 qr. wheat |

250/17 ≈ 14.7 qr. & 125/51 ≈ 2.5 t. & 2000/51 ≈ 39.2 workers | | 2000/51 t. iron |

Table 4 shows the iron subsystem. In this subsystem, 640/51 ≈ 12.5 workers produce
a net output of 8 tons iron. In other words, 80/51 ≈ 1.57 workers are embodied in each ton iron.

**Table 4: Iron Subsystem**
**Inputs** | | **Output** |

8/17 ≈ 0.5 qr. & 4/17 ≈ 0.2 t. & 64/17 ≈ 3.8 workers | | 64/17 qr. wheat |

56/17 ≈ 3.3 qr. & 28/51 ≈ 0.5 t. & 448/51 ≈ 8.8 workers | | 448/51 t. iron |

The quantity flows shown in Tables 3 and 4 add up to the quantity flows in Table 1.
In these subsystems are thought of as operating side-by-side, total quantity flows are
as in the observed economy. No assumptions on returns to scale are needed for this decomposition
into subsystems.

**4.0 A Decomposition of Another Set of Quantity Flows**
I now want to consider another set of quantity flows that might be observed. Suppose these quantity flows are as in
Table 5. The net output of this economy consists of 500 quarters wheat. For ease of calculation, I have used the
same coefficients of production as in Table 2. They very well could be different since gross outputs vary from those in Table 1. If coefficients of production vary, so do returns to scale.
Anyways, in this example, the net output of iron is zero. But even so, one can find here a subsystem for producing iron.

**Table 5: Another Set of Quantity Flows**
**Inputs** | | **Output** |

1250/17 ≈ 73.5 qr. & 625/17 ≈ 36.8 t. & 10000/17 ≈ 588.2 workers | | 10000/17 qr. wheat |

250/17 ≈ 14.7 qr. & 125/51 ≈ 2.5 t. & 2000/51 ≈ 39.2 workers | | 2000/51 t. iron |

Consider the quantity flows shown in Table 6.
The net output is 500 quarters wheat and -8 tons iron.
This is not a Sraffian subsystem. It is unbalanced.
But these quantity flows are constructed from the same coefficients of production
as manifested in Table 5.

**Table 6: Unbalanced Quantity Flows**
**Inputs** | | **Output** |

1242/17 ≈ 73.1 qr. & 621/17 ≈ 36.5 t. & 9936/17 ≈ 584.5 workers | | 9936/17 qr. wheat |

194/17 ≈ 11.4 qr. & 97/51 ≈ 1.9 t. & 1552/51 ≈ 30.4 workers | | 1552/51 t. iron |

Now consider what quantity flows result from subtracting those in Table 6 from those in Table 5. Table 7 results.
This is the same subsystem for producing iron as shown in Table 4. Even though the net output of iron in
the observed quantity flows in Table 5 is zero, one can still find in them an iron subsystem.

**Table 7: The Iron Subsystem Again**
**Inputs** | | **Output** |

8/17 ≈ 0.5 qr. & 4/17 ≈ 0.2 t. & 64/17 ≈ 3.8 workers | | 64/17 qr. wheat |

56/17 ≈ 3.3 qr. & 28/51 ≈ 0.5 t. & 448/51 ≈ 8.8 workers | | 448/51 t. iron |

**5.0 Results**
No assumptions on returns to scale are made in analytically decomposing the observed quantity
flows into subsystems.

Given observed physical quantity flows, one can ask how much more labor would be employed if
net output of the economy was increased by a small quantity of a specified commodity. In other words,
each commodity has an *employment multiplier*, to use the jargon of Leontief analysis.
If non-constant returns to scale do not prevail, the error in this calculation will become
more pronounced as the specified quantity increases.
An insight behind the differential calculus is that, for continuous functions,
a small enough variation has an approximately linear effect.

**Reference**