"It is the *whole* process of production that must be called 'human labour',
and thus causes all products and all values. Marx and Ricardo used 'labour' in
two different senses: the above and that of *one* of the
factors of production ('hours of labour' or 'quantity of labour' has a meaning only
in the latter sense). It is by confusing the two senses that they got mixed up
and said that value is proportional to quantity of labour (in second sense)
whereas they ought to have said that it is due to human labour
(in first sense: a non-measurable quantity or not a quantity at all)."
-- Piero Sraffa (D3/12/11:64, as quoted by Kurz and Salvadori)

"I shall begin by giving a short 'estratto' of what I believe is the essence
of the classical theories of value, i.e. of those which incluce W. Petty, Cantillon,
Physiocrats, A. Smith, Ricardo and Marx. This is not the theory of any one of them.
I state it of course, not in their own words, but in modern terminology, and
it will be useful when we proceed to understand their portata (delivery capacity)
from the view of our present inquiry. It will be a sort of 'frame', a machine into
which to fit their own statements in a homogeneous pattern, so as to be able
to find what is common in them and what is the difference with the later
theories."
-- Piero Sraffa (D3/12/4:12 as quoted by Pasinetti and by Kurz and Salvadori and by )

**1.0 Introduction**
This post outlines a coherent theory to build on in trying to understand capitalist
economies. I thought originally of trying to explain the unoriginal points
in this post without any algebra. I wish I could be more terse. But I find that too difficult. No reason exists
that all uses of this theory should be fully formalized.

This post contains an implicit criticism of other theories based on
what is not here. The data are objective, and the approach is not that
of methodological individualism. I do not draw any well-behaved supply and demand
functions. Nor do I assume that the labor market, for example, clears.
I do not calculate any marginal quantities.
Unobservable preferences are not referred to, and no utility functions
are maximized. I do not see why in extending this approach one needs to
refer to such fictions.

**2.0 Givens From A View From Outer Space**
Consider a capitalist economy during a single production cycle, which
I take as a year. One can observe the following:

- A column vector,
**q**, of the gross output of produced commodities.
Each commodity is measured in physical units (tons, kilowatts, etc.), and
is the output of some process of production.
- The row vector,
**a**_{0}, of labor coefficients of the production processes.
The coefficient *a*_{0,j} is in physical units (for example, labor
units per ton), where labor units are such that total employment over the year is unity.
Different concrete activities are weighted by relative wages to reduce labor units
to a common abstract labor unit. The product (*a*_{0,j} *q*_{j})
is the proportion of total employment allocated to the *j*th industry.
- The square Leontief input-output matrix,
**A**. Each element *a*_{i,j}
is in physical units (for example, kilograms per ton). The quantity (*a*_{i,j} *q*_{j})
is the physical units of the *i*th commodity used in producing the (gross) output of the *j*th process.
The Leontief matrix is assumed to characterize a viable economy that can operate year after year at the same
level. For simplicity, assume that each commodity enters directly or indirectly into the production
of all commodities. That is, all commodities are Sraffian basic commodities.
- The wage,
*w*, where the wage is the proportion of the price of the net output obtained by
the workers.
- The column vector,
**d**, of the commodities on which the wage is spent.

In this formulation, each industry produces a single commodity, and a single process is operated in each industry.
Since the data is from a snapshot of the whole economy, in a sense, no assumptions are made on returns
to scale. Each industry is operating at whatever scale is observed, with observed inputs being obtained
from other industries.

Certain variables are implicit in these definitions of the givens. Net output is related to the
the given technique and gross output:

**y** = **q** - **A** **q** = (**I** - **A**)**q**

where **y** is the column vector of net output, and **I** is the appropriately
sized identity matrix. The net output is that part of gross output that can be consumed,
while leaving available the reproduction of the capital goods used up in producing it
so as to continue production in the next year at the same level and with the same
technique. One can invert the above relation to find gross output from net output:

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

The assumptions ensure that the Leontief inverse exists.

The unit of measurement for labor is chosen such that total employment, at the observed level
of gross outputs, is unity:

**a**_{0} **q** = **a**_{0} (**I** - **A**)^{-1} **y** = 1

The numeraire is the net output:

**p** **y** = 1

where **p** is a row vector of prices.
The total wage is equal to the price of the commodities on which wages are spent:

*w* **a**_{0} **q** = **p** **d**

For what it is worth, the physical composition of wages by industry is **d** **a**_{0}.

**3.0 Social Relations Between Workers**
The data reflect a certain allocation of labor over the industries comprising the economy. A
notion of vertical integration is needed to think about how each commodity can be produced
in a
sustainable
way.

For example, suppose a consumer buys one additional automobile of a specified make. To leave the production
apparatus unchanged, workers will have to produce an automobile in Michigan, tires outside of Akron, steel
from iron ore in Pittsburgh, and so on, with transport workers ensuring everything gets shipped appropriately.
(Actually, this example probably reflects my understanding of the structure of production many decades ago.)
This can all be expressed in algebra:

*v*_{j} = **a**_{0} (**I** - **A**)^{-1} *e*_{j}

where *e*_{j} is the *j*the column of the identity matrix.
The integrated labor units per unit of the *j*th commodity, *v*_{j}, is both an employment
multiplier and the labor embodied in the commodity. I have explained it above as a matter
of laborers working in parallel in the given year. But one can also think of it as a matter
of notionally summing some labor in this year and over past years under the assumption that
the observed technique has been used forever in the past at this scale.

One can also apply this algebra to certain collections of commodities. The
labor embodied in commodities purchased out of wages, *V*, is:

*V* = **a**_{0} (**I** - **A**)^{-1} **d**

The labor embodied in the remaining commodities, *S*, which are produced by
the workers but in the control of those owning the means of production are:

*S* = 1 - *V* = 1 - **a**_{0} (**I** - **A**)^{-1} **d**

The labor embodied in capital goods, *C*, is:

*C* = **a**_{0} (**I** - **A**)^{-1} **A** **q**

Certain ratios of these labor values have often been expressed. For example,
the ratio, *e*, of the labor spent during the year to produce commodities not paid to workers
to the labor spent to produce the commodities purchased out of wages is:

*e* = *S*/*V*

If one assumes wages are totally consumed, the above ratio has something to do with the maximum
possible rate of growth. The ratio, *occ*, of the labor embodied in capital goods to
the labor expended in the year is:

*occ* = *C*/(*V* + *S*)

**4.0 Prices As Relations Between Things**
Those making decisions about which processes to operate and at what levels
have done so with certain expectations about being able to sell the produced commodities
at certain prices.
They also have expectations about costs. Observed market prices
are almost certainly such that some of these expectations are being disappointed, and
some will need to alter their plans. Such alterations include changing the
scale at which certain processes are operated in some industries, disinvesting
in some industries and investing in other industries, and replacing processes
of production (perhaps with newly discovered processes) in various industries.

As a heroic simplification, consider what prices must be
to be consistent with the data. Such prices of production are
based on the assumption that the allocation of the labor force
seen in the data is socially necessary. Perhaps, if the data were to
be repeated year after year, market prices would circulate around
prices of production or approach them in a 'gravitational' process.

Assuming competitive markets, in some sense, prices of production
satisfy the following equation:

**p** **A**(1 + *r*) + *w* **a**_{0} = **p**

where *r* is the rate of profits. In this equation, wages are paid out
of the product at the end of the year.

As a matter of mathematics, the assumptions in this model of circulating capital
are sufficient to determine prices of production and the rate of profits. They
still would be sufficient if one allowed for fixed capital and for land in
a model of extensive rent. Likewise, one could allow for limitations in competition
by assuming known ratios of the rate of profits among industries. Certain issues
arise in determining prices of production and the rate of profits in models
of intensive rent and of general joint production. More than one solution may arise
in some cases.

These extensions to joint production require the modeling of the choice of technique.
I am not sure how I can do this analysis without assumptions on returns to scale.
I do see that I can restrict distribution such that the observed technique is cost-minimizing
at the observed scale.

It is easier to solve for the wage as a function of the rate of profits:

*w* = 1/{**a**_{0} [**I** - (1 + *r*)**A**]^{-1} **y**}

The above is a monotone decreasing function. The maximum wage of unity occurs when the rate of
of profits is zero. The maximum rate of profits occurs at a wage of zero, and that maximum is 1/*occ*.
Prices of production
are:

**p** = **a**_{0} [**I** - (1 + *r*)**A**]^{-1}/{**a**_{0} [**I** - (1 + *r*)**A**]^{-1} **y**}

**4.1 An Aside On Vertical Integration**
I repeat the above equation for prices of production:

**p** **A**(1 + *r*) + *w* **a**_{0} = **p**

This can be rewritten as:

**p** **A***r* + *w* **a**_{0} = **p**(**I** - **A**)

Or:

**p** **A**(**I** - **A**)^{-1}*r* + *w* **a**_{0}(**I** - **A**)^{-1} = **p**

Or:

**p** **H***r* + *w* **v** = **p**

where I am going to call **H** the vertically integrated Leontief matrix. (Pasinetti probably has another name.)
The row vector **v** is a vector of labor values. The solution of this system of equations is the same as above.
When the rate of profits is zero, prices of production are equal to labor values.

**5.0 Conclusion**
The above has outlined a logical theory for describing a capitalist economy. The starting
data are, in principle, observable and close to what can be obtained from the National Income
and Product Accounts (NIPA).

This data allow one to examine how the labor force is allocated in a sustainable
economy. The proportions of workers that are (re)producing certain aggregates (wage
goods and the remainder) or embodied in certain aggregates (the capital goods used
up in production) are noted. Presumably, individual commodities may be produced with
extremes of labor-intensive methods, but these differences could come close to averaging
out in the aggregate. One might want to extend wage goods to include, for example,
commodities consumed by, for example, retired workers, students, the unemployed, and
those unable to participate in the labor force. One might also want to look at
other aggregates such as luxuries spent by those obtaining income produced by the
workers but not paid out as (generalized) wage goods and investment goods used
in expanding the economy.

These aggregates can also be evaluated with prices.
I have drawn a few connections between prices of production and the labor embodied
in these aggregates. The maximum rate of profits is the multiplicative inverse of the organic composition of capital.
In formulating a system of equations for prices of production in vertically integrated terms, I find labor values useful.
Otherwise, this
post has drawn no connection between prices of production and the labor embodied
in these aggregates. Can one pass easily between prices of production
and labor values? The mathematics of
eigenvalues and eigenvectors
is useful for exploring the theory behind this question.
Whatever you think of the answer, including difficulties arising with joint production
and of empirical results,
seems to be independent of the validity of anything in the above post.

I have talked about some of the conditions needed to sustain the operation of a capitalist
economy, while only looking at the data for a single production period. Presumably, the gross
output for the next year will be at a different level and mix. Such a change in scale in
operation can be expected to alter the vector of labor coefficients and the Leontief
input-output matrix.

Non-produced commodities that are available only as a finite stock that are
used up in production (for example, oil, ores, and certain minerals) have been abstracted
from. I have also ignored physical limitations imposed by bounds on throughput and
sinks. The latter issue can be explored by the theory of joint production where one
does not impose the assumption of free disposal.

Half a century ago, some economists demonstrated that economists a century and a half ago
were mistaken. Jevons, Menger, and Walras made fundamental mistakes that cannot be
fixed and are built into the work since then that builds on them. The political economy
that had been developed in the century before them provides an alternative that is
worth updating and building on.