Sraffa's Reproduction Prices versus Prices of Production: Probability and Convergence

Abstract

The article argues that the reproduction prices introduced in the first chapter of Sraffa's book are not necessarily compatible with the profit equalising prices that form the substance of the book. It uses probabilistic arguments about how probable it is that reproduction prices will approximate to profit equalising prices. By use of random matrix techniques, it shows that the solutions space associated with prices of production is similar to that associated with classical labour values. In the latter part of the article, a random sample of reproduction schemes is simulated over time, under assumptions of capital movement, to see whether such systems dynamically converge on profit equalising prices. It is found that some converge and some fail to converge.

Full text

Sraffa’s reproduction prices versus prices of
production: probability and convergence.
W. Paul Cockshott
September 29, 2017
School of Computing Science, 18 Lilybank Gardens
University of Glasgow, Glasgow, G12 8QQ
william.cockshott@glasgow.ac.uk
Abstract
The paper argues that the reproduction prices introduced in the first chap-
ter of Sraffa’s book are not necessarily compatible with the profit equalising
prices that form the substance of the book.
It uses probabilistic arguments about how probable it is that reproduc-
tion prices will approximate to profit equalising prices. By use of random
matrix techniques it shows that the solutions space associated with prices of
production is similar to that associated with classical labour values.
In the latter part of the paper, a random sample of reproduction schemes is
simulated over time, under assumptions of capital movement, to see whether
1

such systems dynamically converge on profit equalising prices. It is found
that some converge, and some fail to converge.
1 Reproduction prices
In the first pages of (Sraffa, 1960) a simple two product economy producing corn
and iron is introduced with the following structure:
280 wheat + 12 iron →400 wheat
120 wheat + 8 iron →20 iron
In volume I of Capital(Marx, 1887) Marx used a price theory in which com-
modity prices are taken to be proportional to the labour required to make the com-
modity. He used this price theory to argue for a theory of surplus value, according
to which workers were exploited of the full value created by their work. In Chapter
20 of the second volume(Marx, 1974) he introduces reproduction schemes, matri-
ces of inter-sectional flows of commodities that had to occur if the economy was
to reproduce itself.
The matrices have 4 column vectors:
C constant capital, his term for expenditure on capital goods and raw
materials.
V varible capita, his term for expenditure on wages.
S surplus or profit.
O output
We give an example in Table 1 of a 2 x 4 matrix, with the row labeled I representing
the production of capital goods and raw materials, and row II the production of
2

C V S O
I £100 £50 £50 £200
II £100 £150 £150 £400
Table 1: A stationary state specified in money
consumer goods. In Marx’s tables all quantities are in terms of money rather than
in terms of use values1. For accounting reasons the relation O=C+V+Smust
hold.
Further ∑C=O1that is to say consumption of capital goods equals their pro-
duction, and ∑(V+S) = O2. Together this implies that sector I of the economy
must trade O1−C1in capital goods for C2worth of consumer goods produced in
sector II. So we have an equilibrium equation
C2=V1+S1
This is the equilibrium condition of an economy in a stationary state where
it simply reproduces itself neither growing nor shrinking.The basic analysis in this
paper will assume this stationary state. Real economies may grow or shrink, but the
rate at which they do this is typically quite small. A developed industrial economy
like the US can go long periods in which the rate of growth averages only 3% a
year or less, so analysis of price systems in a stationary state is a reasonable first
approximation.
Although it is not done by Marx, one can in principle construct a dual table like
Table 2in tons or consumer goods (corn) and tons of capital goods(coal). In this
1They thus differ from the technology matrices of Morishima(Morishima, 1973), though, as we
shall see there is an underlying relationship between the two.
3

Coal Corn wage Corn Profit Ooutput
I 10 20 20 20 ton Coal
II 10 60 60 160 ton Corn
Total 20 ton 80 ton 80 ton
Table 2: A stationary state specified in tons matter
case the first column represents the coal used up productively by the two industries,
and next come the consumer goods (corn) consumed by the workers and employers
in the two sectors. Again we have the requirement that the total consumption and
total production of each good must balance, 160 tons of corn, and 20 tons of coal.
It is clear from this table that the coal industry must sell 10 tons of coal to the
corn industry and get back in return 40 tons of corn, which in turn implies that the
relative price of a ton of coal must be 4 times the price of a ton of corn. Refering
back to the first table and comparing it with the second we see that indeed the price
of a ton of coal was £4 and but a ton of corn cost only £2.50. The important point
here is that given the physical table, the relative prices necessarily follow.
The example is artificial in that in practice sectors I and II would each pro-
duce a whole vector of outputs, but given the constants of proportionality between
the elements of these two vectors, the exchange relation between them establishes
relative sectoral prices.
Marx later extends the scheme to 3 sectors, by dividing consumer goods into
necessities(IIa) which are assumed to be bought out of wage incomes and luxu-
ries(IIb) which are bought out of property incomes. If we retain the label II for
necessities and use III for luxuries, we have the three way trade between sectors in
Fig 3.1.
The tables are given in money terms, much as modern national accounts are,
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but the assumption explicitly remained that these quantities of money are propor-
tional to quantities of labour((Marx, 1974) Chapter 21, Section 7). But in principle
other pricing structures are possible so long as they allow the trade pattern in Fig
3.1. In what follows we will show that the reproduction schemes themselves im-
ply a disctinct set of price configurations and that these price configurations only
partially overlap with those presupposed by either labour values or prices of pro-
duction.
2 Prices of production
In Volume III of Capital(Marx, 1894b) Chapter 9, Marx introduces a distinct model
of prices, which he terms production prices. He points out that the prior assumption
of a constant ratio s
vaccross industries whilst the ratio c
vvaried between industries
would lead to the profit rate s
c+vto vary between industries. He hypothesised
that this would be unstable and that as a result of capital movements into high
profit industries, a uniform average rate of profit would be attained. The resultant
production prices would preserve an equal rate of profit but the side effect would
be that prices would diverge from labour values, being systematically higher in
industries with a high organic composition of capital. The relatively higher prices
in high organic composition industries would then enable them to earn the average
profit rate.
Whilst some commentators argued that prices of production tended to under-
mine his prior arguments about labour values and exploitation (Hilferding, 1951;
Samuelson, 1973; Steedman, 1981), the basic hypothesis of a law of an equal
rate of profit was accepted until the publication of the pioneering econophysics
5

work Laws of Chaos(Farjoun and Machover, 1983). This argued on probabalistic
grounds that the distribution of prices was more likely to follow a simple labour
value model than a price of production model. More recently Greenblatt(Greenblatt,
2014) has also proposed a stochastic model in which labour values appear as an
emegent property along with a spread of profit rates.
Multiple empirical studies (Cockshott and Cottrell, 1997, 1998; Fröhlich, 2013;
Sánchezc and Montibeler, 2015; Shaikh, 1998; Zachariah, 2006) have indicated
that production prices are not systematically better at predicting actual market
prices than simple labour values. It has also been shown that the Marx’s basic
assumption that the rate of profit is the same in high and low organic compostion
industries is not born out empirically today(Zachariah, 2006, 2009), whatever may
have been the case in the 19th century. However this empirical work does not
help us to say whether the observed relationship between labour values/production
prices and market prices is ’close’. The studies reproduced in Table 5 show mean
absolute errors of the order of 10% between labour values/production prices and
observed prices. But is 10% close or distant?
We can only say that if we have some a-priori estimate of just how close we
should expect the market price vector to be labour values/production prices in the
absence of the operation of a law of value, or Marx’s law of the equalisation of
profit rates.
In what follows a new probabalistic technique using reproduction schemes is
developed to evaluate these empirical results. The basic intuition is that one can
systematically count what fraction of possible reproduction schemes are consistent
with prices of production or labour values.
As such they enable us to numerically evaluate the a-priori likelyhood of labour
6

values/prices of production being within, for example, 10% of market prices.
3 Constraints on reproduction schemes
In Tables 1 and 2 we showed that from the physical flow between sectors one could
work out the relative sectoral prices. The aim here is to show how one can start
out from a physical flow pattern for a 3 sector economy and deduce the relative
sectoral prices that must correspond to it.
We will use G, for Goods, to stand for our 3x3 matrix of flows of goods in kind
such that the first column corresponds to the in-kind flows of capital goods that
Marx denotes by his C column vector, the second column to the in-kind flows of
wage goods corresponding to the column vector V, and the last column to the flows
of luxuries denoted by the column vector S.
We stipulate that all elements are positive non-zero and that each column of G
adds to 1, ie, the elements of each column in Gare expressed as fractions of the
total output of the coresponding sector. In other words we normalise the columns.
A concrete example is given in Table 3.
We denote the elements of Gas gi,jfor i,j: 1..3.
If pis a 3 element price vector for capital, wage and luxury goods then
In order to have only 3 prices when in fact each sector makes a wide variety of
goods, we assume that the prices are index prices defined over bundles of capital,
wage and luxury goods. Given the actual physical flows in Gthen the trade pattern
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Physical flow table
Coal Corn Caviar outputs
I 16047 2801 14151 20004 ton coal
II 464 11898 3573 20017 ton corn
III 3493 5318 2286 20011 ton caviar
Totals 20004 20017 20010
the equivalent G matrix
0.80 0.14 0.71
0.02 0.59 0.18
0.17 0.27 0.11
Table 3: For the purposes of studying the relation between physical flows on sec-
toral prices it is convenient to to express the flow elements as numbers between 0
and 1. We can do this by normalising a physical flow table, dividing each column
element by the total of the column.
in Fig 3.1 establishes price constraints:
p1g3,1 =p3g1,3
p3g2,3 =p2g3,2
p2g1,2 =p1g2,1
(3.1)
Where Pis a 3 element price vector whose elements are written pi. For example
given the Gmatrix in Table 3 we can use the above equations to solve for the
relative prices deriving :
P= [2.123, 0.352, 0.524]
from which we can derive the corresponding monetary relations given in Table
4.
Note that since the first equation fixes the ratio p1/p3and the next fixes p2/p3
8

Const Wages Profits
capital
C V S O
I £34067 £986 £7415 £42468
II £986 £4188 £1872 £7046
III £7415 £1872 £1198 £10485
total £42468 £7046 £10485 £60000
Table 4: An example of a 3 sector economy in a stationary state. The sector II now
produces wage goods and sector III luxuries. This should be read in conjuction with
Figure 3.1. Note the symmetry of the table around the diagonal corresponding to
the trade pattern in the figure. This monetary table is derived from the Table 3 by
solving equation set 3.1.
I
II III
v1 c2 c3 s1
s2
v3
Figure 3.1: Three way inter sector trade. So for example sector II sells sector I
wage goods worth v1 and buys back in return means of production c2.
9

then this implies p1/p2is also fixed, so we have to interpret the last of the three
equalities as a constraint on what kind of physical flow matrix is compatible with
inter sector trade. The price constraints set by the Gmatrix define market clearing
prices for a system in which all sectors are self financing, that is to say, there
is no credit provided by one sector to another. This was an implicit assumption
of Marx’s analysis in Volume II of Capital. But these reproduction constraints
themselves impose restrictions on the structure of the G matrix. Not all normalised
Gmatrices are compatible with self financed simple reproduction.
Vol I, Vol II and Vol III of Capital actually provide three distinct price models
which partially overlap. In Figure 3.2 we illustrate the volumes of configuration
space that we are interested in. Reproduction schemes define, by equation 3.1,
a set of market clearing price configurations - the large circle. Smaller circles
denote the volumes of configuration space compatible with prices of production
and labour values. Not all configurations that are compatible with labour values or
prices of production are compatible with simple reproduction. By being compatible
with prices of production we mean that the prices derived from Equation 3.1 result
in rates of profit that are equal, or very nearly equal, in all sectors. By being
compatible with labour values we mean that the prices from Equation 3.1 lead to
nearly equal ratios of wages to profits in each sector.
The experiment reported in section 4 allows us to estimate the degree of overlap
between these sets.
3.1 Derivation of g2,1
Let us first examine how the structure of the G matrix is constrained by reproduc-
tion.
10

labour
values
prices of
production
market clearing
reproduction
prices
no credit
non reproduction
or credit based prices
Figure 3.2: The set of price systems, this study is restricted to market clearing
reproduction prices without credit.
Given g3,1,g1,3,g2,3 ,g3,2,g1,2 we can derive g2,1 as follows:
p1/p3=g1,3/g3,1
p3/p2=g3,2/g2,3
p1/p2=p1
p3
×p3
p2
=g1,3
g3,1
×g3,2
g2,3
but from the original trade relation we have
p1/p2=g1,2/g2,1
so
g1,2/g2,1 =g1,3
g3,1
×g3,2
g2,3
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and
g2,1 =g1,2
g1,3
g3,1 ×g3,2
g2,3
(3.2)
Alternatively the constraint can be expressed in terms of elements of the other
two columns:
g2,3 =g2,1 ×g1,3
g3,1
×g3,2
g1,2
(3.3)
or
g1,2 =g2,1 ×g1,3
g3,1
×g3,2
g2,3
(3.4)
Taken along with our constraint that the columns of Gsum to 1, we have a
4 constraints on the 9 elements of the matrix leaving only 5 degrees of freedom
to the configuration space of reproduction schemes. That is to say that simple
reproduction schemes are samples drawn from an underlying 5 dimensional vector
space. Given such a space we can systematically sample it.
4 First Experiment
A programme was developed that created successive random samples of the con-
figuration space of reproduction schemes. First the elements of Gwere assigned
random values >0and <1such that the totals on columns 2 were each 1, and
the expected value of each element was 1
3. Then with equal probability one of the
equations 3.2 to 3.4 was used to over-ride the previous random variable assign-
ment to one of the elements. This constraint however is not guaranteed to satisfy
the condition that the column must sum to 1, but that is achieved by subsequently
altering the diagonal elements of the matrix to ensure that all columns sum to 1.
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0.0001 0.01 1 100 10000
Organic comp dept 1
0.001
0.01
0.1
1
10
100
1000
Organic comp 2 or 3
organic comp 2
organic comp 3
Figure 4.1: Spread of relative organic compositions over the entire sample set with
the subsampling technique.
The diagonal elements do not enter into inter-sector trade and hence can be altered
without disturbing the relations established in equations 3.2 to 3.4.
The mean of Gover 120,000 samples to two decimal places was
0.40 0.28 0.32
0.25 0.40 0.28
0.35 0.32 0.40
This implies that the expected values for the organic compositions of capital,
for reproduction schemes meeting equation 3.2 will differ between departments.
This means we are not encountering a simple situation of uniform expected organic
compositions. This can be seen in the distribution of relative organic compositions
in Figure 4.1.
For each reproduction scheme configuration the market price vector was set by
constraint 3.1.
Labour values were computed as follows
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(a)
0.0001 0.01 1 100 10000
Organic composition dept 1
0.001
0.01
0.1
1
10
100
1000
10000
Organic composition dept 2
0.0001 0.01 1 100 10000
Organic composition dept 1
0.001
0.01
0.1
1
10
100
1000
Organic composition dept 3
Dept 3
(b)
0.001 0.01 0.1 1 10 100 1000 10000
Organic composition dept 1
0.001
0.01
0.1
1
10
100
1000
10000
Organic composition dept 2
0.001 0.01 0.1 1 10 100 1000 10000
Organic composition dept 1
0.001
0.01
0.1
1
10
100
1000
Organic composition dept 3
Figure 4.2: Plot of the relative departmental organic compositions of reproduction
schemes in which market clearing prices were within 10% of : (a) labour values; (b)
prices of production. Note that the characteristic ’bow tie’ configuration for the left
plots also appears in the overall sample in Figure 4.1, but that whilst the first and
third quadrants are empty here, samples were present in these quadrants in Figure
4.1. This indicates that these quadrants of configuration space are incompatible
with either prices of production or labour values.
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v1=g1,2/(1−g1,1 )
that is to say divide the real wage in dept I by its net output. The assumption
made is that the labour used in each sector is proportional to the flow of wage goods
consumed.
v2=g2,2 +v1g2,1
v3=g3,2 +v1g3,1
At the end of this we have vas vertically integrated labour coefficients, derived
from wages.
∑v=3
The last is a normalisation condition used to ensure that under all prices mod-
els, the sum of prices is the same. If we do not apply this, we would have a sum
of values <3in effect ignoring surplus value. But we are free to apply this linear
rescaling to vsince the assumption Marx makes is that the rate of surplus value
is the same everywhere. The implicit assumption here is that the real wage is the
same in all sectors.
An iterative estimation is used for prices of production. We first set all prices
to 1. Then we repeatedly perform the following steps.
Set rto one plus the rate of profit.
r←1+p3
p2+p1
(4.1)
15

This works because the physical output of each industry is unity by virtue of using
a normalised G. Next set a new estimate of the price vector np.
np←(r×(p2gι,2 +p1gι,1 )) (4.2)
Finally we normalise the sum of prices to be 3, the same as before.
p←3×np
∑np (4.3)
Runs were made with many reproduction schemes. The cummulative total
number of resulting reproduction schemes was recorded along with the number of
schemes that conformed either to labour values or to prices of production. Results
are shown in Tab 5.
Conformance of either labour values or prices of production was determined by
measuring whether their mean absolute deviation (MAD) from market prices were
below a specified threshold. The experiment used a 10% threshold as that is of the
right order for the best that is observed empirically. In addition the program records
the mean of the MADs between exchange values and the two pricing theories.
4.1 Results
From the random sampling of reproduction schemes it appears that the mean spread
of prices of production from market clearing prices is smaller than the mean spread
of labour values. This is incompatible with a number of empirical studies as shown
lower in the table where the relationship is the reverse. However the observed
spreads of both values and production prices from market clearing prices are lower
16

Table 5: Relative frequencies and spreads of prices of production and labour values
as computed from sample of 100000 reproduction schema. and compared with
empirical studies.
Labour value Production price Production price
versus market price market price labour value
Fraction of schemes with MAD <10% 0.44% 0.44% 6.84%
Mean MAD 28% 23.7% 14.8%
Empirical MAD
China(Sánchezc and Montibeler, 2015) 14.2% 16.5% 12.0%
USA(Ochoa, 1989) 10.3% 12.6% 16.9%
Spain(Sanchez and Nieto Ferrandez, 2010) 12.2% 18.8% 19.0%
Germany (1978)(Zachariah, 2006) 16.0% 22.6%
France (1980) (Zachariah, 2006) 12.0% 18.2%
than found in our a-priori estimation. This may be a result of the empirical data
typically using longer price vectors with 30 to 60 elements rather than just 3, with
resultant reversion to a mean. This was essentially Farjoun’s argument(Farjoun,
1984) for why empirical dispersions of prices to market values would be smaller
than those obtained in toy examples using reproduction schemes.
5 Discussion
Sraffa (Sraffa, 1960) showed that given:
1. An assumption of an dqual rate of profit.
2. A technology matrix.
3. A specification of the real wage,
it was possible to deduce a price system that would reproduce both the material
conditions of production and the class distribution of income. This paper shows
17

that the Gmatrix, a use value dual of Marx’s reproduction schemes can also define
a price system that will reproduce the material conditions of production and the
class distribution of income.
The Gmatrix plays both the role of Sraffa’s technology matrix and his real
wage, but Marxian reproduction schemes do not necessitate a uniform rate of profit,
nor do they require that prices are proportional to labour values. Reproduction
schemes can exist with these properties, but Table 5 shows that both labour value
conforming schemes and price of production conformant schemes make up a small
portion of the possible schemes. Even with a very lax definition of conforming,
being within 10% of, less than 1
200 th of all schemes meet this criterion. It would
appear that labour value conforming reproduction schemes are as common as price
of production conforming ones. If one looks at Figure 4.2 showing where the
conforming instances occur in the planes of relative surplus value between sectors,
the pattern is almost identical in both cases, with many of the same data points
appearing in both rows.
From Sweezy onwards it has been conventional for Marxian economists to
present individual example reproduction schemes that either have prices propor-
tional to labour values or prices given by an equal rate of profit. The statistical
analysis here shows that in doing so, economists have been using what are, on
a-priori grounds, rare exceptions to prove rules.
Three sector reproduction schemes, however, capture something additional that
is missing in Sraffa, the fact that different social classes have different consumption
patterns. Marx dealt with the more general case where the capitalists divide their
expenditure in some fixed proportion between necessities and luxuries, what would
in modern terms be called a Leontief demand function. The analysis here has
18

taken the simpler assumption that capitalist expenditure is exclusively on luxuries.
Similarly we neglect that some commodities, for instance coal, may have been a
means of production, a wage good, and have been bought by capitalists to heat
their houses.
The simplification is arguably valid, since one could in principle divide the
coal industry into 3 sub industries, one supplying factories, one supplying workers
cottages, and one supplying mansions. These sub industries would then be statisti-
cally aggregated into sectors I, II or III. But the inter-sectoral constraints may have
implications for the feasibility of attaining prices of production.
Reproduction prices represent a static macro-economic equilibrium condition.
So long as there is no growth in production and no change in technology and no
movement of capital between sectors, reproduction prices will keep the economy in
an equilibrium. They are market clearing prices given the technology and income
distribution. On the other hand, the alternative concept of equilibrium present in
Volume III of Capital (Marx, 1894a) and further developed in Production of Com-
modities (Sraffa, 1960) assumes capital mobility between sectors. Borkiewicz’s
criticism(Hilferding, 1951) of Volume III was based on arguing that the procedure
presented for transforming labour values to prices of production was statically in-
compatible with reproduction prices. But the dynamic question remains open. If
you start of in a macroeconomic equilibrium with reproduction prices operating,
but with divergent profit rates as shown in Table 4, then can capital movements
produce a new equilibrium with a price structure that both achieves reproduction
and profit rate equalisation?
On the one hand the structure of reproduction is so finely balanced, with such
intricate interdependence between the elements of the reproduction table that per-
19

haps any movement in capital would throw the whole system into a catastrophic
crisis. Alternatively one may argue that even if one keeps technology and labour
supply, and money capital constant, the system has still got some degrees of free-
dom left in terms of the relative sizes of three sectors.
One can see that capital movement is very likely to result in a change in the
class distribution of income. A movement of capital in or out of sector II means
a bigger or smaller real wage, and in consequence reduces or increases the real
quantity of luxuries being consumed by employers. So a movement into row 2 of
the table must go along with balancing changes in columns 2 and 3, but whether
these will be dynamically achievable is harder to say. It may depend both on the
adjustment process and on the initial starting structure of the table.
6 Second Experiment
In order to investigate the dynamic process of capital movement from initial re-
production states, a second experiment was carried out. Like the first experiment it
used a sample of reproduction schemes, prepared in the same way as in the previous
experiment. It combined these with rules for capital mobility, for price adjustment,
possible buffer stocks and adjustment of sectoral outputs. The time evolution of
the economies represented by the initial reproduction schemes was then evaluated
for 150 time steps.
Initialisation AGmatrix is prepared as in the first experiment. An initial price
vector is derived and a resulting initial monetary reproduction scheme is derived.
From an assumed money wage of £2 a initial vector of labour allocation λis de-
20

rived. In conjunction with the labour vector the Gmatrix is used to derive a linear
production function for each sector. Each sector is allocated sufficient cash to pay
wages and buy means of production at current prices and the current scale of pro-
duction.
Simulation cycles start at the point where production has just finished, so the
firms in each sector have a stock equal to what was produced, plus any unsold stock
from the previous period. Stocks of goods held are recorded in the A, for available,
matrix.
Capital allocation rule Let sbe the sector with the highest rate of profit. For
each sector x6=sif the rate of profit in xis more than 1% below the rate in
sthen sector xwill transfer 1% of its money capital to sector x. Each sector
divides its money capital into constant and variable capital in the same ratio as its
final allocation in the previous period. We thus get new column vectors Vt,Ctfor
variable and constant capital for time t.
Wage and labour rule Wage rates are then set such that
wt←∑Vt
∑λt−1
and the new wage rate and new is used to reallocate labour so that
λt←Vt
wt
.
21

Prices sectors I and II The total requirement for means of production for each
sector, given λtis then determined using the production functions. If this exceeds
the total stocks of means of production held by all sectors then we have a sellers
market in means of production whose prices rise to a market clearing level.
p1←∑Ct
∑ιAι,1
otherwise if stocks exceed requirements, we have a buyers market and the price of
means of production is reduced by 3%. The price of wage goods is then set as
p2←w∑λ
∑ιAι,2
Sectors then pay wages and workers spend their wages on the output of sector II at
the current p2. Each sector then purchases its requirement of means of production
from sector I at price p1.
Demand for luxuries For sectors I and II we now know their total sales and
their total cost of production. By subtracting purchases from sales we get their
profits which are assumed to be entirely spent on luxuries. For the capitalists of
sector III we have the odd situation that as Marx points out, their profits are self
financing. Whatever they spend on luxuries will return to them as additional profit.
The simulation thus adopts the parsimonious assumption that their expenditure on
luxuries will remain constant in money terms. The price of luxuries is then set to
clear the market given the physical stocks available.
22

Production Production takes place constrained either by the available labour in
each sector or the available means of production, as per the linear production func-
tion. If labour is the limiting factor this may result in some unused stock of means
of production which are carried over to the next period.
6.1 Results
Figure 6.1 shows the results of the simulation in terms of the initial and final stan-
dard deviations of the rate of profit. A simulation run is represented as a point
whose x position is given by the starting spread of its profit rate and its y position
by its terminating profit rate spread. A point on the 45◦diagonal represents a sys-
tem that has undergone no profit rate convergence during the simulation. A point
close to the x axis indicates a system that has undergone convergence.
One can clearly see that the simulated systems fall into two distinct clusters -
one just below the 45◦line, and one close to or below the 1% line. Provided that
profit rates are within 1% they are taken to have converged, since only discrepan-
cies bigger than this are assumed to trigger capital flows.
Detailed examination of the final sectoral output figures for the simulations
run showed that many simulated economies had undergone a drastic contraction in
terms of physical output. Since the amount of money circulating does not change
during the simulation, rises in prices obscure this effect if one looks only at the fig-
ures for output in money terms. We define an economy to be healthy under capital
movement if the final value of output measured in the prices operating at time t0are
>98% of the starting value of output. We define an economy as having collapsed
if output is less than 50% of its starting value. One can see in Figure 6.1 that there
is no particular relationship between the economy being healthy and its profit rate
23

Sector I II III
Collapsing 2.16 0.59 1.18
Healthy 1.23 0.65 1.88
Converging 1.60 1.31 2.21
Non converging 1.96 0.47 1.15
Table 6: Geometric mean of initial organic compositions by sector and group for
the economies simulated in Figure 6.1.
converging. Some of the economies whose profit rates equalise are healthy and
some are collapsing. Conversely some healthy economies retain dispersed profit
rates even in the presence of capital movements that, according to accepted theory,
should result in an equalisation of the rate of profit.
Table 6 does show however that the collapsing economies tend to be charac-
terised by greater sectoral disparities in organic composition, and higher organic
compositions in sector I. Systems that do not converge their rates of profit are char-
acterised by particularly low organic compositions in sector II.
7 Discussion
The first experiment shows that only a very small fraction of possible self reproduc-
ing capitalist economies are characterised by equal rates of profit. Similarly only
a very small fraction of possible reproduction schemes have price structures close
to labour values. The existing literature on the transformation problem relates to
either logical or temporal transition between the small subset of the value confor-
mant reproduction schemes and the small subset of price of production conformant
schemes.
The second experiment indicates that one can not simply assume that the mech-
24

Figure 6.1: When simulated over time, some reproduction schemes can converge
towards an equal rate of profit. However the population of schemes forms two
distinct clusters, one capable of converging and one which does not converge.
Schemes which show no convergence over time would lie on a line at 45◦going
through the origin of this plot. Healthy models are those in which GNP remains
constant or grows, collapse models are those whose GNP has fallen by more than
50% at the end of the simulation.
25

anism that is supposed to bring about an equal rate of profit will, in general, work.
For some starting points, combinations of technology and distributions of income,
the hypothesised convergence mechanism fails. In these cases the system either
remains healthy with a continuing spread of profit rates, or the economy shrinks
catastrophically.
The exact nature of the dynamics that produce this result are at present unclear,
but it appears that in the cases of catastrophic contraction, the problem arises due
to insufficient means of production being produced, which acts as a constraint on
all subsequent output. If the economy moves to a labour distribution where more
means of production would be used by the current distribution of the labour force
than it can produce, then clearly it must undergo contracted reproduction.
In the case of simulated economies that fail converge on a uniform rate of profit,
one hypothesis is that if sector II has a particularly low organic composition of cap-
ital, then a movement of capital into sector II leads to a net increase in the demand
for labour power. This raises wages and increases demand for sector II, so rather
than the price of necessities falling consequent on inward capital movement, wage
goods may rise in price. Another possibility is that the distribution of profit rates
may undergo oscillations. Further investigation into detailed trajectories of prices
and profit rates of individual sectors would be required to test these hypotheses.
8 Model and reality
We know that real capitalist economies do not often go into catastrophic collapse
due to inadequate production of means of production, though the collapse of in-
dustrial production in the former USSR after conversion to capitalism may be an
26

Table 7: Mean price and value vectors.
Capital goods Wage goods Luxuries
Mean labour values 0.95 1.11 0.93
Mean production price 1.08 1.02 0.90
Mean market clearing price 1.00 0.92 1.07
instance of this. Why is this?
It may be that some version of the Anthropic Principle (Barrow et al., 1988) is
in operation. We do not see these collapses because the collapses are history sen-
sitive, and the economies starting out in technological and income configurations
that would result in collapse are eliminated. That may apply to the former socialist
economies suddenly exposed to a profit maximising principle, they contracted until
the technical structure of the economy changed. The end result would be that at
any given time, the population of capitalist economies would have been purged of
those with technical structures that would lead them to collapse under free capital
movement.
Alternatively the basic market clearing price mechanism that is used in the
model may not be realistic. The model basically assumes unit elasticity, a 1%
fall in output, other things being equal, raises prices by 1%. Perhaps capitalist
economies are only stable against collapse given non-linear price responses.
Instead of looking at the problem of collapse, consider that a substantial frac-
tion of healthy models fail to attain an equal rate of profit. This is less of a problem
since it accords with what we observe in reality. We know that typical capitalist
economies have a dispersion of profit rates(Fröhlich, 2013).
All reproduction schemes meeting the constraints described in Section 3 de-
fine a set of market clearing prices for economies with no credit operations. Real
27

economies have credit and therefore the set of actual market prices we observe will
be less constrained than is implied by reproduction schemes. However, reproduc-
tion schemes do have the virtue that they allow us to generate a large sample of
simple economies and associated price structures sans any assumptions about the
underlying price mechanism of the economy. They allow us to explore the space of
possible self reproducing economies and the price structures associated with them.
The input output tables used in empirical studies are approximations to systems
of simple reproduction. They are only approximations, since they depict economies
that are typically growing, but the growth rate is typically small, and the conven-
tions associated with the construction of input output tables impose similar balance
constraints to those seen in reproduction schemes. The existence of credit transfers
between industries in the IO tables, will however introduce a complication absent
in the simple marxian schemes.
Using unbiased samples from the space of reproduction schemes we can de-
termine the a-priori probability of different pricing theories. That is to say, the
probability that such pricing theories would be true if real economies were dis-
tributed with equal probability over all possible positions in configuration space.
We are assuming, in effect, that if economies undergo a random walk through con-
figuration space, the probability of their transiting from one macro-state to another
is proportional to the volume occupied by these macro-states.
The macro-state defined by market prices being within 10% of labour values
has a similar volume to the macro state with market prices with 10% of prices of
production. A priori, we should expect a reproducing economy to be this close to a
labour value conformant configuration as to a price of production conformant one.
If, on the other hand, there is some bias in the random walk, so that economies
28

end up closer to either of these pricing systems than one would a-priori expect,
then this is analogous to evolution in a space with a potential defined over it. The
discrepancy between observed and a-priori probability distributions should then
enable one, to estimate, via some appropriate negative exponential law, the depth
of potential wells. Conversely one could say how strong the potential field would
have to be to produce a world in which a either labour values or production prices
were the operational laws. Even without a deeper analysis though, it is appears
from these results that the assumption of prices of production as an operational
law implies a weaker potential well favouring it than need be assumed for labour
values. The expected a-priori dispersions of labour values are wider than those for
prices of production. The fact that this is not what is empirically observed implies
that the potential well associated with prices of production is weaker than that
associated with labour values. Possibly this an effect of labour being more mobile
than capital. It is easier for steel workers to move into catering jobs than to convert
steel mills into restaurants. Alternatively, the obstacles to profit rate equalisation
shown in the second experiment may act as a frustrating factor effectively reducing
the potential well around prices of production.
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