Values Law Values Metric

by

P Co ckshott A Cottrell

Research Rep ort

RR Novemb er

Values Law Values Metric

W Paul Co ckshott and Allin F Cottrell

septemb er

Abstract

It is argued that the metric space of exchanging commo dities is noneuclidean

and characteristic of a system governed by a conservation law The p ossible can

didates for what is conserved in commo dity exchange are reviewed with reference

to inverted inputoutput matrices of the British economy Strong evidence is pre

sented that the conserved substance is lab our The arguments of Mirowski and

others regarding the appropriateness of such physicalist arguments are discussed

What is meant by the law of value

The phrase law of value is little used by Marx but p opular among his followers

It has no precise denition of the typ e that one would exp ect for a scientic law

Laws such as Ho okes law or Boyles law have a concise denition that any chemist or

physicist could rep eat but it is doubtful if anywhere in the Marxist literature there

exists a comparable denition of the law of value

On the basis of what Ricardo and Marx wrote on the theory we would advance

the following as a reasonable denition

The law of value states that value understood as the labour time social ly necessary to

produce a commodity is conserved in the exchange of commodities

The advantages of this denition are that it is cast in the normal form of a sci

entic law it is empirically testable it has a precise meaning and it emphasizes the

fundamental Marxian prop osition that value cannot arise in circulation

In order to justify this formulation of the law we will rst take a new lo ok at

what Marx Chapter Section called the valueform and then review the

growing b o dy of empirical evidence that justies the law



Department of Computer Science University of Strathclyde and Department of Economics Wake

Forest University

RR

Metric spaces

1

Instead of arguing ab out the valueform or exchangevalue in Hegelian terms we will

use geometric concepts This approach we b elieve enables us to p ose the problem

of exchangevalue with greater generality and at the same time greater concision It

will b e necessary to b egin with a few denitions

A metric space S d is a space S together with a realvalued function d S S

which measures the distance b etween pairs of p oints p q S where d ob eys the

following axioms

Commutation

dp q dq p

Positivity

dp q if p q

Selfidentity

dp p

Triangle inequality

dp q dp r dr q

Examples of metric spaces

Euclidean space This is the familiar space of planar geometry If p and q are two

p oints with co ordinates p p and q q resp ectively then the distance b etween

1 2 1 2

these p oints is given by the pythagorean metric

q

2 2

d

1 2

where p q i It extends to multidimensional vector spaces as

i i i

q

2 2

2

d

n

1 2

2

Manhattan space Socalled after the Manhattan street plan the metric is simply the

sum of the absolute distances in the two dimensions

d j j j j

1 2

1

In the p ostface to the second edition of Capital Marx p noted that he had co quetted

with the mo de of expression p eculiar to Hegel in the chapter on the theory of value

2

This is also known as a Minkowski metric

Values Law Values Metric

'$

x

y

&%

Figure Equality set in Euclidean space

x

y

Figure Equality set in Manhattan space

Equality op erations in metric spaces

Let us dene two p oints q r S to b e equal with resp ect to p if they are equidistant

from p under the metric d Formally

q r if dp r dp q

p

Given an equality op erator E and a memb er q of a set S we can dene an equality

subset that is to say the set whose memb ers are all equal to q under E The equality

set of q under using the Euclidean space metric is shown in Figure while

p

Figure shows the corresp onding equality set under the Manhattan metric

Commo dity bundle space

What it may b e asked has all this to do with value Well value is a metric on

commo dities To apply the previous concepts we dene commodity bund le space as

follows A commo dity bundle space of order is the set of pairs ax by whose elements

are a units of commo dity x and b units of commo dity y A commo dity bundle space

of order is the set of triples ax by cz whose elements are bundles of a units of x

RR

corn

6

p q r s

-

iron

Figure Points equidistant with e iron f corn from a iron b corn in Manhattan

space

b units of y c units of z and so on

Consider for example the commo dity bundle space of order comp osed of bundles

of iron and corn The set of all p oints equidistant with e iron f corn from a iron

b corn under the Manhattan metric is shown in Figure

We have a distinct equality op erator for each p oint p p iron p corn

p 1 2

in our corniron space Let us consider one particular equality op erator that which

denes the equality set of p oints equidistant from the origin Whichever metric

(00)

we take so long as we use it consistently each p oint in the space b elongs to only

one such equality set under the given metric These equality sets form an ordered

set of sets of the space It follows that any of the metrics could serve as a system of

valuation conceived as a partial ordering imp osed up on all bundles This is shown in

Figure Both the diamonds and the conventional circles are in the relevant space

circles the diamonds are circles in Minkowski or Manhattan space

We now advance the hyp othesis that if the elements of a set of commo dity bundles

are mutually exchangeablethat is if they exchange as equivalentsthen they form

an equality set under some metric If this is valid then by examining the observed

equality sets of commo dity bundles we can deduce the prop erties of the underlying

metric space

The metric of commo dity bundle space

What is the metric of commo dity bundle space The observed sets of exchangeable

bundles constitute the isovalent contours or isovals in commo dity bundle space We

nd in practice that they are straight linesknown to economists as budget lines

Values Law Values Metric

corn

6

'$





-



iron



&%

Figure The ordering of equality sets under p ossible metrics

corn

6

isovals



9

-

iron

Figure Observed form of the isovals in commo dity bundle space

see Figure Note that these extend b eyond the axes Why we may ask are they

not circles centered on the origin Commo dity space clearly has a nonEuclidean

and for what it is worth a nonManhattan geometry but why Before attempting an

answer to this question it will b e useful to make some preliminary p oints

We will call commo dity bundle spaces ob eying the observed metric of exchange

value as displayed in the economists budget lines commodity value space whereas

a commo dity bundle space ob eying a Euclidean metric we will call commodity vector

space Although our examples have applied to spaces of order two the argument

can b e extended to arbitrary hyp erspaces There is something very particular ab out

the metric of commo dity value space namely d j j where and are

x y

constants This metric o ccurs elsewherefor instance in energy conservation

Consider Figure the graph of p osition versus velo city for a b o dy thrown up and

then falling All p oints on the tra jectory are freely exchangeable with one another in

RR

Altitude

up

down

Vertical velo city

Figure Points in phase space traversed by a pro jectile thrown upward in a gravita

tional eld

Altitude

6

-

2

v

Figure Points in the space of altitude velo city squared traversed by the particle

shown in previous gure

the course of the timeevolution of the system They may therefore b e treated as an

equivalence set This do es not lo ok like the equivalence set of commo dity value space

until we square the velo city axis This yields the diagram in Figure which lo oks

very much like the budget line in Figure By squaring the velo city axis we obtain

a measure prop ortional to what the physicists term kinetic energy But this kinetic

energy is only revealed through its exchange relation with height Physics p osits a

onedimensional substance energy whose conservative exchange b etween dierent

forms underlies the phenomena

Conjugate isovals

Lo oking more closely at the metric we have deduced for commo dity value space we

can see that our representation of the equality sets as budget lines is only half the

story

Values Law Values Metric

corn

6

P

Q

R

-

iron

R

Q

P

Figure Conjugate pairs of isovals

Let and in the metric d j j Taking the p oint Q

x y

in Figure we can show its equality set with resp ect to the origin as the line P QR

along with its extension in either direction All such p oints are at distance from the

origin But by the denition of the metric the p oint Q is also at distance

from the origin There thus exists a second equality set on the line P Q R on the

opp osite side of the origin In general for a commo dity bundle space of order n there

will b e a conjugate pair of isovals forming parallel hyp erplanes of dimension n in

commo dity vector space

If the p ositive isoval corresp onds to having p ositive net wealth its conjugate cor

resp onds to b eing in debt to the same amount There is an obvious echo of this in the

practice of doubleentry b o okkeeping the eect of which is to ensure that for every

credit entry there exists a conjugate debt entry

Points on an isoval and its conjugate are equidistant from the origin but not

exchangeable with one another If I have a credit of dollar I will not readily

exchange it for a debt of dollar This is reected in the fact that p oints on an isoval

may not b e continuously deformed to a p oint on its conjugate isoval whereas they

may b e continuously deformed within the isoval In other words the isovalent set is

top ologically disconnected

Contrast this with what o ccurs on a Euclidean metric The p oints Q and

p

Q lie on a circle of radius along which we may uninterruptedly move

from one to the other The disconnected character of the isovalent set in commo dity

value space b ecomes understandable once we realize that this space is a pro jection of

a one dimensional space into an n dimensional one As such its unit circles comprise

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disjoint planes corresp onding to the two disjoint p oints of the unit circle in onespace

It is this characteristic of b eing multidimensional pro jections of onespace that marks

conservative systems

Implications for value theory

If value were just a matter of providing an ordering or ranking of combinations of

go o ds then a Euclidean or indeed any other metric would pass muster It is some

additional prop erty of the system of commo dity pro duction that imp oses this sp ecic

metric characteristic of a system governed by a conservation law This ts in rather

nicely with the lab our theory of value where so cial lab our would b e the emb o died

substance conserved during exchange relations which in turn provides us with some

justication for casting the law of value in the form of a classical conservation law

So far however this is merely a formal argument the form of the phenomena

is not inconsistent with a conservation relation To justify our formulation we must

explain why the phenomena are such as to conform to a linear conservation law

show that such a law holds empirically and rule out other p otential value

substances as alternatives to lab our

Why commo dity space is nonEuclidean

Spatial metrics are so much part of our mo de of thought that to imagine a dierent

metric is conceptually dicult Most of us have diculty imagining the curved space

time describ ed by relativity theory Euclidean metrics b eing so ingrained in our minds

Conversely when lo oking at commo dities a nonEuclidean metric is so ingrained that

we have diculty imagining a Euclidean commo dity space

But it is worth the eort of trying to imagine a Euclidean commo dity space what

we referred to earlier as commo dity vector space By bringing to light the implicit

contradictions of this idea we get a b etter idea of the underlying reasons why value

takes the particular form that it do es

Is a Euclidean metric for commo dity space internally consistent In commo dity

bundle space of order the Euclidean isovals take the form of circles centered on the

origin In higherorder spaces they take the form of spheres or hyp erspheres We

assume in all cases that some linear scaling of the axes is p ermitted to convert them

into a common set of units Let us supp ose that the economic meaning of these

isovals is that given any pair of p oints p q on an isoval the bundle of commo dities

Values Law Values Metric

represented by p will b e exchangeable as an equivalent with the bundle represented

by q

If the state of an economic agent is describ ed by his p osition in this commo dity

bundle space then the set of p ermissible moves that can b e made via equivalent

exchanges is characterized by unitary op erators on commo dity vector space The set

of equivalent exchanges of p is fjpj u such that juj g ie the radiuspreserving

3

rotations of p Mathematically this is certainly a consistent system

But economically such a system would break down It says that I can exchange

one appropriately dened unit of corn for one unit of iron or for any equivalent

1 1

p p

combination such as iron corn But then what is to stop me carrying out

2 2

the following pro cedure

1 1

p p

Exchange my initial unit of corn for iron plus corn

2 2

1 1

p p

iron for corn giving me corn Now sell my

2 2

p

2

p

Add my two bundles of corn together to give a total of of corn in

2

total

I end up with more corn than I had at the start so this cannot b e a set of

equivalent exchanges Within the context of the Euclidean metric the second step

is illegal since it involves op erating up on one of the co ordinates indep endently But

in the real world commo dities are physically separable allowing one comp onent of

a commo dity bundle to b e exchanged without reference to others It is this physical

separability of the commo dities that makes the observed metric the only consistent

one

The existence of a commo ditypro ducing so ciety in which the individual comp o

nents of the wealth held by economic agents can b e indep endently traded selects out

of the p ossible value metrics one consistent with the law of value In a hyp othetical

so ciety in which commo dity bundles could not b e separated into distinct comp onents

and exchange ob eyed a Euclidean metric the lab our theory of value could not hold

However there are several p ossible conservative value systems consistent with the ob

served metric That which is conserved in exchange might b e something other than

so cially necessary lab our time

3

A very similar mo del is used in one of the standard formulations of quantum theory to describ e

p ossible state transformations von Neumann

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Table Price regressions UK

constant

lab our value

pr of pro d

N

2

R

Figures in parentheses are t ratios All variables in log

arithmic form Data source Central Statistical Oce

Evidence for lab our value conservation

Following the work of Shaikh there is now a considerable b o dy of econometric

evidence in favour of the prop osition that relative prices and relative lab our values

are highly correlated or in other words in favour of the law of value as dened ab ove

Petrovic Ochoa Valle Baeza Co ckshott Cottrell and Michaelson

forthcoming The general pro cedure in these studies has b een to use data from

national inputoutput tables to calculate the total lab our content of the output of

each industrial sector and then to run a crosssectional regression of the aggregate

money price of output sector by sector on total lab our content Shaikh

explains the details of the pro cess and also oers a theoretical argument in favour

of a logarithmic sp ecication of the pricevalue regressions These studiesutilizi ng

data from the United States Italy Yugoslavia Mexico and the UKhave pro duced

2

remarkably consistent results with R s of well over It is also noteworthy that

there is very little dierence in predictive p ower over prices b etween lab our values

and prices of pro duction

4

Our own ndings from UK data are presented for reference in Table In the

published inputoutput tables the lab our input is expressed in Equation uses

lab ourvalue gures calculated on the assumption of a dummy wagerate of p er

hour for all industries This is equivalent to assuming that any wage dierentials across

industries reect dierential rates of valuecreation p er clo ck hour Equation is the

4

For further details regarding these estimates see Co ckshott Cottrell and Michaelson forthcoming

Values Law Values Metric

same as except for the exclusion of the oil industry which is an outlier in the price

value regressions presumably due to the high rent comp onent in the Ricardian sense

in oil extraction Equation which again excludes the oil industry uses lab our

value gures calculated using wages data from the New Earnings Survey to convert

backwards from wages to hours for each industrya correction relative to equation

if and only if interindustry wage dierentials are the pro duct of extraneous factors

and do not reect dierential rates of valuecreation Finally equation substitutes

prices of pro duction calculated via a recursive pro cedure for lab our values again

excluding oil

As can b e seen from the equation estimates simple lab our values pro duce an

2

R of nearly p ercent when the oil sector is excluded and the dummy uniform wage

5

is adopted Prices of pro duction improve on this p erformance only marginally

Alternative value bases empirical evidence

As remarked earlier however the question arises as to whether one could pro duce

equally go o d results using something other than lab our time as the basis of value

The empirical answer to this question seems to b e negative as shown in Table For

the purp oses of these regressions we used the Leontief inverse of the UK inputoutput

tables Central Statistical Oce Table to calculate the total direct plus

indirect electricity content oil content and iron and steel content of the output of

each industrial sector Using the same metho dology as in Table based on Shaikh

we then regressed aggregate price on these various values b oth singly and

in combination with lab our values in logarithmic form The sample size is for

each of these regressions the electricity industry b eing excluded from the equations

including electricitycontent and similarly for oil and iron and steel

From equations and it can readily b e seen than none of the alterna

2

tives taken alone p erforms anything like as well as lab our The highest R at is

obtained for electricity content as against for lab our in equation of Table

Equations and show how the alternatives p erform when entered along

side lab our values enabling us to address the question of whether the alternatives

contain any indep endent information or in other words oer any marginal predictive

p ower over prices when lab our content is given Only oil content passes this test

5

It should b e noted that due to data limitations our prices of pro duction are calculated on a ow

basisthey are prices consistent with the equalization of the rate of prot on the total ow outlay on

current inputs Ochoa has calculated prices of pro duction on a sto ck basis for the USA and

nds them to b e slighly less well correlated with actual prices than are simple lab our values

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Table Regressions of price on lab our values

and some alternative valuebases

constant

lab our

electricity

oil

iron and steel

2

Adjusted R

Figures in parentheses are t ratios All variables in logarithmic form Data

source Central Statistical Oce

From the t ratios in parentheses b elow the co ecient estimates it can b e seen that

while lab our content retains its statistical signicance in all cases electricity content

and iron and steel content b ecome statistically insignicant in the presence of lab our

content The fact that oil content contains some indep endent information regarding

prices is presumably linked to the element of rent in the price of oil The North Sea

elds are not marginal which means that the lab our time taken to extract North Sea

oil is less than the so cially necessary amount on a world scale The price of oil b eing

determined on the world market UK oil will then sell at a price ab ove that which

corresp onds to its particular lab our content

Table oers another p ersp ective on this issue It rep orts the co ecients of

variation standard deviation divided by the mean across the sectors in the UK

inputoutput tables for x content p er s worth of output where x equals lab our

electricity oil and iron and steel resp ectively This is the basic information supplied by

6

the inputoutput tables in Tables and it is worked up into regression format but

it is worth considering raw Clearly to the extent that x is conserved in exchange

one will nd a relatively small co ecient of variation for x content p er of sales

From the second column of Table we see that the co ecient of variation is almost

four times as large for electricity as for lab our with those for oil and iron and steel

6

That is x content p er s worth of output is multiplied by the total monetary value of output to

yield total x content on which the total monetary value of output is then regressed in log form

Values Law Values Metric

Table Co ecients of variation

for x content p er of output

Co ecient CV relative

x of variation to lab our

lab our

electricity

oil

iron and steel

Source Calculated from Central Statistical

Oce Table Lab our gures calcu

lated recursively by authors

b eing greater still

The theoretical problems of alternative value bases

Apart from the fact that alternative candidates for that which is conserved in com

mo dity exchange show a relatively p o or p erformance empirically compared to lab our

time there are also some theoretical problems with such alternative value systems

The rst problem is denitional Consider for example the system in which the

value of a commo dity is dened as the amount of oil used to pro duce it directly or

indirectly What then is the value of oil itself We must either say it is unity or it

7

is the amount of oil required to produce a unit of oil which must b e less than one

barrel p er barrel for a viable oil industry

In the latter case we nd that if we use the normal metho d of computing the

P

value of an industrys inputsthat is v v x where v is the p er unit value of

j i ij i

commo dity i and x the amount of the ith commo dity needed to pro duce one unit

ij

of commo dity j we get a recursive denition of the value of oil whose xed p oint is

8

a value of This then means everything else takes on a value of zero Value would

then b e trivially conserved in all exchange transactions

In the former case we give an ad ho c denition of the value of oil as unity This is

7

As suggested by Farjoun and Machover pp in the course of their discussion of

p ossible alternative value systems

8

The recursive pro cedure is as follows where x is the commo dity selected as the valuebase In

the rst round calculate the value of each commo dity as its direct x content In the nth round

n calculate the value of each commo dity as the sum of the values of its inputs as

calculated at round n Recursion terminates when the values change by less than some preset

small amount from one round to the next If the oilvalue of a barrel of oil is less than one barrel

then after a few rounds of this pro cess all values will start shrinking monotonically

RR

arbitrary since it involves treating oil dierently from other inputs such as electricity

or corn but we do have something empirically testable If oil values are conserved

then barrel of oil should purchase commo dities that required barrel of oil for their

pro duction We know this is not true in practice for were it so the revenue obtained

from selling barrel of oils worth of corn would b e entirely consumed in purchasing

the oil and other means of pro duction required to grow it Prots would have to b e

zero

How one may ask is this problem avoided in the case of lab our time In the

history of so cialist economic thought two solutions have b een prop osed Ro db ertus

in eect argued that lab our time has a value of unity but that lab our is sold

by the worker to the capitalist at less than its value Ro db ertus subscrib ed to a

basically Malthusian iron law of wages according to which wages could never rise

ab ove subsistence level for any extended p erio d Alternatively one could ground the

claim that lab our is sold b elow its value on the argument that their monop oly of the

means of pro duction enables the capitalist class to enforce unequal exchange on the

workers Marx of course solved the problem in a dierent way by distinguishing

b etween lab our the activity and lab ourp ower the capacity to work According to

Marx it is lab ourp ower not lab our that is sold by the worker to the capitalist and

lab ourp ower sells at its valuethat is the total lab our time necessary to pro duce

and repro duce the workers capacity to work

Neither of these strategies makes much sense for a commo dity such as oil In

relation to Ro db ertuss variant there is no reason to supp ose that the oil industry

is forced to sell its pro duct b elow value while in relation to Marxs it is hard to

9

see how a parallel distinction b etween oil and oilp ower could b e motivated The

fundamental p oint in the background to these ob jections is that commo dities such as

oil electricity and so on are just ordinary pro ducts of capitalist industry There is

no go o d reason to single any one of them out for asymmetrical treatment Lab our

or lab ourp ower on the other hand is the only commo dity that is a essential to

the working of any capitalist economy while b not itself pro duced under capitalist

conditions of pro duction Put dierently the agent selling oil is an ordinary capitalist

facing other capitalists at par but the agent selling lab our is a worker separated from

the means of pro duction and hence facing the capitalist at a disadvantagea p oint

9

Marxs distinction draws attention to the imp ortant p oint that having hired lab ourp ower for a

day the actual amount of lab our extracted by the capitalist is not yet determined this will dep end

on the outcome of struggles over the length of the working day and the intensity of lab our

Values Law Values Metric

10

made by Adam Smith almost as emphatically as by Karl Marx

Value substance versus eld

It will not have escap ed the readers notice that there is a physicalist avour to

our argument Mirowski has recently had a go o d deal of inno cent fun with

the prop ensity of economists to emulate the queen of the sciences His critique is

directed mainly against the utility theorists but he do es devote some attention to

Marx accusing him of vacillating b etween a eld and a substance theory of value and

in the context of the transformation problem of having one conservation principle

to o many

The last accusation is valid but contrary to the Sraans we b elieve that on

b oth empirical and theoretical grounds see Farjoun and Machover it is the

equalization of the rate of prot that must go Mirowskis accusation with regard to

the contradiction b etween eld and substance theories is relevant to our formulation

however since it may app ear that we have used a substance denition of value in our

theoretical discussion and then a eld theory for our empirical test We will attempt

to show that the distinction b etween eld and substance theories is more subtle than

Mirowski suggests and that the empirical tests in the literature are not invalidated

by this distinction

By the eld version of value theory Mirowski means the denition of value as cur

rent so cially necessary as opp osed to historically emb o died lab our He takes as his

formal mo del the nowstandard mathematical account of the determination of lab our

values as oered by authors such as Morishima and Steedman But it is

a little unfair to pro ject these twentiethcentury formulations based up on the math

ematics of inputoutput tables back onto Marx Marx gave no precise mathematical

formulation of the concept of so cially necessary lab our The standard mo dern formu

lation is just one among several p ossible denitions of so cially necessary lab our and

it involves some very unrealistic assumptions If these assumptions are dropp ed and

the mo del made more realistic the distinction b etween eld and substance theories

vanishes

10

For this reason we disagree with the conclusion reached by Ro emer namely that Marxists

have no go o d reason to b e interested in the exploitation of lab our It is quite true as Ro emer states

that any viable industry must satisfy the condition that its pro duct x requires for its pro duction a

total of less than one unit of x But this do es not mean as he infers that there is nothing sp ecial

ab out lab our from a theoretical p oint of view

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The standard metho d of deriving lab our values from the linear inputoutput equa

tions is based up on the assumption that pro duction takes place instantaneously Marx

did not assume this indeed he devoted much of Volume II of Capital to analysing

the turnover times of capital Any pro cess of determination of prices must op erate

in time through actual pro duction pro cesses It is so cially necessary that the steel

11

used in the keel of a ship completed to day was pro duced a year or two earlier The

so cially necessary lab our in steel pro duced a year ago may dier from that which go es

into steel to day but only the former can aect the value of the ship No real pro cess

allows instantaneous information transfer and market economies are no exception to

this rule If one were to lo ok for a physical analogy applying the Morishima equations

under technological change would b e like trying to solve an electro dynamic problem

with electrostatics

The value of the ship will b e aected by the value of steel when it was purchased

assuming it was not purchased unnecessarily early It will also b e aected indirectly

by the value of steel at a still earlier p erio d when steel was purchased to make the

to ols used to build the ship Generalizing v the value of commo dity j is aected by

j

v the value of commo dity i at a series of past dates These eects will b e mediated

i

by coupling co ecients k k corresp onding to the fraction of v that is made up

1 2 j

of the v s at times t t Thus if by v we mean the comp onent of v that

i j i j

is due to the input of commo dity i b oth directly and indirectly we have a dierence

equation of the form

v k v k v

j it 1 it1 2 it2

Expressed in continuous terms this corresp onds to a dierential equation of the form

2

dv dv v d

j i i

v

1 i 2 3

2

dv dt dt

i

which gives us a highly nonlinear eld theory There is no contradiction b etween this

and the substance theory

Are we then justied in using what are basically the Morishima equationsthat

is the wrong eld equationsin our verication of a substance theory of value Yes

b ecause for the constants k we have k k k and similarly

1 2 3 2 3

will all b e very much smaller than One can get a feel for their likely scale by

1

examining the Leontief inverse of the UK inputoutput table Even when we take two

11

We owe this p oint to Alan Freeman

Values Law Values Metric

P

k We highly crosslinked industries like steel and shipbuildi ng we nd

i

i

are thus entitled to assume that although there will b e some errors in our estimation

of values due to using linear eld equations these will b e small relative to noise As

a conservation law the law of value is sto chastic and obviously do es not hold to the

same precision as natural conservation laws though it should b e noted that on an

appropriate scale these to o are sto chastic due to quantum eects

Conclusion

We have argued that several dierent metrics for the valuation of bundles of com

mo dities are p ossible in principle most of them logically incompatible with the idea

that any scalar quantity is conserved in exchange But the fact that individual com

mo dities are separable and separately tradable imp oses one particular metric cor

resp onding to what we called commo dity value spaceand this metric is consistent

with a conservation law This formal argument do es not in itself prove that any iden

tiable substance is in fact conserved nor do es it establish the credentials of lab our

time as prime candidate for conservation That is an empirical matter and we have

shown that the conservation of so cially necessary lab our time holds as a fairly close

approximation in fact Alternative candidates for conservation in exchange fare much

less well empirically and b esides involve theoretical problems that can plausibly b e

circumvented in the case of lab our The RicardianMarxian law of value may b e

given a precise denition on which moreover it turns out to b e valid

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