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