A Note on the Use of Vector Space Metrics

It is argued that the vector space measures used to measure closeness of prices and labour values are invalid because of the observed metric of commodity space. An alternative vector space within which such measures do apply is proposed. It is shown that commodity exchange can be modeled by the application of unitary operators to this space. In the recent literature relating to measuring the closeness of price of production vectors to value vectors [2, 3, 4, 5]it has been taken as given that the use of vector space measures is appropriate. I wish to point out that this is at least questionable. 1. THE VECTOR SPACE PROBLEM Vector spaces are a subclass of metric space. A metric space is characterized by a positive real valued metric function δ(p,q) giving the distance between two points, p,q. This distance function must satisfy the triangle inequality δ(p,q) ≤ δ(p,r) + δ(q,r). In vector spaces this metric takes the form: (1.1) δ(p,q) = √ ∑(pi−qi) We have argued elsewhere[1] that the metric of commodity space does not take this form. Let us recapitulate the argument. Conjecture 1.1. Commodity space is a vector space. Assume that we have a commodity space made up of two commodities, gold and corn and that 1oz gold exchanges for 100 bushels of corn. We can represent any agent’s holding of the two commodities by a 2 dimensional vector c with c0 being their gold holding and c1 being their corn holding. Given the exchange ratio above, we can assume that (1,0) and (0,100) are points of equal worth and assuming that commodity space is a vector space thus (1.2) δ((0,0),(1,0)) = δ((0,0),(0,100)) Date: 5.March 2004. 1 A NOTE ON THE USE OF VECTOR SPACE METRICS 2 This obviously does not meet equation 1.1 but if we re-normalise the corn axis by dividing by its price in gold, we get a metric (1.3) δc(p,q) = √ (p0−q0) +( p1−q1 100 )2 which meets the equation we want for our two extreme points: (1.4) δc((0,0),(1,0)) = δc((0,0),(0,1)) If this is our metric, then we can define a set of commodity holdings that are the same distance from the origin as holding 1oz of gold. Let us term this U the unit circle in commodity space: (1.5) U = {a ∈U : δc((0,0),a) = 1} Since these points are equidistant from the origin, where the agent holds nothing, they must be positions of equal worth, and that movements along this path must not alter the net worth of the agent. Let us consider a point on U, where the agent holds 1 √ 2 oz gold and 100 √ 2 bushels of corn. Would this in reality be a point of equal worth to holding 1 oz of gold? No, since the agent could trade their 100 √ 2 bushels of corn for a further 1 √ 2 oz gold and end up with √ 2oz of gold. Thus there exists a point on U that is not equidistant from the origin, hence equation 1.3 can not be the form of the metric of commodity space and thus conjecture 1.1 falls, and commodity space is not a vector space. 2. THE METRIC OF COMMODITY SPACE The metric actually observed in the space of bundles of commodities is: (2.1) δb(p,q) = ∣∣∑αi (pi−qi)∣∣ where p, q are vectors of commodities, and αi are relative values. The ’unit circle’ in this space actually corresponds to a pair of parallel hyperplanes on above and one below the origin. One such hyperplane is the set of all commodity combinations of positive value 1 and the other, the set of all commodity combinations of value -1. The latter corresponds to agents with negative worth, i.e., net debtors. Because of its metric, this space is not a vector space and it is questionable whether measures of similarity based on vector space metrics are appropriate for it. However it is possible to posit an underlying linear vector space of which commodity space is a representation. A NOTE ON THE USE OF VECTOR SPACE METRICS 3 3. COMMODITY AMPLITUDE SPACE We will now develop the concept of an underlying space, commodity amplitude space, which can model commodity exchanges and the formation of debt. Unlike commodity space itself, this space, is a true vector space whose evolution can be modeled by the application of linear operators. The relationship between commodity amplitude space and observed holdings of commodities by agents is analogous to that between amplitudes and observables in quantum theory. Let us consider a system of n agents and m commodities, and represent the state of this system at an instance in time by a complex matrix A, where ai j represents the amplitude of agent i in commodity j. The actual value of the holding of commodity j by agent i , we denote by hi j an element of the holding matrix H. This is related to ai j by the equation ai j = √ hi j. 3.1. Commodity exchanges. We can represent the process of commodity exchange by the application of rotation operators to A. An agent can change the amplitudes of their holdings of different commodities by a rotation in amplitude space. Thus an initial amplitude of 1 in gold space by an agent can be transformed into an amplitude of 1 in corn space by a rotation of 2 . Borrowing Dirac notation we can write these as 1|gold>, and 1|corn>. A rotation of π 4 on the other hand would move an agent from a pure state 1|gold> to a superposition of states 1 √ 2 |gold> + 1 √ 2 |corn> . Unlike rotation operators in commodity space this is value conserving since on squaring we find their assets are now £ 2 gold + £ 1 2 corn. The second conservation law that has to be maintained in exchange is conservation of the value of each individual commodity, there must be no more or less of any commodity after the exchange than there was before. This can be modeled by constraining the evolution operators on commodity amplitude space to be such that they simultaneously perform a rotation on rows and columns of the matrix A. Suppose we start in state: A = ( 1 0 0 2 ) ,H = ( 1 0 0 4 ) Where agent zero has £1 of gold and no corn, and agent one has no gold and £4 of corn. We can model the purchase of £1 of corn by agent zero from agent one by the evolution of A to: