Arbitrage-free Barter and the Capitalist Process

AbstractThe present paper is intended as a request for comment. Starting from a formalization of capitalistbarter as synchronous, transparent, and above all transitive barter system, we show that a givensquare matrix represents such a transitive barter system if and only if it is constructed in a specificmanner and is exclusively determined by the valuation of the commodity nodes in the correspond-ing barter graph. We then discuss the conditions that a newly introduced money commodity mustsatisfy in such a model in order to be accepted by the barter partners as a means of synchronizingtwo or more otherwise asynchronous barter transactions. In doing so, it emerges that the discrete-time barter model developed thus far does not suffice to derive such a condition and must thereforebe extended by an additional, time-dependent component. To incorporate the required time-depen-dence into the model, the concept of the value of a commodity proves indispensable. We define it– in contrast to prevailing macroeconomic traditions – as the share of the total portfolio obtainablethrough the sale of one unit of the commodity. This time-dependent concept of value then makes itpossible to grasp the capitalist process as a continuous struggle against the permanent devaluationof commodities, mathematically formalized in a system of first-order linear differential equationswhose system matrix is precisely the barter matrix, and whose solution – owing to the special struc-ture of that matrix – can be given in closed form. The paper concludes with the derivation, from thisdifferential equation, of the aforementioned condition to which the money commodity must be sub-jected if it is to enjoy the kind of performative acceptance it does in reality. PS.: Up to and including Section2, the German word Tausch is translated as barter. After the introduction of money into the model– that is beginning with Section 3 – it is then translated as exchange.