Entropy and Accounting

Abstract

In this paper we establish an equivalence between accounting numbers and information. The information numbers are denominated in entropy which is a measure of uncertainty and is a function of probabilities only. Mutual information is defined as the reduction of entropy when an information source X is available. We state that the log of one plus the accounting rate equals to the sum of the risk free rate and mutual information. Three conditions are sufficient: (i) long run decision frame; (ii) arbitrage free prices; and (iii) accounting valuation approximating discounted cash flows while the discount rate is the sum of risk free rate and mutual information. We derive two versions of the equivalence; they differ only in the measure of mutual information. One version is for when states of the world are observable and mutual information is the reduction of classical or Shannon entropy. If all possible states of the world are observable (and contractible), then all transactions can be consummated in the marketplace. Hence, firms, and the accounting for firms, might not arise. We extend the results by modeling state unobservability using quantum measurement techniques. The equivalence relation remains the same with quantum measurement probabilities substituted for classical probabilities and mutual information is the reduction of quantum entropy. The classical state observable case is seen to be a special case of the quantum equivalence.