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The principle of least effort is believed to be a universal rule for livingsystems. Its application to the derivation of the power law probabilitydistributions of living systems has long been challenging. Recently, a measureof efficiency was proposed as a tool of deriving Zipf s and Pareto s lawsdirectly from the principle of least effort. The present work is a furtherinvestigation of this efficiency measure from a mathematical point of view. Theaim is to get further insight into its properties and usefulness as a metric ofperformance. We address some key mathematical properties of this efficiencysuch as its sign, uniqueness and robustness. We also look at the relationshipbetween this measure and other properties of the system of interest such asinequality and uncertainty, by introducing a new method for calculatingnon-negative continuous entropy.

Study of a measure of efficiency as a tool for applying the principle of least effort to the derivation of the Zipf and the Pareto laws