Where does efficiency collide?

An allocation of goods and resources is efficient when no one can be made better off without making at least one other person worse off: this is Pareto efficiency, the state welfare economics treats as the mark of a distribution with no wasted gains from trade. Debreu's 1954 Valuation equilibrium and Pareto optimum proved a formal link between competitive price equilibria and such optima, one half of what are now called the fundamental theorems of welfare economics. The referent is a property of a whole distribution of resources across people, judged only by whether any costless improvement remains, and it says nothing about heat and work in an engine, the running time of an algorithm, or output per unit input on a factory floor.[1]

Unit

One allocation of resources across people, ranked only as efficient or not

Recovery means

Not applicable

Measured by

Assessed against the fundamental theorems: whether a feasible reallocation could help someone at no one's cost

The efficiency of a heat engine or process is the ratio of useful work output to the energy supplied to it, a pure number between zero and one that records how much of the input is converted rather than dissipated. Carnot's 1824 Réflexions sur la puissance motrice du feu showed that this ratio has an upper bound fixed only by the temperatures between which the engine works, the Carnot limit that no real engine can exceed. The referent is a physical ratio of work to heat governed by the laws of thermodynamics, not a property of a resource allocation, a measure of how an algorithm scales, or a plant's output relative to its peers.[2]

Unit

A dimensionless ratio of useful work out to energy in, from zero to one

Recovery means

The unconverted input leaves as waste heat to the cold reservoir and cannot be fully recovered as work

Measured by

Computed as work output divided by heat or fuel energy input, and compared against the Carnot bound

The efficiency of an algorithm is how its running time or memory grows as the size of the input grows, described in asymptotic terms such as big-O so that constant factors and small cases fall away. Hartmanis and Stearns's 1965 On the computational complexity of algorithms put this on a formal footing, measuring the resources a machine needs as a function of input length and grouping problems by how those resources scale. The referent is a growth rate of computational cost, a mathematical property of a procedure, not a ratio of work to heat, a property of an economic allocation, or a firm's benchmarked output.[3]

Unit

A growth rate of time or space as a function of input size, in big-O terms

Recovery means

Not applicable

Measured by

Derived by analysing the algorithm's steps, or estimated empirically by timing runs across input sizes

Technical or productive efficiency asks how much output a unit produces from its inputs compared with the best that similar units achieve, so a fully efficient unit sits on the observed frontier and an inefficient one falls short of it. Farrell's 1957 The measurement of productive efficiency framed efficiency as distance from a best-practice frontier, and Charnes, Cooper, and Rhodes's 1978 Measuring the efficiency of decision making units turned this into data envelopment analysis, a method that scores each unit against a frontier built from the others. The referent is a benchmarked ratio of outputs to inputs for an organisation or process, relative to observed peers, not a heat engine's work ratio, an allocation with no wasted gains, or how an algorithm's cost scales.[4][5]

Unit

A score of outputs over inputs for one unit, relative to a best-practice frontier

Recovery means

Not applicable

Measured by

Estimated by data envelopment analysis or stochastic frontier methods against comparable units