Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Frontier efficiency measurement function

f(y,x,C) = (TE,AE,CE)

Maps a decision-making unit's observed inputs, outputs and costs, and a production or cost frontier estimated for comparable units, to efficiency scores between 0 and 1. A score of 1 places the unit on the estimated frontier. The frontier itself is usually estimated by data envelopment analysis, a linear programming method, or by stochastic frontier analysis, a parametric method that separates inefficiency from random noise.

  • Output-oriented technical efficiency

    TE_O = y / y_max

    Divides the output a unit actually produces by the maximum output that the estimated frontier says is feasible from the same inputs. The shortfall from 1 is the proportion by which output could rise without extra inputs. It measures waste in the production process only and says nothing about whether the right services are produced.

  • Cost efficiency and its decomposition into technical and allocative efficiency

    CE = C_min / C_obs; TE_I = C_tec / C_obs; AE = C_min / C_tec

    Compares three costs of producing the unit's observed output at its input prices: the actual cost, the cost of the technically efficient input bundle found by shrinking all inputs in proportion to the frontier, and the minimum cost with the least-cost mix of inputs. Cost efficiency is the minimum cost over the actual cost; input-oriented technical efficiency is the technically efficient cost over the actual cost; allocative efficiency in this production-frontier sense, the choice of input mix at given prices, is the minimum cost over the technically efficient cost. Cost efficiency equals technical efficiency multiplied by allocative efficiency.

  • Scale efficiency from constant and variable returns frontiers

    SE = TE_CRS / TE_VRS

    Divides technical efficiency measured against a constant returns to scale frontier by technical efficiency against a variable returns to scale frontier for the same unit and orientation. The variable returns score removes the effect of size, so the ratio isolates the part of inefficiency due to operating at a size other than the most productive scale.