Signature
CE = C_min / C_obs; TE_I = C_tec / C_obs; AE = C_min / C_tec
| Inputs | Definition | Unit |
|---|---|---|
C_min | Lowest cost of producing the observed output at the same input prices with the least-cost input mix | currency per period |
C_obs | Observed inputs valued at the unit's input prices, greater than zero | currency per period |
C_tec | Observed inputs reduced in equal proportion to the frontier, valued at the same prices | currency per period |
CE | Minimum cost of the observed output as a share of actual cost | score from 0 to 1 |
|---|---|---|
TE_I | Cost of the proportionally reduced input bundle on the frontier as a share of actual cost, equal to the proportion to which all inputs could be reduced | score from 0 to 1 |
AE | Minimum cost as a share of the cost of the technically efficient bundle, reflecting the input mix given input prices | score from 0 to 1 |
Function
Frontier efficiency measurement function
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.
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Implementations
Excel
Three efficiency scores from three costs
With actual cost in ActualCost, technically efficient cost in TechEfficientCost and minimum cost in MinimumCost, the three formulas return cost, technical and allocative efficiency.
=MinimumCost/ActualCost; =TechEfficientCost/ActualCost; =MinimumCost/TechEfficientCost
Assumptions
Input prices known and common to the comparison
Allocative and cost efficiency require input prices. Units are compared at their own prices, and differences in prices between units are not treated as inefficiency unless a price efficiency term is added.
Input orientation and proportional reduction
Technical efficiency here reduces all inputs in the same proportion at fixed output. The decomposition CE = TE_I multiplied by AE holds for this input-oriented measure; an output-oriented technical efficiency score is not combined with cost-based measures.
Frontier-literature meaning of allocative efficiency
AE here is the least-cost choice of inputs at given prices. It differs from allocative efficiency across programmes, the distribution of resources among competing uses to best meet the stated objective.
Worked examples
Hospital with technical and input-mix inefficiency
A hospital spends 2,000,000 pounds a year. Reducing all its inputs in proportion to the frontier would cost 1,700,000 pounds, and the least-cost input mix for the same output would cost 1,530,000 pounds. Cost efficiency is 0.765, technical efficiency 0.85 and allocative efficiency 0.9, and 0.85 × 0.9 = 0.765. The figures are illustrative.
C_obs = 2000000; C_tec = 1700000; C_min = 1530000; CE = 0.765; TE_I = 0.85; AE = 0.9
Common errors
Reading input-mix allocative efficiency as programme allocation
A score of 0.9 for AE says that the hospital's input mix costs 10% more than necessary at its prices. It says nothing about whether resources are allocated to the right services or programmes.
Comparing scores across different frontiers
Scores are relative to the frontier of the sample, inputs, outputs and model used. A score of 0.85 in one study cannot be compared with 0.85 in another built from different units or variables.
Sources
Farrell decomposition of overall efficiency
Farrell MJ. The measurement of productive efficiency. Journal of the Royal Statistical Society, Series A (General). 1957;120(3):253-290. Overall (cost) efficiency as the product of technical efficiency and price (allocative) efficiency, measured radially against an efficient isoquant.
Textbook on cost, technical and allocative efficiency
Coelli TJ, Rao DSP, O'Donnell CJ, Battese GE. An Introduction to Efficiency and Productivity Analysis. 2nd ed. New York: Springer; 2005. Farrell's input-oriented measures of technical, allocative and cost efficiency, their decomposition CE = TE × AE, and estimation by data envelopment analysis and stochastic frontiers.
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