Concept Architecture
Concept
Theoretically, the Malmquist Index is a productivity index that measures changes in total factor productivity between two time periods by decomposing productivity growth into changes in technical efficiency and technological progress. It is founded on production theory, distance functions and index number theory, and is commonly estimated using Data Envelopment Analysis (DEA) or Stochastic Frontier Analysis (SFA). The Malmquist Index enables comparison of productivity over time without requiring price information or a specified production function.
Mathematically, the Malmquist Index is calculated as the geometric mean of two productivity ratios based on distance functions evaluated relative to production frontiers in consecutive periods. The index can be decomposed into an efficiency change component (catch-up effect) and a technological change component (frontier shift). An index value greater than one indicates productivity improvement, a value less than one indicates productivity decline and a value equal to one indicates no change.
In practice, the Malmquist Index is widely applied in health economics to evaluate productivity changes in hospitals, healthcare systems, primary care organisations and public health programmes. It supports longitudinal benchmarking, assessment of healthcare reforms, evaluation of technological innovation and analysis of resource utilisation over time.
Purpose
Used to measure changes in productivity over time, distinguish improvements in technical efficiency from technological progress, benchmark healthcare organisations and support longitudinal health economic analyses.
Mathematical Formulae
Primary Formula
Malmquist Index
M = [D?(x???, y???) � D?(x?, y?) ? D???(x???, y???) � D???(x?, y?)]^(1/2)
where:
- D = distance function
- x = input vector
- y = output vector
- t = initial time period
- t + 1 = subsequent time period
Supporting Formulae
Malmquist Index = Efficiency Change ? Technological Change
Efficiency Change
EC = D???(x???, y???) � D?(x?, y?)
Technological Change
TC = [D?(x???, y???) � D???(x???, y???) ? D?(x?, y?) � D???(x?, y?)]^(1/2)
Related Mathematical Methods
- Data Envelopment Analysis
- Distance functions
- Technical efficiency
- Total factor productivity
- Stochastic Frontier Analysis
- Productivity analysis
Example
A hospital has:
- Efficiency Change = 1.04
- Technological Change = 1.03
Malmquist Index
= 1.04 ? 1.03
= 1.071
The hospital therefore demonstrates a 7.1% improvement in total factor productivity, resulting from both improved operational efficiency and technological advancement.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| PRODUCT | =B2*C2 | Calculate the Malmquist Index from efficiency and technological change components. |
| SQRT | =SQRT((B2/C2)*(D2/E2)) | Calculate the geometric mean component of the index. |
| IF | =IF(F2>1,"Productivity Improved",IF(F2<1,"Productivity Declined","No Change")) | Interpret productivity change. |
| AVERAGE | =AVERAGE(F2:F101) | Calculate the average Malmquist Index across organisations. |
| RANK | =RANK(F2,$F$2:$F$101,0) | Rank organisations by productivity growth. |
VBA (Optional)
Automate calculation of Malmquist Indices using DEA outputs across multiple time periods while generating productivity reports, efficiency benchmarking dashboards and longitudinal healthcare performance analyses.
Sources
- Caves DW, Christensen LR, Diewert WE. The Economic Theory of Index Numbers and the Measurement of Input, Output, and Productivity.
- F�re R, Grosskopf S, Lindgren B, Roos P. Productivity Developments in Swedish Hospitals: A Malmquist Output Index Approach.
- Coelli TJ, Rao DSP, O'Donnell CJ, Battese GE. An Introduction to Efficiency and Productivity Analysis.
- Cooper WW, Seiford LM, Tone K. Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-Solver Software.
- Hollingsworth B. The Measurement of Efficiency and Productivity of Health Care Delivery.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes.
Related Concepts (2)
Library
Publications
1
Productivity Growth in the English National Health Service from 1998/1999 to 2013/2014 — Bojke, Castelli, Grašič, Howdon & Street, Vol. 26, No. 5 ed., 2017 (Health Economics)
The York Centre for Health Economics measurement of NHS productivity growth as a chained index of outputs over inputs across 15 years, the standard methodological reference for English NHS productivity analysis.
Journal ArticleView source →
Frequently Asked Questions (6)
What is the Malmquist index?
A productivity measure decomposing efficiency change over time into genuine technical progress versus change relative to an evolving best-practice frontier.
Source: Färe, Grosskopf, Norris & Zhang 1994
What change over time does the Malmquist index decompose?
The Malmquist index is a productivity measure decomposing efficiency change over time. It decomposes the change in productivity between periods into its underlying sources. It separates two components: genuine technical progress, the frontier itself advancing, and change relative to that evolving best-practice frontier, a unit catching up or falling behind. It is a productivity measure, gauging how productivity shifts over time rather than at one moment. It builds on data envelopment analysis, using its best-practice frontier as the reference the index tracks. Splitting productivity change into its sources is what it does. Fare, Grosskopf, Norris and Zhang (1994) set this out.
Source: Fare, Grosskopf, Norris & Zhang 1994
What does the Malmquist index decompose?
The Malmquist index decomposes efficiency change over time, so it breaks that change into genuine technical progress versus change relative to an evolving best-practice frontier. This decomposition of efficiency change defines it. So the Malmquist index is a productivity measure decomposing efficiency change over time into genuine technical progress versus change relative to an evolving best-practice frontier These two components of technical progress and change relative to the frontier are what the Malmquist index separates.
Source: Färe, Grosskopf, Norris & Zhang 1994
What two components does the Malmquist index separate?
The Malmquist index separates genuine technical progress from change relative to an evolving best-practice frontier, so it distinguishes real advancement from a unit's movement toward or away from the shifting frontier, in decomposing efficiency change over time. These two components define it. So the Malmquist index is a productivity measure decomposing efficiency change over time into genuine technical progress versus change relative to an evolving best-practice frontier.
Source: Färe, Grosskopf, Norris & Zhang 1994
What kind of measure is the Malmquist index?
The Malmquist index is a productivity measure, so it is a measure of productivity that decomposes efficiency change over time into genuine technical progress versus change relative to an evolving best-practice frontier. This status as a productivity measure defines it. So the Malmquist index is a productivity measure decomposing efficiency change over time into genuine technical progress versus change relative to an evolving best-practice frontier This building on a best-practice frontier is what links the Malmquist index to the frontier data envelopment analysis identifies.
Source: Färe, Grosskopf, Norris & Zhang 1994
How does the Malmquist index relate to data envelopment analysis?
The Malmquist index relates to data envelopment analysis in that it builds on a best-practice frontier like the one data envelopment analysis identifies: the Malmquist index decomposes efficiency change over time relative to an evolving best-practice frontier, and data envelopment analysis identifies a best-practice frontier all units are measured against. So the Malmquist index extends frontier analysis over time, connected in that both use a best-practice frontier to gauge efficiency.
Source: Färe, Grosskopf, Norris & Zhang 1994
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 31 Mar 2026
Content version: 1.0.0
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