# The Malmquist index

## The Malmquist Productivity Index

The Malmquist index introduced by Caves, Christensen and Diewert(1982) measures the change in productivity of the observation under evaluation by comparing its relative performance with respect to reference technologies corresponding to two different time periods.

Following Fare, Grosskopf, Norris and Zhang (1994) productivity change can be decomposed into efficiency change and technical change under the assumption of a constant returns to scale techncology.

In this example we compute the Malmquist productivity index:

using DataEnvelopmentAnalysis

X = Array{Float64,3}(undef, 5, 1, 2);
X[:, :, 1] = [2; 3; 5; 4; 4];
X[:, :, 2] = [1; 2; 4; 3; 4];

Y = Array{Float64,3}(undef, 5, 1, 2);
Y[:, :, 1] = [1; 4; 6; 3; 5];
Y[:, :, 2] = [1; 4; 6; 3; 3];

malmquist(X, Y)
Mamlmquist DEA Model
DMUs = 5; Inputs = 1; Outputs = 1; Time periods = 2
Orientation = Input; Returns to Scale = CRS
Referene period = Geomean
─────────────────────────
M        EC   TC
─────────────────────────
1  2.0      1.33333   1.5
2  1.5      1.0       1.5
3  1.25     0.833333  1.5
4  1.33333  0.888889  1.5
5  0.6      0.4       1.5
─────────────────────────
M  = Malmquist Productivity Index
EC = Efficiency Change
TC = Technological Change

### malmquist Function Documentation

DataEnvelopmentAnalysis.malmquistFunction
malmquist(X, Y)

Compute the Malmquist productivity index using data envelopment analysis for inputs X and outputs Y.

Optional Arguments

• orient=:Input: chooses between input oriented radial model :Input or output oriented radial model :Output.
• refperiod=:Geomean: chooses reference period for technological change: :Base, :Comparison or :Geomean.
• rts=:CRS: chooses constant returns to scale. For variable returns to scale choose :VRS.
• names: a vector of strings with the names of the decision making units.