malmq {Benchmarking}

Malmquist index

Description

Estimates Malmquist indices for productivity and its decomposition between two periods. The units in the two periods does not have to be exactly the same, but the Malmquist index is only calculated for units present in both periods.

Usage

malmq(X0, Y0, ID0 = NULL, X1, Y1, ID1 = NULL, RTS = "vrs", ORIENTATION = "in", 
    SAMEREF=FALSE, SLACK = FALSE, DUAL = FALSE, DIRECT = NULL, param = NULL, 
    TRANSPOSE = FALSE, FAST = TRUE, LP = FALSE, CONTROL = NULL, LPK = NULL)

Arguments

X0

Inputs of firms in period 0, a K0 x m matrix of observations of K0 firms with m inputs (firm x input).

Y0

Outputs of firms in period 0, a K0 x n matrix of observations of K0 firms with n outputs (firm x input).

ID0

Index for firms in period 0; could be numbers or labels. Length K0.

X1

Inputs of firms in period 1, a K1 x m matrix of observations of K1 firms with m inputs (firm x input).

Y1

Outputs of firms in period 1, a K1 x n matrix of observations of K1 firms with n outputs (firm x input).

ID1

Index for firms in period 0; could be numbers or labels. Length K0.

RTS

Returns to scale assumption as in dea.

ORIENTATION

Input efficiency "in" (1), output efficiency "out" (2), and graph efficiency "graph" (3) as in dea.

SAMEREF

Use the same units for reference technology when comparing two periods. This is not restricted to same units in several timpe periods, but only to pairwise periods comparisons for Malmquist. Default is to use available and possible differnt units in pairwise periods.

SLACK

See dea.

DUAL

See dea.

DIRECT

See dea.

param

See dea.

TRANSPOSE

See dea.

FAST

See dea.

LP

See dea.

CONTROL

See dea.

LPK

See dea.

Details

The order of the units in values is given by the returned value id. This is usefull if the order of units differ completely between ID0 and ID1.

The index for technical changes tc is calculated as sqrt(e10/e11 * e00/e01) where e<s><t> is the efficiency for period s when the reference technology is for period t, i.e. determined from the observations for period t and XREF=X_t, YREF=Y_t, as is the option for the function dea.

The Malmquist index for productivity mq is calculates as sqrt(e10/e00 * e11/e01) and the index for change in efficiency ec is e11/e00. Note that mq = tc * ec.

Value

m

Malmquist index for productivity.

tc

Index for technology change.

ec

Index for efficiency change.

mq

Malmquist index for productivity; same as m.

id

Index for firms present in both period 0 and period 1.

id0

Index for firms in period 0 that are also in period 1.

id1

Index for firms in period 1 that are also in period 0.

e00

The efficiencies for period 0 with reference technology from period 0.

e10

The efficiencies for period 1 with reference technology from period 0.

e11

The efficiencies for period 1 with reference technology from period 1.

e01

The efficiencies for period 0 with reference technology from period 1.

Note

The calculations of efficiencies are only done for units present in both periods.

Author(s)

Peter Bogetoft and Lars Otto [e-mail]

References

Peter Bogetoft and Lars Otto; Benchmarking with DEA, SFA, and R; Springer 2011

See Also

dea

Examples

   x0 <- matrix(c(10, 28, 30, 60),ncol=1)
   y0 <- matrix(c(5, 7, 10, 15),ncol=1)
   x1 <- matrix(c(12, 26, 16, 60 ),ncol=1)
   y1 <- matrix(c(6, 8, 9, 15 ),ncol=1)

   dea.plot(x0, y0, RTS="vrs", txt=TRUE)
   dea.plot(x1, y1, RTS="vrs", add=TRUE, col="red")
   points(x1, y1, col="red", pch=16)
   text(x1, y1, 1:dim(x1)[1], col="red", adj=-1)

   m <- malmq(x0,y0,,x1,y1,,RTS="vrs")
   print("Malmquist index for change in productivity, technology change:")
   print(m$mq)
   print("Index for change of frontier:")
   print(m$tc)