malmquist_index function

Malmquist index

This function calculates the input/output oriented Malmquist productivity index under constant or variable returns-to-scale.

malmquist_index(datadealist,
                dmu_eval = NULL,
                dmu_ref = NULL,
                orientation = c("io", "oo"),
                rts = c("crs", "vrs"),
                type1 = c("cont", "seq", "glob"),
                type2 = c("fgnz", "rd", "gl", "bias"),
                tc_vrs = FALSE,
                vtrans_i = NULL,
                vtrans_o = NULL)

Arguments

• datadealist: A list with the data (deadata objects) at different times, including DMUs, inputs and outputs.
• dmu_eval: A numeric vector containing which DMUs have to be evaluated. If NULL (default), all DMUs are considered.
• dmu_ref: A numeric vector containing which DMUs are the evaluation reference set. If NULL (default), all DMUs are considered.
• orientation: A string, equal to "io" (input oriented) or "oo" (output oriented).
• rts: A string, determining the type of returns to scale, equal to "crs" (constant) or "vrs" (variable).
• type1: A string, equal to "cont" (contemporary), "seq" (sequential) or "glob" (global).
• type2: A string, equal to "fgnz" (Fare et al. 1994), "rd" (Ray and Desli 1997), "gl" (generalized) or "bias" (biased).
• tc_vrs: Logical. If it is FALSE, it computes the vrs bias malmquist index by using the technical change under crs (Fare and Grosskopf 1996). Otherwise, it uses the technical change under vrs.
• vtrans_i: Numeric vector of translation for undesirable inputs in non-directional basic models. If vtrans_i[i] is NA, then it applies the "max + 1" translation to the i-th undesirable input. If vtrans_i is a constant, then it applies the same translation to all undesirable inputs. If vtrans_i is NULL, then it applies the "max + 1" translation to all undesirable inputs.
• vtrans_o: Numeric vector of translation for undesirable outputs in non-directional basic models, analogous to vtrans_i, but applied to outputs.

Returns

A numeric list with Malmquist index and other parameters.

Note

In the results: EC = Efficiency Change, PTEC = Pure Technical Efficiency Change, SEC = Scale Efficiency Change, TC = Technological Change, MI = Malmquist Index

Examples

# Example 1. With dataset in wide format.
# Replication of results in Wang and Lan (2011, p. 2768)
data("Economy")
data_example <- make_malmquist(datadea = Economy,
                               nper = 5, 
                               arrangement = "horizontal",
                               ni = 2, 
                               no = 1)
result <- malmquist_index(data_example, orientation = "io")
mi <- result$mi
effch <- result$ec
tech <- result$tc

# Example 2. With dataset in long format.
# Replication of results in Wang and Lan (2011, p. 2768)
data("EconomyLong")
data_example2 <- make_malmquist(EconomyLong,
                                percol = 2, 
                                arrangement = "vertical",
                                inputs = 3:4, 
                                outputs = 5)
result2 <- malmquist_index(data_example2, orientation = "io")
mi2 <- result2$mi
effch2 <- result2$ec
tech2 <- result2$tc

# Example 3. Replication of results in Grifell-Tatje and Lovell (1999, p. 100).
data("Grifell_Lovell_1999")
data_example <- make_malmquist(Grifell_Lovell_1999,
                               percol = 1,
                               dmus = 2,
                               inputs = 3,
                               outputs = 4,
                               arrangement = "vertical")
result_fgnz <- malmquist_index(data_example,
                               orientation = "oo",
                               rts = "vrs",
                               type1 = "cont",
                               type2 = "fgnz")
mi_fgnz <- result_fgnz$mi 

result_rd <- malmquist_index(data_example,
                             orientation = "oo",
                             rts = "vrs",
                             type1 = "cont",
                             type2 = "rd")
mi_rd <- result_rd$mi
 
result_gl <- malmquist_index(data_example,
                             orientation = "oo",
                             rts = "vrs",
                             type1 = "cont",
                             type2 = "gl")
mi_gl <- result_gl$mi

References

Caves, D.W.; Christensen, L. R.; Diewert, W.E. (1982). “The Economic Theory of Index Numbers and the Measurement of Input, Output, and Productivity”. Econometrica, 50(6), 1393-1414.

Fare, R.; Grifell-Tatje, E.; Grosskopf, S.; Lovell, C.A.K. (1997). "Biased Technical Change and the Malmquist Productivity Index". Scandinavian Journal of Economics, 99(1), 119-127.

Fare, R.; Grosskopf, S.; Lindgren, B.; Roos, P. (1989). “Productivity Developments in Swedish Hospitals: A Malmquist Output Index Approach”. Discussion paper n. 89-3. Southern Illinois University. Illinois.

Fare, R.; Grosskopf, S.; Lindgren, B.; Roos, P. (1992). “Productivity changes in Swedish Pharmacies 1980-89: A nonparametric Malmquist Approach”. Journal of productivity Analysis, 3(3), 85-101.

Fare, R.; Grosskopf, S.; Norris, M.; Zhang, Z. (1994). “Productivity Growth, Technical Progress, and Efficiency Change in Industrialized Countries”. American Economic Review, 84(1), 66-83.

Fare, R.; Grosskopf, S.; Roos, P. (1998), Malmquist Productivity Indexes: A Survey of Theory and Practice. In: Fare R., Grosskopf S., Russell R.R. (eds) Index Numbers: Essays in Honour of Sten Malmquist. Springer.

Grifell-Tatje, E.; Lovell, C.A.K. (1999). "A Generalized Malmquist productivity index". Top, 7(1), 81-101.

Pastor, J.T.; Lovell, C.A.k. (2005). "A global Malmquist productiviyt index". Economics Letters, 88, 266-271.

Ray, S.C.; Desli, E. (1997). "Productivity Growth, Technical Progress, and Efficiency Change in Industrialized Countries: Comment". The American Economic Review, 87(5), 1033-1039.

Shestalova, V. (2003). "Sequential Malmquist Indices of Productivity Growth: An Application to OECD Industrial Activities". Journal of Productivity Analysis, 19, 211-226.

Author(s)

Vicente Coll-Serrano (vicente.coll@uv.es ). Quantitative Methods for Measuring Culture (MC2). Applied Economics.

Vicente Bolos (vicente.bolos@uv.es ). Department of Business Mathematics

Rafael Benitez (rafael.suarez@uv.es ). Department of Business Mathematics

University of Valencia (Spain)