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

malmquist_index function