Measuring Malmquist productivity index: A new approach based on double frontiers data envelopment analysis

Y.-M. Wang, Y.-X. Lan / Mathematical and Computer Modelling 54 (2011) 2760–2771 2761 TheDEA-basedMPI has proven to be a very useful tool formeasuring the productivity changes of DMUs in the past several decades. For instance, Färe et al. [13] applied it to analyze the productivity growth in industrialized countries. Grifell-Tatjé and Lovell [14] used it tomeasure the effect of deregulation on Spanish saving banks. Fulginiti and Perrin [15] employed it to study the productivity changes in agriculture in 18 developing countries. Madden and Savage [16] adopted it to analyze the telecommunication productivity, technology catch-up and innovation in 74 countries. Odeck [17] utilized it to analyze the efficiency and productivity growth of Norwegian Motor Vehicle Inspection Agencies for the time period 1989–1991. Asmild et al. [18] combined it with DEA windows analysis in the study of Canadian banking industry within the period 1981–2000. Liu and Wang [19] took advantage of it to analyze the productivity changes of Taiwanese semiconductor companies in the time period from 2000–2003. Odeck [20] employed it to measure the productivity of the agricultural sector in Eastern Norway between 1987 and 1997. Chang et al. [21] used it to study the productivity changes of US accounting firms before and after the Sarbanes–Oxley Act. Lin et al. [22] used it to discuss the economic performances of the local governments in China. Our literature review reveals that existingMPIs are all proposed from the optimistic DEA point of viewby using optimistic DEAmodels. No attempt has been made to examine the MPI from the pessimistic DEA point of view. This inevitably ignores some very useful information on productivity changes because the MPI values measured from different points of view are hardly the same and none of them can be replaced by each other. More importantly, measuring the MPIs from both the optimistic and the pessimistic DEA points of view can provide a comprehensive assessment and panoramic view of the productivity changes over time. In this paper, we first look into the MPI from the pessimistic DEA point of view and develop corresponding DEA measurement models. We then aggregate the MPIs measured from both the optimistic and the pessimistic DEA points of view into an integrated MPI to reflect the average productivity changes of the DMUs over time. The rest of the paper is organized as follows: Section 2 briefly reviews the optimistic DEA-basedMPI and itsmeasurement models. Section 3 examines the MPI from the pessimistic DEA point of view and proposes corresponding DEA models. Section 4 discusses the aggregation of the MPIs and proposes a double frontiers data envelopment analysis (DFDEA)-based MPI. The proposed models and MPIs are tested with a numerical example and applied to the productivity analysis of the Chinese industrial economy in Section 5. The paper concludes in Section 6. 2. The optimistic DEA-based Malmquist productivity index Suppose there are n DMUs to be evaluated in light of m inputs and s outputs. Denote by xij and y t rj as well as x t+1 ij and y rj the inputs and the outputs of DMUj at time periods t and t + 1, respectively, where i = 1, . . . ,m; r = 1, . . . , s; and j = 1, . . . , n. The optimistic DEA-based MPI requires the solution of the following CCR models (1) and (2) and linear programming (LP) models (3) and (4): Do(x t o, y t o) = Minimize θ (1) subject to n − j=1 λjxij ≤ θx t io, i = 1, 2, . . . ,m, n − j=1 λjyrj ≥ y t ro, r = 1, 2, . . . , s, λj ≥ 0, j = 1, 2, . . . , n, Do(x t+1 o , y t+1 o ) = Minimize θ (2) subject to n − j=1 λjxij ≤ θx t+1 io , i = 1, 2, . . . ,m, n − j=1 λjyrj ≥ y t+1 ro , r = 1, 2, . . . , s, λj ≥ 0, j = 1, 2, . . . , n, Dt+1 o (x t+1 o , y t+1 o ) = Minimize θ (3) subject to n − j=1 λjx ij ≤ θx t+1 io , i = 1, 2, . . . ,m, n − j=1 λjy rj ≥ y t+1 ro , r = 1, 2, . . . , s, λj ≥ 0, j = 1, 2, . . . , n, 2762 Y.-M. Wang, Y.-X. Lan / Mathematical and Computer Modelling 54 (2011) 2760–2771 Dt+1 o (x t o, y t o) = Minimize θ (4) subject to n − j=1 λjx ij ≤ θx t io, i = 1, 2, . . . ,m, n − j=1 λjy rj ≥ y t ro, r = 1, 2, . . . , s, λj ≥ 0, j = 1, 2, . . . , n, where Do(x t o, y t o) and D t+1 o (x t+1 o , y t+1 o )measure the optimistic efficiencies of DMUo (o ∈ {1, 2, . . . , n}) in time periods t and t + 1, respectively, Do(x t+1 o , y t+1 o )measures its optimistic efficiency in time period t + 1 using the production technology of time period t , which is called the growth index of DMUo by Sueyoshi [23], andDt+1 o (x t o, y t o)measures the optimistic efficiency of DMUo in time period t using the production technology of time period t + 1. Based on the above optimistic efficiencies, Färe et al. [3] proposed the following optimistic DEA-based Malmquist productivity index: MPIo(optimistic) = [ Do(x t+1 o , y t+1 o ) Do(xo, yo) · Dt+1 o (x t+1 o , y t+1 o ) D o (xo, yo) ]1/2 , (5) whichmeasures the productivity change of DMUo from time period t to t+1. According to Färe et al. [3], MPIo(optimistic) > 1 indicates productivity progress, MPIo(optimistic) = 1 represents that productivity is unchanged, and MPIo(optimistic) < 1 indicates productivity decline. To eliminate Caves et al.’s assumption that Do(x t o, y t o) and D t+1 o (x t+1 o , y t+1 o ) should be equal to unity and to allow for technical inefficiency, Färe et al. [3] decomposed the MPI in (5) into two components: MPIo(optimistic) = [ Do(x t+1 o , y t+1 o ) Do(xo, yo) · Dt+1 o (x t+1 o , y t+1 o ) D o (xo, yo) ]1/2 = Dt+1 o (x t+1 o , y t+1 o ) Do(xo, yo) [ Do(x t o, y t o) D o (xo, yo) · Do(x t+1 o , y t+1 o ) D o (x o , y o ) ]1/2 . (6) The first component OECo = Dt+1 o (x t+1 o , y t+1 o ) Do(xo, yo) (7) measures the optimistic efficiency change (OEC) of DMUo. OEC > 1 means that the optimistic efficiency of DMUo has improved, while OEC < 1 means that the optimistic efficiency of DMUo has declined. The second component OTCo = [ Do(x t o, y t o) D o (xo, yo) · Do(x t+1 o , y t+1 o ) D o (x o , y o ) ]1/2 (8) measures the optimistic technical change (OTC) of DMUo from time period t to t + 1. Chen and Ali [6] showed that the second component, OTCo, needed further investigation because efficiency frontiers are multi-faced and can have a downward shift in one region and an upward shift in another region. Interested readers may refer to their paper for the discussions. 3. The Malmquist productivity index from the pessimistic DEA point of view The efficiencies measured from the pessimistic point of view are referred to as pessimistic efficiencies. The pessimistic efficiency of DMUo relative to the other DMUs can be measured by the following pessimistic DEA model [24,25]: Minimize φo = s − r=1 μryro (9) subject to m − i=1 ωixio = 1, s − r=1 μryrj − m − i=1 ωixij ≥ 0, j = 1, . . . , n, μr , ωi ≥ 0, r = 1, . . . , s; i = 1, . . . ,m, Y.-M. Wang, Y.-X. Lan / Mathematical and Computer Modelling 54 (2011) 2760–2771 2763 whose dual can be written as Maximize φo (10) subject to n − j=1 xijλj ≥ φoxio, i = 1, 2, . . . ,m, n − j=1 yrjλj ≤ yro, r = 1, 2, . . . , s, λj ≥ 0, j = 1, 2, . . . , n. Both models (9) and (10) are referred to as pessimistic CCR models. They differ from the well-known CCR model (1) in that they minimize the efficiency of DMUo relative to the others within the range of no less than one, whereas the latter maximizes the efficiency of DMUo within the range of zero and one. DMUo is said to be pessimistic inefficient if its pessimistic efficiency is equal to one. All pessimistic inefficient DMUs define an inefficiency frontier. Based on the above pessimistic CCR models, we come up with the following pessimistic DEA models for measuring the values of Do(x t o, y t o),D t o(x t+1 o , y t+1 o ),D t+1 o (x t+1 o , y t+1 o ) and D t+1 o (x t o, y t o): Do(x t o, y t o) = Maximize φ (11) subject to n − j=1 λjxij ≥ φx t io, i = 1, 2, . . . ,m, n − j=1 λjyrj ≤ y t ro, r = 1, 2, . . . , s, λj ≥ 0, j = 1, 2, . . . , n, Do(x t+1 o , y t+1 o ) = Maximize φ (12) subject to n − j=1 λjxij ≥ φx t+1 io , i = 1, 2, . . . ,m, n − j=1 λjyrj ≤ y t+1 ro , r = 1, 2, . . . , s, λj ≥ 0, j = 1, 2, . . . , n, Dt+1 o (x t+1 o , y t+1 o ) = Maximize φ (13) subject to n − j=1 λjx ij ≥ φx t+1 io , i = 1, 2, . . . ,m, n − j=1 λjy rj ≤ y t+1 ro , r = 1, 2, . . . , s, λj ≥ 0, j = 1, 2, . . . , n, Dt+1 o (x t o, y t o) = Maximize φ (14) subject to n − j=1 λjx ij ≥ φx t io, i = 1, 2, . . . ,m, n − j=1 λjy rj ≤ y t ro, r = 1, 2, . . . , s, λj ≥ 0, j = 1, 2, . . . , n, whereDo(x t o, y t o) in (11) andD t+1 o (x t+1 o , y t+1 o ) in (13)measure the pessimistic efficiencies of DMUo in time periods t and t+1, respectively, Do(x t+1 o , y t+1 o ) in (12) measures the pessimistic efficiency of DMUo in time period t + 1 using the production technology of time period t , and Dt+1 o (x t o, y t o) in (14) measures the pessimistic efficiency of DMUo in time period t using the production technology of time period t + 1. The productivity change of DMUo from time period t to t + 1 can then be measured by the following pessimistic DEAbased Malmquist productivity index: MPIo(pessimistic) = [ Do(x t+1 o , y t+1 o ) Do(xo, yo) · Dt+1 o (x t+1 o , y t+1 o ) D o (xo, yo) ]1/2 , (15) 2764 Y.-M. Wang, Y.-X. Lan / Mathematical and Computer Modelling 54 (2011) 2760–2771 which is the geometric mean of the two productivity indices Do(x t+1 o , y t+1 o )/D t o(x t o, y t o) and D t+1 o (x t+1 o , y t+1 o )/D t+1 o (x t o, y t o) measured from the pessimistic DEA point of view. Similar to the definition of Färe et al. [3], MPIo(pessimistic) > 1 indicates productivity progress,MPIo(pessimistic) = 1 implies productivity remains unchanged, andMPIo(pessimistic) < 1 represents productivity decline. The above pessimistic DEA-based MPI can also be decomposed into two components: MPIo(pessimistic) = [ Do(x t+1 o , y t+1 o ) Do(xo, yo) · Dt+1 o (x t+1 o , y t+1 o ) D o (xo, yo) ]1/2 = Dt+1 o (x t+1 o , y t+1 o ) Do(xo, yo) [ Do(x t o, y t o) D o (xo, yo) · Do(x t+1 o , y t+1 o ) D o (x o , y o ) ]1/2 . (16) The first component PECo = Dt+1 o (x t+1 o , y t+1 o ) Do(xo, yo) (17) measures the pessimistic efficiency change (PEC) of DMUo. If PEC > 1, then the pessimistic efficiency of DMUo improved from time period t to t + 1; if PEC < 1, then the pessimistic efficiency of DMUo degenerated. The second component PTCo = [ Do(x t o, y t o) D o (xo, yo) · Do(x t+1 o , y t+1 o ) D o (x o , y o ) ]1/2 , (18) measures the pessimistic technical change (PTC) of DMUo from time period t to t + 1. Similar to the work of Chen and Ali [6], PTCo can be further investigated and discussed. The conclusions will be the same since both the optimistic and the pessimistic DEA models give a high efficiency score to a better DMU. Interested readers are referred to their paper for the conclusions and discussions. 4. Aggregation of Malmquist productivity index When the MPI is measured from different DEA points of view, there is no guarantee that a consistent evaluation conclusion can be achieved. Generally speaking, the MPI values measured from different DEA points of view are not the same, even significantly different or strongly inconsistent. So, there is a clear need to aggregate them into an integratedMPI value for each DMU to produce an overall conclusion. Similar to the geometric mean in formulation (5), we can combine the MPI values measured from both the optimistic and the pessimistic DEA points of view in a geometric mean manner. That is, MPIo(DFDEA) = [MPIo(optimistic) · MPIo(pessimistic)] , (19) which measures the average productivity change of DMUo from both the optimistic and the pessimistic DEA points of view simultaneously and can be further decomposed into MPIo(DFDEA) = [(OECo · OTCo)(PECo · PTCo)] = [OECo · PECo] · [OTCo · PTCo] = ECo · TCo, (20) where ECo = [OECo · PECo] and TCo = [OTCo · PTCo] measure, respectively, the average efficiency change of DMUo and its average technical change over time periods t and t + 1. Since the MPI value defined by the left-hand side of (19) is the integration of the MPI values measured from both the optimistic and the pessimistic DEA points of view, we refer to it as the double frontiers DEA-based MPI value or the DFDEAbased MPI value for short, which is more comprehensive and more realistic than the traditional optimistic DEA-based MPI value and can better and more accurately reflect the productivity changes of the DMUs over time. 5. Numerical example and application In this section, we examine the proposed new approach with a numerical example and then apply it to analyze the productivity changes of the industrial economy of China during the years 2005–2009. Example 1. Productivity measurement with a data set in two years [26]. Consider the data set in Table 1 for six DMUs with two inputs and two outputs. The MPI values measured from the optimistic and the pessimistic DEA points of view are shown in Tables 2 and 3, respectively, where Dt(t, t),Dt(t +1, t +1), Dt+1(t +1, t +1),Dt+1(t, t) are used to represent Do(x t o, y t o),D t o(x t+1 o , y t+1 o ),D t+1 o (x t+1 o , y t+1 o ),D t+1 o (x t o, y t o) for brevity. From Table 2, it is seen that the productivity changes of DMUA,DMUB and DMUE are all bigger than one when measured from the optimistic DEA point of view, while those of DMUC ,DMUD and DMUF are smaller than one. This shows that the Y.-M. Wang, Y.-X. Lan / Mathematical and Computer Modelling 54 (2011) 2760–2771 2765 Table 1 Data set for six DMUs with two inputs and two outputs [26]. DMU Year Inputs Outputs StHr Supp MCPD PPPD A 1985 150 0.2 14000 3500 B 1985 400 0.7 1400