Assessing Productivity Change by DEA: Malmquist Indices…

DEA-based Malmquist productivity index, which was first introduced by Färe et al. (1989), measures the productivity change over time. In multi-input multi-output contexts a unit’s productivity is defined as the ratio of an index of

its output levels to an index of its input levels, and the change over time of this measure reflects the change in the unit’s productivity (Thanassoulis, 2001). The DEA-based Malmquist productivity index can be decomposed into two components: one is measuring the technical change (efficiency change) and the other measuring the frontier shift (boundary shift). Also, the DEA-based Malmquist productivity index can either be input or output-oriented.

Consequently, the Malmquist productivity index can be input-oriented when the outputs are fixed at their current levels or output-oriented when the inputs are fixed at their current levels. This section will illustrate the Malmquist Indices with input-oriented DEA models, because our research considers the possible radial reductions of all inputs when the outputs are fixed at DMUs’

current levels.

As far as Figure 7 is concerned, the Malmquist index measures productivity change can be illustrated graphically in a single output and two inputs two periods DEA model. Let a company operate at point F^t in period t and at point F^t+1 in period t+1; ft and ft+1 are the frontiers which are constructed by DMUs of the period t and t+1 respectively.

Figure 7 Measure The Frontier Shift Over Two Periods

Take the frontier of period t, f ,in Figure 7 as concern, the technical input O

D

Input 1 Input 2

E

F^t ( period t data ) F^t+1 ( period t+1 data )

H I

ft (frontier of period t) ft+1 ( frontier of period t+1)

efficiency of company F is OD/OF^t in period t and OE/OF^t+1 in period t+1.

Thus its productivity change based on the frontier period t is the ratio of the efficiencies, (OE/OF^t+1)÷(OD/OF^t) , in two periods. The productivity change based on the frontier period t+1 is the ratio of the efficiencies, (OI/OF^t+1)÷(OH/OF^t), likewise. The geometric mean of these two ratios is taken to measure the productivity change of this company. Thus the productivity change of the company operating at F^t in period t and at F^t+1 in

period t+1 in Figure 7 is

1/ 2 1/ 2

t+1 t t+1 t t+1 t

OE OD OI OH OI OD OE OD

{[ ] [ ]} [ ] [ ]

OF OF OF OF OF OF OI OH

'Catch-up' 'Boundary shift'

÷ × ÷ = ÷ × ÷

component component

The catch-up term in above equation is a measure of how much closer to the boundary the company is in period t+1 compared to period t. OI/OF^t+1 measures the distance of the company from the t+1 efficient boundary and OD/OF^t its distance from the efficient boundary in period t. If the catch up term is one the company has the same distance in periods t+1 and t from the respective efficient boundaries. If the catch up term is over one the company has moved closer to the period t+1 boundary than it was the period t boundary and the converse is the case if the catch up term is under one.

The boundary shift term measures the movement of the boundary between period t and t+1 at two locations: The ratio OE/OI measures the distance of the two boundaries at the input mix of the company in period t+1. The ratio OD/OH measures the distance of the two boundaries at the input mix of the came company in period t. The boundary shift term is over one because both of the foregoing ratios are over one, which represents productivity gain by the industry and the input levels in period t+1 is lower than in period t. In contrast, if the boundary shift term under one would signal that the industry has registered productivity loss as input level would on average be higher in period t+1 compared to period t, controlling for output. Moreover, the productivity would neither gain nor loss on average while the boundary shift term is equal to one. These concepts would be presented in following mathematical forms.

Suppose there are n DMUs, each DMUj ( j =1,2…,n) produces a vector of outputs y^t_j =(y₁^t_j... )y_sj^t by using a vector of inputs x^t_j =( ...x₁^t_j x_mj^t ) at each time

period t, t=1,…,T. The CCR DEA model can be expressed as (Charnes et al., production frontier (EPF), in time period t; otherwise, if Θ1<1, then DMU k is inefficient. Similarly, by replacing x and y^t_j ^t_j with x and y^t+1_j ^t+1_j the technical

From t to t+1, DMUk’s technical efficiency may change and the EPF may shift. Based upon CCR model, the radial Malmquist productivity index can be calculated via (Färe et al., 1994a, b).

(i) Comparing x^t_j to EPF at time t, namely, calculating Θ1=

1( 1, 1)

θ ⁺ through the following linear program:

,

The Malmquist productivity index (MPI) is defined as (Färe et al., 1992)

1

MPIk, a geometric mean of two Caves et al.'s (1982) Malmquist productivity indices, measures the productivity change between periods t and t+1. Färe et al. (1992) define that MPIk >1 indicates productivity gain; MPIk <1 indicates productivity loss; and MPIk =1 means no change in productivity from time t to t+1. Furthermore, Färe et al. (1992) decompose Malmquist productivity index into two components:

1

The first component TECk=Θ4/Θ1, presenting the catch-up term in above equation, measures the change in technical efficiency. It is a measure of how much closer to the boundary the company is in period t+1 compared to period t.

If TECk is equal to one, the particular DMU k (maybe a company) has the same distance in periods t+1 and t from the respective efficient boundaries. If TECk

is over one, the company has moved closer to the period t+1 boundary than it was to the period t boundary and the converse is the care if the TECk is under one.

The second component FSk= ^{3 1 1/ 2}

4 2

[Θ ×Θ ]

Θ ×Θ measures the technology frontier shift , which is the same with the boundary shit in Malmquist equation, between time period t and t+1. Färe et al. (1992,1994a) point out that a value of FSk greater than one indicates a positive shift or technical progress, a value of FSk less than one indicates a negative shift or technical regress, and value of FSk equal to one indicates no shift in technology frontier.

As the literature review has already mentioned before, one of the advantages of the DEA technique is not having to determine an explicit definition of the production function. Hence in our research, the DEA-based Malmquist indices would be applied to inspect the cross-period productivity change of the music industry from 1997 to 2005. These indices, furthermore, would help researchers to evaluate the effects of different competitive strategies on the performance of the music firms individually, and the whole music industry as well.