Input Orientated Efficiency Measurement Concepts

The convexity constraint ensures that the VRS model take into accounts the variation of the efficiency with respect to the scale size of firm. The measurement of the efficiency of each firm only benchmarked against firms that have similar size. Hence, the inefficient firm is a result of measurement with firm that has a similar size. The convexity constraint is not applied in the CRS model. Therefore, a firm might be compared to the firms, which are larger or smaller size than it. In TE calculation using the VRS model, the value of scale efficiencies for each firm will be obtained. It is derived from the ratio of the TE value both the CRS DEA and VRS DEA. Based on this calculation, it is known that the TE value in the CRS includes two components, one due to scale inefficiency and one due to pure technical inefficiency. The firm is indicated that scale is inefficiencies if there is a difference in the CRS and VRS TE value. Otherwise, the firm only has pure technical inefficiency if there is the same in the CRS and VRS value, and surely has the TE value smaller than one. This concept is illustrated on the Figure 8. Source: Coelli, Rao, Battese, 1998 Figure 6 Calculation of Scale Efficiency in DEA CRS Frontier VRS Frontier J J c M J v K L y x NIRS Frontier In Figure 8 illustrated scale inefficiency using one input and producing one output, under input orientated assumption. There are CRS and VRS frontier, and non-increasing return to scale NIRS. Under CRS condition, the technical inefficiency of the point J is the distance between J and J c JJ c . While, the technical inefficiency of VRS condition would only be the distance between J and J v JJ v . This difference is due to scale inefficiency. This is defined on Equation 3.12. TE CRS = MJ c MJ TE VRS = MJ v MJ SE = MJ c MJ v 3.12 Shortcoming arising from scale efficiency is the value does not indicate whether the firm is in a condition increasing return to scale IRS or decreasing return to scale DRS. An additional DEA problem with non-increasing return to scale NIR S can be determines this condition by changing the N1΄λ = 1 into N1΄λ ≤ 1 as on Equation 3.13. min θ,λ θ, st -y i + Yλ ≥ 0, θx i – Xλ ≥ 0, N1΄λ ≤ 1 λ ≥ 0 3.13 Based on Figure 3.7, the firm is under IRS condition when the NIRS value is not equal to VRS value. This condition is illustrated by point J. While, if both of values are equal then the DRS exist for that firm. This condition is shown by point K.

3.5 Slack

Efficiency measures of DEA model assume that the production function is known Coelli, Rao, Battese, 1998. This is not the case in practice, and the efficient isoquant must be estimated from the sample data. Based on study of Farrell, 1957, there are two methods about this. First, a non-parametric piece- wise-linear convex isoquant, constructed such that no observed point lies to the left or below it see Figure 6. Second, a parametric function, such as Cobb- Douglas form, fitted to the data, also such that no observed point lies to the left or below it. The below efficiency measures are defined in the context of CRTS technology. Source: Coelli, Rao, Battese, 1998 The piece-wise linear form of the non-parametric in DEA model cause a few difficulties in efficiency measurement. This appears because of the section of the piece-wise linear frontiers which run parallel to axis see Figure 7 which do not occur in most parametric functions. Figure 7 illustrates the problem, where the firm using combinations of inputs, point C and D are the two efficient firms, which define the frontier. Point A and B are inefficient firms. T he technical efficiency of firm A and B are 0A΄0A and 0B΄0B, respectively Farrell, 1957. However, the question whether the point A΄ is efficient because at that point the input x 2 used can be reduced by the amount CA΄ and still produce output in the same amount. Excess in the use of inputs is known as input slack input excess. To provide an accurate indication of technical Figure 7 Piece-wise Linear Convex Unit Isoquant S S΄ x 1 y x 2 y