Total-factor energy efficiency indicator

1. Methodology of data envelopment analysis

Data envelopment analysis (DEA) finds the efficient outputs and inputs in a total-factor framework. This technique makes use of information available in considering factors simultaneously. It is a non-parametric method that uses linear programming methods to construct a non-parametric piecewise frontier over the data for an efficiency measurement. DEA does not need to specify either the production functional form or weights on different inputs and outputs.

Efficiency is defined by the difference in the ‘best practice’ production frontier, as measured by DEA. The ‘best practice’ in the frontier is the benchmark to calculate the projected and possible energy saving for those not on the frontier. By comparing the relative practice of various inputs and output in different economies, we can identify the main amount (target) for energy saving and CO2 abatement likely to be found. Thus, the performance of the economies that have the ‘best practices’ can serve as a benchmark to evaluate a particular economy’s energy consumption and CO2 emissions. A similar approach to construct abatement ratios from the total-factor framework can be found in Hu (forthcoming) and Hu and Wang (forthcoming).

This paper uses DEA to find out the input targets for each APEC economy by comparing with the annual efficiency frontier constituted by all the APEC economies in each year. There is an efficiency frontier for each APEC economy in each year constituted by data of all APEC economies in that year. Since it is an input-reducing focus, this paper uses input-orientated measures following Farrell’s (1957) original ideas. In order to pursue overall technical efficiency with energy inputs, our study adopts the constant returns to scale (CRS) DEA model (Charnes et al., 1978).

Our measure of relative efficiency is based on non-parametric techniques (Färe et al., 1994). Let us first define some mathematical notations: There are K inputs and M outputs for each of N objects. For the ith object these are represented by the column vectors xi and yi, respectively. The K×N input matrix X and the M×N output matrix Y represent the data for all N objects. The input set L(yi) for the ith object is defined as L(yi) = {xi: yi ≥ f(xi)}. The efficiency score θ equals the value of the distance function, D(yi, xi) = min {λ: xiλ ∈ L(yi)}

(Shephard, 1970). The set L(yi) can be numerically computed by linear programming using observed data. The input-oriented CRS DEA model then solves the following linear programming problem for object i in each year:

D(yi, xi) = Min θ, λ θ subject to -yi + Yλ ≥ 0,

θ xi – Xλ ≥ 0,

λ ≥ 0, (5)

where θ is a scalar and λ is a N×1 vector of constants.

The value of θ is the efficiency score for the ith object, with 0 ≤ θ ≤ 1. The value of unity indicates a point on the frontier that is hence a technically efficient object, according to Farrell’s (1957) definition. The frontier is a piece-wise linear isoquant, determined by the observed data points of the same year, i.e., all the objects in this study of the same year. The object that constructs the frontier is the ‘best practice’ among those observed objects in that year. The weight vector λ serves to form a convex combination of observed inputs and outputs.

Figure 6 illustrates the efficiency measurement: Each point on Figure 6 represents a combination of inputs that all produce the same output level.

Objects C and D are on the frontier and they cannot maintain the given output level by further reducing their inputs. Objects A and B are hence inefficient objects.

Equation (5) is known as the constant returns to scale (CRS) DEA model (Charnes et al., 1978). This model finds the overall technical efficiency (OTE) of each object. The variable returns to scale (VRS) DEA model (Banker et al., 1984) is further extended with adding following convexity constraint:

N1’λ ≥ 0,

where N1’ is a N×1 vector of ones.

The VRS DEA model decomposes the OTE into pure technical efficiency (PTE) and scale efficiency (SE). That is, OTE = PTE × SE. In order to pursue OTE with energy or CO2 emissions, this study adopts the CRS DEA model.

Furthermore, both output-oriented and input-oriented CRS DEA models generate exactly the same efficiency scores, target inputs, and target outputs.

However, results of a VRS DEA model can be drastically changed by shifting output orientation to input orientation.

Output = 1 B

C A

D A’

B’

Other Inputs/Output

Energy/Output

E

Figure 6 DEA representation of ‘best practice’, target, radial adjustment, and input slacks

2. Slack and radial adjustment

An important issue in efficiency studies is the credibility of the assumption that all production processes can actually reach the best practice production frontier (Zofío and Prieto, 2001). In the present study, when measuring energy efficiency and CO2 abatement, it is assumed that all economies have access to the best practice. This assumption seems to be adequate since only APEC economies are considered. Currently, specialized journals, technological fairs, multi-nationals’ global marketing strategies, etc. guarantee that new innovations are readily available to all economies (Zofío and Prieto, 2001). The international trade agreements among APEC force economies to be more competitive and the pressure of Kyoto Protocol requires updated technologies and improves input usage efficiency.

The f(xi) set in the frontier is the ‘best practice’ production among the observed economies. The inefficient object (economy) could reduce inputs by the amount indicated by the arrow and still remain in the input set L(yi) (Boyd and Pang, 2000). For the ith object, the distance (amount) of it to the projected point on the frontier by radial reduction without reducing the output level, (1-θ)xi, is called ‘radial adjustment’. We can illustrate this from Figure 6. Point B is the actual input set and point B’ is the ideal or best practice input set for object B by reducing the radial adjustment BB’.

More over, the mostly seen piecewise-linear form of the non-parametric frontier causes the second stage to shift from the projected point to a point at the practical minimum level of the inputs on the frontier. When the frontier runs parallel to the axes, this could be a problem. In Figure 6, point A’ is the best practice for object A by reducing the redial adjustment AA’. However, the input level at point A’ could be further reduced to input level at point C so as to maintain the same output level. The reduced amount is called ‘slack’ (by the amount CA’). The best practice for object A is point C, instead of point A’, by reducing the radial adjustment AA’ and slack CA’.

The summation of slack and radial adjustment for inputs is called the amount of total adjustment (‘target’) that could be reduced without decreasing output levels. That is, it is the total amount for any individual input which should be reduced by an object or an economy so as to reach its optimal production efficiency. With respect to any specified input, the above summation is called Input-reducing Target (IRT). The formulas are as follows:

IRTk (i, t) = Slack Adjustmentk(i, t) + Radial Adjustmentk (i, t), (6) where it is in the ith object and the tth year for kth input.

An inefficient object (economy) can save or reduce IRT in kth input, such as energy or CO2 emissions, without reducing the real economic growth. The

adjustments require both a promotion of technology level and an improvement of production process so that OTE is optimized. The total reducing amount is then removed and the output level maintains in the same level when an object or an economy operates at the efficient position on frontier of production.

Point E in Figure 6 indicates that object E has been operating on the frontier of production efficiency. The object E has already reached at the target input level of specified (energy) inputs in production. It can be observed that no amount of total adjustments exists for energy. There is only slack amount of the

“other inputs’ (capital or labor) needs further adjustment, which is DE. After reducing the amount of total adjustment in the other inputs except for energy input, the production efficiency of object E then has the practical minimum input for all inputs, including capital, labor, and energy, on the frontier.

The CRS model may suggest the slack and radial adjustments of any individual input for all objects to be efficient and the amount of target input can either be calculated accordingly. DEA calculation then decides this ‘amount of total adjustments’ for each object for production efficiency analysis.

3. Efficient input-reducing target ratio (IRTR)

Efficiency is generally defined in terms of the ratio with which best practice compares with actual operation. The indicator of a specified input efficiency therefore should be the ratios of the aggregate IRT from Equation (6) to the actual input amount. The amount of total adjustments in that input is regarded as the inefficient portion of actual input amount. For example, based on the target of energy obtained from DEA, we can calculate the IRTR for energy consumption considering other factors simultaneously. The target inputs of an object in a year are found by comparing its actual inputs to the efficiency frontier in that year. The formula is as below:

( , ) ( , )

where it is in the ith economy and the tth year for the kth input.

As Equation (7) shows, the IRTR represents each economy’s inefficiency level of energy consumption. Since the actual practice can be improved to the best practice, the actual input amount is always larger than or equal to the ideal input amount, the minimal value of IRT is zero. Therefore, the value of IRTR is between zero and unity. The total-factor input efficiency (TFIE) index for specified input originally proposed by Hu and Wang (forthcoming) has the following relation with IRTRk:

TFIEk (i, t) = 1 − IRTRk (i, t), (8) where it is in the ith economy and the tth year for the kth input.

A zero IRTR value indicates an object on the frontier with the best total-factor kth input efficiency up to one among the observed economies. A zero IRTR means that no redundant or over-consumed kth input use exists (the amount of target zero) in this economy. An inefficient economy with the value of IRTR larger than zero means otherwise that the kth input should and could be saved or reduced at the same economic growth level. A higher IRTR implies higher kth input inefficiency and a higher input-reducing amount.