Productivity change and Undesirable Output

 

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1. Introduction

Undesirable outputs are often produced together with desirable outputs. Traditional productivity indexes (such as the Törnqvist and Fisher indexes) require prices of all inputs and outputs in order to aggregate to form a total factor productivity index. But the prices of undesirable outputs are not straightforward. one possible solution is to use Malmquist index which does not require price information. However, in the presence of undesirable outputs, Malmquist index may not be computable. Moreover, Malmquist index may be misleading about productivity. To break this deadlock Chung et al. (1997) proposed Malmquist-Luenberger index. But this Malmquist-Luenberger index has weak theoretical basis. It may also not be computable.

Pittman (1981) treated the undesirable output as an input because the relation between an undesirable output and normal output looks like the relation between conventional input and output. Any allowed increase in bad output frees resources to produce more normal output. Cropper and Oates (1992) also followed this approach. They took a production function to include a vector of conventional inputs and the quantity of waste discharges. Waste emissions are treated simply as another factor of production. To cut back on waste discharges will involve the diversion of other inputs to abatement activities-thereby reducing the availability of these other inputs for the production of goods. Reinhard et al. (1999) also modeled pollution as an input in the production function. They used the amount of bad output as a proxy for undesirable environmental repercussions.

I try this approach to compute productivity change. Bad outputs cause cost, not revenue. Productivity analysis measures how well firms produce MORE output with LESS input. If bads are treated as output, the productivity analysis loses its direction. To treat bads as inputs make Malmquist index recover all its virtue. Productivity analysis recover its direction.

I measure productivity changes in the Korean fossil-fuel-fired electric utilities. I try three different methods; Malmquist index not considering bad output, output based Malmquist index and input based Malmquist index considering bad as input.

2. Can Bads Be Input?

1) Problems of Malmquist Index When Bads are Treated As Output

When bads are treated as output, Malmquist index may not be computable and may not be reasonable. Malmquist index is defined by

In Figure 1, Malmquist index is not computable. Chung et al. (1997. p. 235.) already described this problem. The technology is based on t period data as indicated by P^t. The observation being evaluated is from the following period t+1, (x^t+1, y^t+1, b^t+1). Note that (y^t+1, b^t+1) lies outside the technology from the previous period, t, presumably because technical progress has occurred allowing production of more good with less bad than was possible in period t. If we try to compute the mixed period distance function,  D^t_o(x^t+1, b^t+1, y^t+1), its value will be infinite, in which case the Malmquist index is not well defined.

Figure 1.

In Figure 2 and Figure 3, Malmquist index may be misleading about productivity. In Figure 2, point A is more productive than point B since A produces the same goods (y) with less bads (b). But D^t_o(x^t+1, b^t+1, y^t+1) is larger in point B than A, in which case the Malmquist index is misleading. In Figure 3, technology A is more productive than technology B, since technology A produces less bad than technology B under given good output. But D^t+1_o(x^t, b^t, y^t) is smaller in technology B than A, in which case the Malmquist index is misleading.

From these examples, we see that the Malmquist index cannot indicate productivity change when bads are treated as output. So Chung et al. (1997) defined Malmquist-Luenberger index.

Figure 2                                           Figure 3

2). Problems of Malmquist-Luenberger index

Chung et al. (1997) defined Malmquist-Luenberger index by using the relation of output distance function and directional distance function. They replaced output distance function of Malmquist index with directional distance function. After replacement, they choose the direction to be (good, -bad). Malmquist-Luenberger is defined by

This Malmquist-Luenberger index credits firms for reductions in bads and increases in goods. But this Malmquist-Luenberger index has weak theoretical basis. At first, Chung et al. (1997) simply substitute directional distance functions for the output distance functions in the Malmquist index. Next, they choose direction (good, -bad), in which direction output distance function, D_O(x,y,-b), is not defined. The relation between directional distance function and output distance function becomes invalid in the direction of (good, -bad). Output distance function is a special case of the directional distance function. We cannot expand something defined in special case into more general case. Even if we admit this, there is no theoretical explanation why they could and should do this. Moreover Malmquist-Luenberger index may not be defined. For example, in Figure 4, Malmquist-Luenberger is not defined.

Figure 4

3) Bads Are Input

We can treat bad output as input. Firms are using pollution permit. Pollution permit is input. If government limits the amount of pollution and a firm's pollution is below the limit. pollution permit is a free input. If government limits the amount of pollution and a firm's pollution is above the limit, the firm should decrease output or stop producing. In this case, pollution permit is a fixed input. If pollution permit is traded in market, it is a variable input. Firms consider pollution since it causes cost. So pollution is an input in economic sense.

On the contrary, to treat bad as output causes theoretical problems. Output distance function and input distance function lose their meaning. We can still characterize the technology with these functions. But these functions cannot measure productivity as seen in [Figure 2] and [Figure 3].

The economists who treat bad as output impose weak disposability on bad output. They use following equation for weak disposability.

                   l=1, 2, ............., L

But this cannot reflect real technology. Firms want more pollution permit if it is free. If firms can pollute as they want, they can reduce cost by using cheaper capital, cheaper labor, and cheaper fuel. The more pollution permit is always the better. The following equation reflect real technology.

                   l=1, 2, ............., L

If we treat bad as output, we cannot use this equation.

If we can treat bad output as input we can use Malmquist index for the joint production of good and bad outputs. Malmquist index recovers all its virtue. The Malmquist index can be decomposed into two component measures, one accounting for efficiency change (MEFFCH), and one measuring technical change (MTECH). These are

Using Malmquist productivity index and DEA, I compute productivity change.

3. Data and Results

I measure productivity changes in the Korean fossil-fuel-fired electric utilities. I try three different method; Malmquist index not considering bads, output based Malmquist index and input based Malmquist index considering bads as input.

Data were gathered from the statistical data Korean Electric Power Corporation announced. The data is annual data on quantities of outputs and inputs from 10 utilities for the period 1990-1995. But the data of one utility is for the period of 1990-1992. Electricity is measured in Gwh. Capital is the installed generating capacity (kw). Labor is number of employees. Fuel is measured in kcal. Bads (SO₂, NO_x, TSP, CO₂) are measured in tons.

1) Model 1 : Malmquist index not considering bads

Inputs are only normal inputs. Output is electricity. The reference (or frontier) technology in period t is constructed from the data as

P(x, b)={ y:

                 m=1, 2, ..........., M

                  n=1, 2, ............, N

                    k=1, 2, ..........., K        }

This activity analysis model exhibits constant returns to scale and strong disposability of inputs and outputs.

For each observation the distance function in the Malmquist index are computed as the solutions to a linear programming problem. For example, for k'

(D^t_o(x^tk, b^tk, y^tk))^-1 = max θ   s.t.

       m=1, 2, ..........., M

                  n=1, 2, ............, N

                    k=1, 2, ..........., K     

2) Model 2 : Output Based Malmquist Index Considering Bads As Input

Inputs are normal inputs and bads. Output is electricity. The reference (or frontier) technology in period t is constructed from the data as

P(x, b)={ y:

                 m=1, 2, ..........., M

                  n=1, 2, ............, N

                   l=1, 2, ............., L

                    k=1, 2, ..........., K        }

This activity analysis model exhibits constant returns to scale and strong disposability of inputs and outputs.

For each observation the distance function in the Malmquist index are computed as the solutions to a linear programming problem. For example, for k'

(D^t_o(x^tk, b^tk, y^tk))^-1 = max θ   s.t.

       m=1, 2, ..........., M

                  n=1, 2, ............, N

                   l=1, 2, ............., L

                    k=1, 2, ..........., K     

3) Model 3 : Input Based Malmquist Index Considering Bads As Input

Inputs are normal inputs and bads. Output is electricity. The reference (or frontier) technology in period t is the same as "2) Input Based Malmquist Index Considering Bads As Input."

For each observation the distance function in the Malmquist index are computed as the solutions to a linear programming problem. For example, for k'

D^t_i(x^tk, b^tk, y^tk) = max θ   s.t.

       m=1, 2, ..........., M

                  n=1, 2, ............, N

                   l=1, 2, ............., L

                    k=1, 2, ..........., K     

4) Results

From the results, I found interesting aspects of productivity changes in the Korean fossil-fuel-fired electric utilities. I compared the results of Model 1 and Model 2. The result of Model 3 is the same as that of Model 2. Malmquist index is greater in Model 2 than in Model 1. It means that there is significant productivity growth in pollution abatement. There is no significant difference in efficiency change between Model 1 and Model 2. But there is significant difference in technical change between Model 1 and Model 2. This means that there is significant technical change in pollution abatement. Synthetically, there is significant productivity growth about pollution abatement and it is caused mainly by technical progress. (Only one utility (data 1) is exceptional.)

Table 1. Result of Model 1   (1990-1995)

   Mo(y,x | C,S) EC TC 

 1 1.1142  0.9159 1.2164 

 2 1.5397  1.4047 1.0961 

 3 1.5119  1.2578 1.2021 

 4 1.1253  0.9132 1.2322 

 5 1.7301  1.4193 1.2190 

 6 1.3109  1.0956 1.1966 

 7 1.1208  0.9575 1.1706 

 8 1.1315  1.0337 1.0946 

 9 1.4687  1.2375 1.1868 

Table 2. Result of Model 2   (1990-1995)

   Mo(y,x | C,S) EC TC 

 1 0.7711  1.0000 0.7711 

 2 1.6372  1.4047 1.1655 

 3 1.7439  1.3028 1.3386 

 4 1.1638  0.9731 1.1960 

 5 1.8576  1.3536 1.3724 

 6 1.5607  1.0951 1.4251 

 7 1.3665  0.9602 1.4231 

 8 1.2849  1.0494 1.2244 

 9 1.6118  1.1965 1.3471 

References

Chung, Y. H.; Färe, Rolf; Grosskopf, Shawna "Productivity and Undesirable Outputs: A   Directional Distance Function Approach," Journal of Environmental Management; 51  1997 pp. 229-40.

Cropper, Maureen L.; Oates, Wallace E. "Environmental Economics: A Survey," Journal of   Economic Literature;  v30 n2 June 1992, pp. 675-740.

Färe, Rolf; Grosskopf, Shawna; Lovell, C.A.K.; and Pasurka, Carl "Multilateral Productivity   Comparisons When Some Outputs Are Undesirable:  A Nonparametric Approach,"   Review of Economics and Statistics;  v71 n1 February 1989, pp. 90-98.

Färe, Rolf; Grosskopf, Shawna; Norris, Mary; and Zhang, Zhongyang "Productivity Growth,   Technical Progress, and Efficiency Change in Industrialized Countries," American   Economic Review;  v84 n1 March 1994, pp. 66-83.

Färe, Rolf; Grosskopf, Shawna; Tyteca, Daniel "An Activity Analysis Model of the    Environmental Performance of Firms--Application to Fossil-Fuel-Fired Electric   Utilities," Ecological Economics;  v18 n2 August 1996, pp. 161-75.

Haynes, K. E.; Ratick, S.; Cummings-Saxton, J. "Toward a Pollution Abatement Monitoring  Policy: Measurements, Model Mechanics, and Data Requirements," The    Environmental professional V16 n4 1994 PAGES: 292

Pittman, Russell W. "Issue in Pollution Control: Interplant Cost Differences and Economies   of Scale," Land Economics;  v57 n1 Feb. 1981, pp. 1-17.

Reinhard, Stijn; Lovell, C. A. Knox; Thijssen, Geert "Econometric Estimation of Technical   and Environmental Efficiency: An Application to Dutch Dairy Farms," American   Journal of Agricultural Economics;  v81 n1 February 1999, pp. 44-60.

          

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