Intuitive presentation

In the rest of this paper, countries are considered as independent production units, which aim at maximizing the production of desirable outputs such as consumption goods. We focus on characterizing their production process, i.e. the way countries transform inputs into outputs. However, as far as the production of desirable outputs also generates pollution, three categories of factors have to be taken into consideration when analyzing their production process: inputs, desirable outputs and pollutants in the form of undesirable outputs. In this respect, we will make several assumptions. They are formulated below (section 2.2). However, one of them, the weak disposability of the undesirable by-products (pollution) is of crucial importance and requires an intuitive explanation.

In production theory, it is usual to assume that outputs are strongly disposable, which implies that the disposal of any output can be achieved without incurring any costs in terms of reduced production of other outputs. However, the symmetric treatment of outputs in terms of their disposability characteristics looses its justification if one or some of the outputs consist of an undesirable by-product such as pollution (Zaim and Taskin, 2000). When producing units are forced to clean or reduce the undesirable by-product, one has to consider the disposal of undesirable outputs as costly to achieve, as it requires a reduction in the good output. In technical terms, the undesirable output is said to be weakly disposable.

In figure 4.II, let y and b denote respectively the good (desirable) and bad (undesirable) outputs. The output correspondence sets (i.e., the amounts of both desirable and undesirable outputs that can be jointly produced with a given set of inputs) illustrate the difference between weak and strong disposability of outputs.

Fig. 4.II Output correspondence set for strongly -Y^s(x)- and weakly –Y^w(x)- disposable undesirable outputs for a given set of inputs x

If the disposability of outputs is costless (i.e. both outputs are freely disposable), the line segment cd adequately describes the technology since a reduction in b would be possible without giving up any y.

b y

c

O

d

Y^w(x) Y^s(x)

If however the disposability of b is costly, the line segment Od is relevant as any reduction in b requires to forego some of y. As we are interested in finding the cost of the undesirable output in terms of the forgone production of the desirable one, we will therefore model the production process of undesirable outputs as weakly disposable.

The method called Data Envelopment Analysis (DEA) is used to estimate the output correspondence set with undesirable outputs weakly disposable (Y^w(x)). More precisely, the DEA is a non-parametric method that estimates the piecewise linear production frontier according to the joint-production of b and y. As indicated by its name, the DEA method consists in enveloping the observations. Thus most efficient units according to the production of both desirable and undesirable outputs define the best practice frontier. For example, in figure 4.III, units A, B and C define the frontier and are fully efficient while unit E is not efficient as it could produce more desirable output in point E’ without increasing the level of pollution. Note that figure 4.III presents the best practice frontier according to a given vector of inputs.

Fig. 4.III The production frontier defined from the “best” units

Once the best practice frontier is estimated, it appears straightforward to compute the shadow-prices for efficient units. They are equal to the slope of the tangent to the frontier, i.e. to the amount of desirable output that has to be forgone in order to reduce emissions by one unit. The same can be done for inefficient units by projecting them on the frontier as shown in Figure 4.IV. All inefficient units (E for example) are transposed using a constant vector g(g_b, g_y) called the directional output vector.

The direction of the latter corresponds to the chosen efficiency rule, which describes the way the desirable output may be expanded and/or undesirable bad reduced. The choice of the efficiency rule will have an impact on the estimated shadow-price.

b A

E

D C B

E’

y

Overall, the estimated shadow-prices are computed as if all units were perfectly efficient. However, in figure 4.IV, the inefficient unit E can reduce emissions without incurring costs when moving towards the frontier. The amount of zero-cost emissions reduction constitutes thus another important result.

This may be grasped by the efficiency score, which reflects the distance from each inefficient unit to the production frontier¹⁵. By construction, the efficiency scores are equal to or smaller than one. They measure the feasible undesirable output reduction and desirable input increase that a unit could attain keeping the resource usage constant. For country E, the efficiency score is equal to βE=Oβ_E/Oβ_E’

when considering a “horizontal” efficiency rule (undesirable output oriented efficiency score). Thus, if the efficiency score of the unit E is equal to 0.8, unit E can potentially reduce the undesirable output by (1-0.8)/0.8= 0.25, i.e. 25 % of its actual undesirable output level, keeping its desirable output level constant and using the same amount of resources. Efficiency scores as well as shadow-prices may be generalized to the case of units transforming multiple inputs into multiple desirable and undesirable outputs.

Fig. 4.IV: Shadow-prices and efficiency scores

Several studies have attempted to analyze producer environmental performance from an estimated bad output shadow-price. Pittman (1981, 1983) studies 30 pulp and paper plants and estimates the Lagrange multiplier, reflecting the shadow-price, of the restriction that plants have to respect a maximum discharge level of pollutant. He found that shadow-prices vary between plants and hence that control regulations allocated the abatement effort inefficiently. Such an approach remains however limited since it assumes that each plant behaves optimally and discharges exactly the

15 Actually, the prime concern of most studies using the DEA method for efficiency evaluation is the efficiency score. Note that in practice, the best practice frontier and the measures of the efficiency scores are estimated in a single step.

b C

E’ E g

g

βE

β_E’

O

authorized quantity of pollutants (Marklund, 2003). Färe et al. (1993) examine the same data as Pittman but use a Shepard multi-output distance function, which estimates the shadow-prices as the trade-off between good output and bad output by considering that bad outputs are an undesirable by-product of the by-production process. Their result confirms the previous findings of Pittman as the shadow-price estimates vary considerably across plants. Coggins and Swinton apply the same approach to SO2 emissions of coal electric power plants. Their results show that the shadow-prices, or marginal abatement costs, differ across plants and that the average shadow-price in their sample is close to the observed prices of SO₂ allowances between utilities. Färe et al (2002) determine the shadow-prices of the pollution of water by agricultural pesticides with a directional output distance function, which contrary to the Shepard output distance function, allows for a simultaneous expansion of the good output and contraction of the bad outputs instead of a proportional expansion of both bad and good outputs to the frontier.

In the rest of this paper, we will apply the directional output distance function in order to compute the marginal abatement costs. However, contrary to the vast majority of studies, we will not consider microeconomic units but countries. To our knowledge, only Färe et al. (1994) have applied the DEA technique and the Shepard distance function at the level of countries. Their study does not however account for the undesirable by-product of the production process¹⁶.