Environmental emissions and production economics: implications of the materials balance.

In modeling emissions, the literature has usually pursued two approaches. One is to specify an explicit emission function, which yields emissions as a by-product, depending on the level of the desired output. The other is to specify a production function for (desired) output in which emissions play the role of an input (or "quasi-input"). While the emission function appears to be quite straightforward and not to deserve much justification, treating emissions as inputs is less self-evident. It is usually defended in a heuristic way. For instance, a typical formulation states that the treatment of emissions as production inputs "seems reasonable since attempts ... to cut back on waste discharge will involve the diversion of other inputs to abatement activities--thereby reducing the availability of these other inputs for the production of goods. Reductions in E [emissions], in short, result in reduced output" (Cropper and Oates 1992, p. 678).

Recently, the materials balance has been invoked in the discussion on the (proper) modeling of emissions. While early papers on the materials balance have focused on linear frameworks with fixed input-output coefficients (Ayres and Kneese 1969), the recent literature investigated the role of the materials balance in nonlinear production frameworks (Krysiak and Krysiak 2003; Baumgartner 2004; Pethig 2006). In this view, production is essentially the transformation of materials into desired outputs, using some nonmaterial inputs (capital, labor, energy) as a second input type. Due to physical laws, this transformation can never be complete: Some residual inadvertently arises as a by-product, and material input, desirable output, and residual are linked by the materials balance. In addition, the materials balance implies that the marginal product of materials is bounded from above.

This article aims to shed some light on the validity of modeling emissions via an emission function or as a production input, from the point of view of the materials balance. It examines several alternative representations of a given technology and shows that the technology can equivalently be described by (i) a production function with material and nonmaterial inputs and bounded marginal product of the material input, (ii) a well-behaved production function with emissions as an input, and (iii) a well-behaved emission function, if the materials balance is accounted for as an additional condition.

Thus, three different ways of modeling emissions are (formally) introduced and their relationship is clarified. It is proved that the alternative representations of the technology are equivalent and it is described how one can derive one from another. The contribution of this article is to present a formal justification of what seems to exist as inherited knowledge among environmental economists, but has never been proven rigorously. Furthermore, the analysis brings out the significance of the materials balance. The latter is an essential part of the argumentation and constitutes the link between the different models.

The article is organized as follows. The first section introduces the materials balance and derives some fundamental consequences given a production function with material and nonmaterial inputs. The next section investigates the alternative ways of modeling emissions and shows that they are equivalent. In the last section, further implications are derived, the relevance of our findings is discussed, and the relationship to the literature is examined.

Basic Concepts and Relationships

At first, we discuss the materials balance and its implications.

The Meaning of the Materials Balance

Due to basic physical laws, every production process involves the utilization of natural resources (materials). In the production process, material inputs are transformed into some material outputs with attributes possibly different from those of the inputs. These outputs can be classified into desired outputs, which are the ultimate purpose of production, and undesired outputs, which arise as a by-product or residual. The First Law of Thermodynamics, that is, the Law of Mass Conservation, implies the so-called materials balance principle, according to which the mass of the material inputs equals the mass of the desired and undesired outputs:

(1) M = Y + R,

where M = material input, Y = desired output, R = undesired output (residual), all measured in units of mass.

The Second Law of Thermodynamics, that is, the Entropy Law, implies that any incremental unit of material input can only incompletely be transformed into the desired output or, in other words, that some residual inevitably arises (Baumgartner et al. 2001):

(2) dR/dM > 0.

Two simple examples illustrate these properties of production (see Anderson 1987). One is the making of potato chips, where the principal material input is potatoes. The potato skins, which are not desired and usually peeled off early in the production process, arise as an inevitable by-product, and the potato mass included in the desired output is less than the potato mass in the input. Similarly, in the production of aluminum from bauxite ore, the ore is the material input but only a fraction of it is actually usable in the production of aluminum, and the aluminum output will be less than the input of ore. Of course, one can conceive of production processes, which do not imply any (significant amount of) residuals, like the making of confetti from paper. We will neglect these cases and focus on processes that obey condition (2).

In addition to material inputs, production processes involve nonmaterial inputs, which are labor, capital, and energy. These serve to actually perform the transformation process described above. As the above examples suggest, it is reasonable to assume that as more nonmaterial input is added, better utilization of a given amount of material inputs is possible (e.g., by more precise peeling of the potatoes). This implies that more desired output can be produced from a given quantity of the material input (subject to the limits imposed by (1) and (2)). Thus there exist nonlinear production processes, which involve some substitution possibilities between material and nonmaterial inputs. It is this type of process we are concerned with.

Implications of the Materials Balance

Given the considerations of the preceding subsection, we consider a simple production process and assume that an output Y [greater than or equal to] 0 is produced by means of two factors M [greater than or equal to] 0 and N [greater than or equal to] 0, where M represents a material and N a nonmaterial factor like labor, capital or energy. The technology is described by a production function F, i.e.,

(3) Y = F(M, N).

F is supposed to possess the usual properties (see below).

This description is augmented by the materials balance (1) introduced above which can be restated in the form R = M - Y, where R [greater than or equal to] 0 represents a residual, an undesired output, which has to be disposed in the environment. As above, it is assumed that output Y, material M, and residual R are measured by mass. Furthermore, it is assumed that the undesired output is strictly increasing in the material input as stated in (2).

As a direct implication of this set-up, we get Y < M and [F.sub.M](M, N) < 1 for M > 0, where [F.sub.M] denotes the partial derivative of Y w. r. t. M (since M = F(M, N) + R). Thus, the imposition of the materials balance restricts the class of feasible production functions F. In particular, the Inada condition [lim.sub.M [right arrow] 0] [F.sub.M](M, N) = [infinity] is ruled out when the materials balance is accounted for. As noted in earlier literature (Pethig 2003; Baumgartner 2004), this makes popular functional forms like the Cobb-Douglas function inapplicable. An additional straightforward implication of the materials balance constraint is that the frequently postulated property of weak disposability (see Shephard 1970) is not admissible. Weak disposability means that the outputs Y and R can be reduced proportionately at any given level of the inputs (M and N in our case). This is obviously inconsistent with the system (1) and (2).

Modeling the Production Process

Several approaches to the modeling of production and emissions are common in environmental economics. One is to treat emissions as a by-product, whose quantity depends on the quantity of the desired output. Another treats emissions as a production input. We now examine whether and under what conditions these approaches are valid, given the restrictions (1) and (2).

Taking into account the framework and results of the preceding section, we suppose that F is defined on [R.sup.2.sub.+] and satisfies

Condition F

(4a) F is twice continuously differentiable on [R.sup.2.sub.+].

(4b) F(0, N) = 0.

(4c) For every Y > 0 there is (M, N) such that Y = F(M, N).

(4d) 0 < [F.sub.M] < 1 and 0 < [F.sub.N] for (M, N) >> 0,

where the notation (M, N) > > 0

means that M > 0 and N > 0.

(4e) F is strictly concave in (M, N)

for (M, N) >> 0 (i.e. [F.sub.MM] < O,

[F.sub.NN] < 0 and [F.sub.MM] [F.sub.NN] - [F.sup.2.sub.MN] [greater than or equal to] 0).

The production function is thus characterized by positive and decreasing marginal products; the marginal product of material is bounded, and F is strictly concave. Material is a necessary input for production and the production function is not bounded. Differentiability is a regularity condition simplifying the presentation. The undesired output is determined by the materials balance R = M - Y = M - F(M, N).