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).

Now the question arises whether R, the residual to be disposed in the environment, can also be modeled as an input. Using the materials balance, we obtain Y = F(Y + R, N) by replacing the variable M, i.e., output can be interpreted as an implicit function of R and N. Let us call this function G. Then we get

(5) Y = G(R, N).

Using the properties of F, we can prove that G is defined on [R.sup.2.sub.+] and has the usual properties of a production function (all proofs have been relegated to an Appendix):

RESULT A: If, given the materials balance (1), F satisfies Condition F, then a production function G (equation 5) exists and satisfies Condition G,

which is given by

Condition G

(6a)

G is twice continuously differentiable on [R.sup.2.sub.+].

(6b) G(0, N) =0.

(6c) For every Y > 0 there is (R, N)

such that Y = G(R, N).

(6d) 0 < [G.sub.R] and 0 < [G.sub.N] for (R, N) >> 0.

(6e) G is strictly concave in (R, N)

for (R, N) >> 0.

Again, we have positive and decreasing marginal products and strict concavity. Emissions R are a necessary input to production. In other words, G is a production function and R can be interpreted as an input. The material input is then determined by the materials balance M = Y + R = G(R, N) + R. Thus, we obtain a formal justification of the view that emissions can be modeled or interpreted as an input of the production process. Two aspects need to be emphasized: First, the production function G satisfies the standard properties of nonlinear production functions. Especially, it can be of the Cobb-Douglas type (see below). This fact is important compared to the first case in which the production function F has to obey an additional, nonorthodox restriction. Second, since the modeling is based on the materials balance, another input, materials, is in the background which has to be taken into account when describing the technology by means of the function G. Consequences of these observations will be considered in the "Discussion" section.

Since the marginal product of R is always strictly positive, we can invert G for any fixed N and obtain an emission function

(7) R = H(Y, N).

R now depends on output Y and the nonmaterial factor. It is defined on the domain D = {(Y, N) | Y [member of] y(N), N [member of] [R.sub.+]}, where y(N) denotes the range of G for fixed N, i.e., y(N) = {Y = G(R, N) | R [member of] [R.sub.+]}. The range y(N) can be a bounded subset of [R.sub.+] since Y can be limited by N (e.g., in a chemical production process the output Y can be limited by the reactor volume N). We can establish:

RESULT B: If, given the materials balance (1), G satisfies condition G, then an emission function H (equation 7) exists and satisfies condition H,

which is given by

Condition H

(8a) H is twice continuously differentiable on D.

(8b) H(0, N) = 0. For every Y > 0 there is

N such that Y [member of] y(N).

(8c) For every Y > 0 there is N such that Y [member of] y(N).

(8d) 0 < [H.sub.y] and [H.sub.N] < 0 for (Y, N) >> 0.

(8e) H is strictly convex in (Y, N) for Y, N >> 0 (i.e. [H.sub.YY] > 0, [H.sub.NN] > 0 and [H.sub.YY] [H.sub.NN] - [H.sup.2.sub.YN] [greater than or equal to] 0).

In this case, emissions R increase in output Y and they increase at an increasing rate. At the same time, emissions decrease in the nonmaterial input N, and they decrease at a decreasing rate. Thus, H(Y, N) is an emission and an abatement function simultaneously, where the marginal abatement effect of N is decreasing. Intuitively, this reflects the circumstance that in the function G(R, N), the input R can be substituted by N, but only at a decreasing rate. The material input is again determined by the materials balance M = Y + R = Y + H (Y, N).

Up to now, we have proved that the description of the technology by means of (3) and the materials balance (1) also implies the representations (5) and (7). It is not clear whether (5) or (7) also imply the production function (3). Replacing R in (7) by means of the materials balance one can demonstrate:

RESULT C: If, given the materials balance (1), the emission function H satisfies Condition H, then a production function F (equation 3) exists and satisfies Condition F.

Thus, we obtain a "circle" and arrive again at the same production function F we started from. The three representations of the technology discussed here are equivalent.

Using the following terminology

Model F: Y = F(M, N), R = M - F(M, N), Condition F (equation 4),

Model G: Y = G(R, N), M = G(R, N) + R, Condition G (equation 6),

Model H: R = H(Y, N), M = Y + H(Y, N), Condition H (equation 8),

we can state:

THEOREM: Given the materials balance (1), environmental emissions can be treated equivalently as a joint output of M in Model F, as an input in Model G, and as a by-product of Y in Model H.

In other words, we have presented various descriptions of the nonlinear production process and emissions. Furthermore, we have demonstrated how to derive one from another. Since the representations are equivalent, their information content is the same.

Discussion

In this section, we want to discuss the significance of the framework presented above. We consider some implications and clarify the relationship of our approach to the literature.

Consequences and Extension

Firstly, the imposition of the materials balance implies restrictions on the functional form of feasible production functions F(M, N). It is not an easy task to find such functions directly. However, the results of the preceding section allow us to generate this type of production function indirectly: Starting from a production function G(R, N) with standard properties, we can try to recover F(M, N) explicitly. The following example demonstrates this point:

Suppose that Y = G(R, N) = [R.sup.1/2][N.sup.1/2]. Inserting the materials balance R = M - Y and solving for Y yields Y = F(M, N) = (MN + [N.sup.2]/4).sup.1/2] - N/2.

This production function F(M, N) satisfies Condition F (equation 4). In particular, the marginal product [F.sub.M](M, N) is bounded by unity (whereas [lim.sub.R [right arrow] 0][G.sub.R](R, N) = [infinity]. We can similarly derive the corresponding emission function R = H(Y, N) = [Y.sup.2]/N. This example highlights, especially, that the Cobb-Douglas form is admissible for G(R, N) but not for F( M, N).

Even if it is not possible to recover the function F explicitly, using the function G(equations 6) and (1) assures that the basic implications of the materials balance are respected. Thus treating emissions as an input--as it is frequently done in the literature--is not only an admissible, but also a convenient approach, as it avoids finding appropriate functions F.

Secondly, our framework allows us to solve a problem, which is often neglected: The literature in general employs the approach Y = G(R, N) without referring to the materials balance. This formulation makes sense when there is a strictly positive "price" [w.sub.R] of emissions (e.g., a tax). But whenever [w.sub.R] = 0, it is optimal for a firm to use an infinite quantity of R since its marginal product is strictly positive by assumption (except in the case of fixed coefficients, which we do not consider in this paper). Obviously, this implication is unrealistic. In other words, in the usual framework, it is impossible to explain the finite level of emissions in the laissez-faire situation. This contradiction is solved if the materials balance is imposed. Then the level of emissions R is bounded by the quantity M and will therefore be finite if M has a positive price. More specifically, the problem of profit maximization is well defined since R is linked to M. For example, for Model G, we obtain the profit function

(9) [PI] = pY - ([w.sub.M] M + [w.sub.N] N + [w.sub.R]R) = pG(R, N) - ([w.sub.M](G(R, N) + R) + [w.sub.N] N + [w.sub.R] R) = (p - [w.sub.M])G(R, N) - ([w.sub.N]N + ([w.sub.M] + [w.sub.R])R)

where p denotes the price of Y and [w.sub.M] and [w.sub.N] are the prices of M and N. Even if [w.sub.R] = 0, the accounting for the materials balance implies that the firm has to "pay the price [w.sub.M]" for R. We get an analogous result for Model H. Thus our model is able to explain the finiteness of R.

Thirdly, our framework can be extended to the production of a certain type of nonmaterial output, namely final energy. It is the type of energy mentioned above as one of the nonmaterial inputs in the production of desired material outputs. However, final energy (e.g., electricity) is itself in most cases the result of a transformation process, namely the transformation of primary energy (e.g., coal). These energy transformation processes obey, in addition to the materials balance principle, the energy balance principle according to which the primary energy input equals the final energy output plus the transformation loss (all measured in energy units, e.g., Joules). By redefining symbols appropriately (M = primary energy, Y = final energy, R = transformation loss), the conditions (1) and (2) can hence alternatively be interpreted as properties of processes that produce final energy from primary energy (where Condition (1) in this case stands for energy balance, rather than materials balance). These processes, described by Y = F(M, N), also involve substitutability since more final energy can be produced from a given amount of primary energy by using more of the nonmaterial input (thus raising the transformation efficiency). However, as in the case of material production, Y < M and [F.sub.M] < 1 for M > 0.

To capture these restrictions implicitly, one can again use the model Y = G(R, N), thus treating the transformation loss as an input in a standard production function G. Such an approach has been proposed and empirically applied by Welsch (1998) with respect to electricity generation, using a Cobb-Douglas specification for G.

Contribution of the Paper

Finally, we want to relate our framework and findings to the literature. There are few papers dealing with the implications of the materials balance and the representation of the technology. Anderson (1987) presents a model which takes into account material processing, energy use and waste generation. But he considers emissions only as output ([equivalent to] waste) and is primarily interested in the implications of his model for the appropriate specification of production functions. Krysiak and Krysiak (2003) analyze the integration of conservation laws of mass and energy into the usual models of production, consumption, and general equilibrium. Assuming that these physical constraints are linear, they describe the implications, e.g., for the set of possible production plans. The basic idea of their approach is to reduce the dimension of this production possibility set by taking into account the linear constraints. They demonstrate the consequences for the profit function by introducing "effective prices" which reflect the constraints. Their approach is more general than the present one, but they do not deal with alternative possibilities of modeling emissions when the materials balance is taken into account. Especially, they do not consider the problem of treating emissions as an input. To the best of our knowledge Pethig (2006) is the only paper addressing this issue. But his approach is different from ours. He considers a complex production-cum-abatement technology and examines the role of emissions in such a framework. This technology is rather complicated (it is characterized by nine (!) conditions and twelve inputs and outputs).

To sum up, our analysis is new and simple. It has clarified the role of emissions in production processes. Whenever the materials balance is accounted for the way emissions are modeled is a matter of convenience: They can be treated as a joint output or an input, or can be described by an emission function. These representations are equivalent.

Appendix

(A1) F [??} G

We assume that Y = F(M, N) and M = Y + R, and that F satisfies Condition F. Inserting the materials balance, we obtain Y = F(Y + R, N). Since there is at least one solution (R, N, Y) and since 1 - [F.sub.M] [not equal to] 0 for (R, N) >> 0, the Implicit Function theorem (IFT) yields that there is a unique twice continuously differentiable function Y = G(R, N) (cf. Berck and Sydsaeter 1993). It is defined on [R.sup.2.sub.+] since N [member of] [R.sub.+] by assumption, R = 0 for M = 0, and R = M - F(M, N) is convex and increasing in M. We get Y [member of] [R.sub.+] and M [member of] [R.sub.+] since R = 0 implies M = 0 and since M = G(R, N) + R [greater than or equal to] R. Applying the IFT again and using the properties of F we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus G satisfies condition G.

(A2) G [??] H

We assume that Y = G(R, N) and M = Y + R and that G satisfies Condition G. Since [G.sub.R] [not equal to] 0 the IFT yields that there is a unique twice continuously differentiable (inverse) function R = H(Y, N). It is by assumption defined for all (Y, N) [member of] D = {(Y, N) | Y [member of] y(N), N [member of] [R.sub.+]} where y(N) = {Y = G(R, N) | R [member of] [R.sub.+]}. Monotonicity of G implies that y([N.sub.1]) [subset] y([N.sub.2]) for [N.sub.1] < [N.sub.2]. Furthermore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] because of (6c). For N = 0, we may have sup{Y ] Y [member of] y(N)} > 0 or = 0. In the latter case, only Y = 0 is admitted. By definition, we have R [member of] [R.sub.+] and get M [member of] [R.sub.+] since M [greater than or equal to] R. Applying the IFT again and using the properties of G, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, H satisfies Condition H.

(A3) H [??} F

We assume that R = H(Y, N) and M = Y + R and that H satisfies Condition H. Inserting the materials balance, we obtain M - Y = H(Y, N). Since there is at least one solution (M, N, Y) and 1/([H.sub.Y] + 1) [not equal to] 0, the IFT yields that there is a unique twice continuously differentiable function Y = [??}(M, N). It is defined on [R.sup.2.sub.+] since M, N [member of] [R.sub.+] by assumption. Similarly Y, R [member of] [R.sub.+]. Applying the IFT again and using the properties of H, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, [??] satisfies Condition F. Then [??](M, N)= F(M, N) since otherwise we get a contradiction.

The authors thank Rudiger Pethig, an editor (Stephen Swallow) of the Journal, and two anonymous referees for helpful comments.

[Received November 2005; accepted June 2006.]

References

Anderson, C.L. 1987. "The Production Process: Inputs and Wastes." Journal of Environmental Economics and Management 14:1-12.

Ayres, R.U., and A.V. Kneese. 1969. "Production, Consumption, and Externalities." American Economic Review 59:282-97.

Baumgartner, St. 2004. "The Inada Conditions for Material Resource Inputs Reconsidered." Environmental and Resource Economics 29:307-22.

Baumgartner, St., H. Dyckhoff, M. Faber, J. Proops, and J. Schiller. 2001. "The Concept of Joint Production and Ecological Economics." Ecological Economics 36:365-72.

Berck, R, and K. Sydsaeter. 1993. Economists' Mathematical Manual, 2nd ed. Berlin: Springer-Verlag.

Cropper, M.L., and W.E. Oates. 1992. "Environmental Economics: A Survey." Journal of Economic Literature 30:675-740.

Krysiak, F.C., and D. Krysiak. 2003. "Production, Consumption, and General Equilibrium with Physical Constraints." Journal of Environmental Economics and Management 46:513-38.

Pethig, R. 2003. "The 'Materials Balance Approach' to Pollution: Its Origin, Implications and Acceptance." Economics Discussion Paper No. 105-03, University of Siegen.

--. 2006. "Nonlinear Production, Abatement, Pollution and Materials Balance Reconsidered." Journal of Environmental Economics and Management 51:185-204.

Shephard, R.W.. Pethig, R. 1970. Theory of Cost and Production Functions. Princeton, NJ: Princeton University Press.

Welsch, H. 1998. "Coal Subsidization and Nuclear Phase-out in a General Equilibrium Model for Germany." Energy Economics 20:203-22.

Udo Ebert and Heinz Welsch are professors of economics, Department of Economics, University of Oldenburg, Germany.