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

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.