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

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.]

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