تولید، مصرف و تعادل عمومی با محدودیت های فیزیکی

This paper analyzes the consequences of integrating the conservation laws of mass and energy into the microeconomic models of production, consumption, and general equilibrium. We show that abstract models and especially general equilibrium theory are consistent with these physical constraints, but most applied and environmental economic models are not. We analyze the consequences of physical conservation laws for substitution possibilities and show that these constraints limit the number of independent substitution processes but not the value of the substitution elasticities. Finally, we propose a method for integrating physical constraints into static microeconomic models with a minimum of changes.

The use of models in scientific research, whether in economics or in the natural sciences, always implies some abstraction from reality. This leads to the question which aspects of reality should be included into models and which can be safely neglected. In economic modeling, such a question arises with the treatment of physical laws, like the conservation of mass and energy. These laws are successfully used in the natural sciences and in engineering but are not explicitly included in most economic models, although these models describe physical processes like the production or consumption of physically measurable quantities. The discussion whether this neglection leads to substantial problems has started more than 30 years ago and has not found a definite conclusion yet.1 Beginning with [3] and [11], several approaches to include the conservation laws of mass and energy into economic models can be found in the literature. Some of them, like those in [1] and [6], or the process analysis approach in [2], use partial analytic models, others, like those in [3] or [12], concentrate on models of the economy as a whole. But all use highly specialized models, so that they neither provide a general answer to the question whether the neglection of physical conservation laws in economics is problematic, nor supply a broadly applicable method for including these laws into economic models. In this paper, we analyze in a general setting the consequences of including mass and energy conservation into the models of production, consumption, and general equilibrium. In addition, we introduce a method that allows this inclusion with only slight changes to these models. Our results suggest that although physical conservation laws can be seen as implicitly accounted for in general equilibrium theory, they are relevant for applied modeling and especially for environmental economics. We show that most of the functional forms that are commonly used in applied modeling are not consistent with any meaningful set of physical constraints. However, this inconsistency is not caused by too large substitution possibilities, as has sometimes be suggested (see, e.g., [4]), but by allowing for too many independent substitution processes. Furthermore, our results suggest that a physically consistent model for environmental economic analysis has to be rather large, that is, it has to include a large number of environmental goods. Static models with only one or two physically measurable environmental goods, a type of model that prevails in environmental economics, have to be either physically inconsistent or economically uninteresting, since physical consistency implies that the environmental goods in such a model are not used in equilibrium. By these results, we provide at least a partial answer to the debate about the relevance of physical conservation laws for economics. The neglection of physical constraints does not undermine the validity of economic models per se. Nevertheless, their explicit inclusion seems desirable to avoid misspecifications of functional forms in applied modeling and to prevent disregarding important couplings between environmental problems in environmental economics. The paper is organized as follows: first, we model the conservation laws of mass and energy (Section 2) and integrate them into production theory (Section 3) and into consumption theory (Section 4). We then analyze the consequences of these physical constraints for general equilibrium and for general equilibrium with environmental interactions (Section 5). Two examples and a discussion of our results conclude the paper.

The theoretical analysis of this paper served two purposes. One purpose was to rigorously examine the consequences of physical conservation laws for microeconomic analysis in a general setting and thereby to inquire whether these consequences warrant the inclusion of physical laws into economic analysis. The second purpose was to develop a method to include physical constraints into standard microeconomic models in a simple way. Our analysis has shown that microeconomic models are perfectly consistent with the conservation of mass and energy as long as they remain on an abstract level. So are the necessary conditions for physical consistency of production and consumption possibility sets not violated by the assumptions of the standard models. Similarly, all important properties of general equilibrium theory, that is, the existence, stability, and Pareto optimality of general equilibrium, remain valid under physical constraints. However, this optimistic result holds only for abstract models. Most models that specify technology or consumer behavior more detailedly, as do all applied models, are not consistent with physical constraints. Our results have shown that the commonly used functional forms for profit or indirect utility functions violate physical conservation laws. Thus for applied modeling, the explicit inclusion of physical constraints seems to be necessary. But contrary to some suggestions, physical conservation laws do not rule out high substitution possibilities. As we have shown, they do not impose a bound on the elasticities of substitution, but limit the number of such elasticities that can be chosen independently from each other. So the point because of which the common functional forms are physically inconsistent is not that they imply a too high degree of substitution, but that they allow for too many independent substitution processes. So physical conservation laws affect substitution possibilities, but they do not pose a “physical bound” on substitution. Indeed, physical consistency can be seen as a generalized form of homogeneity of degree zero: a function is homogeneous of degree zero only if it is constant along rays through the origin, which are hyperplanes of dimension one that contain the origin; similarly, a functional form can only be consistent with r linearly independent physical constraints if it is constant on a hyperplane of dimension r that contains the origin. Since the commonly used functional forms are inconsistent with any meaningful set of physical constraints, new forms have to be sought if physical consistency is required. As an alternative method to the search for new, and probably rather inconvenient, functional forms, we have proposed to include physical constraints into economic models by the use of “effective prices”. These prices consist of the price of a good corrected for the prices of its physically complementary goods. We have shown that on the basis of these effective prices, the standard microeconomic models can be used without further concessions to physical consistency and that these prices arise naturally in physically constrained partial and general equilibrium models. Thus the proposed method has the advantage of leaving the standard methodology nearly untouched, only the interpretation of the prices has to be changed. Summing up these points, the criticism that the neglection of physical constraints in economic theory automatically leads to invalid results is not supported by our analysis. However, the explicit inclusion of physical constraints into economic models is desirable, since it helps to avoid model misspecifications. The use of effective prices provides a simple method for this.