Asymmetric Stages, Ridgelines and the Economic Region for the Two-Variable-Factor Production Function Model

Abstract

In an excellent article appearing in a 1972 issue of this Journal, Vaughn and Pfouts [10] address the pedagogically troublesome matter of of production stages and production. Their paper provides: (i) proof that stage symmetry holds only for functions homogeneous of degree one, (ii) an alternative definition of production stages which yields symmetric stages irrespective of whether the homogeneity property is satisfied, and (iii) demonstrates that production may occur in a supposedly economic stage of production if the traditional definitions of production stages are used [10, 403]. In particular Vaughn and Pfouts assert the following: (1) Under the traditional definition of production stages, linear homogeneity is necessary and sufficient for stage symmetry [10, 403]. (2) The traditional geometric representation of isoquants and ridgelines is ambiguous with regard to the boundary between stages I and II for the respective factors [10, 403]. (3) The typical and generally invalid textbook approach to resolving this ambiguity is to appeal to symmetry. The appeal is along the line that since stage I for factor one corresponds to stage III for factor two and vice versa and since stage III is intuitively uneconomic, then it follows that stage I is as well [10, 403]. (This generally invalid appeal to symmetry is also often made to concomitantly argue that the area bounded by the ridgelines is uniquely and exclusively stage II for both factorsa criticism not formally raised by Vaughn and Pfouts.) (4) Stage II for the two-variable-factor model should be redefined as the range in which the marginal products of both factors are positive [10, 405], i.e., the area circumscribed by the ridgelines. Vaughn and Pfouts argue that this redefinition of stage II is desirable to overcome confusion owning to the traditional definition; i.e., it is possible under the traditional definition to falsely identify production as economic, or show economic production as uneconomic [10, 405]. Finally, Vaughn and Pfouts suggest that under their revised definition, When both marginal products are positive, production is in an economic phase .. . [10, 405]. This author alleges that Vaughn and Pfouts are correct with regards to points (1), (2) and (3) above.' However, the fact that traditional textbook mappings of the isoquants and ridgelines leave