Some remarks on the theory of homogeneous production functions

Abstract

This issue of the recently established and rapidly growing Lecture Notes Series arose (as the author states) from an amalgamation of research on homogeneous production functions, published during the past two years in a variety of journals. It is here put together in a connected form to provide a rigorous and carefully formulated account of a theory for an important class of production functions. Although the exposition is mathematical, it is clearly and understandably put forth with many examples, enabling a limitedly skilled reader of mathematics to easily comprehend the definitions, assumptions and deductions made, yet being a significant account for mathematical economists. Turning now to the specific content of these lecture notes, E i c h h o r n prepares the way for his treatment of homogeneous production functions by an introductory chapter on the fundamental notions which underlie the subsequent exposition. Two alternatives are open to him for interpretation of the production function: "Conception A", where the allocated inputs of the factors of production are necessarily fully applied, and "Conception B", where excess inputs of some factors relative to those of others are disposable. At some places he chooses the first alternative, in the tradition of German production theorists. The axioms for the concept of a production function are quite general, requiring only that the output set for an input vector be bounded and that there exists a cubic subdomain of the nonnegative input vectors within which outputs are nondecreasing for nondecreasing input vectors, thus permitting the production function to be decreasing (by set inclusion or scalar value) when some inputs are increased excessively relative to others. The choice of "Conception A" is not trivial for the theoretical structures which will result. The preference of the reviewer is for "Conception