Introduction

Throughout this chapter and the next the production function is assumed to be single valued, everywhere continuous, and well defined over the range of inputs yielding nonnegative outputs. A second assumption, which increases generality without sacrificing content, is that the production function belongs to a class of functions with continuous first and second partial derivations. Third, it is assumed that the inputs are continuously variable, i.e. the inputs are real variables defined over the nonnegative domain of real numbers.

A fourth assumption, which is sometimes not invoked, concerns the unchanging character of the production function and of the inputs. Specifically, unless otherwise stated, the production function is given and fixed; there is neither technological progress nor retrogression. Similarly, the inputs remain unchanged in character and are assumed to be homogeneous within themselves. Finally, our attention is chiefly focused upon the production of a single commodity and the behavior of a single-product firm.

As mentioned in chapter I, it is both convenient and instructive to examine the behavior of output when only one input is varied, all other inputs being held constant. In the conventional terminology of economics such investigations are called short-run analyses; the inputs held constant are called ‘fixed inputs’, the one allowed to vary is called a ‘variable input’. Otherwise, one may simply regard such investigations as the study of cross-sections of the production surface or of the behavior of a single partial derivative.