Measuring Returns to Scale in the Aggregate, and the Scale Effect of Public Goods

WE PROPOSE to study here the relationship between externalities, public goods, and returns to scale. The ideas behind this connection are certainly not new. Indeed, Marshall frequently spoke of external economies and diseconomies of scale. He had in mind situations in which expansion of one firm conferred external benefits on the industry and led to increased efficiency of the agggregate operations. The quantitative relationship has been explored from time to time in a number of specific contexts. For example, Arrow [1] showed that the presence of learning by doing (a public good) introduced an element of increasing returns into the aggregate production relationships. We are after a general quantitative relationship. Our first task is to construct a general measure of returns to scale for multiproduct technologies. A natural starting point is the theory of homogeneous functions (see Henderson and Quandt [8] for a general discussion of this subject). The degree of returns to scale of a homogeneous function is naturally measured by its degree of homogeneity. This measure is used as a matter of course by economists (see for example, Intriligator [9]) and can be extended to multiproduct technologies. We certainly want our measure to agree with this one for homogeneous functions. However, it is very unlikely that the aggregate production technology can be represented by a function which is homogeneous of any particular degree. Thus, we must find a measure which will apply to more general functions. In the next section we develop a measure which has some intuitive appeal and can be shown to be the "ideal" measure under some circumstances. It can be thought of as a generalization of the well known "elasticity of production" used by many authors (see, e.g., Carlson [2], Frisch [4] or Johansen [10]). We will show that our measure is a "natural" one from several different points of view. Having developed a measure, we will use it to quantify the returns to scale effect of public goods (or bads). This is done in Sections 5 and 6.