The Elasticity of Scale and the Shape of Average Costs

TLDR

Returns to scale can be defined either along rays through the origin or along expansion paths where costs are minimized; while both yield the same elasticity at minimum‑cost points, the change in elasticity along expansion paths—rather than rays—determines the shape of average cost curves. The paper defines and analyzes the elasticity of scale along expansion paths to clarify its role in determining average cost curves. The study shows that assumptions about elasticity along rays or the slope of the technically optimal surface are neither necessary nor sufficient for producing classical U‑shaped average cost curves.

Abstract

different concepts of returns to scale. The first, which is most widely used for defining returns to scale,' is the relative increase in output as all input quantities are increased proportionally along a ray through the origin. The second, which is the more relevant concept for micro-economic analysis, is the increase in output relative to costs for variations along the expansion path where input prices are constant and costs are minimized at every output.2 As it turns out, these two concepts yield equal measures for the Elasticity of Scale3 e at every minimum-cost point in the input space. However, the change in E with output is generally different for the two concepts,4 since expansion paths do not coincide with rays, unless the production function is homothetic.5 But it is the change of E along the expansion path-and not along the raywhich determines the shape of the average cost curve (AC), and which is therefore relevant for the analysis of firms and industry behavior under competition.6 These relations are stated and analyzed below. It is shown that assumptions made in the literature7 which are related to the behavior of E along rays, or to the slope of the technically optimal surface (where returns to scale are locally constant), are irrelevantbeing neither necessary nor sufficient for implying classical U-shaped average cost curves. The following definitions and results are used here: