The Elasticity of Scale, the Shape of Average Costs, and the Envelope Theorem - Revier, B Charles F. (American Economic Review, 2017)
In a widely cited note in this Review (1975), Giora Hanoch drew attention to the distinction between two different concepts of returns to scale, one concerning the relative change in output for equiproportionate changes in all inputs along a ray from the origin, and the other concerning the change in output relative to costs along the expansion path. Hanoch demonstrated that the two concepts give equal measures for the point-elasticity of scale, ?, at any point on the expansion path for any production function. However, if the production function is nonhomothetic, the rate of change in with output along a ray is generally not equal to its rate of change along the expansion path. Hanoch also demonstrated that the shape of the average cost curve depends upon the change in along the expansion path, not along a ray, and that the assumption of a downward-sloping technically optimal surface (where -=1), with - I below it, is neither necessary nor sufficient for classical U-shaped average cost curves. This note utilizes the envelope theorem to provide an alternative derivation of Hanoch's results concerning the relationship between changes in along a ray and changes in along the expansion path. This approach gives greater intuitive insight into Hanoch's conclusions. It allows graphical illustration of the results, with obvious pedagogical rewards. In addition, the derivation provides one significant new result, namely, a general proposition that the rate of change in ? along a ray must be algebraically less than its rate of change along the expansion path at a point where the ray and the expansion path intersect.