INTERNAL PRODUCTION AND COST FUNCTIONS IN CONNECTION WITH ECONOMIES OF SCALE

Abstract

A well-known duality principle between production and cost functions states that any concept defined in terms of a production function has a "dual" definition in terms of the associated cost function and vice-versa. We study duality relationships of two kinds. We define several notions of internal economies of scale (in terms of the cost function) for large and small inputs and find their duals in terms of the respective production function. For example, we prove that the average cost increases indefinitely o output goes to infinity, if and only if the production function "f" grows at infinity slower than any positively sloped linear function. On the other hand, for a technology to exhibit increasing initial economies of scale it is necessary and sufficient that "f" be majorized by any positively sloped linear function for all small inputs. These results allow us to exactly describe all technologies that have U-shaped average cost curves. Strict monotonicity and continuity of the production function are the strongest assumptions we make. Problems of this kind cannot be solved using calculus or the Shephard's duality result. The direct approach developed in this paper does nor require differentiability or concavity (convexity).