The Shape of Ray Average Cost and Its Role in Multioutput Scale Economies: Some Generalisations

Establishing a link between the so-called “neoclassical” and “axiomatic” approaches to production theory, we deal with some central and unresolved issues concerning scale economies in multi-output technologies. First, we reformulate Panzar and Willig’s (1977) result on the duality between primal and dual scale elasticity measures, which helps pointing out the unknown role played in this regard by the monotonicity of the local degree of homogeneity of the technology. Second, under a general representation of a convex technology - allowing for non-differentiability of the cost function and multiple optima - we determine the shape of the ray average cost function. Third, in the same setting, we determine an unambiguous relationship between cost scale elasticity and cost scale efficiency, and therefore between local and global scale economies. Fourth, we develop a complete map of values taken by primal and dual scale elasticities and point out that the equality between returns to scale and scale economies local measures breaks down in a convex technology at points where the cost function is not differentiable. These general results are then applied to simplification and solution of some theoretical and computational problems featured by some important models for the estimation of scale economies, such those of Baumol and Fisher (1978), Färe and Grosskopf (1985) and Sueyoshi (1999).