Output frontiers, variable returns to scale and input transfer elasticities

Convex output set is a common hypothesis in production theory and especially in empirical applications. This often seems to be taken for granted without any reference to returns to scale. However increasing returns to scale have implied a necessary and fundamental revision of general equilibrium theory (see Heal G;1999). This paper focuses on the relation between returns to scale and output frontier shape. We show that output frontier shape for a technology with a fixed input endowment and production functions exhibiting variable returns to scale depends on the difference between relative variation marginal products of production functions. Local concavity or convexity of output frontier are determined through several general, useful formulations : (i)convex combination of partial marginal product elasticities of each production function, which could be shared into partial variable returns to scale level and its returns to scale variability elasticity, (ii) input transfer substitution elasticity between production units, or its inverse, (iii) input transfer marginal product elasticity. When returns to scale are non-variable, these formulations determine output frontier shape, not only locally but on the whole definition domain. These results, based only on technical characteristics of technology without market intervention, should help research, not only on output frontier, but also concerning equilibria between consumption and production, when production activities exhibit variable returns to scale --because an efficient equilibrium, if it exists, postulates a solution on the output frontier.