Discussão dos retornos à escala nos contextos das funções de produção e de custo

The article deals with the cost function at a mathematical level that only requires knowledge of differentials, but except for that, it keeps rigor at a high level. It only states theorems that require long proofs. The article justifies the existence of the cost function, points out its properties, and shows how it relates with the production function in the sense that one is the dual to the other. The article discusses partial and scale elasticities, both in the context of production and cost functions. Whenever profit is maximized, one is the reciprocal of the other. The cost function has not a defined form in the sense that it can be deduced from the axioms of production theory. But the articles points out the plausibility of the form that resembles an open U. The existence of the demand function for a given level of production requires the unicity of the solution of the minimization problem. Whenever the solution is unique, one indicates how to obtain the demand from the cost function. Constant return to scale is not compatible with a competitive economy, and if it prevails net profit is zero, after all factor of production gets its share. The paper extends the cost function to several products, and emphasizes that only by pure luck the minimum average cost shows up if just one observation is surveyed. The article prepares the reader for a more advanced exposition of the cost function, and the text references some of them, and also is useful for econometric research.