Duality between the Short Run Profit and Production Function

Duality in microeconomic theory takes an important role in the theoretical work and in empirical research. The dual description of convex set implies the unique structure of microeconomic phenomena in the consumer and producer theory. In this article the unique structure of microeconomic phenomena is analysed on the problem of duality between the short run production and profit function which describe the firm's technology. The primal problem includes short run profit maximisation, in which the choice variable is the quantity of labour and the normalised short run profit function is derived. The dual model includes minimising the sum of variable costs and maximum profit. Special attention is given to the analysis of the first order necessary condition of both optimisation problems which imply duality between the Hotelling's lemma that generates the labour demand function in the short run and its dual which generates the inverse demand function for labour. It is shown how the dual of Hotelling's lemma generates the solution of the primal problem, and how the Hotelling's lemma generates the solution to the dual problem. Finally, the comparative static results that accompany the short-run production theory, important from an empirical standpoint, are especially emphasised.