# Extension of a new duality theorem in linear programming - Application to the breakdown of long-run marginal costs

*Axel Pierru* ()

Working Papers from HAL

**Abstract:**
In some cases, the objective function of a linear programming problem can be split up into distinct elementary functions, all of which giving rise to a specific interpretation. The theorem we will reformulate was originally derived by Babusiaux (2000) in such a context. The aim was to define a marginal cost in CO2 emission for each of the refined products of an oil refinery. The initial objective function resulted from the addition of a classic operating cost function and a CO2-emission cost function. The author showed that at the optimum, under some assumptions, this marginal cost had an average-cost structure. In other words, the wellknow duality property was extended to each one of the elementary functions, for which it is possible to define elementary dual variables. We reformulate and generalize this theorem. We show that it is valid in any basic feasible solution. Moreover, we provide a simple interpretation of this result. We then show, with a capital budgeting example, how to break down a long-run marginal cost into a marginal operating cost and a marginal equivalent investment cost. This decomposition allows us an in-depth analysis of the formation of longrun marginal costs. This theorem, with the associated concept of elementary dual variable, should give rise to a sizable number of applications in capital budgeting modeling.

**Date:** 2002-04

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