# Présentation d'une propriété nouvelle des multiplicateurs de Lagrange et application à la fonction de coût d'une firme avec produits liés

*Axel Pierru* ()

*Revue d'économie politique*, 2006, vol. 116, issue 5, 683-695

**Abstract:**
We show a simple but novel property of Lagrange multipliers in non-linear constrained optimization: at the optimum, the product of the objective function value with its scale elasticity is equal to the sum of the products of every Lagrange multiplier with the right-hand side coefficient and the scale elasticity of the corresponding constraint equation. In other words, when the value of a constrained optimization problem is considered as a function of the right-hand-side coefficients of the constraints, a general form of the Euler?s theorem holds. We apply our result to the determination of the cost function of a multiproduct firm, when using the process analysis approach. In this case, a vector-valued production function is assumed to exist, each element in which being the production function (or, more generally, the demand constraint) of a given output. In perfect competition, the cost function of the firm is equal to the sum of the products of every marginal cost, the scale elasticity and the value of the corresponding output production function. This leads to a cost-allocation rule among products which is particularly relevant. Under certain assumptions, the cost allocated to a given product is equal to the average cost of a decentralized production of this sole product.

**Keywords:** Lagrange multipliers; cost function; multiproduct firm; marginal cost; scale elasticity; Euler's theorem (search for similar items in EconPapers)

**Date:** 2006

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