The Producer’s Minimum Cost Function

For given rates of output U and prices p_i of the factors of production, the rates xi at which the factors are used satisfy equation (2) and minimize (5), if the process is organized instantaneously for minimum cost. Necessary conditions for this minimum property are (6)$$ \begin{gathered} {p_{i}} = \,\lambda .\frac{{\partial \psi }}{{\partial {x_{1}}}}\quad \left( {i = 1,{\kern 1pt} \,2,...,\,n} \right) \hfill \\ \psi \left( {U,{x_{1}},\,...,\,{x_{n}}} \right) = 1 \hfill \\ \end{gathered} $$where the Lagrangian multiplier λ may depend in general upon U, p₁,..., p_n.