# Derivative of shadow price in LP objective coefficient

Consider the standard LP with value function $$$$\Omega(\mathbf{A}, \mathbf{b}, \mathbf{c}) = \max_{\mathbf{x}} \left\{\mathbf{c} \cdot \mathbf{x} \,|\, \mathbf{A}\mathbf{x} \leq \mathbf{b}, \mathbf{x} \geq \mathbf{0}\right\}$$$$ and given parameters of suitable dimensions. Supposing primal non-degeneracy, we know the $$k$$-th Lagrange multiplier $$\lambda_k = \partial \Omega / \partial b_k$$ has the shadow price interpretation.

Are there any general results about $$\partial \lambda_k / \partial c_\ell$$, perhaps under additional assumptions about the parameters?

• Let $z=\textbf{cx}$ the profit of $n$ products with $m$ constraints, where $1\leq k\leq m$. If you raise the bound $b_k$ by one unit, then the optimal value $z^*$ raises by $\lambda_k$. I can post an example if you want. Jan 31, 2023 at 18:37
• Thank you for the comment, but I have already noted this in the question. Jan 31, 2023 at 18:42
• Sorry . I don't see that any additional assumption has to be made. Jan 31, 2023 at 18:44
• You seem to answer a different question than I asked: I am interested in the derivative of $\lambda_k$ in the $\ell$-th coefficient in $\mathbf{c}$ of the value function $\Omega$. Jan 31, 2023 at 18:46