Consider the standard LP with value function \begin{equation} \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\} \end{equation} 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?