Consider an LP $\max_x c^t x$ s.t. $Ax \leq b, x\geq 0$. Let the dual problem be $\min_v b^t v$ s.t. $A^\top v \geq c,v\geq 0$. It is stated in standard text book that the shadow prices of constraints in the primal problem are exactly $v^*$, the solution to the dual problem.
May I ask for a formal and rigorous proof of why this is the case? Any references are greatly appreciated. I did extensive search online but in almost every lecture note/references it is only mentioned that $v^*$ is the shadow price without proving it. Some might use the graphical method for an interpretation but I am looking for a completely rigorous mathematical proof.
To be more rigorous: let $V(\beta)=\max_{x\geq 0}c^t x$ s.t. $Ax\leq b$. Prove that there exists an non-empty neighborhood of $b$ such that for every $\beta$ in the neighborhood, $V(\beta)=V(b)+v^\top (\beta-b)$.
