How can I prove $∇U(x).D_m x(p,m)= \text{shadow price}$?

Question

Why inner multiplication of the gradient of utility function in derivative of demand function with respect to income is equal to shadow price? This is the equation which is given but I don't know where it comes from, I would be grateful if sb can give me a hint

Answer 1

Assuming certain regularity conditions, the first order conditions for $$ \max_{x, \lambda} U(x) - \lambda (p \cdot x - m) $$ are \begin{align*} &D_{x}U(x(p, m)) - \lambda p = 0 \\ \text{and} \quad & p \cdot x(p, m) - m = 0. \end{align*} Moreover $x(p, m)$ will be differentiable with respect to $m$ at $(p, m)$, and this fact together with the second equation implies $$ p \cdot D_{m}x(p, m) - 1 = 0. $$ So, by the first equation, $$ D_{x}U(x(p, m)) \cdot D_{m}x(p, m) = \lambda p \cdot D_{m}x(p, m) = \lambda. $$ The derivative $D_{x}U(x(p, m))$ of the utility function at $x(p, m)$ corresponds to $\nabla U(x)$ in your notation.