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

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

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.
• I'm not sure I understand what you mean. Is the time spent working viewed as an entry of the consumption decision $x$, and would then overtime correspond to a restriction of $x$ that must be satisfied in addition to the normal budget constraint? – induction_is_a_laddah Nov 10 '19 at 16:38