# Shadow Price vs Marginal Cost

Is there a difference between the two? If so can you intuitively explain it? I see internet resources saying these are equivalent, but I thought I remember someone telling me once shadow price was different than marginal cost, but I may be misremembering it. Bonus points if you can place the definition in the context of a non-equilibrium setting.

Marginal cost is always the same as the shadow price in the cost minimization problem \begin{eqnarray*} \min_x && w \cdot x \\ s.t. && f(x) = y. \end{eqnarray*} In optimum the shadow price (Lagrange multiplier) belonging to the condition $f(x) = y$ is the marginal cost.