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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.

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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.

However there are other optimization problems where the shadow price is something else, for example in the utility maximization problem

\begin{eqnarray*} \max_{x,y} && U(x,y) \\ s.t. && p_x \cdot x + p_y \cdot y = m \end{eqnarray*}

the shadow prices correspond to the marginal utility of money. (One could argue that this is the reciprocal of the marginal cost of producing more utility.)

Neither of these examples had anything to do with equilibrium, just individual optima.

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