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As said in the title, why is real wage divided by marginal product of labor often called real marginal cost?

In mathematical formula, $mc_t = (W_t/P_t)/MPN_t$ where $MPN_t$ refers to marginal product of labor, $\partial Y_t/\partial N_t$ and $N_t$ is amount of labor. $W_t$ refers to nominal wage, and $P_t$ refers to price level.

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This relation can be obtained if we realize that Total Cost can be written as a function of output (which in turn is a function of input factors), but also, that Total Cost equals total payments to factors of production. In real terms,

$$TC=TC(Q) = (w/p)N + (r/p)K$$ with the payment rates exogenous. But $Q = Q(N,K)$. Differentiate both sides by $N$:

$$\frac {\partial TC(Q(N,K))}{\partial N} = w/p $$

$$\Rightarrow \frac {\partial TC(Q(N,K))}{\partial Q}\cdot \frac {\partial Q(N,K)}{\partial N} = w/p$$

$$\Rightarrow MC \cdot MPN = w/p \Rightarrow MC = \frac {w/p}{MPN}$$

One can perform the same exercise to obtain the analogous relation with respect to capital:

$$MC = \frac {r/p}{MPK}$$

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