# why is real wage divided by marginal product of labor often called real marginal cost?

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.

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:

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