Example

Let $P(L,K)=L^{\frac{1}{2}}K^{\frac{1}{3}}$ be a production function with production factors labor $L$ and capital $K$.

Then $P'_L(L,K)=\frac{1}{2}L^{-\frac{1}{2}}K^{\frac{1}{3}}$ is called the Marginal Physical Product of Labor and denoted by $MPP_L(L,K)$.

At an input of 100 units of labor and 1000 units of capital the marginal physical product of labor is

$MPP_L(100,1000)=\frac{1}{2}100^{-\frac{1}{2}}1000^{\frac{1}{3}}=\frac{1}{2}$,

which means that at a constant input of 1000 units of capital the extra output as a result of one extra unit of labor at an input of 100 units of labor is approximately $\frac{1}{2}$ units.

Furthermore, $P'_K(L,K)=\frac{1}{3}L^{\frac{1}{2}}K^{-\frac{2}{3}}$ is called the Marginal Physical Product of Capital and denoted by $MPP_K(L,K)$.

At an input of 100 units of labor and 1000 units of capital the marginal physical product of labor is

$MPP_K(100,1000)=\frac{1}{3}100^{\frac{1}{2}}1000^{-\frac{2}{3}}=\frac{1}{30}$,

which means that at a constant input of 100 units of labor the extra output as a result of one extra unit of capital at an input of 1000 units of capital is approximately $\frac{1}{30}$ units.