2.2 Marginal Products

Another word for the output of a production function is the total product — i.e., $f(L,K)$ is the “total amount produced” using $L$ hours of labor and $K$ units of capital.

The partial derivatives of a production function therefore represent the marginal product generated by increasing the level of an input. For example, if the two inputs for a production function are labor and capital, the marginal product of labor ($MP_L$) is the amount by which the total product increases per additional hour of labor, holding $K$ constant: \(MP_L = {\partial f \over \partial L} = \lim_{\Delta L \rightarrow 0} {f(L + \Delta L, K) - f(L, K) \over \Delta L}\) Likewise, the marginal product of capital ($MP_K$) is the amount by which the total product increases per additional unit of capital, holding $L$ constant: \(MP_K = {\partial f \over \partial K} = \lim_{\Delta K \rightarrow 0} {f(L, K + \Delta K) - f(L, K) \over \Delta K}\) Visually, can be seen as the slopes of tangent lines in the $L$ and $K$ directions at a point along the production function:

It’s important to bear the units of marginal products in mind. In each case, they measure the additional output per additional input. So the $MP_L$ is measured in units of output per hour of labor, and the $MP_K$ is measured in units of output per unit of capital.