# Is it possible to derive the marginal product of an input using a transformation function?

I'm using a transformation function $F(\cdot)$ to describe a production set $y = (x, z, L, K)$, where $x$ and $z$ are private goods denoted by positive numbers, $L$ is labour input, $K$ is capital input and both are denoted by negative numbers. Here I'm just following Mas-Colell (1995), page 128, but changing the labels of the variables in the production set.

In this case, if $F(y) = 0$, I think I can claim that the marginal rate of transformation (MRT) of good $x$ for good $z$ is

$$MRT_{xz} = \frac{\partial F / \partial x}{\partial F / \partial z}$$

and also that the marginal rate of technical substitution (MRTS) of labour ($L$) for capital ($K$) is

$$MRTS_{LK} = \frac{\partial F / \partial L}{\partial F / \partial K}$$

In this same setting, can I also claim that the marginal product of labour ($L$) with respect to the private good $x$ is

$$MP_L = \frac{\partial F / \partial L}{\partial F / \partial x}$$

• This seems to be a $MRS_{L,x}$ (even though that it does not make so much sense). I believe that $MP_L=\partial F/\partial L$ – Yorgos Jan 2 '17 at 16:22