According to marginal productivity theory, every input is paid the value of its marginal product. This means that the entire product will always be handed out to those who work on it. In other words, the sum of the marginal products add up exactly to the total output. There is thus neither a surplus nor a deficit left at the end.

This proposition can be proved by using Euler’s Theorem. It suggests that if a production function involves constant returns to scale (i.e., the linear homogeneous production function), the sum of the marginal products will actually add up to the total product.

This can be proved by the total differentiation theorem. Now, if we have the function z = f(x, y) and that if, in turn, x and y are both functions of some variable t, i.e., x = F(t) and y = G(t), then

So the effect of a change in t on z is composed of two parts: the part which is transmitted via the effect of t on x and the part which is transmitted through y. Thus, the latter is represented by the expression (∂f/∂y) (∂y/∂t).

Here (dy/dt) shows the change in y produced by the increment in t and (∂f/∂y) is the resulting change in z produced by each unit of this change in y.

It follows from a linear homogenous production function

P = g (L, C), where we have, for any k,

kP = g(kL, kC)

If we now take the total derivative of kP with respect of k [i.e., setting kP=z, kL=h, kC=y, and k=t in our formula for (dz/dt)] we get

Since this result holds for any value of k it is must also be valid for k = 1 so that