EDIT: I have since found a proof that proves the proposition I was questioning, so this is correct.

Hey folks,

I'm currently reading a textbook that wants to give a theoretical underpinning before some empirics are unleashed, motivating to use structural estimation.

In the beginning, the author defines production technology of a single final good as being produced using capital and labor in a linear homogeneous production function:

Q = f(K,L)

The author then imposes positive first partial derivatives and negative second partial derivatives (second in the same argument, so f_LL and f_KK < 0). He then writes that "linear homogeneity implies that f_KL > 0" i.e. the cross-derivative with respect to labor and capital is positive.

However, the production function f(K,L) = K L is linear homogeneous but does not exhibit a positive cross-derivative, so it seems odd to say that linear homogeneity implies it. Naturally, this technology of production does not allow for second-order derivatives unequal to zero whatsoever, so this production technology is not one that would fit his full description, but is there something inherently incompatible with strictly(!) negative second-order-derivatives in the same argument and a zero (or negative) cross-derivative or can anyone think of an example that contradicts the demands on technology he gives?

Of course I understand that most standard CD or CES technology structures satisfy his condition, but it seems odd to me to declare that linear homogeneity implies it rather than just imposing it. I'd have no problem assuming it, to be honest; some complementarity between labor and capital is, at the end of the day, not unreasonable. Just seems odd to say its implied by linear homogeneity.

EDIT: I have since found a proof that proves the proposition I was questioning, so this is correct, I believe.