# Substitutive or supportive factors of production

Given the function: $y=A K^\alpha L^\beta, A,\alpha,\beta >0$

$y=aK+bL^{0.5}, a,b>0$

Decide whether capital and labour are substitute or supportive factors of production. How to solve this question? To answer your possible questions i want to highlight the fact that this is neither my homework assignment nor anything i need to submit, furthermore i have simply no idea how to answer this question that is why i am not able to show any attempt to solve this question.

EDIT: My ideas where to somehow look for the slope of MRTS curve if negative this will imply perfect substitutes if not existing then theoretically it should imply supportive factors, however the answer key states that:

if $\frac{\Delta MP_L}{\Delta K}>0 \rightarrow$ supportive

if $\frac{\Delta MP_L}{\Delta K}<0 \rightarrow$ substitues

if $\frac{A\beta K^\alpha L^{\beta-1}}{A\beta \alpha K^{\alpha -1} L^{\beta-1}}>0 \rightarrow$ subsitutes.

And here i am utterly lost.

Let's find the Marginal Product of Labor, $MP_L$: \begin{align} MP_L \equiv \frac{\partial Y(K,L)}{\partial L} &=\frac{\partial }{\partial L}\left( aK + b L^{0.5} \right) \\ &= 0.5\cdot b \cdot L^{-0.5} \end{align} Now finding $\frac{dMP_L}{dK}$ is trivial: \begin{align} \frac{dMP_L}{dK} &= \frac{d}{dK} \left( 0.5\cdot b \cdot L^{-0.5}\right) = 0 \end{align} We see that it does not depend on the capital. So a change in $MP_L$