What is the the return to scale of per capita production function?

If $$Y=f(AL,K)$$ is CRS, $$a_k+a_L=1$$ by the Euler Theorem.

However, I wanted to know the return to scale of $$y=f(1,k)$$ (i.e. $$Y$$ divided by $$AL$$). I tried $$z=p/AL$$ , then $$py=f(p,pk)$$, differentiate w.r.t $$p$$:

$$y=f(p)+f(pk)\cdot k$$

substitute $$p=1$$:

$$y=f(p)|_{p=1}+f(k)\cdot k|_{p=1}$$

I stopped here, because I don't know the sign of $$f(p)|_{p=1}$$.

• You might want to consider taking some time to format your question and make it more readable :) – Art Sep 20 '19 at 2:47