17.5.20 Chain Rule Example, Economics Word Problem

First, just set up the problem and what we are looking for. 
We have 

and we are trying to find;

Starting with , we start with 



and what is partial-P over partial-L? - Well, capital-P isn't a direct function of capital-L, we'll therefore need the chain rule to get through the intermediate variables inside 


Going throw the terms inside the brackets one-by-one,

  l we don't know -- if we were given a functional form for  we could probably find it, but for now we'll just leave it as 

  is pretty simple, given , it's just .

   isn't known. Given a functional form of  we could probably solve for it -- but now we'll just leave it at is.

 We are told to assume that k is independent of L & h. Which implies that .

Great, so we can rewrite the above as;     

which further simplifies to the follow,

Now let's find 



Now let's break open 


Now we're going to go through each one of those four terms inside the bracket;

 isn't known to us now. If we were given a functional form for P, we could probably find it easily. But for now we leave as is. 

 - since we are told that , this implies that  

 isn't known to us now. Again, if we were given a function form for P, finding this term wouldn't be too hard. But for now we leave it as is. 

, since we were told in the question that k is independent of h (and L).

This means that the above terms simplifies into the following;