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;
        


        











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