The attached paragraph is from "The Application of Linear Programming to Team Decision Problems."
I do not understand how the profit depends on the capital limit $k$ (whose definition is also confusing). Is $k$ the maximum capital the company can access?
This is my understanding currently:
cost for production (alloted) = $a$
cost for promotion (alloted) = $b$
amount produced (can only produce based on unit cost for production) = $xa$
amount that can be sold (effectively how well did promotion work?) = $yb$
additional capital cost. This is the amount raised but at an immediate cost = $(1+f)$
how much profit (this is effectively sold subtracted by money spent) = $\min(xa+yb)-a-b$
Now this profit can be used for more production if it's postive and above a certain threshold $k$. Then the profit is: $\min(xa+yb)-a-b$
If negative, capital has to be raised. Let profit be less than $k$. So, $a+b<k$. The cost of this capital raise will be $(1+f)(a+b-k)$. The profit is then: $\min(xa+yb)-a-b-(1+f)(a+b-k) = \min(xa+yb)-a-b-a-b+k-f(a+b-k)$