# What is the meaning of $k$ in this paragraph from “The Application of Linear Programming to Team Decision Problems” by Radner?

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) = $$\min(xa+yb)-a-b$$

1. Now this profit can be used for more production if it's postive and above a certain threshold $$k$$. Then the profit is: $$\min(xa+yb)-a-b$$

2. If negative, capital has to be raised. Let profit be less than $$k$$. So, $$a+b. The cost of this capital raise will be $$(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)$$