I am currently reading the book "Microeconomics: Principles and Analysis" by Cowell on my own. I'm reading the externalities chapter, and i found an interesting example:
There are just two firms: firm 1 is a polluter and firm 2 the victim. Firm 2 (the victim) makes an offer of a side-payment or bribe to firm 1. The bribe is an amount that is made conditional upon the amount of output that firm 1 generates: the greater the pollution, the smaller is the bribe; so we model the bribe as a decreasing function β(⋅).
The optimization problem is
My question is how did they arrive at those FOC's?
UPDATE:The second part of this optimization is to look at the problem from firm 1 perspective, it follows like this: Now look at the problem from the point of view of firm 1. Once the victim firm makes its offer of a conditional bribe, firm 1 should take account of it. So its profits must look like this
This is from F.A.Cowell - Microeconomics - Principles and Analysis p.444-445