Some More Really Bad Math*

Given the following assumptions:

1. As average total compensation a increases linearly, the number of talented people y increases proportionally (The You Get What You Pay For Theorem);
2. As the number of idiots i increases, the rate of hiring more idiots p increases exponentially (The Law of Critically Stupid Mass);
3. As average total compensation a increases linearly, the rate of retention u increases exponentially and tends to infinity (The Fur Lined Rut Axiom);
4. Productivity q is directly proportional to the number of talented people y and inversely proportional to the number of idiots i (The Law of No Free Lunch).
   

Find the value of i which maximizes the following equation, an expression of corporate value:
You must show all your work. No credit will be given for incomplete answers.

*A mathematician of medium stature has gladly endorsed this problem as "Terrible."