The math seems right, but I don't understand why you
would assume that K is constant. It should be an increasing function of
A, no? My intuition is that as ability increases, people have to do more
and more costly stuff to differentiate themselves.
If K is increasing in A, then it's not at all clear that as A rises more people will signal.
This is definitely plausible in theory. But in the real world, academic standards actually seem to fall at a given education level as education levels rise. A B.A. ain't what it used to be; neither is a high school diploma. Better students respond to stagnant or falling standards by acquiring additional degrees. So if anything my model understates the response of diploma completion with respect to ability.