That's not exactly what I'm asking.
Why do you think that the Argent Savant gets an improved caster level at all, even on levels where he gains extra spells per day? I'm looking for the rule that provides that key bit of information. Otherwise, by the RAW, no prestige class allows the caster to improve his caster level even if he gains extra spells per day.
(edit)
Hmm...I think I have found my answer. In some of the PrCs, they mention that you gain an extra caster level in the section on extra spells per day, in the description of that class ability. However, the Archmage doesn't have that line, and that's the PrC I was looking at when I came up with the question. I flipped to the front of the section to look for a general rule on improving caster level, but there is none.
So it seems that the rule is included in the descriptive text of each class. Of course, the omission from the Archmage means that the Archmage doesn't gain caster levels when he gets extra spellcasting levels.
Had they defined "spellcasting level" somewhere, this wouldn't be a problem. I notice that the FAQ seems to think that "spellcasting" has been defined and includes extra spells known, which isn't true (actually, I'm pretty sure I caused that FAQ entry to be added, since it matches an email conversation I had with CustServ a while back). Rather than add errata that defines these things broadly and generally, they rely on patch-up rules. How odd.