Gaining the Master's Approval.

I believe the only option to raise my connected: The Master’s rank is a liberate for Mr.Pages. So the question is should I pick the expensive option for three cp or the cheap option for one cp? I mean long term which is more effective?

Expensive. Much more effective.

You can always pay a bit of Fate and do the Mr Wine’s Revels storylet. I think that’s more efficient, but more expensive in the literal sense of the word.
I’m also told that the Nephrite Lens has a rare option that gives a point of Connected: Masters, though I can’t personally attest to that.

In terms of Mr Pages, garnering all the items needed for his “rare books” and ten thousand Proscribed Materials both take roughly 340 actions, but the rare books give 3cp whereas the Materials only give 1.

The revels only works up to level 4 though.

But by that point you can grind at the Carnival.

The Carnival is a poor option for increasing Connected: Masters. That option is an “either way” luck challenge with a lucky result gaining a loss of Wounds and 30 Cryptic Clues and an unlucky result losing at least 1cp of Masters. You only gain Connected: Masters on rare success.
Unless this has changed since the last time I tried it.
.
edited by Nigel Overstreet on 8/6/2012

The Carnival is a poor option for increasing Connected: Masters. That option is an “either way” luck challenge with a lucky result gaining a loss of Wounds and 30 Cryptic Clues and an unlucky result losing at least 1cp of Masters. You only gain Connected: Masters on rare success.
Unless this has changed since the last time I tried it.
.
edited by Nigel Overstreet on 8/6/2012[/quote]

I lost my Connected: Masters trying to grind at the Carnival once. Never tried it again.