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I read the answer on Why did Tywin allow Shae to be a witness at the trial? that argues that Tywin was teaching Tyrion a cruel lesson when he had Shae testify against him at the trial.
Tywin claims that he wouldn't really have had Tyrion executed, though that was at crossbow-point and he very likely could have been lying to save his own skin.
Would he really have had Tyrion executed?
It's quite different between the book and the show, but I think overall the sentiment can be summarised as follows:
In the show, it's shown that the entire situation was a manipulation of Jaime, to move him to make the offer to return to Casterly Rock in exchange for Tyrion's life. The whole thing was a show of how well he has judged Jaime's character, and will gain him back as the heir to House Lannister. The deal he made there is to send Tyrion to the watch, and since the fundamental purpose is to get Jaime back, if the price is to not execute Tyrion and secretly ship him off to Castle Black, he still has an incentive to go through with the plan; it would just be secret instead of Tyrion making a formal plea. The deal between Jaime and him is the important one.
Now that's not as explicit in the book. In the book, as he's on the toilet, he tells Tyrion that his intent is still to send him to the Wall, but he needed Mace Tyrell's "consent". Tyrion "never heard his father speak the words that condemned him". Jaime speaks to Cersei after the trial and says this all his father wants to do is "force me from the Kingsguard, and send me back to Casterly Rock". The overall run of events might be there, but Jaime adamantly demands that he is not leaving the Kingsguard here, so that particular plot point can either be missing, or it's running implicitly in the background and Jaime doesn't want to reveal it to Cersei. I don't think it's the latter - if it is the latter, Jaime has no incentive to free Tyrion and change the course of events.
You can look at it in many ways from there:
I think it would depend on whether Tywin considered Tyrion to have been broken to his will or not - by learning the harsh lessons and consequences of dallying with whores and, more importantly, in defying your father and family patriarch.
If Tyrion was still defiant, then being able to execute him would be a regrettable, but simple solution. A sacrifice of a piece on in the Game of Thrones.
On the other hand, a compliant Tyrion is an asset - he's smart, capable and he's still a Lannister - heirs and marriage are useful, but not so useful that it's worth have a disobedient child. You can see that in his dealings with Cersei and Jaime - he will tolerate minor disobedience, hate and fear, but not to the point that it harms the Lannister dynasty.
If he considered Tyrion a threat to the family, or of no further use or even if his death has more benefits than otherwise he would have him executed.
It's difficult to tell if Tywin planned to execute him - he's more pragmatic than that, but he certainly would have if he felt it was necessary.
