There is something very strange about a counterfactual conditional in which the counterfactual is logically impossible. There are no possible worlds in which something logically impossible happens, so we can't ask which of the possible worlds in that subset is most plausible. The subset is empty.
"If saving were bigger than investment, what would happen to income?"
That's a bit like asking: "If 6 were 9, what would happen to income?".
(Assume closed economy, "saving" means "national saving", etc.)
People's actions have to all add up, but their planned actions don't have to. I can plan to buy 6 apples from you, and you can plan to sell 9 apples to me. And if our plans don't add up, at least one of us will be unable to carry out his plans. We can ask what would happen next if that were the case. But we can't ask what would happen if I buy 6 apples from you and you sell 9 apples to me. It doesn't make sense. It's logically impossible.
Now, if you asked instead: "If planned saving were bigger than planned investment, what would happen to income?", that question makes sense. We are not being asked to imagine something that is logically impossible.
"Planned investment" is a simple concept. It's how many newly-produced investment goods we plan to buy. "Desired investment", or "quantity of investment goods demanded" are equally good ways of saying the same thing.
In a Keynesian model with deficient aggregate demand, where quantity sold is determined by quantity demanded, "planned saving" is a trickier concept. It means "the income we expect to earn from the sale of newly-produced consumption and investment goods, minus the quantity of newly-produced consumption goods we plan to buy". It can be slightly misleading to talk about "desired saving" rather than "planned saving", because the income we expect to earn in these models is (usually) less than the income we desire to earn. Because we expect we can't sell as many newly-produced consumption and investment goods as we desire to sell.
This is what I think Paul Krugman should have said, or meant to say, with [my edits]:
"We’re not forgetting that in the end [planned] savings must equal [planned] investment; we’re just doing the first step in a comparative statics exercise. Suppose that I want to ask how, say, a fall in housing wealth affects the economy. First I ask how much this would increase [planned] savings, holding GDP constant; then I ask how much GDP has to fall to restore the equality between [planned] savings and [planned] investment.
In the end, of course, [planned] savings must equal [planned] investment, which is why GDP must fall from Y1 to Y2."
Actual saving and actual investment are always equal. They aren't just equal in the end, or when GDP adjusts to make them equal. They are the same thing. But planned saving and planned investment only become equal over time if people adjust their expectations and plans to make them equal.
No big deal.
Please, let's all just stop talking about "saving". Talk about the demand for stuff, and its flip-side, the demand for money. That's where the action is.
I'm beginning to forget where this argument all started.