Yeah, Paul Allen might have a 10G$ superyacht, but he's worth 14.8T$, the majority in securities, bonds and so on, I presume.

I assume you mean 10M$ (million, 10^6) and 14.8B$ (billion, 10^9)? I doubt that Paul Allen is worth the entire US GDP. :-)

As for what his wealth is doing, say the majority of it is invested in securities, bonds, etc. How much actual investment is being done with that? "Investment" in the economic sense doesn't mean owning shares of stock, or bonds; it means applying resources to the creation of new forms of wealth, instead of to the consumption of existing forms of wealth. How efficiently is Paul Allen's money being used for that purpose? Since we're talking about Paul Allen, it may actually be used fairly well, but as I said before, he's an exception. I think most rich people do not really care how their money is being used, and don't spend much time or effort trying to use it efficiently. Why would they? Most of them are not entrepreneurs or investors; they're just people who happen to have a pile of money.

you assume Say's Law holds true

Even if I agree that it's true for the sake of argument (see below for further comment on that), it doesn't justify this claim:

Symmetry in your argument would require that growth by demand should also be "exponential"

No, because growth by demand is limited by supply. There was no demand for air travel in 1900, not because people could not have used it, but because it didn't exist. Once it existed, it created a whole new category of demand. To some extent that demand displaced other demand (some people now made trips by air that they would previously have made by car or train), but most of that demand was new demand that increased the total GDP. But that growth was only made possible by the existence of a new source of supply. New sources of supply are only created by investment, not consumption; that's the difference.

Second, this is the classical economic framework, which doesn't work.

Only according to all the non-classical economists whose jobs depend on saying that it doesn't work.

You need to factor in savings and debt, and then Say's Law breaks

In the short run, yes. But what is the savings and debt for? All these things do, in effect, is transfer resources through time: by saving now I put aside resources for later use; by taking on debt now I expend resources that I don't currently have, and pay for it later with excess resources. None of this changes the fact that ultimately, whatever is consumed must be produced.

Well, actually you have debt now, which can be easily be a problem. But my cursory, layman's understanding is that it's a much less severe problem, since d(GDP)/dt > 0 now.

Yes, it is, but why? As I said above, debt means you're using up resources now that you don't have (you borrow them from someone else) with the expectation that you will have excess resources in the future to pay it back. Where will the excess resources in the future come from?

The best outcome is that you have used the debt to fund investments that create new sources of supply, so you have a lot more wealth in the future and can easily afford to pay back the debt.

Another possible outcome is that you keep on increasing your tax base (this is the Krugman solution, referred to by someone else in another post in this thread), so that when the debt comes due you have a lot more people available to tax to pay it back.

The third possible outcome is that you inflate the debt away, by denominating it in your own currency instead of in actual goods. This is how the US government has historically dealt with its debt; US debt is denominated in dollars, and the US can print as many dollars as it likes, so effectively it can make its own debt as small as it likes by making dollars worth less.

For example, in the Krugman article, he mentions the US debt from World War II, and says that we never paid it back; it just grew increasingly irrelevant as the tax base grew. In other words, he says it was the second outcome. But he fails to mention that the US dollar has inflated by almost 1200 percent since World War II, i.e., by a factor of 12, whereas the tax base has grown by at most a factor of 3. So the third outcome has by far the largest effect.