In Young's case the exponent on ideas is one, and progress looks like log(log(researchers)). (You need to pay a fixed cost to make the good at all in a given period, so only if you go above that do you make positive progress.) See Section 2.2.

Peretto (2018) and Massari and Peretto (2025) have SWE models that I think do successfully avoid the knife-edge issue (or "linearity critique"), but at the cost of, in some sense, digging the hole deeper when it comes to the excess variety issue.

Thanks!

And yeah, that's fair. One possible SWE-style story I sort of hint at there is that we have preferences like the ones I use in the horses paper; process efficiency for any given product grows exponentially with a fixed population; and there are fixed labor costs to producing any given product. In this case, it's clear that measured GDP/capita growth will be exponential (but all "vertical") with a fixed population. But if you set things up in just the right way, so that measured GDP always increases by the same proportion when the range of products increases by some marginal proportion, it will also be exponential with a growing population ("vertical"+"horizontal").

But I think it's hard to not have this all be a bit ad-hoc / knife-edge. E.g. you'll typically have to start out ever less productive at making the new products, or else the contribution to real GDP of successive % increases in the product range will blow up: as you satiate in existing products, you're willing to trade ever more of them for a proportional increase in variety. Alternatively, you can say that the range of products grows subexponentially when the population grows exponentially, because the fixed costs of the later products are higher.

A bit tangential, but I can't help sharing a data point I came across recently on how prepared the US government currently is for advanced AI: our secretary of education apparently thinks it stands for "A1", like the steak sauce (h/t). (On the bright side, of course, this is a department the administration is looking to shut down.)

(FYI though I think we've chatted about several new varieties issues that I think could come up in the event of a big change in "growth mode", and this post is just about one of them.)

No worries, good to hear!

Thanks! People have certainly argued at least since Marx that if the people owning the capital get all the income, that will affect the state. I think more recent/quantitative work on this, e.g. by Stiglitz, has generally focused on the effects of inequality in wealth or income, rather than the effects of inequality via a high capital share per se. But this isn't my area at all—ask your favorite LLM : )

The reference point argument is also about consumption inequality rather than what gives rise to it. My guess would be that if we all really get radical life extension and a huge quantity of amazing goods and services, that will probably for most people outweigh whatever jealousy comes with the knowledge that others got more, but who knows.

In any event, my guess would be that even if the marginal product of labor stays high or rises following full automation, most people's incomes will eventually come not from wages, but from interest on whatever investments they have (even if they started small) or from redistribution. And full automation could well trigger so much redistribution that income inequality shrinks, since it will remove one motivation for letting income inequality remain high today, namely that unlike with robots, taxing productive people too much can discourage them from working as much.

Great to hear, thanks!

As for the prediction—fair enough. Just to clarify though, I’m worried that the example makes it look like we need growth in the new good(s) to get this weird slow GDP growth result, but that’s not true. In case that’s the impression you got, this example illustrates how we can have superexponential growth in every good but (arbitrarily slow) exponential growth in GDP.

Here's an example in which utility is additively separable, $u_n(.)$ is identical for all goods, the productivity and quantity of all goods grow hyperbolically, and yet GDP grows exponentially.

Ok, fair enough--thanks for getting me to make it clearer :). So I guess the disagreement (if any remains, post-retitling/etc) is just about how plausible we think it is that the technological advances that accompany full automation will be accompanied by further technological advances that counterintuitively slow GDP growth through the "new-products-Baumol" mechanism illustrated here. I don't think that's so implausible, and hopefully the note I'll write later will make it clearer where I'm coming from on that.

But this post isn't aiming to argue for the plausibility, just the possibility. It seems to me that a lot of discussion of this issue hasn't noticed that it's even a theoretical possibility.

not being on track to produce Good 2 only happens in your model specifically because you define automation to be a thing that takes Good-2 productivity from 0 to something positive... Automation is usually understood to be something that increases the productivity of something that we could already produce at least a little of in principle

Okay, I'm happy to change the title to (a more concise version of) "the ambiguous effect of a technological advancement that achieves full automation, and also allows new goods to be introduced on GDP growth" if that would resolve the disagreement. [Update: have just changed the title and a few words of the body text; let me know.]

On the second point: in practice I don't think we have additively separable utility, and I don't know what you mean by "extracting this from our utility function". But anyway, if I'm understanding you, that is wrong: if your utility function is additively separable with an upper bound in each good, say $u\left(x\right)={\sum }_{n}max(0,1-1/x_n)$, a technological shift can yield superexponential growth in the quantity of each n but exponential GDP growth. I'll write up a note on how that works this evening if that would be helpful, but I was hoping this post could just be a maximally simple illustration of the more limited point that Baumol-like effects can slow growth even past the point of full automation.