Commenter John Handley points out (correctly) that Paul Romer is talking about real output, while in my previous post, I talk about the price level. The problematic extrapolation from RGDP growth rates doesn't really depend on using real output or the price level (specifically, the physical analogy I wrote down); I was attempting to connect the result to a previous result about the price revolution of the 1500s (or so).
So I went back and re-worked the result in terms of real output (R) with real growth rate ρ (i.e. R ~ exp ρ t). I also made use of some stuff from the section on the AD-AS model in the paper. Using the market (information equilibrium relationship) P : N ⇄ M we can show
k M ≤ R ≡ N/P
And the picture we produce is very similar to the price level one and has the same interpretation -- simply extrapolating back with constant real growth rate ρ doesn't capture the fact that there probably wasn't a monetary economy in the past:
What's also interesting is if we assume information equilibrium and use the "money mediated AD-AS model" from the paper, i.e.
N ⇄ M ⇄ S
we can show that, if k ≡ kn/ks where kn is the IT index of the market N ⇄ S and ks is the IT index of the market M ⇄ S, we have the information equilibrium relationship R ⇄ S with IT index ks. That means
R ~ exp ρ t ~ exp ks σ t
where σ is the growth rate of aggregate supply. Therefore ρ = ks σ. The real rate of growth ρ is only proportional to the rate of growth σ of aggregate supply widgets. See the derivation in the postscript below.
We don't know if ks is changing or even what its value is (except that it is of order kn in order for k ~ 1 empirically). It could be greater than one or less than one. Therefore so-called real output doesn't necessarily represent real widgets, but rather some conversion of physical widget units to money units. We have R ≡ N/P = N/(dN/dM) so the units of real output are the units of M (i.e. dollars).
An argument about real growth isn't an argument about physical widgets, so saying 2% real growth extrapolated backwards to the year 1000 is a small number doesn't tell you anything about the number of physical widgets. If ks > 1, then σ < 2% and the number of physical widgets in the year 1000 could be much greater than would be surmised from a tiny value for R.
Of course, this is a model dependent result. And that is the point -- Romer's argument that real growth has been accelerating is also model dependent.
PS Here are more details of the derivation (in long hand):