"No model can have a competitive equilibrium with price-taking behavior and partially excludable nonrival goods.
If you are not an economist, this would be a model in which someone who has a monopoly on an idea can charge for its use, but somehow is unable to influence the price that users have to pay, which should sound implausible at least. If you are an economist, you know that there is a very simple argument based on Euler’s theorem that proves this type of model is impossible."
That seems wrong to me. Maybe I don't understand it; or maybe it is wrong.
Here is a very simple growth model:
Initially, one acre of land produces one ton of wheat per year. No labour, all land identical, fixed supply of land, constant returns, perfect competition, no funny stuff.
Then I come up with an idea that lets one acre of land produce two tons of wheat per year. My idea is non-rival (just because one landowner uses my idea doesn't mean another landowner can't use it too). My idea is excludable (I patent my idea, so nobody can use it without my permission). If you like, we can assume my idea is only partially excludable, because my patent only works in Canada, and I can't stop non-Canadians using my idea.
How much can I charge landowners for using my idea? The demand curve for the use of my idea is perfectly elastic at a price of one ton of wheat per acre per year, then goes vertical at the total number of acres in the world (or in Canada, if it's only partially excludable). If I set the price at more than one ton, nobody will use my idea; if I set the price at less than one ton, everyone will use my idea. So I will set the price for using my idea at one ton of wheat per acre per year (or maybe a smidgen less, if you want to be picky).
Then David Andolfatto comes up with a new idea, that is better than mine. David's idea lets you grow three tons of wheat per acre per year. David can also charge one ton of wheat per acre per year. If he charges more than one ton, people will use my idea instead (I won't charge more than one ton, even if David charges more than one ton, because nobody would use my idea if I charged more than one ton); if he charges less than one ton, everyone will use his idea. So David charges one ton (or maybe a smidgen less), and everyone uses David's idea and stops using mine.
[Update: I'm assuming Bertrand competition between me and David (where we each set price taking the other's price as given). If instead we collude, or play Cournot (each setting quantity, taking the other's quantity as given), we can earn higher profits than this.]
Then Glenn MacDonald comes up with an even better idea, that lets you grow four tons per acre.
And so on. Productivity grows at one ton per acre per new idea. The person with the newest idea earns one ton of wheat per year for every acre in the world (or every acre in Canada, if it's only partially excludable).
That still looks like a competitive equilibrium to me. The person with the newest idea faces a demand curve that is perfectly elastic up to the quantity where he captures the whole market. Just like the demand curve facing an individual wheat farmer in a perfectly competitive market for wheat. The individual farmer can set a price for his wheat above the market-clearing price P* if he wants, but he will sell no wheat if he does this; and he can set a price for his wheat below the market-clearing price P* if he wants, and everyone in the whole world will want to buy only his wheat if he does this, but he won't maximise his profits if he does this.
Here (I think) is where Euler's theorem kicks in. Sure, landowners earn the marginal product of land under the second-newest idea, which is less than the marginal product of land under the newest idea. But that's just an integer problem (is that the right word, in math-speak?). In the limit, as ideas get smaller and smaller, or productivity gets larger and larger, that difference matters less an less. It doesn't matter at all if there's a continuous flow of tiny new ideas. And it doesn't make any difference to the efficiency of equilibrium anyway, if land is in perfectly inelastic supply (Henry George and all that). And it doesn't affect what I said about the demand curve for the newest idea being perfectly elastic.
[Update: see jonathan's comments below.]
What is logically wrong with my counterexample to what Paul Romer said?
I'm crap at math (it took me some time before I vaguely remembered reading about Euler's theorem in Mark Blaug's book on history of thought). I don't do growth theory (unless I have a co-author to help). I drove across Minnesota once (Typepad says that's how you spell it). But me and Minnesota economics ... have issues.
Maybe I'm just signalling my membership of the Western club?
No. I'm pissed because somebody said something (in words) on the internet that seems to me to be wrong, and backed it up with mathy theory that I don't understand. And either I'm wrong, and I learn something new when people explain it to me, or I'm right, and I further my personal agenda of seeking fame and fortune on the internet.