"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.
Worthwhile Canadian Initiative 
Hi rsj
no, you’re addressing a different question to the one I am asking. I am only asking about why replication arguments prove aggregate production functions must be CRS in rival inputs. But Nick reckons the replication argument does not rule out IRS in rival inputs in aggregate, only DRS.
Luis,
I think Nick is wrong!
Ha!
The argument shows that the function is degree 1 in one input. But it has a second input. The total degree is the sum of the inputs. IRS means degree > 1. The only way it’s not degree > 1 is if that second input is degree 0 — e.g. is a constant. The assumption of growth theory is that patents are a choice variable such that increasing the quantity of patents increases total production. Therefore the total degree has to be > 1.
I do not get the semantics here.
The “technology” refers to the production function, right? Not to its arguments/variables. In Af(K,L) the technology isn´t “A” but e.g. one day, with one technology, 2f(K,L) and another day, with another technology, 3f(K,L).
Obviously, if the f(K,L) part (and, thus, the technology Af(K,L)) has constant returns to scale, you can´t assign all the payoff to the variables and at the same time give something to the owner of the function.
You can, however, assign a payoff equal to its marginal contribution if you enter “technology” (or, in your case, incremental change in productivity compared to the previous technology?) as an argument into the production function.
That might make analytical sense, depending on what you want to do, but is it conceptually sound?
PS: “you can´t assign all the payoff to the variables” should read “you can´t assign the variables their marginal product times the amount of the variable” – or however it is you should put it.
“The “technology” refers to the production function, right? ”
No, that’s exogenous growth. This is an endogenous growth function, in which useful and excludable knowledge is an input into the production function just as labor and capital are.
As to whether it is sound, well, is a production function sound? Certainly it’s more sound than exogenous production function in which knowledge is treated as an endowment, but you may have your own usefulness threshold.
rsj,
so it’s “The argument shows that the function is degree 1 in one input.” that I am disputing.
Do you accept that IRS in rival inputs can exist at the firm level? If you are ruling that out, then fine, we’re done.
If IRS can exist at the firm level, do you accept that when adding more inputs it matters whether you are duplicating firms or increasing scale of existing firms?
If so, you can duplicate earth and only double output whilst still having IRS in aggregate, when the scale of firms in increased.
@Luis, You seem to be relying on “economies of scale”. That doesn’t happen by mere duplication of (rival) inputs. There has to be some (nonrival) reorganization as well to accomplish it.
Jeff, not sure. Just doing the same but larger ought to be possible over some range without needing to invoke anything akin to the generation of new nonrival knowledge.
Luis, Not new nonrival knowledge, old nonrival knowledge. Common sense (or “what every schoolboy knows”) about how to re-do an org chart when growing a firm so as to maximize efficiency. E.g. adding new hires under existing managers rather than hiring new managers for them (up to a point anyway). Etc. etc. etc.
jonathan: I think I see your point, but I think we could argue it both ways. Here’s one example which seems to go the other way (I just got a new computer): How many copies of Windows 7 does my university use? “One per professor” seems to be the “natural” answer. Microsoft gives us a site licence, but I wonder if a university with twice the number of profs pays twice the site licence? Is that a “competitive” price structure?
@Nick 02:39 PM, Two word counterargument as to competitive pricing environment: Microsoft, Windows. 😉
Luis,
I don’t understand the point you are trying to make — are you arguing that
1. The aggregate production is not IRS, so that the argument that a competitive equil. doesn’t exist does not hold?
2. The aggregate production is IRS, but can be shown to be IRS with a different line of reasoning than earth-doubling argument?
In principle, you can try to argue that if we add the second earth, then trade with the two earths creates an environment in which output is more than doubled in rival inputs. The difficulty with that argument is that the relative prices are going to be the same so how would the two earths trade together? But that’s really a distraction because we only need to show that total output is not less than double in order to get the IRS result we want.
Looking at an individual firm is a distraction. Even if one firm in the economy has IRS in rival inputs, the aggregate production function might not, because there might be diminishing returns on producing the intermediate inputs needed by the increasing returns firm, so the aggregate function might be CRS. You really need to look at the whole earth doubling.
Ugh. Isn’t this just the Schumpeter view of innovation versus the “Arrow” view of it? It’s been awhile so I can’t give the references but the way I remember is this (in my own words):
Let’s say an “idea” represents a reduction in the costs of production.
Schumpeter says ideas are non-excludable. So you need monopoly power to incentivize idea production. The gain to a monopolist from inventing lower cost production technique is then that little rectangle between the new and old marginal cost curves. The monopolist comes up with new ideas until the size of that little rectangle is equal to the (fixed) cost of coming up with the new idea.
Arrow (it may actually be someone else) says that competition can drive innovations. Say you got lots of firms producing a homogeneous good. Lots of firms, homogenous good, so it’s “competition”. But they do get to set prices. Somehow one firm ends up with lower marginal costs than the other firms. It does limit pricing, setting price at the level of the second lowest marginal cost. It gets the whole market, makes a positive profit. If you’re strictly Marshallian that is actually a “short run competitive equilibrium” (we DO teach in Econ 101 that in short run firms can make profit). Now suppose all firms start with same marginal cost. They can all pay some fixed cost for a chance to get their marginal cost lowest of all for a period and so make positive profits. Some more assumptions here. Then it becomes a competitive race where you better innovate or leave (maybe you need some costs of staying in the market). Here competition drives innovation.
I was under the impression that the “Arrow” (again I’m not 100% sure that Arrow came up with it but for whatever reason that’s who I associate it with) model was textbook, if I had time I’d track down the 195x citation. The Boldrin and Levine (2008) paper that Romer is hating on is pretty much a dressed up version of it.
Romer seems to be upset that Boldrin and Levine are calling this “perfect competition”. Which is sort of right, because it’s not. It’s limit pricing. But it doesn’t make the model wrong. Romer seems to be upset that there’s some insinuation that “perfect competition” models can explain innovation even when ideas are non-excludable. He’s right, there’s some equivocation going on. There is a difference between “Bertrand competition with limit pricing” and “perfect competition”. IO folks have done this to death. But the Arrow/Boldrin&Levine model does suggest that you can have innovation even under competitive market structures. Macro, and growth, folks, should be aware of it. They probably are; come on, do you really think Levine never heard of limit pricing before? So yep, there’s some unwarranted rhetoric in that paper and Romer is right in general even if he’s nit picky in the particulars.
Did you get lost in that? I did, a bit. Romer is sort of right, and exactly right about some things, but then he’s not right or exactly right about other things, but that’s because the guys he’s criticizing are wrong about some things too.
But anyway. This kind of an issue could be one of those “methodological debates” that take place in top journals all the time. Except that they don’t. Which is why Romer wrote a nasty piece in a non-peer reviewed issue of AER and then posted all over his blog rather than write a “Comment” paper. But let’s pretend for a moment that these kind of issues do actually get hashed out in the top econ journals (we’re economists, we’re good with the “let’s suppose” unrealistic assumption). Even then the striking thing about Romer’s post is that some other guys published a paper which had a blatant mathematical mistake in it, it was pointed out to them, they re-submitted it anyway without correction, it got published and everybody danced a dance of joy. That’s is “Mathiness”. (I can actually give a few similar instances). The whole griping about innovation and perfect competition and economies of scale is a side issue, or at least Romer should’ve put it in a different blog post.
Rsj
There is a proposition – all agg prod funvts must be IRS in rival inputs – and a proof – earth duplication. That’s the thing I am disputing and only that. My reasoning is IRS at firm level as explained above
notsneaky: good comment. I had to read it twice, but I think I get what you are saying.
“Arrow” wouldn’t be “Samuelson” would it? (I’m just guessing.)
“There is a difference between “Bertrand competition with limit pricing” and “perfect competition”.”
The (rather good , IMHO) debate between me and jonathan here does seem to come down to what we mean by “competition”. My reading of Romer is that he identifies “competition” with “inputs get paid their value Marginal Products”, while I think that W=VMP is a consequence of competition plus some other assumptions. (For example, we don’t get W=VMP when there are taxes. Or if firms don’t know what VMP actually is.) People can have different concepts of “competition”, and it’s not obvious which one is most useful.
I still don’t get what Romer means by “mathiness”, even though it seems he has tried hard to explain it. But his own post here looks like what I would call “mathiness”. Saying “A model of the economy like X is impossible because…Math!”. It’s just an open invitation for anyone like me to try to come up with a counterexample.
Funny thing is, I too don’t like the amount of math in economics. And I think it sometimes obscures rather than clarifies. But from the cases I have seen, the people who write mathy papers that I think are wrong seem to me to be confusing themselves rather than trying to confuse other people. Like that whole Minnesota sign wars thing. They were just as confused when they said it in plain English.
Luis, rsj, I think the Earth example works even at the firm level. If I may elaborate: Let’s say a firm decides to grow itself by X percent. Let’s say the markets for its inputs (costs) and outputs (prices) remain constant. CEOdumb says “all we need to do is increase every single part of our firm by X percent, using exactly the same methods we used to build the existing firm, and we’ll automatically increase our profit by X percent! It’s easy, we know how to do this, because we’ve already done it once.” This is the “duplicate Earth” proposal. Once done as described, there is essentially a second firm X percent the size of the original firm operating alongside the original firm in parallel. Note: EVERYTHING is duplicated (at scale X/100), from having separate delivery trucks, to a separate sales force, to separate management. Completely “siloed” as they say. (How you hire X/100th of a 2nd CEO is a technical issue, but you get the idea.) And it works! Using nothing more than what they already knew how to do, they increased profit by X percent. CRS.
To do ANYTHING more efficient than that requires MORE than the simple rival-input increase described above. Efficiencies of scale DON’T just happen automatically on their own as some kind of emergent phenomenon. It may APPEAR that way because they are so common-sensical that people left to their own devices usually just implement them. (Although reference military organization where no one is allowed to do anything without being ordered explicitly.) But in an honest model, you can’t just “sneak in” supposed “common sense”. Sharing delivery trucks requires executive decision-making, planning, organizing. Having an existing sales force take on work coming from a new “division” takes executive commandment, negotiation, re-budgeting. Folding new hires into existing teams requires training managers on managing larger teams. Even aside from the training in specialization (of work) where the big gains from scaling up come from. In other words, there is a huge amount of non-rival “smarts” involved, even if a lot of it is “common sense”, freely available.
Hi Jeff.
No, with an IRS firm, increasing the scale of firm is not the duplication scenario. That would be creating another firm of the same scale.
Luis, All I am saying is that “increasing the scale of the firm” does not get you beyond CRS unless you add some non-rival “smarts”. Are you maybe getting hung up on the term “innovate” as including mundane things like a firm’s internal reorganization? Even just ordering divisions within a firm to “talk to each other” rather than siloing is using a non-rival “innovation, namely language.
See latest Vollrath post, suggests Nick might be right to say point of Earth duplication argument is to rule out diminishing returns to scale in rival inputs
I would like a clearer understanding of what alternatives about returns to scale would also fit with the argument that if you have 2) you cannot have 3) [these numbers refer the post above]. For example, if there are either IRS or DRS when changing scale of firms, as opposed to duplicating firms at same scale, would the argument Romer is making still work?
b.t.w Nick, I don’t think you are on Twitter but somebody there suggested to Romer that he took your blogs in the wrong spirit (when he wrote you are a neo-marshallian trying to blow smoke over the issue) and Romer said he accepted that.
I should have written “to rule out diminishing returns to scale in rival inputs at the aggregate level”
I can believe that there are DRS at the firm level, but sensible to argue that this simply means when economy adds inputs it does not do it by increasing the scale of firms experiencing DRS. OTOH if there is sometimes IRS in rival inputs at firm level, would expect to see some IRS at aggregate level too.
@Luis, The Vollrath post looks to me to be saying the opposite of what you say. Namely, that virtually everyone accepts “1. Output is constant returns to scale in rival inputs” except those who abandon (1) (McGrattan and Prescott), who say “decreasing”. I.e. no one says “increasing”. And Vollrath’s argument “How could it possibly be that a duplicate Earth produces less than the actual Earth?” applies equally as well to “more”. Because exact duplicate.
Why is there resistance to taking on the “more smarts” (non-rival) explanation for IRS? Why constant search for IRS in rival inputs? There’s almost a “begging the question” quality to it.
And as to “getting IRS by not paying rival inputs their MP”, that directly contradicts the price-taking assumption. In real world terms, your parts suppliers would say f___ off, i.e. refuse to sell (no soup for you).
Again I come back to the possibility that the confusion (over no IRS in rival inputs) is that a lot of non-rival inputs are free and maybe thus not even being “counted” (by debaters) as inputs at all? Things like language, math, and yes the existence of money itself. But also process improvements developed and implemented (informally) by production workers themselves, as well as transportation infrastructure (public roads), which accounting departments certainly don’t track (on the input side). It’s not intellectually honest to reject/ignore these as production inputs (thereby attributing all RS to rival inputs) just because businesses don’t formally account for them on their ledger sheets and/or because they were not paid for. (This is my Obama “you didn’t build that” argument. :-/)