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Perhaps I'm misunderstanding, but I'm not sure why a landowner would earn any more than 1 ton of wheat per year (assuming there is no labor input), regardless of how many new ideas there are. How could a landowner not pay for use of the second newest idea? Can he use this patent for free just because it's not the latest one?

are you and Romer using the same definition of competitive equilibrium?

Did anybody else read the related post on Volrath's (excellent) blog? I need somebody to put me out of my misery, when it comes to idea that any aggregate production function must be CRS in rival inputs, because of replication arguments. It seems to me that if you allow IRS in rival inputs at the firm level, then what happens at the aggregate depends on whether you are giving each firm more inputs or whether you are duplicating firms without changing their scale. I am sure I have missed the point, can anybody help?

FYI, the blog referenced by Romer here:

https://growthecon.wordpress.com/2015/06/05/market-power-versus-price-taking-in-economic-growth/

makes a very clear and easy to understand argument with almost no math

ah rsj you look like the person to help me

Put your model in continuous time. Suppose there is a unit interval of land, with productivity A(t) (using state of the art tech). At time t there is an invention that raises productivity to A(t) + dA, and this tech stays best for a time interval dt, when it is supplanted by a better invention.

Then the land-owners earn A(t) for time interval dt, while the inventor earns dA for this time interval. Thus total income to land-owners in this interval is A(t)*dt, and total income of inventors is dA*dt. Tech growth rate g = (dA/dt)/A is constant.

As you point out, for dt large this is not a competitive equilibrium, because land-owners are not earning their marginal product.

What happens if we take the limit as dt -> 0? Now we have continuous innovation. But the problem is that the innovators have zero income in the limit!

To see this, note that total income of land-owners is the integral of A(t)dt over time, which integrates to a positive quantity (average A). The total income of innovators, though, is the integral of dA(t)dt over time. This integrates to zero!

This is because dAdt is the product of TWO things that are converging to zero. i.e. you can write it as:

dAdt = (dA/dt) dt^2 = gA (dt)^2

where g is the constant growth rate.

So the limiting economy doesn't violate Romer's comment. The process converges to a competitive equilibrium with no profits to inventors.

Luis,

The assumption is that you have an *aggregate* production function in two inputs, F(P, X). X is the rival goods (land, people), and P is the excludable non-rival goods (patents).

Now duplicate the earth, by putting an exact copy of the old earth next to it. The production function *aggregates* across both earths, doubling the rival inputs but leaving the patents the same. But the output of the two duplicate earths is the 2*the output of the single earth.

F(P, 2X) = 2F(P, X).

So the production function is CRS in X but must be increasing returns to scale overall. Because the production is constant returns to scale, you can take the partial with respect to X and, by euler's theorem, get

F(P,X) = partialF_x*X + "profits", where profits is just the residual (but is positive).

Therefore there doesn't exist a competitive equilibrium.

In Nick's argument, I take it that the "mathiness" he is employing is to assume, for some reason, that all patents except the newest one are free (receive no profits), and then assume that the latest patent receives a payment "close" to the other patents (close to zero), because it's innovation is very small.

rsj:

The new idea fully supplants the old one.

A farmer has the choice of using old tech with productivity A, or new tech with productivity A + dA. Then if the new tech is priced at any level above dA, say dA + eps, the inventor of the old tech can undercut it by offering a price in the interval (0,eps). This is worthwhile because the idea is already invented, and it doesn't cost anything to sell someone the right to use the idea.

The highest price that won't be undercut in this way is dA, so this is the equilibrium price.

Of course, we can look around and see different technologies and observe that the second best technology is not free, it is worth whatever gain is obtained from adopting that technology over just using land. The landowner, faced with the choice of technology A or B or C, that each allow him to earn an extra 2, 3, or 4 tons of wheat, is willing to pay up to an extra 2, 3, or 4 tons of wheat for using each technology. That's the indifference point of the landowner when deciding which technology to use. Just because a technology isn't the best doesn't mean that it's worthless and available to be used for free.

Jonathan,

It does not. See above. You are playing fast and loose with differentials here -- translate the problem to integers. Or, if you want differentials, tech A gives an advantage of dA, and that is the price. Tech B gives an advantage of dB and that is the price. Whether dA > DB is beside the point. Saying that, in the limit, the differential is "zero" and therefore the technology is free is the height of mathiness.

Here's another way of thinking about why the price of new tech is the difference in productivity from the old tech, which hopefully will clarify things.

Consider a 3-person game. The players are farmer, inventor1, and inventor2.

The farmer's choice is who to buy from. The inventors choices are to set a price for their technology.

Then the Nash equilibrium is for the inventor with the less productive tech to set P = 0, and the more productive inventor to set a price equal to the difference in productivity. Any choice by the farmer is a NE, but it makes sense to assume he buys the more productive technology (by a limiting argument).

Why is this the Nash Eqm? Well, clearly no inventor will set a negative price. If the inventor with the better tech sets a lower price, the farmer will buy from him, but he can do better by raising his price. If the inventor with the better tech sets a higher price, the other inventor will undercut him. So this is a NE.

This clearly isn't competitive. But if we let the productivity difference go to zero (which is what you do when you take the limit as dt -> 0), then it converges to a competitive eqm with zero profits for inventors.

jonathan: OK. But the demand curve facing an inventor is still perfectly elastic, in either discrete or continuous time, which does violate Romer's comment.

And so the question is: do inventors get paid *their* marginal product? And as far as I can see, that depends on whether David would have had his 2003 idea (3 tons per acre) if I hadn't had my 2002 idea (2 tons per acre). If David's 2003 idea depends on my 2002 idea, and if I can't charge David for using my idea to develop his own, then David rips off my marginal product. But if David would have come up with his idea anyway, in 2003, then David gets paid his marginal product.

Is that right?

rsj: If David charges 1 ton to use his idea, then if I charge more than zero, everybody will use David's idea. In Bertrand equilibrium between me and David, David charges 1 ton, and I charge 0. David and I would need to collude, to share the market between us, to get anything better.

Jonathan,

We're talking about competitive equilibria here, not Nash Equilibria. You can find Nash equilibria with monopolistic competition, which is the point of the example.

But yes, that's a good point -- if we solve this as a Nash equilibrium game, we get a (non-competitive) equilibrium in which only the latest tech is used. If we try to solve this by determining what is the marginal gain by putting a unit of land into production with a given tech, then there is a different price vector. That alone should be a proof that the competitive equilibrium doesn't exist.

rsj: see my update. We need to distinguish between Bertrand-Nash and Cournot-Nash equilibria.

I think that this can be considered a particular case of a general rule:

- A monopolist is in the same situation as a perfect competitor if: a) the utility of his product is not subjected to decreasing marginal utility (or if the only decrease is from "have utility" to "no utility"); AND b) the product has the same utility (or the same reserve price) to all potential customers.

Miguel: that sounds right to me.

jonathan @10.41 That sounds right to me. You are talking about Bertrand-Nash.

Nick,

I'm not sure how a perfectly elastic demand curve violates Romer's comment. This still isn't a competitive eqm, because farmers aren't earning their marginal product.

As to your question -- I think you're right about why inventors are not paid their marginal product.

Suppose there was some cost of invention, say you pay a cost C(p) for a probability p of an invention that is an advance dA above the existing tech. Then it's pretty clear that you don't get the optimality properties of a competitive equilibrium. This is because inventors don't capture the benefits of advancing the technology frontier and therefore allowing FUTURE innovations to raise productivity still higher.

An inventor WOULD take this into account if there was just one inventor who owned all the patents. Then he wouldn't let his past patents compete against the current state of the art, and would capture all of the accumulated benefits of tech progress. Thus he would take the future benefits into account when deciding current expenditures on invention.

No model can have a competitive equilibrium with price-taking behavior and partially excludable nonrival goods

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.

Maybe silly, but worth a shot: By introducing land you now have a (excludable) rival good?

Nick,

Thinking about it a little more, I think I understand how to fit it into your model.

If inventors were paid their marginal benefit it would look like this:

year 1: Nick earns 1
year 2: David earns 1 AND Nick earns 1
year 3: Glenn earns 1, David earns 1, Nick earns 1

etc. You don't just capture the marginal benefit of your innovation in the first year, but also the benefit in every subsequent year from the fact that your innovation enabled future progress.

Been a while since I did micro, but price-taking firms make 0 'super-normal' profits, but normal profits still exist as a minimum to justify the firms existence (basically a salary for the owner/manager, we could restrict this to being his marginal product), if we make some assumption about simply being an owner of a firm provides utility in itself (it's nice not being a wage-slave) then why would this not provide some incentive to innovate & form a new firm? If this is true, then price-taking and at least some degree of innovation is compatible, what am I missing here?

Nick,

Yes, I agree re: Bertrand-Nash equilibria. But I still accuse you of mathiness using games with differentials.

Ask yourself, if you insist on making a limit-based argument, what would the final economy look like as you take the limit of an infinite number of technologies? E.g. Your production function is now F( P, t, X), where t is a (positive) real number indexing the productivity gains of version t of the technology.

And your function would look something like this:

F(P, t, X) = tPX

It looks to me like you've got an increasing returns to scale function!

How could you get rid of the increasing returns to scale function and make it constant returns to scale? Well, you want F(P, t, X) = kX.

So on the one hand, you want new technologies to be more valuable, so F(P, t, * ) is an increasing function of t, but on the other hand, you want it to be a constant. It's only by playing fast and loose with differentials that you achieve this (the technology gets better, but *really slowly*).

So I think the fundamental result holds, and the multiple tech scenario is just a distraction. The core issue, whether techs come in many flavors or just one flavor, is that the function is going to be increasing returns to scale.

Britonomist,

You're talking about another (rival) input into firm production, like entrepreneurial capital or something.

Romer is talking about a non-rival good, like ideas that anyone can use.

Jonathan, I don't see the problem though. I come up with a new idea that will improve the productivity of producing X; I have no means of protecting this idea however & my production process will be completely transparent, meaning all other firms will be able to immediately replicate it. Does that remove my incentive to go into business? Of course not! I'll still get to run a business, and be heralded as a great innovator, this gives me utility and an incentive in the absence of super-normal profits.

For the record, I definitely agree that a lack of super-normal profit incentive will significantly reduce innovation. But reduce it to absolutely nothing? No way.

jonathan @11.27.

Agreed. You are assuming there that Glenn's idea depends on David's, which depends on mine. And if so, and if ideas are excludable, I could charge David and Glenn for using my idea.

And then the problem is that landowners only get paid 1, while the MP of land under the newest idea is 4. And if new land could be produced (contra my assumption) that would mean there is underinvestment in new land, unless Glenn cuts the producers of new land a special deal.

Right?

I think it's very sneaky to talk about land being fixed, since the idea of marginal product is to add one more unit, at least conceptually. But perhaps it was Nick Van Rowe who was making this post.

But a general principle I wish could be kept in mind is that if your argument is going to depend on taking an infinite limit of things, then see what is the final object that your limits converge to. Then see if your limits of equilibria converge to an equilibria on the final object. Then see if that final equilibria is a competitive equilibria. It's a very long road to travel one, when we have a beautiful 2 earth argument in front of us.

This talk of invention is getting me thinking about tech companies.

Nick,

Right.

The basic point is that, no matter the assumptions, this economy is not competitive. Either farmers earn less than the marginal product of their land, or inventors earn less than the marginal product of their research.

The only exception would be if research were continuous and not costly at all (up to an upper bound). Then the latest tech would be free, and this economy would basically be a model with exogenous tech progress.

jonathan: OK. But if Henry George puts a tax of one ton of wheat per acre of land, is that economy not "competitive"? Individual landowners still face a perfectly elastic demand curve for their land, but it's below the Present Value of MP.

But yes, we are now arguing semantics.

rsj: Marginal product is still defined, regardless of whether the input is in perfectly inelastic supply. Ricardo and all that.

jonathan: "I'm not sure how a perfectly elastic demand curve violates Romer's comment."

Paul Romer said: "... 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,..."

Is Paul Romer going to say that an individual wheat farmer who faces a perfectly elastic demand curve for his wheat is able to influence the price that buyers of his wheat have to pay? I don't think so.

Nick,

A competitive economy is one in which every agent is a price-taker. That means they take a price P as given, and can buy or sell any quantity at that price.

In your example, inventors are NOT price takers. They face a demand curve that is elastic up to the quantity Q=1, but they can't sell more than that. So the demand curve is not perfectly elastic. With a perfectly elastic demand curve, if you set P = P* - epsilon, you get infinite demand. Here inventors see demand Q = 1.

This is important. If inventors were price takers, they would want to raise their production above Q=1 at any price P>0, because the marginal cost is zero. The only solution would be P = 0.

Instead of an elastic demand curve, the inventors face a kinked demand curve. Viewed in the standard (P,Q) space, it's flat (locally elastic) at P = A' - A up to Q=1, and then drops vertically (inelastic) to zero at Q=1.

The solution to their profit-maximization problem is to set P = A'-A and Q=1.

Nick,

I think the key here is "idea" -- e.g. there is a non-rival good, and the claim is that you can't both have monopolies on non-rival goods while at the same time being price takers for them. The non-rival good is important to the "two earth" argument, because if you duplicate the earth, you are duplicating all the rival goods (twice as much wheat), but are leaving the ideas the same. There is no production function specific effect to "duplicating" a non-rival good, since it's already ubiquitously available. Mathematically, this is crucial to arguing that the resulting production function has increasing returns to scale. If it were a rival good, such as labor, then the resulting production function would be constant returns to scale and the argument would not follow through.

Or what Jonathan said.

I guess I'm the only one who is utterly in love with the simple and beautiful "two earth" thought experiment.

jonathan: under that definition, nobody is ever a price taker. If an individual firm in a competitive market cuts its price below the market equilibrium it can only sell a finite quantity, which is much bigger than it wants to sell, true, but still finite. Or take standard Bertrand duopoly, with identical goods. There's a kink where the individual demand curve hits the market demand curve.

There are two ways to define "quantity" for a non-rival good: number of different songs; number of copies of each song (assuming copies have zero MC). In the equilibrium of my model, everyone buys a copy of the newest and best song, which is efficient. And we also get the efficient number of different songs. (But if land wasn't in perfectly inelastic supply, we wouldn't get the efficient quantity of land, true.)

rsj: I thought the 2 earth thing was neat too.

Nick,

I know you don't like the Walrasian Auctioneer, but the standard definition is as I describe. The Auctioneer names a price vector P, and every agent takes this as given and acts as though it could buy or sell any quantity at this price.

By the standard definition then, this economy is not competitive, since the inventors are not price-takers.

(When I talk about "quantity" in the above, I mean the sale of the right to use the patent. This is what has zero marginal cost.)

===

Also, rsj makes a good point above. I would phrase it like this:

The aggregate production function is

Y = F(A,L) = AL

where L is land and A is the productivity of the technology being used.

Now in a competitive equilibrium, the rental price of land will be r = F_L = A, and there's no income left over to pay the inventors.

rsj

it's just the bit about any aggregate production function necessarily being CRS in rival inputs that I do not understand.

IRS in rival inputs can exist at the firm level, right? An an economy of N such firms has an aggregate production function? Or maybe it doesn't. Because in such an economy when scaling inputs it matters whether you are scaling firms or duplicating firms. Which is my problem with the duplicating earth argument. Because I think you can have IRS in rival inputs, and still only double outputs when duplicating units of production without changing their scale. So when you write "the production function *aggregates* across both earths" that's the bit I am objecting to - supposing doubling earths is like doubling the size of a firm. Maybe I am getting muddled because I have in mind an economy, made up of IRS firms, for which an aggregate production function does not exist?

Or maybe I am talking rubbish. Is Sum over i for i=1 to N f(X^2_i) an aggregate production function?

jonathan: yep, I think I must be going a bit Marshallian? Individual firms know they can't sell more than total market demand at that price.

I get the bit about factor payments adding up to total product under CRS and W=MPL. But look, if the government puts a tax on something, then factor payments will be less than MPL and less than total product under CRS. Do taxes suddenly eliminate price-taking competitive behaviour? No.

Luis: as I understand it, the replication argument is an argument against decreasing returns to scale, but it doesn't prevent increasing returns to scale. But under IRS, sum MPX.X > Y, so you can't have all inputs being paid their Marginal Products.

ah, well that would explain my confusion!

[Sure I get the point inputs can't be paid marginal product under IRS - and a host of other production functions (O-ring etc.) it was just that one bit about duplicate earth and CRS that I was caught on]

Nick: Are you assuming that farmers have to pay inventors PER UNIT of output produced using their technology, or just pay a one-time fee for the technology?

I was thinking about the latter. But then it makes sense for one farmer to pay the cost, and then rent everyone's land and farm it using the best technology. In this case, since Y = AL, the rental rate will be r = A, and we get the adding up problem.

But a per-unit fee, like a per-unit tax, would affect the rental rate, and so everything would add up. Is that what you have in mind?

jonathan: I was assuming each farmer paid a fee per acre per year. So a farmer with 200 acres pays double a farmer with 100 acres. (That would be the same as a fee per ton of wheat, in my simple model.)

It wouldn't be profit-maximising for me to charge all farmers the same fixed fee, regardless of how many acres they farm. Charging a fee per acre maximises my profits.

Nick: Interesting. Unless I'm mistaken, I think that your per-unit pricing assumption violates the conditions under which Romer proves his statement.

Reading the original mathiness paper, it looks like the argument in question is in the 2nd paragraph on page 3. It goes like this:

Production is F(a,x), where x are rival inputs. F is HOD 1 in x. Therefore, by Euler's theorem, the entire output is spent paying for the rival inputs (x), and so nothing is left over to pay the nonrival inputs (a).

This argument assumes that the firm maximizes:

profits = F(a,x) - w*a - p*x

and takes p and w as given.

In your setup, the firm problem is:

profits = (A+zdA)*L - rL - zpL

where z is the decision whether to buy the new technology, which raises productivity by dA.

This is clearly a different setup than assumed by Romer. In particular, you're assuming that the total amount paid for the new technology dA DEPENDS ON the amount of land L. Romer's formulation seems not to allow for this.

jonathan: Ah, that is interesting. Yep, I can see that if the inventor charged a fixed price *per farmer*, regardless of acreage, you would end up with only one monopoly farmer (or maybe a small number of Cournot oligopoly farmers). All the other farmers would lend sell him their land and live off the interest from the loan they gave him to buy their land.

But that wouldn't be profit-maximising for the inventor.

But maybe that's the root of the disagreement? (I haven't read David Andolfatto's paper to see what he assumes; scared to even look at it, because it's probably got lotsa math.)

The way I teach "Euler's theorem" in ECON 1000: IRS means downward-sloping LRATC curve, which means MC < ATC, which means you can't have both P=MC and P=ATC. Each firm is gonna want to get bigger, or else exit, so you can't have a LR equilibrium with lots of firms.

But I confess, I did have a frustrating time once trying to explain this to PhD economists, so maybe Paul Romer does have a point.

Wouldn't the rest of the world be flooding Canada with wheat (or at least their customers), or are you assuming they do have a way to exclude this?

Lord: no. The cost of wheat would be the same. Canadian farmers would pay 1 ton rent on the land, plus 1 ton payment for my idea. ROW farmers would pay 2 tons rent.

I think Romer view's P as the quantity of technologies or patents. Because the rival goods are CRS, they can all be aggregated into a single firm, which then licenses some quantity of P. A patent to do X better, a patent to do Y better, etc. P doesn't measure how many licenses are sold for a single patent, but how many different patents are sold.

You need at least this to put P into the production function as a variable which would cause IRS, otherwise P is just a constant. IRS is the key here, and it comes by being able to increase the number of patents (invest in research), not the number of patent licenses.

As I read all this, I have a nagging question. Who is going to eat all that wheat?

And, even assuming no foreign competition, don't diminishing returns kick in?

Luis,

The replication argument *proves* increasing returns to scale under the assumption that increasing P also increases total output. You have, by euler's theorem,

f(P, X) = Residual(X,P) + const(P)*X

All you need to get IRS is to show that the derivative of f with respect to P is always > 0. I.e. if you increase P you get more output.

"mathiness" comes in when you assume that P is not a variable for purpose of scale determination (e.g. it is a constant), but at the same time you want to assume that P is somehow a choice variable for purposes of modeling behavior.

Romer is interested in models of endogenous technical change, so P is an input into the production process, and whenever that happens, unless there is a ceiling on technology so that further investment yields no gains, you must have IRS and therefore cannot have a competitive equilibrium.

Nick: I have a hard time calling per-acre pricing "competitive".

The usual way to set up a firm problem with two inputs would be something like:

max F(K,L) - rK - wL

But what you're doing is making the price of one input (technology) DEPEND ON the amount of the other input used. So it's like you have:

max F(K,L) - r(L)*K - wL

This doesn't seem competitive, because you're letting a price depend on a choice variable.

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 A*f(K,L) the technology isn´t "A" but e.g. one day, with one technology, 2*f(K,L) and another day, with another technology, 3*f(K,L).

Obviously, if the f(K,L) part (and, thus, the technology A*f(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

https://growthecon.wordpress.com/2015/06/13/what-assumptions-matter-for-growth-theory/

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. :-/)

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