# Market Power versus Price-taking in Economic Growth

NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

I’m sure you’ve been breathlessly following along with the discussion on “mathiness” that Paul Romer kicked off (see here, here, here). Romer used several growth models to illustrate his point about “mathiness”, and his critique centered around the assumption of price-taking by firms and/or individuals in these papers. His argument was that these papers used “mathiness” as a kind of camouflage for their price-taking assumptions. Romer argues that the reasonable way to understand growth is to allow for market power by some firms and/or individuals over their ideas.

From what I can see, the heart of this is about replication. What Romer has asserted is that any aggregate production function must have constant returns to scale (i.e. be homogenous of degree one) in its rival inputs. The mental exercise here is the following. Imagine that tomorrow there was a perfect replica of the Earth floating next to this one. What would be the output of the alternative Earth? It would be exactly the same as output here, right? It would have to be. It is an exact replica.

What we did was replicate the rival inputs (land, capital, people, education, etc..) and held constant the ideas/plans/technologies. As the output of the alternative Earth is exactly equal to the output of the current Earth, the production function has constant returns to scale with respect to the rival inputs, the things we duplicated. We doubled Earth, and got double the output. It follows that if I doubled Earth’s rival inputs and doubled the ideas/plans/technologies, then I’d get more than double the output. In short, I’d have increasing returns to scale.

What does this imply about market structure? Write down a production function that depends both on rival inputs (${X}$, standing in for all the stocks of capital and people and land) and non-rival inputs (${A}$, standing in for ideas/plans/technologies),

$\displaystyle F(A,X). \ \ \ \ \ (1)$

What the replication argument says is that this function has constant returns to scale in terms of ${X}$, or ${\lambda F(A,X) = F(A,\lambda X)}$. This in turn implies that the following must be true

$\displaystyle F(A,X) = \frac{\partial F(A,X)}{\partial X} X. \ \ \ \ \ (2)$

The above says that total output can be calculated as the marginal product of rival inputs (the derivative) times the total amount of rival inputs, ${X}$. This is just Euler’s theorem for homogenous functions.

What we also know has to be true about this production function is that the total amount paid to all the factors of production – rival and non-rival – can only add up to total output. In other words, we’ve got

$\displaystyle F(A,X) = Profits + wX \ \ \ \ \ (3)$

where ${Profits}$ are whatever we pay (possibly zero) to the owners of the ideas/plans/technologies. ${w}$ is the “wage” paid to a rival factor ${X}$. If we had lots of rival factors, then we’d have lots of these terms with things like wages, rents paid to owners of land, rents paid to owners of capital, etc.. etc..

Both the two expressions I’ve shown have to hold, and this is where we get to the problem. If we want to assume that there is price-taking, then all the rival factors would be paid their marginal product. If they were underpaid, then other firms could pay them more, and use all the inputs from the original firm. But if wages are equal to marginal products, then ${w = \partial F(A,X)/\partial X}$. And if this is true, then the only possible way for the second expression to hold is if ${Profits = 0}$. If rival factors of production are paid their marginal products, there is nothing left over to pay out as profits.

If you have ${Profits>0}$, then you must have that ${w<\partial F(A,X)/\partial X}$, or rival factors of production are paid less than their marginal product. And the only way for this to be the equilibrium outcome is if there is not price-taking. If other firms could pay more, they would, and would equate the wage and marginal product. So positive profits imply some kind of market power (possibly a patent, or a legal monopoly, or some kind of brand identity that cannot be mimicked) for firms.

Romer’s 1990 paper argues that this second situation is the only one that makes sense for explaining long-run growth. If ${Profits}$ did equal zero, then no one would bother to undertake innovative activity. What would be the point? So firms that innovate must earn some profits to incent them to undertake the innovation. This doesn’t mean they are gouging people, by the way. The positive profits may simply be sufficient to offset a fixed cost of innovating. But once you accept that innovation takes place in large part as a deliberate economic activity, Romer’s argument is that this inevitably implies that firms have some market power and rival factors are not being paid their marginal products. You have to be careful here. Romer is not arguing that this is how the world should work. He’s arguing that this is how it does work.

This framework makes it easier to understand what is going on in papers that assume price-taking or perfect competition. Take the Solow model, which implicitly has price-taking by firms. In the Solow model, technology ${A}$ just falls out of the sky, and no deliberate activity is necessary to make it grow. So ${Profits=0}$, because there is no one to remunerate for innovating. Hence we can have price-taking by firms.

Learning by doing, a la Ken Arrow, makes a similar assumption. Arrow doesn’t have ${A}$ exactly fall out of the sky. ${A}$ is strictly proportional to ${X}$ in a learning-by-doing model, so it grows only as fast as ${X}$ grows. But similar to Solow, no one has to take any deliberate effort to make this happen. It’s a pure externality of the production process, and no one even realizes that it is occurring, so no one earns any profits on it.

Note that this concept is pretty crazy in terms of the replication argument. Arrow’s learning by doing model implies that when the alternate Earth shows up, we more than double output because all those additional rival factors generate some kind of ….. well, it’s not clear exactly how this is supposed to work. Presumably you’d have some kind of gains from trade type argument? The two Earth’s could trade with each other, and so we could let Earth 1 produce Lego and Earth 2 produce Diet Coke. But remember, these Earth’s are identical, so relative prices are identical, and so there isn’t any incentive to trade in the first place.

What of more modern models of price-taking and growth? I mentioned the McGrattan/Prescott (2010) paper in the last post, and effectively they assume that ${F(A,X)}$ is constant returns to scale over both ${A}$ and ${X}$. Formally, ${\lambda F(A,X) = F(\lambda A, \lambda X)}$. This means that the production function is decreasing returns to scale with respect to rival inputs, and

$\displaystyle F(A,X) > \frac{\partial F(A,X)}{\partial X} X. \ \ \ \ \ (4)$

Now, given this, we could easily have price-taking (${w = \partial F(A,X)/\partial X}$) and still have ${Profits > 0}$.

But does this assumption make sense? Well, what happens when the alternate Earth shows up? In the MP setting, when the alternate Earth arrives total output across our two planets is less than double what we produce today. But alternate Earth is an exact replica of our planet. So how could it possibly produce less than us? Or maybe alternate Earth produces the same amount, but its arrival somehow made us less productive here on the original Earth?

MP aren’t exactly after a model of endogenous growth, but Boldrin and Levine (2008) explicitly write down a model that is meant to show that perfect competition is compatible with firms/people making deliberate innovation decisions. It’s taken me a few days to get my head around how their work fits (or does not fit) in with Romer’s. BL don’t write a model that uses a standard production function, so it’s difficult to map it into the terms I’ve used above.

In the end, though, a (the?) key point is that BL assume that ideas are in fact rival goods. A working paper version of this paper mentions the following in the abstract: “We argue that ideas have value only insofar as they are embodied in goods or people, …” By assuming that ideas have no productive value by themselves, the production function is essentially just ${F(X)}$, and is constant returns to scale in the ${X}$ rival inputs. Hence price-taking is something that could happen. Innovation in BL means providing more inputs (i.e. better inputs) into the production function, raising ${X}$. BL assume that the profits accruing to ideas themselves are zero. BL is similar to a model like Lucas (1988), where all innovation is embodied in human capital.

In BL, the incentives to innovate (i.e. to accumulate a new kind of input) come because you own a rival good that is scarce. Innovators in BL are like landlords in a classic Ricardian model. They have a fixed factor of production, and they earn rents on it. If those rents outweigh the cost of coming up with the idea in the first place (producing the 1st copy), then people will innovate.

Does the BL version make sense? It depends on how you conceive of technological progress. Is it embodied (and hence rival) or not (and so it is non-rival)? If all technological progress is embodied, then it is possible that all firms or persons are price-takers. But if any deliberate technological progress is non-rival (disembodied), then there are at least some firms or people with market power. Note that this doesn’t mean that all markets are imperfect, but firms that own non-rival ideas and have some ability to exclude others from using them (e.g. a patent) will charge more than marginal cost.

The important difference here is the all vs. any, I think. Everyone could be price-takers if all technology is embodied (and hence rival). That is a strong condition. It means there is literally no such thing as a non-rival idea. One way to think about this is kind of the opposite of the replication argument. What if tomorrow everyone who knew Linux was wiped off the face of the Earth? Would Linux be gone? Would we have to wait for some new pseudo-Torvalds to arrive and re-write it? I don’t think so. Someone could figure it out by reading manuals left behind. Would they learn it quickly? Maybe not. But the idea of Linux is clearly non-rival. And so long as there are any non-rival ideas that are useful, then if you want there to be economic incentives to produce them, there has to be some market power that allows firms to capture those rents.

By the way, BL use their model to argue that intellectual monopolies (like patents, copyright, etc..) may be counter-productive in fostering innovation. That can be true even if you have non-rival ideas. The fact that profits exist for non-rival ideas don’t require that intellectual monopolies be made eternal and absolute. Within any Romer-style model there is some sweet spot of IP protection that fosters innovation without incurring too much deadweight loss due to the monopolies provided. We certainly could be well past that sweet spot in reality, and be over-protecting IP with patents that are too strong and/or too long. But if you eliminated all IP protection, then the Romer-style setting would tell you that we would effectively shut down innovation in non-rival ideas, as they could not otherwise be compensated.

## 21 thoughts on “Market Power versus Price-taking in Economic Growth”

1. In eq 4, if F(.)<df/dx*x, and df/dx=w, then F(.)<w*x, so profits must be negative (not positive), right?

• The first inequality should be the other way around. F() > df/dx*x. Which means that I’ve got equation 4 in the post backwards. Argh! Good catch.

2. Dear Dietz,

Thanks for this consistently fascinating and informative blog and the great public good you are providing. I am in the business of trying measure intangible assets and this blog is very thought-provoking in that regard where you have expressed the competition modelling dilemma incredibly clearly.

Can I try a different replication example to test my understanding of your argument? Rather than thinking about a new earth (a great example) let me try this: suppose I take my UK factory and put it in China. Can I get the same output? The replication argument says, I think, that I can install the same capital and, let us say, the same labour (suppose I employ immigrant Chinese unskilled labour in repetitive tasks in the UK and such labour is replicable). But I can also use the same ideas from the UK, so that in fact I have CRS in the K and L, but IRS in A.

In practice though, I am not sure the example always stands up. The classic UK (anecdotal) case is Brompton Bikes who make folding bikes in West London (if you fly to Heathrow you will fly over their plant) and tried to outsource manufacture in China. They had to come back to West London, despite the expense, since they simply couldn’t get the precision that the folding bikes need (so that they fold correctly).

My interpretation, if I understand correctly, is this is may be counter-example. The production process needs K and L, but also knowledge in the production process. And for some reason that knowledge is actually not replicable, it is in fact rivalrous. Now, it might be that it is non-rival within the UK factory so that the UK factory can be expanded using the same knowledge, but is rivalrous between the UK and China. Or the knowledge might be span-of-control type knowledge which would have diminishing returns even within the UK. Note too that you sometimes hear similar stories about pharma production, namely that the production process of the drug itself is very specialised and cannot be undertake except close to where researchers are for example.

If this is right, I wonder if might be useful to distinguish between general ideas and commercialised ideas. The latter are ideas that are used in the production process (which might be a production line, or WalMart’s knowledge of retail cross-docking) which are in fact rival?

Yours,

• It’s a weird argument to get your head around, definitely. So let’s say Brompton Bikes puts a plant in China. Replication simply says that if I double the rival inputs, I get 2x the output. Now, managerial talent is rival, so I’ve created a replica of the Brompton management. Tacit skills in building bikes are rival (they are embodied in workers), so I’ve created a replica of the Brompton Bikes workers as well. And input networks and infrastructure are rival, so I’ve created a replica of those as well. Everything that is *rival* is replicated. If I truly did that, I’d get 2x as much output. At that point, if I then *also* doubled the *non-rival* inputs (a better bike design) then I’d get more than 2x the output.

What the actual Brompton experience suggests is that it is actually really hard to perfectly replicate rival goods. It’s hard to find workers like those in West London, and managers that understand the factory, and etc. etc. etc.. It’s actually for that reason that I like the Earth example, because it avoids this issue.

The thing is, we don’t have to be able to actually replicate production for the argument to work. If there is one unique manager that makes Brompton work the way it does, then he/she is un-replicable in reality. That doesn’t mean they aren’t replicable in theory. And all Romer’s argument requires is replication in theory.

• dear Dietz,
Many many thanks for this very thoughtful reply. The extent to which intangibles are embodied in capital and labour is interesting when it comes to rivalry. Let me think about it some more since my immediate reaction is that if I teach a manager to do calculus that knowledge is not embodied in the manager. Does that mean that embodiment and rivalry is tied up with codified v tacit?
Thanks again and best,
Jonathan

3. it’s a minor quibble, but I think your introductory analogy is confusing. We might expect economies of scale when, for example, doubling the size of my car factory. Not when building a replica car factory on another continent. You might increasing returns if production was somehow integrated across the two planet Earth, but not from two separate planets operating in isolation.

• The replica will have exactly the same output as your original, right? I’ve doubled machines, labor, and raw materials. That’s just constant returns.

So if I were to double productivity also (in addition to doubling the machines, labor, and raw materials) I’d necessarily get more than twice as many cars.

The IRS comes not from replicating the rival goods, it comes from doubling rival *and* non-rival goods. But if that didn’t come across in the post then I didn’t write it clearly.

• the fault is no doubt mine.

Suppose you had an economy that consisted of N firms that possess an increasing returns to scale technology in rival inputs, each firm employs the same quantity of rival input X. Output is N*f(X). No non-rival inputs.

At the aggregate, you could add inputs to this economy by adding more firms at the same scale (N+a)*f(X) or you could add inputs to this economy by increasing size of existing firms N*f(X+b). It matters how inputs are added.

So I didn’t see why the doubling Earth thought experiment means you must have CRS in rival inputs. It seems to me that if you allow IRS in rival inputs, you would still only get twice output from two earths.

I am probably making a fool of myself – this is why I post pseudonymously.

4. >>Note that this concept is pretty crazy in terms of the replication argument. Arrow’s learning by doing model implies that when the alternate Earth shows up, we more than double output …

I think that will not always be the case. To achieve that goal, we need to assume that the two earths are producing non-overlapping knowledge, ie A = A_1 + A_2. Right ?

5. Regarding “no incentive to trade” between the identical Earths, didn’t Krugman’s prize-winning trade work show the opposite? Essentially that trade in and of itself is a value (incentive)? That like the (Mallory?) Everest quip, we trade with them just “because they’re there”? (I could well be wrong about Krugman; I thought that was the gist though.)

• I’d suggest that agglomerations, where we trade because specialization allows for higher productivity, is a form of non-rival good. So duplicating Earth doesnt by itself cause that.

6. First, as I should have said up front in the last comment: thanks for the great education around growth theory math!

I also have to take some small issue however with the last sentence about “could not otherwise be compensated”. Maybe I’m coloring outside the lines (of growth theory) or nitpicking, but doesn’t that have to say “paid” instead to be accurate? I mean there are other “compensations” to innovating besides financial remuneration, “just for fun”, “for my own satisfaction/curiosity”, etc. I do some of that on my blog, for instance, not that it’s worth all that much, but still :-). The point is that people are incented by things other than money. Yes?

Thanks again.

• Yes, paid is more correct. There are probably plenty of inventors who chase the idea rather than the money.

7. I don’t see why your thought experiment implies HOD 1.

It’s true that replicating the Earth would at LEAST double economic output, since we can always just reproduce the same pattern of production on the new Earth. But maybe we could do even better with a different pattern of production across the new two-world system, for instance by taking greater advantage of local agglomeration economies.

• Okay. Two earths so far away that trade is impossible. Or if you like, local agglomeration economies are a form of non-rival good, where no one owns the patent. Regardless, an exact duplicate would produce 2x the output.

8. Reblogged this on The curious incident of DD on the Web and commented:
I was traveling when Paul Romer’s “mathiness” paper and blog post started a debate about how economic theory built with mathematical tools can be manipulated for purposes that are more political than scientific. There has been a ton of discussion about this, and for now I want to highlight this piece by Dietz Vollrath. If you assume that the relevant production function is well-defined and differentiable, his arguments make many things about this controversy very clear.

Meanwhile, Paul Romer has come back to this topic many times, and his latest, as of now, http://paulromer.net/the-norms-of-politics-ferguson-and-ehrlich/ , pursues the science / politics point by bringing in two figures whom he criticizes for twisting words for political purposes and not admitting it when caught.

9. Thanks a lot. A pretty good blog to help me get the idea of Romer(1990). Additionally, thanks for your introduction of BL(2008). To my knowledge, disembodied technological change and embodied technological change could coexist in reality, being two distinct engines of economic growth as well as capital accumulation maybe.

10. Your description of all innovation being embodied seems a bit confusing. Let me see if I have it right.

Innovation is embodied to the extent that it is entierly the product of some person/thing which can only contribute “once” to production.

In other words for all innovation to be embodied the following would have to be true:

Suppose an fiber optic cable magically connected us to Earth2, a world with exactly the same total production (and population and resources etc..) as earth but with a different set of ideas/plans etc.. So maybe Earth2 knows a way to purify aluminum more efficently than we do but doesn’t know how to make carbon fiber as well as we do but overall it all balances out. Without the magic cable the total production of Earth + Earth2 would be just twice that of earth.

If innovation was entirely embodied the total production wouldn’t increase just because we could both search each other’s internet and read the (completely unenforceable) patents from the other planet. Or at least it would take as much time and money to look up ideas on the other planets internet as it would to produce equally good ones ourselves.

Wait wouldn’t the idea that all innovation is embodied imply that per capita production should never increase (assuming linear production in rival goods)?

• I agree it was confusing. I’m still struggling to come up with the right analogy/story.

In your story, though, Earth2 has duplicated the rival factors (land, capital, people) *and* added to the stock of non-rival ideas. So yes, output on Earth + Earth2 would be more than 2*Earth, but that is because of the extra ideas. If Earth2 had only the exact same ideas as Earth, then output would be 2*Earth.

And yes, if all innovation is embodied, then I think growth would go to zero, because of diminishing marginal returns to those rival goods.