# Understanding Diffusion Models of Growth

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

There has been a recent bloom of research that studies the diffusion of ideas and economic growth. Alvarez, Buera, and Lucas (2013), Lucas (2009), Lucas and Moll (2014), and Perla and Tonetti (2014) are some of the most prominent examples. In each case, firms or individuals learn new techniques after meeting other firms or individuals with better ideas. The papers show the assumptions under which this type of diffusion or imitation process will lead to constant, sustained growth.

I’ve been trying to get my head around what these models teach me about the process of economic growth. I’m going to use Perla and Tonetti (PT) as a specific example in this post, but that’s only because I need an example, and it was the last one I read.

Here’s a quick verbal summary of the model of imitation in PT:

There are a bunch of risk-neutral firms, each with an individual level of productivity. The firm can produce using its own productivity level, or it can shut down for a period and search for better productivity to imitate. If it searches, it is randomly matched with another firm that is producing, and imitates that firms productivity level for free. Searching firms compare the expected value of productivity they get from imitating to the cost of shutting down, and only low-productivity firms search and imitate. Firm productivity is Pareto-distributed. The average level of productivity is rising over time because low-productivity firms imitate high-productivity terms. Because of the specific nature of the Pareto, the distribution remains Pareto even as the average rises.

The PT model delivers sustained growth through this search and match process, while maintaining a distribution of firm-level productivities. But that isn’t due to the economics of imitation, that is due to the specific mathematical structure assumed. To see this, compare the PT model to my newly created model of growth:

There are a bunch of risk-neutral villages, each with an individual level of Tecknologie. The village can consume what it produces, or it can sacrifice all of that production as a sacrifice to the glorious Hephaestus, God of craftsmen, in the hopes that he will bestow on them insight into a new type of Tecknologie. Hephaestus is fickle, like many of the Gods, and his ways are mysterious to mortals. Hence, if the village offers the sacrifice, the new Tecknologie that Hephaestus grants them is unknown, but is equal to the Tecknologie in one of the other villages around them. Villages compare the expected level of Tecknologie from Hephaestus to the cost of sacrifice, so only low-Tecknologie villages perform sacrifices. Village Tecknologie is Pareto-distributed. The average level of Tecknologie is rising over time because low-Tecknologie villages sacrifice and are blessed by Hephaestus. Because of the specific nature of the Pareto, the distribution remains Pareto even as the average rises.

These models are mathematically identical. With a sophisticated use of search-and-replace I could rewrite PT to be a paper on the growth implications of Hephaestus worship in ancient Greece.

The point is that we can call “draw productivity from a Pareto distribution matching currently producing units (DPFAPDMCPU)” anything we want. PT call it “imitation”. In my little story I call it “blessings from Hephaestus”. You could call it “R&D”, or you could call it an “externality” or “diffusion” if you wanted. DPFAPDMCPU is just an assumption about how innovations arrive.

This isn’t to say that DPFAPDMCPU is wrong, or even a bad assumption to make. Every growth model makes some kind of unsupported assumption about how productivity arrives. Solow assumed that productivity grew exponentially, which led to constant growth in steady state. Aghion and Howitt said new innovations arrive as a Poisson process, but the productivity bump you get is always the same. In expectation, or if you have lots of sectors, you get constant growth in steady state. In a standard Romer model, the productivity bump you get from innovation is proportional to the effort you put into R&D, and growth is constant in steady state.

PT isn’t really a model of imitation and growth. It is a model of DPFAPDMCPU and growth. And DPFAPDMCPU has a clever implication, which is that the distribution of firm (or village) productivities stays Pareto forever even though we have all this churning in the distribution going on. That’s something that other assumptions about how innovations arrive can’t capture.

And PT get this distinction. This paragraph is from their conclusion:

This paper contributes an analytically tractable mechanism for analyzing growth and the evolution of the productivity distribution, with both the evolution of the productivity distribution and the technology adoption decision jointly endogenously determined in equilibrium. Thus, we can analyze the effect the productivity distribution has on adoption incentives, the effect of adoption behavior in generating the productivity distribution, and the corresponding growth implications of this feedback loop. We develop a solution technique that obtains closed-form expressions for all equilibrium objects—including the growth factor—as a function of intrinsic parameters.

Here they’ve dropped any use of the word “imitation” and talk about a generic process of “technology adoption”, which could be anything from R&D to Hephaestus-worship. PT state they have figured out how to use DPFAPDMCPU as the mathematical structure to model the arrival of new technologies to adopt, all while still ending up with a constant growth rate.

The question now is why or when DPFAPDMCPU is a better choice than other structures. In what situations, or for what types of products, or in what markets, is it reasonable to think of DPFAPDMCPU as the way that innovations arrive?

“Imitation” or “diffusion” doesn’t seem to cut it as motivation. If we take imitation seriously, then the DPFAPDMCPU structure has several issues:

• Searching firms are randomly matched with producing firms. Why random? If you’re searching for someone to imitate, then wouldn’t you search for someone with particularly high productivity? The firms are assumed to have perfect foreknowledge of the distribution of productivity, so how come they do not know which firms are the best to imitate?
• Why is matching one-to-one? If you can imitate a firm, then why can’t all of us imitate one firm? Why can’t we all imitate the best firm?
• Search costs resources, but imitation is free. That is, the searcher has to give up production to look for someone to imitate. But once they match, they can copy the productivity level for free. So productivity techniques are absolutely non-excludable. But knowing that imitation is happening, why wouldn’t high-productivity firms hold out and demand some kind of side-payment for being imitated?

In short, I’m struggling at this point to see the specific economic context for these models of diffusion/imitation that use DPFAPDMCPU or something similar. Am I missing some kind of obvious examples here? If I am, is there a reason to think that most of the innovation that occurs is due to non-excludable imitation?

Lots of models deliver a prediction of constant growth in steady state, so why are these that use the DPFAPDMCPU assumption a better description of why that happens? I think this literature would benefit from providing a clearer answer to that question.

# Embedded Ideas and Economic Growth

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

In the last post I laid out three conditions that could when describing how economic growth worked, and said we had to pick two. In short, I argued that we should pick (1) constant returns to scale in rival inputs, and (2) non-rival ideas earn some part of output. This meant that (3), rival inputs earn their marginal product, had to be done away with.

One particular complaint about my characterization of the issue is that it did not address the concept of ideas “embedded” in people (or things). Something like “the ability to solve differential equations” is a skill that is embedded in some people, and not in others. Once you think about embedded ideas, the argument goes, then you have to consider those ideas “rivalrous”. The person who knows differential equations can’t be two places at once. The idea of solving these equations is just along for the ride in this persons head, so the idea is rivalrous as well. The strongest form of this argument would then say that it only makes sense to think of all ideas as embedded and rivalrous, and by dropping non-rival idea completely we can maintain the idea that all rival inputs earn their marginal product.

I think this is wrong. Yes, some non-rival ideas or skills are embedded in people or things. I don’t have any problem with that. But an embedded skill is still non-rival. Tomorrow someone new is going to learn to solve differential equations and that will have absolutely zero effect on my ability to solve them. I can copy the skill over and over and over, teaching the whole world to solve differential equations, and that will not diminish the ability in anyone else. That’s the definition of non-rivalry.

Saying a skill or idea is embedded in a person (or thing) is a statement about exclusion only. People who did not take the right math classes are excluded from solving differential equations. Those of us who know how to do it own an excludable skill. But it is still a non-rival skill.

Non-rival ideas or skills can even be uniquely embedded in a person or thing. Usain Bolt is uniquely capable of running 100 meters in 9.58 seconds. There has never been anyone in recorded history to run 100 meters this fast. Despite that, running a 9.58 is a non-rival skill. If this year Justin Gatlin runs a 9.58, that isn’t because he took away the skill from Bolt, which would mean the skill was rival. They could both run this fast; it is a non-rival skill.

Saying a skill is unique is a statement about exclusion, not rivalry. If no one ever again runs 100 meters as fast Usain Bolt did, that doesn’t mean running a 9.58 is a rival skill, it means that running a 9.58 is an exceptionally excludable skill. So excludable that it is impossible. But still non-rival.

Making non-rival skills hard to copy doesn’t change their non-rivalry. The fact that teaching everyone in the world how to solve differential equations would be very, very time-consuming doesn’t make this a rival idea. High costs of time or resources to create copies of skills make those skills highly excludable, but not rivalrous.

World class athletes are still probably the best example here. Roger Federer has a set of highly exclusive – and yet non-rival – skills. It is almost impossible to copy Federer’s skill set. I certainly could not, even if I had started training at age 4. But Djokovic and Nadal, after years and years of grueling training and practice, have copied enough of them that they can now beat Federer (sometimes). The skills of playing world class tennis are embedded and highly exclusive. But they are still non-rival.

So what does this have to do with growth theory? The non-rivalry of ideas or skills allows for continuous economic growth. But it is the excludability of those ideas or skills that provides incentives for individuals to create them or learn them.

Romer originally focused on non-rival ideas that were incredibly easy to copy, like software, books, or blueprints. Being easy to copy, these things are not easily excludable, and hence it would be hard to earn rents on them without some kind of protection. Things like patents or copyrights give these easy-to-copy ideas excludability. Those intellectual property rights provide the incentive for people to create new easy-to-copy ideas.

Boldrin and Levine focus on non-rival ideas that are incredibly hard to copy, like the embedded skills of solving differential equations or playing world class tennis. The sheer effort involved in copying makes these ideas highly excludable. The owners can earn rents even without explicit property rights over the idea or skill. Roger Federer doesn’t own a patent on world class tennis playing. It’s just nearly impossible to copy his skill.

In both situations, growth will arise because of the acquisition of new non-rival ideas or skills. In both situations, that acquisition occurs because the exclusivity of the idea or skill allows them to earn rents on it, and those rents are sufficient to offset the costs of inventing or acquiring it in the first place.

Where I think BL went wrong is in claiming that embedding skills or ideas in people or machines makes them rival. They used that term incorrectly. Embedding makes skills or ideas excludable, even though they are still non-rival. Once they claimed that some ideas were rival, they had to contort themselves into arguing that non-rival ideas don’t earn any rents ever to satisfy the “pick 2 of 3” conditions I laid out in the last post. If you want to accuse BL of “mathiness”, then it would be because they mis-matched the language (rivalry) with the math (excludability).

For his part, Romer has probably over-stated the importance of monopoly power over ideas. Yes, a patent gives you monopoly power over an idea. And without that patent, an easy-to-copy idea would most likely not be produced. But some ideas or skills are hard to copy, and the people who hold them do not necessarily need a monopoly over them in order to earn rents. Some ideas are hard enough to copy that you can earn rents even though you face some Cournot-style competition from the few others capable of copying you (i.e. Federer, Nadal, Djokovic). Romer doesn’t really need strict monopoly power, he just needs rents to accrue to idea owners.

The ultimate point is that the world can make sense with (a) non-rival ideas/skills, (b) that are embedded and highly excludable, (c) with Cournot-style competition among owners of the ideas/skills, and yet still satisfy Romer’s conditions that (d) we have constant returns to scale in rival inputs and (e) positive payments to non-rival ideas/skills. (b) and (c) are not incompatible with (d) and (e). But saying that non-rival ideas simply don’t exist doesn’t make any sense to me.

Last point. Given the last post, we know that such a world would require that rival inputs (raw labor, capital, land) earn less than their marginal product. The rents earned by owners of those embedded non-rival skills have to come from somewhere. How do I square that with the wages earned by someone with an embedded skill, like Federer, or someone who knows how to solve differential equations?

The important point here is to not confuse someone’s total reported “wages” with the return earned by their rival input. My total paycheck is some combination of a return to my rival input (i.e. time) and the return to my non-rival, embedded and excludable skills (i.e. teaching 1st-year grad macro). The fact that UH does not separately compensate me for these inputs doesn’t mean that my wage is being paid only for rival inputs. Some of my paycheck is rents I earn for providing a scarce, embedded, excludable but non-rival set of inputs. Some of my paycheck is compensation for my rival input, time. What the conditions I laid out last post say is that this compensation for rivalrous time is below the marginal product of my time.

# What Assumptions Matter for Growth Theory?

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

The whole “mathiness” debate that Paul Romer started tumbled onwards this week while I spent four days in a car driving from Houston to Quechee, Vermont. I was able to keep up with several new entries (Harford, Rowe, Andolfatto, Romer) regarding the specifics of growth theory when it was my turn in the passenger seat. I also had running around in my head a series of e-mails I shared with Pietro Peretto, who helped clear up a lot of questions regarding this debate (The usual disclaimer applies – Pietro is not responsible for anything stupid I say here).

Somewhere along I-40 and I-81 I was able to get a little clarity in this whole “price-taking” versus “market power” part of the debate. I’ll circle back to the actual “mathiness” issue at the end of the post.

There are really two questions we are dealing with here. First, do inputs to production earn their marginal product? Second, do the owners of non-rival ideas have market power or not? We can answer the first without having to answer the second.

Just to refresh, a production function tells us that output is determined by some combination of non-rival inputs and rival inputs. Non-rival inputs are things like ideas that can be used by many firms or people at once without limiting the use by others. Think of blueprints. Rival inputs are things that can only be used by one person or firm at a time. Think of nails. The income earned by both rival and non-rival inputs has to add up to total output.

Okay, given all that setup, here are three statements that could be true.

1. Output is constant returns to scale in rival inputs
2. Non-rival inputs receive some portion of output
3. Rival inputs receive output equal to their marginal product

Pick two.

Romer’s argument is that (1) and (2) are true. (1) he asserts through replication arguments, like my example of replicating Earth. (2) he takes as an empirical fact. Therefore, (3) cannot be true. If the owners of non-rival inputs are compensated in any way, then it is necessarily true that rival inputs earn less than their marginal product. Notice that I don’t need to say anything about how the non-rival inputs are compensated here. But if they earn anything, then from Romer’s assumptions the rival inputs cannot be earning their marginal product.

Different authors have made different choices than Romer. McGrattan and Prescott abandoned (1) in favor of (2) and (3). Boldrin and Levine dropped (2) and accepted (1) and (3). Romer’s issue with these papers is that (1) and (2) are clearly true, so writing down a model that abandons one of these assumptions gives you a model that makes no sense in describing growth.

If there is a sticking point with McGrattan and Prescott, Boldrin and Levine, or other papers, it is not “price-taking” by innovators. It is rather the unwillingness to abandon (3), that factors earn their marginal products. Holding onto this assumption means that they are forced to abandon either (1) or (2).

From Romer’s perspective, abandoning (1) makes no sense due to replication. How could it possibly be that a duplicate Earth produces less than the actual Earth? Abandoning (2) also does not make sense for Romer. We clearly have non-rival ideas in the world. Some of those non-rival ideas are remunerated in some way, whether there is market power or not. So (2) has to be true.

The “mathiness” comes from authors trying to elide the fact that they are abandoning (1) or (2). McGrattan and Prescott have this stuff about location, which is just to ensure that (1) is false. Lucas (2009), as Romer explained here, is abandoning (2), and asserts that this is something we know as a result of prior work. It’s not.

Regardless, once you’ve established the properties that you think are true, now you can talk about market power or the lack of it. Romer, taking (1) and (2) as given, asks how non-rival inputs could possibly be earning output. They are costless (or close to costless) to copy, so how is it possible for them to earn anything? Romer says that non-rival ideas must be excludable, to some extent, in order to earn the output we see them earning in reality.

A patent or copyright is one way of giving a non-rival idea some exclusivity. If that patent is strong, then it gives the owner a monopoly on the idea, and hence they can exert some market power over that idea. Market power, in this case, means that the owner can charge any price they want and still be in business. They may set a price that maximizes profits, or not. Whatever. They will not lose all their business if they raise the price.

But even if the exclusivity of the non-rival idea is not complete, and the owner doesn’t have absolute market power, this doesn’t mean the non-rival idea earns nothing. Let’s say that an idea is non-rival, but copying is somewhat difficult. Reverse engineering an iPhone, for example, is non-trivial. So perhaps no single firm owns an idea outright, but there are only limited firms that can use the idea. These firms engage in some kind of Cournot game, which means that they all earn profits, but any single firm cannot charge any price they want. If they charge slightly more, they will lose all their business to other firms. In this case the non-rival idea earns some output (i.e. the profits to those firms), but no firm has full market power.

The lack of full market power here is fully compatible with (1) and (2) being true, and (3) being false. The issue with Boldrin and Levine isn’t that they allow people to compete with the innovator immediately, it’s that they dismiss the whole idea of non-rival ideas and abandon (2). For what it’s worth, Boldrin and Levine are not guilty of mathiness, in my mind. They are really clear that they deny such a thing as a non-rival idea exists. I don’t agree with them, but they don’t try to hide this.

Aside #1: What does all this have to do with Euler’s Theorem? This theorem is the reason (1), (2), and (3) cannot all be true at once. This was implicitly what I was saying in my last post. The production function is ${Y = F(R,N)}$, where ${R}$ are rival inputs and ${N}$ are non-rival. If the function is constant returns then ${\lambda Y = F(\lambda R,N)}$. Take derivatives of both sides with respect to ${\lambda}$, and you get ${Y = R F_R(\lambda R,N)}$. Evaluate at ${\lambda = 1}$ without losing anything, or ${Y = R F_R(R,N)}$, meaning that total output equals rival factors times their marginal products. This holds, no matter what we say about how factors are paid, for a function CRS in rival inputs.

If I then say that each rival input ${R}$ gets paid a wage/return equal to its marginal product, this means that the payments to ${R}$ are exactly equal total output, ${Y}$. So there is nothing left over to pay owners of non-rival inputs. The only way to pay non-rival inputs anything is to force the wage/return to be less than ${F_R}$. Or to dismiss the assumption that the function is CRS with respect to rival inputs in the first place.

Aside #2: Yes, I spent four days driving from Houston to Quechee. Rules for long car trips with kids. First, no food in the car. Second, when the car stops, everyone pees. Third, stop every 2-2.5 hours, without fail.

# 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.