# 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.

# Scale, Profits, and Inequality

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

After my post last week on inequality, I got a number of (surprisingly reasonable) responses. I pulled one line out of a recent comment, not to call out that particular commenter, but because it encapsulates an argument for *not* caring about inequality.

Gates and the Waltons really did probably add more value to humanity than the janitor at my school.

The general argument here is about incentives. Without the possibility of massive profits, people like Bill Gates or Sam Walton will not bother to innovate and create Microsoft and Walmart. So we should not raise taxes because those people deserve, in some sense, the fruits of their genius. More important, without them innovating, the economy wouldn’t grow.

But if we take seriously the incentives behind innovation, then it isn’t simply the genius of the individual that matters for growth. The scale of the economy is equally relevant. In any typical model of innovation and growth, the profits of a firm are going to be something like Profits = Q(Y)(P-MC), where (P-MC) is price minus marginal cost. Q(Y) is the quantity sold, and this depends on the aggregate size of the economy, Y.

The markup of price over marginal cost (P-MC), is going to depend on how much market power you have, and on the nature of demand for your product. This markup depends on your individual genius, in the sense that it depends on how indispensable people find your product. Apple is probably the better example here. They sell iPhones for way over marginal cost because they’ve convinced everyone through marketing and design that substitutes for iPhones are inferior.

The scale term, Q(Y) does not depend on genius. It depends on the size of the market you have to sell to. If we stuck Steve Jobs, Jon Ive, and some engineers on a remote island, they wouldn’t earn any profits no matter how many i-Devices they invented, because there would be no one to sell them to.

People like Gates and the Waltons earn profits on the scale effect of the U.S. economy, which they did not invent, innovate on, or produce. So the “rest of us”, like the janitor mentioned above, have some legitimate reason to ask whether those profits are best used in remunerating Bill Gates and the Walton family, or could be put to better use.

There isn’t necessarily any kind of efficiency loss from raising taxes on Gates, Walton, and others with large incomes. They may, on the margin, be slightly less willing to innovate. But if the taxes are put to use expanding the scale of the U.S. economy, then we might easily increase innovation by through the scale effect on profits. Investing in health, education, and infrastructure all will raise the aggregate size of the U.S. economy, and make innovation more lucrative. Even straight income transfers can raise the effective scale of the U.S. economy be transferring purchasing power to people who will spend it.

Can we argue about exactly how much of the profits are due to “genius” (the markup) and how much to scale? Sure, there is no precise answer here. But you cannot dismiss the idea of taxing high-income “makers” because their income represents the fruits of their individual genius. It doesn’t. Their incomes derive from a combination of ability and scale. And scale doesn’t belong to individuals.

The value-added of “the Waltons” is particularly relevant here. Sam Walton innovated, but the profits of Walmart are almost entirely derived from the scale of the U.S. (and world) economy. It’s the presence of thousands and thousands of those janitors in the U.S. that generates a huge portion of Walmart’s profits, not the Walton family’s unique genius.

Alice Walton is worth around \$33 billion. She never worked for Walmart. She is a billionaire many times over because her dad was smart enough to take advantage of the massive scale of the U.S. economy. I’m not willing to concede that Alice has added more value to humanity than anyone in particular. So, yes, I’ll argue that Alice should pay a lot more in taxes than she does today. And no, I’m not afraid that this will prevent innovation in the future, because those taxes will help expand the scale of the economy and incent a new generation of innovators to get to work.

# Is Capital Important?

There is kind of a disconnect in teaching economic growth. We spend a lot of time telling students about the Solow model and capital accumulation, but at the same time the general consensus among growth economists is that total factor productivity is more important to understanding levels of output per worker.

Why do we think that capital isn’t terribly important to levels of output per worker? Basically, because the correlation between capital per worker and output per worker is low – or rather, we assume that it is low. Here’s a way of thinking about this in terms of simple regressions. If I was interested in how important capital per worker was in explaining output per worker, I could run this regression for a sample of countries (${i}$)

$\displaystyle \ln{y}_i = \beta_0 + \beta_1 \ln{k}_i + \epsilon_i \ \ \ \ \ (1)$

where I’ve put output per worker (${y_i}$) and capital per worker (${k_i}$) in logs. Logs keep countries with very small or very big values of capital per worker from being so influential, and in logs this regression will have an obvious interpretation for the coefficient ${\beta_1}$.

If I run this regression, I’ll get some estimated coefficient ${\hat{\beta}_1}$, which is the elasticity of output per worker with respect to capital per worker. Moreover, I could look at the R-squared of this regression. This R-squared will tell me what fraction of the variance of log output per worker (${Var(\ln{y}_i)}$) is explained by variation in log capital per worker (${Var(\ln{k}_i)}$). The R-squared is really what I want; it’s the answer the question “How important is capital in explaining differences in output per worker?”. The coefficient by itself doesn’t tell us that answer.

Now, there are some big problems with this regression. Most importantly, it is almost certainly the case that ${\ln{k}_i}$ is correlated with ${\epsilon_i}$, the residual. The residual captures things like technology levels, institutions, human capital, etc.. etc.. and capital per worker tends to be large when these things are “big”, meaning that they have a big positive effect on output per worker.

So that means we cannot trust our estimate ${\hat{\beta}_1}$, and cannot trust our value of R-squared. It’s worth writing out what the “true” R-squared is if we in fact had the right estimate of ${\beta_1}$. I’ll pre-apologize for the fact that this involves a lot of steps, but I’m writing them all out so it is easier to follow.

$\displaystyle \begin{array}{rcl} R^2 &=& \frac{{\beta}_1^2 Var(\ln{k}_i)}{Var(\ln{y}_i)} \\ \nonumber &=& \beta_1 \frac{Cov(\ln{k}_i,\ln{y}_i)}{Var(\ln{k}_i)}\frac{Var(\ln{k}_i)}{Var(\ln{y}_i)} \\ \nonumber &=& \beta_1 \frac{Cov(\ln{k}_i,\ln{y}_i)}{Var(\ln{y}_i)} \\ \nonumber &=& \frac{Cov({\beta}_1\ln{k}_i,\ln{y}_i)}{Var(\ln{y}_i)} \\ \nonumber &=& \frac{Cov({\beta}_1\ln{k}_i,{\beta}_1 \ln{k}_i + \epsilon_i}{Var(\ln{y}_i)} \\ \nonumber &=& \frac{ Var({\beta}_1\ln{k}_i) + Cov({\beta}_1 \ln{k}_i,\epsilon_i)}{Var(\ln{y}_i)}. \nonumber \end{array}$

The last line is identical to what Pete Klenow and Andres Rodriguez-Clare (1997, and KRC hereafter) use to evaluate the importance of capital in explaining cross-country output per worker differences. In other words, KRC are just looking for an R-squared. But as they point out, they cannot simply run the regression I proposed above and get the R-squared from that, because almost certainly ${\hat{\beta}_1 \neq \beta_1}$.

Rather than run the regression, KRC suggest that we use some alternative means of estimating ${\beta_1}$. They propose using the share of total output that gets paid to capital. Why? Because under perfect competition and constant returns to scale, that share should be precisely equal to ${\beta_1}$. In data from the U.S., capital’s share of output is usually something between 0.3–0.4, and KRC use ${\hat{\beta}_1 = 0.3}$. The rest of their data (${\ln{k}_i}$ and ${\ln{y}_i}$) is exactly the same data that one would use to run the regression. The only thing they are doing differently is plugging in their outside estimate of ${\hat{\beta}_1}$. What KRC find is that their R-squared is about 0.30, or that only 30% of the variation in log output per worker across countries is accounted for by variation in capital per worker across countries. This is a big reason why growth economists don’t think capital is of primary importance in explaining cross-country differences in output per worker.

It’s interesting to consider, though, what could rescue capital as an important explanatory variable. KRC use the idea that capital’s share in output is equal to ${\beta_1}$ under perfect competition and constant returns to scale. But what if there is not perfect competition and/or constant returns to scale? There is a neat little relationship that holds if we assume that firms are cost-minimizers. That is

$\displaystyle s_K = \frac{\beta_1}{\mu} \ \ \ \ \ (2)$

where ${s_K}$ is capital’s share in output (which KRC say is about 0.3) and ${\mu \geq 1}$ is the markup over marginal cost for firms. ${\mu = 1}$ only under perfect competition, and if there is imperfect competition or increasing returns to scale then markups are greater than one, meaning that the price charged by firms is greater than their marginal cost. From this we see that capital’s share may understate the value of ${\beta_1}$ if ${\mu>1}$. In particular, if there are increasing returns to scale at the firm level (i.e. fixed costs) but perfect competition (i.e. free entry/exit) then ${s_K}$ still measures the payments to capital accurately, but ${\mu}$ will be greater than one as firms with increasing returns need to charge more than marginal cost in order to cover the fixed costs.

Practically, if ${\beta_1 = 0.55}$, meaning that ${\mu = 1.83}$, or a markup of 83%, then the R-squared for capital goes to one. That is, with ${\beta_1 = 0.55}$, capital perfectly explains the varation in output per worker. Even with ${\beta_1 = 0.45}$, the R-squared is 0.67, meaning capital explains 2/3 of the variation in output per worker. So a relatively slight adjustment in the value of ${\beta_1}$ changes the conclusion regarding capital’s importance for output levels.

The issues with this line of thinking are (1) if there are increasing returns to scale at the firm level, why don’t we see increasing returns to scale at the aggregate/country level? (2) even if capital explains most of the variation in output per worker, there isn’t any data showing that savings rates actually vary across countries meaningfully. The differences in capital are probably the result of different technologies/institutions, and so those are the more fundamental source of variation.

# Technology and Scale

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

This is a neat write-up by Matt Ridley regarding some research done by anthropologists Michelle Kline and Rob Boyd (ungated original paper here). They collected information on marine foraging technology used by 10 different Pacific Islander tribes at the time they first met Western explorers/colonizers. According to Ridley they assigned scores not only for the number of tools but also for their complexity. “A stick for prying clams from the reef, for example, counted as one techno-unit, whereas a bamboo crab trap with a baited lever counted as 16, because it comprised 16 working parts, each a technology in its own right.” The actual paper can give you a more detailed idea of the method.

The big take-away is that the higher the population, the more complex the technology being used. Hawaii, with 275,000 people, had seven times as many tools and those were of twice the complexity of those in Malekula, which only had 1,100 people. Further, the size of the network mattered. Island tribes that had more connections with other tribes also tended to have more tools and tools of higher complexity.

This is precisely what goes into our standard models of technological innovation. We tend to say something like $\dot{A}/A = \theta L/A$, so that the growth rate of technology is increasing in population, as the anthropologists found, but decreasing in the level of technology itself. Moreover, that population L need not be limited to a country, but is really the population of those economies that are integrated enough to share ideas. Regardless, the idea that technological change is positively related to population size can seem counter-intuitive the first time you encounter it. But Kline and Boyd’s study gives a really nice demonstration of the power of scale. Simply put, more people means more chances for someone to have an “Aha!” moment, and more people tinkering around with existing ideas.

A model like this has the implication that long-run technological change is proportional to the rate of population growth (of integrated economies). In other words, long-run living standards depend positively on the population growth rate. Population growth may instill some drag on living standards because of fixed resources and/or lower capital/labor ratios, but ultimately the positive effect of population growth on technology wins out.

I’m absolutely saving this paper to use next time I teach growth at any level.