# 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.

## 21 thoughts on “What Assumptions Matter for Growth Theory?”

1. As I was driving to Paris from Chicago, I realized that Romer is making some good points but in a garbled way. It’s not by accident that his follow ups are always better than his initial posts. In this case, mathiness is lack of logical consistency and that is not the same thing as a law of physics.

• No, you must not be following the debate closely enough (and/or are deliberately tring to obfuscate), because that is not all that Romer means by “mathiness”. More like “deliberately hiding a thesis’s fatal logical flaw(s) under layers of difficult math”.

• Thanks for the not needed clarification! In his last clarification however Romer’s point was about logical inconsistency. He seems to be claiming this is done intentionally and thus mathiness is used to cover it up. Except those who He accuses of logical inconsistency wouldn’t agree with him! Andolfatto does a good job responding to this issue. This blog does a better job than Romer by supporting his criticism of logical inconsistency- intentional or not!

On a long drive, thank god for pee breaks!

• As I get older the pee breaks turn into more of a “stretch my back” breaks.

I think you’re right, that there is a set of logical problems with the price-taking models Romer goes after. But I’m not sure its the actual price-taking that is the issue.

2. Wait
we can duplicate the world?

Isn’t it true that all factors of production are somehow constraint and that with this in mind everything will eventually have diminishing product of scale?

• It’s a thought experiment. You don’t actually have to be able to do it to draw the conclusion about returns to scale.

3. Isn’t K a non-rival good in this debate? Pretty clearly, there are returns to K. And no, ,the argument that non-human K requires human K is (increasingly) not valid.

• No, capital is rival. My computer is K. If I’m using, you cannot use it. So it’s rival. Rivalry isn’t about returns, it’s about whether my use of the input prevents you from using the input. So my computer could easily have increasing returns to scale (i.e. if I got a computer twice as powerful, I’d be four times as productive), and still be rival.

4. DV:
a) Greetings from (elsewhere in) the Upper Valley
b) I have paid little attention to the growth literature since my first year in graduate school, i.e., pretty much the literature that Romer kicked off decades ago with his dissertation. A production function with ideas as one of the inputs looks to me to be related to (older) discussions about production frontiers, with ideas moving the frontier out. Can you point me to something reasonably basic that either relates these together or explains why they are different? How or why is it useful combine a family of production functions into a single one by adding the argument of ideas? I am missing much of the point of these discussions, at least as they relate to the substantive points at issue* because I don’t really see the point of formulating things this way.

Thanks

*I understand 1-3 above and your discussion about them, but I am missing something about the underlying substance of the papers under consideration, and would like to understand what the disagreement is about, beyond the mathiness issues and the alleged differences in ideology (i.e., perfect vs. imperfect competition and any policies that might follow).

• Paul – thanks. I’m enjoying the cool weather – Houston is already up to basically 100% humidity these days.

I can’t think of a source off-hand on this, but I don’t know that we’ve strayed too far from what you know. A PPF tells us the amounts of two different goods that we can produce (guns versus butter, classically). And an increase in rival inputs (capital, labor) or non-rival inputs (innovations in production) would push “out” the PPF, allowing more of one or both goods to be produced.

Growth theory dropped the concern with tradeoffs between the two goods, which was most useful when talking about trade anyway. So now just think of a PPF that only is concerned with one good – “output”. The one-dimensional PPF lies along a line. More rival inputs or non-rival ideas push the PPF “out”, allowing us a higher level of output. The assumptions I laid out in 1-3 are just about how much that one-dimensional PPF expands when rival inputs are added.

Behind this you can imagine that there is a whole set of decisions being made about how to allocate our production between guns, butter, and other outputs. So one way to think about growth theory is that it has divided up the problem into two parts. First, study the PPF that produces “output”, this blob of “stuff”. Second, people decide how to divide the “stuff” up into cars and computers and apples and iPads. And growth theory has run off to study the first while not worrying (as much) about the second.

5. Some of the mathiness argument seems to include elements similar to pharmaceutical firms who tweak their blockbuster meds to maintain economic rents. Here, ironically, some with market power tweet their price taking models to maintain journal and ‘market’ dominance.
(We don’t always need more cowbells.)

• That may be an accusation you could level at all kinds of research. One of the frustrations may be that papers that simply add “more cowbell” are treated as ground-breaking.

6. Dietz: that clarifies it for me. This bit in particular:

“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.”

Yep.

• Great. I just saw your latest posts on the questions here – I missed them when writing the last post.

7. Dietz, how does this model fit with your three statements:

Assume a farmer growing tomatoes. At first, he waters them whenever it comes to his mind. Then he has an idea: Watering them in the evening will give the water more time to soak into the ground then doing it when it is sunny. He now waters his plants only in the evening which yields him bigger tomatoes. It seems to me that all rival inputs (land, water, labor) stay the same but output still increases. Would (1) not be true in that case?

• Fits perfectly. (1) still holds here. If you doubled land, water, and labor, but kept watering randomly, you’d double output. That is, you’ve doubled rival inputs, but left the non-rival input (watering technique) unchanged. Night-time watering is an improvement in technique, raising output while keeping rival inputs constant.

• Thanks, Dietz. I think your 1st statement must then mean that increases in rival inputs can raise output but are not necessary to raise output (as in my example). Now, which of the three statements is then not true in my example?

8. Great post. But isn’t the choice between these ‘assumptions’ an empirical question? I hope we have moved beyond thought experiments…