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.