Believe those who are seeking the truth. Doubt those who find it. Andre Gide

Friday, June 19, 2015

Competitive Innovation

In my previous post I took a crack at understanding Paul Romer's mathiness critique. As far as the post related to Romer (most of it was devoted to Brad DeLong's careless embellishment of the idea), my assessment was that Romer is basically just frustrated that his ideas have not swept away the competition in the field of growth theory. I do not believe that the academics Romer called out are defending indefensible positions for the sake of academic politics. To me, the present situation looks more like a healthy competition between theories of growth emphasizing different mechanisms. This is just what one would expect in a field where the forces at work are complicated and the answers to important questions are hard to come by.

Judging by his reply to my post here, it seems that Romer has a rather low opinion of me. Evidently, I am an "Euler-theorem denier." Admittedly, this is not the worst thing I've ever been called. But in addition to this, I am apparently motivated to deny the truth of this mathematical proposition because doing so signals my commitment to an academic club of serpents that includes the likes of Nobel prize-winning economists Bob Lucas and Ed Prescott. Paul, you flatter me.

I want to take some time here to address the specific charge leveled against me by Romer:

Andolfatto’s brazen mathiness involves a verbal statement about a mathematical model that flies in the face of an impossibility theorem. No model can do what he claims his does. No model can have a competitive equilibrium with price-taking behavior and partially excludable nonrival goods.

Romer's proposition is stated clearly enough. Now all we have to do is check whether it's valid or not. If I can produce a counterexample, then I will have shown Romer's proposition to be invalid. Let me now produce the counterexample.

Consider an economy where people combine raw labor (n) with skill (x) to produce labor services (e) according to the formula e = xn. Interpret n as hours worked over given period of time and assume, for simplicity, that everyone has the same n. On the other hand, people generally differ in their skill level, x. People with greater skills (or skill sets) are represented by larger values of x.

What makes people operating in the same environment more or less skilled than one other? Well, it could be several things. Consider two laborers, one of which is endowed with a physical tool that doubles labor productivity. Then we can write x = 1 for the one laborer and x = 2 for the other. Now consider two entrepreneurs, one of which is possessed with a "mental tool" (an idea, or some general know-how) that doubles labor productivity. Then we could similarly write = 1 for the one entrepreneur and x = 2 for the other.

Physical and mental tools can differ along an important dimension called "rivalry." Economists call physical objects "rival"(or "subtractable") goods because a physical tool can be in the possession of only one laborer at any given time. The same need not be true for a mental good like an idea. Suppose that "knowing how to perform a task at level x" requires knowledge of a certain recipe. Unlike a physical good, an idea is not subtracted from your mind if you share it with someone else. If I teach you my calculus tool, this in no way diminishes my capacity to use the same calculus tool (or what amounts to the same thing, a perfect replica of my calculus tool). Economists call goods with this property "non-rival" or "non-subtractable" goods.

Alright then, let's proceed to the next step. Assume that the aggregate production function takes the form Y = aE, where a > 0 is a scalar and E represents the aggregate labor input measured in "efficiency units." That is, E equals the sum of e = xn over a population of size N. Let me normalize n = 1 and define X as the sum of x over population N. In this case, E = XN.

Notice that the aggregate production function exhibits constant returns to scale in E. The function is also linear in N. Of course, the function displays increasing returns in X and N together. That is, if we double both X and N, output Y is more than doubled.

The question Romer might ask at this point is "where are the books in this economy?" What he means by this is why can't the smartest person in this economy (the one with the largest x) not publish his knowledge in a recipe book and sell it to the others for a huge gain? This is an excellent question. The answer to it is that knowledge is not always very easily communicated and absorbed by all members of society in this manner. To the extent that knowledge transfer is difficult, the smartest guy in the room has a temporary monopoly over the idea that generates the highest x. Personally, I am surprised that Romer is so offended by this assumption of a missing market. Any university professor knows that distributing the course text book does not, by itself (re: Lucas quote 2009), lead to a growth of his/her students' knowledge base. Some types of knowledge can be sold, but other types of knowledge must be absorbed through effort, not through simple purchase. It is an empirical question as to which type of knowledge transfer mechanism is more or less important.

Next, I want to impose perfect competition. I do this not because I am wedded to the idea that this is how one must model the economy. I do it because Romer claims that it cannot be done.

But wait a minute...how can I assume perfect competition when the smartest person has a monopoly over his/her x? The answer is simple. The people in this economy are competing over the supply of efficiency units of labor e. There is no monopoly over the supply of labor services, e. Under perfect competition, the equilibrium price of an efficiency unit of labor is given by w* = a.  Of course, the measured wage per unit of raw labor (w*x) differs across people according to their skill x. The person with the highest x commands the highest price per unit of raw labor.

I should like Paul to note that Euler's theorem does indeed hold in my economy. The entire output is exhausted as payments for labor services (measured in efficiency units). Would people ever devote costly effort to learn something that would give them a higher x in this competitive economy? Sure, why not? Suppose that some raw labor time can be moved away from work and into a learning activity (l). The opportunity cost of this time is the person's foregone wage w*xl. The benefit is the expected value of learning something new (a higher x).

Anyone with an elementary training in economics can plainly see that the model described above has a competitive equilibrium with price-taking behavior and partially excludable non-rival goods. Ergo, Romer is wrong.

I am led to speculate how Romer might object to my counterexample. I suspect he might claim that the x in my model is not really a non-rival good. I get the feeling he might be defining a non-rival good as an idea that can be costlessly disseminated and instantaneously absorbed throughout the population (unless the use of the idea is protected by patent, etc.). If this is the case, then the argument would seem to be one of semantics.

In any case, I have made my point. Romer's proposition above is invalid. One can model a competitive equilibrium with price-taking and partially excludable non-rival goods. This should not be taken as a criticism against Romer's preferred modeling strategy. I'm actually a fan of his research program. My complaint with Romer's mathiness critique is that it ascribes unseen and unknown ulterior motives to a class of economists that find it fruitful to view growth through the lens of competitive equilibrium.

***
PS. I see that Nick Rowe has offered a thoughtful response to Romer's "whack-a-mole" comment.

1. The problem is that in this model, "books" (and all learning) are entirely free. That's the "perpetual motion" problem to which Romer is alluding. You've accounted for learning time, but not learning's monetary cost (teachers getting paid). (The paragraph on books above is a false dichotomy fallacy -- since an author/teacher cannot get a "huge gain", he/she is assumed to get zero gain. Anyone should have a problem with that assumption.)

1. Jeff,

You have to get your logic straight here. Romer made a nonexistence claim. I produced an example of an economy that he claimed could not exist. His proposition is false.

Now, maybe you don't like some of the assumptions I made in my model and that's fine. But you can't sit here and tell me I'm wrong about Romer's proposition because you don't like one of my assumptions. Romer claimed that there were *no* set of assumptions that could deliver said result. OK?

Next, there are not any "books" in my model. One could hand-wave and assume there are books. But I assume that books alone do not transmit knowledge--one has to devote effort to read and assimilate what's written down. One could easily modify the model in a way that requires a person to learn from a professional teacher. The teacher would share in the NPV of the enterprise. With all due respect, I think you are completely wrong here in every dimension of your criticism. But thanks for the comment! :)

2. You're dropping the "macro" requirement in a claimed macroeconomy model? You could similarly claim that a radiometer is a perpetual motion machine if you claim that ignoring light energy is a valid assumption. It's not.

3. Jeff, I don't think that's the proper analogy. The assumption I make--that there is no market in ideas--is a legitimate one and it corresponds to reality for many types of ideas. I assume that for this class of ideas, people must devoted learning effort to absorb. I'm not sure why you find this idea so difficult to absorb, but the fact that you do demonstrates my point! :)

4. David,

Thanks for the honest acknowledgment that that is in fact an assumption built into your model here -- no market for ideas (even with excludability). I understand the idea; what I can't absorb is how that can even be characterized as a macroeconomic "model" of the U.S. (or any large) economy. It seems completely "untethered" to me, like the rental agent's "model" of reservations in the clip below. For instance, Nick R. just stated an estimate of something like 50% of the actual economy nowadays being a "market in ideas" (in response to a comment in his blog).

At any rate, this looks to me to be the crux of the disagreement, and thus the right area to focus on (for any hope of consensus developing). Thanks again for confirming so clearly that assumption in your model.

5. Jeff,

If I recall, Nick was talking about the expense of training, teaching, etc. All of that is possible in my model. In fact, in my diffusion paper with Glenn, all people with superior know-how are "teachers" in the sense that with some effort, we can learn from them (via imitation in my paper).

What is not permitted in my model is me walking into my macro class at the beginning of the semester and selling the knowledge associated with the semester's class material. There is no formal market in that because it's not possible--not yet, anyway.

Again, this is not meant to disregard knowledge transfer through (e.g.) the sale of patents, etc. It is important. But it's not the only thing.

Now, I want you to agree or disagree on this for me: Is what Romer said true or false? (I'm not asking if it's reasonable or not, that's a separate issue). Thanks.

6. In the model, there are no teachers of macro classes to begin with, because there is no market for it (no one is paid to teach). There is also no market for books -- no one is paid to write them. No market for research of any kind of course. What is it you do again? ;-)

I think Romer's take on what "macroeconomic model" means is correct, and yours is not, as above. Yes, I agree that what Romer said about impossibility is true, under the meanings of terms he's using. Because I've duplicated "the math" on my own and understand the argument.

7. Jeff, it is an easy matter to extend the model in that way.

Introduce a second set of agents called "teachers" endowed with one efficiency unit of time. Teachers value consumption and leisure.

For simplicity, suppose that the agents I described earlier in my model do not devote time to learning effort. Instead, they use some of the output they produce to invest in teaching services (which are priced competitively at real wage w* = a).

Everyone does the obvious cost-benefit calculation. Viola. We have your teachers getting paid to teach.

If you don't get it, I'll stop here because I'm afraid there's nothing much else I can do. I do appreciate the civil nature of your objections, however. Cheers!

8. I get it. But any payments to teachers is money that will necessarily be reducing the wages of workers below their marginal product (somewhere). That's the impossibility proof. I guess we could go through another iteration of you trying to show me mathematically what you mean by this proposed extension, but I can guarantee that you can't get around the impossibility proof. So it would just be another iteration of whac-a-mole. Because that is what "proof" means in this context (math, logic). This is not a courtroom nor debate-team standard (of proof).

But, as Obama once said, please proceed (if you'd like). I am game. Layout the math of teachers being paid in this model. I will respond as time allows.

9. This comment has been removed by the author.

10. Jeff, first of all, not that the technology is linear here. The wage *rate* of workers will remain the same. They just earn less wage income in my model as they devote some time to learning instead.

If we introduced a separate set of agents called teachers, then our workers would work the same amount of time and earn the same wages. But now they'd have an investment opportunity. They could use some of their earnings to pay teachers for their services (which could be produced according to a separate production technology). The supply of teaching services could be modeled as a competitive market.

Maybe you're just too hung up on aggregate production functions.

11. David, As mentioned before, understood that you are already accounting for the *time* for learning (away from work) (first paragraph).

I suspect the problem will arise accounting for "use some of their earnings" in the second paragraph (together with the requirement of equating aggregate demand and supply).

So as I said, please proceed (with the math). I promise to respond, as time allows.

12. I've tried to think ahead a bit (anticipate your model, make my own model), and I've had an insight. Once you introduce "any x_i having a cost" into the model, the problem for such a model is even more basic than any accounting or equating I mention above, and independent of any rivalry/non-rivalry considerations. So I'll put this out there in the interest of possibly saving you some time, depending on your reaction:

Once any x_i has a price in the ideas market, and a worker buys it to use at work, it's economically just like any other tool that a worker buys to use at work (computer, shovel) with the employer's permission (again regardless of any rivalry/non-rivalry considerations). And as such it has become an input to production with marginal product equal to marginal cost. If the employer pays the worker his or her marginal product but does not reimburse him or her the cost of x_i in addition, then x_i is an input into production that is not paid it's marginal product. Violating the "competitive equilibrium" condition.

13. Also, as to "aggregate production functions": how do you propose creating a mathematical macroeconomic model without modeling economic output as a function(s) of production inputs?

14. I meant to say "aggregate production function." The alternative is to specify multiple production functions that do not aggregate in the usual way. This is completely an aside, however. You might want to check the comment of anonymous below.

15. Really, what I am seeing is what is colloquially known as hand-waving. Do not aggregate in the usual way? "Missing markets" as acceptable? In a *macro*economic model?

Show the math. Thanks.

16. Sure Jeff. Here you go. I whipped this example up in just a few minutes. I hope it helps. http://www.sfu.ca/~dandolfa/competitive%20innovation.pdf

17. I see, thanks. As anticipated, there is an input on the "future period" side that is not being paid its MC(=MP), namely "y". (Future-side output is still being allocated totally to the efficiency units of labor only.) And you certainly can't argue that accounting for the price on the "present period" side only, with a zero price on the "future period" side, is sufficient without blatantly violating the price-taking requirement, right?

My sweetmeats remain uninjured. Another round?

18. Jeff,

Teaching services do not get paid in the future period because teachers only work in the present period in this model.

My point is demonstrated and there is no need for another round.

19. That's like saying you don't need to pay parts suppliers for the parts, because the parts were manufactured in a prior period (and everyone got paid then). As I said, it breaks the price-taking requirement.

20. "That's like saying you don't need to pay parts suppliers for the parts, because the parts were manufactured in a prior period (and everyone got paid then)."

Yes, that's right. If you only need parts in one period, you buy them in that period.

"As I said, it breaks the price-taking requirement."

I have no idea how paying for parts in the period you need to purchase them "breaks the price-taking requirement." All the agents in the model I described above are price-takers. It's in the math that you insisted on seeing. Seems to me that you are the one now "hand waving" objections, instead of demonstrating the proof of your claims mathematically.

21. Everyone got paid who participated in *making* the parts (all prior period). No one paid for the parts themselves in the future period, as inputs to the future production process. It's like the parts company took a loan to make its payroll, then got stiffed/robbed by it's customer (the parts "buyer"). Actually that's not a bad analogy for real-life student-loan difficulties today.

So you're either 1) not paying for an input to a production process (breaking the "competitive equilibrium" condition), or 2) changing prices between the prior period (price y) and the future period (price zero) (breaking the price-taking requirement).

Clearer now?

22. You are confused.

People do get paid for the "parts" in the future, because these parts are embedded in their future human capital, which earns the market price.

Please just work through the math. No need to speculate.

2. I think Romer would be right to say your x is not non-rival, and it does not require any such exteme definition of "rival". For non-rivalry there needs to be some spillover possible between the amount of the good possessed by one person and the amount used by another. I see none here. Each person builds his own x. It's just human capital.

To make it paritally non-rival -- and get a very Romerish model -- make the cost of aquiring a unit of x a decreasing function of the aggregate X in the economy. Then suppose I choose, in some year, to invest part of my time in increasing my x. My marginal product in that year is the normal function of the time I spend in current production; plus the discounted value of the increments to my future production; plus the discounted value of the increments to everyone else's future production (net of the learning effort they expend) induced by the fact that my decision increased aggregate X and thus reduced their learning costs. I don't get paid that last increment. So in this case, Euler's theorem and non-rivalry are made consistent by some people not earning their true marginal product.

I'm not sure whose semantics I'm supporting here. When Romer speaks of "competitive equilibrium" he always seems to have in mind that all markets exist, and in my example there is a missing one -- we can't transact over my X-augmentation. But it is certainly not weird to call this a competitive economy.

1. Michael,

You say: "For non-rivalry there needs to be some spillover possible between the amount of the good possessed by one person and the amount used by another."

But if you look up the definition of non-rivalry, it makes no mention of spillovers. And even if it did, then we're just arguing semantics, and Romer had no right to launch into his Quixotic crusade.

2. "then we're just arguing semantics,"

I'd guess about 50% of economics is arguing semantics.

3. You were remarkably polite David. I'm still not Canadian enough.

Loved your revenge of the https://londonhistorians.wordpress.com/2011/03/08/to-the-little-gentleman-in-the-black-velvet-waistcoat/

Did you see Robert Waldman's counterexample? http://rjwaldmann.blogspot.ca/2015/06/costly-r-in-model-of-competitive_18.html

Sort of similar to yours?

Setting aside Paul Romer: I was wondering if ideas aren't subject to decreasing returns at the macro level, if new ideas are cumulative (we have to learn the prerequisite before we can understand the new stuff), and if learning takes time, and if we have finite lives. I think we might end up with something a bit like the Solow growth model, Past the golden rule, we spend too much time learning and have too little time left in our lives to produce consumption goods. (Is this idea old hat? I'm a short run guy myself.)

1. Nick,

I left a reply at your post, but it seems not to be there anymore? In any case, thanks--I had never heard of the little gentleman in the black velvet waistcoat. :)

I think your posts raised legitimate questions about the concept of rivalry. I have a hard time understanding why PR feels compelled to attack these lines of inquiry with such vicious pettiness. He might just be having a bad month, so I'm willing to cut him some slack.

Thanks for pointing me to the Waldman post. Indeed, his model looks virtually identical to my own. But then, I wrote down my model in 1990 (the model that PR disparaged as a "graduate student exercise"). Of course, the idea was not original with me. I adapted it from the work of Jovanovic and MacDonald on "Competitive Diffusion."

With respect to your aside, I'm not sure what to think. My feeling is that different types of knowledge have different properties and are learned in different ways. The Romer approach makes sense for a large class of technologies/ideas. But one can legitimately ask, the way Lucas did, whether the Romer mechanism is central, or whether other mechanisms might be more important. I'm not sure. I don't work too much on growth these days. That darn \$4.5T balance sheet consumes most of my time! :)

2. Nick,

I left a reply at your post, but it seems not to be there anymore? In any case, thanks--I had never heard of the little gentleman in the black velvet waistcoat. :)

I think your posts raised legitimate questions about the concept of rivalry. I have a hard time understanding why PR feels compelled to attack these lines of inquiry with such vicious pettiness. He might just be having a bad month, so I'm willing to cut him some slack.

Thanks for pointing me to the Waldman post. Indeed, his model looks virtually identical to my own. But then, I wrote down my model in 1990 (the model that PR disparaged as a "graduate student exercise"). Of course, the idea was not original with me. I adapted it from the work of Jovanovic and MacDonald on "Competitive Diffusion."

With respect to your aside, I'm not sure what to think. My feeling is that different types of knowledge have different properties and are learned in different ways. The Romer approach makes sense for a large class of technologies/ideas. But one can legitimately ask, the way Lucas did, whether the Romer mechanism is central, or whether other mechanisms might be more important. I'm not sure. I don't work too much on growth these days. That darn \$4.5T balance sheet consumes most of my time! :)

3. David: your comment is still there. You left it on a different post. (I now have 3 posts on the rivalry thing.)

4. Ah, I will go read them all now! :)

5. Re the cumulative nature of innovation potentially leading to decreasing returns see Ben Jones' "The Burden of Knowledge and the Death of the Renaissance Man: Is Innovation Getting Harder?" http://www.kellogg.northwestern.edu/faculty/jones-ben/htm/BurdenOfKnowledge.pdf

4. David,

I don't think Romer's counter-arguement would be that x is actually a rival good, clearly it's not. I think he'd claime the model isn't really one of perfect competition.

He'd probably call it monopolistic competition, people competing to sell differentiated goods. Differing in the amount of x the goods come with.

I don't know, that's my first guess.

1. He might complain that I am assuming that some markets are missing. But these missing markets would be missing in model under perfect or imperfect competition.

In any case, he would have promoted the discussion in a much better way by respectfully asking people just what they had in mind, instead of asserting that what they say is impossible and that they are driven to say things to satisfy some ulterior motive.

2. David,
Yours is a very clear counterexample. It reminds me of how a price-taking equilibrium can exist with some firms more productive than others so long as the more productive firms either facing some capacity constraints or what is the essentially the same thing a decreasing returns technology. This is elementary undergrad economics. Of course a more sophisticated way of putting it is that of a missing market.
More to the point – I think what is missing in the debate is that there is no need for the superior technology concerned to literally diffuse to the entire population at no cost to wipe out any meaningful pure profits for the innovator. Patent or no patent, I couldn’t make a smartphone no matter how hard I try to learn to make one. All that’s needed is that there are people who can acquire the knowhow quickly enough and at a low enough cost. Romer’s model is best thought of as an idealization of such a scenario. The debate, like what you said, would be more useful if people spell out explicitly what they think is more or less applicable to reality, instead of accusing others for having ulterior motives, and the like.
C

5. I like how clear you made your counterexample and I really appreciate how you took the time to keep explaining your point in the comments.