Monday, August 28, 2017

The replication argument

Increasing or decreasing returns to scale?

Now that the book is out and I'm back from vacation, I can start up the regular blogging again. While I was on vacation, I read Miles Kimball's post on decreasing returns to scale (via unlearningecon). It opens:
There is no such thing as decreasing returns to scale. Everything people are tempted to call decreasing returns to scale has a more accurate name. The reason there is no such thing as decreasing returns to scale was explained well by Tjalling Koopmans in his 1957 book Three Essays on the State of Economic Science. The argument is the replication argument: if all factors are duplicated, then an identical copy of the production process can be set up and output will be doubled. It might be possible to do better than simply duplicating the original production process, but there is no reason to do worse. In any case, doing worse is better described as stupidity, using an inappropriate organizational structure, or X-inefficiency rather than as decreasing returns to scale.
I think this is an excellent example of a case where economics takes a cold logical argument, and attempts to apply it to real world data. Essentially, Kimball is saying there is no such thing as decreasing returns to scale because of logic (the replication argument [1]) therefore everything that appears as decreasing returns to scale must be something else. That something else, however, relies on a particular model of the underlying microeconomics that we don't necessarily understand with a good empirical model (organizational structure, "stupidity").

In physics, sometimes we don't understand the particular micro theory well enough but are able to make an effective macro theory by using micro theory to narrow down a form and fit parameters. The classic example is chiral perturbation theory (which is so classic that when you say Effective Field Theory without specifying any more detail, the assumption is that you're talking about chiral perturbation theory). In that case we don't understand quark physics enough to describe nuclei and hadron interactions in terms of the quark theory (QCD).

In another (possible) example, Einstein's gravity may actually not be a real force but rather an entropic force (here, here) and therefore Einstein's description is an effective theory where we don't understand the real micro theory (e.g. is there quantum gravity?).

In the economics example, we don't necessarily understand the underlying microeconomics that yield decreasing returns to scale, but we can begin to understand them as an effective theory of decreasing returns to scale. Kimball's claim translated into physics would have him saying there is no such thing as a gravitational field, it's all gravitons. Not only would physicists continue to use Einstein's equations without knowing the quantum theory of gravity, but Kimball the physicist could be completely wrong because it may turn out there is no such thing as a graviton because gravity is an emergent effective theory.

I think the point I am trying to make here is that the underlying micro theory of production by humans organized in firms is not some well-established empirically accurate theory. But Kimball is making assumptions about it that may turn out to be incorrect. I can illustrate the converse with what turns out to be a formally equivalent argument in physics: the Gibbs paradox.

The Gibbs paradox is about the entropy of an ideal gas: the first formula derived was a decent effective description but had issues with a theoretical replication argument. If you doubled the amount of an ideal gas, you more than doubled the entropy using Gibbs formula. It was a problem, but in this case it was a problem that involved an otherwise empirically successful micro theory (statistical mechanics of atoms). Because physicists had been successful with the micro theory, you could take the replication argument seriously. If physicists had been ignorant of the underlying micro theory, there would have been no reason to think this was a problem (maybe entropy wasn't extensive, i.e. "constant returns to scale"). Maybe whatever matter was made of had this property in terms of Boltzmann's definition of entropy? With 20/20 hindsight, we know it wasn't correct [2] but was statistical mechanics a foregone conclusion? If it started disagreeing with empirical data, like many other ideas in physics, it would've been thrown out. How does Kimball know X-efficiency isn't going to be thrown out by empirical studies?

The thing is that organizations and economic forces are at their heart social systems. There is no particular reason that doubling all the means of production should yield at least double the output, especially if we're including things like money. Two people working on a project doesn't necessarily double the output or divide the time it takes in half. Why? I don't know. It's probably complicated.

However, let me close with an explicit example of a plausible social model that could manifest decreasing returns to scale: trust. Trust has decreasing returns to scale. The bigger a group of humans, the less trust there is among them. As trust decreases, contracts inside a firm need to be more explicitly specified resulting in additional costs (Coase and the theory of the firm). It's true I may never be able to build a microfounded model of agent trust, but I could build an effective theory based on sociology studies.

In the end, it doesn't make sense to say: "There is no such thing as decreasing returns to scale, you should call it lack of trust, which always happens in human systems and always leads to decreasing returns to scale." Decreasing returns to scale (if that model is true) is as fundamental to production as the production inputs if it always happens. It is true that maybe we could discover an alien species where trust doesn't decrease as you increase the size of the social group (e.g. the Borg). At that point we might have to rewrite the theories. But much like we can't logically deduce the existence of aliens, using logic to say there's no such thing as decreasing returns to scale without an empirically accurate theory that tells us the underlying micro theory assumptions are sensible.



[1] The replication argument is used to argue that there can't be increasing returns to scale either.

[2] The solution was found in recognizing atoms of the same type are actually indistinguishable, so the many identical states where you exchange one atom for another are over-counted in Gibbs' entropy formula.


  1. The problem with your argument regarding trust is that we could simply reduce the size of "teams" in the company until the trust issue is not a factor. If groups of ten are most efficient then organise your 100 person company into 10 groups of 10 people with little connection between them.

    The little empirical evidence I have found also suggests that most firms do not face decreasing returns to scale which suggests the theoretical argument you are critiquing is right.

    1. I am sure there are multiple plausible models one could propose that could cause increasing or decreasing returns to scale -- and that is my point: why make *any* assumption about returns to scale (decreasing, constant, or increasing) in a production function? Let empirical data tell you rather than inventing just-so stories about trust.

      For example, I can take your example and ask why intra-group trust increasing isn't overtaken by inter-group trust falling? Maybe trust is increased among the 10 people, but the trust between each pair of the 10 groups of 10 is *even worse* than the case of 100 people as a single organization?

      This is actually a plausible mechanism behind factionalization.

      And if this does in fact happen, we've now created a "puzzle" for e.g. an empirical finding that sees constant or increasing returns to scale.

      The question I am asking above is why assume anything based on just-so stories? Why do you need to have a preconception before looking at the data?

      Just measure the empirical data and look at the result.

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    3. If trust between our subgroups is a problem we organise things such that there is zero connection between our subunits of 10 people. We can always double our output by exactly doubling our production process and having no connection between the two different units.

      That means at worst we have constant returns to scale. This isn't some theoretical argument we could put this into practice, and franchising is an example of businesses doing exactly that.

      It isn't much of a puzzle that some empirical fields find increasing returns because a minority of fields find themselves in that situation and when they do it is typically because a factor of production cannot be duplicated without that factor increasing in price. But even in that case the returns to scale are decreasing on an industry wide level and not for individual firms.

      Of course it would be a good idea to look at the data but since the focus of much of microeconomics on decreasing returns to scale isn't justified empirically it makes sense to critique that approach theoretically.

    4. Yes, your "franchise" model may well deliver constant returns to scale, and I'm sure there are other just-so stories that can plausibly explain constant returns to scale. But are all businesses franchises? No, so you're going to need another just-so story to explain constant returns to scale for some other business model.

      I also think you are mis-characterizing the empirical data on returns to scale: there is not evidence *for* constant returns to scale, but a lack of evidence *against* it, e.g.:

      It remains a fine assumption in a model, but this finding does not tell us that any "just-so" story that results in constant returns to scale is valid -- nor does it validate the replication argument.

      This is a question of scientific methodology. As Feynman says, you should be leaning over backward to reject your models. The present state of the data -- which says we can't be sure constant returns to scale isn't present in the empirical data -- means that we should be completely agnostic about the mechanism causing it (because it may well not be happening, and there might be increasing or decreasing returns to scale ... e.g. this paper finds returns to scale might be slightly decreasing).

      The problem with a lot of economics research is that people seem too eager to propose mechanisms or assume something is true. What if returns to scale are found to be slightly decreasing (per Fernald's paper)? What does that do to the replication argument held up as pure logic?

      One time everyone thought waves had to travel in some medium, and therefore thought it eminently logical that light waves travel in some medium and called it the aether. This was bad science. We should have been leaning over backwards to reject our intuitions -- there was no evidence for or against the aether, so we should have been agnostic.

      We exist in the same situation with regards to constant returns to scale and our aether is the replication argument. Proper scientific methodology would lead us to be agnostic about the mechanism (all those just-so stories). There is no way that a finding that we can't reject constant returns to scale leads to a rejection of the trust mechanism for decreasing returns to scale I describe above. I proposed it as an example of how you could invent a just-so story to explain anything -- even why an argument like the replication argument could be wrong. It was supposed to prompt leaning over backwards to reject inventing just-so stories that aren't scientific. It was supposed to help us say "whereof I do not know, thereof I will not speak".

      Unfortunately we live in a world of "male answer syndrome" that seems to be worse in economics -- a male dominated field. Men seem to answer "yes" (or sometimes "no") to the question of whether there are constant returns to scale when they should say -- if they were real scientists -- "I don't know".

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    6. The argument you are critiquing is not saying firms will always get constant returns to scale. It is saying that there is always a way for them to get at worst constant returns to scale. If they aren't they are not being rational.

      So of course decreasing returns to scale are no problem for this argument at all.

      It seems incorrect to me to say that this argument is representative of anything in the field of economics as a whole. At least 80% of economics teaching up to the graduate level assumes decreasing returns to scale with little empirical or theoretical justification.

      In addition many economists seem to take models that assume decreasing returns and apply them well outside the scope conditions you talked about in a previous post.

      I would say the argument we are discussing is equivalent to Einsteins thought experiment that shows there can be no length contraction for an object moving at the speed of light in the direction perpendicular to the direction of travel. There are many cases of using logic to arrive at conclusions about the real world and this one seems no different. It isn't bad science to do so. Would you seriously suggest agnosticism about that argument that Einstein made?

      It is basically just an matter of logic that if we take all the inputs that produce 10 output and exactly double them while having no connection between them we will produce double as much. It seems to me that the burden should be on the people arguing that somehow it isn't possible to do that.


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