Sunday, August 27, 2017

Scope conditions

I think I should credit Noah Smith for the phrase "scope conditions" I use in my book. My use of the phrase has been a bit of a tongue-in-cheek jibe ever since Noah ascribed a phrase I had never used to physicists:
I have not seen economists spend much time thinking about domains of applicability (what physicists usually call "scope conditions"). But it's an important topic to think about.
A discussion of scope conditions referencing two of Noah's posts is here.

Being a much more widely read blogger with a much bigger platform at Bloomberg View, he has since made "scope conditions" or "theory scope" more ubiquitous than the more technical physics terms "limit", "scale", or "region of validity".

But I've been attempting to get economists to see and understand this idea for awhile now. The earliest documented evidence is from a comment on Scott Sumner's blog from 2011 (using the physics term "limit"):
My opinion here, but I think finding a theory that reduces to both a monetarist theory and a Keynesian theory in various limits or under specific constraints may be a key to understanding macroeconomics and more focus should be in that direction (unless it has already been done! I haven’t been able to find anything). Both theories appear to save the phenomena in particular regimes so the “correct” macroeconomic theory should reduce to each in particular limits.
Not that I have any issue with "scope conditions" — it's a really good phrase for this concept. It's just that I'd never used the phrase as a physicist. I do have a personal story about the concept however.

During my thesis defense (aka the final exam at UW, where you present the research in your thesis to your committee and then spend an hour or longer responding to the toughest questions about it your committee can think of), I was asked where the model of quark physics I was using was valid (its region of validity). Normally this would be an easy question because the particular assumptions usually yield direct "scope conditions". You assume the speed of light is large and so the model is valid for velocities small compared to the speed of light: v << c.

However, my model lacked a specific property of the underlying theory (you could call it the "microfoundations") quantum chromodynamics (QCD) called confinement — you never see an individual quark, they always come in "colorless" combinations of three red-green-blue or two red-antired.

The problem is that confinement has not been proven analytically from QCD yet, only shown via some experimental and computational methods. So it is much more difficult to translate an assumption that confinement doesn't matter into a specific scope condition. So you can probably guess I struggled with the question in my thesis defense.

I've thought about that question off and on over the past 12 years (almost to the day in August of 2005). My best answer now is that the scope is the same scope as the deep inelastic scattering model: Q² >> 1 GeV² (probing the nucleus at less than 1 fm length scale). Asymptotic freedom (confinement disappears at high energy) of QCD means that confinement doesn't have an effect at high energy except for the input scale for DGLAP evolution (the equations used to change the energy scale of quark and gluon distributions). However, until confinement is understood analytically, this particular scope condition is going to be more hand waving than science.

So in the end I completely agree with Noah: scope conditions are important to think about.

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