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Lumifer comments on How much does consumption affect production? - LessWrong
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That is an excellent question, but it requires an additional piece: good for what purpose?
Right. In this case, to answer the question, "If I decide to reduce my lifetime consumption of chicken by one, should I expect the long term production of chicken to drop by ~1, ~0, or something in between?" Which is of demonstrated interest to the authors I am critiquing.
That question seems to have a simple answer: your decision will not affect the long-term production of chicken.
Now I suspect that the real question is "If X million people decide to stop eating chicken, what would happen to the long-term production?" That is a much more complicated question which I don't think can be answered by moving or bending the supply and demand curves under the ceteris paribus assumption. One reason is that it's scale-dependent: different magnitude of X gives different answers. If X is small, its effect would be swamped by other factors (e.g. the growing prosperity in the developing world which generally leads to more people eating meat) and at the other end, obviously, if everyone stops eating chicken the production would drop to zero and chicken will become extinct.
Let X and Y have a causal relationship of the following form: if X is set to a given value x, Y takes on a value kx + z, where z is the value of a random variable Z which is independent of X.
What is the expected change in Y caused by a change in X of size dx?
It is k dx.
It does not matter how much the change in X is "swamped" by the variability of Z. It does not matter if it is so large that the change in Y consequent on that change in X cannot possibly be observed. The expected change is still k dx.
Of course, but you are assuming a linear relationship (y =kx + z) and I am unwilling to assume one.
Everything is linear, to first order. We are, after all, considering small changes in X. dx will only have no effect if the gradient of the actual function expressing the causal effect on Y is zero. It still has nothing to do with the magnitude of all the other effects on Y independent of X.
You are talking math and I'm talking empirics. You don't know the true function (or even whether it exists), all you can do is make approximations and build models (which, to quote George Box, are always wrong but sometimes useful).
A couple of posts up you said " It does not matter if it is so large that the change in Y consequent on that change in X cannot possibly be observed" -- and that's where we disagree. It doesn't matter in math, it does matter in reality because you don't have access to the underlying function.
Those models will say what I said: if delta Y/delta X is measurable for large enough delta X, dY/dX cannot be zero at every point throughout such a change. The average value of dY/dX over that range will be delta Y/delta X. That includes both the model that coincides with reality and all the others.
But no one is talking about every point. If you can only detect delta Y for large changes in X, it does NOT imply that small changes in X will also cause some changes in Y. Step functions are common in reality.
In this context, chicken production utilizes economies of scale. This means that a "unit" of chicken production is quite a large chicken factory. To change the number of chicken factories, you need a sufficiently large change in the relevant factors -- if the demand drops by 1 chicken, the number of factories won't change.
And if you don't know where the step is, this applies.
I think we've come full circle here.
OK, so I argue option A, you state option B, and the articles I link argue option C.
I agree it's a complicated question (in that it requires lots of information to answer precisely and accurately). If you had no empirical data to work with, what would be your best guess/expectation? Also if your answer is proportionally different than in the 'single chicken' case, I'd be curious to know why.
If I had no empirical data, I would not be making any guesses in this case.
The "single chicken" case is below the noise floor. Empirically speaking, the consequences are undetectable. And for "many chicken", how many matters -- I don't think there is a straightforward linear case here.
OK so you have no prior for large cases, you have no prior about the relationship between large cases and small cases, and your guess for small cases is "zero impact".
My prior for large cases is 1:1 impact, my prior is that the impact in large cases is proportionally similar to the impact in small cases, and therefore my prior for small cases is 1:1 impact.
Let's be clear about distinguishing between the map and the territory. To what do your priors apply?