Stability and counterfactual conditionals

I'm writing this just to try to get my own thoughts straight, on a topic I know nothing about. And because commenter Ram asked me to. Read at your own risk.

I have a theory about water levels. According to my theory, the level of water in Lake Ontario will always be exactly the same on both sides of the Canada/US border.

Suppose it weren't the same. Suppose it were higher on the Canadian side. Then pressure would be higher on the Canadian side, and water would flow from high to low pressure, which would equalise the levels.

That was a counterfactual conditional statement (actually four counterfactual conditional statements). According to my theory, there never has been a time and never will be a time at which the water level is higher on the Canadian side of the border. I am asking what would happen if something were to happen which my theory says has never happened and will never happen. And my counterfactual conditional statements explain why it has never happened and will never happen. Because if it started to happen it would immediately stop happening and go back to where it was before.

But it is only a counterfactual relative to my theory. In the real world, there may indeed be times when the water level is higher on the Canadian side, due to wind or waves or rainfall, or something else that my simple theory assumes away. But my counterfactual conditional statements assume away wind and waves and rainfall too. It is my theory, and not the real world, that says it is counterfactual.

And yet, at the same time, that counterfactual conditional thought-experiment is an integral part of my theory. It explains why my theory is true. It explains why the water cannot be higher on the Canadian side, by supposing it were higher. Which is a bit of a paradox.

I stole this example from Michael Parkin, who used it as an example of a stability experiment ages ago. I've always liked it. And anybody reading this will also probably see what I am doing as checking to see if an equilibrium (same water level both sides of the border) is stable (does it return towards equilibrium if it's initially away from equilibrium).

Most people think of "equilibrium" as meaning "at rest", or "constant", or "balance of opposing forces", or something like that. But economists nowadays don't think of "equilibrium" like that. We don't even (necessarily) mean "quantity supplied equals quantity demanded". We don't even (necessarily) mean "an attractor towards which things will move regardless of initial conditions". There can be moving, non-market-clearing, history dependent, equilibria. All we really mean is: "that which the theory says will happen". The word "equilibrium" is almost redundant. But it's not totally redundant.

It's a useful word to use if we are checking to see whether something is or is not compatible with the theory: "No, that can't be an equilibrium, because there are $100 bills nobody is picking up" or "because there would be an excess demand for peanuts".

And it's a useful word if we are asking about stability. Because there we ask a counterfactual conditional question about what would happen if we were out of equilibrium.

But stability thought-experiments, like all counterfactual conditionals, are paradoxical. The theory itself denies the antecedent. So how can it be part of the theory? I think this is why some economists refuse to talk about stability. But understanding why Lake Ontario is the same level both sides of the border is as important as understanding that it is the same level both sides of the border. Stability is about what keeps it the same level both sides of the border. It's about what's stopping deviations from equilibrium from even starting, not what is happening when there is a deviation from equilibrium. Because there never are any deviations, according to the theory. It's about what is stopping incipient deviations from becoming actual deviations from equilibrium.

Comments

[Min]

"But it is only a counterfactual relative to my theory."

Is that really how to think about it? Should it even be a counterfactual, according to your theory? That is, it is within the scope of your theory, since your theory says what would happen if it were the case. In fact, your theory is better off **not** claiming that it will never occur, because that is not what your theory states. Then if instances are observed when it does occur, and what your theory says would happen actually happens, then your theory has passed a test. OTOH, if your theory claims that it would never happen, and it does, your theory has flunked a test.

"According to my theory, the level of water in Lake Ontario will always be exactly the same on both sides of the Canada/US border."

That is a crappy theory.

"Suppose it were higher on the Canadian side. Then pressure would be higher on the Canadian side, and water would flow from high to low pressure, which would equalise the levels."

That is a much better theory. The proposition that the water level will be the same on both sides of the lake does not follow from the second theory, and is not only unnecessary but detrimental. :)

[Peter N]

You can certainly define equilibrium to be "that which the theory says will happen", but if you do it using it to describe system behavior becomes a tautological. It has no real explanatory power. It also is a fruitful source of useless controversy when two people argue using different definitions. This, of course, already happens.

Economic terminology has developed a very unfortunate lack of precision and a tendency to use existing technical terms in a private sense. Equilibrium is the poster child for this.

Ideally equilibrium with respect to x would mean a place where the gradient of x is 0. Define your function x, and you're done.

[Min]

I hope my previous note was clear. I see that I altered what I meant by your theory while I was thinking. :(

All your theory needs is a description of what would happen if the system is not in equilibrium. It does not need a statement that the system is always in equilibrium. In fact, it is better not to make that claim.

Especially if you want to talk about stability. To talk about stability you want to talk about what happens when the system is perturbed.

[scepticus]

Thanks, I think this at least tried to begin to answer my question/objection to the prior article.

Of course: " Lake Ontario has the lowest mean surface elevation of the lakes at 243 feet (74 m)[2] above sea level; "

So the lake equilibrium is only a local one. The brief analysis in this article reasonably decides to ignore the sea level issue, because the level of the sea is not relevant to the lake surface in recent memory.

A local minima. A higher surface level equilibrium exists, if we temporarily raise sea level by 244 feet, then quickly lower it again.

[Jason]

I had been wondering what was really meant by equilibrium in your earlier post:

http://worthwhile.typepad.com/worthwhile_canadian_initi/2013/12/a-simple-story-about-reversibility-of-causation.html

If I read this correctly then all the theory is saying (in terms of the speed S and needle N analogy) when you write a N = b S is that there are economies in equilibrium with stable N and stable S, but their relationship to one another (i.e. the coefficients a and b) are dependent on history.

When you experience a shock, for example, the new "equilibrium" is c N(t+1) = d S(t+1) but getting there involves moving farther away from equality in the equation a N(t) = b S(t) (since e.g. S actually moves the "wrong" way).

http://informationtransfereconomics.blogspot.com/2013/12/a-delicate-balance-part-1.html

[marris]

> moving, non-market-clearing, history dependent, equilibria

Is this the same thing as a disequilibrium? Or does it satisfy some conditions that make it not-a-disequilibrium?

[Gene Callahan]

Min, Nick's theory is a model. A model has not "flunked" a test when it does not exactly match the real world, anymore than my subway map "flunks" a test because it doesn't show any rats or trash on the tracks. The model is good or bad in that it helps us to think about real phenomena, not because it is exactly like them -- in that case, we wouldn't need the model!

[Min]

Gene Callahan: "Min, Nick's theory is a model"

Scientific theories are testable. The proposition that the water level is always the same is unnecessary. Nick's wrestling with the question of counterfactuals that are not actual counterfactuals, but would be if the water lever were actually always the same illustrates one problem with including it in the theory.

Consider Newton's Laws of Motion. What if the First Law were "Every body perseveres in its state of rest, or of uniform motion in a right line"? Then we would not have heard of Newton, would we?

[Phil Koop]

Hmm. I don't think you are using "counterfactual" in quite Ram's sense; your use of an equilibrium example obscures how counterfactuals are used to A) distinguish causes from associations and B) allow for the possibility of multiple causes.

So, let's consider a different lake level theory, in which variations in level actually occur (so there is something to explain), and the theory is that these are caused by geographic variations in atmospheric pressure. You can use counterfactuals to state mathematically that pressure causes level differences rather than the other way around. You can also use them to test your theory in the face of multiple causes (the position of the moon raises level more at one end than the other) or confounding causes.

Larry Wasserman makes good use of counterfactuals to explain Simpson's paradox: http://normaldeviate.wordpress.com/2013/06/20/simpsons-paradox-explained/.

[Gene Callahan]

"Consider Newton's Laws of Motion."

Good point Min. They give us a model. But when we look around us, the model isn't true! Things moving slow down, not continue moving forever! (You might see N. Cartwright, How the Laws of Physics Lie, on this point in general.) Your idea of how models work and what they are for is "falsified" by actual scientific practice.

[Min]

OK, here is an analogy: Kepler's Laws of Planetary Motion and Newton's Law of Gravity.

We start with Kepler's Laws, which describe the heliocentric motion of the planets. You also have Ptolemaic epicycles, which produce more accurate ephemerides, but are much more complex and ad hoc. We take Kepler's Laws as our model, which fit our observations to within one degree of arc.

Later we get Newton's Law of Gravity, which not only explains the force of the sun upon a planet, but also the forces of other planets, which produce a slightly different orbit for the planet. Even though we cannot solve the n-body problem, we can produce more accurate ephemerides. **We no longer take Kepler's Laws as our model of planetary motion.**

However, Newton derived Kepler's Laws as a special case when there are only the sun and one planet. Thus, for convenience we may use Kepler's Laws as an approximation.

In the case of the water level of Lake Ontario, we may start with a model that say that the water level is the same everywhere on the lake. It fits our observations to within a few metres.

Next, we add the law of gravity, which tells us that when two points on the lake are at different levels, water will tend to flow from the higher point to the lower point. As Nick say, this explains why the water level is approximately the same across the lake. **But now we discard the first model, even as we discarded Kepler's Laws.** It is the failure to do so that produces the problems with so-called counterfactuals and paradoxes. We may still take our first model as a special case and as an approximation, with no logical problems.

Here it is as a special case.

If the lake is closed, with neither input nor output, and

if the amount of water in the lake remains the same, and

if the only force acting upon the water is gravity, and

if the water across the lake is all at the same level,

then the water will remain at that level across the lake.

That is rather different than saying that the water is always at the same level.

[Min]

Moi: "Consider Newton's Laws of Motion."

Gene Callahan: "Good point Min. They give us a model. But when we look around us, the model isn't true! Things moving slow down, not continue moving forever!"

That's why I misquoted Newton's First Law. What Newton's actual first law says is that if things do not remain at rest or in uniform motion, then they are being acted upon by a force or forces. (Now, OC, our model is Einstein's theory of General Relativity, with Newton's Laws as an approximation.) The First Law is not testable per se, it is a preamble to the other two. Scientific laws do not pretend to be absolute truth, to be the last word. They are always vulnerable to disproof.

[Majromax]

From a mathematical point of view, what Min's getting at is also the "Leading order solution" from perturbation theory.

In that mathematical technique, the first approximation is that "everything small is zero, and everything big is in balance." This can be done after-the fact (having a more complete system and systematically eliminating small terms) or, more interestingly, a priori -- assume some terms are small/negligible, build the resulting leading-order model, and the reintroduce the small terms.

The latter is more along the lines of economic theory. There, reintroducing the small terms ends up being the "counterfactual" of Nick's story: to leading order the water is level on Lake Ontario, and if that level is perturbed slightly then the fluctuations still remain small. If the analysis shows that introducing a small term drives the model away from the leading order solution, then that small term isn't really small and should have been included to begin with[1].

The leading order solution doesn't have to be "nice" -- it can be static or not, small or big. Hyperinflation, for example, is perfectly fine as a leading-order outcome of a model. But "stability" here is also the right word, because it reflects the (sometimes very mathematically precise) notion that the excluded factors remain less important than the included factors

[1] -- Sometimes this is even domain-dependent. For example, to leading order when throwing a ball we can ignore the decay of gravity with elevation, unless the initial velocity is comparable to escape velocity.

[Nick Rowe]

Peter N and marris: suppose I have a simple supply and demand model of the price of apples. There is an equilibrium P* where the two curves cross. Now suppose that the supply and demand curves are moving over time. That means P* is moving over time, so I have a moving equilibrium.

Thanks everyone for the other good comments. I don't have anything much useful to say in reply.

[marris]

Thanks for the response Nick.

But why would P* be "non-market-clearing"? It seems that we have clearing at the intersection price and quantity. Or is this a statement about clearing at a quantity that is too low (lower than what you'd get with better monetary policy)?

[Nick Rowe]

marris: P* (by definition) is the market-clearing equilibrium price.