Friday, June 26, 2015

Ramsey model and the unstable equilibrium of a pendulum

Also in Romer's Advanced Macroeconomics is the Ramsey-Cass-Koopmans model (here is Wikipedia's version). It has some of the same flavor as the Solow model, but it has a rather silly (from this physicist's perspective) equilibrium growth path:

We are expected to believe that an economy not only will start out (luckily) somewhere on the path from F to E in the diagram above (you can extend F back towards the origin), but will in fact stay on that (lucky) path until reaching E at which point it will stay there (with a bit of luck).

This is a bit like a believing a damped pendulum, given just the right swing from the just the right height, will go all the way around and just come to a stop so that it is "bob-up" like in this picture from Wikipedia:

My first reaction to seeing that growth phase diagram was to laugh out loud. Economists couldn't be serious ... could they? Now it isn't strictly impossible, but the likelihood is so small that the tiniest air current will cause it to fall back to one of the more normal equilibria:

But the phase diagram from the Ramsey-Cass-Koopmans model is basically equivalent to the phase diagram of a damped pendulum near one of its unstable equilibria:

So basically, according to the Ramsey-Cass-Koopmans model, all economies head towards being all capital or all consumption. Who thought this was a good model?

Now there is some jiggery-pokery in the model -- economists include "transversality conditions" that effectively eliminate all other possible paths. If I eliminate all other paths besides the ones that lead to the unstable equilibria in the pendulum case, I get magic pendulum that stands on its head too!

1. Two asides:

Years ago a book I read mentioned that back in the early 1900s there was a metaphor of the economy as a pendulum at which small boys were throwing rocks.

Parlor trick: Hold a two to three foot stick in front of you and get it swinging as a pendulum. Tell people that you can stop it swinging without touching it. Or ask if anybody knows how to do that. Aside from physicists and maybe gymnasts, dancers, and martial artists, few people will know. The trick is to wait until the tip of the stick begins its fall and then to side step so that it falls straight down. Mohammed moves to the mountain.

2. Suppose you reversed the direction of time, in a physical system. How would that affect stability?

What's the difference between pendulums and people? People have plans and expectations about the future, that affect their current actions.

1. Nick, I might have known you'd be all over a pendulum post! Ha! (In fact I almost left a comment here earlier about you and your past pendulum analogies).

Although I confess I'm not drawing a connection between your 1st and 2nd paragraphs here. Is there one? Are you implying that pendulums don't care which way time runs but people do? That seems true for ideal (no air resistance) pendulums, but aren't there other physical systems for which that's not true (like the universe as a whole)... at least if we think of reversing time like a movie reel ... but then I've been reading this book and the author get's into this very topic, and in fact has co-authored a paper with his grad student in which they surmise a mirror image universe on the other side of the big bang running backwards in time (but still from low entropy (big bang) to high (t=+inf or t=-inf)). I'm only on page 15, so I can't give you any more details. Lol.

2. Hi Nick,

It depends on the system under consideration ... In general PT is a good symmetry (parity and time reversal) for electrically neutral Hamiltonian systems. In this case, however, we have a non-conservative damping force that would cause the system to acquire ever more energy -- the damping force comes from thermodynamics and is dependent on the arrow of time.

But the fact that individual people's expectations matter in a macroeconomic system is still a model assumption -- the SMD theorem says that all that matters is homogeneity of degree zero, Walras' law and zero prices = infinite demand. Nothing in there about expectations. A representative agent gets around this, but then representative agents can't trade assets with themselves to adjust to an expected future ...

3. Jason. Let's start with a much simpler model. Robinson Crusoe is alone on his island with a fixed stock of rum K, knows he will live for T years, has a per-period utility function U(C) where C is consumption of rum, and discounts future utility at rate rho. Solve for his consumption of rum today.

Your math is much better than mine (at least, I hope to God it is.) How would you solve it?

I would solve it backwards. He plans to have K(T)=0 (he drinks the last drop just before he dies).

Now suppose he suddenly learns that he will live one extra period. How does that change dT affect C(0)? He will reduce C(0). The future (strictly, his current beliefs about the future) affects the present.

Pendulums aren't like that.

I know from long experience that you mathy physics guys like to jump right in at the deep end of economics. You see the mathy economics model and think: "Hey, I understand this!". And in one sense you do, but in another equally important sense you don't. This happens all the time. And it is especially likely to cause problems when it comes to stability. (Like the sign wars episode.)

You will hate this advice, and reject it, but I'm going to give it anyway. Start with an intro text.

4. Now complicate the model.

Assume the stock of rum grows at rate r, where r is a decreasing function of K, minus what he consumes.

Now take the limit of that model as T approaches infinity.

5. Really short version: pendulums don't say to themselves: "Hey, at time t=T in the future I'm gonna want to be at point K(T)=0, so I had better move by an amount C(0) today, if I'm going to get to that point!"

(Of course, whether an economy that contains multiple people, even if they are identical, behaves exactly like Robinson Crusoe, is an important question, that is swept under the carpet by R-C-K sorts of models. Can they coordinate their expectations about what everyone else will do? But we can't even hope to think about that question until we learn the difference between Robinson Crusoe and a pendulum.)

6. And you are really BS'ing when you talk about the SMD theorem and Walras Law, which is talking about a totally different stability question than the one you are talking about here. That's about whether prices approach the *market clearing* vector if we start out with excess demand or supply. This R-C-K model just assumes prices are always at market-clearing, and the question is whether the path of that market-clearing equilibrium approaches some stationary state over time.

7. My apologies Jason. I should have been more polite. I think you are a smart guy, and it's good you are exploring this stuff, but like many physics guys, you have to watch out for hubris, when outside your field.

8. Hi Nick,

I expect the smackdowns every once in awhile, and I don't think anything you've said has been rude. That's a bit of education that one usually has to pay for.

However, my point was that the RCK saddle path seems unstable to shocks in the real world. The intuition I was borrowing was that although the pendulum the equations say the inverted solution is an equilibrium, from our everyday experience we know that equilibrium is hard to achieve ... not just because the equilibrium itself is unstable, but that paths in phase space infinitely close to the saddle path head off to distant lands.

If Robinson Crusoe is hit by a positive shock to his health at t = T, he dies of dehydration instead of cancer. If he's hit by a negative shock, he wishes he'd have drank those few drops of water at t = T - 1. This naturally leads to two solutions that we know really well from human experience: carpe diem (all consumption) and Ebenezer Scrooge (all savings).

It's true that the representative agent should expect shocks to average to zero (rational expectations), but for any given agent the chance that the shocks average to exactly zero is vanishingly small. And because those shocks end up being slightly positive or slightly negative, you should fall off the saddle path.

The RCK model makes sense for an ensemble of agents (hit by individual shocks), though! Some end up investing nearly all of their wealth, some end up consuming it.

9. Nick, you write:

"pendulums don't say to themselves: "Hey, at time t=T in the future I'm gonna want to be at point K(T)=0, so I had better move by an amount C(0) today, if I'm going to get to that point!""

This pendulum does. But then it's had a feedback control path wrapped around it, so it's phase diagram will look different (i.e. it's a different physical system).

Sorry, I couldn't resist!

10. Nick,

You also said:

"Really short version: pendulums don't say to themselves: 'Hey, at time t=T in the future I'm gonna want to be at point K(T)=0, so I had better move by an amount C(0) today, if I'm going to get to that point!'"

Yes, but how would physics look if they did?

I can't quite see the difference between pendulums that want to obey the laws of physics (model-consistent, rational expectations), pendulums that have to obey the laws of physics (classical mechanics) and pendulums that on average happen to follow the laws of physics and given an enormous number of bits on their causal horizon do so with vanishingly small variance ~ ℓp (entropic gravity).

11. (That is to say the equations are the same in each case.)

12. Jason: your math is better than mine. Take the original RC problem. Assume U(t)=log(C(t)). That gives you:

C(t)=C(t+1)(1+rho)

You know that K((t+1)=K(t)-C(t)

You know that K(0) is pinned down by history at K*.

That gives you an infinite number of paths for K(t)

But if you pin down K(T)=0 (because RC doesn't want to die with rum undrunk), you get a unique path for K(t), as a function of K*, rho, and T

Now change it to K(t+1) = (1+r(t))K(t) - C(t), where r(t) is a decreasing function of K(t), like r(t) = K(t)^0.5. Then take the limit as T goes to infinity. That's basically RCK. There's (usually) a unique path towards which it converges, regardless of K(0).

Introducing uncertainty doesn't change it much.

But introducing multiple RC's who trade with each other, maybe using money, where each one's choice becomes a function of what he expects the other to choose, and it's not at all obvious whether it's stable.

13. Tom: I've been taught more physics than you and Jason combined have been taught economics. By the same good teachers who taught Stephen Hawkin. And they gave me very good grades. But I understand little physics, beyond the basics.

14. should be r(t) = K(t)^-0.5

15. Nick, my smartass comment was 100% a product of the image of that amazing inverted tripple pendulum popping into my head upon reading your comment. If there's a point in my comment about economics I assure you I don't know what it is. (C:

16. @Nick

Your Crusoe example contains a logical contradiction. Crusoe "knows he will live for T years" but then "he suddenly learns that he will live one extra period." If he does learn that, then he did not know when he would die.

17. Hi Nick,

I think LAL below hits on something that's making us talk past each other.

I don't disagree that economics of real humans makes us choose the saddle path -- my point is that when it does, it becomes most of the model. The whole phase diagram becomes mostly a footnote to the transversality conditions.

My response kind of got away from me and became a post of its own:

http://informationtransfereconomics.blogspot.com/2015/06/the-importance-of-transversality.html

Tom -- I used your pendulum link as an example in that post.

3. I think you should pay more attention to the transversality conditions...there is a lot more economic content to them than you are realizing...https://en.wikipedia.org/wiki/Inada_conditions

1. in case you miss it, they are spookily close to guaranteeing cobb-douglas production functions...so at the end of the day transversality conditions vs information transfer...who cares....

2. when i see this model i take it as evidence that fairly relaxed assumptions about the aggregate production function can guarantee a stable growth path even in a highly unstable dynamical system...now is a rational agent/expectations framework the right assumption for learning the most we can from the model? maybe not...can you see an interesting way to plug in the information model??? then science might happen...for once...

3. I'm having something close to a thought ... Could the Inada conditions be iff the information transfer model like some ergodicity theorem?

4. Hi LAL,

I think that may be the key point here (and why there appears to be confusion) -- I was actually saying all of the economics of the RCK model was in the conditions ... I ended up writing a whole post about it:

http://informationtransfereconomics.blogspot.com/2015/06/the-importance-of-transversality.html

... it goes part way towards exploring your thought as well ...

4. I think there are lots of things to not like about this model, but I'm not sure this is one of them.

F is the point at which current consumption is at the level which maximises the discounted utility of expected current and future consumption.

At any point higher than F, if household consumption changes at the rate implied by period to period maximisation, then eventually the economy will run out of capital. So expected utility from these paths must take into account that consumption at some point in the future will have to be less than is implied by the Euler condition, simply due to the inability to produce stuff. Taking this into account, the discounted expected utility from any point above F is actually less than that for F.

Shocks may well cause F to jump around. But there will always be a point which is the highest point up the k(0) line, compatible with the expectation that the capital stock will always be sufficient to produce the planned level of consumption.

1. Hi Nick,

I completely agree -- my point was that the conditions that keep the model on the saddle path (as you describe) end up becoming the whole model and the RCK differential equations become superfluous ...

I ended up writing a whole post about it:

http://informationtransfereconomics.blogspot.com/2015/06/the-importance-of-transversality.html

5. I am Responding to an IdiotJanuary 2, 2016 at 8:41 PM

Good God this crap is still going?

1. Without more information about the idiocy in question it is difficult to know who the idiot is.

There is a follow up that gets at the issue more clearly which might help.

http://informationtransfereconomics.blogspot.com/2015/06/the-importance-of-transversality.html