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A possible "explanation" can be found in the physics lit on turbulent flows. What your describing is the same phenomenon as the criticality around the Reynold's number in fluid mechanics.

Unfortunately, even in the native subject, no one really understands turbulence... so you're out of luck. Still, your problem is much simpler than fluid flows, so if you build your own Reynold's-like number for this problem and then try to parameterize that flow with it... it might work.

Nick,

I haven't thought about your post or this paper enough to know how relevant it is, but just for the sake for its title ...

A Traffic Jam Theory of Recessions
Jennifer La'O
http://ideas.repec.org/p/red/sed013/412.html
or from here

I read the abstract... looks like exactly what Nick is proposing.

Some traffic modeling notes:
* Real drivers have a reaction time - so their buffer is certain to be shortened. If they are inflexible about buffer size, they will definitely decrease speed by *more* than the car infront did so that the buffer can be restored to original size.
* Many roads have multiple lanes. A slow down in one lane results in drivers switching lanes. The switch in lane can dramatically cut into the next car's buffer. This can create a jam in both lanes. Perhaps assets classes can be substituted for lanes. Hence slow downs in any one kind of value flow could impact all other value flows?

Amusing. Behavioral economics has no unified model (Noah Smith). Nick comes around, let me explain driving and macro with the same model. Not a coincidence: nick know about cars and economics. I assume he likes to drive too.

"Imagine a large number of cars forever circling around a very large roundabout."

With or without a theory of disutility?

There's a similar concept in electrical engineering when there's an impedance mismatch, say between a source and a load.
http://en.wikipedia.org/wiki/Standing_wave_ratio

I'm not sure that you can model a liquidity crisis just by looking at the quantity of money. There are two aspects to liquidity - the liquid assets you hold and your ability to borrow. I think disruption to the latter is much more relevant to a modern liquidity crisis.

Traffic modeling isn't a terrible approach.

Ultimately, people drive more slowly when there is a smaller gap in front of them to the next car. But even if they're the only car on the road, they drive at some fixed maximum speed.

Let's take the simplest, stupidest model and say that it varies linearly with traffic density: speed(density) = max_speed*(1-density/max_density), where max_density is bumper-to-bumper traffic.

Traffic flow is density*speed (cars/meter * meters/second = cars/second). So plug that in and you get:

d(density)/dt = density*speed = density*(1-density) (where I'm calling maximum speed and density "1" for simplicity). That means that traffic flow is maximized for some intermediate density, here half the maximum.

More interestingly, this differential equation causes even smooth initial conditions to sometimes evolve into shocks. Above the half-max density mark, a small perturbation evolves into an abrupt shock (stop-and-go traffic) because the cars in the more-dense part slow down, becoming even *more* dense. This gets all sorts of pretty pictures in the fundamental diagram of traffic flow.

Applying this to a simplified economy isn't too bad. Instead of physical space in front of a car, you have liquidity possessed by an agent. At infinite liquidity, an agent is still limited in consumption by other factors (especially permanent income), and at zero liquidity an agent is limited by liquidity such that it cannot spend at all until it receives income (the car ahead moves forward).

In this view, monetary policy acts to lengthen the road, while keeping the number of cars the same. Above the critical density (illiquidity), doing so increases traffic flow (economic activity). Below critical density (full liquidity), what happens depends on what agents actually do when unconstrained: some but not all future spending can be profitably brought forward, and "maximum speed" itself depends on permanent income which is then a matter of expectations.

@BS Economist:

> A possible "explanation" can be found in the physics lit on turbulent flows. What your describing is the same phenomenon as the criticality around the Reynold's number in fluid mechanics.

Er, I'd be extremely careful with that analogy. I've done fluid modeling, and it's nothing like economics. Most importantly, fluid is modeled as a continuum evolving in a low-dimensional (2D/3D) space. Economics is much less of a continuum, and it evolves in a much, much higher-dimensional space.

Turbulence is a great example of "chaos happens" when a locally well-defined system is allowed to freely-evolve, but drawing quantitative comparison is something I'd be exceptionally skeptical of.

I like the analogy but it sounds a bit wrong.

The traffic conductor is the central bank. When the cars are driving too fast around the roundabout he starts blowing his whistle, waving his arms, jumping into the traffic and shouting at the cars to stop. This causes multiple crashes and some of the smaller cars spin off into the ditch by the side of the road. Seeing the cars burning by the side of the road causes the drivers of the other small cars to be more fearful and cautious, so they drive slower. Usually when there are lots of small cars on the road they can, by the sheer weight of their numbers, cause the average driving speed to increase at a steady rate - but when there are less small cars on the road and more small cars burning in the ditch, the average speed is largely determined by the drivers of the big expensive cars instead, which take up more space and are much heavier and sturdier than the small cars. The drivers of these big expensive cars tend to change the speed at which they drive based on the speed they think the other big car drivers are going to drive at in the near future. Sometimes they think the other big car drivers are going to speed up, so they speed up, and sometimes they think they are going to slow down, so they slow down. Occasionally the remaining small car drivers can temporarily cause the average speed to increase by engaging in risky driving - leveraging their small size by erratically speeding up, beeping, swerving around etc. This sometimes excites the big car drivers for a while and they play along by also speeding up and driving in a risky way. But after a while people tend to realise that the small cars are just pretending to be big cars, and the game comes to an abrupt end as the big cars suddenly slow down, causing multiple crashes with more little cars spinning off into the ditch. Now and again the big car drivers decide to slow down so much that all the cars start to grind to a halt. This causes the traffic conductor to panic and he starts blowing his whistle, waving his arms, and shouting at the cars to speed up again. But whereas he can quite easily cause a crash when all the cars are driving fast, it's much harder for him to get them to speed up when they have stalled, and sometimes his antics seem to have no effect at all. In a last-ditch attempt to get the cars moving again the traffic conductor sometimes makes a solemn promise to all the drivers, shouted through his megaphone, that once they start moving again he'll let them get up to a faster speed than usual, before he causes them to crash again. He refers to this as a Nominal Car Speed Target (NCST) - The idea being that the drivers will anticipate a higher average speed in future, and so speed up today. However the big car drivers are all waiting for each other to speed up first, and there's not much impetus from the small car drivers as so many of them are burning in the ditch. So not much happens really. Sometimes you get little bouts of reckless driving which speed things up for a bit but that's about it. The drivers come to think of it as the new normal.

"And if observing that ripple causes people to want to increase the size of the buffer zone, and reduce velocity, the ripple will get bigger as it spreads."

Doesn't all this come down to a Bayesian inference problem involving variance estimates? I keep a buffer zone between my self and the driver ahead (a reserve of money) because I believe there's an element of unpredictability in their behaviour (in the relation between my current receipts and my desired spending). If the driver ahead temporarily slows down I may see this as just an instance of the unpredictability I'd already assumed, which doesn't call for any permanent change in my own behaviour. But I may also take the other driver's slowing as an indication their behaviour is in fact more erratic than I'd assumed to that point (the receipts/spending relation are more variable), in which case I'll probably seek to maintain a larger buffer zone (a larger precautionary reserve of money) in the future. The system equilibrium will change to the extent people are inclined to revise their beliefs regarding the overall variability they're facing in response to particular instances of displacement-from-mean.

Luis Enrique: very good find! A very interesting paper. More of a keynesian twist/interpretation than my more monetarist interpretation, but she is way ahead of me.

Nick: The size of the roundabout can expand if drivers get too close! The traffic authorities are happy with this because it improves safety. They just don't let it expand too much because then other cars come on and then it has to expand again. The trouble is that all the drivers take the ability for it to expand in a crisis for granted and they follow too closely.

Nick,

"The whole economy slows down as V falls everywhere, unless the central bank increases M in proportion, to keep MV constant."

The way central banks are set up (buy bonds low lower market interest rates, sell bonds to raise market interest rates) they trade liquidity for velocity.

In your car example imagine the central bank trying to introduce more cars onto the road that have smaller buffer zones - will this increase the velocity of the cars on the road, decrease it, or have no effect?

On an infinitely long road this will have no effect, but imagine instead a racetrack. Will more cars on the racetrack with smaller buffer zones increase or decrease the velocity of the cars going around?

Take a look at the latest post on monetary policy by Cochrane : http://johnhcochrane.blogspot.com/2014/03/the-sign-of-monetary-policy-part-ii.html

anonym: OK. I just looked at it. FTPL, plus perfectly flexible prices, plus perfectly credible inflation target, does indeed imply:

"In this model, to raise (expected) inflation, the Fed and Treasury agree to a higher inflation target, and then the Fed raises rates."

A higher inflation target causes higher nominal interest rates. Yep. But that does not mean that if inflation is below target, and you want to get it up to the target, an increase in nominal interest rates will cause inflation to rise.

Re Cochrane:

Seems trivial that if you announce a higher inflation target and then delay the whole tightening cycle - that rates will then end up at a corresponding higher level. But you can't raise rates if you're below rather than above your previous inflation target. That would just send an ambiguous signal.

Isn't this about the same thing again as that Kocherlakota debate?

JKH: Yes.

I may have misread Cochrane, but it looked like a there's supposed to be the same primary budget surplus/deficit but more (a lot more) borrowing. I took that to imply a lot of government spending. And that would lead to inflation and higher rates.

Otherwise he's just confusing discounting rate with inflation.

I also think the single rate (both declared and bond rate) is too simplified.

Nick, I mentioned in a previous post about queuing theory, packets and internet routers, which may give you the math you're looking for. One interesting comparison is the Alpha/Beta bank, as you allude to in your post ...
Alpha Bank = "core" ISPs (e.g. AT&T): core routers, core packet buffer sizes
Beta Bank = "metro" ISPs (your local ISP): "metro" routers, "metro" packet buffer sizes, packets

Alpha bank "money" = fixed core packet buffer sizes
Alpha bank "credit" = flexible core packet buffer sizes
Beta bank "money" = packets
Beta bank "credit" = flexible metro packet buffer sizes or packets

So who is the Alpha bank or the Beta bank?
In some ways the Alpha bank seems to be in charge because at first glance all local ISPs "have to" communicate to each other through the core ISP. The core ISP tries to control the volume of YouTube videos by using a packet policy. Metro ISPs can trade/borrow packet buffer space from each other in the core network or from the core under the paternal control of Mother Bell.
But, the Beta banks are also very powerful. As Comcast gets bigger, their packets are produced and consumed only on their own network, minimizing the Core ISP's role. Comcast also strikes their own interconnection agreements with other metro ISPs to "trade" (i.e. settle/net) packets, perhaps by putting up collateral. Packet intermediaries may also enable "trading" of packets by using their balance sheet to guarantee future packet delivery (another form of packet settlement). These actions dampen the "packet multiplier effect" of the core.
But, although Comcast can reduce its average reliance on the core, it still has a marginal exposure. When the core tightens packet policy, Comcast immediately passes on the cost to users, because occassionally Comcast still needs the marginal bandwidth at the core.
Core ISPs do not multiply the packets, but they are very good at showing who's in charge occasionally. And metro ISP's customers care about the margins. That "lost" packet may be a Youtube video of dancing cats, or it may be 911 call.

In this Wicksellian roundabout, you'd have cycles of varying sizes, many of which would be huge. Additionally, many of the paths would be indistinguishable. It would look locally like this:

But only globally like this:
http://www.doingitwrong.com/wrong/1435_ATT00048.jpg

That is to say you could look at the velocity flow through the entire roundabout as a random flow of money and focus on an infinitesimal bit of the roundabout. This would lead to a Navier-Stokes like equation with net convection related to V:
http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations

I like jt26's commment above about packet routing. This information theory model of money is based on Shannon's theory of communication; it may be the the zeroth order analysis of the jt26's packet theory of money:

http://informationtransfereconomics.blogspot.com/2014/03/how-money-transfers-information.html

It effectively leads to a model of V:

http://informationtransfereconomics.blogspot.com/2013/07/the-information-transfer-model-and.html

Nick,

Perhaps this is a way to model it. Let's say that Arnold buys haircuts from Bernard and that Bernard buys haircuts from Christine, and Christine buys haircuts from Arnold. All haircuts are paid for by cash after which the cash is deposited in the bank.

Arnold, Bernard, and Christine are all three trying to keep enough cash on hand to pay for the haircuts. All three have an ideal length of hair and deviations from this length are costly: they don't want their hair to be too long and they don't want their hair to be too short.

At the start of each period, Arnold, Bernard and Christine withdraw some money to pay for today's haircut and perhaps to pay for tomorrow's haircut. There is however a limit to how much can be withdrawn and this limit depends on how much Arnold, Bernard, and Christine leave on their bank account.

The ATM however can malfunction with a certain probability and this probability is drawn from a Beta-Binomial distribution but where alpha and beta are unknown, but for which Arnold, Bernard, and Christine have common priors.

Arnold, Bernard, and Christine try to keep enough cash on hand to smooth their consumption of haircuts and to minimize the cost over the benefit received of all haircuts. How much cash they will keep on hand depends on the probability of the ATM malfunctioning.

For some values of p Arnold, Bernard, and Christine will try to keep more cash on hand than is in the bank and thus cannot withdraw enough. In that circumstance Arnold, Bernard, and Christine will have to change how much they cut their hair this period so they will have enough money to cut their hair the next period.

Now if the ATM malfunctions for Arnold, then there will be a ripple effect on all three of them as the next period Arnold will want to withdraw more and thus leave less to withdraw for Bernard and Christine.

If the ATM malfunctions enough for one of them, then the other two will want to hold a larger buffer as well and this is your ripple effect I think.

In short I think you can model this as a Bayesian game. Two periods are probably enough, and I think it is even enough if just one actor faces a malfunctioning ATM. The ripple effect you can capture by making how much cash everyone can keep on hand depend on the not withdrawn total (perhaps with rationing if the demand exceeds what can be withdrawn) and consumption smoothing will do the remainder of the work.

I hope this makes some sense and captures what you are trying to model? I mean you just want to show that this ripple effect exists right?

This discussion would not be complete without a mention of Ripple, which attempts to take this 'buffer' and make it usable at a higher order level. The 'cash kept on hand' becomes usable *as money* by other aspects of the economy. While it's possible there are higher order 'buffers' (due to savings/investment patterns and risk preference)...Ripple can actually be used to use these macro patterns as a pathway for the economy to absorb shocks as well.

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