## Thursday, May 22, 2014

### Causality in the information transfer framework

 It doesn't matter where or how the central bank injects liquidity. Image from wikimedia commons.

A quick note on causality because it's come up in a variety of forms recently (I'll use the interest rate model for concreteness):

1. Can you be sure that the relationship between interest rates and the monetary base will hold if the central bank does X?
2. Does lowering interest rates cause the monetary base to rise or vice versa?
3. You say that interest rates can be lowered by the central bank printing currency, but that's not how open market operations work in real life, so the central bank can't lower interest rates by printing currency.
The information transfer model is based on information theory, but also behaves like a kind of generalized thermodynamics. Let's say we have two variables: money m and interest rate r. The idea is that if you change one variable (r1 → r2), then the other variables respond (m1 → m2) not because they are "caused" to do so by "forces", but rather because it is overwhelmingly statistically probable that if r2 becomes true of the macrostate, then the microstates will be in a macrostate described by m2. This is the basic idea behind entropic forces. Money doesn't make its way from bank vaults into a person's hand because of changes in incentives or utility because you changed r and m (well, at least in this model [1]). It makes its way there because the state with the money in a person's hand is far more likely than the state with the money in the vault given the macroeconomic observation of the system in the state (r2, m2). It is not important how the money got there from a macroeconomic perspective.

Imagine bacon cooking in the kitchen. In the diffusion process, the smell fills the house. I do not need to know the actual trajectories of the molecules, or even how much kinteic energy they leave the bacon with. It doesn't matter if the bacon is cooked in the kitchen or in one of the bedrooms -- the smell will fill the whole house.

The thing that is important is that the economy is a large system. In that case, because of the law of large numbers, I can know that the result will converge to the mean and fluctuations around it will be small. Flipping a fair coin 5 times has a lot of uncertainty in the final outcome (4 heads, 1 tail? 2 heads, 3 tails? 5 tails?). Flipping a fair coin 5 million times does not (2.5 million ± 2200 heads).

The law of large numbers means we know that the final state (r2, m2) will be realized with high probability. But it actually doesn't have to be! It could end up as (r2, m1) or (r2, m3). That's why saying r1 → r2 causes m1 → m2 is problematic. The final state m2 is not a foregone conclusion -- there will be some statistical fluctuation around it.

The law of large numbers also lets us know that the reverse causality works. If m1 → m2 then we will have r1 → r2. That's because if the most probable macrostate (r2, m2) is consistent with r2 if we change r, then it is also the most probable macrostate with m2 if we change m. If m1 → m2 but r1 → r3, then (r3, m2) would be the most probable macrostate consistent with m2. This state would have to be more probable than (r2, m2) since it is the most probable macrostate consistent with m2. The only way this is logically consistent with is if r2 = r3. See this link for more on this subject.

1. If the model is correct, then yes.
2. Yes. Both. Or neither. Doesn't matter.
3. If the model is correct, the particulars of the process do not matter.
[1] This represents a huge break with traditional economics. It also makes traditional economics seem kind of silly if you try to use the language of utility in thermodynamics ... The pressure of an ideal gas falls because an atom has diminishing marginal utility of extra volume. Ha! This break is also interesting because it means that all the properties of supply and demand are emergent. Atoms don't have feelings about pressures and volumes and the idea gas law isn't even true for an individual atom. Bringing this insight back to economics: diminishing marginal utility is not a property of an individual economic agent, but rather an ensemble of agents. Supply and demand is not about incentives, but rather a property of ensembles of people performing market transactions.

1. "Money doesn't make its way from bank vaults into a person's hand because of changes in incentives or utility because you changed r and m (well, at least in this model [1]). It makes its way there because the state with the money in a person's hand is far more likely than the state with the money in the vault given the macroeconomic observation of the system in the state (r2, m2). It is not important how the money got there from a macroeconomic perspective.""

Ok but this macro observation isnt in line with the reason you are applying information theory to markets. If I understand corectly it is "to come up with universal economic relationships, not accidents of history". The observations are not universal becuase you are observing a system which operates on limited principles on how monetary policy is operated. As soon as monetary policy is operated in a different manner you should see that monetary policy can be effective even when MB/ngdp is high and therefore high MB/ngdp is a symptom of deterioration of monetary policy efficiency relative to the economy it is interacting with.

"Imagine bacon cooking in the kitchen. In the diffusion process, the smell fills the house. I do not need to know the actual trajectories of the molecules, or even how much kinteic energy they leave the bacon with. It doesn't matter if the bacon is cooked in the kitchen or in one of the bedrooms -- the smell will fill the whole house."

But the smell of bacon might not transmit if you have a good extractor in your kitchen. If you dont put on the extractor then you will get a stronger smell in your house than if you did. If you cooked bacon outdoors and the wind was travelling in one direction the neighbour downwind could get the smell of bacon on his clothes on the clothesline while your nextdooor neighbour upwind doesnt get any smell of bacon. If the wind changed direction you would get the opposite effect.

"The law of large numbers means we know that the final state (r2, m2) will be realized with high probability. But it actually doesn't have to be! It could end up as (r2, m1) or (r2, m3). That's why saying r1 → r2 causes m1 → m2 is problematic. The final state m2 is not a foregone conclusion -- there will be some statistical fluctuation around it."

I hope I understood previous paragraph correctly. We dont know where the economy will be in the long run. We have no idea if the final state (whatever it may be) will be even reached at all. We know the Roman empire and all the monetary actions it undertook had limitations and led to its collapse. We can mostly count on the short run.

"3. If the model is correct, the particulars of the process do not matter."

I will make a process to show you that it does matter. If 1mil of base money is expanded through asset purchase of wealthy individual through investment bank or 1mil base is directly credited to accounts of 1000 unemployed people you will get completely different effects on ngdp for a given increase in MB. Wealthy people spending is lower proportion of their wealth compared to poorer people. Thyerefore whatever increase in wealth comes from this OMO to the wealthy will result in less spending compared to poorer people.

Even if we assume the wealthy and poor have the same ratio of spending to wealth you still get different effects because asset purchases will increase asset prices to create a different change in wealth to directly crediting bank accounts.

1. The observations are not universal because you are observing a system which operates on limited principles on how monetary policy is operated.

Ah, but what I am saying is that information theory governs the behavior of the macroeconomy, not institutions. If monetary policy were conducted in a different way that and the theory failed to fit it in, then the theory would be incorrect.

But the smell of bacon might not transmit if you have a good extractor in your kitchen ...

Sure. If you change the conditions, things will be different. But you still don't need to know about the molecules carrying the scent of bacon and the diffusion process still operates.

We dont know where the economy will be in the long run.

The "long run" in the model here is on the order of months. The interest rate market appears to reach its new information equilibrium faster than the quarterly reporting of NGDP. In the example, the transition r1 → r2 and m1 → m2 happens almost instantly.

I will make a process to show you that it does matter.

The markets in question are set up by humans, so they could easily break them down and replace them with different markets. I go into that a little at this link in terms of the labor market. The key question to ask is: what is sending information and what is capturing it? When considering the price level and interest rates, the model here assumes that aggregate demand is sending the information and the monetary base is capturing it (as a stand in for the aggregate supply, see here).

I hope I've explained the model and the equations enough so that if you feel something is missing, you can take these tools and show it.

One side issue is that macro data is uninformative. I am not sure your experiment would show up in the empirical data. Heck, the data barely chooses the currency component of the base over the full monetary base.

Also, check out Scott Sumner on the question of who gets the money first:

http://www.themoneyillusion.com/?p=20216

2. I like your approach to economic modeling. I think maybe if you added a couple things it could be really good.

1. Thanks!

As I noted in the reply above, I don't own the equations -- if you feel something is missing you can take these tools and show it.

3. That points to you using patial differential equations. I have seen ODEs in econ.

What would the typical space x,y,z components be for the use of PDE in econ?

4. Maybe I should have called x,y,z space components independent variables.

5. If the diffusion model were correct, then trickle down economics would work, right? That idea, if not the name, has been around since the 19th century, and it has never, to my knowledge, been shown to work. Instead, there is a money sink, such that money naturally flows upwards in society. This is indicated in at least two ways: 1) workers are never paid enough to consume all of what they produce; 2) higher classes have a lower propensity to consume (spend money) than lower classes. Now, if investment were adequate there could be a steady flow. However, in history investment has often been inadequate. There may well be entropic reasons for that. I. e., there is no perpetual motion money machine, either. Anyway, it does seem that it is easier to lend money to people in the middle classes than to invest it. Eventually the lending machine seizes up, as well. That is what happened leading up to the Great Depression, and what happened with our financial crisis, as well. The debt machine also seems to have seized up in the ancient Middle East, which is why those societies had jubilees (mass debt forgiveness) once in a lifetime. Japan also had mass debt forgiveness in more recent centuries, but I do not know the details.

Anyway, if the purpose of the CB injecting liquidity into the economy is to prime the money pump, injecting it near the money sink is suboptimal, n'est-ce pas?

1. Not necessarily:

http://arxiv.org/abs/cond-mat/0002374

Random markets can lead to Pareto ('heavy tail') distributions of wealth.