## Friday, June 27, 2014

### Towards Arrow-Debreu-McKenzie equilibrium, part N of N

After the results of this post on the macroeconomic partition function, I'm abandoning Arrow-Debreu-McKenzie equilibrium. Not because it is hard, but because it is likely meaningless for macroeconomics.

Let's look at what the ADM equilibrium says with regards to a partition function in thermodynamics. It effectively says there exists some set of occupation numbers so that the energy of the system is the total energy, or more generally, there exists a microstate consistent with an observed macrostate. The SMD theorem then tells us that there are only limited properties of that microstate that survive to the macrostate. In some sense, the SMD theorem should be intuitive: if you have a system with N degrees of freedom, but is described by n << N degrees of freedom at the macro scale, then the subset of properties of N degrees of freedom that follow as properties of the n degrees of freedom, is likely to be smaller.

The other consequence of the SMD theorem should also be intuitive. If your macro system appears to be described by n << N degrees of freedom, then it seems highly likely that among the total number of microstates, large subsets of the microstates are going to be described by a given macro state -- i.e. the equilibrium (the microstate satisfying macro constraints) is not going to be unique. For example, in an ideal gas, you can reverse the direction of the particle velocities and obtain another equilibrium (actually, all spatial, rotational and time-reversal symmetries lead you to other equilibria).

The reason economists think ADM is useful is probably due to their obsession with initial endowments. The ADM theorem goes part way to answering the question: Which set of prices let households and firms reach their final desired endowments given their initial endowments? The theorem says that there exists a set of prices that do that, and that is good to know! But these prices clear the markets in period two and they've finished their job [1]. This is a bit like worrying about how the energy gets redistributed to each atom of in gas when two gasses are mixed.

In an economy, the equilibria are more restricted than energies among the atoms in a gas and it's not trivial to show that they exist (or that they are Pareto efficient). I'm not knocking ADM. However, the existence seems meaningless for a real economy. As soon as a new product is invented, you're heading to another equilibrium. As soon as someone gets paid, you're heading for another equilibrium (if that someone would like to have more goods and services instead of holding cash). In reality, there may be a detailed balance that keeps the equilibria in an equivalence class described by e.g. a given NGDP growth rate. But that's the rub! Macroeconomics is the study of the behavior of those equivalence classes, not the instances of them!

That is to say macroeconomics is the study of the properties of ensemble averages (equivalence classes of microstates). Or another way, what we're interested in is:

$$\langle P \rangle = \langle a m^{a-1}\rangle = \frac{\sum_{i} a_{i} m^{a_{i}-1} e^{-a_{i} \log m}}{\sum_{i} e^{-a_{i} \log m}}$$

$$= \frac{\sum_{i} a_{i} m^{-1}}{\sum_{i} e^{-a_{i} \log m}} = \frac{1}{m} \frac{\sum_{i} a_{i}}{Z(\log m)}$$

not the particular configuration of the $i^{th}$ market.

This is not to say the individual configurations are meaningless in general. You might have very small number of markets. You might have a strongly interacting system. You might care about the effect of some policy or other in a particular market. But inasmuch as you are studying macroeconomics, the existence of an ADM equilibrium does not help you reach understanding.

[1] Footnote added 10/3/2014: David Glasner quotes Franklin Fisher: "To only look at situations where the Invisible Hand has finished its work cannot lead to a real understanding of how that work is accomplished.", which is similar to my sentiment.

1. Jason, I've only grasped a fraction of this series (and I'm glad you closed it out with the Nth out of N installments: just for completeness' sake), but it was still interesting to me. Has this series altered your opinion of DSGE type models for the purpose of analyzing macro economies at all? Do they depend on the ADM?

I've only heard of partition functions and statistical mechanics this year when I got curious about how Planck came up with his famous idea about energy being quantized (I never took a class in thermodynamics or quantum mechanics). Here's one of the series I looked in in a lazy attempt to educate myself:

On my todo list is to revisit all these posts of yours to understand them better.

Also, does your "exogenous" solution to the price level DE fit into this analysis at all? I'm still super curious how the solution can move from endogenous to exogenous and how we might be able to tell which is which while they are happening.

"Initial endowments" is a phrase I've first encountered here. How are they different than initial conditions?

1. I've only given up on trying to prove an ADM equilibrium exists in this model ... I can't say that it's really changed anything about how I look at dSGE (they only really depend on ADM equilibrium in the sense that you know a DSGE model has at least one equilibrium).

I'm still curious as to how the hyperinflation solution fit or how they change back and forth too. In a thermodynamics situation you'd actually be doing different things (adiabatic expansion, isothermal expansion) ... Natural processes generally don't switch from one to another.

However, I think the thermal bath analogy helps. If the economy is in a "money supply bath" it undergoes hyperinflation as opposed to the usual "adiabatic" or isentropic expansion for the price level.

And yes initial endowments are basically initial conditions ... I read that in a few places and thought that was the economic term.

2. O/T: I thought this was interesting for historical reasons (Krugman links to it today):

He talks about a market in information, but I don't think that's related to your approach, is it?

1. Yeah, Wren-Lewis has done a couple of interesting posts lately.

But no that use of information is different ... It more has to do with how can you tell a fluctuating sequence has changed its mean than how do bits flow around an economy.

3. O/T: I've been recommending your blog to a particular commenter (John D.) at pragcap for a while. He finally "took the bait" (maybe). Here's the thread if you're interested:

http://pragcap.com/a-cheat-sheet-for-understanding-the-different-schools-of-economics/comment-page-1#comment-179034

1. Also, both Scott and Dustin got back to you at themoneyillusion.

2. Thanks.

I don't have any major issues with what John is saying. In a sense he's coming at the problem from a pessimistic viewpoint about eventually getting a handle on economics where I'm a bit more optimistic.

Also -- it appears this blog is getting a citation in the next update to their paper on the information transfer model. I should really get to work putting something on the arXiv.

And I responded to Dustin and Scott. I would say that I agree that product quality may matter, but given the results that ignore product quality don't seem to be wildly off it seems that product quality doesn't matter very much.

3. re: citation: congratulations!

re: your response to Dustin and Scott. I liked that. It allowed me to finally understand the full significance of the "vinegar" question/quote in your original post... maybe I'm just slow, but I read it twice at the time and still didn't completely get it. :D

4. Mark Buchanan is more negative on Arrow Debreu than I am:

http://physicsoffinance.blogspot.com/2014/08/arrow-debreu-derangement-syndrome.html

For him, it's the unrealistic assumptions. For me, it's more the fact that it would be useless from a practical point of view even if the assumptions were realistic.

Comments are welcome. Please see the Moderation and comment policy.

Also, try to avoid the use of dollar signs as they interfere with my setup of mathjax. I left it set up that way because I think this is funny for an economics blog. You can use € or £ instead.