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.