Friday, September 18, 2015

Two deep looks into microfoundations

This post represents some theoretical musings on my part ... so it's probably a bit "out there" from a typical social science or economics background. I make no claim to originality either.

Here are two recent looks into the idea of microfoundations (in economics and sociology). 
The Neoclassical Synthesis and the Mind-Body Problem (David Glasner) 
Microfoundations and mechanisms (Daniel Little)
Both writers delve into what microfoundations mean. 

Little's main point is about the relationship between microfoundations and mechanisms and his "preliminary answer" is that microfoundations are mechanisms acting on micro states. Mathematically, this is an expansion of the macro operator expectation in the macro state |Ω⟩ in a micro basis |i⟩

⟨Ω|Ô|Ω⟩ =  Σi ⟨Ω|Ô|i⟩⟨i|Ω⟩

Little then says there are some issues with this, that maybe mechanisms don't imply a level. And in the formulation above, they don't. The basis |i⟩ doesn't have to be micro states. It could be any sort of intermediate state (agents, firms, institutions)

⟨Ω|Ô|Ω⟩ =  Σx ⟨Ω|Ô|x⟩⟨x|Ω⟩

The key requirement for this to be generally true is that |x⟩ is a complete basis. To me, individual agents seems like a complete basis relative to e.g. firms because while firms may buy and sell from each other, firms also produce goods that individual agents consume. That is to say the entirety of economic activity can be stated as a bunch of data about individuals but not necessarily a bunch of data about firms [2]. 

Glasner takes issue with the representative agent model that simply asserts the equivalence of macro observables and the outcomes of micro mechanisms acting on micro degrees of freedom. If you do this, he says, you leave out the fact that macro observables might emerge from the interactions between micro degrees of freedom. Mathematically, Glasner's point is that the representative agent implies the ensemble average of an operator is no different from the single agent expected value, that

⟨Ω|Ô|Ω⟩ ~ ⟨1|Ô|1⟩

where ⟨1|Ô|1⟩ = ⟨2|Ô|2⟩ = ... which obviates the difference between the n-agent macro state |Ω⟩ and the single agent micro states |1⟩, |2⟩, ... and e.g. "unintended consequences" from the interaction between |1⟩ and |2⟩ are left out. Basically, the representative agent approach assumes the macro observable determined by the operator Ô is diagonal. Actually, we can derive the representative agent model from my formulation of Little's definition of microfoundations:

⟨Ω|Ô|Ω⟩ =  Σi⟨Ω|Ô|i⟩⟨i|Ω⟩

⟨Ω|Ô|Ω⟩ =  Σi ⟨i|Ω⟩⟨Ω|Ô|i⟩

[correction] The representative agent model is that the macro states are exactly the same as (identified with [1]) some micro state (the representative agent) so we can change out Ω for some j (the representative agent)

⟨Ω|Ô|Ω⟩ =  Σi ⟨i|j⟩⟨j|Ô|i⟩

⟨Ω|Ô|Ω⟩ =  Σi  δij ⟨j|Ô|i⟩

⟨Ω|Ô|Ω⟩ =  ⟨j|Ô|j⟩

[end correction]

In contrast, the information equilibrium approach can be seen as a different application of my formulation of Little's microfoundations

⟨Ω|Ô|Ω⟩ =  Σi ⟨i|Ω⟩⟨Ω|Ô|i⟩

⟨Ω|Ô|Ω⟩ =  tr |Ω⟩⟨Ω|Ô = tr Ô|Ω⟩⟨Ω| 

⟨Ω|Ô|Ω⟩ =  tr Ô exp(- Â log m)

where I identified the operator |Ω⟩⟨Ω| ≡ exp(-  log m) where Â picks off the information transfer index of a micro market. This is the partition function approach. And we are saying the macro state Ω is directly related to a maximum entropy distribution (partition function):

Z = tr |Ω⟩⟨Ω|

The partition function approach is basically a weighted sum over (Walrasian) micro markets that produces macro observables ... borrowing Glasner's words, we are: "reconciling the macroeconomic analysis derived from Keynes via Hicks and others with the neoclassical microeconomic analysis of general equilibrium derived from Walras."

What about the emergent representative agent? Well, if Â is diagonal in its representation in terms of microstates, then

⟨Ω|Ô|Ω⟩ =  tr Ô exp(- Â log m)

⟨Ω|Ô|Ω⟩ =  Σi ⟨i|Ô exp(- Â log m)|i⟩

⟨Ω|Ô|Ω⟩ =  Σi ⟨i|Ô|i⟩ exp(- ai log m)

If we take m >> 1 (a large economy) the leading term is min {ai} that we'll call a₀ so that

⟨Ω|Ô|Ω⟩ ≈ ⟨0|Ô|0⟩ exp(- a₀ log m)

⟨Ω|Ô|Ω⟩ ~ ⟨0|Ô|0⟩

This has the analogy of the ground state (at zero temperature) in physical system [3].

...

Footnotes:

[1] For example, as Glasner says, "the business cycle is not the product of the interaction of individual agents, but is simply the optimal plan of a representative agent"

[2] As an aside, the idea of having consumers and firms seems like a strange basis that isn't necessarily complete (and actually includes two different levels as firms are made of consumers).

[3] For bosons, this would be a Bose-Einstein condensate. So the representative agent is like a bunch of molecules in the same state behaving as a single entity.

5 comments:

  1. Sigh, yet another notation for me to learn to talk about inner product spaces...I want to ask intelligent questions like what allows you to commute the stuff in the sum for the derivation of the rep agent...but I don't know what the stuffs are called...and I don't understand what my multiplication is without the vertical bars...

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    1. The basics are here:

      https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation

      Something like ⟨i|Ω⟩ is an ordinary (potentially complex) number, so commutes with matrices (operators). But there is an error, which I am going to fix.

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  2. Although I am math challenged compared to the usual crowd here, this does not sound promising:

    "So the representative agent is like a bunch of molecules in the same state behaving as a single entity.

    Does not sound to me like a great start to microfounding an economy. Perhaps a study of "entangled consumers" might be more apropos?

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    Replies
    1. I'm actually not sure that you could put agents all in a single state -- the physical interpretation is right for bosons (as I mention), but is problematic for agents. I think a better interpretation is the single agent, like the gas made of a single giant molecule:

      http://informationtransfereconomics.blogspot.com/2015/08/giant-molecules-and-representative.html

      Delete