Wednesday, August 12, 2015

A trivial maximum entropy model and the quantity theory

This is a trivial maximum entropy model, but I still think the result is interesting.

Let's take an economy with d = 100 dimensions subjected to a budget constraint Σ m = M(t) where t is time, m is the (randomly chosen) allocation of the total amount of money M to each dimension (these could be "markets", "firms" or even "agents"). This is effectively this picture except d = 3 here (the Ci are the dimensions):

If we take the derivative of the ensemble average d/dt log 〈m〉(assuming a uniform distribution), we get something (blue paths) that roughly follows d/dt log NGDP (yellow path):

The paths obviously converge to d/dt log M (black path) ... it doesn't even depend on d because 〈m〉= α M with constant α even if we don't have d >> 1 since

d/dt log α M = d/dt (log M + log α) = d/dt log M

If d >> 1 then we have α ~ d/(d + 1) → 1, but this isn't terribly important. What this does show is that

d/dt log M ~ d/dt log NGDP

k M ~ NGDP

and we have a simplistic quantity theory of money based on randomly allocating money to different markets.


  1. What happens if the allocation of money to markets (or agents) follows a Pareto distribution? Thanks. :)

    1. Hi Bill,

      The Pareto distribution is the maximum entropy distribution for when E(log x) is fixed and also x > x_min so in a sense you'd specify log x and x_min instead of x_max and x_min for a uniform distribution.

      However with a Pareto distribution it would be harder to maintain a budget constraint.

    2. Hi, Jason,

      Thanks much. :)

      Did you feel the quake? It was a couple of miles from me.

    3. You're welcome.

      And I'm up in Seattle, so it was a bit farther away. USGS was saying it was a 4.0?

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