## 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?

4. This comment has been removed by the author.

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.