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.

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

ReplyDeleteHi Bill,

DeleteThe 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.

Hi, Jason,

DeleteThanks much. :)

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

You're welcome.

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

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