A thousand random markets

In this post [1], I set up a framework with a large number of markets $p_{i}: n_{i} \rightarrow s_{i}$ mediated by money so that we obtain (for the individual markets) the differential equations

$$
\frac{dn_{i}}{dm} = a_{i} \frac{n_{i}}{m}
$$

The solution to these differential equations are

$$
\frac{n_{i}}{n_{i}^{ref}} = \left( \frac{m}{m^{ref}} \right)^{a_{i}} + c_{i}
$$

In  [1], I made the approximation that the $a_{i}$ could be replaced by their average $\bar{a}$ and therefore the sum of the markets obeyed the approximate differential equation

$$
\frac{dN}{dm} \simeq \bar{a} \frac{N}{m}
$$

where

$$
N = \sum_{i} n_{i}
$$

Now I ask the question: how well does this work? First, here is the sum of 10 random markets (with 10,000 random evaluations, blue points) where we take $n_{i} \sim m^{a_{i}}$ with a uniformly distributed $a_{i} \in [0,1]$. The approximate aggregate differential equation has solution $N \sim m^{\bar{a}}$ (shown in red):

A region that represents 10% variation is shown in gray. We can see that this solution works pretty well, but I found that if we summed up 1000 random markets a systematic deviation for higher values $a_{i}$ and higher/lower values of $m$ begin to show up. Here are the results for $a_{i} \in [0, 0.5]$, $a_{i} \in [0,1]$, $a_{i} \in [0,2]$, and $a_{i} \in [0,4]$ which have $\bar{a} = $ 0.25, 0.5, 1.0 and 2.0, respectively.

A systematic deviation appears for small/large values of $m$ that is more apparent for larger values of $a_{i}$.  The source of this is not mysterious: for larger values of $m$, $m^{\text{max } a_{i}}$ tends to dominate while for smaller values of $m$, $m^{\text{min } a_{i}}$ tends to dominate. Still, for 10% shifts from the reference point $(m^{ref}, N^{ref})$, it remains a remarkably good approximation.

The markets with high values of $a_{i}$ would, in the long run, come to dominate the economy (e.g. if the market for apples went as $n_{apples} \sim m^{4}$, the entire economy would quickly become just apples). This doesn't appear to happen in diversified economies [2] (it might be true of e.g. oil-based economies), which implies there is a constraint on the values of the $a_{i}$. The interesting thing is that this constraint appears to make the macro formulation (the aggregate market) more accurate than the sum of individual markets -- i.e. there is an enforcement mechanism that makes the individual markets behave more like the average in diversified economies.

Is there an effect due to

$$
a_{i} = \frac{\log \nu_{i}}{\log M}
$$

so that we should use

$$
\bar{a} = \frac{\log \bar{\nu}}{\log M}
$$

instead? Does that then have a relationship with $\kappa (M, N)$? A uniform logarithmic distribution is related to e.g. Benford's law. I will look into all of this in a future post.

[2] This is almost a circular definition: diversified economies are economies that haven't had one commodity or product take over their economy. However, it seems that diversified economies stay diversified -- that is the sense of the statement.