Tuesday, September 29, 2015

A random walk inside the simplex

I thought I'd do something similar to this post for the idea behind these two posts ([1], [2]): that most of the points in a high dimensional space are near the boundary. Here is an example random walk over d = 3 goods (3 dimensions) restricted to the simplex bounded by the budget constraint Σ ci ≤ 1:


And here is the resulting "total consumption", i.e. the sum of the consumption of each good (1 means the ensemble of agents spends all of their money in that time period):


If you increase the number of goods to d = 20, you get a similar result, but closer to the boundary:


The deviations from Σ ci = 1 represent "recessions" where the ensemble of agents saved (reduced consumption) more than usual.

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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.