Re-allocating
Geometric
Brownian
Motion
# RGBM

# wealth distribution

model

model

Interactive simulation of \(N\) wealth trajectories over
\(dt\) time steps following the model detailed in

Wealth Inequality and the Ergodic Hypothesis: Evidence from the
United States

Adamou, Berman, Peters (2019)

The paper describes an economy in which the change in an individual's wealth follows a geometric brownian motion, with an added term that accounts for the re-distribution of wealth. Every year everyone contributes a proportion \(\tau\) of his/her wealth to a central pot, and then the pot is split evenly across the population.

Let \(x(t)\) describe the wealth of an individual at time \(t\), the change in wealth is given by: $$dx = x(\mu dt + \sigma dW) - \tau(x - \langle x \rangle_N) dt$$ where \(W\) is a Wiener process with mean \(\mu\) and standard deviation \(\sigma\), \(\langle x \rangle_N\) represents per-capita wealth and \(\tau\) is the re-allocation parameter.

You are encouraged to try out different values for \(\tau\) and explore the wealth distributions it generates. Further details on the RGBM model are found in this blog post.

Wealth levels over time

Each individual's wealth is shown as a fraction of maximum wealth. We can see that as \(\tau \to -1\), the
wealth distribution becomes more unequal. Importantly, wealth *can* become negative, but the plot only
shows positive wealth individuals.

Histogram of wealth levels

In this plot we discretize wealth levels in a number of bins, and show the number of individuals per bin.