Re-allocating Geometric Brownian Motion

# wealth distribution 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

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.

100 100 0.1 0.08 0.18 0.01 20

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.