How wealth inequality naturally spreads in an unconstrained market¶
This notebook shows how wealth distributes in an open market where a set of agents trades without any mechanisms in place to ensure a situation of extreme inequality. The system is modelled in the following way:
A fixed number of agents is initialized either with the same amount or a random amount of wealth within a certain range.
At each iteration two agents will be randomly picked up and will exchange a random amount.
The maximum amount exchanged will be a fraction $\Delta w$ of the poorest agent's available wealth.
No negative wealth is allowed. If a set of agents reaches a wealth of zero then they will no longer be involved in outgoing transactions until they receive a positive amount again.
This approach is based on the Affine Wealth Model paper published by B. Boghosian et al. in 2016.
Simulation class with animations¶
Market initialization¶
Initialize the market and plot the initial wealth distribution.
Set init_random_amount=True to initialize the wealth of each agent with a random amount in [75, 125], False to initialize everyone's wealth to 100.
Market simulation without wealth tax¶
Simulate a number of random transactions without a wealth tax and show the final wealth distribution
It can be easily seen that wealth quickly stashes up in the pockets of a very small group of agents, while most of the other agents end up piling up in the lowest bucket.
The situation is the same even if we initialize the agents with random wealth instead of $100 each.
But why?¶
If transactions are random, how come wealth concentrates in such a drastic fashion?
Think for a moment of being at a casino where you play a certain initial amount (let’s say $100). On each hand a coin flip will occur: if the coin lands on heads then you win back 20% of the initial amount you played, otherwise you lose 17% of the amount you played.
At first sight, it seems like a quite advantageous game: assuming that you’ve got a fair coin, on each transaction you’ll have:
$$ 0.50 \times $20 - 0.50 \times $17 = $1.50 $$
So the net balance is positive.
What happens if, however, the reasoning becomes more like “what if I stay at the table for 10 games?”. Then, assuming that half the times you win and half the times you lose, your net gain will be:
$$ (1.2)^5 \times (0.83)^5 \times \$100 = \$98.02 $$
Therefore, we played \$100, but we’re statistically likely to get back \\$98.02 after 10 games — therefore we’re playing a game with a statistical a net loss. You can apply the same principle to calculate your final gain if you stay at the table for 100 games (it’ll be \$81.84), or 1000 games (it’ll be \\$13.45).
Open markets without wealth redistribution are similar to this casino example: you win some and you lose some, but the longer you stay in the casino, the more likely you are to lose. The difference is that a free market is a casino where players can never leave.
Market simulation with wealth tax¶
Run the simulation again, but this time apply a wealth tax $0 \leq \chi \leq 1$ to each transaction.
Given two agents $A$ (sender) and $B$ (receiver), with respective total wealth $w_A$ and $w_B$, that transact an amount $x$, the wealth tax correction applied to $x$ will be:
$$ \begin{equation} x \leftarrow \begin{cases} x \mbox{ if } w_A \lt \bar{w} \\ x + \chi(w_A - \bar{w}) \mbox{ if } w_A \geq \bar{w} \end{cases} \end{equation} $$
Where $\bar{w}$ is the average wealth owned by all the agents.
Note that the wealth tax is a share of the difference between the total wealth owned by A and the average wealth, and it's not proportional to the transacted amount.
This helps model a fairer redistribution system.
Let's first run a simulation with an aggressive 25% wealth tax on each transaction made by a rich.
That's better from the perspective of wealth equality, and also very stable over time, but now the distribution is perhaps too static to be realistic. After 250k iterations the richest one will still have at most 3-4x of their initial amount, and, since the mean wealth is quite static and variance is quite narrow under this scenario, they're very unlikely to make more money.
This may seem an acceptable scenario for some socialists, but a liberal may argue that it doesn't provide sufficient financial incentives for the most brilliant to invest in their skills.
Let's try and adjust the wealth tax to a more modest 5% instead:
What if we try again with a 1% wealth tax, but this time peg it to the median wealth instead of the average.
Let's run a few more simulations with the median instead of the average to see if it affects the final distribution.
Wealth tax = 25% and pegged t