## Sunday, November 20, 2016

### Inequality and maximum entropy

What does maximum entropy have to say about economic inequality? I am going to put forward a possible understanding of how inequality impacts macroeconomic fluctuations (aka the business cycle).

First we have to be precise with what we mean. One could say that the maximum entropy distribution of income should be a Pareto distribution, which can be highly unequal. However, the uniform distribution is also a maximum entropy distribution of income. How can that be? Constraints. The Pareto distribution is the maximum entropy distribution of x given a constraint on the value of ⟨log x⟩ (the average value of log x). The uniform distribution is the maximum entropy distribution given x is (for example) constrained to be 0 ≤ x c.

It is true that income appears to be Pareto distributed (at least in the tail, per Pareto's original investigations). Does this mean that inequality is an optimal equilibrium (i.e. maximum entropy)? Not exactly; it really just means that Pareto distributions of income are probably easiest to achieve given random processes. Let's build a simple model to look at the effects of inequality.

Let's say each agent (in the simulations below, I use N = 100 agents) has an income  that is drawn from a Pareto distribution, and can choose to spend some fraction of it in each time period on consumption. However, total consumption is limited (limited resources). This creates a bounded region in N-dimensional space. For three agents (N = 3), it might look something like this:

Over time, people adjust their consumption by a few percent (I used Brownian motion with σ = 5%), so the economy follows some random walk (blue squiggly line) bounded by this N–1 simplex (2-simplex for N = 3). You could add recessions (correlated motion among the dimensions) or growth (expanding the region as a function of time), but these are not necessary for my point here. We will compare the Pareto distributed income with uniform income which would be bounded by a symmetric simplex (yellow):

If we look at random walks inside each N-dimensional space bounded by these simplices (as I did here), we can see a key difference in the instability (variance) in the total consumption. Here is a typical time series path (blue is Pareto, yellow is uniform):

Note that these two time series were given the exact same set of log-linear shocks. It also manifests in a larger variation in the growth rate (average here is zero by construction, but as mentioned above, it could be positive), again blue is Pareto, yellow is uniform, and gray is the actual set of shocks applied to both series (which closely matches the uniform distribution):

What is happening here? Well, the Pareto distribution effectively turns the N-dimensional region into an n << N dimensional region. The variation in a few of the large-income dimensions becomes much more important. We could also imagine they'd become more susceptible to coordination (e.g. panic/financial crisis) since it is easier to obtain a correlation among n << N agents than N agents.

What is also interesting is that one can easily visualize a couple of policies that address inequality in this framework. A progressive income tax or a progressive consumption tax would act as a uniformly-distributed (soft) bound on the allowed consumption states (income tax affects the bound itself, while consumption tax impacts the realized values of consumption). A universal basic income (UBI) would act as a lower bound. With these two policies, you could create an allowed region for the Pareto bound, making it more likely to be close to a uniform distribution sandwiched between the two. The figure below shows this with a 100% marginal income tax rate (red bound) and a universal basic income (green bound); the constrained Pareto bound is in blue.

I am not saying this is exactly what is happening in reality. This is just a possible understanding in terms of the maximum entropy/information equilibrium framework.

Footnotes

 This could easily be reconstructed in terms of wealth inequality or whatever form of economic inequality you prefer.

1. Cool. This is an intuitive and interesting result. Indeed, much easier to coordinate n<<N agents (recession). One trick, though, is that the agents aren't necessarily individuals... also extends to companies, where a limited number of individuals acts to make decisions coordinating their employees. Thinking about maximum entropy as a desirable condition for the economy takes some reflection... in some sense, taking the population and homogeneously distributing ourselves in mud huts across the liveable portions of the globe would maximize instantaneous entropy, but quality of life wouldn't exactly improve. Hard to get recessions in that state of the world, but also hard to launch weather satellites. Seems like there's always going to be a trade off in how much coordination risk we accept in exchange for coordination benefit. From a quality of life perspective, though, I agree that something in between uniform and pareto would be preferred... you want to find a way to reward those who use their natural talents to benefit the aggregate, and yet given a certain level of wealth, it seems only moral to attempt to provide a certain base level of living standard. Interesting boundary between philosopy/ethics and economics/policy to consider.

1. Cheers. I would agree there would definitely be a trade. For example, you miss out on the positive fluctuations (which in the case of growth compound). It also would probably be costly to enforce a uniform distribution. What would be better is simply a Pareto distribution with a different shape parameter leading to a lower Gini coefficient (see here). The uniform distribution was more of a easy comparison case rather than a particular policy prescription (the last figure shows policies that would make a Pareto economy more like a uniform one).