Some results on income distributions and total utility

Check my math

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I played around with some math regarding income distributions (though wealth distributions would apply here as well), Gini coefficient, and total utility. There’s not really a point to all of this, but I make some observations at the end.

I’m going to model utility functions using CRRA utility:

Note: I actually use gamma instead of eta for the rest of this post.

When you actually try to measure peoples risk aversion parameter gamma, it typically lands between 1 and 2, with most results clustering around 1.5.

I’m going to model income distributions using either Pareto or lognormal distributions. Lognormal is a pretty good fit for income distributions in general, while Pareto is generally a better fit for the tail of wealthy individuals.

The lognormal distribution

Gini Coefficient

Wikipedia gives us a formula for the Gini coefficient of a lognormal distribution:

\(G = erf(\frac{\sigma}{2})\)

Where erf is the error function.

For the Pareto distribution we have to assume alpha is greater than 1 but it’s:

\(G = \frac{1}{2\alpha - 1}\)

Lognormal distribution with gamma = 1

With gamma set at 1, everyone’s utility function just becomes ln(c). We can calculate the total utility of the population by integrating over everyones utility:

\(U = \int_0^\infty \ln(x) \frac{1}{x \sigma \sqrt{2 \pi}} \exp(- \frac{(ln(x)-\mu)^2}{2 \sigma^2}) dx\)

Which simplifies to:

\(U = \mu\)

Notice that the median of the lognormal distribution is exp(mu), so this expression is equivalent to ln(median). Also note also that there is no dependence on G.

Pareto distribution with gamma = 1

Now let’s do the same exercise with the Pareto distribution:

\(U = \int_{x_m}^\infty \ln(x) \frac{\alpha x_m^\alpha}{x_m^{\alpha + 1}} dx \)

Which once again simplifies nicely:

\(U = \ln(x_m) + 1\)

The median of the Pareto distribution is:

\(x_m 2^{1/\alpha}\)

So this is roughly proportional to the log of the median of the Pareto distribution, once again with no dependence on G.

Now let’s consider more risk-averse utility functions when gamma > 1 (technically some of the results also apply to situations where gamma is less than 1).

Lognormal distribution with gamma > 1

Same excercise with a CRRA utility function:

\(U = \int_0^\infty (\frac{x^{1 - \gamma} - 1} {1 - \gamma}) (\frac{1}{x \sigma \sqrt{2 \pi}}) \exp(- \frac{(ln(x)-\mu)^2}{2 \sigma^2}) dx\)

With some help from Wolfram Alpha we get:

\(U = \frac{1}{1 - \gamma} \exp(\frac{1}{4} (\gamma - 2)(-4 \mu + (\gamma - 2) \sigma^2))\)

Notice that as you increase sigma, the expression inside the exponent increases (the gamma terms get squared in front of sigma, so it doesn’t matter if gamma is greater than or less than 2) and utility decreases overall (since the constant in front is negative). So sigma increases Gini coefficient and lowers utility.

Pareto distribution with gamma > 1

Last one I promise:

\(U = \int_{x_m}^\infty (\frac{x^{1 - \gamma} - 1} {1 - \gamma}) \frac{\alpha x_m^\alpha}{x_m^{\alpha + 1}} dx \)

This gives us:

\(U = x_m^{2 \alpha} (\frac{1}{\gamma - 1} - \frac{\alpha x_m^{1 - \gamma}}{(\gamma - 1)(\alpha + \gamma - 1)})\)

We can rewrite this in terms of the Gini coefficient:

\(U = x_m^{\frac{1}{G}} * x_m (\frac{1}{\gamma - 1} - \frac{\alpha x_m^{1 - \gamma}}{(\gamma - 1)(\alpha + \gamma - 1)})\)

So increasing the Gini coefficient lowers utility.

Conclusion

I don’t really know what to conclude from this, though I’m somewhat surprised that the degree of inequality doesn’t come up at all with log utility. The total utility simplifying to the log of the median is also interesting. This reminds me of the Kelly criterion, where an agent is modeled as having logarithmic utility in money and maximizes the median of terminal wealth. A social planner in this instance would end up acting like a Kelly bettor maximizing median income.

When we add more risk aversion, inequality becomes important again. In the Pareto case, a low Gini coefficient (low inequality) makes it easier to increase total utility by boosting median (or mean) income since the lower the inequality the higher the power in the first term.

I’m not sure if any of this will end up being useful, but I’m going to post it here for future reference.