# Hyperbolic discounting — The irrational behavior that might be rational after all

When I was in grad school I occasionally overheard people talk about how humans do something called “hyperbolic discounting”. Apparently, hyperbolic discounting was considered irrational under standard economic theory.

I recently decided to learn what hyperbolic discounting was all about, so I set out to write this blog post. I have to admit that hyperbolic discounting has been pretty hard for me to understand, but I think I now finally have a good enough handle on it to write about it. Along the way, I learned something interesting: Hyperbolic discounting might be rational after all.

## Rational and irrational discounting

##### Rationality of hyperbolic discounting

The problem with this story is that it only works if you assume that the interest rate is constant. In the real world, the interest rate fluctuates.

Before taking on the fluctuating interest rate scenario, let’s first take on a different assumption that is still somewhat simplified. Let’s assume that the interest rate is constant but we don’t know what it is, just as we didn’t know what the hazard rate was in the previous interpretation. With this assumption, the justification for hyperbolic discounting becomes similar to the explanation in the blue and pink plots above. When you do a probability-weighted average over these decaying exponential curves, you get a hyperbolic function.

The previous paragraph assumed that the interest rate was constant but unknown. In the real world, the interest rate is known but fluctuates over time. Farmer and Geanakoplos (2009) showed that if you assume that interest rate fluctuations follow a geometric random walk, hyperbolic discounting becomes optimal, at least asymptotically as $\tau \rightarrow \infty$. In the near future, you know the interest rate with reasonable certainty and should therefore discount with an exponential curve. But as you look further into the future, your uncertainty about the interest rate increases and you should therefore discount with a hyperbolic curve.

Is the geometric random walk a process that was cherry picked by the authors to produce this outcome? Not really. Newell and Pizer (2003) studied US bond rates in the 19th and 20th century and found that the geometric random walk provided a better fit than any of the other interest rate models tested.

## Summary

When interpreting discounting as a survival function, a hyperbolic discounting function is rational if you introduce uncertainty into the hazard parameter via an exponential prior (Souza, 2015). When interpreting the discount rate as an interest rate, a hyperbolic discounting function is asymptotically rational if you introduce uncertainty in the interest rate via a geometric random walk (Farmer and Geanakoplos, 2009).