# Impatience and aggregate risk

Imagine two populations of asexually reproducing people (asexual reproduction is where each child comes from a single parent, not a couple). In the first population, each person has a 50 per cent chance of having no children, and a 50 per cent chance of having two children. If there is no relationship between the outcomes for each person (i.e. they face idiosyncratic risk) and the population is large, we would expect the population to remain relatively constant over time.

In the second population, each person also has a 50 per cent chance of having no children, and a 50 per cent chance of having two children. However, in this case, the population faces an aggregate risk so every person in the population has the same outcome - either zero or two children. At any time in the future, the expected population (the mean of all possibilities) is constant, just like the first. However, most outcomes lead to the extinction of the population, and as time goes on, this will almost surely occur. The only reason the mean outcome is constant is the small probability of a very large population (doubling every generation).

This difference between aggregate and idiosyncratic risk was used by Arthur Robson and Larry Samuelson (ungated version here) to partly explain the problem that the rate of time preference (also called the discount rate, or level of patience) estimated in many theoretical evolutionary papers is lower than we see - often around a couple of per cent per year compared to empirical findings that people discount the future at rates of 10 per cent per year or higher. People are far more impatient than many theoretical models predict.

Theoretical evolutionary estimates of the rate of time preference are typically tied to the rate of population growth and the chance of death, as was the case in work by Ronald Fisher, Ingemar Hansson and Charles Stuart and Alan Rogers. If population is growing, investments in the future are worth less. More obviously, death before reproduction erases all future benefit, so a higher probability of death should lead you to discount the future more. But with long-term population growth through our evolutionary history being near flat and death rates each year a couple of per cent at most, theoretical calculations often come out at around two per cent per year.

To resolve this problem, Robson and Samuelson argue that idiosyncratic and aggregate risk affects the optimal discount rates. In a population with idiosyncratic risk, the rate of time preference should approximate the population growth rate and the death rate. But in a population with aggregate risk, the expected population growth rate is not what we should look at.

To see why, look at the example of aggregate risk at the beginning of this post. Although the expected population is constant, the reality is that the population will either double every year, or it will plunge to zero. If it plunges to zero and everyone is wiped out, the rate of time preference does not matter. But what if the aggregate risk did not strike. The population would double every year. As a result, people should discount for the one scenario that matters - the annual doubling - and have a higher discount rate than they would have in the idiosyncratic risk scenario.

This argument holds with less extreme examples. In another (excellent) paper (ungated version here) in which Robson and Samuelson survey the literature on the evolution of preferences, they work through an example similar to the one above, but where people have one or two children. Again, the population with the aggregate risk would be expected to have a higher rate of time preference.

Putting this theory into context, human populations through our evolutionary history would have faced a lot of aggregate risk, such as change in climate, drought and crop failures. So although population growth is near zero, their rate of time preference would be higher.

One way to think about it is to picture population growth as occurring in a saw-tooth manner - increases and then sudden drops. The population growth rate during those periods of increase is more relevant for the rate of time preference than the fact that the sudden drops through droughts and disasters reduce long-term population growth to near zero. The rate of population growth during the good times is what matters.

Ultimately, this paper is one of my favourite in economics. It has a great idea that is not immediately obvious (nor intuitive), but once you wrap your mind around it, it makes a lot of sense.