### Two Futures

I found this article from 1998 online, ‘The Simple Economics of Easter Island‘, by James Brander and M. Scott Taylor.

The authors consider some small South Pacific islands, especially Easter Island, and have built a simple mathematical model that they claim helps explains the island’s growth and decline in human population from its settlement around the year 400 AD until the arrival of Europeans in the 18th Century.

The model is a version of a predator-prey system with people as the hunter and forest as the prey. The forest would naturally grow exponentially at a rate of about 4% per decade, but its actual growth rate is constrained by a ‘carrying capacity’, the maximum area of trees the island can sustain when it is completely covered in forest. As the forest cover increases toward the island’s carrying capacity, the growth rate of additional forest slows down.

People harvest the forest as one of their resources. In the absence of any harvesting, the human population would decline 10% per decade. Population is gained in proportion to the amount of resource harvesting.

The equations in the model are shown below

‘Resource’ is the number of (acres of) trees on the island and ‘Population’ is the number of people on the island. ‘b’ is the fraction of people involved in harvesting trees (so, ‘b x Population’ is the number of people involved in harvesting trees) and ‘a’ is their ‘harvesting productivity’, according to the authors.

Why does the harvesting rate depend on the number of trees? One reason is that the higher the number of trees, the easier it is to harvest them (and vice-versa). As the number of trees drops, workers must go further from their homes to access the remaining trees and use more time to bring the wood back to the village.

The authors suggest that 40% of the population is involved in harvesting resources. “Various pieces of evidence suggest that the resource sector probably absorbed somewhat less than half the available labor supply. A value of 0.4 for B is probably in the reasonable range.”

These equations and parameter values reproduce what the authors claim was the basic population dynamic over 1500 years of Easter Island history. A small population grew to about 10,000 people by the year 1200 and then slowly declined to around 3000 people by the time of first contact with Europeans.

I am interested in the values of Population, Resources and ‘harvesting Resources’ if we run the simulation for a long time. For the case of b=40%, the Population will eventually settle down to around 4790 people and the resource reaches about 6250 acres of trees (just over half the carrying capacity) if we run the calculations long enough. Thus, there are about 1.3 acres of trees per person. The long-term harvesting rate is about 0.025 acres (about 10 meters by 10 meters) per person per year.

What if the value of ‘b’ (the portion of the population involved in resource harvesting) was some other value, like 20% or 80% or 100%? What would happen to the Population, and Resources per person?

The graph below shows the long-term (that is, ‘equilibrium’) Population for various values of ‘b’.

If ‘b’, the fraction of people who are involved in harvesting resources, is less than about 20%, the population eventually falls to 0, as the rate of losing population always exceeds the rate of gaining population. For this model, the largest sustainable population is about 4800 people when ‘b’ is 42% and it falls to about 3166 as ‘b’ rises to 100%.

One measure of the prosperity of the Easter Island society is the number of acres of forest per person (that is, the quantity ‘Resources’ divided by ‘Population’). The forest serves not only as a source of wood, but also as a habitat for birds and nuts the islanders can eat, so more trees per person is, in itself, a measure of how prosperous the society is.

The graph below shows the long-term (‘equilibrium’) ratio of Resources / Population for various values of the ‘b’ parameter. When 40% of the population is involved in harvesting resources, the Population eventually reaches its largest possible long-term value. That point is marked with a dark blue dot on the graph.

Raising the portion of the population harvesting resources above 42% causes the equilibrium population to decrease and also causes the Resources/Population to decrease. When 100% of the population is involved in harvesting resources, the population settles down to about 3166 and the Resources/Population to about 0.8. (That point is marked with an open circle.)

There is another value of ‘b’ that also results in a long-term population of 3166 and that is when the portion of people harvesting resources is around 26.3%. In that case, the Resources/Person is about 3.

Another measure of the prosperity of the society is the per-person rate of harvesting resources (‘harvesting Resources’ / Population). The forest is a source of wood, which is useful for making houses, canoes, tools and fire. So, the amount of forest cut down per time tells how much wood is available per person. Interestingly, for all of the different ‘b’ values above 20%, the long-term ‘Harvesting per Person’ is exactly the same: 0.025 acres per person per year.

So, depending on the value of ‘b’, two different Easter Islands are possible. In one, the portion of the population harvesting resources is at or below 40%, the resources per person is 1.3 or more, and the forest covers at least half of the island.

In the other version, the portion of the population harvesting resources is above 40%, the resources per person is below 1.25, and the forest covers less than half of the island.

Which version of the island would you rather live on?

The good news is that, in this model, is it possible to reversibly switch from one version of Easter Island to the other, although it can take several hundred years for the system to settle down and the population can change (read: drop) dramatically during the transition.

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