This equilibrium point is globally asymptoti- cally stable, the proof of which relies on a simple
application of a theorem due to Kolmogorov relating to planar systems of this type see May,
1973; Edelstein-Keshet, 1988. Beginning from any interior initial condition, the system will con-
verge to the steady state. Depending on parameter values, the steady state will either be a node or a
spiral which will force the system to converge to the equilibrium either monotonically or through a
series of damped oscillations. Of interest to Brander and Taylor is that for certain parameter
values representative of the situation on Easter Island, the system will exhibit transitory oscilla-
tory behavior which manifests itself in overshoot and collapse. Fig. 6 shows the human population
and resource stock trajectories for an initial con- dition of 40 humans landing on Easter Island with
the resource stock at carrying capacity. The units for the resource are a matter of scaling. Brander
and Taylor 1998 choose a carrying capacity of 12 000 units for convenience. The remaining
parameters were b = 0.4, f =
4
, r = 0.04, a = 0.00001, and b − d = − 0.1.
As previously noted, the archaeological record indicates the first presence of humans at around
400 AD. The population increases which is ac- companied by a decrease in resource stock. The
population and available labor peaks at around 1250 AD corresponding to the period of intense
carving in the archaeological record. The popula- tion subsequently declines due to resource deple-
tion. The model predicts a population of about 3800 in 1722, close to the estimated value of 3000.
The model thus gives a reasonable qualitative picture of what may have happened to the culture
on Easter Island. The culture became very pro- ductive and able to undertake the construction of
major monuments, i.e. the labor force increased thus making L
M
large enough to complete such a large scale project. The population subsequently
declined due to resource degradation which left the small population who knew nothing of the
origin of the great monuments to meet the Dutch ships in the 18th century.
5. A closer look at Easter Island population decline
In this section we examine more closely the nature of the population collapse and how this
depends on the way behavior is modeled. It takes 600 years for population to drop from 10 000 to
3800. Compare this to populations doubling every
Fig. 6. Population and resource stock trajectories for Easter Island model from Brander and Taylor 1998.
40 years at present. The authors argue that this slow rate of change may be one of the reasons
that institutional change did not occur on Easter Island. Alternatively, this slow decline, if recog-
nized, might have given the islanders time to respond. Thus, two possibilities could arise: there
was a slow decline and institutional adaptation was possibly prevented by, for example, insuffi-
cient ecological understanding or conflicts be- tween competing groups as Brander and Taylor
suggest; or the decline occurred faster than the model suggests so that there was no time for
institutional adaptation.
The nature of the decline depends on the way behavior is modeled. First, consider the labor
allocations in the economy. By equating total supply and demand and solving for L
H
and L
M
: LabS = L
H
a S
[ L
H
= b L
19a L1 − b = L
M
19b we see that a constant proportion, b, of the labor
force is directed towards producing bioresource goods, while the remaining portion of the labor
force, 1 − b, directs its labor towards the produc- tion of manufactured goods. This implies that as
the bioresource stock is depleted and becomes more expensive to produce, individuals continue
to consume the same amount of manufactured goods and consume less and less bioresources.
The population could be starving, yet the utility maximizing strategy is to keep the proportion of
labor directed to each activity constant. Eq. 19b implies that, for the Brander and Taylor parame-
ter set, when the bioresource reaches its most degraded state between 1400 and 1500, manufac-
turing output will still be more than 60 of its maximum. This does not seem consistent with the
archeological record. Also note that the behav- ioral model is identical to the simple constant
proportion approach used in the Tsembaga model where this assumption tended to stabilize the
system.
The issue here is that Cobb – Douglas utility functions allow for unlimited substitution between
goods without affecting utility. Based on this model, the optimal strategy in the face of a re-
source good shortage is to increase consumption of cheaper manufactured goods. Such a model
may be reasonable in a static, context but presents problems in a dynamic context. To be realistic
one must consider limits to substitution in con- sumption. It turns out, as shown below, that this
makes a considerable difference in the model dynamics.
One way to introduce the possibility for struc- tural change in the economy is to modify the
utility function. I do so by utilizing a Stone – Geary type utility function which assumes that
there is a minimum amount of bioresource goods subsistence level at which utility is zero, i.e.:
Uh, m = h − h
min b
m
1 − b
20 where h \ h
min
. The essential difference between 20 and 10 is that in the latter there are limited
substitution possibilities
between bioresource
goods and manufactures. Such limits to substitu- tion in both production and consumption have
received attention in the literature, and we will see their importance again here.
As before, we can determine the optimal con- sumption of resources, but now there is a corner
solution as a result of the subsistence requirement. Maximizing Uh, m subject to the income con-
straint interior solution results in the following indirect demand functions:
h = 1 − bh
min
+ wb
p
h
21a m = 1 − b
w − p
h
h
min
p
m
. 21b
Now we have that the optimal consumption level of h consists of a price dependent and a price
independent portion. This is more realistic as it says to spend excess income on certain propor-
tions of h and m only after meeting minimum nutritional requirements. Eqs. 21 only make
physical sense when
p
h
5 w
h
min
22 but this condition will always be satisfied if h \
h
min
. Substituting Eq. 15 for p
h
into Eq. 22 and assuming as before that w = 1 and p
m
= 1, we see
that the condition for the system to make physical sense reduces to
h
min
5 a
S 23
which simply says that if the demand h
min
can be met at the present work level, use the optimality
conditions given by Eqs 21 to divide excess capacity to the tasks of producing m and h.
If Eq. 22 is not met, the optimality conditions do not apply. Common sense suggests that if
people are trying to meet minimum nutritional requirements, they would produce all the biore-
source goods possible corner solution, i.e.
h = aS 24
Combining 21 with this corner solution defines the optimal consumption program. Finally, the
optimal consumption program and the produc- tion functions given by Eqs. 13a and 13b, can
be used to compute the amount of labor the population should devote to producing biore-
source goods and manufactured goods:
L
H
= Á
à Í
à Ä
L1 − bh
min
a S
+ b L
if h
min
5 a
S L
otherwise 25a
L
M
= Á
à Í
à Ä
1 − bL 1 −
h
min
a S
if h
min
5 a
S otherwise
25b Notice that in contrast to the original model, the
division of labor is no longer fixed. As the price of bioresource goods increases, labor is shifted out
of the production of manufactured goods into the bioresource sector — i.e. there is structural
change in the economy. This is analogous to allowing c to be dynamically determined by Eq.
5 in the Tsembaga model. We focus our atten- tion on the effect that the subsistence require-
ment, h
min
, has on the model. If we take h
min
= 0,
we retrieve the original model for which we know there is a globally stable equilibrium point at
N = 4791.7 and S = 6250 for parameters chosen by Brander and Taylor. We can use pseudo-ar-
clength continuation to investigate the nature of this equilibrium point as h
min
is varied. Fig. 7 is the result of this exercise.
Graph a is a bifurcation diagram that plots the long-run equilibrium population for the model
for different values of h
min
. The heavier line not the very heavy line made up of circles for lower
values of h
min
indicates that these long-run equi- libria are stable; but trajectories approaching
them may involve fluctuations. The lighter por- tion of the curve for higher values h
min
indicates that the equilibria are unstable, the system will
never approach them. Rather, for value of h
min
above the Hopf bifurcation point near 0.0177, the system will approach a stable limit cycle whose
amplitude is shown by the very heavy line ema- nating from the curve of fixed points. Graph b
shows what this limit cycle looks like in phase space for h
min
= 0.02 as well as representative
trajectory for the Brander and Taylor parameter set and initial condition. The existence of a bifur-
cation point and limit cycle causes the model to fluctuate even more dramatically than in the case
investigated thus far.
To illustrate how my model suggests a different interpretation of why institutional change may
not have occurred, I set h
min
= 0.03 and b = 0.1,
with all other parameters unchanged. This choice of parameters implies individuals have a very low
preference to consume more bioresources once they have met their minimum demand. Fig. 8
shows the results for this case.
Graph a shows the population trajectories for the original model and the modified model as well
as the manufacturing output for the modified model labeled as 1, 2, and 3, respectively. Graph
b shows how the structure of the economy evolves over time. For the first 400 years, the
structure of the economy remains fairly stable with approximately 33 of the labor force work-
ing in the bioresource sector and the remainder in the manufacturing sector. As bioresources become
more scarce, the economic structure begins to change and labor is shifted into the bioresource
sector until all of the population is working in this sector by 1430 AD. This causes the complete
cessation of manufacturing activities as shown by curve 3 in graph a. This abrupt cessation of
manufacturing material culture seems more con- sistent with the archeological record than the
Brander and Taylor model.
Fig. 7. a Bifurcation diagram for Easter Island model with subsistence. b Phase plane showing limit cycle and representative trajectory for h
min
= 0.02.
Further, this abrupt cessation of manufacturing suggests, as previously mentioned, a different in-
terpretation for why institutional adaptation did not occur. From around 1000 up until 1430, the
population shifts labor into the bioresource sec- tor. By doing so, per capita intake of bioresources
can be maintained above h
min
. The population peaks near 1450 and subsequently crashes to half
this peak in 150 years. Such change might have been far too rapid for institutional responses to
occur. The ability of the population to increase its work effort and maintain its material well-being
hides the feedback from the resource base. When change finally does occur, the resource base is so
degraded that the change is rapid and dramatic, precluding any hope of institutional response. A
present day parallel is overcapitalization in fisheries. Thus, a relatively small change in model
assumptions
produces a
quite different
interpretation.
6. Discussion