The
Economics of
Modeling
Resource Sustainability in Flash
Introduction
User’s Guide References
Math
Appendix Download Exercises
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to Model
Easter
Island has become a metaphor for Malthusian resource overexploitation and
societal collapse. The stylized facts begin
with the arrival of fewer than 50 Polynesian settlers in 400 A.D. on the
resource rich island densely forested by a slow growing species of palm
tree. The population peaks at 10,000
inhabitants in 1300 A.D., at which time the carving of monumental statues (Moaii) begins. The palm
forest is mined for materials used in statue building and by 1550 can no longer
support the population which subsequently crashes. When Dutch explorers visit the island in 1722,
population estimates range from 2,000 to 3,000.
The explorers find no evidence of the technical capacity to construct
the monumental Moaii, insufficient wealth to support
the substantial artisan class required to build them, and no memory among the
islanders of how the monuments were moved to their platforms.
Dalton,
Coats, and Taylor (2006) provide a detailed review of the Easter Island
literature. Bahn
and Flenley (1992), and Diamond (2005) suggest the Easter
Island story portends the collapse of modern society due to environmental
degradation and population overshoot. More
recent research has challenged the “ecocide” interpretation of the Easter
Island story. Hunt and Lipo (2006) and Wilmshurst et. al. (2011) present results from improved radiocarbon
dating techniques that compress the settlement chronologies for Easter Island
and East Polynesia. Their research indicates
Easter Island was settled much later in 1200 A.D. The later settlement date suggests that
monumental building activity commenced soon after colonization.
Brander and
Taylor (1998) build a model of renewable resource use on Easter Island based on
simple Ricardian and Malthusian principles
encountered by students in a typical upper division natural resource economics
course. The model provides an
understanding of simple dynamic economic systems while analyzing a parable of
one of the most fascinating aspects of sustainability: population overshoot and
societal collapse. The Brander and
Taylor model has spawned extensive economic research into the dynamic performance
of renewable resource based economies.
For example, Dalton and Coats (2000) incorporate institutions that allow
society to make informed resource harvesting decisions in the face of
increasing resource scarcity. Pezzey and Anderies (2002) extend
the basic model to include subsistence consumption and Erickson and Gowdy (2002) investigate a model in which fertility
responds to technological innovation.
Authored in
Adobe Flash CS4, the Easter Island Modeling Software (EIMS) allows instructors
and students to explore Brander and Taylor’s (1998) Easter Island model
interactively by changing any of its original six parameters and plotting the
resulting dynamic and time paths of each model variant they create. Users can explore a rich variety of
oscillating and direct paths to steady state equilibrium, or investigate the
economics of limit cycles and extinction.
EIMS also supports the incorporation of critically depensated
resource growth functions and the analysis of Dalton and Coats (2000)
institutional response model of
EIMS is
very simple to use. Sliders allow the
adjustment of all biological, economic, and demographic parameters of Brander and
Taylor’s (1998) base case model (BT) to analyze the performance of the dynamic
system under alternative scenarios. The
adjustable BT model parameters include:
Biological Parameters:
r = intrinsic growth
rate of the resource
K = carrying capacity
of the resource
Economic Parameters:
= harvest technology
= resource
taste (share of labor force employed in harvesting)
Demographic
Parameters:
bd = net birth rate
= fertility response to changes in the
average product of labor in harvesting
The BT
model parameter default values are given below:
r 
0.04 
K 
120000 

0.00001 

0.4 
bd 
0.1 

4.0 
The
software loads the model with these parameters and defaults to the Phase Diagram view, drawing the blue dL/dt=0 and green dS/dt=0 isoclines as shown
in Figure 1. The steady state
equilibrium (S*, L*)
is also marked in Figure 1. To change to
Time Paths view (Figure 2), select the corresponding checkbox.
Figure
1: Phase Diagram 
Figure
2: Time Paths 
To adjust any parameter of the model, simply click and drag that parameter’s slider thumb to the left
or right. The parameter’s value changes
in the textbox immediately to the right of the slider. For example, suppose we want to investigate
the effect of a higher resource intrinsic growth rate (r).
Figure 3
shows the BT base case model with r = 0.08.
The dS/dt=0 isocline steepens as
a higher resource intrinsic growth rate allows a given resource stock (S) to
support higher human populations in the steady state.
Figure 3: BT model with r = 0.08
To view the
dynamic path to steady state equilibrium associated with the case shown in Figure
3, select an initial resource stock (S_{0}) and human population (L_{0}). The default starting values for resource
stock and population, S_{0 }= 12000 and L_{0} = 40, are taken
from Brander and Taylor (1998), and are calibrated to fit the presettlement
resource carrying capacity of the island and the original number of settlers
arriving in roughly 400 A.D. Users may
change these initial values by simply typing new values in the corresponding
text boxes.
Click the
“Path” button. An animated dynamic path
cycles toward the steady state equilibrium intersection of the two
isoclines.
Figure 4
shows the oscillating dynamic path to steady state equilibrium for the BT model
with r = 0.08. The software records the
steady state equilibrium values of resource stock (S*
= 6234) and human population (L* = 9610).
Figure 4: Dynamic path to equilibrium; r =
0.08
To view the
Time Paths of the resource stock (S) and population (L)
for this version of the model, check the Time
Paths checkbox and click the Path button
again.
Figure 5
shows the Time Paths of resource
stock and population from 400 A.D., the year of settler arrival determined by
the archaeological record, to 2000 A.D.
Figure 5: Time Paths of S and L for r = 0.08
model variant
For
comparison purposes, Figures 6 and 7 show the Phase Diagram and Time Paths
views associated with the BT base case model with r = 0.04. (These diagrams match
Figures 2 and 3 from Brander and Taylor’s (1998) article). It is clear from Figures 5 and 7 that a
higher resource intrinsic growth rate supports a more robust population that
recovers after 1700 A.D. whereas with the lower growth rate of the base case
model, population continues it’s decline until
reaching its steady state equilibrium value (L*
= 4805).
Figure
6: BT Base Case
Phase Diagram 
Figure
7: BT Base Case
Time Paths 
Many other
experiments may be carried out by simply using the sliders to change
the model’s parameterization. The
parameter sliders allow changes within the following ranges:
Parameter 
Minimum
Value 
Maximum
Value 
r 
0 
0.4 
K 
0 
15000 
a 
0 
0.00005 
b 
0 
0.8 
bd 
0.15 
0 
f 
0 
8.0 
Critical Depensation
The Easter
Island Modeling Software allows users to specify a critically depensated resource growth function with a minimum viable population
of the type:
where r = intrinsic growth rate, S =
resource stock, S_{0} = minimum viable population, and K = carrying
capacity. The third Biological Parameters slider, here shown adjusted to S_{0}
= 2350, allows minimum viable populations between zero and 5000.
The dS/dt=0 isocline becomes a quadratic with a critically depensated growth function as shown in Figure 8.
Figure 8: Minimum Viable Population (S_{0}
= 2350)
Figures 9
and 10 show the Phase Diagram and Time Paths views of the model in the
presence of critically depensated growth. Note that the resource crashes in this model
variant as the population (L) increases
dramatically followed by a precipitous collapse that is reinforced when the
resource stock drops below its minimum viable population.
Figure
9: Phase Diagram
view of the model with critically depensated growth (S_{0} = 2350) 
Figure
10: Time Paths
view of the model with critically depensated growth (S_{0} = 2350) 
Incorporating Institutions
.
Assume that
dS/S < 0, that is, society is effectively mining
its renewable resource, consuming it at a rate greater than its replenishment
rate. In this case, a negative
institutional response parameter ( < 0), would increase harvest
employment, as harvesters exhibit the classic tragedy of the commons, racing to harvest and thereby eliminating
any scarcity rent. Alternatively, a
positive institutional response parameter ( > 0), perhaps due to property
rights in the renewable resource, would reduce harvest employment in the
presence of declining resource stocks.
An intermediate case, ( = 0), corresponds to the BT base
case “consumption rights” model.
The EIMS
software includes an Institutional Parameter slider that allows users to
specify and fine tune the strength of institutions. The allowable institutional parameter range
is between 15, extreme common access rights, and +25, strong personal
ownership rights.
Consistent
with Dalton and Coats (2000) model, three types of institutions are possible:

Type
of Institutions 
< 0 
Common
access rights 
= 0 
Consumption
rights; Brander & Taylor base case 
> 0 
Personal
ownership rights 
Figure 11
shows the isoclines for a personal ownership rights model with strong property
rights ( = 20).
Note that the dL/dt=0 isocline pivots
around the steady state resource stock (S* =
6234) and becomes curvilinear at high resource stock values.
Figure 11: Institutional Response Parameter ( = 20)
Strong property rights
The
pivoting dL/dt=0 isocline puts an
earlier brake on population growth, dampening the oscillations and ensuring a
smooth dynamic path to equilibrium as shown in Figures 12 and 13. These figures correspond to Figures 6 and 2 from
Dalton and Coats (2000).
Figure 12: Phase Diagram view of DaltonCoats’ personal ownership
institutions model ( = 20) 
Figure
13: Time Paths
view of DaltonCoats’ personal ownership institutions model ( = 20) 
Bahn, Paul, and John Flenley,
Easter Island, Earth Island, Thames
and Hudson, 1992.
Brander, James,
and M. Scott Taylor, “The simple economics of
Dalton, Thomas, R. Morris Coats, and Leon
Taylor, “Economics and the Easter Island Metaphor,” Rapa Nui Journal 2006, 20, pp. 97110.
Diamond,
Jared, Collapse: How Societies Choose to Fail
or Succeed, Viking Penguin, 2005.
Ensley,
Douglas, and Barbara Kaskosz, Flash and Math Applets: Learn by Example, www.flashandmath.com, 2009.
Erickson,
Jon, and John Gowdy, “Resource use, institutions, and sustainability: A tale of two Pacific
island cultures,” Land Economics 2002, 76, pp. 345354.
Hunt, Terry,
and Carl Lipo, The
Statues that Walked: Unraveling the Mystery of Easter Island, Simon &
Schuster, 2012.
Hunt, Terry,
and Carl Lipo, “Late
Colonization of Easter Island,” Science 2006, 311, pp. 16031606.
Pezzey, John, and John Anderies,
“The effect of subsistence on collapse
and institutional adaptation in populationresource societies,” Economics
and Environment Network, Australian National University Working Paper, 2002
Schaefer,
M. B. "Some
Considerations of Population Dynamics and Economics in Relation to the
Management of Marine Fisheries." Journal of the Fisheries
Research Board of
Wilen, James, “Common Property Resources and the Dynamics of Overexploitation: the
Case of the North Pacific Fur Seal,” 1976, unpublished manuscript.
Wilmshurst, Janet, Terry Hunt, Carl Lipo, and Atholl Anderson, “Highprecision radiocarbon dating shows recent and rapid
initial human colonization of East Polynesia,” Proceedings of the National Academy
of Sciences 2011, 108, pp. 18151820.
The BT
model is an adaptation of Wilen’s (1976) model in
which fishing effort harvests the North Pacific Fur Seal. In the BT model, the renewable resource stock
is the forestsoil complex (S), which, having attained its carrying capacity (K)
without human exploitation, is subsequently harvested by the Easter Island settlers
(L) after their arrival in 400 A.D.
The change
in the forestsoil complex (dS/dt)
through time is the difference between natural growth G(S) and harvest H(S,L):
(1)
where S = resource stock (forestsoil
complex) and L = human population. (Note:
Time subscripts have been omitted for clarity of exposition.) To capture the constrained environment of an
island ecosystem, the model assumes a logistic growth function G(S):
(2)
where r = intrinsic resource growth rate of
the forestsoil complex, and K = carrying capacity. The resource harvest H(S,L)
is specified as a stock dependent Schaefer (1957) harvest production function
(3):
(3)
where =
the state of harvest technology and = the share of labor employed in harvesting. Note that L_{h} = bL is the quantity of labor employed
in harvesting the renewable resource.
Substituting (2) and (3) into (1) gives the the
forestsoil complex dynamics:
(4)
Human
population dynamics are determined by the net birth rate (b – d) plus a
Malthusian fertility response ()
to per capita resource consumption (H/L = ):
(5)
Setting
equations (4) and (5) equal to zero determines the isoclines dS/dt = 0 and dL/dt = 0 shown in the EIMS software base case:
(6a) 

(6b) 

The
intersection of equations (6a) and (6b) reveals the steady state values forestsoil
complex S* = 6,234, and population L* = 4,805.
Dalton and
Coats (2000) modify the BT model by including an institutional parameter ()
that allows the labor employed in harvesting to respond to the relative change
in the resource stock (dS/S), an indicator of future
resource scarcity:
(7)
Incorporating
Dalton and Coats institutions modifies the resource dynamics (dS/dt) and population dynamics (dL/dt) as shown in equations (8)
and
(9):
(8)
(9)
Note that
equations (8) and (9) collapse to equations (4) and (5) when = 0, so the BT base case model is a
special case of Dalton and Coats more general model.
To
incorporate critically depensated resource growth and
a minimum viable population (S_{0}), equation (10) is substituted for
equation (2).
(10)