The Economics of Easter Island:

Modeling Resource Sustainability in Flash

                                                                                                                            

Introduction       User’s Guide      References    

 

Math Appendix       Download Exercises      Go to Model    

             

 

 

Introduction

 

Easter Island has become a metaphor for Malthusian resource over-exploitation 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 Easter Island.  This website includes a user’s guide to the software, bibliographic references, mathematical appendix, and a set of exercises available for download.  Future model enhancements will include the subsistence consumption model of Pezzey and Anderies (2002) and Erickson and Gowdy’s (2002) innovation-enhanced fertility model.

 

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User’s Guide

 

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:

          b-d    =      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

b-d

-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 (S0) and human population (L0).  The default starting values for resource stock and population, S0 = 12000 and L0 = 40, are taken from Brander and Taylor (1998), and are calibrated to fit the pre-settlement 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

b-d

-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, S0 = minimum viable population, and K = carrying capacity.  The third Biological Parameters slider, here shown adjusted to S0 = 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 (S0 = 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 (S0 = 2350)

 

Figure 10: Time Paths view of the model

with critically depensated growth (S0 = 2350)

 

 

 

Incorporating Institutions

 

Dalton and Coats (2000) ask if institutional arrangements could have saved the Easter Island civilization from collapse.  Their model introduces an institutional response parameter (), which captures society’s response to expected resource scarcity.  One predictor of future resource scarcity is dS/S, the relative change in the renewable resource stock.  If dS/S > 0, society is harvesting and consuming less than the annual growth of the resource and future resource cost will reflect only harvest cost.  However, if dS/S < 0, society is consuming the resource at a rate that exceeds its annual growth and future resource scarcity values will rise to reflect the user cost of the resource.  Dalton and Coats incorporate the institutional response parameter () in the harvest employment equation as follows:

 

.

 

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 Dalton-Coats’ personal ownership institutions model ( = 20)

 

Figure 13: Time Paths view of Dalton-Coats’ personal ownership institutions model ( = 20)

 

 

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References

 

Bahn, Paul, and John Flenley, Easter Island, Earth Island, Thames and Hudson, 1992.

 

Brander, James, and M. Scott Taylor, “The simple economics of Easter Island: A Ricardo-Malthus model of renewable resource use,” American Economic Review 1998, 88, pp. 119-138.

 

Dalton, Thomas, and R. Morris Coats, “Could institutional reform have saved Easter Island?” Journal of Evolutionary Economics 2000, 10, pp. 489-505.

 

Dalton, Thomas, R. Morris Coats, and Leon Taylor, “Economics and the Easter Island Metaphor,” Rapa Nui Journal 2006, 20, pp. 97-110.

 

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. 345-354.

 

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. 1603-1606.

 

Pezzey, John, and John Anderies, “The effect of subsistence on collapse and institutional adaptation in population-resource 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 Canada, 1957, 14, pp. 669-81.

 

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, High-precision radiocarbon dating shows recent and rapid initial human colonization of East Polynesia,” Proceedings of the National Academy of Sciences 2011, 108, pp. 1815-1820.

 

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Mathematical Appendix

 

 

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 forest-soil 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 forest-soil complex (dS/dt) through time is the difference between natural growth G(S) and harvest H(S,L):

 

(1)                                                                                                                                     

 

where S = resource stock (forest-soil 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 forest-soil 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 Lh = bL is the quantity of labor employed in harvesting the renewable resource.  Substituting (2) and (3) into (1) gives the the forest-soil 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 forest-soil 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 (S0), equation (10) is substituted for equation (2).

 

(10)                                                                         

 

 

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