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1. Easter Island Dynamics
Here, you will use Excel to simulate what happened on various islands like Easter Island, which differ in some of their ecological parameters, when 50 human pioneers land on their shores and settle them. You will simulate what happens to S (a renewable, but exhaustible natural resource) and L (the population). The change in S will be given by G-H where G is natural growth and H is the harvest.
G is given by the logistic growth function: G = rs (1-S/K)
where r is the ‘unlimited’ growth rate of the resource, and K is the carrying capacity of the island. Our time step will be a decade (ten years).
H will be given by a harvest function: H=aLS
We will take the value of α to be 0.000004.
For both simulations, we will take the initial value of population (L0) to be 50, and the initial value of the resource stock (S0) to be at carrying capacity.
Population will grow according to the Malthusian growth rate –p+fH/L
where p is the death rate without the natural resource. Take p = 0.1 and f = 4.
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a) Start off with simulating what would happen on an Island in which the carrying capacity is K=5,000. The growth rate is r=0.04 (4% growth per decade).
Use L and S to calculate G and H. Use G and H to calculate the change in S, which you can add to the current value of S to find S in the next time period. Use H/L to calculate the population growth rate, which you can then use to calculate L in the next time period.
Where does human and tree population head in the long run (simulate at least 200 time periods)?
b) Now change the carrying capacity to be 12,000. Is there any change in long-term outcomes?
c) What happens if you vary the initial population size on both types of islands (a and b)? are long-term outcomes affected? Explain.
d) Discuss the difference in your answers to questions (b) and (c) in terms of the concept of the steady state.
e) Now change the growth rate r to be much faster, r=0.35 (35% per decade), with carrying capacity at 12,000. What change do you see in the dynamics, if any?
2. Technological progress on Easter Island: In the Malthusian agricultural land based economy, we saw that technological improvement in farming (captured by the parameter T), led to long term increases in human population but not in income per capita.
In the Easter Island model, which parameter represents the technological “fishing” capacity of humans? If this parameter is increased (a one-time increase), how does that impact the long term level of income (harvest) per capita? How does it impact the “fish” population? Finally, for extra credit, how does it impact the human population in the long term?
3. Substitution: Find one historical story in which an economic substitute was found for an important resource (either non-renewable or renewable but scarce) for the production economy of that time. Be prepared to briefly describe in class (3 minutes), what the scarce resource was, why it was important, and how the substitute was found. Use as many numbers as possible.
4. Simulating the two-sector model
Simulate the two-sector model in Excel.
L = LM + LF
F = T LFβ A (1 – β ) (Food production)
σ = LF / L (Definition of σ)
M = γ T LM (Urban-based manufactures)
m = M / L = γ T ( 1 – σ ) (Per capita manufactures)
Tt+1- Tt= ψ m (Productivity growth)
n = GR(L) = ρ σ (Population growth)
In Excel, include columns for L, F, σ, T (technology), M, m. In all equations, refer to fixed cells (using $ signs) which have the parameter values for f* (take a value of 2), ψ (0.3), ρ (0.05), β (0.3), γ (0.9), Z (100) and the starting values for population and technology (T) which you will vary.
Simulate the model for some 100 time periods, and see what possible long term outcomes for σ are possible, as you vary the initial levels of population and technology (each one in its turn, while keeping the other one fixed, initial population at 200, and initial technology at 2).
What are the possible outcome values for sigma in the long run? Do you see tipping point behaviour, in the sense that there are threshold values for initial levels of population and of technology (theta) that change the outcome?
Can you try to analytically solve the model and find algebraically the value of the tipping point for initial levels.
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