### Natural Resource Economics

```Department of Economics
Roma Tre University
Natural Resource Economics
Instructor:
Prof. Luca Salvatici
[email protected]
Natural Resource Economics - a.a.2014/15
Lecture 20
Nonrenewable resources:
–
Extraction costs
–
Excel exercises
Natural Resource Economics - a.a.2014/15
Extraction costs: optimal control model
T
max
y ,T , A
Π [y ] =
∫
( py − ( ay + cy
2
)) e − rt dt
0
.
x = − y , x ( 0 ) = S , x (T ) = A ≥ 0
1.
2.
3.
4.
Assume A = 0: write the current value Hamiltonian
Derive the FOCs
What is going to be y*(T)?
Draw the marginal and average cost curves
Economia delle risorse naturali - a.a. 201011
3
FOCs
∗ ∗
μ
2 μ 0
0
0 ∗ 0
∗
∗
∗
∗ ∗ =0
0 0 Natural Resource Economics - a.a.2014/15
FOCs
Natural Resource Economics - a.a.2014/15
What is going to be y*(T)?
Natural Resource Economics - a.a.2011/12
6
Marginal and average costs
Economia delle risorse naturali - a.a. 201011
7
Optimal control
Economia delle risorse naturali - a.a. 201011
8
Optimal control: graph
y≤4
Economia delle risorse naturali - a.a. 201011
9
Discrete time version
2-2sol.xls
Natural Resource Economics - a.a.2014/15
EXCEL example: extraction costs
• Mineral stock: X(0) = 1 and ten years from now the mine is going to be taken
by the Government
• Find the optimal extraction path (Y*(t), t = 0, 1, …, 9) to maximize the
present value of the following profit function:
Parameters:
Mineral Price p = 1 = prezzo del minerale
c = 0.5 = average extraction cost
δ = 0.05 = interest rate
π
t
=



p
−
cY
X
(t ) 
(t ) 
Y
(t )
Natural Resource Economics - a.a.2014/15
11
EXCEL example: solution
 0,5Y (t ) 

Max∑ 0,95 1 −
Y (t )

X (t ) 
t =0


subX t +1 = X t − Yt
9
t
What if there was a salvage value?
compito1-Es3.xls
Natural Resource Economics - a.a.2014/15
```