Optimal rotation age (ORA)
• Dynamic optimization problem
• Long discussed in economic literature
• Shorter rotation
– benefit arrives earlier
– earlier replanting opportunity
– planting more frequent
– timber yield is lower
Forest scientist's ORA
• doesn't like cutting down trees …
• Maximizes sustained gross yield
Max[f (T) / T]
T
• Solution
f (Tg )  f '(Tg )  Tg
"Economic" forest scientist's
ORA
• takes into account planting cost
• maximizes sustained net yield
Max[f (T) / T  (W / P)  L / T]
T
• Solution
f (Tn )  (W / P)  L  f '(Tn )  Tn
Forest economists' ORA
• Maximize profit
• Economic Literature:
– Maximizing present discounted value
over one cycle (Von Thünen, Irving
Fisher)
– Maximizing internal rate of return
(Boulding)
– Maximizing present discounted value
over infinite cycles
Max NPV over 1 Period
Max[e
 r T
T
 f (T)  (W / P)  L]
FOC
(r)  e
 r T1
 f (T1 )  e
r  f '(T1 ) / f (T1 )
 r T1
 f '(T1 )  0
Max Internal Rate of Return
ri  r e rT f (T) (W / P)L 0
ri  T  ln  f (T)  P /(W  L) 
1
Max[T  ln  f (T)  P /(W  L) ]
1
T
FOC
f '(Ti ) / f (Ti )  T  ln  f (Ti )  P /(W  L) 
1
i
Max NPV over all Periods

Max[ P  f (T)  e
 rT
 W  L 1 e
Max[ P  f (T)  e
 rT
 W  L  / 1  e
T
T
 rT
 rT
 e
]
FOC
f '(T )  r  f (T )
 r   P  f (T )  e
 rT
 W  L  / 1  e
 rT


 rT 2

 ... ]
Optimal Rotation Age
Ti < T < T1 < Tg < Tn
ORA - Assumptions
• future prices, wages, interest rates are
known
• future technologies (yields, input
requirements) are known
• growth rate initially increasing later
decreasing (I.e. cubic growth function)
ORA - Example
f (t) = b*t^2 + a*t^3
a = -1/800
b = 0.2
W[age] = 16
L[abor] = 25
P[rice] = 20
r = 0.06 = 6%
Growth Function
Timber f(t)
700
600
500
400
300
f (t) = 0.2 t2 - 1/800 t3
200
100
20
40
60
80
100
Time (t)
120
140
ORA Example, Results
(see Mathematika output)
Maximization Criterion
Optimal Rotation Age
Gross sustainable yield
80.00
Net sustainable yield
81.21
Net present value of one
period
Net present value of all
periods
Internal rate of returns
29.56
28.62
26.94