Some new equations presented at
NCSU 2011-06-28
Peter Lohmander
1

max ( x(t ),...)   m(t )  x(t )  F (t )  B(t )  P(t )  dt
x ( t ),...
0
In general,
 (.)
Is strictly concave, with a maximum
that gives the ”ideal climate”.
Subject to dynamic equations and constraints.
dm
0
dt
?
m(t )  m  1
?
2

max  ( x(t ),...)   m(t )  x(t )  F (t )  B(t )  P(t )  dt
x ( t ),...
0
If we replace the nonlinear objective function
by a linear objective funcion, this could
lead to extreme variations and make sustainability impossible.
dm
0
dt
?
m(t )  m  1
?
3
With “complete rotation forestry”:
_
x(T )
h
T
4
In case of continuous harvesting:
h(t )  E (t )  k
E (t )  k  h(t )
E  k h
dE
 1
dh
5
In general:
_
_
dF





   k  h


 dt 
_
 dF 
F ( )  F (0)  

 dt 
6


F ( )  F (0)   h  k 


_
dF ( )
_

dh
_
_
dx
dF
_
dh

_
dh
for T  
7


d
x


_


dh

lim

0
_
  

d
F


_


d
h


_
8
 _
d x
 _
dh

lim
0
_
  

dF 
 _ 
 dh 
Focus on replacing coal by biomass!
The carbon stored in the forest is
almost irrelevant compared to the
substitution effect in the long run.
9
Forestry with maximum harvest in
relation to forestry with maximum
carbon in the forest
10
Derivation of a “carbon optimal” harvest rule using a
simple functional form that can be generalized.
x(t )  a  bt  ct
a  bt  ct
h
t
_
_
2
2
1
h  at  b  ct
11
_
dh
2
 at  c  0
dt
2
_
d h
3


2
at

0
2
dt
12


h
d
2


 0   at  c 

 dt


a 2
t
c
_
t t 
*
a
c
13
a
a
x(t )   a  b
c
c
c
*
a
x(t )  2a  b
c
*
a
a
h(t ) 
bc
c
a
c
_
*
_
h(t * )   ac  b  ac
_
h(t * )  2 ac  b
14
Numerical example:
x(t )  a  bt  ct
2
x(t )  40  10t  0.1t
2
15
First: Let us maximize the average
harvest level based on rotation
forestry:
a
t 
c
*
40
t 
 400  20
0.1
*
16
x(t )   a  bt  c (t )
*
*
* 2
x(t )  40  10(20)  c(20)
*
2
x(t )  40  200  0.1(400)
*
x(t )  120
*
17
a
*
h(t )  *  b  ct
t
_
40
*
h(t ) 
 10  0.1(20)
20
_
_
*
h(t )  2  10  2
_
*
h(t )  6
*
18
Second: Let us maximize the stock
level of the forest:
x(t )  a  bt  ct
2
x(t )  40  10t  0.1t
2
19
dx
 b  2ct  0
dt
2
d x
 2c  0
2
dt
 dx

  0    b  2ct 
 dt

b 10
t

 50
2c 0.2
20
x(t )  40  10(50)  0.1(2500)
x(t )  40  500  250
x(t )  210
_
210
h(t ) 
 4.2
50
21
Optimal combination of carbon
storage and timber production:
- Joint production and rational
adaption to different conditions
Z  w1( x)  w2( x)
22
3 2
 ( x)  3x  x
4
23
3 2
 ( x)  3x  x
4
4
1 2
( x)  x  x
3
9
24
Numerical example:
Z  w1 ( x)  w2  ( x)
3 2
1 2

4
Z  w1  3x  x   w2  x  x 
4 
9 

3
25
Optimal forest management decision
rule with two objectives:
dZ
3 

4 2 
 w1  3  x   w2   x   0
dx
2 

3 9 
2
d Z
6
2


w

w

0
1
2
2
dx
4
9
26
 dZ

 0 

 dx

3
4
2
*
*
3w1  w1 x  w2  w2 x  0
2
3
9
4
2  *
3
3w1  w2   w1  w2  x
3
9 
2
4
3w1  w2
*
3
x 
3
2
w1  w2
2
9
27
4
3w1  w2
*
3
x 
3
2
w1  w2
2
9
28
• How should forestry be adapted to
roundwood net price changes and carbon
storage subsidy changes?
• (OR: Why should you not manage all forest
stands in exacctly the same way?)
dZ
3 

4 2 
 w1  3  x   w2   x   0
dx
2 

3 9 
2
dZ
3
4
2
d
Z *

 

*
*
d
   3  x  dw1    x  dw2  2 dx  029
dx
 dx   2 
3 9 
4
3w1  w2
*
3
x 
3
2
w1  w2
2
9
30
2  
4 3
3
3  w1  w2    3w1  w2 
*
dx
2
9  
3 2


2
dw1
2 
3
 w1  w2 
9 
2
4
 w2
*
dx
3

2
dw1  3
2 
 w1  w2 
9 
2
*
dx
 0 for w2  0
dw1
31
dx*
dw2
dx*
dw2
43
2  
4 2
 w1  w2    3w1  w2 
32
9  
3 9

2
2 
3
 w1  w2 
9 
2
4
w1
3

2
2 
3
 w1  w2 
9 
2
*
dx
0
dw2
for
w1  0
32
The End
33