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Some new equations presented at NCSU 2011-06-28

Peter Lohmander

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( ),...

0

max ( ( ),...) ( ) ( ) ( ) ( ) ( )x t

x t m t x t F t B t P t dt

0dm

dt ?

Subject to dynamic equations and constraints.

( ) 1m t m ?

In general, (.) Is strictly concave, with a maximumthat gives the ”ideal climate”.

3

( ),...

0

max ( ( ),...) ( ) ( ) ( ) ( ) ( )x t

x t m t x t F t B t P t dt

0dm

dt ?

( ) 1m t m ?

If we replace the nonlinear objective function by a linear objective funcion, this could lead to extreme variations and make sustainability impossible.

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_ ( )x Th

T

With “complete rotation forestry”:

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In case of continuous harvesting:

( ) ( )

( ) ( )

1

h t E t k

E t k h t

E k h

dE

dh

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In general:

__dF

k hdt

_

( ) (0)dF

F Fdt

7

_

( ) (0)F F h k

_

( )dF

d h

_ _

_ _

d x d Ffor T

d h d h

8

_

_

_

_

lim 0

d x

d h

d F

d h

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Focus on replacing coal by biomass! The carbon stored in the forest is almost irrelevant compared to the substitution effect in the long run.

_

_

_

_

lim 0

d x

d h

d F

d h

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Forestry with maximum harvest in relation to forestry with maximum carbon in the forest

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Derivation of a “carbon optimal” harvest rule using a simple functional form that can be generalized.

2( )x t a bt ct 2_ a bt ct

ht

_1h at b ct

12

_

2

_2

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0

2 0

d hat c

dt

d hat

dt

13

_

2

2

*

0d h

at cdt

at

c

at t

c

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*

*

_*

_*

_*

( )

( ) 2

( )

( )

( ) 2

a ax t a b c

c c

ax t a b

c

a ah t b c

cac

h t ac b ac

h t ac b

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Numerical example:

2( )x t a bt ct 2( ) 40 10 0.1x t t t

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First: Let us maximize the average harvest level based on rotation

forestry:

* at

c

* 40400 20

0.1t

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* * * 2

* 2

*

*

( ) ( )

( ) 40 10(20) (20)

( ) 40 200 0.1(400)

( ) 120

x t a bt c t

x t c

x t

x t

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_* *

*

_*

_*

_*

( )

40( ) 10 0.1(20)

20

( ) 2 10 2

( ) 6

ah t b ct

t

h t

h t

h t

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Second: Let us maximize the stock level of the forest:

2( )x t a bt ct 2( ) 40 10 0.1x t t t

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2

2

2 0

2 0

0 2

1050

2 0.2

dxb ct

dt

d xc

dtdx

b ctdt

bt

c

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( ) 40 10(50) 0.1(2500)

( ) 40 500 250

( ) 210

x t

x t

x t

_ 210( ) 4.2

50h t

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Optimal combination of carbon storage and timber production:- Joint production and rational adaption to different conditions

1 2( ) ( )Z w x w x

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23( ) 3

4x x x

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2

2

3( ) 3

44 1

( )3 9

x x x

x x x

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Numerical example:

1 2

2 21 2

( ) ( )

3 4 13

4 3 9

Z w x w x

Z w x x w x x

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Optimal forest management decision rule with two objectives:

1 2

2

1 22

3 4 23 0

2 3 9

6 20

4 9

dZw x w x

dx

d Zw w

dx

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* *1 1 2 2

*1 2 1 2

1 2*

1 2

0

3 4 23 0

2 3 94 3 2

33 2 9

43

33 22 9

dZ

dx

w w x w w x

w w w w x

w wx

w w

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1 2*

1 2

43

33 22 9

w wx

w w

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• 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?)

1 2

3 4 23 0

2 3 9

dZw x w x

dx

2* * *

1 2 2

3 4 23 0

2 3 9

dZ d Zd x dw x dw dx

dx dx

30

1 2*

1 2

43

33 22 9

w wx

w w

31

* 1 2 1 2

21

1 2

3 2 4 33 3

2 9 3 2

3 22 9

w w w wdx

dww w

* 2

21

1 2

*

21

43

3 22 9

0 0

wdx

dww w

dxfor w

dw

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* 1 2 1 2

22

1 2

* 1

22

1 2

*

12

4 3 2 4 23

3 2 9 3 9

3 22 9

43

3 22 9

0 0

w w w wdx

dww w

wdx

dww w

dxfor w

dw

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The End