1

CM

Optimal Resource Control Model&

General continuous time optimal control model of a forest resource, comparative dynamics and CO2

storage consideration effects

Peter Lohmander

Presented in the seminar series of Dept. of Forest Economics,

Swedish University of Agricultural Sciences, Umea, Sweden, Thursday 2008-09-18, 1400 – 1500 HRS

Marginally updated 111215

2

First some

General optimal control theory

3

( )

0

max , , ( ),

( ) ( )

( , , ) (0)

T

u st

J F x u s ds S x T T

u s s

x f x u t x x

4

( )

0

( , ) max ( ), ( ), ( ),

( ) ( )

( ( ), ( ), ) (0)

T

u st

V x t F x s u s s ds S x T T

u s s

xx f x s u s s x x

s

5

( )

( , ) max ( ), ( ), ( ( ), ) ( )

( ) ( )

t t

u st

V x t F x s u s s ds V x t t t t t

u s s

6

( )

( , ) max ( ), ( ), ( ( ), ) ( )

( ) ( )

u tV x t F x t u t t t V x t t t t t

u t t

7

Taylor expansion:

( ( ), ) ( , ) ( , ) ( , ) ( )x tV x t t t t V x t V x t x V x t t t

8

( ( ), ) ( , ) ( , ) ( , , ) ( , ) ( )x tV x t t t t V x t V x t f x u t t V x t t t

( )

( , ) max , , ( , ) ( , ) ( , , ) ( , ) ( )

( ) ( )

x tu t

V x t F x u t t V x t V x t f x u t t V x t t t

u t t

9

( )

0 max , , ( , ) ( , , ) ( , ) ( )

( ) ( )

x tu t

F x u t V x t f x u t V x t t t

u t t

10

( )

( )0 max , , ( , ) ( , , ) ( , )

( ) ( )

x tu t

tF x u t V x t f x u t V x t

tu t t

11

0

( )lim 0t

t

t

12

( )

0 max , , ( , ) ( , , ) ( , )

( ) ( ) ; ( , ) ( , )

x tu t

F x u t V x t f x u t V x t

u t t V x T S x T

13

*( ) ( ( ), )xt V x t t

14

( , , , ) ( , , ) ( , ) ( , , )x xH H x u V t F x u t V x t f x u t

( , , , ) ( , , ) ( ) ( , , )H H x u t F x u t t f x u t

15

( )

0 max ( , , , ) ( , )

( ) ( )

x tu t

H x u V t V x t

u t t

16

( , )tV x t u( , )tV x t

( )0 max ( , , , ) ( , )

( ) ( )

x tu t

H x u V t V x t

u t t

is not a function of

Hence, we may exclude from the optimization.

17

( )x t ( )u tis optimized via

We assume an unconstrained local maximum.

18

0x txH V

x tx xtH V V

x xH V

19

The adjoint equation:

xH

20

Terminal boundary condition:

( )

,( ) ( ),x

x x T

S x TT S x T T

x

21

0, (0)

, ( ) ( ),X x

x H x x

H T S x T T

22

The maximum principle:Necessary conditions for

*uto be an optimal control:

* * * *0

* * *

* * *

( , , ) , (0)

( ( ), ( ), ( ), ) , ( ) ( ),

( ( ), ( ), ( ), ) ( ( ), ( ), ( ), ) ( ), 0,

X x

x f x u t x x

H x t u t t t T S x T T

H x t u t t t H x t u t t t u t t T

23

Now, a more specific

Optimal control model

24

2

1

21 2 1 2 3max

trt

t

J e f f t x k k t u k u dt

25

0 1 2x g g x g t u

1 1 2 2( ) ; ( )x t x x t x

26

21 2 1 2 3 0 1 2

rtH e f f t x k k t u k u g g x g t u

1 2 32 0rtHe k k t k u

u

1 2 1rtH

e f f t gx

27

Determination of the time derivative of via the adjoint equation:

H

x

1 2 1rte f f t g

1 1 2rtg e f f t

28

1 2 3 1 2 32 0 2rt rtHe k k t k u e k k t k u

u

1 1 2 3 1 22rt rtg e k k t k u e f f t

1 1 2 3 1 22rte g k k t k u f f t

1 1 1 1 2 2 1 32rte g k f g k f t g k u

29

Determination of the time derivative of via 0H

u

1 2 30 2rtHe k k t k u

u

30

1 2 3 2 32 2rt rtre k k t k u e k k u

1 2 3 2 32 2rte r k k t k u k k u

2 1 2 3 32 2rte k k r k rt k ru k u

31

The time derivative of as determined via 0H

u

must equal the time derivative of as determined via the adjoint equation:

2 1 2 3 3 1 1 1 1 2 2 1 32 2 2k k r k rt k ru k u g k f g k f t g k u

3 1 3 3 1 2 1 1 1 2 1 2 22 2 2k u g k k r u k r k g k f k r g k f t

3 3 1 1 1 2 1 2 1 22 2k u k r g u k r g k f k r g f t

32

Definition:

2 1r r g

Observation:This differential equation can be used to determine the optimal control function ( )u t.

2 1 2 2 1 2 2 23 3

1 1

2 2u r u k r k f k r f t

k k

33

Solution of the homogenous differential equation:

Let

2 0h hu r u

mt

hu Ae

2 0 ; 0mt mtm r Ae Ae

2m r2( ) r t

hu t Ae

34

Determination of the particular solution:

2 1 2 2 1 2 2 23 3

1 1

2 2p pu r u k r k f k r f tk k

Let 1 2pu z z t

2 2 1 2 1 2 2 1 2 2 23 3

1 1

2 2z r z z t k r k f k r f t

k k

2 1 2 2 2 1 2 2 1 2 2 23 3

1 1

2 2z z r z r t k r k f k r f t

k k

35

2 2 2 2 23

1

2z r k r f

k

22 2

3 2

1

2

fz k

k r

2 1 2 1 2 2 13

1

2z z r k r k f

k

22 1 2 1 2 2 1

3 2 3

1 1

2 2

fk z r k r k f

k r k

36

21 2 1 2 2 1 2

3 3 2

1 1

2 2

fz r k r k f k

k k r

1 2

1 123 2 2

1

2

f fz k

k r r

( ) ( ) ( )h pu t u t u t

37

Conclusion:The optimal control function is:

21 2( ) r tu t Ae z z t

38

Determination of the optimal stock path equation:

0 1 2x g g x g t u

39

The optimal control function is introduced in the differential equation of the stock path equation:

20 1 2 1 2

r tx g g x g t Ae z z t

40

As a result, the differential equation of the optimal stock path equation is:

21 0 1 2 2

r tx g x g z g z t Ae

41

Solution of the homogenous differential equation:

1 0h hx g x

Let nt

hx Be

1 0 ; 0nt ntn g Be Be

1n g

1( ) g thx t Be

42

Determination of the particular solution:

21 0 1 2 2

r tp px g x g z g z t Ae

Let: 20 1 2

r tpx c c t c e

21 2 2

r tpx c r c e

43

2 2 21 2 2 1 0 1 2 0 1 2 2

r t r t r tc r c e g c c t c e g z g z t Ae

2 21 0 1 1 1 2 2 1 0 1 2 2

r t r tc c g c g t c r g e g z g z t Ae

1 1 2 2c g g z

2 21

1

z gc

g

44

1 0 1 0 1c c g g z

2 20 1 0 1

1

z gc g g z

g

2 20 1 0 1

1

z gc g g z

g

2 20 1 0

1 1

1 z gc z g

g g

45

2 2 1c r g A

2 1 2 21 2

Ac A g r c

g r

( ) ( ) ( )h px t x t x t

46

Conclusion:The optimal stock path equation is:

1 20 1 2( ) g t r tx t Be c c t c e

47

Determination of ,A B via the initial and terminal conditions

1 1 2 2( ) ; ( )x t x x t x

2

10 1

1 2

( )r t

g t ex t Be c c t A

g r

2 1

1 1

2 2

1 2

1 0 1 11 2

2 0 1 21 2

r tg t

r tg t

ex Be c c t A

g r

ex Be c c t A

g r

48

2 1

1 1

2 2

1 2

1 0 1 11 2

2 0 1 21 2

r tg t

r tg t

eA e B x c c t

g r

eA e B x c c t

g r

49

2 1

1 1

2 2

1 2

1 2 1 0 1 1

2 0 1 2

1 2

r tg t

r tg t

ee

g r x c c tA

x c c tBee

g r

50

2 1

1 1

2 1 2 2

1 2 1 1

2 2

1 2

1 2

1 2 1 2

1 2

r tg t

r t r tg t g t

r tg t

ee

g r e eD e e

g r g ree

g r

51

2 1 1 2 2 2 1 1

1 2

r t g t r t g te e e eD

g r

52

2 1 1 2 2 2 1 1( ) ( )

1 2

r t g t r t g te eD

g r

53

1 1

1 2

1 0 1 1

2 0 1 2

g t

g t

x c c t e

x c c t eA

D

54

1 2 1 11 0 1 1 2 0 1 2

g t g tx c c t e x c c t eA

D

55

2 1

2 2

1 0 1 11 2

2 0 1 21 2

r t

r t

ex c c t

g r

ex c c t

g rB

D

56

2 1 2 2

2 0 1 2 1 0 1 11 2 1 2

r t r te ex c c t x c c t

g r g rB

D

57

2 1 2 22 0 1 2 1 0 1 1

1 2

r t r te x c c t e x c c tB

g r D

58

Determination of ( )t via 0H

u

and ( )u t

1 2 3 1 2 32 0 2rt rtHe k k t k u e k k t k u

u

21 2( ) r tu t Ae z z t

21 2 3 1 2( ) 2 r trtt e k k t k Ae z z t

21 3 1 2 3 2 3 2 1( ) 2 2 2 ;r trtt e k k z k k z t k Ae r r g

59

Conclusion:The optimal adjoint variable path equation is:

11 3 1 2 3 2 3( ) 2 2 2 g trtt e k k z k k z t k Ae

60

Determination of the optimal objective function value:

2

1

2

1 2 1 2 3( ) ( ) ( )t

rt

t

J e f f t x t k k t u t k u t dt

61

2

1

2

1 2 1 2 3( ) ( ) ( )t

rt

t

J e f f t x t k k t u t k u t dt

2

1

( )t

t

J K t dt

21 2 1 2 3( ) ( ) ( )rtK e f f t x t k k t u t k u t

62

2 10 1

1 2

( ) r t g tAx t e Be c c t

g r

1 20 1

1 2

( ) ;g t r t Ax t c c t Be De D

g r

21 2( ) r tu t z z t Ae

63

1 2 2 22

1 2 0 1 1 2 1 2 3 1 2g t r t r t r trtK e f f t c c t Be De k k t z z t Ae k z z t Ae

21 2 1 2 3( ) ( ) ( )rtK e f f t x t k k t u t k u t

64

1 2 1 2

2 2

2

2

2 2 2

21 0 1 1 1 1 2 0 2 1 2 2

21 1 1 2 1 2 1 2 2 2

23 1 3 1 2 3 1

2 23 1 2 3 2 3 2

223 1 3 2 3

( )

( )

g t r t g t r trt

r t r t

r t

r t

r t r t r t

Ke f c f c t f Be f De f c t f c t f Bte f Dte

k z k z t k Ae k z t k z t k Ate

k z k z z t k z Ae

k z z t k z t k z Ate

k z Ae k z Ate k A e

65

1 2

2 1 2

21 0 1 1 3 1 1 1 2 0 1 2 2 1 3 1 2

2 22 1 2 2 3 2 1 1 1 3 1

223 2 2 2 3 2

( ) 2

( ) 2

2

rt

g t r t

r t g t r t

Ke f c k z k z f c f c k z k z k z z t

f c k z k z t f B e f D k A k z A e

k A e f B te f D k A k z A te

66

1 2

2 1 2

0 1

22 3 4

25 6 7

rt

g t r t

r t g t r t

Ke w w t

w t w e w e

w e w te w te

67

1 2

2 1 2

0 1

( ) ( )22 3 4

(2 ) ( ) ( )5 6 7

rt rt

g r t r r trt

r r t g r t r r t

K w e w te

w t e w e w e

w e w te w te

68

0 0

0 1 2

3 1 2

0 1

22 3 4

5 6 7

s t s t

s t s t s t

s t s t s t

K w e w te

w t e w e w e

w e w te w te

69

Now, we determine ( )I t , the antiderivative of K.

IK

t

. Below, we exclude the constant of integration.

70

0

0

1 2

0

3

1 2

0 1 20 0 0

2 20 0

2 3 430 1 2

5 6 72 23 1 1 2 2

1

( )

( ) 2 2

( )

1 1

( ) ( )

s ts t

s t s ts t

s ts t s t

e tI w w e

s s s

s t s t e ew e w w

s s s

e t tw w e w e

s s s s s

2 1( ) ( )J I t I t

71

Code section for calculation of J (in JavaScript):

var r2 = r-g1;var z1 = 1/(2*k3)*(f1/r2 + f2/r2/r2 - k1);var z2 = 1/(2*k3)*(f2/r2 - k2);var c0 = 1/g1*(z1+(z2-g2)/g1 - g0);var c1 = (z2-g2)/g1;

72

var sysdet = Math.exp(r2*t1)/(g1-r2)*Math.exp(g1*t2) - Math.exp(r2*t2)/(g1-r2)*Math.exp(g1*t1);

var A = ( (x1-c0-c1*t1)*Math.exp(g1*t2) -(x2-c0-c1*t2)*Math.exp(g1*t1) )-/sysdet;

var B = ( Math.exp(r2*t1)/(g1-r2)*(x2-c0-c1*t2) – Math.exp(r2*t2)/(g1-r2)*(x1-c0-c1*t1))/sysdet;

73

var D = A/(g1-r2);var w0 = f1*c0 + k1*z1 + k3*z1*z1;var w1 = f1*c1 + f2*c0 + k1*z2 + k2*z1 + 2*k3*z1*z2;var w2 = f2*c1 + k2*z2 + k3*z2*z2;var w3 = f1*B;var w4 = f1*D + k1*A + 2*k3*z1*A;var w5 = k3*A*A;var w6 = f2*B;var w7 = f2*D + k2*A + 2*k3*z2*A;

74

var s0 = - r;var s1 = g1 - r;var s2 = r2 - r;var s3 = 2*r2 - r;

75

var AntidK2 = w0*(1/s0*Math.exp(s0*t2)) + w1*(Math.exp(s0*t2)*(t2/s0 - 1/(s0*s0))) + w2*(Math.exp(s0*t2)*((s0*s0*t2*t2-2*s0*t2+2) /(s0*s0*s0))) + w3*(1/s1*Math.exp(s1*t2)) + w4*(1/s2*Math.exp(s2*t2)) + w5*(1/s3*Math.exp(s3*t2)) + w6*(Math.exp(s1*t2)*(t2/s1 - 1/(s1*s1))) + w7*(Math.exp(s2*t2)*(t2/s2 - 1/(s2*s2))) ;

76

var AntidK1 = w0*(1/s0*Math.exp(s0*t1)) + w1*(Math.exp(s0*t1)*(t1/s0 - 1/(s0*s0))) + w2*(Math.exp(s0*t1)*((s0*s0*t1*t1-2*s0*t1 + 2) /(s0*s0*s0))) + w3*(1/s1*Math.exp(s1*t1)) + w4*(1/s2*Math.exp(s2*t1)) + w5*(1/s3*Math.exp(s3*t1)) + w6*(Math.exp(s1*t1)*(t1/s1 - 1/(s1*s1))) + w7*(Math.exp(s2*t1)*(t1/s2 - 1/(s2*s2))) ;

var Integral = AntidK2 - AntidK1;

var J = Integral;

77

General observations of optimal solutions:

First we consider the following objective function. Note that the stock level does not influence the objective function directly in this first version of the problem.

2

1

( ( ))

( )

trt

t

J e R u t dt

x F x u

78

The Hamiltonian function is:

( ) ( ( ) )rtH e R u F x u

rtu

He R

u

79

When we have an interior optimum, the first order derivative of the objective function with respect to the control variable is 0. As a result, we find that the present value of the marginal profit,

( ( ))rtue R u t

is equal to the marginal resource value at the same point in time,

( )t

0 rtu

He R

u

.

80

The adjoint equation says that

H

x t

Since x

HF

x

, we know that xF

81

Now, we make the following definition:

, consists of many different parts.

0

( ) ( )x

F x G z dzThe resource, with total stock

x

Different parts of the resource have different relative growth. A particular part of the resource,

z , has relative growth ( )G z .

82

We may define the relative growth of that particular unit as

( )

( )

y z

y z

( ( )y z is the stock level of that particular unit and

y

is the growth.)

83

When we integrate,

0

( ) ( )x

F x G z dzwe arrange the different particular resource units in such a way that

( ) 0 ,xxF x x

84

( )( )

( )

y zG z

y z

( )( ) ( )

( )x

y xF x G x

y x

( )

( )

y x

y x

( )

( )

y x

y x

85

Observation:The relative decrease of the marginal resource value over time is equal to the relative growth of the marginal resource unit.

( )

( )

y x

y x

86

Example with explanation from forestry:

If the marginal cubic metre grows by 5% per year (and will give 0.05 new cubic metres next year) then the value of the marginal cubic metre this year is 5% higher than the value of the marginal cubic metre next year.

87

The value of an already existing cubic metre must be higher than the value of a cubic metre in the future year since the cubic metre that we already have may give us more than one cubic metre in the future year, thanks to the growth.

88

The relative marginal resource value decrease is higher

in case the relative marginal growth

( )

( )

y x

y x

is higher.

( )

( )

y x

y x

89

The relationship between the speed of change of the relative marginal

resource value and direct valuation of the stock level

90

Here, we add

2

1

( )t

rt

t

e W x dtto the objective function.

91

As a result, we get:

2

1

( ( )) ( )

( )

trt

t

J e R u t W x dt

x F x u

92

The Hamiltonian function becomes:

( ) ( ) ( ( ) )rtH e R u W x F x u

93

These derivatives are obtained:

rtu

He R

u

rtx x

He W F

x

94

The adjoint equation gives a modified result:

H

x t

rtx xe W F

rtx

x

e WF

95

rtxe W y

y

rtxe Wy

y

96

Observation:

>0 implies that

becomes more negative if xW increases.

97

Explanation and connection to forestry:

0xW means that the stock gives an “instant contribution” to the objective function.

One such case is if the forest owner continuously gets paid for the amount of stored CO2 in the forest. In such a case, one cubic metre that exists now (and that will not be instantly harvested),

will give contributions in this period and in the following period.

98

One cubic metre that will exist in the next period, but not already in this period, will not “be able” to give the contribution during this period. As a result, the fact that we get “instant contributions”, directly from the stock, means that the relative marginal value of the stock falls faster over time.

99

Software for the optimal control model

http://www.lohmander.com/CM/CM.htm

100

101

102

2

1

21 2 1 2 3max

trt

t

J e f f t x k k t u k u dt

0 1 2x g g x g t u

1 1 2 2( ) ; ( )x t x x t x

103

J 1216344

t x u Lambda

0 3100 143 139

5 2920 138 132

10 2761 132 126

15 2628 124 120

20 2533 115 114

25 2485 104 108

30 2500 90 103

104

Optimal strategy and dynamic effects of

the rate of interest

105

Optimal Objective Function Values

0

200

400

600

800

1000

1200

1400

1600

1800

J_3% J_5% J_7%

Alternative

Ob

ject

ive

Val

ue

(Rel

evan

t C

urr

ency

)

106

Optimal Stock Path

0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35

Time (Years)

Op

tim

al S

tock

(M

m3s

k)

x_3%

x_5%

x_7%

107

Optimal Control Path

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30 35

Time (Years)

Op

tim

al C

on

tro

l (M

m3s

k)

u_3%

u_5%

u_7%

108

Optimal Shadow Price Path

0

50

100

150

200

250

0 5 10 15 20 25 30 35

Time (Years)

Sh

ado

w P

rice

(R

elev

ant

Cu

rren

cy)

SP_3%

SP_5%

SP_7%

109

Comparisions with alternatives that are not optimal:

30

.05 61

0

1000 3*106 106 1.123229489·10tN e dt

30

.05 52

0

1000 3*86 86 9.914723644·10tN e dt

110

Optimal strategy and dynamic effects of

the slope of the demand function

111

112

Optimal Objective Function Values

0

200

400

600

800

1000

1200

1400

1600

J_-4 J_-3 J_-2

Alternative

Ob

ject

ive

Val

ue

(Rel

evan

t C

urr

ency

)

113

Optimal Stock Path

0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35

Time (Years)

Op

tim

al S

tock

(M

m3s

k)

x_-4

x_-3

x_-2

114

Optimal Control Path

0

20

40

60

80

100

120

140

160

180

200

0 5 10 15 20 25 30 35

Time (Years)

Op

tim

al C

on

tro

l (M

m3s

k)

u_-4

u_-3

u_-2

115

Optimal Shadow Price Path

0

50

100

150

200

250

300

0 5 10 15 20 25 30 35

Time (Years)

Sh

ado

w P

rice

(R

elev

ant

Cu

rren

cy)

SP_-4

SP_-3

SP_-2

116

Comparisions with two alternatives:

30

.05 63

0

1000 2*106 106 1.297807679·10tN e dt

30

.05 64

0

1000 2*(181 5* ) (181 5* ) 1.397224592·10tN e t t dt

117

Optimal strategy and dynamic effects of

the terminal condition

118

119

Optimal Objective Function Values

1100

1120

1140

1160

1180

1200

1220

1240

J_2500 J_2800 J_3100

Alternative

Ob

ject

ive

Val

ue

(Rel

evan

t C

urr

ency

)

120

Optimal Stock Path

0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35

Time (Years)

Op

tim

al S

tock

(M

m3s

k)

x_3100

x_2500

x_2800

121

Optimal Control Path

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30 35

Time (Years)

Op

tim

al C

on

tro

l (M

m3s

k)

u_3100

u_2500

u_2800

122

Optimal Shadow Price Path

0

50

100

150

200

250

0 5 10 15 20 25 30 35

Time (Years)

Sh

ado

w P

rice

(R

elev

ant

Cu

rren

cy)

SP_3100

SP_2500

SP_2800

123

Optimal strategy and dynamic effects of

continuous stock level valuation

124

125

Optimal Objective Function Values

0

200

400

600

800

1000

1200

1400

1600

1800

J_f1=0 J_f1=5 J_f1=10

Alternative

Ob

ject

ive

Val

ue

(Rel

evan

t C

urr

ency

)

126

Optimal Stock Path

0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35

Time (Years)

Op

tim

al S

tock

(M

m3s

k)

x_f1=5

x_f1=0

x_f1=10

127

Optimal Control Path

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30 35

Time (Years)

Op

tim

al C

on

tro

l (M

m3s

k)

u_f1=5

u_f1=0

u_f1=10

128

Optimal Shadow Price Path

0

50

100

150

200

250

300

0 5 10 15 20 25 30 35

Time (Years)

Sh

ado

w P

rice

(R

elev

ant

Cu

rren

cy)

SP_f1=5

SP_f1=0

SP_f1=10

129

Optimal strategy and dynamic effects of

continuous stock level valuation in combination with variations of

the the slope of the demand function

130

131

Optimal Objective Function Values

0

500

1000

1500

2000

2500

J_f_-3 J_f_-2.3 J_f_-1.6

Alternative

Ob

ject

ive

Val

ue

(Rel

evan

t C

urr

ency

)

132

Optimal Stock Path

0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35

Time (Years)

Op

tim

al S

tock

(M

m3s

k)

x_f_-2.3

x_f_-3

x_f_-1.6

133

Optimal Control Path

0

20

40

60

80

100

120

140

160

180

200

0 5 10 15 20 25 30 35

Time (Years)

Op

tim

al C

on

tro

l (M

m3s

k)

u_f_-2.3

u_f_-3

u_f_-1.6

134

Optimal Shadow Price Path

0

50

100

150

200

250

300

350

400

450

500

0 5 10 15 20 25 30 35

Time (Years)

Sh

ado

w P

rice

(R

elev

ant

Cu

rren

cy)

SP_f_-2.3

SP_f_-3

SP_f_-1.6

135

Optimal strategy and dynamic effects of

the growth function

136

137

Optimal Objective Function Values

1200

1205

1210

1215

1220

1225

1230

1235

1240

1245

1250

J_-.01 J_.001 J_.01

Alternative

Ob

ject

ive

Val

ue

(Rel

evan

t C

urr

ency

)

138

Optimal Stock Path

0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35

Time (Years)

Op

tim

al S

tock

(M

m3s

k)

x_.001

x_.01

x_-.01

139

Optimal Control Path

0

20

40

60

80

100

120

140

160

180

0 5 10 15 20 25 30 35

Time (Years)

Op

tim

al C

on

tro

l (M

m3s

k)

u_.001

u_.01

u_-.01

140

Optimal Shadow Price Path

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30 35

Time (Years)

Sh

ado

w P

rice

(R

elev

ant

Cu

rren

cy)

SP_.001

SP_.01

SP_-.01