1 cm optimal resource control model & general continuous time optimal control model of a forest...
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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