Optimal continuous natural resource extraction
with increasing risk in prices and stock dynamics
Professor Dr Peter Lohmander
http://www.Lohmander.com
[email protected]
BIT's
5th Annual World Congress of
Bioenergy 2015
(WCBE 2015)
Theme: “Boosting the development of green bioenergy"
September 24-26, 2015
Venue: Xi'an, China
1
Lohmander, P., Optimal continuous natural resource extraction with increasing risk in prices
and stock dynamics, WCBE 2015
Abstract
• Bioenergy is based on the dynamic utilization of natural resources. The dynamic supply
of such energy resources is of fundamental importance to the success of bioenergy. This
analysis concerns the optimal present extraction of a natural resouce and how this is
affected by different kinds of future risk. The objective function is the expected present
value of all operations over time. The analysis is performed via general function multi
dimensional analyical optimization and comparative dynamics analysis in discrete time.
First, the price and/or cost risk in the next period increases. The direction of optimal
adjustment of the present extraction level is found to be a function of the third order
derivatives of the profit functions in later time periods with respect to the extraction
levels. In the second section, the optimal present extraction level is studied under the
influence of increasing risk in the growth process. Again, the direction of optimal
adjustment of the present extraction is found to be a function of the third order derivatives
of the profit functions in later time periods with respect to the extraction levels. In the
third section, the resource contains different species, growing together. Furthermore, the
total harvest in each period is constrained. The directions of adjustments of the present
extraction levels are functions of the third order derivatives, if the price or cost risk of
one of the species increases.
2
Case:
We control a natural resource.
We want to maximize the expected present
value of all activites over time.
Questions:
• What is the optimal present extraction level?
• How is the optimal present extraction level
affected by different kinds of future risk?
3
Source:
http://www.nasdaq.com/
2015-09-13
4
Source:
http://www.nasdaq.com/
2015-09-13
5
Source:
http://www.nasdaq.com/
2015-09-13
6
Probability
density
7
• In the following analyses, we study the solutions to maximization
problems. The objective functions are the total expected present values. In
particular, we study how the optimal decisions at different points in time
are affected by stochastic variables, increasing risk and optimal adaptive
future decisions.
• In all derivations in this document, continuously differentiable functions
are assumed. In the optimizations and comparative statics calculations,
local optima and small moves of these optima under the influence of
parameter changes, are studied. For these reasons, derivatives of order
four and higher are not considered. Derivatives of order three and lower
can however not be neglected. We should be aware that functions that are
not everywhere continuously differentiable may be relevant in several
cases.
8
The profit functions used in the analyses are functions of the revenue
and cost functions.
The continuous profit functions may be interpreted as approximations
of profit functions with penalty functions representing capacity
constraints.
The analysis will show that the third order derivatives of these
functions determine the optimal present extraction response to
increasing future risk.
9
Optimization in multi period problems
The multi period problem
We maximize Z , the total expected present value. Rt (.) and Ct (.) denote discounted revenue and cost functions in period t .
Now, we introduce a three period problem. In period 1, x1 , the extraction level, is determined before the stochastic event in period 2 takes place. In period
2, the outcome of the stochastic event is observed before the extraction level is period 2, x2 , is determined. With probability  , the discounted price in
period 2 increases with h in relation to what was earlier assumed according to the revenue function. With probability (1   ) , the discounted price in
period 2 decreases by h . In the first case, we select x2  x21 and in the second case, we select x2  x22 . The resource available for extration in period 3, x3 ,
is of course affected by the decisions in period 2. If x2  x21 , then x3  x31 . If x2  x22 , then x3  x32 .
Z
R1 ( x1 )  C1 ( x1 ) 
   R2 ( x21 )  hx21  C2 ( x21 )   (1   )  R2 ( x22 )  hx22  C2 ( x22 ) 
   R3 ( x31 )  C3 ( x31 ) 
 (1   )  R3 ( x32 )  C3 ( x32 ) 
We may also study the effects of risk in the resource volume process, growth risk, with the same basic structure. Then, g serves as the risk parameter. With
some probability, the volume increases by g and with some probability, the volume decreases by g , in relation to what was earlier expected.
10
Let us study a special case:
max Z
subject to
 x1   x21  x31  A  g
 x1   x22  x32  A  g
We note that we have five decision variables. In period 1, we only have one decision, the optimal extratction level, x1 . In period 2, we have two alternative
optimal extraction levels, x21 or x22 , depending on the outcome of the stochastic event. In period 3, the optimal extration level x31 or x32 , is conditional on
all earlier extraction levels and outcomes.
We may instantly solve for x31 and x32 .
x31  A   x1   x21  g
x32  A   x1   x22  g
11
 t (.)  Rt (.)  Ct (.)
Z
 1 ( x1 ) 
   2 ( x21 )  hx21   (1   )  2 ( x22 )  hx22 
   3 ( x31 )
Z
 1 ( x1 ) 
 (1   )  3 ( x32 )
   2 ( x21 )  hx21 
 (1   )  2 ( x22 )  hx22 
   3 ( A   x1   x21  g )  (1   )  3 ( A   x1   x22  g )
12
Three free decision variables and three first
order optimum conditions
Optimization:
We have three first order optimum conditions since two of the five decision variables can be determined via the constraints and the other decision variables.
The first order optimum conditions are:
dZ
dZ
dZ
 0,
 0 and
 0 . These may be expressed as:
dx1
dx21
dx22
13
dZ

dx1
d 3 ( A   x1   x21  g )
d 3 ( A   x1   x22  g )
d 1 ( x1 )
 
 (1   )
0
dx1
dx3
dx3
 d 2 ( x21 )

d 3 ( A   x1   x21  g )
dZ
 
 h   
0
dx21
dx3
 dx2

 d 2 ( x22 )

d 3 ( A   x1   x22  g )
dZ
 (1   ) 
 h   (1   ) 
0
dx22
dx3
 dx2

14
Let us differentiate the first order optimum conditions with
respect to the decision variables and the risk parameters:
:
2
d 2Z *
d 2Z
d
Z
*
*
dx

dx

dx
1
21
22
dx12
dx1dx21
dx1dx22
d 2Z

dg  0
dx1dg
2
2
2
2
d 2Z
d
Z
d
Z
d
Z
d
Z
*
*
*
dx1 
dx21 
dx22 
dh 
dg  0
2
dx21dx1
dx21
dx21dx22
dx21dh
dx21dg
2
2
2
2
d 2Z
d
Z
d
Z
d
Z
d
Z
*
*
*
dx1 
dx21 
dx22 
dh 
dg  0
2
dx22 dx1
dx22 dx21
dx22
dx22 dh
dx22 dg
15
The effects of increasing future price risk:
Now, we will investigate how the optimal values of the decision variables change if h increases.
d 2Z
 0
dx21dh
d 2Z
 (1   )  0
dx22 dh

1
2


 0 
 dx1*  

1

*
 D   dx21     dh 
2
 dx22*  


1 
  dh 
 2 
16
2
2
2
 d 2 1





d

x
d
 3  x32  




d



3
31
2
3
 E 






2
2 
2
2

dx3
dx3
 dx3  
 dx1

 2

 2



2
2
2
2
  d  3  x31  
 1 d  2  x21   d  3  x31  



0
 D   



2
2
2

2
dx
2
dx
2
dx
3
2
3






   d 2 3  x32  
 1 d 2 2  x22   2 d 2 3  x32   
0

 



2
2
2
dx3
2
dx3
  2

 2 dx2
 
17
0
  d 2 3  x31  


2
dx3
 2

  d 2 3  x32  


2
dx3
 2

2
2
2

d

x
d
 3  x31  
1
1

2  21 




2
2
2  2 dx2
2
dx3

*
1
dx

dh
1

2
0
 1 d 2 2  x22   2 d 2 3  x32  



2
2
2
dx
2
dx
2
3


0
D
   d 2 3  x31    1   1 d 2 2  x22   2 d 2 3  x32   

 
  

2
2
2
*
dx3
dx2
2
dx3
dx1
1   2
 2  2




dh
D    d 2 3  x32    1 d 2 2  x21   2 d 2 3  x31    1  


  
2
2
2
  2
dx
2
dx
2
dx
3
2
3

 2  
 
18
dx1* 

dh 8 D
2
2
2
2
 d 2 3  x31   d 2 2  x22 



d

x
d

x
d

x
d
 3  x31   






3
32
3
32
2
21
2
2







2
2
2
2
2
2
dx3
dx3
 dx3
 dx2
  dx3
 dx2
 
Simplification gives:
2
2
2
2

dx
  d  3  x31   d  2  x22    d  3  x32   d  2  x21   






2
2
2
2
dh 8 D  dx3
 dx2
  dx3
 dx2
 
*
1
A unique maximum is assumed.
D 0
19
Observation:
  d 2 2  x21   d 2 3  x32    d 2 2  x22   d 2 3  x31   
 dx1* 
sgn 

 

 
  sgn  
2
2
2
2
 dh 
 dx3
  dx2
 dx3

  dx2
We assume decreasing marginal profits in all periods.
d 2 2 .
dx2
2
d 2 3 .
dx3
2
0
0
20
The following results follow from optimization:
 h  0   x21  x22    x31  x32  
 d 3 2
0 

3
 dx2
 d 3 2
0 

3
 dx2
 d 3 2
0 

3
 dx2
  *
d 3 3
dx1

0


3
0

dx3

dh


3
*

d 3
  dx1
 0   
0
3
dx3
   dh
*


3
dx
1

d 3

0


 0
3
dh


dx3

21
 d 3 2
0 

3
 dx2
 d 3 2
0 

3
 dx2
 d 3 2
0 

3
 dx2
 d 3 2
0 

3
 dx2
 d 3 2
0 

3
 dx2
 d 3 2
0 

3
 dx2
  *
d 3 3
dx1
 0 
3
0

dx3

dh

   dx1*
d 3 3
 0   
0
3
dx3
   dh
*


3
dx
1

d 3
0



0

dh


dx33

  *
d 3 3
dx1
 0 
3
0
dx3
   dh
 *

3
   dx1
d 3

0
0
  
3
dx3
   dh
*


3
dx
1

d 3
0



0

dh


dx33

22
The results may also be summarized this way:

 d 3 2   d 3 3 
d 3 3
*

dx
0  

 0  
1
3
3 
3 
0
dx3
dx
dx

 2   3 

dh
  *
3
3
 d 2

d 3
  dx1
0 
 0
0
 

3
3
dx3
 dx2

  dh
*


dx
3
3
1
 d 3 2

 d 3 2   d  3 
d 3
0


0 
0  

 0 

dh
3
3
3 
3 


dx3
 dx2   dx3 
 dx2

 d 3 2
0 

3
 dx2
23
2  dK 
d


2
d 3K
dL  d 



0
2
2
2
dLdP
dP
dP
 dK 
E    E 
 decreases
dL


if the risk in P increases.
The expected future marginal
resource value decreases from
increasing price risk and we should
increase present extraction.
24
2  dK 
d


2
d 3K
dL  d 



0
2
2
2
dLdP
dP
dP
 dK 
E    E 
 increases
dL


if the risk in P increases.
The expected future marginal
resource value increases from
increasing price risk and we should
decrease present extraction.
25
Some results of increasing risk in the price
process:
• If the future risk in the price process increases, we should increase
the present extraction level in case the third order derivatives of
profit with respect to volume are strictly negative.
• If the future risk in the price process increases, we should not change
the present extraction level in case the third order derivatives of
profit with respect to volume are zero.
• If the future risk in the price process increases, we should decrease
the present extraction level in case the third order derivatives of
profit with respect to volume are strictly positive.
26
The effects of increasing future risk in the volume process:
Now, we will investigate how the optimal values of the decision variables change if g increases.
We recall these first order derivatives:
dZ

dx1
d 3 ( A   x1   x21  g )
d  3 ( A   x1   x22  g )
d 1 ( x1 )
 
 (1   )
0
dx1
dx3
dx3
 d 2 ( x21 )

d 3 ( A   x1   x21  g )
dZ
 
 h   
0
dx21
dx3
 dx2

 d 2 ( x22 )

d 3 ( A   x1   x22  g )
dZ
 (1   ) 
 h   (1   ) 
0
dx22
dx3
 dx2

27
The details of these derivations can be found in the
mathematical appendix.
28
2  dK 
d


2
d 3K
dL  d 



0
2
2
2
dLdV
dV
dV
 dK 
E    E 
 decreases
 dL 
if the risk inV increases.
The expected future marginal
resource value decreases from
increasing risk in the volume
process (growth) and we should
increase present extraction.
29
2  dK 
d


2
d 3K
dL  d 



0
2
2
2
dLdV
dV
dV
 dK 
E    E 
 increases
dL


if the risk inV increases.
The expected future marginal
resource value increases from
increasing risk in the volume
process (growth) and we should
decrease present extraction.
30
Some results of increasing risk in the volume
process (growth process):
• If the future risk in the volume process increases, we should increase
the present extraction level in case the third order derivatives of
profit with respect to volume are strictly negative.
• If the future risk in the volume process increases, we should not
change the present extraction level in case the third order derivatives
of profit with respect to volume are zero.
• If the future risk in the volume process increases, we should decrease
the present extraction level in case the third order derivatives of
profit with respect to volume are strictly positive.
31
The mixed species case:
• A complete dynamic analysis of optimal natural resource
management with several species should include decisions
concerning total stock levels and interspecies competition.
• In the following analysis, we study a case with two species, where the
growth of a species is assumed to be a function of the total stock level
and the stock level of the individual species.
• The total stock level has however already indirectly been determined
via binding constraints on total harvesting in periods 1 and 2.
• We start with a deterministic version of the problem and later move
to the stochastic counterpart.
32
 is the total present value. xit denotes harves volume of species i in period t .
 it ( xit ) is the present value of harvesting species i in period t .
Each species has an intertemporal harvest volume constraint.
H t denotes the total harvest volume in period t.
These total harvest volumes are constrained in periods 1 and 2,
because of harvest capacity constraints, constraints in logistics or other constraints,
maybe reflecting the desire to control the total stock level.
33
Period 1
Period 2
Period 3
max    11 ( x11 )   21 ( x21 )   12 ( x12 )   22 ( x22 )   13 ( x13 )   23 ( x23 )
s.t.
 x11   x12  x13  C1
 x21   x22  x23  C2
x11  x21  H1
x12  x22  H 2
34
The details of these derivations can be found in the
mathematical appendix.
35
Some multi species results:
With multiple species and total harvest volume constraints:
Case 1:
If the future price risk of one species, A, increases, we should now harvest less of this species (A) and
more of the other species, in case the third order derivative of the profit function of species A with
respect to harvest volume is greater than the corresponding derivative of the other species.
Case 3:
If the future price risk of one species, A, increases, we should not change the present harvest of this
species (A) and not change the harvest of the other species, in case the third order derivative of the
profit function of species A with respect to harvest volume is equal to the corresponding derivative of
the other species.
Case 5:
If the future price risk of one species, A, increases, we should now harvest more of this species (A)
and less of the other species, in case the third order derivative of the profit function of species A with
respect to harvest volume is less than the corresponding derivative of the other species.
36
CONCLUSIONS:
• The properties of the revenue and cost functions, including capacity
constraints with penalty functions, determine the optimal present
response to risk.
• Conclusive and general results have been derived and reported for
the following cases:
• Increasing risk in the price and cost functions.
• Increasing risk in the dynamics of the physical processes.
37
Optimal continuous natural resource extraction
with increasing risk in prices and stock dynamics
Professor Dr Peter Lohmander
http://www.Lohmander.com
[email protected]
BIT's
5th Annual World Congress of
Bioenergy 2015
(WCBE 2015)
Theme: “Boosting the development of green bioenergy"
September 24-26, 2015
Venue: Xi'an, China
38
Mathematical Appendix
presented at:
The 8th International Conference of
Iranian Operations Research Society
Department of Mathematics
Ferdowsi University of Mashhad, Mashhad, Iran.
www.or8.um.ac.ir
21-22 May 2015
39
OPTIMAL PRESENT RESOURCE EXTRACTION
UNDER THE INFLUENCE OF FUTURE RISK
Professor Dr Peter Lohmander
SLU, Sweden, http://www.Lohmander.com
[email protected]
The 8th International Conference of
Iranian Operations Research Society
Department of Mathematics
Ferdowsi University of Mashhad, Mashhad, Iran.
www.or8.um.ac.ir
21-22 May 2015
40
41
Contents:
1.
2.
3.
4.
5.
6.
Introduction via one dimensional optimization in dynamic
problems, comparative statics analysis, probabilities, increasing
risk and the importance of third order derivatives.
Explicit multi period analysis, stationarity and corner solutions.
Multi period problems and model structure with sequential
adaptive decisions and risk.
Optimal decisions under future price risk.
Optimal decisions under future risk in the volume process
(growth risk).
Optimal decisions under future price risk with mixed species.
42
Contents:
1.
2.
3.
4.
5.
6.
Introduction via one dimensional optimization in dynamic
problems, comparative statics analysis, probabilities, increasing
risk and the importance of third order derivatives.
Explicit multi period analysis, stationarity and corner solutions.
Multi period problems and model structure with sequential
adaptive decisions and risk.
Optimal decisions under future price risk.
Optimal decisions under future risk in the volume process
(growth risk).
Optimal decisions under future price risk with mixed species.
43
• In the following analyses, we study the solutions to maximization
problems. The objective functions are the total expected present values. In
particular, we study how the optimal decisions at different points in time
are affected by stochastic variables, increasing risk and optimal adaptive
future decisions.
• In all derivations in this document, continuously differentiable functions
are assumed. In the optimizations and comparative statics calculations,
local optima and small moves of these optima under the influence of
parameter changes, are studied. For these reasons, derivatives of order
four and higher are not considered. Derivatives of order three and lower
can however not be neglected. We should be aware that functions that are
not everywhere continuously differentiable may be relevant in several
cases.
44
Introduction with a simplified problem
Objective function:
y( x)  P  f ( x)  g ( x, h)   (1  P)  f ( x)  g ( x, 0) 
Definitions:
x
Present extraction level
y ( x)
Expected present value (expected discounted value) of present and future extraction
f ( x)
Economic value of present extraction
h
Risk parameter
P
Probability that the expected present value of future extraction is affected by the risk parameter h
g ( x, h )
Expected present value of future extraction (in case the expected present value of future extraction is affected by risk parameter h )
g ( x,0)
Expected present value of future extraction (in case the expected present value of future extraction is not affected by risk parameter h )
45
The objective function can be rewritten as:
y( x)  f ( x)  Pg ( x, h)  (1  P) g ( x,0)
Let us maximize
y ( x) with respect to x . The first order optimum condition is:
dy df ( x)
dg ( x, h)
dg ( x, 0)

P
 (1  P)
0
dx
dx
dx
dx
Optimal values are marked by stars.
x* is assumed to exist and be unique.
d2y
0
dx 2
How is the optimal value of
x , x* , affected by the value of the risk parameter
Differentiation of the first order optimum condition with respect to
x* and
h , ceteres paribus?
h gives:
2
d2y
 dy  d y
d    2 dx* 
dh  0
dxdh
 dx  dx
d2y *
d2y
dx  
dh
dx 2
dxdh
 d2y 


dxdh 
dx*


dh
 d2y 
 2
 dx 
46
 d2y
 
 dx* 
 d2y 

0

sgn

sgn
 2
 




 dx
 
 dh 
 dxdh  
d2y
d 2 g ( x, h )
P
dxdh
dxdh
P0
 dx* 
 d 2 g ( x, h) 
sgn 
  sgn 

 dh 
 dxdh 
Result:

 0 if

dx* 
  0 if
dh 

  0 if

d 2 g ( x, h )
0
dxdh
d 2 g ( x, h )
0
dxdh
d 2 g ( x, h )
0
dxdh
How can this be interpreted?
Our objective function was initially defined as:
y( x)  f ( x)  Pg ( x, h)  (1  P) g ( x,0)
Let us define marginally redefine the optimization problem:
y ( x)  f ( x)  PK ( L( x), h)  (1  P) K ( L( x),0)
47
K ( L( x), h)  g ( x, h)
Here, K replaces g and we have the function L ( x ) that represents the resource available for future extraction as a function of the present extraction level.
With growth and/or without growth, we usually find that:
dL
0
dx
d2 f
0
dx 2
d 2K
0
dL2
The first order optimum condition then becomes:
dy df  dK ( L( x), h)
dK ( L( x), 0)  dL
 P
 (1  P)
0

dx dx 
dL
dL
 dx
A special case is when there is no growth of the resource. Then,
dL
 1 .
dx
Then, we get:
df  dK ( L( x), h)
dK ( L( x), 0) 
P
 (1  P)

dx 
dL
dL

This means that the expected marginal present value of the resource used for extraction should be the same in the present period and in the future. Then, if
d 2 K ( L, h )
 0 , and the value of the future risk parameter h increases, this makes the expected marginal present value of future extraction, dK ( L( x), h)
dLdh
dL
increase.
48
2
d y
df
 0 , the only way to make the first order optimum condition hold, is to reduce the present extraction level, x* .
Then,
also has to increase. Since
2
dx
dx
d2 f
 0 , df increases if x is reduced. Then, the expected marginal present values of present and future extrations can again be set equal.
Since
2
dx
dx

 0 if

*
dx 
  0 if
dh 

  0 if

d 2 K ( L, h )
0
dLdh
d 2 K ( L, h )
0
dLdh
d 2 K ( L, h )
0
dLdh
This illustrates the earlier found result:

 0 if

*
dx 
  0 if
dh 

  0 if

d 2 g ( x, h )
0
dxdh
d 2 g ( x, h )
0
dxdh
d 2 g ( x, h )
0
dxdh
49
Probabilities and outcomes:
Now, let us more explicitly define increasing risk and derive the conditional effects on the optimal value of x. In the next period, the outcome of a stochastic
variable, s , will be known. This stochastic variable can represent different things, such as growth, price, environmental state etc.. More explicit cases will be
defined in the later part of this analysis.
The original objective function was:
y( x)  P  f ( x)  g ( x, h)   (1  P)  f ( x)  g ( x, 0) 
Now, we get this objective function:
I
y ( x)  f ( x)    ( si ) ( x, si ,  (h, si ))
i 1
y ( x) is the sum of the expected present values of present and future extraction, before future stochastic outcomes have been
observed. y ( x) is a function of the extraction level x in the first period, period 1.
The objective function
50
Increasing risk:
Definitions:
 ( si )
Probability that the stochastic variable takes the value si in period 2.
(The decision concerning x is taken in period 1, before si is known.)
su  sv
Two particular values the stochastic variable s . su  sv
h
During a ”mean preserving spread”, su decreases by h and sv increases by h . h  0 .
_
_

   ( su )   ( sv )  0
E (ds )
Expected change of s as a result of a mean preserving spread. E (ds)   ( su )h   ( sv )h  0
 (h, si )
The change of si as a result of a mean preserving spread.
 ( x, si ,  (h, si ))
Expected present value of future extraction when the value si is known. (Of course, x and h are also known.)
51
Probability
density
52
i
 (h, si )
1
.
u
.
v
.
I
0
.
-h
.
+h
.
0
Remark:
An almost identical analysis could be made with even more general mean preserving spreads, such that: E (ds)   ( su )hu   (sv )hv  0 . Then, su would
be reduced by hu and sv would be increased by hv .
 (su )hu   (sv )hv and
hu  ( sv )

hv  ( su )
In such a case, we would not need the constraint  ( su )   ( sv ) . The notation would however become more confusing and the results of interest to this
analysis would be the same as with the present analysis.
Let us define the function  ( x, s ) , as the expected present value of future (from period 2) extraction as a function of
adjusted by the increasing risk in the probability distribution via the mean preserving spread.
x and of the stochastic variable s ,
su2  su  h
sv2  sv  h
53
 ( x, si )   ( x, si ,  (h, si ))  si i u  i v
 ( x, su  h)   ( x, su ,  (h, su ))
 ( x, sv  h)   ( x, sv , (h, sv ))
 ( x, su )   ( x, su ,  ( h, su ))
2
 ( x, sv )   ( x, sv ,  (h, sv ))
2
First order optimum condition:
d ( x, si ,  (h, si ))
dy df
    ( si )
0
dx dx i
dx
A unique interior maximum is assumed:
d 2 ( x, si ,  (h, si ))
d2y d2 f

  ( si )
0
dx 2 dx 2
dx 2
i
d 2 ( x, si ,  (h, si ))
d2y
  ( si )
dxdh i
dxdh
d 2 ( x, su ,  (h, su ))
d 2 ( x, sv ,  (h, sv ))
d2y
  ( su )
  ( sv )
dxdh
dxdh
dxdh
54
 ( x, su ,  (h, su ))   ( x, su ( su , h))   ( x, su  h)
2
 ( x, sv ,  (h, sv ))   ( x, sv ( sv , h))   ( x, sv  h)
2
2
2
d 2 y _  d  ( x, su2 ( su , h)) d  ( x, sv2 ( sv , h)) 




dxdh
dxdh
dxdh


d2y
Can the sign of
be determined ? (We remember that h  0 .)
dxdh
2
2
d 2 y _  d  ( x, su2 ( su , h)) d  ( x, sv2 ( sv , h)) 




dxdh
dxdh
dxdh


2
2
d 2 y _  d  ( x, su2 ) dsu2 d  ( x, sv2 ) dsv2


 dxds
dxdh
dh
dxds
dh




2

d 2 ( x, sv2 )
d 2 y _  d  ( x, su2 )

1 
1 


 dxds

dxdh
dxds


2
2
d 2 y _  d  ( x, su2 ) d  ( x, sv2 ) 
 



dxdh
dxds
dxds


 (su  sv )  (h  0)    su
2
 sv2

55
 d2y 
 d 3 ( x, s) 
sgn 
  sgn 

2
 dxdh 
 dxds 
 d2y 


*
dxdh
dx
 2 
dh
d y
 2
 dx 
The sign of this third order derivative
determines the optimal direction of
change of our present extraction level
under the influence of increasing
risk in the future.
 dx* 
 d 3 
sgn 
  sgn 
2 
dh
dxds




56
How can these results be interpreted?
2  d 
d


d 3
 dx 

dxds 2
ds 2
d
is the derivative of the expected present value of future (from period 2) extraction as a function of x with respect to the present
dx
 d 
d2 

3
d
d
dx 

extraction level. If
>0, then
is a strictly convex function of the stochastic variable. Then, Jensen’s inequality tells us that the expected

2
2
dx
dxds
ds
d
d 3
value of
increases if the risk of the stochastic variable increases. Hence, if the risk increases and
 0 , it is rational that x* increases.
2
dx
dxds
We note that
d 3
d 3
*
Furthermore, if the risk increases and
 0 , x decreases. If the risk increases and
 0 , x* remains unchanged.
2
2
dxds
dxds
57

d 3 ( x, s)  
()  0 means that the marginal value of the resource used for present extraction increases (is unchanged) (decreases) in relation to the
dxds 2  

expected marginal present value of the resource used for future extraction, in case the future risk increases.
Then, it is obvious that the present extraction should increase (be unchanged)(decrease) as a result of increasing risk in the future.
d 3 ( x, s)
d 3 ( x, s)
Obviously,
is of central importance to optimal extraction under risk. In the next sections, we will investigate how
is affected by multi
dxds 2
dxds 2
period settings and dynamic properties such as stationarity in the stochastic processes of relevance to the problem. Different constraints such as extraction
volume constraints may also affect the results, in particular in multi species problems.
In order to discover the true and relevant effects of future risk on the optimal present decisions, it is necessary to let the future decisions be optimized
conditional on the outcomes of stochastic events that will be observed before the future decisions are taken. The lowest number of periods that a resource
extraction optimization problem must contain in order to discover, capture and analyze these effects is three.
For this reason, the rest of this analysis is based on three period versions of the problems. With more periods than three, the essential problem properties
and results are the same but the results are more difficult to discover because of the large numbers of variables and equations. Earlier studies of related
multi period problems have been made with stochastic dynamic programming and arbitrary numbers of periods. Please consult Lohmander (1987) and
Lohmander (1988) for more details.
58
Contents:
1.
2.
3.
4.
5.
6.
Introduction via one dimensional optimization in dynamic
problems, comparative statics analysis, probabilities, increasing
risk and the importance of third order derivatives.
Explicit multi period analysis, stationarity and corner solutions.
Multi period problems and model structure with sequential
adaptive decisions and risk.
Optimal decisions under future price risk.
Optimal decisions under future risk in the volume process
(growth risk).
Optimal decisions under future price risk with mixed species.
59
Marginal
resource value
In period t
Expected marginal
resource value
In period t+1
60
Marginal
resource value
In period t
61
Marginal
resource value
In period t
62
Probability
density
63
Contents:
1.
2.
3.
4.
5.
6.
Introduction via one dimensional optimization in dynamic
problems, comparative statics analysis, probabilities, increasing
risk and the importance of third order derivatives.
Explicit multi period analysis, stationarity and corner solutions.
Multi period problems and model structure with sequential
adaptive decisions and risk.
Optimal decisions under future price risk.
Optimal decisions under future risk in the volume process
(growth risk).
Optimal decisions under future price risk with mixed species.
64
Optimization in multi period problems
The multi period problem
We maximize Z , the total expected present value. Rt (.) and Ct (.) denote discounted revenue and cost functions in period t .
Now, we introduce a three period problem. In period 1, x1 , the extraction level, is determined before the stochastic event in period 2 takes place. In period
2, the outcome of the stochastic event is observed before the extraction level is period 2, x2 , is determined. With probability  , the discounted price in
period 2 increases with h in relation to what was earlier assumed according to the revenue function. With probability (1   ) , the discounted price in
period 2 decreases by h . In the first case, we select x2  x21 and in the second case, we select x2  x22 . The resource available for extration in period 3, x3 ,
is of course affected by the decisions in period 2. If x2  x21 , then x3  x31 . If x2  x22 , then x3  x32 .
Z
R1 ( x1 )  C1 ( x1 ) 
   R2 ( x21 )  hx21  C2 ( x21 )   (1   )  R2 ( x22 )  hx22  C2 ( x22 ) 
   R3 ( x31 )  C3 ( x31 ) 
One index corrected
150606
 (1   )  R3 ( x32 )  C3 ( x32 ) 
We may also study the effects of risk in the resource volume process, growth risk, with the same basic structure. Then, g serves as the risk parameter. With
some probability, the volume increases by g and with some probability, the volume decreases by g , in relation to what was earlier expected.
65
Let us study a special case:
max Z
subject to
 x1   x21  x31  A  g
 x1   x22  x32  A  g
We note that we have five decision variables. In period 1, we only have one decision, the optimal extratction level, x1 . In period 2, we have two alternative
optimal extraction levels, x21 or x22 , depending on the outcome of the stochastic event. In period 3, the optimal extration level x31 or x32 , is conditional on
all earlier extraction levels and outcomes.
We may instantly solve for x31 and x32 .
x31  A   x1   x21  g
x32  A   x1   x22  g
66
 t (.)  Rt (.)  Ct (.)
Z
 1 ( x1 ) 
   2 ( x21 )  hx21   (1   )  2 ( x22 )  hx22 
   3 ( x31 )
Z
 1 ( x1 ) 
 (1   )  3 ( x32 )
   2 ( x21 )  hx21 
 (1   )  2 ( x22 )  hx22 
   3 ( A   x1   x21  g )  (1   )  3 ( A   x1   x22  g )
67
Three free decision variables and three first
order optimum conditions
Optimization:
We have three first order optimum conditions since two of the five decision variables can be determined via the constraints and the other decision variables.
The first order optimum conditions are:
dZ
dZ
dZ
 0,
 0 and
 0 . These may be expressed as:
dx1
dx21
dx22
68
dZ

dx1
d 3 ( A   x1   x21  g )
d 3 ( A   x1   x22  g )
d 1 ( x1 )
 
 (1   )
0
dx1
dx3
dx3
 d 2 ( x21 )

d 3 ( A   x1   x21  g )
dZ
 
 h   
0
dx21
dx3
 dx2

 d 2 ( x22 )

d 3 ( A   x1   x22  g )
dZ
 (1   ) 
 h   (1   ) 
0
dx22
dx3
 dx2

69
Let us differentiate the first order optimum conditions with
respect to the decision variables and the risk parameters:
:
2
d 2Z *
d 2Z
d
Z
*
*
dx

dx

dx
1
21
22
dx12
dx1dx21
dx1dx22
d 2Z

dg  0
dx1dg
2
2
2
2
d 2Z
d
Z
d
Z
d
Z
d
Z
*
*
*
dx1 
dx21 
dx22 
dh 
dg  0
2
dx21dx1
dx21
dx21dx22
dx21dh
dx21dg
2
2
2
2
d 2Z
d
Z
d
Z
d
Z
d
Z
*
*
*
dx1 
dx21 
dx22 
dh 
dg  0
2
dx22 dx1
dx22 dx21
dx22
dx22 dh
dx22 dg
70
Contents:
1.
2.
3.
4.
5.
6.
Introduction via one dimensional optimization in dynamic
problems, comparative statics analysis, probabilities, increasing
risk and the importance of third order derivatives.
Explicit multi period analysis, stationarity and corner solutions.
Multi period problems and model structure with sequential
adaptive decisions and risk.
Optimal decisions under future price risk.
Optimal decisions under future risk in the volume process
(growth risk).
Optimal decisions under future price risk with mixed species.
71
The effects of increasing future price risk:
Now, we will investigate how the optimal values of the decision variables change if h increases.
d 2Z
 0
dx21dh
d 2Z
 (1   )  0
dx22 dh

1
2


 0 
 dx1*  

1

*
 D   dx21     dh 
2
 dx22*  


1 
  dh 
 2 
72
2
2
2
 d 2 1





d

x
d
 3  x32  




d



3
31
2
3
 E 






2
2 
2
2

dx3
dx3
 dx3  
 dx1

 2

 2



2
2
2
2
  d  3  x31  
 1 d  2  x21   d  3  x31  



0
 D   



2
2
2

2
dx
2
dx
2
dx
3
2
3






   d 2 3  x32  
 1 d 2 2  x22   2 d 2 3  x32   
0

 



2
2
2
dx3
2
dx3
  2

 2 dx2
 
73
0
  d 2 3  x31  


2
dx3
 2

  d 2 3  x32  


2
dx3
 2

2
2
2

d

x
d
 3  x31  
1
1

2  21 




2
2
2  2 dx2
2
dx3

*
1
dx

dh
1

2
0
 1 d 2 2  x22   2 d 2 3  x32  



2
2
2
dx
2
dx
2
3


0
D
   d 2 3  x31    1   1 d 2 2  x22   2 d 2 3  x32   

 
  

2
2
2
*
dx3
dx2
2
dx3
dx1
1   2
 2  2




dh
D    d 2 3  x32    1 d 2 2  x21   2 d 2 3  x31    1  


  
2
2
2
  2
dx
2
dx
2
dx
3
2
3

 2  
 
74
dx1* 

dh 8 D
2
2
2
2
 d 2 3  x31   d 2 2  x22 



d

x
d

x
d

x
d
 3  x31   






3
32
3
32
2
21
2
2







2
2
2
2
2
2
dx3
dx3
 dx3
 dx2
  dx3
 dx2
 
Simplification gives:
2
2
2
2

dx
  d  3  x31   d  2  x22    d  3  x32   d  2  x21   






2
2
2
2
dh 8 D  dx3
 dx2
  dx3
 dx2
 
*
1
A unique maximum is assumed.
D 0
75
Observation:
  d 2 2  x21   d 2 3  x32    d 2 2  x22   d 2 3  x31   
 dx1* 
sgn 

 

 
  sgn  
2
2
2
2
 dh 
 dx3
  dx2
 dx3

  dx2
We assume decreasing marginal profits in all periods.
d 2 2 .
dx2
2
d 2 3 .
dx3
2
0
0
76
The following results follow from optimization:
 h  0   x21  x22    x31  x32  
 d 3 2
0 

3
 dx2
 d 3 2
0 

3
 dx2
 d 3 2
0 

3
 dx2
  *
d 3 3
dx1

0


3
0

dx3

dh


3
*

d 3
  dx1
 0   
0
3
dx3
   dh
*


3
dx
1

d 3

0


 0
3
dh


dx3

77
 d 3 2
0 

3
 dx2
 d 3 2
0 

3
 dx2
 d 3 2
0 

3
 dx2
 d 3 2
0 

3
 dx2
 d 3 2
0 

3
 dx2
 d 3 2
0 

3
 dx2
  *
d 3 3
dx1
 0 
3
0

dx3

dh

   dx1*
d 3 3
 0   
0
3
dx3
   dh
*


3
dx
1

d 3
0



0

dh


dx33

  *
d 3 3
dx1
 0 
3
0
dx3
   dh
 *

3
   dx1
d 3

0
0
  
3
dx3
   dh
*


3
dx
1

d 3
0



0

dh


dx33

78
The results may also be summarized this way:

 d 3 2   d 3 3 
d 3 3
*

dx
0  

 0  
1
3
3 
3 
0
dx3
dx
dx

 2   3 

dh
  *
3
3
 d 2

d 3
  dx1
0 
 0
0
 

3
3
dx3
 dx2

  dh
*


dx
3
3
1
 d 3 2

 d 3 2   d  3 
d 3
0


0 
0  

 0 

dh
3
3
3 
3 


dx3
 dx2   dx3 
 dx2

 d 3 2
0 

3
 dx2
79
2  dK 
d


2
d 3K
dL  d 



0
2
2
2
dLdP
dP
dP
 dK 
E    E 
 decreases
dL


if the risk in P increases.
The expected future marginal
resource value decreases from
increasing price risk and we should
increase present extraction.
80
2  dK 
d


2
d 3K
dL  d 



0
2
2
2
dLdP
dP
dP
 dK 
E    E 
 increases
dL


if the risk in P increases.
The expected future marginal
resource value increases from
increasing price risk and we should
decrease present extraction.
81
Some results of increasing risk in the price
process:
• If the future risk in the price process increases, we should increase
the present extraction level in case the third order derivatives of
profit with respect to volume are strictly negative.
• If the future risk in the price process increases, we should not change
the present extraction level in case the third order derivatives of
profit with respect to volume are zero.
• If the future risk in the price process increases, we should decrease
the present extraction level in case the third order derivatives of
profit with respect to volume are strictly positive.
82
Contents:
1.
2.
3.
4.
5.
6.
Introduction via one dimensional optimization in dynamic
problems, comparative statics analysis, probabilities, increasing
risk and the importance of third order derivatives.
Explicit multi period analysis, stationarity and corner solutions.
Multi period problems and model structure with sequential
adaptive decisions and risk.
Optimal decisions under future price risk.
Optimal decisions under future risk in the volume process
(growth risk).
Optimal decisions under future price risk with mixed species.
83
The effects of increasing future risk in the volume process:
Now, we will investigate how the optimal values of the decision variables change if g increases.
We recall these first order derivatives:
dZ

dx1
d 3 ( A   x1   x21  g )
d  3 ( A   x1   x22  g )
d 1 ( x1 )
 
 (1   )
0
dx1
dx3
dx3
 d 2 ( x21 )

d 3 ( A   x1   x21  g )
dZ
 
 h   
0
dx21
dx3
 dx2

 d 2 ( x22 )

d 3 ( A   x1   x22  g )
dZ
 (1   ) 
 h   (1   ) 
0
dx22
dx3
 dx2

84
d 2Z

dx1dg
d 2 3 ( A   x1   x21  g )
d 2 3 ( A   x1   x22  g )
 
 (1   )
2
dx3
dx32
d 2 3 ( A   x1   x21  g )
d 2Z
  
dx21dg
dx32
d 2 3 ( A   x1   x22  g )
d 2Z
  (1   ) 
dx22 dg
dx32
85
With more simple notation, we get:
d 2Z

dx1dg
d 2 3 ( x31 )
d 2 3 ( x32 )
 
 (1   )
dx32
dx32
d 2 3 ( x31 )
d 2Z
  
dx21dg
dx32
d 2 3 ( x32 )
d 2Z
  (1   ) 
dx22 dg
dx32
We have already differentiated the first order optimum conditions with respect to the decision variables and the risk parameters:
d 2Z *
d 2Z
d 2Z
*
dx1 
dx21 
dx22*
2
dx1
dx1dx21
dx1dx22
d 2Z

dg  0
dx1dg
d 2Z
d 2Z
d 2Z
d 2Z
d 2Z
*
*
*
dx1 
dx21 
dx22 
dh 
dg  0
dx21dx1
dx212
dx21dx22
dx21dh
dx21dg
d 2Z
d 2Z
d 2Z
d 2Z
d 2Z
*
*
*
dx1 
dx21 
dx22 
dh 
dg  0
2
dx22 dx1
dx22 dx21
dx22
dx22 dh
dx22 dg
86
Now, we will investigate how the optimal values of the decision variables change if g increases. ( dh  0 .)
2
2
2
2
d Z *
d Z
d Z
d Z
*
*
dx1 
dx21 
dx22  
dg
2
dx1
dx1dx21
dx1dx22
dx1dg
2
2
2
2
d Z
d Z
d Z
d Z
*
*
*
dx1 
dx21 
dx22  
dg
2
dx21dx1
dx21
dx21dx22
dx21dg
2
2
2
2
d Z
d Z
d Z
d Z
*
*
*
dx1 
dx21 
dx22  
dg
2
dx22 dx1
dx22 dx21
dx22
dx22 dg
87
2
2
2
 d 2 1





d

x
d
 3  x32  




d



3
31
2
3


E







2
2 
2
2

dx
dx
2
dx
2
dx
3
3
 3 
 1







2
2
2
2




d

x
d

x
d

x







1

3
31
2
21
3
31


0
 D    



2
2
2

2
dx3
2 dx2
2
dx3






2
2
2
   d  3  x32  
 1 d  2  x22   2 d  3  x32   
0

 



2
2
2
dx3
2
dx3

 2 dx2
 
  2

 

 dx1*  
 D   dx21*   

 dx22*  






1
2
d 2 3 ( x31 )
d 2 3 ( x32 )
 (1   )
2
dx3
dx32

d 2 3 ( x31 ) 
 
 dg
2
dx3



d 2 3 ( x32 ) 
 (1   ) 
 dg
2
dx3


 
 dg 
 







88
*
1
dx

dg
  d 2 3 ( x31 )  d 2 3 ( x32 ) 



2
2
2
dx
2
dx
3
3


  d 2 3  x31  


2
2
dx
3


  d 2 3  x32  


2
2
dx
3


  d 2 3 ( x31 ) 


2
2
dx
3


 1 d 2 2  x21   2 d 2 3  x31  



2
2
2
dx
2
dx
2
3


0
0
 1 d 2 2  x22   2 d 2 3  x32  



2
2
2
dx
2
dx
2
3


  d 2 3 ( x32 ) 


2
2
dx
3


D
89
Let us simplify notation:
d 2 2 (.)
U (.) 
dx2 2
d 2 3 (.)
W (.) 
dx32
 W ( x31 )  W ( x32 ) 
dx
1
 
dg  8 
*
1
 W ( x31 ) 
  W ( x32 ) 
W ( x31 ) 
U ( x
2
)


W ( x31 ) 
21
0
W ( x32 ) 
0
U ( x
22
)   2W ( x32 ) 
D
90
  W ( x31 )  W ( x32 )  U ( x21 )   2W ( x31 ) U ( x22 )   2W ( x32 )  


dx1*
1 
2


  W ( x31 )  W ( x31 )  U ( x22 )   W ( x32 ) 

dg 8 D 
   W ( x32 )  U ( x21 )   2W ( x31 )    W ( x32 ) 



Now, we simplify notation even further:
u j  U ( xij )
w j  W ( xij )
   w1  w2   u1   2 w1  u2   2 w2  


dx1*
1 


  w1   w1   u2   2 w2 

dg 8 D 
   w2   u1   2 w1 )    w2 ) 



91
    w1  w2   u1   2 w1  u2   2 w2    w1   w1   u2   2 w2    w2   u1   2 w1 )    w2 ) 
We once again simplify notation to the following expression (where all variables appear in the same order as before and all indices are removed):
 = a(w - x)(u + bbw)(s + bbx) - (abw)(bw)(s + bbx) + (abx)(u + bbw)(bx)
This expression can instantly be simplified to:
 = a(suw - x(b2 w(s - u) + su))
This can be rearranged to:
 = a(su(w-x) - x(b2 w(s - u)))
 = a  su(w-x) + b 2 wx(u-s) 
Now, we slowly move back to our original notation:
92
 =   u1u 2 (w1 -w 2 ) +  2 w1w 2 (u1 -u 2 ) 
 =   U(x 21 )U(x 22 )  W(x 31 )-W(x 32 ) 


 =  



+
 2 W(x 31 )W(x 32 )  U(x 21 )-U(x 22 ) 
d 2 2 (x 21 ) d 2 2 (x 22 )  d 2 3 (x 31 ) d 2 3 (x 32 ) 



dx2 2
dx2 2  dx32
dx32 
d 2 3 (x 31 ) d 2 3 (x 32 )  d 2 2 (x 21 ) d 2 2 (x 22 ) 
+



2
2
2
2
dx3
dx3
dx
dx

2
2

2








Observations:
dx1*


dg 8 D
We already know that D  0 .
93


 d 2 2
  d 2 3

0
 0      x21  x22    x31  x32  
  g  0   

2
2
 dx2
  dx3


Results:
  d 3 2
   dx1*
  d 3 3


0


0
 0
 
   
 
3
3

  dx3
   dg
  dx2
  d 3 2
   dx1*
  d 3 3


0


0
 0
 
   
 
3
3

  dx3
   dg
  dx2
  d 3 2
   dx1*
  d 3 3

 0  
 0    
 0
 
3
3

  dx3
   dg
  dx2
  d 3 2

  d 3 3
 dx1*


0


0

 0
 
 
 

3
3
 dg

  dx3

  dx2
  d 3 2
   dx1*
  d 3 3


0


0
 0
 
   
 
3
3

  dx3
   dg
  dx2
  d 3 2
   dx1*
  d 3 3


0


0
 0
 
   
 
3
3

  dx3
   dg
  dx2
  d 3 2
   dx1*
  d 3 3

 0  
 0    
 0
 
3
3

  dx3
   dg
  dx2
94
2  dK 
d


2
d 3K
dL  d 



0
2
2
2
dLdV
dV
dV
 dK 
E    E 
 decreases
 dL 
if the risk inV increases.
The expected future marginal
resource value decreases from
increasing risk in the volume
process (growth) and we should
increase present extraction.
95
2  dK 
d


2
d 3K
dL  d 



0
2
2
2
dLdV
dV
dV
 dK 
E    E 
 increases
dL


if the risk inV increases.
The expected future marginal
resource value increases from
increasing risk in the volume
process (growth) and we should
decrease present extraction.
96
Some results of increasing risk in the volume
process (growth process):
• If the future risk in the volume process increases, we should increase
the present extraction level in case the third order derivatives of
profit with respect to volume are strictly negative.
• If the future risk in the volume process increases, we should not
change the present extraction level in case the third order derivatives
of profit with respect to volume are zero.
• If the future risk in the volume process increases, we should decrease
the present extraction level in case the third order derivatives of
profit with respect to volume are strictly positive.
97
Contents:
1.
2.
3.
4.
5.
6.
Introduction via one dimensional optimization in dynamic
problems, comparative statics analysis, probabilities, increasing
risk and the importance of third order derivatives.
Explicit multi period analysis, stationarity and corner solutions.
Multi period problems and model structure with sequential
adaptive decisions and risk.
Optimal decisions under future price risk.
Optimal decisions under future risk in the volume process
(growth risk).
Optimal decisions under future price risk with mixed species.
98
The mixed species case:
• A complete dynamic analysis of optimal natural resource
management with several species should include decisions
concerning total stock levels and interspecies competition.
• In the following analysis, we study a case with two species, where the
growth of a species is assumed to be a function of the total stock level
and the stock level of the individual species.
• The total stock level has however already indirectly been determined
via binding constraints on total harvesting in periods 1 and 2.
• We start with a deterministic version of the problem and later move
to the stochastic counterpart.
99
 is the total present value. xit denotes harves volume of species i in period t .
 it ( xit ) is the present value of harvesting species i in period t .
Each species has an intertemporal harvest volume constraint.
H t denotes the total harvest volume in period t.
These total harvest volumes are constrained in periods 1 and 2,
because of harvest capacity constraints, constraints in logistics or other constraints,
maybe reflecting the desire to control the total stock level.
100
Period 1
Period 2
Period 3
max    11 ( x11 )   21 ( x21 )   12 ( x12 )   22 ( x22 )   13 ( x13 )   23 ( x23 )
s.t.
 x11   x12  x13  C1
 x21   x22  x23  C2
x11  x21  H1
x12  x22  H 2
101
Consequences:
x21  H1  x11
x22  H 2  x12
x13  C1   x11   x12
x23  C2   x21   x22
x23  C2   ( H1  x11 )   ( H 2  x12 )
102
  11 ( x11 )   21 ( x21 )  12 ( x12 )   22 ( x22 )  13 ( x13 )   23 ( x23 )
  11 ( x11 )   21 ( H1  x11 )  12 ( x12 )   22 ( H 2  x12 )
 13 (C1   x11   x12 )   23 (C2   ( H1  x11 )   ( H 2  x12 ))
103
Now, we move to a stochastic version of the same problem.  is the expected total present value under the influence of stochastic future events and
optimal adaptive decisions. With probability  , the discounted price of species 1 increases by h in period 2 and with probability (1   ) , the price
decreases by the same amount. We define this a ”mean preserving spread” via the constraint   (1   )  1 .
2
xitp =Harvest volume in species i , at time t , for price state p
A: Consequences for harvest decisions in periods 2 and 3 of a price increase of species 1 in period 2:
Consequences for harvest decisions for species 1:
If the price in period 2 of species 1 increases by h , then we harvest x12  x121 in period 2. In period 3, we get the conditional harvest x13  x131 .
Consequences for harvest decisions for species 2:
If the price in period 2 of species 1 increases by h , then we harvest x22  x221 in period 2. In period 3, we get the conditional harvest x23  x231 .
104
B: Consequences for harvest decisions in periods 2 and 3 of a price decrease of species 1 in period 2:
Consequences for harvest decisions for species 1:
If the price in period 2 of species 1 decreases by h , then we harvest x12  x122 in period 2. In period 3, we get the conditional harvest x13  x132 .
Consequences for harvest decisions for species 2:
If the price in period 2 of species 1 decreases by h , then we harvest x22  x222 in period 2. In period 3, we get the conditional harvest x23  x232 .
105

 11 ( x11 )   21 ( H1  x11 )
   12 ( x121 )  hx121   22 ( H 2  x121 )    13 ( x131 )   23 ( x231 )  
 (1   )  12 ( x122 )  hx122   22 ( H 2  x122 )    13 ( x132 )   23 ( x232 )  
x131  C1   x11   x121
x231  C2   ( H1  x11 )   ( H 2  x121 )
x132  C1   x11   x122
x232  C2   ( H1  x11 )   ( H 2  x122 )
106
1

2
Z  2  2 11 ( x11 )  2 21 ( H1  x11 )
  12 ( x121 )  hx121   22 ( H 2  x121 )    13 ( x131 )   23 ( x231 ) 
  12 ( x122 )  hx122   22 ( H 2  x122 )    13 ( x132 )   23 ( x232 ) 
x131  C1   x11   x121
x231  C2   ( H1  x11 )   ( H 2  x121 )
x132  C1   x11   x122
x232  C2   ( H1  x11 )   ( H 2  x122 )
107
Z  2  2 11 ( x11 )  2 21 ( H1  x11 )
  12 ( x121 )  hx121   22 ( H 2  x121 )
  12 ( x122 )  hx122   22 ( H 2  x122 )
  13 ( x131 )   23 ( x231 )
  13 ( x132 )   23 ( x232 )
x131  C1   x11   x121
x231  C2   ( H1  x11 )   ( H 2  x121 )
x132  C1   x11   x122
x232  C2   ( H1  x11 )   ( H 2  x122 )
108
Z  2  2 11 ( x11 )  2 21 ( H1  x11 )
  12 ( x121 )  hx121   22 ( H 2  x121 )
  12 ( x122 )  hx122   22 ( H 2  x122 )
  13 (C1   x11   x121 )
  23 (C2   ( H1  x11 )   ( H 2  x121 ))
  13 (C1   x11   x122 )
  23 (C2   ( H1  x11 )   ( H 2  x122 ))
109
Now, there are three free decision variables and three first
order optimum conditions:
dZ
 2 11' ( x11 )  2 21' ( H1  x11 )
dx11
  13' (C1   x11   x121 )
  23' (C2   ( H1  x11 )   ( H 2  x121 ))
  13' (C1   x11   x122 )
  23' (C2   ( H1  x11 )   ( H 2  x122 ))
0
dZ
'
  12' ( x121 )  h   22
( H 2  x121 )
dx121
  13' (C1   x11   x121 )
'
  23
(C2   ( H1  x11 )   ( H 2  x121 ))
0
dZ
'
  12' ( x122 )  h   22
( H 2  x122 )
dx122
  13' (C1   x11   x122 )
'
  23
(C2   ( H1  x11 )   ( H 2  x122 ))
0
110
 2 11'' ( x11 )  2 21'' ( H1  x11 )

 2 ''

   13 (C1   x11   x121 )

 2 ''




(
C


(
H

x
)


(
H

x
))
23
2
1
11
2
121


2
''
   (C   x   x )

13
1
11
122
 2 ''

   (C   ( H  x )   ( H  x )) 
23
2
1
11
2
122 

  13'' (C1   x11   x121 )



''
  23 (C2   ( H1  x11 )   ( H 2  x121 )) 
  13'' (C1   x11   x122 )



''
  23 (C2   ( H1  x11 )   ( H 2  x122 )) 
  (C1   x11   x121 )

D
  '' (C   ( H  x )   ( H  x )) 
23
2
1
11
2
121 

  12'' ( x121 )   22'' ( H 2  x121 )

 2 ''

    13 (C1   x11   x121 )

 2 ''

    23 (C2   ( H1  x11 )   ( H 2  x121 )) 
0
0
  12'' ( x122 )   22'' ( H 2  x122 )

 2 ''




(
C


x


x
)


13
1
11
122
 2 ''




(
C


(
H

x
)


(
H

x
))
23
2
1
11
2
122


''
13
  13'' (C1   x11   x122 )



''


(
C


(
H

x
)


(
H

x
))
23
2
1
11
2
122 

111
D11
D12
D13
D  D21
D22
0
0
D33
D31
D  D11D22 D33  D12 D21D33  D13 D22 D31  0
 d 2Z 


dx
dh
 11   0 
 d 2Z   

1
 dx121dh   1
 d 2Z   


 dx122 dh 
112
*
 dx11
  0 
 *  
 D   dx121    1 dh 
* 
 dx122

  1 dh 
0
D12
1 D22
*
1
dx11

D11
dh
D21
D31
D13
0
0 D33
D12 D33  D13 D22

D12 D13
D
D22
0
0
D33
U  D12 D33  D13 D22
113
''
''



(
x
)


12 122
22 ( H 2  x122 )
''
  13 (C1   x11   x121 )
  2 ''

U 
   13 (C1   x11   x122 )



''
  (C   ( H  x )   ( H  x )) 
23
2
1
11
2
121  

   2 '' (C   ( H  x )   ( H  x )) 
23
2
1
11
2
122 

''
''



(
x
)


12 121
22 ( H 2  x121 )
''
  13 (C1   x11   x122 )
  2 ''


   13 (C1   x11   x121 )

  '' (C   ( H  x )   ( H  x ))  
23
2
1
11
2
122  

   2 '' (C   ( H  x )   ( H  x )) 
23
2
1
11
2
121 

114
Assumptions:
''
 13''  0   23
0
In order to produce strong and relevant results, we assume that:
 13'''  0   23'''  0
and even
 13'''  0   23'''  0
In general, one should expect that
13'''
13'''
 1  '''
12'''
12
1 and
 23'''
 23'''
 1  '''
 22'''
 22
1 , since the effects of volume increases on
the marginal profit level are usually less dramatic in the long run than in the short run. In the long run, there is more time available to adjust infrastructure
capacity, logistics, labour force and industrial capacities to large volume changes.
  0  0
Consequence:
115
  12'' ( x122 )   22'' ( H 2  x122 ) 
U     





  12'' ( x121 )   22'' ( H 2  x121 ) 
    





_
U
_
U
U

U

''
''
  12'' ( x121 )   22
( H 2  x121 )    12'' ( x122 )   22
( H 2  x122 ) 
''
''
  12'' ( x121 )   12'' ( x122 )    22
( H 2  x121 )   22
( H 2  x122 ) 
116
Observations:
 x121  x122 

 H2  x121  H2  x122 
*
 dx11

 _ 
sgn 
  sgn   U 


 dh 
_
 dx 
 dx 
 
sgn 
  sgn  
  sgn  U 
 
 dh 
 dh 
*
21
*
11
117
Multi species results:
CASE 1:
*
 dx11
      dh  0 

'''
12
'''
22
*

dx21
 0
dh

CASE 2:
*
 dx11
      dh  0 

'''
12
'''
22
*

dx21
 0
dh

CASE 3:
*
 dx11
      dh  0 

'''
12
'''
22
*

dx21
 0
dh

CASE 4:
*
 dx11
      dh  0 

'''
12
'''
22
*

dx21
 0
dh

CASE 5:
*
 dx11
      dh  0 

'''
12
'''
22
*

dx21
 0
dh

118
Some multi species results:
With multiple species and total harvest volume constraints:
Case 1:
If the future price risk of one species, A, increases, we should now harvest less of this species (A) and
more of the other species, in case the third order derivative of the profit function of species A with
respect to harvest volume is greater than the corresponding derivative of the other species.
Case 3:
If the future price risk of one species, A, increases, we should not change the present harvest of this
species (A) and not change the harvest of the other species, in case the third order derivative of the
profit function of species A with respect to harvest volume is equal to the corresponding derivative of
the other species.
Case 5:
If the future price risk of one species, A, increases, we should now harvest more of this species (A)
and less of the other species, in case the third order derivative of the profit function of species A with
respect to harvest volume is less than the corresponding derivative of the other species.
119
Related
analyses,
via
stochastic
dynamic
programming,
are found here:
120
Many more references, including this presentation, are found here:
http://www.lohmander.com/Information/Ref.htm
121
122
OPTIMAL PRESENT RESOURCE EXTRACTION
UNDER THE INFLUENCE OF FUTURE RISK
Professor Dr Peter Lohmander
SLU, Sweden, http://www.Lohmander.com
[email protected]
The 8th International Conference of
Iranian Operations Research Society
Department of Mathematics
Ferdowsi University of Mashhad, Mashhad, Iran.
www.or8.um.ac.ir
21-22 May 2015
123