CM

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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

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First someGeneral optimal control theory
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Taylor expansion
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is not a function of
Hence, we may exclude
from the optimization.
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is optimized via
We assume an unconstrained local maximum.
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The adjoint equation
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Terminal boundary condition
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The maximum principle Necessary conditions for
to be an optimal control
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Now, a more specific Optimal control model
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Determination of the time derivative of
via the adjoint equation
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Determination of the time derivative of
via
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The time derivative of
as determined via
must equal the time derivative of
as determined via the adjoint equation
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Definition
Observation This differential equation can be
used to determine the optimal control function
.
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Solution of the homogenous differential equation
Let
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Determination of the particular solution
Let
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Conclusion The optimal control function is
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Determination of the optimal stock path equation
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The optimal control function is introduced in the
differential equation of the stock path equation
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As a result, the differential equation of the
optimal stock path equation is
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Solution of the homogenous differential equation
Let
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Determination of the particular solution
Let
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Conclusion The optimal stock path equation is
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Determination of
via the initial and terminal conditions
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Determination of
via
and
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Conclusion The optimal adjoint variable path
equation is
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Determination of the optimal objective function
value
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Now, we determine
, the antiderivative of K.
. Below, we exclude the constant of integration.
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Code section for calculation of J (in JavaScript)
var r2 r-g1 var z1 1/(2k3)(f1/r2
f2/r2/r2 - k1) var z2 1/(2k3)(f2/r2 -
k2) var c0 1/g1(z1(z2-g2)/g1 - g0) var c1
(z2-g2)/g1
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• var sysdet Math.exp(r2t1)/(g1-r2)
• Math.exp(g1t2) - Math.exp(r2t2)
• /(g1-r2)Math.exp(g1t1)
• var A ( (x1-c0-c1t1)Math.exp(g1t2)
• (x2-c0-c1t2)Math.exp(g1t1) )
• /sysdet
• var B ( Math.exp(r2t1)/(g1-r2)(x2-c0-c1t2)
  Math.exp(r2t2)/(g1-r2)(x1-c0-c1t1))
• /sysdet

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var D A/(g1-r2) var w0 f1c0 k1z1
k3z1z1 var w1 f1c1 f2c0 k1z2 k2z1
2k3z1z2 var w2 f2c1 k2z2
k3z2z2 var w3 f1B var w4 f1D k1A
2k3z1A var w5 k3AA var w6 f2B var w7
f2D k2A 2k3z2A
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var s0 - r var s1 g1 - r var s2 r2 -
r var s3 2r2 - r
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var AntidK2 w0(1/s0Math.exp(s0
t2)) w1(Math.exp(s0t2)(t2/s0 -
1/(s0s0))) w2(Math.exp(s0t2)((s0s0
t2t2-2s0t22) /(s0s0s0)))
w3(1/s1Math.exp(s1t2))
w4(1/s2Math.exp(s2t2))
w5(1/s3Math.exp(s3t2))
w6(Math.exp(s1t2)(t2/s1 - 1/(s1s1)))
w7(Math.exp(s2t2)(t2/s2 - 1/(s2s2)))
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var AntidK1 w0(1/s0Math.exp(s0t1))
w1(Math.exp(s0t1)(t1/s0 - 1/(s0s0)))
w2(Math.exp(s0t1)((s0s0t1t1-2s0t
1 2) /(s0s0s0)))
w3(1/s1Math.exp(s1t1))
w4(1/s2Math.exp(s2t1))
w5(1/s3Math.exp(s3t1))
w6(Math.exp(s1t1)(t1/s1 - 1/(s1s1)))
w7(Math.exp(s2t1)(t1/s2 - 1/(s2s2))) var
Integral AntidK2 - AntidK1 var J Integral
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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.
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The Hamiltonian function is
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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,
is equal to the marginal resource value at the
same point in time,
.
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The adjoint equation says that
Since
, we know that
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Now, we make the following definition
The resource, with total stock
, consists of many different parts.
Different parts of the resource have different
relative growth. A particular part of the
resource,
, has relative growth
.
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We may define the relative growth of that
particular unit as
(
is the stock level of that particular unit and
is the growth.)
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When we integrate,
we arrange the different particular resource
units in such a way that
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ObservationThe relative decrease of the
marginal resource value over time is equal to
the relative growth of the marginal resource
unit.
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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.

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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.
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The relative marginal resource value decrease is
higher
in case the relative marginal growth
is higher.
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The relationship between the speed of change of
the relative marginal resource value and direct
valuation of the stock level
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Here, we add
to the objective function.
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As a result, we get
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The Hamiltonian function becomes
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These derivatives are obtained
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The adjoint equation gives a modified result
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Observation
gt0 implies that
becomes more negative if
increases.
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Explanation and connection to forestry
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.
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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.
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Software for the optimal control model
http//www.lohmander.com/CM/CM.htm
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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
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Optimal strategy and dynamic effects of the
rate of interest
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Comparisions with alternatives that are not
optimal
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Optimal strategy and dynamic effects of the
slope of the demand function
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Comparisions with two alternatives
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Optimal strategy and dynamic effects of the
terminal condition
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Optimal strategy and dynamic effects of
continuous stock level valuation
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Optimal strategy and dynamic effects of
continuous stock level valuation in combination
with variations of the the slope of the demand
function
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Optimal strategy and dynamic effects of the
growth function
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