3. Optimal control model of dynamics of forest regimes
Many authors have used optimal control theory for modelling forest stands Anderson, 1975;
Clark, 1976, pp. 263 – 269; Sethi and Thompson, 1981, pp. 287 – 294; Synder and Bhattacharyya,
1990; however, these models are based on a traditional production function, which includes
only the transformation function. The concept of the transaction function, and hence, resource
regime, is missing from these models. An optimal control model of the full forest production pro-
cess, which is described by the non-separable
7
across time transformation and transaction functions, is developed in this section, and the
dynamic path of optimal resource regime for a given forest and user group environment is
examined.
I assume, for simplicity and clarity of analysis, a composite forest product which comprises all
timber and non-timber products, with a net value per unit area, at time t, of Vt. The timber is
removed on a rotation period, whereas NTFPs are removed continuously. Hence, Vt represents
the sum of the net value of standing timber at time t and the net value of all NTFPs removed
and available to be removed up until time t. As Vt is the net value, all costs, such as regenera-
tion, harvesting and land rent, except the cost of a resource regime the transaction cost, are ac-
counted for in this formulation of Vt. The cost of the resource regime P
r
R will be treated sepa- rately. It is also assumed that there are only two
production factors: time
8
and the resource regime. It is further assumed that, in the absence of the
effect of the resource regime, the rate of change of value is represented by the logistic function: dV
dt = mV[1 − VV
a
] = fVt, where m is the positive growth parameter and V
a
is the asymp- totic value of V, and fVt is the growth func-
tion of the value of the composite product. Due to the non-separable nature of transaction
and transformation functions, resource regime ar- rangements the transaction function will affect
the growth rate of forests as well as the net value of the composite product available to the legal
right holder. As per the definition of the transac- tion function, the net value available to the legal
right holder will be the product of the natural net value
Vt times
the transaction
function GRt, t. It is also assumed that the effective
growth rate
9
will also be the product of the natu- ral growth rate times the transaction function. In
this context, policy makers or forest managers will likely design and modify forest regimes in such a
way that the legal right holder can maximise the net returns from forests. Hence, the policy maker
or forest manager’s problem is to maximise:
T
[VtGRt, t − P
r
R]exp − rt dt subject to dVdt = f[Vt]G[Rt, t],
where f[Vt] = mVt[1 − VtV
a
], GRt, t = dtR
a t
1 − R
b t
and V0 = V .
This is a standard optimal control problem, in which Vt is the state variable, and Rt, which is
bounded within 0
+
and 1
−
, is the control vari-
7
Renewable resources such as forests grow and regenerate over economically relevant periods of time. The growth of
these resources depends upon the existing growing stock, which is clearly affected by resource regime arrangements.
Hence, the transformation and transaction functions work simultaneously, or are non-separable. Non-renewable re-
sources are formed by geological processes that typically take millions of years, thus for practical purposes, these can be
treated as having a fixed stock. In such cases, transformation and transaction functions can be treated as separable. How-
ever, even in the case of non-renewable resources, the quantity retrievable for use is related to the transformation process.
Hence, in the broader perspective of the transformation pro- cess, the transaction and transformation functions will be
non-separable even in the case of a non-renewable resource.
8
In the case of forest resources, the period of production is lengthy, and ‘time’ itself acts as a factor of production Nau-
tiyal, 1988, p. 335. Hence, it is common practice to have forest production models in terms of time.
9
The natural growth rate means the growth rate attainable by the forest in the complete absence of any interference. But,
as formulated, the resource regime arrangement may alter the total existing volume growing stock which will also alter the
growth rate, resulting in the effective growth rate instead of natural growth rate.
able. This optimal control problem can be solved by a standard method. The current value
Hamiltonian of this optimal control problem can be written as:
H = [VtGRt, t − P
r
Rt] + l
tfVtGRt, t 1
where lt is known as the co-state variable, which is the marginal valuation of the state
variable Vt, and is also known as the shadow price of the state variable. In the case of the
current value Hamiltonian, lt gives the current marginal value of the state variable at time t. I
also assumed that the objective function and the function which is describing the law of motion
of the state variable fVtGRt are con- cave, hence the necessary first order conditions
of the optimal control problem will also be the sufficient conditions Lambert, 1985, p. 175.
First order conditions are given next.
3
.
1
. Optimality condition HR = VGR − P
r
R + lfGR = 0. 2
In the remaining text, the subscript is used to denote the partial derivative with respect to a
variable given in subscript. Hence, Eq. 2 gives:
l = P
r R
− VG
R
fG
R
. 3
3
.
2
. Co-state 6ariable l
condition lt = rl − HV = rl − G − lf
V
G. 4
This equation gives the motion of the co-state variable. The equation can be written as:
lt1l + f
V
G + Gl = r. 4a
Eq. 3 gives the shadow price of the state vari- able, and is equal to the marginal net value per
unit of available growth due to marginal change in the resource regime. Eq. 4a gives the mo-
tion of the shadow price. The first term in Eq. 4a gives the relative rate of change of the mar-
ginal valuation the shadow price of Vt, the second term gives the value available from the
growth of the state variable, and the third term gives the relative available value from one unit
of the state variable with respect to the shadow price. The right hand side gives the psychic cost
due to time preference. A common understand- ing of the nature of forest growth f
6
being quite high in the early and middle stages of a
forest, as compared to r, indicates that the rela- tive marginal value of the stand Vt will de-
crease at a decreasing rate along the optimal path. However, even though the value is mar-
ginally decreasing, the forest is not cut because the value of available growth is greater than the
value obtained by cutting the forest Donnelly and Betters, 1991.
3
.
3
. Law of motion Vt = fG = mV[1 − VV
a
]dR
a
1 − R
b
. 5
On substituting the value of the growth function f from Eq. 2,
Vt = [P
r R
− VG
R
lG
R
]G 6
The co-state variable equation Eq. 4 and the law of motion equation Eq. 6 give the mo-
tion of the co-state and state variables, respec- tively. The standard practice is to study the
paths of the state variable and co-state variable or the path of the state variable and control
variable. Hence, one option is to analyse the optimal path of the state and co-state variables.
However, even though these two first order dif- ferential equations Eqs. 4 and 6 are au-
tonomous time does not appear explicitly, both equations contain the control variable Rt in
some form. Hence, the analysis of the motion of the state and co-state variables, with the help of
a phase diagram of these two equations in the present form, is not feasible. In addition, the
main aim of this paper is to integrate natural and social systems, and to study the impact of
these systems on the evolution of optimal re- source regimes. Hence, I focus upon developing
an equation of the dynamics of an optimal re- source regime that can aid in evaluating the im-
pact of the dynamics of natural and social factors upon the dynamics of forest regimes.
On differentiating Eq. 3 with respect to t, and substituting the value of l, and lt in Eq. 4
and further substituting the value of Vt from Eq. 5, the following is achieved:
− P
r R
fG
R
t =
P
r R
− VG
R
G
R
[ f
t
+ fr − f
V
G] 7
Since, GRt = dtRt
a t
1 − Rt
b t
, and
G
R
= G[aR − b1 − R].
When G
R
is differentiated with respect to time, the following is achieved:
G
R
t1G = [{aR − b1 − R}
2
− aR
2
+ b1 − R
2
]Rt +
[{aR − b1 − R} ×
{1ddt + atlnR +
btln1 − R}] +
[at1R −
bt11 − R]. 8
On substituting the value of G
R
t in Eq. 7, I get
h
2
Rt =
[h
1
{VG
R
− P
r R
P
r R
}{r + f
t
f − f
6
G}] −
1ddth
1
+ at[ − h
1
lnR − 1R] +
bt[{11 − R} − h
1
ln1 − R] 9
where h
1
= {aR − b1 − R}.
and h
2
= [h
1 2
− aR
2
+ b1 − R
2
]. Eq. 9 gives the rate of change of the optimal
resource regime. The solution to two non-linear differential equations of motion Eqs. 6 and 9,
subject to the initial conditions, which will pre- scribe the initial values of the two SEFs a and b,
the scaling factor d, the state variable, and the transversality
10
conditions, will give the unique optimal path of the state variable Vt and the
control variable Rt. However, as stated earlier, the main objective of this paper is not to find a
unique path for given initial and transversality conditions, but rather to develop an understand-
ing of the interactions between natural and social systems, and their impact on the dynamic path of
optimal forest regimes. Hence, I examine Eq. 9, and evaluate the role of natural and socio-eco-
nomic factors in the dynamic path of optimal resource regimes.
In brief, Eq. 9 can be written as: Rt
= function V, f, P
r
, r, dt, at, bt. Hence, the change in the optimal resource regime
will depend upon neither the natural factors, nor the social factors, but rather upon both of the
factors as well as upon their interactions.
11
As stated earlier, the resource regime R repre- sents the degree of exclusion of the local user
group and varies from 0 to 1. Hence, a positive rate of change of resource regime means more
exclusion or a shift towards a private regime, and a negative rate of change means a move towards
less exclusion or a community regime. As per Eq. 9, four terms contribute to the rate of change of
forest regimes. The first term comprises the state variable Vt, growth function f , transaction
function and transaction costs, and one term each represents the contribution of the rate of change
of the scalar function, the heterogeneity of the
10
Normally, the forest manager’s main tasks are to develop a management plan for a fixed period or to find out an
optimal rotation and develop a management plan for this rotation period. In the first case, the time horizon is fixed, and,
normally, there is no limit on the terminal value of the state variable VT. Hence, the terminal condition is that VT
remains free. This terminal condition gives lT = 0 as a transversality condition. The latter case of determining the
optimal rotation is the free terminal-time problem, in which T is not specified in advance. In this case, the maximum principle
includes the transversality condition that the Hamiltonian, at time T, HT, is equal to zero. Depending upon the objective
of the forest manager, both of these two transversality condi- tions with initial boundary conditions can be used to deter-
mine the unique optimal path of state and control variables.
11
However, this result is an outcome of the non-separable nature of the transaction and transformation functions which,
I believe, represents the real growth process of renewable resources such as forests. In the case of separable transaction
and transformation functions, the change in optimal resource regime will be independent of natural factors and dependent
upon the socio-economic factors only.
user group, and the dependence of the user group, respectively. The actual contribution of each term
will depend upon the initial conditions, and the aggregate rate of change of the resource regime
will depend upon the relative contribution of each term. However, an understanding of the main
contributors of each term will provide a broad framework to policy makers and forest managers
in designing and modifying forest regimes as per the requirements of change in natural and social
systems. The impacts of natural and socio-eco- nomic factors on the rate of change of resource
regime are discussed next.
3
.
4
. Natural and socio-economic factors and the dynamics of optimal forest regimes
Natural factors influence the path of the opti- mal resource regime through marginal relative
return VG
R
− P
r R
P
r R
} and relative growth r + f
t
f − f
6
G. The marginal relative return is the relative rate of change of net benefit to the cost of
the resource regime due to a marginal change in the resource regime. Thus, if the change in the net
benefit, due to a change in resource regime, is higher as compared to the resource regime cost,
the rate of change of the resource regime will be higher. In neo-classical economic analysis, the
contribution of this term will be independent of the socio-economic conditions of the user group.
However, in my formulation, even the contribu- tion of this term will depend upon socio-economic
factors because the value of the composite product V and the transaction cost P
r
are sensitive to socio-economic factors. The compo-
nents of the composite product may change with time depending upon the socio-economic condi-
tions and preferences of the group. For example, in the early stages of economic development, the
local people are dependent upon forests for their livelihood and V will include timber, as well as
many non-timber forest products which may not have market value. With economic development,
many non-timber forest products may not be valuable to the user group anymore, and hence
will be excluded from V. Thus, when V includes many high value products, and its value is very
high, the rate of change of the resource regime will be highly sensitive to the values of V. As
mentioned throughout this paper, the transaction cost is mainly dependent upon the two socio-eco-
nomic factors. Hence, as user groups pass on to the next economic growth phase, the exclusion
cost may be reduced due to a reduction in depen- dence, but co-ordination cost may increase due to
an increase in heterogeneity. Thus, the overall impact of the relative marginal valuation term will
depend upon the relative changes in V and trans- action costs. If the change in value is higher than
the change in transaction costs due to a marginal increase in exclusion, the resource regime should
be modified towards more exclusion private. If the change in values is less than the change in
transaction cost due to marginal increase in exclu- sion, the resource regime should be modified to-
wards less exclusion community. In other words, in less developed and highly forest-dependent
communities, smaller changes in the forest regime in opposing directions than that of economically
expected may cause high economic and welfare losses. At the same time, larger marginal costs of
the resource regime will logically lead to smaller changes in the resource regime. If there is high
cost of change in the resource regime, it will not be optimal to change it. The relative growth term
r + f
t
f − f
6
G implies that the higher rate of time preference and the higher rate of change of
the growth function with respect to time will lead to a higher rate of change of the optimal resource
regime, and the effect of the rate of growth func- tion with respect to time and volume are in oppo-
site direction. Hence, fast growing forests high f
t
f will require rapid changes in forest regimes as compared to slower growing forests. However,
the growth function, f, represents the growth of V. The terms, f
t
and f
6
will depend upon the composition of V, and hence upon the socio-eco-
nomic factors of the user group. The dynamics of the rate of time preference, and hence its impact
upon the dynamics of optimal forest regimes, is a controversial issue. Strict neo-classical economists
are unwilling to deviate from equating the rate of time preference to the real market rate of return
on man-made capital investments. Ecological economists are of the view that man-made capital
and natural capital are not substitutes Costanza
and Daly, 1992, and thus, I subscribe to the view that the real rate of interest is not the correct
measure of the rate of time preference for natural capital. Kant 1999 demonstrates that traditional
forest-dependent communities normally have a lower rate of time preference for forest resources
as compared to industrialised communities.
12
Hence, the impact of the rate of time preference will also vary with the socio-economic environ-
ment of the user group. The rate of change of the optimal forest regime will be lower in the case of
traditional communities, who have a lower rate of time preference, as compared to economically de-
veloped communities.
In addition to these impacts of socio-economic factors through interactions with natural factors,
the socio-economic factors also independently contribute to the rate of change of optimal forest
regimes. As the transaction function is defined in terms of heterogeneity and dependence, it is natu-
ral that the optimal resource regime will vary with the variation in these two socio-economic factors.
The change in optimal resource regime is directly proportional to the rate of change in heterogene-
ity and dependence. However, the increase in heterogeneity will drive the optimal resource
regime towards a private regime, while an increase in dependence will drive towards a community
regime. The scaling factor d, which normalises the achievable maximum value of the transaction
function, and the maximum value can be different under
different socio-economic
environments. Therefore, the two socio-economic factors will
also affect the optimal resource regime through the scaling factor. This analysis demonstrates that
socio-economic factors are critical to the optimal- ity of the total production process of forest re-
sources,
and that
non-inclusion of
the socio-economic environment, and its interaction
with natural factors, will result in economic inefficiencies.
The continuous rate of change in the resource regime may be questionable from the practical
aspects of designing forest regimes. It is under- stood that to make continuous changes in the
resource regime is not feasible, and this is the main reason why a specific solution to the two
motion equations has not been attempted in this paper. However, the broad outcome of the model
— the optimal resource regime arrangements not remaining stationary and changing over time ac-
cording to changes in socio-economic factors — is very critical for efficient forest management
decisions, and can be used by forest managers to improve the efficiency of forest management.
Forest managers should include SEFs in their set of management variables, and necessary amend-
ments should be made to the resource regime either to increase or decrease the exclusion of
local communities as and when required due to changes in socio-economic factors.
4. Final comments
Kategori