R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
The optimal rotation period for the firm
Now, let us come back exclusively to forestry. As we have
already seen, for forests created using plantation programmes, it
is important to study the optimal ways to increase the utility
from the forests. Let us now concentrate on the dynamic aspects
of forestry plantation.
In particular we want to find when is the best time to harvest a
tree or a stand of trees. In practice, it depends on many factors,
such as the final use of the harvested forest, whether we begin
with bare land or a mature forest, cost, productivity and demand
conditions, etc. If the plantations are made for pulp to make
paper, the size of the stems and trunks are not important, but the
total volume is. For timber, large trees are important. Similarly,
for energy purposes, the total volume is important.
For simplicity, let us assume that we start with a tract of trees
where trees have to be planted first. Let us study about the
optimal time to cut a stand of trees of uniform age and growth
characteristics. Once the stand is harvested, saplings will be
replanted and the cycle is continued. We seek an economically
efficient rotation period over an infinite number cycles of
planting, harvesting, replanting, and harvesting.
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R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
We choose the optimal rotation period that maximizes the
present value of a stream of net benefits. This involves choosing
a date to harvest and replant time after time. We assume that the
harvesting and replanting occur without delay – at the same
period of time – say one year.
Two types of direct costs are associated with forest revenues.
First, the actual costs of managing the forest – costs of planing,
silviculture, harvesting, storage, transportation to market, and so
on. The second cost is the interest foregone while waiting to
harvest the trees – the money that could be obtained if the trees
were cut sooner in time and re-invested in growing more trees
on the land and in alternate enterprises. The value of the land on
which the trees are grown is an indirect cost. Land receives the
residual income – the rent – from raising and marketing the
forest, the income remaining after all planting, maintenance and
harvesting costs have been paid.
Suppose that it costs a fixed amount of $ D to plant a unit of
land (say a sq. metre) and Rs. C per unit volume of wood to
harvest the trees. the cost in present value of the first "round" in
the infinite cycle is,
D  CV (T1  T0 )e  r (T T )  D  CVe  r (T T )
1
0
2
1
0
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
where, T0 is the date of planting and T1 is the date of harvesting.
The term V (T1  T0 ) in the RHS refers that volume of wood V is
a function of the time period (T1  T0 ) . The term e r (T T )
1
0
transfers harvest costs from the harvest date to the beginning of
the "cycle" – back to the planting date. If the price per unit of
harvested wood is p, then the profit in present value (land
owner's
benefit)
from
planting
and
harvesting
is:
 p  C Ver (T T )  D .
1
0
After harvesting, the land is replanted at cost D, and a new
round is undertaken. This process is repeated infinitely. then the
complete present value of net benefits is:
W
 p  C Ve  r (T T )  D
e  r (T T )  p  C Ve  r (T T )  D 
e  r (T T )  p  C Ve  r (T T )  D 
1


0
1
0
2
1
2
0
3
2
 
Now, we can assert that the intervals (T1 – T0), (T2 – T1), (T3 –
T2) etc. must be the same. Suppose that we have somehow
arrived a the value of T1. Now the remaining problem can be
made identical to the original problem by re-labeling dates (i.e.,
re-labeling T1 to T0, T2 to T1, etc.). Once another cycle is
completed, the problem is the same as before. this is because the
horizon is always infinity.
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R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Now, let the equal intervals be written as I. This is called the
rotation period. With this new notation, we have,
W



 p  C Ve  rI  D
e  rI  p  C Ve  rI  D 
e  2 rI  p  C Ve  rI  D 
 


 p  C Ve
 p  C Ve


 rI

 D 1  e  rI  e  2 rI  
 1 
D
 rI
1  e 
 rI
We have to choose I such that W is maximized. The first order
condition for maximization is dW/dI = 0. After some algebraic
simplifications, this condition leads to the following result.
 p  C  rVe
 rI
 V e
 rI



V   rV

re  rI  p  C Ve  rI  D
1  e  rI
 r  *
 
W
  p  C 
Thus, the optimality condition is,
dV
r
 rV 
W*
p  C
dI
4

R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
where W* is the optimized value of the net present value of
profits for all periods. It is the maximum value obtainable for
the land by using its forestry. Note that rW* is the return per
period of the maximum value of the land, or the flow value of
the land in its optimal land use.
Obviously, the maximum value obtainable for the land by using
it for forestry should be compared with other uses to decide
whether the land will be used for forestry or not. If the land
remained less used in the past, it may be profitable now to use it
for forestry. However, if the land could be advantageously for
housing or for agriculture, its opportunity value will be larger.
Note that the above optimality condition states dV/dI as a linear
equation with V as its slope and (rW*/p-c) as its intercept.
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R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
dV
 r

 rV  
W *
dI
 p  C 
dV/dI
dV
 rV
dI

r
*
W
 p  C  
V(I*)
V(I1)
V(Imax)
V(I)
V(Imax)
V(I1)
*
Wood V(I )
Volume
V(I)
I*
I1
I
6
Imax
x
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Let us assume that we know the opportunity value W of land.
Then the straight line in the above figure can be drawn. We
know the variation of dV/dI with respect to I. Hence, the optimal
I can be determined by considering the parabolic behaviour of
dV/dI with respect to I is used in conjunction with the equation
dV
r
 rV 
W*.
p  C
dI
In the figure, the straight line intersects the parabola in two
places. Optimal I is determined by the intersection points. The
large is the I, the larger is the volume and hence we choose the
larger of the two solutions. Note that if we know the algebraic
equation of V(I) in terms of I, we could have incorporated in the
derivation to get optimal values of I. We can use the algebraic
equation in the LHS of
dV
r
 rV 
W * to get the optimal
p  C
dI
I. The result will be the same as that of the graphical solution.
Suppose that r = 0. This means that there is no discounting of
future harvests, or there is no rate of return on alternative
investments. Obviously, when we do not care about future, we
shall allow the trees to reach their maximum size, and then
harvest them. This can be readily seen from the optimality
condition,
dV
r
 rV 
W * . When r = 0, dV/dI = 0, and
p  C
dI
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R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
the parabolic relation of dV/dI = 0 only at I = 0 or I = Imax. This
means, in every harvest, we wait till the trees reach their
maximum volume and then cut them. Note that with
discounting, I* < Imax, because we associate some positive value
for the value of future harvests. (The opportunity value is given
rW *
by the RHS of the optimality condition: rV 
is the
p  C
opportunity value of holding the trees (rV) and the land
 rW * 

 for another instant of time instead of harvesting them


p

C


now.)
Suppose now that r > 0 but W = 0. This means that land has no
implicit value; the opportunity cost of using the land for
alternative use is zero. May be it is because land is available in
unlimited supply. Our optimality condition now becomes
dV
 rV * . This is a straight line passing through the origin with
dI
slope r. This cuts the parabola at I1, which is the optimal rotation
period under this assumption (W = 0). Note that, , I* < I1 < Imax.
I* is the shortest rotation period because it considers the
opportunity costs of deferring the harvest when we associate a
positive discount rate for the future, and we associate a positive
value for land.
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R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Let us now study how optimal I will change when we change
some parameters of the model.
Suppose that there is an increase in the planting costs, D. When
D is higher, the planner will do well to reduce the importance of
D incurred in the second and future periods. (He cannot avoid
incurring D in the first period.) This can be done by increasing I.
Hence, a rise in planting (set up) costs will lead to a longer
rotation period.
Let us now look at this result more mathematically. We have the
optimality condition:


dV
r
 r 
 1 
 rV 
W*  
  p  C Ve  rI  D 
 rI
p  C
dI
1  e 
  p  C 
We need to find dI/dD now. To get this, we totally differentiate
both the sides. On LHS, I is the only variable, while D is the
only variable in RHS. Thus,
  dV
  rW 

 rV dI 

dD

I  dI
D   p  C  

9
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
dV/dI
V(I*)
V(Imax)
V(I)
V(Imax)
*
Wood V(I )
Volume
V(I)
I*
Imax
I
10
x
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Let us focus on the LHS. We know that between I* and Imax (in
the figure in the previous page), dV/dI is decreasing and hence,
  dV 

 is negative. But, during the same period, V is
I  dI 
increasing and hence dV/dI is still positive. (dV/dI becomes
negative only after Imax. Hence the expression,
  dV

 rV  is

I  dI

negative. Even between I = 0 and I = I*, the rate of growth of V
is much sharp compared to the rate of growth of dV/dI. Hence,
we can safely claim that
  dV

 rV  is negative.

I  dI

Let us now come to the RHS.


  rW 
 r   
 1 
 rI


p

C
Ve

D

  


1  e  rI  
D   p  C  
  p  C   D 
 r   1 
 

 rI


p

C

 1  e 
 r   e rI 
 
  rI



p

C

  e  1
The above expression is negative.
  rW 


D   p  C    ve
dI


  ve . Thus, the optimal
Thus,
dD   dV
  ve
 rV 

I  dI

rotation period increases as the setup costs increase.
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R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
What will happen when forest plots differ in their accessibility
to markets, i.e., when C is increased? When costs increase,
profits decrease. With a positive discount rate, people tend to
maximize their profit in earlier periods and hence the optimal
rotation period will increase. This can easily be seen in our
mathematical derivation.
Totally differentiating the optimality condition, we have,
  dV
  rW 

 rV dI 

dC

I  dI
C   p  C  

The RHS simplifies to 
rD
dC . Note that both the
 p  C  1  e  rI


numerator and denominator are positive and hence RHS is
negative (as it has a negative sign). Hence, dI/dC is positive,
meaning that the optimal rotation period will reduce as price
increases.
Let us now see what happens when the discount rate is
increased. Intuitively, a higher discount rate means more
impatience. We associate lesser importance to future revenues.
There is a desire to get results sooner. This results in the
reduction of optimal rotation period. Let us use the optimality
condition to substantiate our claim.
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R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Totally differentiating the optimality condition yields,
  dV
  dV
  rW 


 rV dI  
 rV dr  
dr ,



I  dI
r  dI
r  p  C 


which simplifies to,
  dV
W* 
rI 
rI 


 rV dI 

1  rI
dr  V 1  rI
dr
 p  c  e  1
I  dI

 e  1
LHS is negative and RHS is positive. Hence, dI/dr < 0. Thus,
the optimal rotation period decreases for higher interest rates.
This can also be seen using the parabolic variation of dV/dI
with respect to I, and the linear relationship of dV/dI as given by
the optimality condition. For a higher discount rate, the line will
have a larger intercept and a larger slope, reducing the
equilibrium rotation period I (see the figure in the next page).
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R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Larger slope for
higher discount rate
dV/dI
larger
intercept
Imodified
Ioriginal
I
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R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
This sensitivity analysis with respect to r brings us to an
important point. Normally, private firms will use a higher
discount rate than the socially optimal one. (Private firms will
take additional risk and adjust the social rate by the risk.) Also,
there is a difference in the perception of society and a private
firm in viewing the forest. A private firm will view the forest as
a source of income from the trees, while the society will value
the recreational use and biodiversity also. This makes the
optimal rotation period for a private firm to be different from
that for a society. These divergences suggest a role for
governmental intervention – generally in the form of taxes.
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R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Let us now see different tax regimes on forests and how they
affect the optimal rotation period.
A tax per ton harvested
If  is the tax rate, then imposition of this tax will increase costs
from C to C+. As we have seen earlier, this will increase the
rotation period – trees will be cut when they are slightly older,
and will have slightly more volume.
A site-use tax
A tax per acre (levied whenever land is brought into forestry
use) will essentially increase the initial fixed cost D. The
effective setup cost will now increase to D+. As seen before,
the effect will be to increase the optimal rotation period.
16