3.5 Forestry Economics
A. Characteristics of forest resources
•
•
•
•
•
•
Forests are multi-functional
o timber, fuelwood, food, water for drinking and irrigation, stocks of genetic
resources, and other forest products
o variety of services such as removal of air pollution, regulation of
atmospheric quality, nutrient cycling, soil creation, habitats for humans
and wildlife, watershed maintenance, aesthetic and other amenity
functions
Forests are productive capital assets. They occupy valuable land – an opportunity
cost.
Long time lags from seedling to full maturity
Harvesting does not involve a regular cut of incremental growth. Single stands
are usually felled in their entirety – although selective harvesting of mature trees
is also carried out in some areas.
old growth, managed forests, plantations
plantations make up 5% of all forest area but 35% of all roundwood (wood in the
rough)
V(t) = wood volume available for commercial harvest, as a function of age of stand.
CAI = Current Annual Increment (Like a marginal product curve) Incremental periodic
change in volume
MAI = Mean Annual Increment (Like an average product curve) Average growth up to
age t
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t = age of stand.
V(t)
t
tx
tm
dv(t)/
dt
CAI
MAI
t
tx
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tm
2
Biologic growth plotted as a function of the stock of wood is not a logistic growth curve,
but it looks similar to the one we used for fish:
dV/dt
V(t)
B. A single rotation forest model
Unlike the fishery, or like typical production functions, one cannot harvest incremental
growth –
the whole tree is taken at once. So, we do not have a continuous problem
here… so, the question becomes:
Q:
what is the optimal age at which to harvest?
→ is it the maximum of the MAI?
→ is it the maximum of the CAI?
→ what factors must be considered?
MAI and CAI alone are physical, and do NOT take into account costs (of harvest,
maintenance, regeneration) or benefits (of harvest, other uses).
Maximize the Present Value of land:
max e − rt (TB − TC ) 
t
•
•
Start first with only 1 forest rotation (plant-harvest-sell), and then derive the
optimal t for multiple rotations.
The 1 rotation model is referred to as the “Fisher Model”.
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P = market price per unit volume of (“stumpage value”) standing trees
C = cost to harvest per unit volume
V ( t ) [ P − C ] = net value of trees at harvest
D = a fixed regeneration cost
Assume:
• V(t) is known
• All trees are the same age
• P, C, D do not change overtime
• r = discount rate
• best use of land is forestry
max PV = e − rt {V ( t ) [ P − C ] − K }
(1)
t
Taking the derivative w.r.t. time, our first order condition is:
−r [V ( t )( P − C ) − D ] + V ′ ( t )( P − C ) = 0
(2)
Rearrange this to get the Fisher Rule:
V ′ ( t )( P − C )
=r
V ( t )( P − C ) − D
•
The optimal harvest age will satisfy this equation.
•
Economic interpretation:
•
But, there is a problem with this rule – what is it?
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(3)
4
C. Faustmann Rule
•
•
•
Assume an infinite number of rotations.
Soil fertility, real prices and costs all remain constant over time.
How does taking future land use into account alters our optimal harvest age.
Sequence of times,
T0 ⇒ 1
T1 ⇒ 1
T3 ⇒ 2
At
So,
T0 , T1 , T2 ,...
st
planting
st
harvest, 2st planting
nd
harvest, 3rd planting
T1 − T0 , T2 − T1 , T3 − T2 , are rotation ages, corresponding to the ages of
the stands when they are harvested.
Under assumptions of constant real prices, costs and productivity,
(V ( t ) ) , it is the
case that all rotation periods are the same for each rotation. Why?
So,
e.g.
Ti = iI
where
i = 1, 2,3 and I is the rotation interval
T1 = I , T2 = 2 I , T3 = 3I
and all
Ti − Ti −1 = I
Let W denote the value of the forest land.
W = ( P − C )V (T1 − T0 )e − r (T1 −T0 ) − D 
+ e − r (T1 −T0 ) ( P − C )V (T2 − T1 )e − r (T1 −T0 ) − D 
+ e − r (T2 −T0 ) ( P − C )V (T3 − T2 )e − r (T1 −T0 ) − D 
(4)
+ ...
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We can substitute I for Ti-Ti-1
W = ( P − C )V ( I )e − r ( I ) − D 
+ e − r ( I ) ( P − C )V ( I )e − r ( I ) − D 
+ e −2 r ( I ) ( P − C )V ( I )e − r ( I ) − D 
(5)
+ e −3r ( I ) ( P − C )V ( I )e − r ( I ) − D 
+ ...
This can be written as:
Now, our problem can be written as:
W = ( P − C )V ( I )e − r ( I ) − D 
 ( P − C )V ( I )e − r ( I ) − D 


 (6)
−r (I ) 
−r (I )
−r (I )
+e
− D 
+ e

( P − C )V ( I )e
 −2 r ( I )

−r (I )
( P − C )V ( I )e
− D  + ...
e
Note that after the first term we have the same infinite sequence as in Equation (5),
except that it is discounted by one rotation interval.
Rewriting Equation (6):
W = ( P − C )V ( I )e − r ( I ) − D  + e − rIW
(7)
Solve for W:
W = ( P − C )V ( I )e
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−r(I )
1 

− D  
− rI 
−
1
e


(8)
6
Forest owner maximizes W by choosing the rotation interval I such that dW
dI
= 0.
Taking the derivative w.r.t. time, the first order condition is:
∂W (1 − e
=
∂I
− rt
) e [V ′ ( I )( P − C ) − V ( I )( P − C ) r ]
(1 − e )
− rI
− rt 2
(9)
[( P − C )V ( I )e− rI − D]e − rt r
−
=0
(1 − e− rt ) 2
Rearrange this to get the Faustmann Formula
V ′ ( I )( P − C )
r
=
V ( I )( P − C ) − D 1 − e − rI
(10)
Now compare (10), the Faustmann Formula with(3), the Fisher Rule.
The Faustmann Formula takes into account the opportunity costs associated with future
use of the land.
r
>r
−δ t
1− e
because
1 − e −δ t < 1
Will the optimal harvest age be higher or lower under the Faustmann compared to the
Fisher formula?
USER COST – a harvest at any time interval I will affect future income stream of all
future harvests.
So now, the interpretation:
Rewrite (10) for optimal I*
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V ′ ( I * ) ( P − C ) = r V ( I * ) ( P − C ) − K  +
r V ( I * ) ( P − C ) − D 
(
1 − e − rI
*
)
Benefit from delaying the harvest one more year:
Cost of delaying the harvest one more year:
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D. Comparative Statics
D.1 Changes in the Interest Rate, r
δ
1 − e −δt
*
i = i1
i = i2
i = i3
t1
t2
V ′ ( I )( P − C )
r
=
V ( I )( P − C ) − D 1 − e − rI
t3
(11)
Steeper line is the LHS of the Faustman Formula.
Flatter lines show the RHS of Faustmann.
Intersection show optimal harvest age.
We see that as
What if
r ↑, I ↓ .
δ = 0 ? So there is no discounting of the future and all periods are alike…
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lim  r  = 1
r → 0 1 − e − rI  I
(by L’Hopital’s Rule)
so, (11)becomes:
V ( I )( P − C ) − D
V ′ ( I )( P − C ) =
I
which is maximizing the average annual economic rent.
In general, as
r ↑, I ↓ .
D.2 Changes in P, C, and D
How will changes effect the optimal rotation age?
Rewrite the LHS of the Faustmann formula:
V ′ ( I )( P − C )
V ′( I )
=
V ( I )( P − C ) − D V ( I ) − D
P −C
(
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)
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V ′ (t )
V ( t ) − [ D /( P − C )]
i = i0
I
if
( P − C ) ↑⇒ I ↓↑
(
V ′ (t )
V ( t ) − [ D /( P − C )]
shifts _____ )
P ↑⇒ I ___
C ↑⇒ ___
D ↑⇒ I ___
What if
D = 0? (i.e. naturally regenerating?)
D.2 Taxes
•
Severance or royaty tax:
⇒
↑C
like
a tax/ton harvested
• Site-use tax – a tax per acre of land each time land is brought into forestry use
•
⇒ like ↑ K
profit tax
⇒
*
t% on V , This will not change the optimal rotation interval. The tax cannot be
shifted by altering the rotation interval
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E. The Hartman Model
→
Multiple Use Model
The conventional Faustmann Model does not reflect the value of services provided
directly by the standing forest.
i.e. amenity values
wildlife habitat
erosion control
water cycle maintenance
∴ Select the optimal rotation age to provide the combined net benefits from timber and
other services of standing timber.
a ( n ) = Net benefit flow of an acre of standing stock at age n
Present Value from stand is the discounted sum of
a ( n ) over all future harvest
rotation cycles:
T
ψ (T ) = ∫  a ( n ) e − rn dn (1 − e − rT )
−1
0
ψ (T ) = present value of net amenity benefits
T=
n=
rotation age
r= interest rate
age of tree (stand)
T
Select
to maximize the combined present value from timber harvest and amenity
services of standing stock:
φ* =
max
{W (T ) + ψ (T )}
T
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max 
− rn
− rT −1 
=
W (T ) + ∫  a ( n ) e  dn (1 − e ) 
T 

0
T
φ * = overall asset value of an acre of land
W (T ) = Timber value (market) as defined by Faustmann Model
So the overall value is reflected in both the commercial timber value and the standing
timber value.
F.O.C.
V ′ (T ) + a (T ) = r V (T ) + φ * 
φ * is the optimal value of the land including both timber and non-timber values
T*
As with the Faustmann Solution,
should be at that age which the increase in value
from a marginal delay in harvest equals the opportunity cost of that delay. Now, in the
multiple-use formulation, delaying the harvest provides the benefit of:
increased timber growth and,
a (T ) , the flow of amenity value over the period
of delay.
Timber is harvested when the full asset value of land AND timber is equal to the rate of
return available from comparable market investments at r.
[NO real difference between Faustmann and Hartman Models in economic
interpretation.]
a ( n ) is increasing or
decreasing with age. There are likely multiple ages meeting criteria (if a ( n ) is NOT
We have no a priori information by which to determine whether
linear).
F. Characteristics of the Hartmann Solution
In general:
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•
TH* will be somewhere between the Faustmann age and that age which would
max present value of amenity services alone.
•
TH* will depend upon the total amenity benefits of a harvest cycle relative to the
net timber revenues, and upon the relative growth rates in amenity and timber
values.
i.e.
•
In areas where forage and increased water flow are important, amenity values
may decline with stand age. Thus,
TH
will be less than the Faustmann
TH
t*.
•
If amenity values increase with age then
•
If amenity values are not sensitive to age, there will be no difference in the
optimal rotation age, no matter how large the amenity values are.
will be greater than Faustmann
t*.
Note: The Hartmann Multiple-Use Model implies that the average supply of timber may
be greater than the Faustmann Model.
a) Higher average stocking level necessary to maintain amenity services also
allows for greater sustainable harvest. i.e. the
TH* is closer to the
maximum sustainable yield.
b) Multiple-use may justify timber planting/harvesting on some lands that
would be uneconomical to manage timber alone. So, amenity services in
the short-run are seen to reduce timber harvesting, but in the long-run may
justify higher average levels of harvesting…(closer to the Forest
Manager’s goal…)
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