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Optimal Rotation
Optimal Rotation
Biological vs. Economic Criteria
What age should we harvest timber? Could pick the age to yield a certain size Or could pick an age where volume in a stand is
maximized Or pick an age where the growth rate is maximized
Our focus will be on finding the rotation that maximizes economic returns
How do we find that?
Determine the age that maximizes the difference between the present value of future revenues and future costs
We first simplify the problem Only interested in commercial returns Only one type of silvicultural system-Clearcutting (even-
aged) Start with an existing timber stand
Volume and Value Increase with Age
Volume or value of timber ($/ha/yr or m3/ha/yr)
Age (years)
volume
value
Harry Nelson 2011
Average growth and marginal (incremental) growth (m3/ha/yr)
Age (years)
Average growth
Marginal or incremental growth
Average and Incremental Growth in Value and Volume
Relationship Between Maximum Marginal Growth and Average Growth in Value
Volume or value of timber ($/ha/yr or m3/ha/yr)
Age (years)
Volume or Q(t)
Value or p(t)
Average growth and marginal (incremental) growth ($/ha/yr)
Marginal or incremental growth in value or ∆p
Average value of the stand or p(t)/t
Key idea is to weigh the marginal benefit of growing the stand another year against the marginal cost of not harvesting The marginal benefit of waiting to harvest a year is the increase in value of the stand
The marginal cost is what you give up in not harvesting now is the opportunity to invest those funds-or the opportunity cost
As long as you earn a higher return “on the stump”, it makes sense to keep your money invested in the timber
When the rate falls below what you can earn elsewhere, then harvest the timber and invest it where it can earn the higher return
Optimal Rotation for a Single Stand
T*
Rate of growth in the value of timber (%/yr)
i
Change in value/Total value
or ∆p/p(t)
Optimal Rotation for a Single Stand
Introducing Successive Rotations
In the previous example only considered the question of how best to utilize capital (the money invested in growing the timber stand)
We now turn to the problem of deciding the optimal rotation age when we have a series of periodic harvests in perpetuity We assume each rotation will involve identical
revenues and costs And we will start off with bare land
p
60 120 180 240
Perpetual Periodic Series– (pg. 129 in text)
What then is the present value of a series of recurring harvests every 60 years (where p=Revenues-Costs)?
Optimal Rotation for a Series of Harvests
p p p
Harry Nelson 2010
V0=p
(1 + r)t - 1
Vs=p
(1 + r)t - 1
This is the formula for calculating the present value of an infinite series of future harvests.
Pearse calls this “site value”. It can also be called “Soil Expectation Value (SEV)”, “Land Expectation Value (LEV)”, or “willingness to pay for land”.
If there are no costs associated with producing the timber, Vs then represents the discounted cash flow-the amount by which benefits will exceed costs
Associated Math Harry Nelson 2011
Land Expectation Value
Present value of a series of infinite harvests, excluding all costs
Evaluated at the beginning of the rotation
Vs=p
(1 + r)t - 1
So if I had land capable of growing 110 m3/ha at 100 years, and it yielded $7 per m3, evaluated at a discount rate of 5% that would give me a value of $5.90/ha
Vs=p
(1 + r)t* - 1
So in order to maximize LEV the goal is to pick the rotation age (t*) that maximizes this value.
Identifying optimal age can be done by putting in different rotation ages and seeing which generates the highest value
Associated Math Harry Nelson 2011
At 90 years, only 109 m3/ha and worth $6 per m3, but LEV is higher-$8.20
Calculating Current Value and Land Expectation Value at Different Harvest Ages
LEV maximized at 50 years
Harry Nelson 2011
Vs(t*)=P(t*)
(1 + r)t* - 1Vs(t*+1)=
P(t*+1)
(1 + r)t+1* - 1
=r
1 -(1+r)-t
∆P
P(t)
Comparison with Single RotationHarry Nelson 2011
The problem now becomes determining what age given successive harvests
The idea is still the same-calculate the benefit of carrying the timber stand another year against the opportunity cost
The difference here is that instead of evaluating only the current stand you now look at the LEV, which takes into account future harvests
=
Incremental growth in value or ∆p/p(t)
Incremental increase in cost or r/1-(1+r)-t
Annual costs & returns
Rotation age (t)
=r
1 -(1+r)-t
∆P
P(t)
This result-where the marginal benefit is balanced against the marginal cost of carrying the timber-is known as the Faustmann formula
You end up harvesting sooner relative to the single rotation
The economic logic is that there is an additional cost-land.
By harvesting sooner is that you want to get those future trees in the ground so you can harvest sooner and receive those revenues sooner
T*
Faustmann Formula
Modifying the MathHarry Nelson 2011
Vs=p
(1 + r)t* - 1+
a - c
r
The formula can be modified to include other revenues and costs
Here recurring annual revenues and costs are included in the 2nd term
Vs=p
(1 + r)t - 1
Reforestation-Cr
Commercial thinning -
net revenue (NRt)
0 20 50 80
P = (1 + r)80 *Cr + (1 + r)60*Cpct+ (1 + r)30*NRt
+ NRh
Imagine you have a series of intermittent costs and revenues over the rotation -how do you calculate the optimal rotation then?
Pre-Commercial Thin -Cpct
Harvesting -
net revenue (NRh)
You can compound all the costs and revenues forward to a common point at the end of the rotation-this then becomes p
Further ModificationHarry Nelson 2011
Impact of Different Factors
Interest rate Higher the interest rate the shorter the optimum
rotation
Land Productivity Higher productivity will lead to shorter rotation
Prices Increasing prices will lengthen the optimal rotation
Reforestation costs Increase will increase the optimal rotation length
Growth in value without amenity values
Growth in value with amenity values
Rotation age
Rate of growth in the value of timber (%/yr) Growth in value with
amenity values
Rotation age
“Perpetual rotation”
i or MAR
Amenity Values and Non-Monetary Benefits
Harry Nelson 2011
In this case you’d never harvest
How Does the Rule Affect Harvest Determination?
How does the rotation rule apply when we extend it to the forest?
Start with the assumption of a private owner maximizing value
Imagine applying the optimal rotation age to two types of forests
In one forest all the stands are the same age so all the harvest would take place in one year with no harvests until the stands reached the optimal age again
Harry Nelson 2011
“Normal” forest
In another forest the stands are divided into equal-sized areas and there is a stand for each age class-so that each year one stand is harvested
In this case the harvest levels would be constant (assuming everything else such as prices and costs remained constant)
Harry Nelson 2011
Why Private Harvest Levels Are Unlikely to be Constant
Stands vary in size and productivity
Markets are changing
So harvest levels are likely to fluctuate
May also be specific factors that influence the owner (size constraints, etc.)
Regulating Harvests on Public Land
Harvest rules on public land have historically been concerned with maximizing timber yield
Historic concern has been that cyclical markets would lead to variations in harvesting, employment, and income for workers
Goal has been to smooth out harvest levels and maintain harvests in perpetuity
Harvesting policies in Canada
Sustained yield (or non-declining even flow) has been preferred approach as it was originally seen as contributing to community stability and maintaining employment
Established on basis of growth rate for a given age
Usually done as a volume control (AAC determination) Alternative is area control
Several Important Consequences
Where mature forests exists affects the economic value of forestry operations
Can be long-term effects on timber supply
Changes how we evaluate forestry investments
Fall Down Effect
Historically transition from old growth (primary forest) to sustained yield
This approach yields the “fall-down” effect
Hanzlick formula-based on proportion of old growth and mean annual increment associated with average forest growth AAC = (Qmature /T*) + mai
where Qmature equals amount of timber greater than harvest age T*
Fall Down Effect
Harry Nelson 2011
Allowable Cut Effect
Cost of improving the stand -$1000 per hectare
Result-doubling of growth (an additional 995 cubic metres)
Standard cost-benefit: Discounted Benefit: $13,187/1.0558=$778 Cost: $1000 So NPV =-$222; B/C = 0.78
Introducing ACE
If you can take additional volume over the 58 years… ($13,187/58) Then it looks quite different
Using a formula-the present value of a finite annuity
NPV = ($13,187/58)*((1.05)58-1)/.05*(1.05)58
Or $4,546
Using ACE as an incentive
Experience with ACE