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 ID: 543463
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Slide1
Optimal Rotation
Optimal RotationSlide2
Biological vs. Economic Criteria
What age should we harvest timber?Could pick the age to yield a certain sizeOr could pick an age where volume in a stand is maximizedOr pick an age where the growth rate is maximizedOur focus will be on finding the rotation that maximizes economic returnsSlide3
How do we find that?
Determine the age that maximizes the difference between the present value of future revenues and future costsWe first simplify the problemOnly interested in commercial returnsOnly one type of silvicultural system-Clearcutting (even-aged)Start with an existing timber standSlide4
Volume and Value Increase with Age
Volume or value of timber ($/ha/yr or m3/ha/yr)
Age (years)
volume
value
Harry Nelson 2011Slide5
Average growth and marginal (incremental) growth (m3/ha/yr)
Age (years)
Average growth
Marginal or incremental growth
Average and Incremental Growth in Value and VolumeSlide6Slide7
Relationship Between Maximum Marginal Growth and Average Growth in Value
Volume or value of timber ($/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)/tSlide8
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 costAs 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 returnOptimal Rotation for a Single StandSlide9
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 StandSlide10
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 perpetuityWe assume each rotation will involve identical revenues and costsAnd we will start off with bare landSlide11
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 2010Slide12
V
0
=
p
(1 + r)
t
- 1
V
s
=
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, V
s
then represents the discounted cash flow-the amount by which benefits will exceed costs
Associated Math
Harry Nelson 2011Slide13
Land Expectation Value
Present value of a series of infinite harvests, excluding all costsEvaluated at the beginning of the rotation
V
s
=
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 Slide14
V
s
=
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-
$8.20Slide15
Calculating Current Value and Land Expectation Value at Different Harvest Ages
LEV maximized at
50
years
Harry Nelson 2011Slide16
V
s(t*)
=
P(t*)
(1 + r)
t*
- 1
V
s(t*+1)
=
P(t*+1)
(1 + r)
t+1*
- 1
=
r
1 -(1+r)
-t
∆P
P(t)
Comparison with Single Rotation
Harry 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
=Slide17
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 FormulaSlide18
Modifying the Math
Harry Nelson 2011
V
s
=
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 termSlide19
V
s
=
p
(1 + r)
t
- 1
Reforestation-C
r
Commercial thinning -
net revenue (NR
t
)
0
20
50
80
P =
(1 + r)
80
*C
r
+
(1 + r)
60
*C
pct
+
(1 + r)
30
*NR
t
+
NR
h
Imagine you have a series of intermittent costs and revenues over the rotation -how do you calculate the optimal rotation then?
Pre-Commercial Thin -C
pct
Harvesting -
net revenue (NR
h
)
You can compound all the costs and revenues forward to a common point at
the end of the rotation
-this then becomes p
Further Modification
Harry Nelson 2011Slide20
Impact of Different Factors
Interest rateHigher the interest rate the shorter the optimum rotationLand ProductivityHigher productivity will lead to shorter rotationPricesIncreasing prices will lengthen the optimal rotationReforestation costsIncrease will increase the optimal rotation lengthSlide21
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 harvestSlide22
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 valueImagine applying the optimal rotation age to two types of forestsIn 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 2011Slide23
“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 harvestedIn this case the harvest levels would be constant (assuming everything else such as prices and costs remained constant)
Harry Nelson 2011Slide24
Why Private Harvest Levels Are Unlikely to be Constant
Stands vary in size and productivityMarkets are changingSo harvest levels are likely to fluctuateMay also be specific factors that influence the owner (size constraints, etc.)Slide25
Regulating Harvests on Public Land
Harvest rules on public land have historically been concerned with maximizing timber yieldHistoric concern has been that cyclical markets would lead to variations in harvesting, employment, and income for workersGoal has been to smooth out harvest levels and maintain harvests in perpetuitySlide26
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 employmentEstablished on basis of growth rate for a given ageUsually done as a volume control (AAC determination)Alternative is area controlSlide27
Several Important Consequences
Where mature forests exists affects the economic value of forestry operationsCan be long-term effects on timber supplyChanges how we evaluate forestry investmentsSlide28
Fall Down Effect
Historically transition from old growth (primary forest) to sustained yield This approach yields the “fall-down” effectHanzlick formula-based on proportion of old growth and mean annual increment associated with average forest growthAAC = (Qmature /T*) + maiwhere Qmature equals amount of timber greater than harvest age T*Slide29
Fall Down Effect
Harry Nelson 2011Slide30
Allowable Cut Effect
Cost of improving the stand -$1000 per hectareResult-doubling of growth (an additional 995 cubic metres)Standard cost-benefit:Discounted Benefit: $13,187/1.0558=$778Cost: $1000So NPV =-$222; B/C = 0.78Slide31
Introducing ACE
If you can take additional volume over the 58 years… ($13,187/58)Then it looks quite differentUsing a formula-the present value of a finite annuityNPV = ($13,187/58)*((1.05)58-1)/.05*(1.05)58Or $4,546Slide32
Using ACE as an incentiveSlide33
Experience with ACE