Optimal Rotation-Pt 2
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Transcript of Optimal Rotation-Pt 2
Optimal Rotation-Pt 2
February 27, 2014
John Maynard KeynesBorn: June 5, 1883, Cambridge, United KingdomDied: April 21, 1946, East Sussex, United Kingdom
Adam SmithBorn: June 5, 1723, Kirkcaldy, United KingdomDied: July 17, 1790, Edinburgh, United Kingdom
David RicardoBorn: April 18, 1772, London, United KingdomDied: September 11, 1823, Gatcombe Park, United Kingdom
Alfred MarshallBorn: July 26, 1842, Bermondsey, London, United KingdomDied: July 13, 1924, Cambridge, United Kingdom
Ronald CoaseBorn: December 29, 1910, Willesden, London, United KingdomDied: September 2, 2013, Chicago, Illinois, United States
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Post lecture slides before class Identify equations and graphs to be used in problem
sets Examples of how to use them Better labeling of graphs
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 6% that would give me a value of $42.26/ha
Harvest Age (ys)
Volume (m3/ha)
Net Revenues/m3 Value ($/ha) LEV
10 29 0 $0 $0.0020 46 0 $0 $0.0030 61 0 $0 $0.0040 74 1 $74 $32.7150 85 2 $170 $50.2460 94 3 $282 $57.6570 101 4 $404 $58.4080 106 5 $530 $54.9790 109 6 $654 $49.17
100 110 7 $770 $42.26110 109 8 $872 $35.12120 106 9 $954 $28.30130 101 10 $1,010 $22.13140 94 11 $1,034 $16.76150 85 12 $1,020 $12.25160 74 13 $962 $8.57170 61 14 $854 $5.65180 46 15 $690 $3.39
Calculating Current Value and Land Expectation Value at Different Harvest Ages
LEV maximized at 70 years
Harry Nelson 2011
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.
This can be done in a spreadsheet 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-$49.17
Reforestation-Cr
Commercial thinning -
net revenue (NRt)
0 20 50 80
Imagine you have a series of intermittent costs and revenues over the rotation along with annual costs and revenues -how do you calculate the optimal rotation then?
Pre-Commercial Thin -Cpct
Harvesting -
net revenue (NRh)
Further ModificationHarry Nelson 2011
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
For periodic costs and revenues over the rotation:
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
For problem setHarry Nelson 2011
For problem setHarry Nelson 2011
Vs=p
(1 + r)t* - 1+
a - c
r
Recurring annual revenues and costs can be are included in a 2nd expression
Adding Carbon
It is not the biological side that makes C accounting complex-it is the market side.
• Baseline
• Leakage
• Buffer
• Harvest
What happens if we manage for Carbon?
Carbon payment schemes pay for either C sequestered or avoided C emissions.
In forestry focus has been on sequestration (trees are efficient C storage mechanisms)
Impact of Different Factors
Interest rate Higher the interest rate the shorter the optimum
rotation
Impact of Different Factors
Interest rate Higher the interest rate the shorter the optimum
rotation Land Productivity
Higher productivity will lead to shorter rotation
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
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
What if there are other values?
Incremental growth in value or ∆p/p(t)
Rotation age (t)T*
i*
Annual costs & returns
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
Assessing Risk
No selection
Selection
No infestation
No infestation
infestation
infestation Selection worksSelectio
n didn’t
workP=0.2
P=0.2
P=0.8
P=0.8
P=0.3
P=0.7
p. 124 in text
Cost of outcome: S1 Cost of outcome: S2
Cost of outcome: NS2
Cost of outcome: NS1
Cost of outcome: S3
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
From Chapter 8, 163-65
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