# Optimal Rotation Optimal Rotation.

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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
Harry Nelson 2011 Volume and Value Increase with Age Volume or value of timber (\$/ha/yr or m3/ha/yr) value volume Age (years)

Average and Incremental Growth in Value and Volume
Average growth and marginal (incremental) growth (m3/ha/yr) Marginal or incremental growth Average growth Age (years)

Relationship Between Maximum Marginal Growth and Average Growth in Value
Volume or value of timber (\$/ha/yr or m3/ha/yr) Value or p(t) Volume or Q(t) Average growth and marginal (incremental) growth (\$/ha/yr) Average value of the stand or p(t)/t Marginal or incremental growth in value or ∆p Age (years)

Optimal Rotation for a Single Stand
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
Rate of growth in the value of timber (%/yr) Change in value/Total value or ∆p/p(t) i T*

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

Optimal Rotation for a Series of Harvests
Harry Nelson 2010 Optimal Rotation for a Series of Harvests What then is the present value of a series of recurring harvests every 60 years (where p=Revenues-Costs)? p p p p 60 120 180 240 Perpetual Periodic Series (pg. 129 in text)

Harry Nelson 2011 Associated Math 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”. p V0= (1 + r)t - 1 p Vs= (1 + r)t - 1 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

Land Expectation Value
Present value of a series of infinite harvests, excluding all costs Evaluated at the beginning of the rotation p Vs= (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

Harry Nelson 2011 Associated Math So in order to maximize LEV the goal is to pick the rotation age (t*) that maximizes this value. p At 90 years, only 109 m3/ha and worth \$6 per m3, but LEV is higher-\$49.17 Vs= (1 + r)t* - 1 This can be done in a spreadsheet by putting in different rotation ages and seeing which generates the highest value

Harry Nelson 2011 Calculating Current Value and Land Expectation Value at Different Harvest Ages LEV maximized at 70 years

Comparison with Single Rotation
Harry Nelson 2011 Comparison with Single Rotation 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 P(t*) P(t*+1) Vs(t*)= = Vs(t*+1)= (1 + r)t* - 1 (1 + r)t+1* - 1 ∆P r = P(t) 1 -(1+r)-t

Faustmann Formula ∆P r = P(t) 1 -(1+r)-t
Annual costs & returns Incremental growth in value or ∆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 Incremental increase in cost or r/1-(1+r)-t T* Rotation age (t)

Harry Nelson 2011 Modifying the Math The formula can be modified to include other revenues and costs Here recurring annual revenues and costs are included in the 2nd term p a - c + Vs= r (1 + r)t* - 1

Harry Nelson 2011 Further Modification Imagine you have a series of intermittent costs and revenues over the rotation -how do you calculate the optimal rotation then? Commercial thinning - net revenue (NRt) Harvesting - net revenue (NRh) Reforestation-Cr Pre-Commercial Thin -Cpct 80 20 50 You can compound all the costs and revenues forward to a common point at the end of the rotation-this then becomes p + + P = (1 + r)80 *Cr + (1 + r)60*Cpct (1 + r)30*NRt NRh p Vs= (1 + r)t - 1

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

Amenity Values and Non-Monetary Benefits
Harry Nelson 2011 Amenity Values and Non-Monetary Benefits Growth in value without amenity values “Perpetual rotation” Rate of growth in the value of timber (%/yr) Growth in value with amenity values Growth in value with amenity values i or MAR Rotation age Rotation age In this case you’d never harvest

How Does the Rule Affect Harvest Determination?
Harry Nelson 2011 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)

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

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*

Harry Nelson 2011 Fall Down Effect

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