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 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 ppp Harry Nelson 2010

V0=V0= p (1 + r) t - 1 Vs=Vs= p 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 2011

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

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 =

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 Math Harry Nelson 2011 Vs=Vs= p (1 + r) t* 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=Vs= p (1 + r) t - 1 Reforestation-C r Commercial thinning - net revenue (NR t ) 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 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 = (Q mature /T*) + mai  where Q mature 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/ =$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