Optimal Economic rotation for Even-Aged Stands

1. Optimal Economic rotation for Even-Aged Stands

2. Lab Wednesday!!! • Computer lab in JOUR106 • Completing our activity will be the homework for that week (due the following Thursday)

3. Introduction • Optimal economic rotation is a classic problem in resource economics • Optimal economic rotation can also be used to determine the size of the forest inventory of each age class that must be carried to maintain a target level of timber production • Optimal economic rotation traditionally examined from the perspective of a private present value maximising firm • But, non-timber benefits considered by a public lands forester planning optimal economic rotations, can be incorporated into the analysis.

4. Introduction • The decision problem is one of comparing annual growth in timber value against cost of holding the timber for an extra year • If holding the timber for another year would lead to a net loss, we should harvest now • Assumptions • the only relevant benefit is revenue from timber. • harvesting involves clearfelling (clearcutting) the stand, i.e. trees are all of the same age. • the stand consists of a single species (monoculture). • costs and returns can be predicted with certainty. • for the present, ignore planting, establishment and maintenance costs of a forest.

5. Stumpage value • Stumpage value S(A) is the maximum price competitive buyers would be prepared to pay for the timber of age, A, standing in a forest • S(A) = R - C • Note that Klemperer (1996, pp. 212-215) uses the abbreviation Hy to represent S(A).

6. Timber Volume, Stumpage Value and Stand Age * Total standing volume of timber, Q(A), typically increases over time according to a sigmoid function. • MAI for a given age = slope of ray drawn from origin of top panel to Q(A) • * MAI = Q(A)/A • Max MAI is where ray is tangential to Q(A)

7. Timber Volume, Stumpage Value and Stand Age The stumpage value of timber S(A) is likely to increase with age: 1. increasing timber volume; 2. increased timber quality; 3. lower harvest and transport costs per unit of timber. Eventually, mortality may exceed new growth and increase in timber value such that S(A) declines with age S(A)/A has the same relationships as MAI

8. Optimum Economic Rotation for a Single Crop • A forester wanting to select the harvest age that maximises profits must consider the marginal benefits and marginal costs of growing the timber crop one more year • MC is the return that could be earned on the proceeds of a harvested timber stand, i.e. the opportunity cost of capital, MAR (the discount rate), r • The marginal benefit of growing the stand one more year is: • S = S(A+1) - S(A) • This is amount by which timber value will increase if stand is left to grow for another year • Geometrically, it is equivalent to the slope of the S(A) curve.

9. Optimum Economic Rotation for a Single Crop • The incremental growth rate in the value of the stand, S/S(A), represents the rate of return of the forest owner’s capital tied up in the timber crop • S/S(A) declines as the stand ages, since for a sigmoid relationship between S(A) and A, S declines while S(A) increases. S Klemperer calls S/S(A) the timber value growth percent S

11. Optimum Economic Rotation for a Single Crop • S/S(A) does not include the bare land value • Land is a capital asset that could be sold and reinvested at r • Thus investors need to make an adequate return on the timber and land capital • Forest value growth percent = (S + L) / (S(A) + L(A)) • We assume that the most efficient use of the land is under forest and we can ignore L(A)

12. Optimum Economic Rotation for a Single Crop • The owner could harvest the timber crop, and invest the proceeds at an interest rate r (MAR) • So economically optimal rotation for a single rotation tree crop is where: • S/S(A) = r • or • S = r S(A)

13. Optimum Economic Rotation for a Single Crop • S/S(A) = r or S = r S(A) • Economically optimum rotation length: • will increase with higher and more prolonged stand growth (the higher S/S(A) for any stand age); • is inversely related to the rate of interest, i.e. for lower interest rates the incremental rate of growth can be lower.

14. Simple Example • Suppose we plant a forest today and that it will grow at the constant rate of 20 m3/yr. Also suppose that the value of this timber is $100/m3, regardless of the age of the trees. The next best investment alternative returns 5% per annum. What is the optimal harvest age for a single rotation of tree crops? • At the optimal rotation age, we know S/S(A) = 5% • We also know S = $2,000 • So S(A) at the optimal harvest age must be $40,000 [S/r = S(A)] • The optimal harvest age is $40,000/$2,000 = 20 years. • ‘Real world’ problems can be accommodated in spreadsheet software

15. Equivalence of Optimal Rotation and Maximisation of NPV (p. 151-52) • NPV of single forest rotation is • Vo = S(A) / (1+r)A • Optimal rotation where Vo = 0 • Lab exercise • S(A+1) – S(A) = 0 (1+r)A+1 (1+r)A (working not shown here) • S = r S(A) corresponds to the optimal rotation rule for a single tree crop rotation.

16. Economically Optimal Rotation for Land Managed Perpetually for Timber Production (p. 162) • The single rotation did not account for the value of land tied up in forestry • To factor this into the analysis, assume for now that the best use of the land is for forestry • NPV =

17. Economically optimal rotation (perpetual) • The notes show how this equation can be transformed to • Vs = S(A) / ((1+ r)A -1) • which is equivalent to PV perpetual periodic series, Vo = Vt / ((1 + r)t -1) • This equation calculates site value or land expectation value (LEV), i.e. value of bare land • The rotation that maximises Vs is the economically optimum rotation for land managed perpetually under forest crops

18. Economically optimal rotation (perpetual) • Vs maximised where Vs = 0 Vs A** Rotation age, A

19. Economically optimal rotation • Can be rearranged such that we get the relationship • This equality will be satisfied at the economically optimal rotation • This implies balancing the proportionate change in stumpage value from delaying harvest for another year with the marginal cost, including the annual cost of holding the land • This formula is known as the Faustmann formula

20. Economically optimal rotation • How does this differ from the single rotation case? • The incremental or opportunity cost is now greater • [instead of S/S(A) = r] • This accounts for the incremental cost of delaying the harvest of all future rotations – opportunity cost of land • Will the economically optimal rotation under perpetual tree crops be longer or shorter than for a single crop?

21. Economically optimal rotation • Incremental cost is asymptotic to r at long rotation lengths, A • A is often long, so optimal rotation for continuous forestry often not much less than for a single crop

22. An example from today’s review questions • Question 4 from Optimal economic rotation review questions..\Review question answers\10_optimal rotation.xls

23. Examining how changes in management parameters can affect the economically optimal rotation • The optimal economic rotation is affected by changes in factors such as: • productivity of the land • reforestation costs • stumpage prices • the interest rate • And non-timber values, e.g. • grazing • payments for ecosystem services (e.g. carbon sequestration and salinity amelioration) • How can we estimate how these factors affect the economically optimum rotation?

24. With our trusty incremental cost and incremental benefit diagram, of course!

25. Example Q 1b • What is the impact on the economically optimal rotation if stumpage prices increase and there is no expectation of them increasing further in the foreseeable future?

26. Example Q2b • The theory thus far has assumed no cost is incurred establishing new crops following harvest. If reforestation costs must be incurred at the outset of each rotation, what is the impact on the economically optimal rotation length?

27. Example Q4 • What will be the effect of an increase in the interest rate on the economically optimum rotation?

28. Example Q6

29. Example Q8 • 8. What will be the effect of a constant annual management cost or constant annual tax on land value on the economically optimal rotation? • The present worth of an infinite series of annual costs, m, is m/r – a constant that is independent of stand age or the volume of timber harvested • This is subtracted from the land expectation value of a site, but will not change the optimal rotation • This is because the age which maximises land expectation value will also maximise land expectation value minus a constant, m/r • It is not possible to illustrate this outcome with our simple diagrammatic approach

30. Summary of Faustmann • Optimal economic rotation, A**, will increase if S(A) or r decreases • Optimal economic rotation, A**, will decrease if S(A) or r increases • Typically, the formula is applied to timber plantation monocultures with little value other than for timber • With mixed species planting, there is little information available as to likely yields, harvesting is more likely to be by selective logging, and longer rotations lead to greater yield and price uncertainty • In these cases, estimates of site value become less reliable.

31. Maximising MAI • The rotation that maximises MAI will maximise the volume of timber harvested over an infinite time horizon • But it ignores economic considerations, like the net price of timber, the cost of replanting and the discount rate. • That is, maximising MAI does not ensure socially efficient investment of capital. • Often the rotation that maximises MAI will be longer than the economically optimal rotation • However, if the discount rate is low, stumpage prices increase substantially with increasing stand age or if non-timber values are sufficiently high, economically optimum rotations can exceed the rotation that maximises MAI