Only the forest-rent and soil-rent rotation lengths are affected by
changes in prices or costs. A higher price shifts the curve {Q1/[Q-
wE/P]} to the left because a higher price increases the size of the
denominator, making the value of the curve smaller for any t. The
result is a shortened optimal rotation for both criteria as the curve will
cross its equality condition at an earlier period. A higher regeneration
cost shifts the curve to the right, causing an increase in the optimal
rotation age. Finally, increased interest rates shift those functions in-
ward, causing shorter rotations.

Extensions of the Faustmann Model

The primary use of the Faustmann model is to gain a better under-
standing of the effects that various market perturbations and assump-
tions will have on landowner behavior. Although market situations
differ from the idealized assumptions used in the discussion presented
above, this does not alter the usefulness of the basic results. What is im-
portant is that the Faustmann model presents a standard neoclassical
approach to management that is internally consistent with general
economic analysis.

Some authors have criticized the general applicability of the
Faustmann model in an operational sense. Duerr and others (1956)
discuss the difficulty of finding easy marking guidelines for
fieldworkers to use, whereas Grainger (1968) discusses the fact that
modern forestry organizations need more accurate accounting to value
their land than the Faustmann formulation offers. Neither argument is
particularly persuasive and both can lead to incorrect conclusions when
indiscriminately applied.

The rest of this report discusses attempts to broaden the model by
deriving the comparative static results of the formulation (Chang 1983;
Clark 1976; Graham-Tomasi 1983; Howe 1979; Hyde 1980; Jackson
1980; Morgan 1974) or extending the model to more accurately reflect
real world conditions. Extensions of the model have examined the ef-
fects on the optimal timber rotation of such factors as: externa
(Berck 1981; Bowes 1983; Calish and others 1978; Hartman 1976;
Nguyen 1979; Strang 1983), risk and uncertainty (Kao 1982; Martell
1980; Nostrom 1975; Reed 1984; Routledge 1980), changing price and
cost assumptions (Bare and Waggener 1980; Gregersen 1975; Hardie
and others 1984; McConnell and others 1983; Newman and others
1985), taxation (Chang 1982; Klemperer 1979; Pearse 1967), and im-
perfect markets (Murphy and others 1977; Nautiyal and Fowler 1980).

16


Faustman n (1849) and Pressler and Ohlin (Lofgren 1983). Extensions
of the problem include (1) taxes, (2) once and for all changes in
biotechno logy, and (3) regularly improving biotechnological change.

Kao, Chiang. 1982.
Optimal stocking levels and rotation under risk. Forest Science
28(4):7 11-719.
A probabilistic dynamic programming model is constructed to evaluate
rotation and thinning under a criteria of ..aximized mean annual incre-
ment. Increasing variances of the growth prediction lead to shorter op-
timal rota tions, lower stocking levels, and lower mean annual
increments.

Kao, Chiang. 1984.
Optimal stocking levels and rotation under uncertainty. Forest
Science 30(4):921-927.
Using a maximum sustained-yield criterion, the paper models the op-
timal stocking control when the growth function of a species is initially
unavailable and the decisionmaker is risk neutral. As with risk (Kao
1982), uncertainty leads to lowered stocking levels and lowered mean
annual increments.

Kemp, Murray C.; Long, Ngo V. 1983.
On the economics of forests. International Economic Review
24(1): 113-131.
Using land and labor as inputs, discusses optimum rotation length in
terms of a welfare maximizing central planner. In a number of cases
where labor is redundant, the Faustmann rotation is asymptotically op-
timum. If land is redundant, the Jevons (one period) rotation is
asymptotically optimum. In other cases, it may be optimal to diversify
rotation Lengths to keep both land and labor occupied.

Kemp, M urray C.; Moore, Elvin J. 1979.
Biological capital theory: a question and a conjecture. Economic
Letters 4(2):141-144.
It is shown that for single-tree plots, Faustmann’s linear utility func-
tion, or even a strictly convex function, the optimum rotation is a peri-
odic constant. For a strictly concave function, however, the cutting
times on the plots across the forest spread out, leading to a sustained-
yield harwest scheme.

Klemperer, W. David. 1979.

Inflat-ion and present value of timber income after taxes. Journal
of Forestry 77(2):94-96.

39


Discusses the effect and characteristics of the land opportunity cost
and its importance in the determination of the optimal rotation. If the
function for the marginal value growth has a steep negative slope, the
difference between the present net worth and the Faustmann rotation
will be small. However, the flatter the slope, the greater the effect on
the rotation.

Hall, Dale O. 1983.
ancial maturity for even-aged and all-aged stands. Forest

Sci ence 29(4):833-836.
Generalizes the Bentley-Teeguarden (1965) present net worth formula-
tion to the case of an all-aged forest stand. In contrast to Chang
(19839, the stand is viewed immediately after rather than prior to har-
vest. This procedure seems to work better than the use of yield tables
for stand simulation.

Hardie, lan W. 1984.
Comparative rents for farmland and timberland ina subregion of
the South. Southern Journal of Agricultural Economics 16(4):45-
53.
Uses the framework developed in Hardie and others (1984) to com-
pare the potential rents of forest and agricultural lands. With 50-year
projections of costs and prices, the expected dynamic equilibrium
results with respect to the behavior of the optimal rotation age is
achieved. For high-site land, forestry compares quite favorably with
agriculture at all reasonable discount rates while on lower sites the dis-
count rate assumes a larger role in determining the relative rents.
Results call into question the validity of relying on forestry as only a
residual land use.

Hardie, lan W.; Daberkow, Julie N.; McConnell, Kenneth E. 1984.
A timber harvesting model with variable rotation lengths. Forest
Science 30(2):511-523.
In order to include changing prices and costs, the Faustmann formula
is modified into a dynamic optimization framework, Optimal rules are
developed for rotation determination where an eventual steady-state
condition is assumed. In comparisons of rotation lengths and present
values when changing prices are forecast, ignoring the prices leads to
rotations that are too short and to large revenue losses. Also, changing
intere st rates of the basic Faustmann formula to cover value changes
will not work to get closer to optimal rotation lengths.

Hartman, R. 1976.
The harvesting decision when the standing forest has value.
Economic Inquiry 14(1):52-58.

Qn


stant prices, had been chosen. Later rotation lengths are found to
decrease over time to a steady state as the relative cost effect decrease
A dynamic optimization model is presented that reflects these theoreti-
cal results.

Nguyen, D. 1979.
Environmental services and the optimum rotation problem in
forest management. Journal of Environmental Management
8(2):127- 136.
Builds on the Hartman (1976) model, with a modified selection harvest-
ing system so that the current total benefits from a stand never drop to
zero but rather to some minimum level. The optimal rotation
lengthens in the presence of environmental services but generally not
to infinity as the optimum harvest time must lie on the rising part of
the growth curve.

Nostrom, Carl J. 1975.

A stochastic model for the growth period deci:

Swedish Journal of Economics 77(3):329-337.
For a one-period harvest with price fluctuations as the only form of un-
certainty, the paper develops a model for examining the rotation
decision. The optimal stochastic policy leads to a higher present value
than if uncertainty is ignored. It is unclear from reading the article but
it appears the increase comes about from shorter rotations in the
stochastic case.

n in forestry.

Pearse, Peter H. 1967.
The optimum forestry rotation. Forestry Chronicle 43(June):178-
195,
Uses a classical derivation of the Faustmann equation based on the Gaf-
fney (1957) study to show that the rotation giving the highest land rent
is the optimum. Land rent is defined as the annualized value of the in-
finite stream of like rotations. Since land is the fixed factor in the
forestry enterprise, its return must be maximized. At the optimum, the
change in marginal stumpage value must equal the annual opportunity
cost plus the interest on the growing stock at the optimum. Other is-
sues examined include: (1) fixed annual costs, (2) precommercial and
commercial thinning, and (3) property and severance taxes.

Reed, William J. 1984.
The effects of the risk of fire on the optimal rotation of a forest.
Journal of Environmental Economics and Management 11(2):180-
190.

Develops a Faustmann type model where fires occur with a Poisson dis-

tribution. With a zero salvage value for the remaining stand, the op-


Maximization of the Rate of
Growth of Capital (V6)

This criterion is attributed to Boulding (1955) and commonly
known by such terms as “internal rate of return" (IRR) or "rate of
return” (ROR) maximization. It assumes that capital, not land, is fixed
and it returns all rents to the capital investment and maximizes the rate
of return on that investment. It assumes that the returns can be rein-
vested at this maximal rate forever and thus has an initial formulation
similar to the Faustmann formulation of (9). The equation is maxi-
mized with respect to the interest rate, however. The optimal formula-
tion is:

it

(23) V6 = max(i) {[pQ(T)e" - wEV/(1 eet} =0

where i = IRR.

Solving for i and then taking the first derivative of that expression,
we have:

(24) i* = max(T){(1/T)In[pQ(T)/wE]} = Ov/Q(T)
(25) pQr

where the star signifies the maximum IRR. The rotation length is op-
timal when the rate of growth is maximized. This is then the maximum
rate of return of the IRR. Ifi* > r, the IRR criterion will give a
shorter rotation than the PNW criterion. Ifi* < r, then no investment
will be made as a negative PNW arises.

i*pQ(T)

There are many conceptual problems with the use of this
criterion. The primary problem is that maximizing the IRR maximizes
something besides the owner’s wealth. Although there may be a rela-
tion between the two, it is not necessarily direct. IRR maximization as-
sumes that the amount of land available for forestry is infinite and that
access to all capital markets is closed. Thus, the owner will continue to
invest all the returns from the original investment and never consume
from the
earnings.

When the IRR is greater than the interest rate, the potential total
earnings from investment by this criterion are infinite. Since infinite
returns do not exist, market forces must act to raise the interest rate
and drive the present value of the investment to zero. Thus, this
criterion leads to either an absurd conclusion, infinite wealth incre-
ment, or a trivial one, zero wealth increment (Hirshleifer 1970).

10


The Forest Service, U.S. De-
partment of Agriculture, is dedi-
cated to the principle of multiple
use management of the Nation's
forest resources for sustained
yields of wood, water, forage,
wildlife, and recreation. Through
forestry research, cooperation
with the States and private forest
owners, and management of the
National Forests and National
Grasslands, it strives—as di-
rected by Congress—to provide
increasingly greater service to a
growing Nation.

USDA policy does not permit discrimination because of
race, color, national origin, sex or religion. Any person
who believes he or she has been discriminated against in
any USDA-related activity should write immediately to
the Secretary of Agriculture, Washington, D.C. 20250.




Nostrom (1975) examined uncertainty with respect to price, and
Kao (1982 and 1984) looked at risk and uncertainty in the growth func-
tion to see how the optimal rotation was affected. Using the MSY
criterion, Kao found that increasing uncertainty in the growth function
leads to a shorter rotation, Nostrom, using the PNW criterion, found
that ignoring price uncertainty leads to lower net present values and an
uncertain change in the optimal rotation

Work in the general area of value uncertainty has been hampered
by the relative intractability of the problems to simple mathematical
solutions. Therefore, it is necessary to use large amounts of computing
effort in order to solve the dynamic optimization problem for specific
cases. Kao found that solving the deterministic case took just 3 seconds
of computing time, whereas the uncertain cases took from 235 to 365
seconds. As faster, more powerful computers become available, it can
be expected that more research into this topic will be performed.

A tinal form of uncertainty that has been extensively examined in
the silvicultural literature, but rarely with regards to optimal rotation
effects, is the effect of fires or other natural catastrophes that can
either alter or destroy the forest’s growing stock. Martell (1980), Reed
(1984), and Routledge (1980) examined the rotation effects by propos-
ing various stochastic modeling schemes to explain the occurrence of
fires. Using a simple probability structure, Routledge (1980) found
that if the burned stand could be harvested without value loss after the
fire, the optimal rotation would be lengthened from the no fire case.
This result pertains because the land-rent term in the Faustmann solu-
tion (equation 16) is decreased due to occasional premature harvests;
thus, a lower growth rate is necessary to maintain the basic equality.
Risk of stand loss beyond salvage, however, fairly substantially
decreases the rotation length.

Martell (1980) used a Markov decision model with constant fire
probabilities to determine the general effects of fire and found that
shortened rotations would occur. Using an analysis with fire inde-
pendent of stand age and the destruction total, Reed (1984) treated the
average rate at which fires occurred as a risk factor that could be added
to the interest rate. Thus, the shortened rotation length is due to a
higher effective interest rate. In tests of more complex fire cases, he
found the optimal rule less clear,

Market Failure - Externalities

A major complaint leveled at all the economic criteria used for rota-
tion determination is that insufficient attention is given to the non-
market benefits derived from growing timber. Critics claim that since
timber is often the only output from the forest producing direct dollar

aa


Discrete analysis shows that inflation will not affect the real returns
from timber investments with property or yield taxes. Investments with
capital-gain taxes are affected by inflation, leading to an overestimate
of real returns if inflation is ignored. Inflation bears more heavily on
short-term investments because the purchase cost makes up a greater
share of the basis.

Lofgren, Karl G. 1983.
The Faustmann-Ohlin theorem: a historical note. History of Politi-
al Economy 15(2):261-264
Recalls how Bertil Oblin, while still a student, independently derived
the Faustmann formulation for optimal rotation determination in
order to be admitted into a graduate seminar led by Eli Hecksher.
Needless to say, Ohlin was admitted to the seminar.

Lofgren, Karl G. 1985.
Effect on the socially optimal rotation period in forestry of biotech-
nological improvements of the growth function. Forest Ecology
and Management 10(2):233-249,

Analyzes how genetic progress affects the optimal rotation period and

the value of the forest land. A one-time change in genetic quality is

similar to a price change and leads to a shorter rotation and a higher
land value. A regularly improving technology leads to an initially
shorter rotation that increases over time. The final conclusion is that it
is not the improved growth as such but the known possibilities of future
improvements of the growth function which motivate a shortening of
the optimal rotation period.

Lyon, Kenneth §. 1981,

Mining of the forest and the time path of the price of timber. Jour-
nal of Environmental Economics and Management 8(4):330-345.
With Hotelling’s (1931) theory of mine depletion as a starting point, an

optimization model is used to analyze the rate of growth of price in
situations of zero and increasing transport cost for the cutting of old-
growth timber. With zero costs, the standard result of price increase
(p) equal to interest (r) holds only in the case of no growth or regenera-
ton. If the old growth is growing, p< r. If regeneration costs are con-
sidered, p>r. For the model with transport costs, the first case has the
same answer as the no-cost case only if it is the growth of net price
(price - marginal cost) that is compared to r. However, with a con-
tinuously increasing cost function, the increase in p<r. Finally, at a
stationary point, price will no longer grow and the Faustmann situation
is optimal.

40


The Optimal Forest Rotation:
A Discussion and Annotated Bibliography

ABSTRACT

The literature contains six different criteria of the optimal forest rotation:
(1) maximum single-rotation physical yield, (2) maximum single-rotation an-
nual yield, (3) maximum single-rotation discounted net revenues, (4) maxi-
mum discounted net revenues from an infinite series of rotations, (5) maxi-
mum annual net revenues, and (6) maximum internal rate of return. First-
order conditions for maximization show the rotation effects of the criteria.
Various authors have extended basic models and discussed effects of exter
nalities, imperfect markets, changing parameters, and taxes.

Keywords: Forest economies, forest valuation, harvest scheduling.

Introduction

What is the optimum rotation length for a forest stand? This ques-
tion has been debated for more than a century, and numerous compet-
ing theories and practices have been developed in attempts to answer
it. These theories have used combinations of physical, biological, and
economic criteria for determining the optimal rotation. Confusion
over what is the proper criterion continues to this day. At one extreme,
as witnessed by the legal requirement of the National Forest Manage-
ment Act of 1976, the Forest Service, U.S. Department of Agriculture,
must manage national forests on a maximum physical sustained-yield
rotation.” Meanwhile, economists and other critics argue for economic
criteria that maximize discounted net revenues and generally imply
much shorter rotations (Hirshleifer 1974; Hyde 1980; Samuelson 1976).

In the economics literature, the Faustmann (1849) optimal decision
rule for maximizing discounted net revenues is accepted as superior for
society under many conditions (¢.g., Mitra and Wan 1985, 1986). In
order to understand these conditions more clearly, a number of impor-
tant topics related to the model's results have been examined. These
topics include the delineation of the comparative statics of the optimal
solution and the loosening of the assumptions that characterize the op-
timal formulation. By extending the results in these directions, re-
searchers have been able to make much stronger statements about the
behavior of the forest sector in the economy.

"US Department of Agriculture, Forest Service. 1983 rev. The principal laws relating
to Forest Service activities, Agric. Handb. 453. Washington, DC. (p. 450). 591 pp.


Compar és the internal rate of return (IRR) and maximization of net
present value (NPV) criteria. The IRR and NPV analyses assume that
cash-flows will be invested at the rate of return and the discount rate,
respectively. Thus, IRR holds as a criterion only in the situation when
excess amounts of identically productive timberland are available.

Gaffney, M. Mason. 1957.

Concepts of financial maturity of timber and other assets. Agric.

Econ. Inf. Ser. 62. Raleigh: North Carolina State University.

105 pp.
The first definitive analysis of the various methods for gauging finan-
cial mat urity of forest stands. The Faustmann derivation is compared
with the Jevons/Allen/Fisher single rotation, Boulding’s maximization
of internal rate of return, and other variants. In mathematical terms,
the Faustmann rotation is clearly shown as the superior to the other
criteria which make special assumptions or have special intentions.
The marginal conditions for optimality are set, and the basic compara-
tive static results regarding price, costs, and tax effects are analyzed.

Goundry, G.K. 1960.

Forest management and the theory of capital. Canadian Journal of

Political Economics 26(Aug.):439-451.
Compares maximum sustained yield (MSY), maximum discounted net
revenue (present net value), maximum present value of the infinite
stream (land expectation value), and maximum rate of growth of capi-
tal (internal rate of return (IRR)) criteria for rotation length deter-
mination. The expected ordering of optimum rotation lengths is
reported with IRR shortest and MSY longest. The author assumes
that capital is the fixed factor in production and as such, an owner
would want to maximize the return to it rather than the return to the
land. He claims that land is not the fixed factor because it is not
managed intensively enough to justify that assumption. This logic was
shown to be incorrect by Samuelson (1976). The article does include a
good literature review.

Graham-Tomasi, Ted. 1983.
The comparative statics of the Faustmann model of forest manage-
ment. Staff Paper P-83-23. Minneapolis: University of Minnesota
Department of Agriculture and Applied Economics. 32 pp.
The author uses a model that differs from Chang (1983) in the manner
in which regeneration costs are included (Chang more explicitly) and
proves similar results. The model is then used to describe timber sup-
ply in cases of restricted (short run) or unrestricted (long run) entry.
Changes in volume supplied in response to a demand shift depend on
the original rotation age and may increase or decrease, depending on

R2


Comparative Statics

The basic results given by authors who have looked at the effect of
changes in the exogenous parameters of the model, on the optimum
rotation age, and silvicultural effort are shown in table 2. The two most
thorough presentations are by Chang (1983) and Graham-Tomasi
(1983), who show general agreement in their results, though they used
slight ly different formulations in setting up the problem. Both authors
denote volume as a function of not only time but silvicultural effort as
well. Chang uses seedling planting density as a proxy and divides the ef-
fort into site preparation and planting variables while Graham-Tomasi
uses Only a single, general effort variable for his formulation.

The major cause for the uncertain answers in table 2 is that it is
often unclear, due to the use of a general growth function, whether
time and effort can be considered as substitutes or complements and to
what degree. Therefore, it is necessary in those uncertain cases to em-
pirically estimate a stand’s growth response for the absolute determina-
tion of the sign. Also, in the case of changes in the interest rate, as
shown in (16), both factors in the right-hand side are affected, but in
different directions. An increase in the interest rate increases the cost
of holding the stock (rpQ(T)) while lowering the cost of delaying the
harvest (rV5 decreases), since the value of managing the land for tim-
ber (V5) decreases with an increase in the interest rate.

One use for these comparative static results is to help determine the
ultimate supply responses from changes in the exogenous variables.
For instance, Hyde (1980) states that since the optimum rotation age
will generally decrease as price increases, each acre in a managed
forest will produce less volume on a sustained-yield basis. If acreage is
held fixed, the ultimate supply elasticity from a price increase would
therefore be negative. Hyde counters this argument in both the short
and long run, The short-run response will necessarily be positive as
timber owners reduce the rotation length of their stands by increasing,
temporarily at least, the amount of timber cut. The long-run supply
response will likely also be positive as two other factors will come to
bear to increase total production: (1) that increasing land-intensive sil-
vicultural effort becomes profitable, increasing yields on all acres, and
(2) more acres that were previously submarginal or in other uses are
brought into timber production. Binkley (1987) found that with a fixed
land base, the supply curve will be negatively related to price only for
rotation lengths less than the MSY rotation.

Another use of the comparative static results is in the analysis of the
effects of different tax policies on landowner behavior (Chang 1982;
Pearse 1967). The results for the major forestry taxes are:

17


M urphy and others (1977) examined the more common situation
of impe rfect capital markets. In this situation, either the amount of
money that can be borrowed is restricted or a multiplicity of borrowing
and lending rates apply. Their policy criterion was the maximization of
the liquidation value of the firm with a finite 100-year planning
horizon. They found that higher borrowing than lending rates lowered
the optimal rotation length and greatly reduced the liquidation value of
the firm. This result was caused by the need for the landowner to
restrict early borrowing and acreage planted in order to build up suffi-
cient ca pital reserves to finance increased acreage levels.

Summary, Conclusion,
Future Research Areas

One would think that a topic with such a long written history had
been completely described and understood by now, but the theory of
the optimal forest rotation continues to attract new ideas and research.
The majority of research centers on the comparison of the various
criteria for rotation determination and the derivation of the compara-
tive static conditions of the basic model. Although these problems
have been thoroughly covered, relatively scant attention has been
placed on extending the model by challenging its assumptions or ex-
tending them to better reflect actual market conditions. This situation
is changing. Research begun in the last 10 years is the first to examine
many of these extensions. The increasing mathematical sophistication
of both researchers and those using the research make it likely that this
trend will continue.

Likely, new areas of research include changing price scenarios
and the effects of uncertainty on the optimal rotation. Some good
work on these topics has already been done, but both a clearer
generalization of the problem and specific analyses using various
production functions are needed to gain a better understanding of the
changes that uncertainty can cause. Some topics that could be inves-
tigated for optimal behavior could be: uncertainty about the value
parameters, either in a pure stochastic sense (where the mean and the
variance of the parameters are known) or in a Bayesian sense (where
neither the mean nor the variance is known); the assessment of risk on
forestry rotations and whether risk premiums are assigned to the
forestry situation.

Much of the research that is needed to deal with these topics
would necessarily be more applied and computer intensive. Program-
ming methods, which can optimize not only the harvest timing but
other silvicultural treatments as well, are beyond the scope of this
report. The object here is to investigate the theoretical underpinnings

25


tinued price increases for the forseeable future.” The price level, as
shown by Lyon (1981), can rise at a rate greater than, less than, or
equal to the interest rate, depending on assumptions relating to
transportation costs, the rate of volume growth, and the amount of
regene ration.

This information was used by McConnell and others (1983) to ex-
amine the effects of varying real prices and costs on the optimal rota-
tion decision. Their model used a dynamic optimal control framework
in which the land could, at some point in the future, switch from
forestry to agricultural production when the costs of forestry became
too high. They found that if price is increasing at a constant rate (q)
and planting costs (c) are increasing al a faster rate (c >q), the op-
timal rotation is no longer constant but increases over time. Ifc <q,
then the optimal rotation decreases over time. These results pertain
because with costs rising at a greater rate than prices, it is worth it to
the owner to have shorter rotations earlier on because the relative costs
of later harvests will have increased. With prices rising faster than
costs, the owner postpones early harvests in order to receive the price
benefits of delay but then decreases the rotation length due to discount-
ing effects. The net price rise must also be less than the discount rate.
If it does not do so, harvesting would be delayed indefinitely because
the present value of the stand would be continuously increasing.

Hardie and others (1984) extended this model to examine the effect
of ignoring changing prices on the net forest returns. When price
changes were ignored, they fould that large losses resulted as, incon-
gruously, too short rotation lengths were used with the Faustmann
equation, Newman and others (1985) attributed the lengthened op-
timal rotations to two countervailing effects. Rising prices lead toa
shortened rotation, but rising prices in relation to regeneration costs
serve to lengthen the rotation because delay reaps the benefits of
progressively decreasing relative regeneration costs.

A related form of parameter change occurs from genetic or other
biotechnological progress, which causes shifts in the biological produc-
tion function. Lofgren (1983) and Johansson and Lofgren (1985) ex-
amine one-time shifts in the yield function as well as continuing
exponential growth in productivity. Their results are similar to those
found by Newman and others (1985) because continuing yield increases
have the same effect as continuing price increases. Thus, rotation
lengths initially decrease and then gradually increase over time.

4

U.S.Department of Agriculture, Forest Service.1982. An analysis of the timber
situation in the United States: 1952-2030. For. Resour.Rep. 23. Washington, DC.
499 pp. (p. 205)

21


the -elation of the optimum rotation length to the maximum su:
yield rotation. In addition, if the rate of change of price is pos
then rotation length increases but effort decreases.

tained-
iv

Grainger, M.B. 1968.

Problems affecting the use of Faustmann’s formula as a valuation

tool. New Zealand Journal of Forestry 13:168-183.
The per-acre land values of the Faustmann equation may not be ap-
plicable in the situation of a large forest enterprise that is just begin-
ning afforestation operations. Since the Faustmann formula gives a
value nearly twice as large for a large tract of land than a financial ac-
counting, the latter method is recommended for valuation with explicit
coverage of costs. The major problem is that the Faustmann equation
was developed for small ongoing operations and is not applicable to
today’s large forest enterprises which have more intricate cost and
revenue formulations. Other problems stated are delayed planting for
large operations, capitalization procedure, administrative costing, et
but the value of the Faustmann calculation is recognized for compara-
tive purposes.

Gregersen, Hans M. 1975.
Effect of inflation on evaluation of forestry investments. Journal
of Forestry 73(9):570-573.

Elementary article explains how inflation does not affect real returns in

an investment analysis. Problem for forestry investments is that real

timber returns on investments are often compared with nominal alter-
native investments (i.¢., savings deposits) and as a result show poorly.

Gregory, Gordon R. 1972.

Forest resource economics. New York: Ronald Press Co. 548 pp.
Chapter 14 presents a discussion of financial criteria for rotation age.
Soil rent, forest rent, and financial maturity formulations are compared
through graphical analysis. Management objectives of various land-
owner types are also compared.

Guttenberg, Sam. 1953.
Financial maturity versus soil-rent. Journal of Forestry
51(10):714.
Rebuts a paper by Worrell cri ing the financial maturity criterion.
The primary benefit of the financial maturity criterion is ease of use in
the field.

Haley, David. 1966.
The importance of land opportunity cost in the determination of
financial rotations. Journal of Forestry 64(5):326-329.


Table 1.—-Mathematical determination of the optimal rotation
age for loblolly pine by solution of the six optimal conditions

Age Q(T) Q@,  Q(T)/T ,/Q(T)  Q(T)=wE/p i* A/T

4 31 15 7 0.484 ° = 0.25 0.276
6 105 36 17 343, ~ 18 coe +16 +193
8 338 116 = 42 2343 3.05 0.002.125 +152
10-715 «:188 7k 263 453 +079 «10 2127
121,175 23097 +196 1263 s111 08 eee
141,665 = 244118 2147 178 2122.07 2099
16 2,147 241 134-76 (112 }# 4319123] .06 091
18 2,600 226 144 7087 :

20 3,013 206 = 150 068

22 3,383 184 153, 2054

243,707 162,154 73 LON

26 = 3,988 T2{140 Jeol 153 035

28 4.229 «120151 028

30. 4.433 101147 023

32 «4,603 84 143 018

34 4,743, 69 139 015

36 4.855 56134 012

38 4,943 43° 130 009

4o 5,009 33 125 007

42 5,056 23° 120 +005

44 5,085 1h 115 +003

46 5,100 7 110 2001

48 5,101 TI[_0} 106 ° . .

50 5,090 ~6 101 - -- 056 02 +055

p = $0.30 per cubic foot, wE = $100, and r = 5 percent. Connected values
indicate approximate equality of the functions.

Source: Volumes are calculated from equations in Francis Schumacher and
Theodore Coile's "Growth and Yields of Natural Stands of the Southern
Pines." Durham, NC: T.S. Coile Inc. 1960. 115 pp.

TI = MAXIMUM GROSS YIELD
T2 = MAXIMUM SUSTAINED YIELD

1600 + T5= FOREST RENT 7
T6 = INTERNAL RATE OF RETURN pact)
2 7
& 1200 F |
4 |
az |
fo) | |
8 |
= g00} \|
” if
WwW |
= |
G 400 |
> i" |
i | |
aa | |
wE-$100) | |
Oe
0 io T 20 30 40 50

LENGTH OF ROTATION (YEARS)

Figure 1.--Total yield determination of the optimal forest rotation.


that it specifies a different set of productive opportunities for the land-
owner than does the Faustmann formulation.

This criterion assumes that the owner will maximize the PNW of
the forest investment but does not account for future uses of the land
when the rotation is completed. Even though harvesting returns are
received at discrete intervals, their value is discounted at a continuous
rate. Thus the equation includes the exponential function (€) in order
to discount future returns, which may then be compared with the initial
investment.

The formulation is:
(7) V3 = max(T) {pQ(T)eTM! - wE}
The first-order condition is:
(8) V3r = [pQr- rpQ(T)Je"! = 0
(9) pQr = rpQ(T)
(10) Qr/Q(T) =r

The rotation length is optimal when the rate of change of growth
is equal to the market interest rate or when the VMP is equal to the op-
portunity cost of delaying the harvest one period. Neither price nor
costs affects the optimal solution, but the PNW is affected by these fac-
tors. Its major error is that no opportunity cost is associated with the
land itself. Thus, all rents accrue to management and none to the fixed
factor, land.

Maximization of the Discounted Net
Revenues From an Infinite Series
of Like Rotations (V4)

This formulation was derived first by Faustmann (1849) and later
by Pressler, and finally by Ohlin (Lofgren 1983). It is known to
foresters by such terms as "soil rent," "soil expectation value" (SEV), or
"land expectation value" (LEV). It is the superior decision criterion of
those presented for several reasons. It assumes that the site will stay in
forest production in perpetuity so that there is an explicit awareness of
the effect that present decisions have on future possibilities. It op-
timizes the use of all inputs according to their guiding market prices
and leaves any excess returns as a rent to the site. Thus, unlike the
PNW formulation, this one accounts for changes in stumpage price and
unit silvicultural cost in computing the optimal rotation length. The
optimal solution includes a separate term for the land-rental charge.
This value often is not large for long rotations, but its inclusion in the
rotation determination is nonetheless necessary because it assigns a


Analyzes the comparative statics of timber under a very general
framework. Yield is set as a function of rotation length and planting
density, and a Faustmann formulation, including site preparation and
planting costs, is used to examine the following conditions: (1) higher
site preparation costs, (2) higher planting costs, (3) higher stumpage
prices, (4) higher interest rate, (5) ad valorem tax, (6) site value tax,
and (7) yield tax. Uncertainties in the analysis come about due to the
generality of the growth function and thus uncertainty with respect to
the resulting shape of the yield function.

Chang, Sun Joseph. 1984.
Determination of the optimal rotation age: a theoretical review.
Forest Ecology and Management 8:137-147.
Basically, a modernization of earlier works by Gaffney (1957) and
Bentley and Teeguarden (1965). The four major competing criteria
for rotation length (soil rent, present net worth, forest rent, and the
traditional biological model) are presented in such a way that makes
derivation understandable to foresters with as little as one semester of
calculus. The are clear and the explanation of the results is very
sound, A good synthesis of the material.

Clark, Colin. 1976.
Mathematical bioeconom' the optimal management of renew-
able resources. New York: John Wiley & Sons. 352 pp
Chapter 8 presents the Faustmann model and the first-order condi-
tions for optimality. The left- and right-hand sides of the optimality
equation are explained graphically, The book is an excellent introduc-
tion into optimal control techniques and their uses for renewable
resources (particularly fisheries) management.

Comolli, Peter M. 1981.
Principles and policy in forestry economics. Bell Journal 12(1):300-
309.
From the basic Faustmann formulation, casually derives an industry
supply function to analyze policy issues relating to forest taxes. A tax
will raise prices and lower quantities because it acts like an increase in
costs to the producers. An excise tax will raise prices more than a
property tax if land’s share is less than 50 percent of total income.

Comolli, Peter M. 1984.
A proposition on imperfect stumpage markets. Land Economics
60(3): 297-300.
Extends the argument of Nautiyal and Fowler (1980) to that of a mo-
nopolist with imperfect knowledge of future demand. In such a case,
the monopolistic rotation length may not always be shorter than the

31


Bentley, William R.; Teeguarden, Dennis E. 1965.
Financial maturity: a theoretical review. Forest Science 11:76-87.
A more basic explanation of Gaffney (1957) using graphs. The paper
compares the primary models of financial maturity: (a) one rotation
maximum revenue model with and without interest, (b) Faustmann
equation model, and (c) Boulding’s maximization of internal rate of
return. The paper has little math and no proofs but presents a clear
nontechnical exposition of the models in an economic framework.

Berck, Peter. 1979,

The economics of timber: a renewable resource in the long run.

Bell Journal 10(2):447-462.
Analyzes two questions: (1) Have private timber owners over time
depleted stocks at a faster-than-optimal rate? and (2) Have public
owners depleted stocks at a slower-than-optimal rate? Using rational
expectations for future prices, Berck sets up a supply and demand sys-
tem which essentially assumes that supply equals demand throughout
time. He finds that private owners have been using a 5 percent dis-
count rate which is lower than for other industries and lower than the
10 percent rate that he uses as a determinant for past overexploitation.
He hypothesizes that nonmarket goods which public owners value
could explain differences between public and private cutting rates.

Berck, Peter. 1981.
Optimal management of renewable resources with growing
demand and stock externaiities. Journal of Environmental
Economics and Management 8(2):105-118.
Applies optimal control techniques to situations of a renewable
resource. In the general model, an optimal solution occurs when the
rate of change of net price plus the marginal production equals the in-
terest rate in the case of constant costs. With growing demand, a
steady state will be reached as long as demand growth is less than the
interest rate. In the presence of positive externalities, either a subsidy
or a tax can be used to bring about the desired increase in stocks to give
the socially optimal use over time.

Bierman, Harold, Jr. 1968.

The growth period decision. Management Science 14(6):302-309.
Compares present value maximization and internal rate of return
(IRR) solutions. Problems associated with using IRR as an investment
guide with respect to scale and intensity of forest management are dis-
cussed, The presentation leads circuitously to the Faustmann rule
without stating it as such.

28


2. Perfect certainty regarding the stand’s growth function, future
market prices, interest rates, and costs. All are assumed constant.

3. Access to capital markets is unlimited, and mone:
rowed or lent at the same market interest rate (r). This assumption is
dropped for rate of return maximization criterion.

4. Net stumpage price (p) is not a function of tree quality but
rather is Constant per unit of volume produced. This assumption is not
completely necessary, but its inclusion avoids the potential problem of
multiple Optimum rotation ages. Multiple optima can arise when
management objective! favor different end products or when discon-
tinuous jumps in prices create nonconcave sections in the total revenue
function. Also, only timber is valued; returns from other nonmarket
and public goods derived from the forest are ignored. Finally, the tim-
ber is sold in a competitive market; any quantity of timber output can
be sold without changing the market price.

5. Regeneration costs (wE, where w is the cost per unit of silvicul-
tural effort, E) are the only expenses and occur immediately after har-
vesting. Intermediate management costs and returns can be included
without loss of generality. A substantial body of literature has ex-
amined the effects of thinning on optimal management strategies
(Reed 1986).

6. Timber production is a function of time alone or of time and
silvicultural effort. Early analyses (Gaffney 1957; Goundry 1960;
Pearse 1967; Samuelson 1976) describe total merchantable forest
production (F) as a concave function of time (T) only. Thus:

F = O(T)

where Or>0
Orr <0

where the subscripts denote the first and second partial derivatives of
the timber production function.

Later analyses examine the comparative statics of the optimal rota-
tion choice to assess the management tradeoffs inherent in the decision
process. Quantity is not a function of time alone but also of silvicultural
effort (Chang 1983; Graham-Tomasi 1983; Hyde 1980; Jackson 1980;
Morgan 1974). Thus:

= Q(T.E)
where: Orand Qe > 0
Orr and QrE < 0


In the first section of this report, I discuss six alternative criteria for
determining the optimum rotation length: (1) maximum physical yield
of a single rotation, (2) maximum annualized yield of a single rotation,
(3) maximum discounted net revenues of a single rotation, (4) maxi-
mum discounted net revenues from an infinite series of like rotations,
(5) maximum undiscounted net revenues from a regulated forest, and
(6) maximum rate of growth of capital from an in| inite series of like
rotations. The ordering of these criteria assists in their understanding.
The first two formulations give purely biological and physical manage-
ment rules, while the other four combine economic and biological infor-
mation of differing complexities. Excellent surveys of the criteria for
the optimal forest rotation have already been performed (Bentley and
Teeguarden 1965; Chang 1984; Gaffney 1957; Goundry 1960; Pearse
1967; Samuelson 1976), and this section only reflects their results. The
econd section of the report discusses major extensions and develop-
ments of these models and suggests directions for future research, The
final section is an annotated bibliography.

Comparison of Criteria for Rotation
Length Determination

An optimal forest rotation criterion should maximize net economic
benefits to either society or the private forest owner. Since competitive
markets and an absence of externalities are assumed in this discussion,
the interests of individual owners and society will coincide (Anderson
1981). Other goals may be to maximize personal or social utility or
maximize physical (cither market or nonmarket) benefits. These goals
may be useful for various purposes, but net benefit maximization allows
for clear comparison of criteria and has been the general (though often
unstated) goal used in the literature.

Assumptions are necessary to 4 comparative forest rotation analysis.
There are six basic assumptions for the present analysis:

1, Even-aged management. Starting from bare ground, a single
stand of trees is grown and all trees are cut at the same time. The land
has an unchanging growth potential, and the technology available for
growing trces does not change. Finally, the land is capable of being
regenerated (at some fixed cost) instantaneously after harvest. The
rotation starting from bare land is not essential. Faustmann (1849)
showed that whether a regulated forest or a single stand is examined,
the optimal rotation length is unaffected and the per-acre returns are
the same (plus the capitalized value of the currently standing trees).
The even-aged management assumption is important. Chang (1981)
and Hall (1983), among others, have shown that the optimal conditions
will change when an uneven-aged stand is managed



300F TI = MAXIMUM GROSS YIELD
T2= MAXIMUM SUSTAINED YIELD

|
E 2507 T6 = INTERNAL RATE OF RETURN

w 200

g 150} / AERC y

g 100- / any or

YIELD
a
Oo 0
Ne
[sx

T= Qty

L nl

are L
Te T2

L fant iit 1 1
ie) 10 20 30 40 50
LENGTH OF ROTATION (YEARS)
Figure 2.--Marginal and average yield analysis of the optimal forest
rotation.

20F

TI = MAXIMUM GROSS YIELD
=) po, 72 = MAXIMUM SUSTAINED YIELO
= Ie" T3 = PRESENT NET WORTH
pOtry=wE

& POT) WE 14 = SOL RENT

IS- fl TS = FOREST RENT
z T T6 = INTERNAL RATE OF RETURN (i*)
=)
ly 10
a
L
{e)
ws 5
qt
a

° Te, 74)

O° ite) 20 30 40 50

LENGTH OF ROTATION (YEARS)

Figure 3.--Marginal determinations of the optimal rotations by the
six Criteria.


Maximization of Annual Yield (V2)

This Criterion, known to foresters as the culmination of mean an-
nual incre ment (CMAI) or the biologically maximum sustained yield
(MSY), produces the greatest total physical product from a site per-
manently managed for forestry. It maximizes the average annual physi-
cal yield per acre. It is the criterion by which the Forest Service is
mandated to manage its timber. The total production, on a long-term
basis, is greater than that of V1 because MSY implicitly assumes that
the site will continue to produce trees after harvest and thus contribute
to the gross yield. Use of this criterion will allow the achievement of a
country’s desired timber production goals with the least amount of land
devoted to forestry (Goundry 1960). It, like V1, assumes zero costs and
interest rates so the desired formulation is:

(3) V2 = max(T) {pQ(T)/T}

The first-order condition is:
(4) V20 = [pOrT - pQ(T)T? = 0
(5) pOr = pQ(T)/T
(6) Qy/Q(T) = VT

At the optimum, the marginal growth rate is equal to the average
growth rate. The problems with this method are that, again, costs and
interest charges are ignored. Since price cancels out of the optimal
solution, the optimal rotation is unaffected by changes in market condi-
tions. Thus, while this criterion is relatively simple to plan and ad-
minister, it is inflexible in its operation.

Maximization of Discounted Net Revenues
From a Single Rotation (V3)

Such great economists as Fisher, Jevons, and Wicksell have
proposed this model as the optimal criterion for forestry rotation deter-
mination. They saw the problem as being related to the classical
economic problem of optimizing wine aging. The model is often simp-
ly called the present net worth (PNW) model, but Duerr and others
(1956) called it the financial maturity model. They claimed it was
easier to use and only slightly less precise than the Faustmann formula-
tion, which will be described later. Gaffney (1957) showed that the loss
in precision is variable and is influenced by a number of factors, includ-
ing the length of the optimal rotation, market price, and interest rates.
Hirshleifer (1970) states this formulation is not incorrect per se, but


returns, its growth requirements receive the highest priority in manage-
ment. Other goods from which society derives utility (often in common
or public form and thus not directly priced) such as water, recreation,
wildlife and other amenity goods, whose highest value rotation may not
concur with the economic rotation, often are not accounted for in the
final management decision. The Forest Service partially justifies its
use of the MSY criterion in this manner. Since the Forest Service is
obligated to manage national forests for multiple use, it is claimed that
longer rotations will best ensure their multiple objectives.

A few authors have looked at the externality problem that arises in
forestry. Berck (1981) discussed the possible use of taxation or sub-
sidies to ensure the optimal production of the external goods, whereas
Nguyen (1979) used a selection management system and developed op-
timal cutting conditions for that situation.

In response to criticisms of using the MSY criterion to ensure the
production of externalities by Hirshleifer (1974) and Samuelson
(1976), Hartman (1976) first adapted the Faustmann equation to in-
clude nontimber amenity benefits. The formulation became:

(26) V7 = max(T) {[pQ(T)e - wE + f/FineTMdny(1-e)}

where V7 is the solution for the maximized joint production of timber
and external benefits, F(n) is the functional form for the growth of the
amenity values derived from the growing forest, and all other terms are
the same as in the previous equations. The first-order condition gives:

(27) [pOr + F(T)\/[pQ(T) - wE + fF(ayeTMda} = 1/(1-e")

which is very similar to equation (15) from the Faustmann formulation.
It was decomposed by Bowes (1983) to show more clearly the effects of
amenity value on rotation age:

(28) alF(T) Sf) F(n)e""da] + (1 a)[pQr/(pQ(T) - WE)|
= -eTM
where a= S,Fe@dedn/[pQ(T) - wE + [UF jean]

If the nontimber share of benefits (a) is low, the optimal rotation
approaches the Faustmann rotation. If a is high, however, the optimal
rotation will be longer or shorter than the Faustmann rotation, depend-
ing on whether the amenity value increases or decreases with stand age.
The formulation also shows that it may be worthwhile to delay harvest
forever, if the rate of growth of amenity values is high enough.

Calish and others (1978) tested the Hartman (1976) formulation
using derived yield equations for seven nontimber goods in the
Douglas-fir timber region. The individual effect of each of these goods

23


United States
Department of
Agriculture

Forest Service

Southeastern Forest
Experiment Station

General Technical
Report SE-48

The Optimal
Forest Rotation:

A Discussion and
Annotated Bibliography

David H. Newman



forest, if the forest is less than the optimal rotation age, it should be cut
when it reaches that age. If, however, it is older than the optimal age, it
may be optimal to never harvest the stand.

Tait, David E.N. 1987.
The good fairy problem: one more look at the optimum rotation
age for a forest stand. Forestry Chronicle 63(4):260-263.
Questions the differences between economic and physical optimal rota-
tions through the metaphor of the good fairy giving an owner the
choice of initial forest conditions. With the use of a heuristic proof,
the paper shows the difficulty of reconciling a sustained-yield forest
with stands managed for optimal economic rotation lengths.

Thomson, Roy B. 1942.
An examination of basic principles of forest valuation. Bull.6.
Durham, NC: Duke University School of Forestry. 99 pp.

An interesting historical account of the German, British, and

American precedents for forest valuation comparing gross-yield, soil-

rent, and forest-rent doctrines. Some of the economic analysi

flawed, but the paper’s main value is chapter 2, which deals with the his-
torical development of forest valuation.

Walter, G.R. 1980.
Financial maturity of forests and sustainable yield concept.
Economic Inquiry 18(2):327-331.
Applies the Faustmann equation to a situation where rotation lengths
are not constant over time and the forest enterprise may eventually
stop. The optimal rule in this case is the old Faustmann rule with ex-
plicit attention given to the lengthening of the rotation.

Wan, Henry Y., Jr. 1978.
A generalized Wicksellian capital model: an application to
forestry. pp.141-153. In: Smith, V.L., ed. Economics of natural and
environmental resource: ew York: Gordon and Breach.
Using Wicksell’s wine-aging model, the author derives the one-period
optimal condition and a variation of the Faustmann rule. The effect of
a negatively sloped demand curve on rotation is examined with relaxed
competitive price assumptions. The discounted marginal revenue over
current cost must equal the imputed rent, and the optimal rotation
length may change over time.

Worrell, Albert C. 1953.
Financial maturity: a questionable concept in forest management.
Journal of Forestry 51:711-714.

46


where R* and T* are the optimal land rent and rotation length, respec-
lively. Solving for R* gives:

(17) R* = [pQ(T*)eTM - wEy(1-e") = (V4)

Dividing (14) by r yields an infinite stream of annual rental payments
that is just equal to the Faustmann formulation or the LEV of the site.
As shown in (14), this rental payment is exactly equal to the oppor-
tunity cost of the land from the right-hand side of (12).

Binkley (1987) has shown that comparing the maximum sus-
tained-yield rotation with the optimal Faustmann rotation length gives
ambiguous results, depending on the relative costs and the interest
rate. In particular, he shows that with positive profits, all that is neces-
sary is an interest rate less than 1/Tmsy, as with the PNW criterion, to
give a longer Faustmann rotation.

Using a similar analysis, it can be shown that the optimal
Faustmann rotation is bounded from above by the PNW rotation.
Reexpressing equation (11) so that it can be compared with the op-
timal condition for the PNW criterion (equation (8)) yields:

(18) Qy/Q(T) = r{1- wE/pQ(T)/1-e) =r

Rearranging terms, we find that in order for the PNW and Faustmann
rotation to be equal in length, it must be the case that:

1-wE/pQ(T) = 1-77
For the PNW rotation to be shorter, it must be the case that:
wE > pQ(T)eTM

This condition implies losses on forestry investments because dis-
counted revenues are necessarily greater than or equal to costs. Thus,
the Faustmann rotation will never be larger than the PNW rotation,
Q.E.D.


value to the opportunity cost of the land from delaying the harvest one
period.

Fa ustmann first derived the formulation to determine just com-
pensation to forest-land owners when their land was appropriated by
the State for conversion to agriculture. Its importance was quickly
grasped by other workers in the field and led to a rather bitter debate
over its use over other criteria (Thomson 1942), The formulation as-
sumes that after a harvest is completed, the site will be immediately
regenerated, and that this process will be continued into the infinite fu-
ture. The resulting stream of net revenues is discounted back to the
present. Since all the variables are fixed, the optimum rotation age will
also be constant (Heaps and Neher 1979). The formulation is:

(11) V4 = max(T) {[pO(T)e- wE](1 + eT + eT + 2}

max(T) {{pQ(T)eTM! - wEV(1 - eTM)}
The first-order condition is:

(12) V4r = {(1-e")[pQr- rpa(T)]
-1[pQ(T)eTM - wE}}/(1 - eT)?

(13) Qy/[Q(T) - wE/p] = /-e")

(14) pQr = r(pQ(T)- wE)/(1-eTM")

(15) pOr = rpQ(T) + r[pO(T)e! - wEV(1 -e")
= rpQ(T) + r(V4)

The rotation age is optimal when the VMP from letting the stand
grow one more period is equal to the cost of holding the growing stock
(the amount of interest that could be earned by harvesting the stand
and investing the returns at the interest rate for one period) plus the
cost of holding the land (the interest costs of delaying revenues from fu-
ture harvests one period). As the planting costs increase or the price
decreases, the rotation length increases.

Samuelson (1976) has shown that the Faustmann formulation can
be expressed as in equation (6) if an explicit land-rent term is included.
This equation must sum to 0 because the land rent (R) will absorb all
excess profits in a competitive market. Thus:

(16) 0 = max(T) {pQ(T)e"!- wE-R S edt}

pa(T*ye -wE-R*[1-e" yr


Medema, E. Lee; Lyon, Gary W. 1985,
The determination of financial rotation ages for coppicing tree
species. Forest Science 31(2):398-404.
Presents a model for determining the optimal harvest period and num-
ber of harvests for coppicing tree species. The decision rule is a
modified Faustmann solution in which the individual coppice rotations
may decline over time. The solution for the number of cycles to
manage the stand is found through an optimal search program.

Mitra, Tapan; Wan, Henry Y., Jr. 1985.
Some theoretical results on the economics of forestry. Review of
Economic Studies 52(2):263-282.
The paper poses the following general question: Suppose the owner of
a piece of land obtains utility in any time period, and this utility is
determined by the timber content of trees harvested in that period. If
the owner wishes to maximize the discounted sum of utilities from any
initial forest, what pattern of planting and harvesting system should be
used? Systematic examination using linear and concave utility functions
shows the Faustmann formulation to be generally correct.

Mitra, Tapan; Wan, Henry Y., Jr. 1986.
On the Faustmann solution to the forest management problem.
Journal of Economic Theory 40(2):229-249.
Examines optimal solutions to the forest management problem when
future utilities are not discounted. The asymptotic properties of such
solutions indicate: (1) if the utility function is linear, then the
Faustmann solution is optimal, and (2) if the utility function is increas-
ing and strictly concave, an optimal solution converges to the maximum
sustained-yield solution.

Morgan, William F. 1974.
The theory of the firm and the behavior of woodland owners: a
reappraisal based on empirical evidence of owner’s behavior. Dur-
ham, NC: Duke University School of Forestry. 172 pp. Ph.D.
Dissertation.
Possibly the first presentation giving the complete, and correct, com-
parative static derivation of the Faustmann model. The derived results
provide positive tests of landowner behavior with regard to wealth maxi-
mization. The actual rotation ages of the major management types,
derived from Forest Survey data, are consistent with the predicted
results. Farm owners and industry have an apparent average rate of
return of 5 to 5.5 percent. The Forest Service appears to be using no
economic rationale in its management.

42


Calish, Steven; Fight, Roger D.; Teeguarden, Dennis E. 1978.

How do nontimber values affect Douglas-fir rotations? Journal of

Forestry 76(4):217-222.
Using derived equations for seven nontimber yields, the authors com-
pare the land expectation value (LEV), economic optimum rotation,
biological optimum rotation (maximum sustained yield), and joint
economic optimum rotations. Opportunity costs are significant in com-
parisons of MSY with the joint optima for all options at a wide range of
nontimber valuations. Including nontimber yields into the LEV can
either lower or increase the economic optimum rotation length,
depending on the stream of nontimber returns. At moderate valua-
tions of the nontimber returns, the changes in rotation length from the
economic optimum are generally small

Chang, Sun Joseph. 1981.

Determination of the optimal growing stock and cutting cycle for

an uneven-aged stand. Forest Science 27(4):739-744.
Demonstrates that optimal rotation length for uneven-age manage-
ment can be derived from a formulation very similar to the Faustmann
equation. Opportunity cost on the growing stock represents the
regeneration costs of the stand. As with even-aged management, in-
creased interest rates lead to shorter cutting cycles and lower stocking
rates. Here, however, changing prices do not affect the optimal condi-
Uons as the prices are canceled out of the optimal condition, Finally,
with increasing interest rates,uneven-aged management is the
preferable alternative.

Chang, Sun Joseph. 1982.
An economic analysis of forest taxation’s impact on optimal rota-
tion age. Land Economics 58(3):310-323.
A rigorous examination of five types of taxes and their effect on rota-
tion age. They are (1) unmodified property tax, (2) site value tax, (3)
timber tax, (4) yield tax, and (5) severance tax. Capitalizing a tax and
shifting the incidence of the tax forward to higher valued stumpage will
have opposite effects on rotation age. A graphical analysis, based on
the technique first demonstrated by Clark (1976), shows the effects on
the optimal harvest period. Property taxes reduce the rotation age
while yield taxes delay the harvest. Poses questions regarding the con-
cept of tax neutrality and current timber taxation.

Chang, Sun Joseph. 1983.
Rotation age, management intensity, and the economic factors of
timber production: do changes in stumpage price, interest rate,
regeneration cost, and forest taxation matter? Forest Science
29(2):267-278.

on


Discusses the problems of using the financial maturity methodology of
Duerr and others (1956) for optimizing the rotation length. Use of
only the alternative rate of return ignores the opportunity cost of the
land and gives rotation lengths that are too long. The combination of
the optimal rotation and capital will be such that the highest annual
economic rent is received.

Zivnuska, John A. 1949,
Some aspects of the economic theory of forestry. Land Economics
25(2):165-172.
Thoughtful paper on the economic structure of forestry. The long
production process, the single identity of the product and the produc-
tion process, and the presence of different cost trees (virgin vs. second
growth), lead to market stumpage prices that may have little to do with
the costs of production. Results are pronounced short-term variations
in supply in response to market price, while the physical quantity of
standing, timber changes slowly over the long run. Long lags in produc-
tion lead to problems in the maintenance of equilibrium because ex-
tremely long reaction times are required to meet changes in demand.

47


Binkley, Clark S. 1985.
Tine role of capital in long run timber supply. Geneva: Internation-
al Institute of Applied Systems Analysis (IIASA) Working Paper,
26 pp.
Develops a supply function based on the Faustmann optimal rotation
in a manner similar to Hyde (1980). With a fixed technology and land
area, the supply curve is elastic only for rotations longer than the maxi-
mum sustained yield (MSY) and will otherwise be negative for rota-
tions shorter than MSY. Negatively sloped supply can lead to an
unstable equilibrium, which is demonstrated. The capital/output ratio
is related to interest rates.

Binkley, Clark S. 1987.
When is the optimal economic rotation longer than the rotation of
maximum sustained yield? Journal of Environmental Economics
and Management 14(2):152-158.
Exam ines the situation that would cause the Faustmann rotation to be
Jonge x than that prescribed by the maximum sustained yield (MSY).
Cond itions that will bring this result are an interest rate (r) less than
the in verse of the MSY rotation length (1/T*) along with a positive
profit . These conditions are most likely to occur in fast- growing
species such as southern pine or tropical plantation species. An ex-
ample verifies the result. The importance of this result is in determin-
ing the sign of the elasticity of supply in response to a change in
stumpage price.

Bould ing, Kenneth E, 1955

Economic analysis. New York: Harper and Bros.
Chapter 30 discusses maximization of the internal rate of return as the
decision criterion for rotation length determination.

Bowes, Michael D. 1983.

Economic foundations of public forestland management. Discus-

sion Paper D-104. Washington, DC: Resources for the Future.

80 pp.
Presents a model based on Hartman (1976) examining optimal manage-
ment for public ownerships who must account for nonmarket benefits.
He shows that if the nonmarket benefits are growing (declining) with
age, the rotation is lengthened (shortened). The major stated problem
with this type of model is its focus on the single stand, whereas the
manager must manage the whole forest. A full working example using
the Douglas-fir region illustrates the problem. The paper is an excel-
lent te chnical overview of the problem of rotation determination with
and without nonmarket benefits present.

29


and OrrQece > OrEQET

In economic terms, the inclusion of effort allows for the substitution of
labor (£) for capital (T) in the optimization procedure.

Each of these assumptions is quite ri ve. Real world condi-
tions are ignored or altered, calling the validity of the analytical results
into question. The assumptions are made, in part, to ease the mathe-
matical manipulations but more importantly because the world of 19th
century Germany, where the models were first presented, fit many of
these static factor assumptions. The assumptions are continued be-
cause the analytical results are primarily for theoretical demonstrations
and comparisons, which can then be applied under more general
conditions.

Maximization of Gross Yield (V1)

The maximum gross-yield (MGY) criterion has no present theoreti-
cal support. Itis included for historical development and because it
has some plausibility where interest approaches zero, It is an out-
growth of the mercantilist concept of management, in which a
country’s economic policy was guided by the desire to maintain as large
a stock of precious metals as possible. Maximum physical yields were
desired to minimize importing of wood and spending of gold. In one
form or another, it has been used as a rotation criterion until relatively
recently in Germany and other countries (Thomson 1942). It implicitly
assumes a zero interest rate and zero costs, so the formulation is:

(1) V1 = max(T) {pQ(T)}

where max(T) denotes maximization of the expression with respect to
T and V1 is the solution for gross-yield maximization. The price term
is included only to conform with later formulations. Prices have no ef-
fect on the optimal formulation.

The first-order condition is:
(2) Vir = pQr = 0 = Or

The rotation is optimal when the value of the marginal product
(VMP = price times the marginal physical product), or more simply
since price is constant, when the marginal physical product alone is
equal to 0. This criterion is unacceptable because it ignores all costs
involved in growing the stand, interest costs in holding the timber, and
treats the land as if it had no value. What is more, it fails to achieve its
intended goal of maximum total wood production because the
annual production from this model is less than the maximum average
production.


Uncertain or Changing Value
Parameters

A major assumption of the Faustmann formulation is that of perfect
certainty with respect to the. parameters of the model, This assumption
may have been justifiable when dealing with a relatively static economy
like that of 19th century Germany. However, in our present economy,
where new technologies and events continually affect production and
changing market conditions affect demand, it becomes necessary to in-
vestigate the changes in the optimal conditions that will occur in this
dynamic setting or in a situation of uncertainty.

Asa less complex first approach, Gregersen (1975) and Klemperer
(1979) examined the case of simple price inflation (i.e., no change in
real values over time) and its effect on forestry investments. They
found that as long as the inflation is accounted for in the calculations,
either by using real or current values for prices and costs, there is no
real effect on returns or rotation lengths as all costs and returns are
growing at the same rate. Thus, the inflated values in the equation will
simply cancel out. In cases with taxation, however, real distortions may
occur. For instance, in situations with capital gains taxation, short-
term investments are affected to a greater real extent than long-term in-
vestments as the tax-deductible purchase costs make up a greater
percentage of the final sale value of the timber crop.

Although the optimal behavior designated by the Faustmann condi-
tions will hold under static conditions, supply forces are at work that
seem to ensure that real timber prices will rise. Hotelling’s (1931)
theory of the mine states that as the quantity of a nonrenewable
resource decreases, its price will increase over time at a rate equal to
the interest rate. This price rise will continue until the time it reaches
the price of a substitute resource, at which point a steady-state condi-
tion will ensue with the nonrenewable resource essentially used up.

This scenario has been applied and discussed in the forestry litera-
ture with regards to the dwindling supply of the basically non-renew-
able old-growth timber resource and its substitute, second-growth
timber (Berck 1979; Lyon 1981). The theory has been substantiated by
the findings of rising real stumpage prices over the past 100 years.> It
has been used in various timber supply projections that predict con-

3
Manthy, Robert S, 1978. Natural resource commodities: a century of statistics.
Baltimore: Johns Hopkins University Press. 240 pp.

20


Hirshlei fer, Jack. 1970.
Investment, interest and capital. Englewood Cliffs, NJ: Prentice-
Hall. 620 pp.
In chap ter 3, Hirshleifer looks at models of duration and replacement
using the forestry paradigm. Maximization of the internal rate of
return (IRR) leads to the unsatisfactory conclusion of either infinite
or zero profits, depending on the viewpoint. The Fisher/Jevons (one
period) model and the Faustmann model present no basic conflict be-
cause each presents a different specification of productive oppor-
tunities for the land manager. The Faustmann solution’s optimal age
must fall between the IRR and the one-period solution.

Hirshle ifer, Jack. 1974.
Sustained yield versus capital theory. In: Dowdle, Barney, ed.
Economics of sustained yield forestry. Seattle: University of
Washington. 35 pp.
Discusses the economic consequences of sustained yield with non-
declining flow constraints on harvesting. Since timber and timber in-
vestments are capital goods, they should be managed at a rate of return
equal to the return of other capital investments in the economy; i.c., 4
the market interest rate. The use of allowable cut will therefore |
in production decisions.

economic falla

Hotelling, Harold. 1931.

5 of exhaustible resources. Journal of Political
Economy 39:137-175.

Discusses the optimal exploitation of nonrenewable resources, such as

old-growth timber and derives the "golden rule" of resource

exploitation.

Howe, Charles W. 1979.

Natural resource economics. New York: John Wiley & Sons.

350 pp.
Chapte r 11 of this general resource economics text discusses forest
management. The Faustmann model is developed and the effects of six
factors on the optimal rotation age are discussed: (1) increased interest
rate, (2 ) increased price, (3) increased planting costs, (4) severance
tax, (5) annual property tax, (6) increased distance to the mill.

Hyde, William F. 1980,
Tim ber supply, land allocation and economic efficiency. Bal-
timore: John Hopkins University Press for Resources for the Fu-
ture . 245 pp.

Chapte-rs 3, 4, and appendix B present a strong analysis of the timber

produc:tion decision and its effect on timber supply. This analysis is

37


one of the first to present both first- and second-order conditions on
optimality. It also gives a good historical synthesis of the development
of forest economics from the German tradition to present scenarios. It
examines rotation and supply effects that may occur as a result of rising
prices and technological change.

Jackson, David H. 1980.

The microeconomics of the timber industry. Boulder,CO:

Westview Press. 136 pp.
Chapter 2 presents the theory of the private timber firm. The compara-
live statics are performed using a format similar to Chang (1983), but
an artificial logistic growth function is assumed in which planting den-
sily is seen as a partial substitute for time. Because of this substitution
effect, the long-run effect of an interest rate change is to increase rota-
tion age as producers shift to a lower yield function by planting less,
The effect on firm supply of either an increase in price or interest rate
is negative. If, however, planting density and rotation length are com-
plements, a change in price leads to an increase in density, rotation
length, and supply.

Johansson, Per-Olov. 1984.
Disequilibrium cutting rules.
Research 14(3):321-325.

Derives cost-benefit rules for examining whether unemployment and

other market imbalances provide justification for national forest

managers to depart from the Faustmann rotation rule. Because of the
intertemporal nature of the forest production process, it is generally
not socially optimal to increase production in a single period as a result
of unemployment or excess demand.

“anadian Journal of Forest

Johansson, Per-Olov; Lofgren, Karl G. 1983.
Six different results on the properties of the timber supply func-
tion. pp. 169-181. In: Andersson, Ake E,; Olsson, Mats-Olov; Zack-
risson, Uno, eds. Forest sector development: issues and
analysis.Res. Rep. Sweden: University of Umed: 1983:1. 195 pp.
Presents six basic propositions about the harvesting decision under un-
certainty. A simple two-period model is developed to examine the ef-
fects of uncertainty with regards to price level, price variability,
lump-sum taxes, and profit taxes.

Johansson, Per-Olov; Lofgren, Karl G. 1985.
The economics of forestry & natural resources. New York: Basil
Blackwell Inc. 292 pp.
Chapters 4 and 5 discuss the basic optimal rotation period decision and
extensions. A history of the formulation of the problem includes

38


of the general case rather than to become engrossed in specific

method ological solutions. It now appears that since the general theory
has bee n well discussed, it will be necessary to incorporate these
prograrnming methods into the investigation of specific problem areas.

The effects of technical change on forest production and how
forestry fits into a technologically evolving society are important con-
cerns. Unequal access to new technologies may create substantial con-
trasts in management behavior. Owners will manage land for different
objectives as access to new techniques become available. Discerning
the rotation effects from the presence of these new conditions will be
challenging.

A final area for future research is in market imperfections. Un-
equal knowledge of market conditions between buyers and sellers is a
common condition for nonindustrial owners, and inadequate
knowledge can influence the decision to harvest. Oligopolistic or
monopsonistic markets seem to be more prevalent than purely monopo-
listic markets and could be expected to produce substantial disturb-
ances from the theoretical optimum state.

In conclusion, the controversy and debate that Faustmann began
more than 150 years ago between economic and biological criteria for
forest management continues. As society’s uses of its forest resources
expand and evolve, new debates and issues will continue to occur.

26


Contents

Introduction ..

Comparison of Criteria for Rotation Length Determination

Maximization of Gross Yield (V1) ............0..05

Maximization of Annual Yield (V2)...... bees

Maximization of Discounted Net Revenues from a
Single Rotation (V3)..0.0.0.0.0.0c0e ce eeeee ee eee

Maximization of the Discounted Net Revenues from
an Infinite Series of Like Rotations (V4) ...........

Maximization of Annual Net Revenues (V5) ......-.

Maximization of the Rate of Growth of Capitol (V6) ..

Relationships Among the Six Criteria..............

Extensions of the Faustmann Model ..... . eee ences

Comparative Statistics 00.0... eee eee

Uncertain or Changing Value Parameters...
Market Failure - Externalities ...............0..0-
Market Failure - Imperfect Markets..............

Summary, Conclusion, Future Research Areas ...........

Annotated Forest Valuation Bibliography .............

sees 25


timal rule is the same as the Faustmann rule with the interest rate in-
creased by the average probability of fire occurrence. Thus, rotations
are shortened, Also examined are cases with partial salvage and where
the probability of fire is not homogeneous through time.

Reed, William J. 1986.
Optimal harvesting models in forest management - a survey.
Natural Resource Modeling 1(1):55-79.
An extensive survey of the literature of optimal timber harvesting
models for even-aged and uneven-aged stands, single-stand models,
and forest-level scheduling models.

Routledge, R.D. 1980.
The effect of potential catastrophic mortality and other unpre-
dictable events on optimal forest rotation policy. Forest Science
28(3):389-399,
Modifies the Faustmann optimal condition to include the probability
of a catastrophic fire and the degree of salvage after the fire. If salvage
value is low, the optimal rotation will decrease as the probability of fire
increases. With a high salvage value, the optimal rotation lengthens. Ig-
noring risks can lead to large over-estimates of the land expectation
value of the site.

Samuelson, Paul. 1976.
Economics of forestry in an evolving society. Economic Inquiry
14( Dec.):466-492.
Probably the most frequently cited article on forest valuation. The in-
ternal rate of return, one period, and maximum sustained-yield for-
mulations are dismissed as possible criteria for optimal rotation length
determination. Maximizing soil rent leads to the optimum rotation,
and this value can be included in the present net value calculation in a
one-period model. This wide-ranging article looks at the economic ra-
tionale of a number of forestry concerns such as sustained yield and the
bogey of compound interest. An excellent annotated bibliography is
provided.

Strang, William J. 1983.

On the optimal forest harvesting decision. Economic Inquiry

21( Oct.):576-583.
Extends the Hartman (1976) analysis of optimal rotations where the
standing timber itself provides increasing value over time. While the
first-order conditions give a rotation length longer than the standard
one-period or Faustmann solution, the resulting present value must be
compared with the corner solution value of no harvest at all. This step
determines whether a global maximum exists. For a mature standing

AR


Murphy, Paul A.; Fortson, James C.; Bethune, James E. 1977.
Tamber management decision making under imperfect capital
markets. American Journal of Agricultural Economics 59(2):302-
340.

Examines the effect of imperfect capital markets, using dynamic

prog ramming, on the criteria of maximization of liquidation value of

the firm. Ina perfect capital market, the globally optimal rotation is
constant. Under imperfect capital markets, optimum rotations vary.

Naslund, Bertil. 1969.

O ptimal rotation and thinning. Forest Science 15(4):446-451.
Disc usses principles for solving the problem of determining optimal
rotat ion length and other managerial inputs (specifically thinning)
simu Itaneously. Pontryagin’s maximum principle is used to find the
gence ral optimal conditions.

Naut iyal, Jagdish C. 1983.
Towards a method of uneven-age forest management based on the
theory of financial maturity. Forest Science 29(1):47-59.

Advances the work of Duerr and others (1956) by including all relevant

costs inherent in growing a forest stand. Rules are developed for har-

vesting mature trees and for thinning using both economic and biologi-
cal (in the form of potential growth form) criteria. Effects on the
ultimate rotation length of quality variations that influence price are
clearly discussed.

Nautiyal, Jagdish C.; Fowler, Kenneth S. 1980.
Optimum forest rotation in an imperfect stumpage market. Land
Economics 56(2):213-226.
Develops a Faustmann type model where the landowner has a monopo-
listic position in the market such as the state in western Canada. For
the owner, maximization of profit rather than present net value should
be the objective. This change leads to a shorter rotation than the
atomistic solution. Societal optimum lies between these two points. In
a regulated forest, however, the differences between the two ownership
types diminish and the question may not have much relevance.

Newman, David H.; Gilbert, Charles E.; Hyde, William F. 1985.
The optimal forest rotation with evolving prices. Land Economics
61(4):357-394,
Using a dynamic optimization model, adapted from McConnell and
others (1983), examines the case of increasing stumpage prices and
their effects on rotation lengths. With constant costs and exponentially
rising prices growing at a rate less than the interest rate, the initial rota-
tion length is longer than if the static Faustmann rotation, using con-


Another problem with the IRR criterion is that decisions relating
project scale and timing may be incorrect and that changes in prices or
the market interest rate do not affect the optimal rotation or the level
of investment. If the planting costs are equal to zero, then the IRR is
undefined and if intermediate costs and benefits are received, there
will be multiple IRR’s that solve equation (19) (Bierman 1968).

Relationships Among the Six Criteria

Model formulations and basic results for the six optimal rotation
criteria are summarized below in a condensed form so that they can be
more easily referenced and compared. Equation numbers are those
found in the text.

Let:
T = time p = net real stumpage price
rt = real discount rate i = internal rate of return
wE = labor price (wage) times unit regeneration effort

Q(T) = timber growth, a concave function of time with zero
regeneration lag and fixed technology, Or > 0, and Qrr<0

Maximization of gross yield (also: maximum physical yield, MGY)
Q) V1 = max(T) {pQ(T)}
(2) pOr = 0=Qr

Maximization of annual yield (also: CMAI, MSY)
G3) V2. = max(T) {pQ(T)/T}
(5) Or = Q(T)/T
(6) pQr/pQ(T) = 1/T

Maximization of single rotation discounted net revenues (also:
PNW)

(7) V3 = max(T) {pQ(T)e - wE}
(9) pQr = rpQ(T)
(10) QuiQ(T) =r

ll


Table 1 is derived from Schumacher and Coile’s yield equations
for loblolly pine.” It displays the effects of the choice of criterion on
the optimal rotation age. The table assumes a price of $0.30 per cubic
foot, planting costs of $100, and an interest rate of 5 percent. The con-
nected values in the table show the approximate time at which the
necessary equality condition for the optimal rotation age of each
criterion is reached. The rotation length is shortest with the IRR
criterion and longest with the MGY criterion, Use of the MSY
criterion in place of the soil-rent criterion lengthens the rotation by ap-
proximately 20 percent, a significant increase. The soil-rent rotation
length lies between the IRR and PNW rotations.

Figures 1, 2, and 3 depict the information in table 1.yThe optimal
rotations are easily computed graphically. From figure 1, the MGY
rotation occurs where the yield function is at its maximum point (the
slope is equal to 0), or in figures 2 or 3 where the marginal growth func-
tion equals 0.

A ray drawn from the origin to the yield curve in figure 1 gives the
average yield at any point in time. Therefore, the MSY rotation occurs
when this ray is tangent to the yield curve. MSY can also be found in
figure 2 of the graph where the marginal and average yield curves
cross. The forest-rent rotation length is necessarily longer than MSY
as the ray is vertically displaced from the origin by the amount of the
regeneration cost.

The IRR rotation occurs at the inflection point of the yield func-
tion in figure 1 or where the slope of the marginal growth in figure 2 or
figure 3 is equal to 0.

The PNW and Faustmann criteria explicitly contain economic in-
formation in their solutions and so cannot be estimated using physical
yield curves alone. Figure 3 graphs the left- and right-hand sides of the
six optimal solutions so that they can be estimated directly.

The MGY, MSY, and forest-rent criteria share the common
shortcoming that interest costs are ignored. Thus, no rents are
returned to the land, because it is assumed that the opportunity cost of
the land is zero. They therefore imply that capital is costless and there
are no other uses for the land, including the growing of a new stand of
trees. In addition, they assume that the forest owner and society are in-
different between revenues earned now and later.

Francis Schumacher and Theodore Coile. 1960. Growth and yields of natural
stands of the southern pines. Durham, NC: T.S. Coile Inc. 115 pp.


The Optimal Forest Rotation:
A Discussion and Annotated Bibliography

David H. Newman
isiting Assistant Professor of Forest Economics

School of Forestry and Environmental Studies
Duke University
Durham, North Carolina


Broadens the general Faustmann model to the case where environmen-
tal benefits are included as a return in the valuation stream. In the one-
period case, the harvest period is lengthened or even bypassed
completely if the environmental benefits are high enough. In the
Faustmann case, a similar result as in the one-period case is found.
Thus, harvest age is probably increased from the original Faustmann
solution and again potentially delayed infinitely. An important Paper
that has served as a source reference for numerous papers that have fol-
lowed it.

Heaps, Terry. 1981.

The qualitative theory of optimal rotations. Canadian Journal of

Economics 14(4):686-699.
Examines the effect of harvesting and other site quality costs on the op-
timal rotation, Results are similar to changes in regeneration costs in
their effects. Thus, increased hauling costs lead to longer rotations;
higher site leads to shorter rotations. Paper has a good comparative
statics analysis of the general case.

Heaps, Terry. 1984.
The forestry maximum principle. Journal of Economic Dynamics
and Control 7(2):131-151.

When the optimal rotation decision is completely generalized into a

dynamic control framework, the Faustmann rotation is the steady-state

solution of the more general case. The generalizing is accomplished by
loosening of the Faustmann assumptions of constancy by (1) allowing
the net stumpage value to vary with the age of the trees and with the
harvest rate and (2) allowing an arbitrary initial age distribution of the
site.

Heaps, Terry; Neher, Philip A. 1979,

The economics of forestry when the rate of harvest is constrained.

Journal of Environmental Economics and Management 6(4):297-

319.
Using control theory, broadens the Faustmann derivation to the situa-
tion where the standing timber cannot be harvested instantaneously
and is cut over a transition period. When looking at a single rotation,
the harvest goes as a bang-bang solution where harvesting is done at
the maximum rate when the rate of change of value is less than or equal
to the interest rate (r) and 0 when >r. With ongoing forest, assumes
the first harvest period is the transition period, after which the
manager attempts to manage as closely as possible to the Faustmann
rule. When costs are a function of harvest level, it is not optimal to con-
duct the first harvest at a constant rate but rather to increase rotation
length with time.

QR


confirmed Hartman’s results in regards to the shape of the amenity
value benefit stream and its effect on the optimal rotation length.
Using sensitivity analysis to value the nontimber goods, they found the
economic optimum rotation given by the Faustmann equation was
quite robust. When amenity goods were added, the optimal rotation
changed only slightly. Large economic losses, however, generally
resulted from the use of the MSY criterion.

A major problem with the Hartman formulation is that it focuses
the multiple use benefits on a single stand, ignoring the effects of ac-
tivities in adjacent stands on the derived amenity values. Incorporating
the forestwide rotation effects is a linear programming problem that
has been investigated by Bowes (1983).

Market Failure - Imperfect Markets

This general topic has received relatively little research effort, in
part because the applicability of these cases seems somewhat limited.
In imperfect stumpage markets, local, or even regional, monopolies
may exist, but two factors serve to blunt attempts to assert this monop-
oly power. First, where such monopoly power exists, it is most often
held by government, as in western Canada, the Pacific Northwest of the
United States, or in socialist countries where all land is publicly held.
It is not necessarily in the public’s interest to maximize profits from
stumpage sales at the cost of a decrease in total social welfare. Second,
many close substitutes for the monopoly-held stumpage may exist. One
close substitute is similar stumpage supplied from nearby areas. Some
monopoly power may still exist, but the demand curve is fairly elastic
and the shipping costs necessary to import this external stumpage
places an upward bound on any price increase that can be charged.

Nautiyal and Fowler (1980) used a Faustmann model to compare
the rotation length and welfare losses that occur where pure monopoly
power exists. They model the behavior of a pure monopolist, a social
welfare maximizing central planner, and the standard atomistic (com-
petitive) ownerships. In an unregulated forest with a constant area of
managed forest land, they find that the monopolist uses the shortest
rotation and atomistic owners use the longest. The monopolist short-
ens rotations to reduce the amount of stumpage in the market. Forcing
competitive behavior where monopoly power exists is inefficient be-
cause the marginal conditions are violated (value of the marginal
product is less than the marginal cost). The societal optimum, there-
fore, is found to lie between the two extreme cases and closer to the mo-
nopolistic rotation. In the case of a regulated forest, the differences in
rotation length among the three cases virtually disappear, but the or-
dering of the lengths stays the same as for the unregulated cases.

24




ee



competitive optimum rotation, The rotation for the monopolist will be
longer if the rate of price increase is expected to be greater than the
discount rate.

Dasgupta, Partha. 1982.
The control of resources. Cambridge, MA: Harvard University
Press. 223 pp.
Chapter 9 discusses optimal forest management strategies using a
single-tree stand model to derive the Faustmann rule, The author dis-
cusses the effects of nonlinearity of the social benefit rule on optimal
decisions.

Duerr, William A.; Fedkiw, John; Guttenberg, Sam. 1956.

Financial maturity: a guide to profitable timber growing. Tech.

Bull. 1146. Washington, DC: U.S. Department of Agriculture.

74 pp.
Discusses the use of a modified present net worth (PNW) criterion to
determine the optimal rotation age. With the modified criterion, mark-
ing guides can be decided upon in the field by using a simple rate of
growth determination. The appendix compares PNW with the
Faustmann formulation. The authors use a synthetic growth function
and find a relatively small change in the optimal rotation when the
PNW criterion is used. This result was shown by Gaffney (1957) to be
a result of the growth function chosen rather than the general case.

Faustmann, Martin. 1849.
On the determination of the value which forest land and immature
stands pose for forestry. In: Gane, M., ed. Martin Faustmann and
the evolution of discounted cash flow. Paper 42. Oxford, England:
Oxford Institute; 1968. 54 pp.
Written as a reply and extension of an earlier article by von Gehren,
this paper sets forth the basic economic tenets by which timberland
should be valued; that is, the land is worth the present value of an in-
finite series of like rotations. A path-breaking paper that painstakingly
explains the concept that the forest-land value will remain unchanged,
regardless of whether there are trees growing on the site or not, as the
growing tree’s value would necessar-ily be capitalized into any sales
price. Derives the formula used today for calculating land expectation
value. The foreword by Gane is an excellent historical background and
sets the place of Faustmann and his contemporaries within the stream
of early economic thought.

Fortson, James C. 1972.
Which criterion? Effect of choice of the criterion on forest
management plans. Forest Science 18(4):292-297.

32


Markus, R. 1967
Ostwald’s relative forest rent theory. Munich, Germany: Neue
Press.
Critiques the soil-rent theory for (1) considering only the soil, and not
including the growing stock, as fixed and thus deserving of rents, and
(2) considering only the single stand and not the effects on the whole
forest. Develops Ostwald’s relative forest-rent theory with a numeric
example in a not altogether clear manner. The major point seems to be
to include the cost term as a cost of harvesting rather than separating
the two decisions. The result is that with revenues thus lowered, the
rotation is lengthened.

Martell, D.L. 1980.
The optimal rotation of a flammable forest stand. Canadian Jour-
nal of Forest Research 10(1):30-34.
Describes a variation of the Faustmann model that includes a stochas-
tic element for potential catastrophic losses (particularly fires). With
dynamic programming, as the risk of fire increases, the optimal rota-
tion age and soil expectation value (SEV) decrease. A method is given
for gauging the efficiency of fire management activities that reduce the
risk by estimating changes in the SEV.

McConnell, Kenneth E.; Daberkow, Julie N.; Hardie, lan W. 1983.
Planning timber production with evolving prices and costs. Land
Economics 59(3):292-299.
Draws Conclusions for optimal rotation length under conditions of
dynamic price (p) and cost (c) assumptions. With constant real price:
(1) increasing c relative to p increases optimal harvest length, (2)
decreas ing ¢ relative to p decreases optimal harvest length, and (3) un-
certain results with constant c and p changing either way. Assumes
constant technology in the presence of changing costs.

McConnell, Kenneth E.; Phipps, Tim. 1984.
Opti mum forest rotation in an imperfect stumpage market: com-
ment. Land Economics 60(2):215-217.
Comme nt on paper by Nautiyal and Fowler (1980). States that while
compet itive optimum is longer than the social optimum in an imperfect
market, it is uncertain whether the monopolistic optimum is always
shorter than the social optimum. It does hold for a linear demand
curve w here elasticity decreases with an increase in volume. It does not
hold for constant elasticity demand functions for which the social and
monopolistic rotation lengths are the same.

AI



Annofaated Forest Valuation Bibliography

Anderson, F.J. 1976.
Coratrol theory and optimum timber rotation. Forest Science
22(3):242-246.
The net present value maximization problem of Faustmann is com-
pared ~with the optimal control framework for renewable natural
resour<es. The optimal control model gives the same answers about
rotation age as the earlier intuitive formulations. That is, shorter rota-
tions Uhan maximum sustained yield when r >0. The article points out
some Problems of the Faustmann formulation in relation to beliefs on
changi ng prices and other non-steady-state conditions.

Anderson, FJ. 1981.
Optimum forest rotation: comment. Land Economics 57(2):293-
294.
This paper is a comment on a paper by Nautiyal and Fowler (1980).
Anderson felt they implied that the atomistic (competitive) rotation
length is different from the social optimum. He shows that unless
other issues are introduced (externalities, etc.), this cannot be the case
as the marginal conditions of the Faustmann equation ensure
optimality.

Bare, Bruce B.; Waggener, Thomas R. 1980.

Forest land values and return on investment. Forest Science

26( 1):91-96.
The paper examines increasing price scenarios and their effect on the
land expectation value (LEV) of a site. The major point is that increas-
ing fut ure prices leading to higher future values will already be incor-
porated into the LEV; thus return on investment (ROI) will stay the
same. The paper errs in postulating a 60-year rotation for Douglas-fir
under all scenarios. The constant rotation example works for their
compa risons but is basically incorrect because the LEV could be in-
creased by altering the rotation length in a period of rising prices, thus
giving a higher ROI.

Bentley, William R.; Fight, Roger D. 1966.
A zero interest comparison of forest rent and soil rent. Forest
Science 12:460.
Using L’Hopital’s rule (although not cited), the authors mathemat-
ically prove that the forest-rent calculation is a subset of the land expec-
tation walue (soil rent) calculation. Thus, when the interest rate
approaches zero, the soil rent approaches the forest rent.

27


Maximization of Annual Net
Revenues (V5)

This criterion, also known as forest rent, was proposed, in part, to
justify longer rotations than those prescribed by the soil-rent criterion.
It is meant to be used in a fully regulated forest — a forest where equal
annual harvests are made and the land is immediately returned to tim-
ber production. Thus, it is claimed, there are no holding costs on the in-
itial planting costs; these are covered as an operating expense to be
taken from the cash-flow generated by the harvest receipts. This as-
sumption is faulty because it fails to recognize that the decision to in-
vest by planting to start a new rotation is entirely separable from the
harvest decision. Saying that the land will continue to be managed for
timber production is not the sam ying that it has no opportunity
costs. The major effect of the assumption is that it avoids the discount-
ing of future revenues against present costs. As shown by Bentley and
Fight (1966) and others, it is a special case of the soil-rent calculation
under the condition of zero interest rate. Samuelson (1976) called this
criterion the "forester’s maximum sustained (net) yield." This is some-
thing of a misnomer because MSY is generally derived only in biologi-
cal terms, ignoring financial constraints.

The formulation of the problem is similar to equation (3) except
that planting costs are included:

(19) V5 = max(T) {(pQ(T) - wE)/T}

The optimal solution is:

(20) VSr = [pOrT - (pQ(T) - wE)|/T? = 0
(21) Qr = [Q(T) - wE/p)/T
(22) VT = pOr/{[pQ(T) - wE|

At the optimum, the VMP equals the average annual net revenues.
Both prices and costs affect the optimal rotation length. An increase
in p increases the right-hand side of (17) and thus decreases the op-
timal rotation length while an increase in wE lengthens the rotation.
Since the numerator of the right-hand side of (17) is less than that of
(4) but larger than (2), the optimal rotation for the forest-rent
criterion is bounded from below by CMAI and from above by the MGY
rotation,


Newman, Deavid H.

1988. The optimal forest rotation: a discussion and annotated
bibliography. Gen. Tech. Rep. SE-48. Asheville, NC: U.S.
Department of Agriculture, Forest Service, Southeastern Forest
Experiment Station. 47 pp.

The litereature contains six different criteria of the optimal forest
rotation: (1) maximum single-rotation physical yield, (2) maximum
single-rotation annual yield, (3) maximum single-rotation discounted
net revenues, (4) maximum discounted net revenues from an infinite
series of rotations, (5) maximum annual net revenues, and (6) max-
imum inter-nal rate of return. First-order conditions for maximization
show the rotation effects of the criteria. Various authors have
extended basic models and discussed effects of externalities,
imperfect markets, changing parameters, and taxes.

KEYWORDS: Forest economics, forest valuation, harvest scheduling.

Newman, David H.

1988. The optimal forest rotation: a discussion and annotated
bibliography. Gen. Tech. Rep. SE-48. Asheville, NC: U.S.
Department of Agriculture, Forest Service, Southeastern Forest
Experiment Station. 47 pp.

The literature contains six different criteria of the optimal forest
rotation: (1) maximum single-rotation physical yield, (2) maximum
single-rot ation annual yield, (3) maximum single-rotation discounted
net revenues, (4) maximum discounted net revenues from an infinite
series of rotations, (5) maximum annual net revenues, and (6) max-
imum intermal rate of return. First-order conditions for maximization
show the retation effects of the criteria. Various authors have
extended basic models and discussed effects of externalities,
imperfect markets, changing parameters, and taxes.

KEYWORDS: Forest economics, forest valuation, harvest scheduling.


Partial funding for this study was from the
U.S. Department of Agriculture, Forest Service,
Southeastern Forest Experiment Station.

April 1988

Southeastern Forest Experiment Station
P.O. Box 2680
Asheville, North Carolina 28802


Maximization of infinite rotation discounted revenues (also:
Faustmann’s soil rent, LEV, SEV)

(11) V4 = max(T) {{pO(T)eTM! - we] - e"4)}
(13) Qr/{Q(T)- WE yp] = #16")
)

(4) pOr = r(pQ(T) - wE/(1-¢
(15) pOr = rpQ(T) + r(V4)

Maximization of annual net revenues (also: forest rent)
(19) VS = max(T) {(pQ(T) - wE)/T}
(21) Or = [Q(T)- wE/p)T
(22) WT = pOr/[pQ(T) - wE]

Maximization of the rate of growth of capital (also: IRR, ROR)
(23) Vo = max(i) {JpQ(T)e"! - wEV(-e"")} = 0
(24) i* = max(T){(1/T)In[pO(T)WE]} = OVQ(T)
(25) pQr = i*pQ(T)

12