Ecological Economics 31 (1999) 91–106

ANALYSIS

Non-renewability in forest rotations: implications foreconomic and ecosystem sustainability�

Jon D. Erickson a,*, Duane Chapman b, Timothy J. Fahey c, Martin J. Christ d

a Department of Economics, Rensselear Polytechnic Institute, Troy, NY 12180-3590, USAb Department of Agricultural, Resource, and Managerial Economics, Cornell Uni6ersity, Ithaca, NY 14853-7801, USA

c Department of Natural Resources, Cornell Uni6ersity, Ithaca, NY 14853-7801, USAd Department of Biology, West Virginia Uni6ersity, Morgantown, WV 26506, USA

Received 4 May 1998; received in revised form 19 March 1999; accepted 19 March 1999

Abstract

The forest rotations problem has been considered by generations of economists (Fisher, 1930; Boulding, 1966;Samuelson, 1976). Traditionally, the forest resource across all future harvest periods is assumed to grow withoutmemory of past harvest periods. This paper integrates economic theory and intertemporal ecological mechanics,linking current harvest decisions with future forest growth, financial value, and ecosystem health. Results andimplications of a non-renewable forest resource and the influence of rotation length and number on forest recoveryare reported. Cost estimates of moving from short-term economic rotations to long-term ecological rotations suggestthe level of incentive required for one aspect of ecosystem management. A net private cost of maintaining ecosystemhealth emerges and, for public policy purposes, can be compared with measures of non-timber amenity values andsocial benefits exhibiting increasing returns to rotation length. © 1999 Elsevier Science B.V. All rights reserved.

Keywords: Forest rotation; Ecosystem management; Ecological economic modeling

www.elsevier.com/locate/ecolecon

1. Introduction

Ecological economics has distinguished itselffrom traditional economic analysis of the environ-

ment by stressing the essential role of ecosystemservices and the maintenance of ecosystem pro-cesses. Traditional analysis of renewable resourcessuch as forests, fisheries, and agriculture has longstressed the conditions for steady-state systemsof management. The economic study of thesenatural resources, however, has often overlookedcritical intertemporal ecological mechanics relatedto the timing and impact of disturbing naturalsystems.

� Based on a paper presented in a session entitled ‘ModelingSpatial and Temporal Dynamics of Terrestrial Ecosystems’, atthe Fourth Biennial Meeting of the International Society forEcological Economics, Boston University, August 4–7, 1996.

* Corresponding author.E-mail address: erickj@rpi.edu (J.D. Erickson)

0921-8009/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved.

PII: S 0921 -8009 (99 )00040 -3

J.D. Erickson et al. / Ecological Economics 31 (1999) 91–10692

The study of the regeneration of forests undercenturies long harvest sequences is beginning toredefine our understanding of resource renewabil-ity. Traditional financial models of the forest re-source assume perfect renewability in forestgrowth following infinite optimal rotations ofconstant length. Study of forest ecology, however,suggests that rotations affect future growth,product quality, and forest health. For instance,alteration of successional sequences, nutrient cy-cles, and other components of ecosystem functionare influenced by rotation length, harvest inten-sity, and cutting frequency. These cross-harvestinteractions suggest a non-renewable forestgrowth specification, leading to the addition of amarginal benefit of recovery in the traditionaloptimal rotation decision rule.

In this paper, an integrated forest succession,product, and price model for the northern hard-wood forest ecosystem is developed to evaluatethe impact of increasing density of pioneer speciesfollowing disturbance on rotation length and tim-ber profits. For the ecosystem type examined, thesuccess of early successional species in distur-bance–recovery cycles due to short, repetitive ro-tations have the effect of delaying forestdevelopment and entrance into late successional,higher quality, higher return species. Accordingly,an overlooked financial benefit to forest recoveryis specified and estimated for a discrete horizonrotations problem. A non-renewable growth spe-cification has the effect over traditional models oflengthening forest rotations, adjusting profitsdownwards, and valuing the long-term mainte-nance of ecosystem processes. By incorporatingecosystem modeling into an economic framework,a clearer management picture results.

2. The marginal benefit of recovery

For the commercial forest manager, the princi-pal economic question centers on harvest timing.The majority of the economic literature on thisquestion is grounded in the model developed inthe 19th century by the German tax collectorMartin Faustmann (Faustmann, 1849). Faust-mann was concerned with estimating the bare-

land expected profits1 of a forthcoming forest.Assuming land is to remain in forestry, the prob-lem is to solve for the rotation length (T) over aninfinite stream of future profits from harvesting aperfectly renewable resource.

Assuming a continuous-time discount factor(e−dt) and a continuously twice differentiablestand profit function (p(t)), the infinite horizonprofit maximization problem converges to:

Max P=p(t)

edt−1(1)

where:

p(t)=P Q(t). (2)

Stumpage price (P) equals net price per unitvolume. Natural regeneration is assumed, so re-planting costs are assumed zero. In the mostgeneral case of the multispecies, multiqualityproblem, P represents a matrix of stumpage pricesand, likewise, Q(t) models a matrix of timbervolumes across species and quality classes.

Solving Eq. (1) produces the following first-or-der condition, known as the Faustmann formula:

p %(t)=d p(t)+d p(t)edt−1

(3)

From Eq. (3), a single optimal rotation length(T) maximizes net present value (P) by equatingthe marginal benefit of waiting to the marginalopportunity cost of delaying the harvest of thecurrent stand (i.e. interest forgone on currentprofit) plus the marginal opportunity cost of de-laying the harvest of all future stands (i.e. interestforgone on all future profits, often called sitevalue).2

1 The term ‘value’ has been used to represent forest profits(e.g. Clark, 1990) in economics. Here, ‘value’ is reserved forproblems incorporating non-forest amenities and other posi-tive externalities. For example, forest profits include onlyincome from the sale of timber, where forest value wouldinclude non-market goods such as aesthics, biodiversity, orrecreation.

2 If real stumpage prices are assumed to grow at a rate r,then the Faustmann formula simply becomes: p %(t)= (d−r)p(t)+ [(d−r)p(t)]/[e(d−r)t−1]. Eqs. (29)–(33) in the empiri-cal analysis introduces price growth.

J.D. Erickson et al. / Ecological Economics 31 (1999) 91–106 93

Fig. 1. Cubic forest undiscounted profit functions.

Adaptations and expansions to this model in-clude modeling non-timber benefits (e.g. Hart-man, 1976; Calish et al., 1978; Berck, 1981),multiple-use forestry (e.g. Bowes and Krutilla,1989; Snyder and Bhattacharyya, 1990; Swallowand Wear, 1993), stochastic price paths (e.g.Clarke and Reed, 1989; Forboseh et al., 1996),market structure (e.g. Crabbe and Van Long,1989), and uneven aged forestry (e.g. Mont-gomery and Adams, 1995).

All these improvements to the basic Faustmannformula, however, share a strong assumption ofperfect growth renewability—a constant growthfunction (Q(T)) across all future planning peri-ods. In contrast, evidence from the study of forestecology and management indicates a strong rela-tionship between rotation length, rotation fre-quency, and harvest magnitude in a currentmanagement period, with the growth and mainte-nance of the forest in future periods (e.g. Bormanand Likens, 1979; Kimmins, 1987). This is partic-ularly the case where natural regeneration seedsthe new forest, or soil renewability is compro-mised. In the Faustmann framework, this ecologi-cal knowledge implies a forest stand profitfunction dependent on rotation length (T) androtation number (i ), given constant technologyand harvest magnitude.

Growth in merchantable timber volume is typi-cally modeled using a cubic or exponential form,

consistent with stages for rapid growth, biologicalmaturity, and disease and decay (Clark, 1990).Consider a cubic functional form for undis-counted profit at constant prices:

p(t)=b1 t+b2 t2+b3 t3. (4)

Fig. 1 illustrates three plots of Eq. (4) followinga harvest at T0 assuming different parameter val-ues for b1, b2, and b3. Suppose T1A is an optimalFaustmann rotation in the first harvest cycle (i=1). Therefore, a longer rotation in this first cycle(for instance, T1B) would be sub-optimal as itwould decrease the marginal value of waitingbelow the sum of first harvest and future harvestopportunity costs.

However, there may be an additional marginalvariable to consider in the first rotation decision.Suppose rotation length in the first harvest cycleinfluences the form of the functional stand profitfunction in subsequent cycles. For instance, sup-pose the choice of T1A in cycle i=1 results in theprofit function p(Ti=2�T1A) in cycle i=2. Alonger rotation such as T1B, however, results in ahigher profit function p(Ti=2�T1B) in cycle i=2.In this case, a longer first rotation has the benefitof allowing the forest more time to recover fromthe initial cut at T0. Now, waiting until T1B toharvest during the first cycle has the benefit ofshifting the second cycle curve upwards to p(Ti=

2�T1B). A sufficiently long first rotation would

J.D. Erickson et al. / Ecological Economics 31 (1999) 91–10694

result in an identical second rotation profit function.Without taking into account this cross-harvestimpact, the Faustmann solution of T1A would leadto a sub-optimal decision.

To incorporate this interaction between currentharvest length and subsequent profit functionsconsider Eq. (5). The function f(Ti−1, i−1) isadded as a variable to the period i profit function.The level of f(Ti−1, i−1), or ecological impact,depends on the length of the last period’s rotation(Ti−1), and the number of rotations since the firstcut at T0 to take into account any cumulativeimpacts. It influences the cubic function parameters(b1, b2, and b3) of the stand profit function throughan ecological impact represented by the parametersa1, a2, and a3:

p(ti, f(Ti−1, i−1))

= (b1+a1 f(Ti−1, i−1))ti

+ (b2+a2 f(Ti−1, i−1))t i2

+ (b3+a3 f(Ti−1, i−1))t i3 (5)

where:

f(T0, 0)=V, (6)

f(Ti−1, i−1)]V (7)

(f(Ti−1, i−1)(Ti−1

B0 (8)

(2f(Ti−1, i−1)(Ti−12

\0 (9)

(f(Ti−1, i−1)((i−1)

\0 (10)

(2f(Ti−1, i−1)((i−1)2 B0 (11)

and:

a1B0, (12)

a2B0, (13)

a3B0 (14)

for i=1, 2, 3, . . .Stand profit in the current rotation cycle (i ) now

depends on the current rotation length (Ti), theprevious rotation length (Ti−1), and the number ofrotations (i−1) since the pre-disturbance period(i−1=0). The ecological impact function, f(),represents a forest recovery relationship based on

physical and biological parameters. For example,f() might measure the impact on forest regenerationfrom pioneer species rebound (stems/acre), fromsoil nutrient loss (nutrients/m2) or erosion (soildepth), or possibly from a general index of resourcerenewability.

The first-order conditions for f() imply that as theprevious period rotation length (Ti−1) increases, thenegative ecological impact decreases. Also, as thenumber of rotations since the pre-disturbance pe-riod (i−1=0) increases, the ecological impactincreases. An initial condition (V) is assumed whichdefines the level of f() following the initial harvestat T0. This parameter can be considered a foresthealth endowment left from the previous landmanager. In the case of inheriting a mature forestnot previously managed, V could be considered theecological effect on forest growth from periodicnatural disturbances (e.g. wind storms, fires).

Assuming this non-renewable, rotation length-dependent, stand profit specification over an infinitehorizon, the profit maximization problem becomes:

Max

P=p(t1, f(T0, 0))e−dT1+p(t2, f(T1, 1)) e−dT2

+p(t3, f(T2,2))e−dT3+ . . . (15)

Under an assumption of perfect renewability,f(T0, 0)= f(T1, 1)= . . .= f(T�,�)=V, and theprofit maximization problem converges to Eq. (1),from which the usual Faustmann result of a con-stant rotation length in Eq. (3) is obtained.

Under the assumption of partial non-renewabil-ity, however, the selection of the optimal rotationlength set (Ti for i=1, 2, 3, . . .) now considers theimpact on each subsequent period’s profits throughthe addition of a marginal benefit of recovery(MBR). The marginal benefit of recovery in periodi from a rotation length in the previous period i−1is represented as:

MBRi=(f(Ti−1, i−1)(Ti−1

{a1 Ti+a2 Ti2+a3 Ti

3}\0

(16)

Thus, balancing the benefits to recovery fromlonger rotations against the opportunity costs ofdelaying current and future harvests will determinethe optimal rotation set.

J.D. Erickson et al. / Ecological Economics 31 (1999) 91–106 95

Fig. 2. Kimmins’ ecological rotation vs. successional retrogression (Kimmins, 1987).

In the forest ecology literature, Kimmins (1987)outlines the distinction between a Faustmann-typerotation where net present value is maximized,and an ecological rotation, the time required for asite managed with a given technology to return tothe pre-disturbance ecological condition. Fig. 2demonstrates the concept of an ecological rota-tion, and the hypothetical case of rotating beforea successional sequence is completed. Successionis defined as the orderly replacement over time ofone species or community of species by another,resulting from competitive interactions betweenthem for limited site resources (Marchand, 1987).The vertical axis of Fig. 2 delineates a range fromearly successional species (pioneer) to late succes-sional species (climax).

Under a moderate disturbance regime (for in-stance, stem harvesting or selective cutting), T and2T represent two Faustmann rotations. The de-clining path of ‘backwards’ succession is referredto as successional retrogression. For a moderatedisturbance, an ecological rotation is representedby T e, the time when the forest recovers to theoriginal successional condition. A more severedisturbance regime (for instance, whole-tree har-vesting or clear-cutting) is also represented where

a longer ecological rotation (TE) would necessar-ily be required for successional rebound. Ecologi-cal observations also suggest the possibility thatsevere or repeated disturbance could shift thebiotic community into a different domain inwhich the mature (climax) phase of succession isvery different than the pre-disturbance condition(Perry et al., 1989). For instance, a clear-cut of amature forest resulting in the permanent replace-ment by grasslands might be represented in Fig. 2as a path that never rebounds.

While Fig. 2 focuses on a potential decay insuccessional pathways due to short forest rota-tions, a similar diagram could model other ecosys-tem retrogressions. For example, Federer et al.(1989) describe the effects of intensive harvest onthe long-term soil depletion of calcium and othernutrients, and the potential limiting effect onforest growth.

In the next section, a model is developed toinvestigate the ecological mechanisms and eco-nomic consequences behind a rotation-dependentprofit function in the spirit of the Kimmins’ suc-cessional retrogression hypothesis. Knowledge ofthe relationship between rotation length and fu-ture profit functions may influence rotation deci-

J.D. Erickson et al. / Ecological Economics 31 (1999) 91–10696

sions, with both economic and ecological benefits.Furthermore, valuing ecosystem recovery maybenefit non-timber amenities exhibiting increasingreturns in T as described elsewhere (often referredto as the Hartman model after Hartman, 1976).Lastly, the cost and benefits of moving fromeconomic rotations to ecological rotations can beobtained and used for public policy extensions.

3. An ecological economic model of the northernhardwood forest

The northern hardwood forest ecosystem is thedominant hardwood component of the largernorthern forest of the United States, stretchingwest to northern Minnesota, east through NewEngland, south into parts of the PennsylvaniaAppalachians, and north into Canada. It is char-acterized by sugar maple (Acer saccharum),American beech (Fagus grandifolia), and yellowbirch (Betula alleghaniensis) predominance, withvarying admixtures of other hardwoods and soft-woods. A model was developed to account forforest growth, pioneer species introduction, con-version from biomass to merchantable timber andpulpwood, and stumpage price growth. Develop-ment and details of these four components aredescribed in detail in Erickson et al. (1997).

3.1. Growth simulation

The stochastic forest growth simulatorJABOWA developed by Christ et al. (1995) wasused to model succession and growth following aclear-cut in the northern hardwood forest. TheJABOWA model simulates growth of individualtrees on small plots at the forest gap level, builton silvical data for the species of the HubbardBrook Experimental Forest in the White Moun-tains of New Hampshire. ‘Gap’ refers to a hole inthe forest canopy created by the felling of a tree,naturally or otherwise. Christ et al. (1995) devel-oped a version of the model in PASCAL to testthe accuracy of the original Botkin et al. (1972)model predictions against forest inventory data.Model development, parameters, and forest spe-cies characteristics are described in Erickson et al.

(1997). In general, growth algorithms for eachspecies consist of the following components(adapted from Botkin et al., 1972):

Dd=G(s, L, dmax, hmax) · r(L(I, Z))

· h(D, Dmin, Dmax) · S(A, u) (17)

G()=s L{1− [(d · h)/(dmax · hmax)]} (18)

r()=1−e−4.64(L−0.05) {shade-tolerant} (19)

r()=2.24 (1−e−1.136(L−0.08)) {shade-intolerant}(20)

where:

L=I e−kZ (21)

h()=4(D−Dmin)(Dmax−D)

(Dmax−Dmin)2 (22)

S()=1−A/u (23)

Eq. (17) represents the annual change in speciesdiameter at breast height (d). Only growth indiameter is modeled because it will be used topredict merchantable volume (Q) by species andproduct class for estimating the stand profit func-tion in Eq. (1). The function G represents agrowth rate equation for each species under opti-mal conditions, depending on a solar energy uti-lization factor (s), leaf area (L), and maximumvalues for diameter (dmax) and height (hmax).

The remaining right-hand side functions act asmultipliers to the optimal growth function to takeinto account shading, climate, and soil quality.The shading function, r, is modeled separately forshade-tolerant and -intolerant species and de-pends on available light to the tree (a function ofannual insolation (I) and shading leaf area (Z)).The function h accounts for the effect of tempera-ture on photosynthetic rates, and depends on thenumber of growing degree-days (D) and speciesspecific minimum and maximum values of D forwhich growth is possible. Lastly, S is a dynamicsoil quality index dependent on total basal area(A) on the plot and maximum basal area (u)under optimal growing conditions.

Stochastic dynamics of stand growth enter themodel through stem birth and death subroutines.Each year, individual trees competing for lightbecome established on the forest floor, grow, or

J.D. Erickson et al. / Ecological Economics 31 (1999) 91–106 97

Fig. 3. Northern hardwood succession following clear-cut.

die. Species characteristics and chance determinethe dynamics of these birth–growth–death cycles.New saplings randomly enter the plot within lim-its imposed by their relative shade tolerance anddegree-day and soil moisture requirements. Astaller trees shade smaller ones, the amount ofshading is dependent on the species’ characteristicleaf number and area, and survival under shadedconditions depends on the shade tolerance of aspecies. Given this stochasticity, simulation datavary widely with each model run.3

3.2. Successional retrogression

Building on the JABOWA model, the challengeis to incorporate an ecological mechanism to cap-ture Kimmins’ hypothesis of rotation-dependentsuccession and growth. Such a mechanism is evi-dent in the early succession rebound of pioneerspecies. A possible succession of dominant speciesis represented by Fig. 3, adapted from Marks(1974).

During the first 15–20 years following a clear-cut, the recovering forest is dominated by pioneer

species such as raspberry bushes, birches, and pincherry. These fast growing, opportunistic speciesplay a critical role in ecosystem recovery from aclear-cut by reducing run-off and limiting soil andnutrient loss (Marks, 1974). However, their initialdensity will also influence stand biomass accumu-lation and growth of commercial species (Wilsonand Jenson, 1954; Marquis, 1969; Mou et al.,1993; Heitzman and Nyland, 1994).

In this application to the northern hardwoodforest, pin cherry (Prunus pensyl6anica) is as-sumed to be the dominant pioneer species. As aparticularly fast growing, short-lived, shade-intol-erant species with no commercial value, the effectof its growth following a clear-cut on forest suc-cession and future harvest profits can be signifi-cant. Tierney and Fahey (1998) demonstrate theinfluence of short rotations on the survival of itsseeds, and its subsequent germination and growthat very high density in young stands. This forestecology research indicates that pioneer speciesdensities may stabilize at low levels following a120-year or more rotation regime (comparablewith a Kimmins’ ecological rotation). Rotationsat 60-year intervals (closer to a Faustmann eco-nomic rotation) result in increasing pioneer spe-cies densities toward a carrying capacityasymptote.

3 Data specific to defining equations in the remainder of thissection can be obtained from the authors, and are based onten runs (ten 100 m2 plots). This builds an approximately 1/4acre plot, which is subsequently expanded to a full acre byassuming each tree represents four trees per acre.

J.D. Erickson et al. / Ecological Economics 31 (1999) 91–10698

The dependence of the initial density of a pio-neer species (PS) on the previous harvest rotationlength (Ti−1) and the number of previous harvests(i−1) is used to represent the more general caseof successional retrogression from Fig. 2. Thefollowing ordinary least squares model was esti-mated to capture the hypothesis of a rotation-de-pendent ecological impact function proposed inEq. (5). Estimation is based on data from the soilseed bank dynamic modeling results of Tierneyand Fahey (1998). Ecological assumptions andresearch results are reported in Erickson et al.(1997).

PS= f(Ti−1, i−1)=100=V

for Ti\140 years (24)

PS=7342.27−89.18 Ti−1+0.25 T i−12

+550(i−1) for Ti5140 years4 (25)

3.3. Multiproduct, stochastic quality model

The third model component converts biomassoutput from JABOWA into economic output.The financial value of standing timber depends onage, size, species, and quality distributions. Atypical northern hardwood stand can providesawtimber, pulpwood, and firewood. Dependingon the market and the land owners motivations,any combination of these three product classesmay be managed. Stand profit is represented as5:

P(t, PS)=! %

9

S=1

%6

C=1

QS,C [d, M, PS ]"

· Pt (26)

Profit [p(t, PS)] is defined at a year following aclear-cut and before the next (t=0, 1, 2, . . . , T),given initial pioneer species density (PS). Pioneerspecies density influences profitability through in-

troducing significant competition for light andother resourcs in the JABOWA model duringearly stand development. As in Eq. (1), totalstand profit (US$/acre) is the product of a pricematrix (Pt) and merchantable volume (Q) foreight commercial species (S=1–8) and a non-commercial species group (S=9) in each productcategory (C=1–6). Product categories are com-prised of grade 1–3 timber (C=1–3), belowgrade sawtimber (C=4), and hardwood (C=5)and softwood (C=6) pulp. To assign qualityclasses, a random number is generated and as-signed to each stem and compared with classprobability limits as estimated by a generalizedlogistic regression (GLR) model developed byYaussy (1993). The GLR procedure, parameters,and an example are described in Erickson et al.(1997). Firewood output was not considered.

Merchantable volume (Q) is modeled on stemdiameter (d), provided for each tree by a growthsimulation, and merchantable length (M), whichis also modeled on d. The level of initial pioneerspecies density (PS) is predicted from Eq. (25)based on the previous period’s rotation length(Ti−1) and number (i−1). PS influences diametergrowth through the dynamics of the forest growthsimulator, as well as influencing merchantablevolume calculations through impacting forest sitequality. The procedures for converting diameterestimates to merchantable volume by species andproduct class are described in detail in Erickson etal. (1997).

3.4. Parameterization

Integrating the first three components of themodel outlined above, merchantable stand vol-umes were generated at 10-year intervals fromyear 20 to 250, at initial pioneer species densitiesof 0, 10, 20, 50, 100, 200, 500, 1000, 2000, and5000 stems per 100 m2. Volume within each spe-cies, product class, and year was then convertedto profit by multiplying a net price matrix of 1995stumpage prices. The initial distribution of netprices (P0) across product classes and species issummarized in Table 1. Stand profit for each yearwas then summarized across all products andspecies to generate data for p(t, PS) at each PSvalue run.

4 All parameters are significant at a=0.10; R2=0.98; F=65.05.

5 The timber/pulpwood component of the model was pro-grammed in Visual Basic for Microsoft Excel Macros. A19-page appendix including the code is available from theauthors. These procedures were originally developed by theUSDA Forest Service and have also been incorporated in theNE-TWIGS forest growth model. See Miner et al. (1988) for ageneral reference to the TWIGS family of models.

J.D. Erickson et al. / Ecological Economics 31 (1999) 91–106 99

Table 1Initial sawtimber stumpage and pulpwood prices (P0)a

Grade 1Species Below grade Grade 3 Grade 2 Pulp

(US$/thousand board feet) (US$/cord)

650Sugar maple 125 298.30 7471.575Beech 20 38.15 56.3 7

200 7Yellow birch 149.050 99.50289.5 400 7White ash 75 182.30

100Balsam fir 30 53.10 76.2 12100 12Red spruce 76.230 53.10

68.1 80Paper birch 745 56.557150Red maple 116.050 83.00

–Non-commercial – – – 7

a Note: Sawtimber prices in each quality class were calculated from ranges of stumpage prices reported in NYDEC (1995) for theAdirondack region. Within each range: min=below grade price, 33rd percentile=grade 3 price, 66th percentile=grade 2 price, andmax=grade 1 price.

This specification results in a 264×6 explana-tory variable matrix. The following cubic modelwas fitted6:

p(t, PS)= (b1+a1PS)t+ (b2+a2PS2)t2

+ (b3+a3PS)t3 (27)

p(t, PS)

= (7.718−0.0025PS)t

+ (0.219+1.52×10−9PS2)t2

− (0.00082+1.40×10−8PS)t3. (28)

Fig. 4 plots p() at some illustrative PS values.Here p() represents stand profit at 1995 prices.Price growth is taken up separately in Section 3.5.

3.5. Price growth (Pt)

The influences on stumpage prices at the foreststand level are complex. They might include: tim-ber quality, volume to be cut per acre, loggingterrain, market demand, distance to market, sea-son of year, distance to public roads, woods laborcosts, size of the average tree to be cut, type oflogging equipment, percentage of timber speciesin the area, end product of manufacture,landowner requirements, landowner knowledge of

market value, property taxes, performance bondrequirements, and insurance costs (NYDEC,1995). At the macroeconomic level, exports, millstocks, and aggregate demand are typically ex-planatory variables (Luppold and Jacobsen,1985). Emerging effects on northeast stumpageprices include increasing substitution of recycledfibers in paper making, board feet restrictions onremovals in the Northwestern United States, andcontinued growth in global wood demand.

For the purposes of this model, the Pt matrixwill depend on an initial price distribution at t=0(see Table 1), and algorithms for growth in threeproduct classes. As a stand matures, it is assumedto enter three stages of product development: (1)pulpwood, (2) low quality sawtimber, and (3) highquality sawtimber. To illustrate, Fig. 5 plots arepresentative model run. Here prices are assumed

Fig. 4. p(T, PS) at five initial pioneer species (PS) densities.

6 Assessment: a1, b2, a2, and b3 all significant at aB0.005;b1 and a3 significant at a=0.073 and 0.209, respectively;R2=0.58; F=59.19.

J.D. Erickson et al. / Ecological Economics 31 (1999) 91–106100

Fig. 5. Sugar maple stumpage and hardwood pulp value,PS=0, 1995 US$/acre.

an exponential growth rate of r(t)=1%. At tL,the stand shifts into a low quality sawtimberphase (below grade and grade 3), and the expo-nential growth rate increases to 3%. As thestand continues to mature, high quality timberbecomes more prevalent until a time tH isreached when timber prices grow at a maximumrate more characteristic of high quality timber.

As continued short rotations enhance pioneerspecies abundance, species competition pushescommercial species development further into thefuture, thus delaying the entrance into higherquality product classes. To capture this succes-sional retrogression hypothesis, a shift variable(Di) is assumed to add years to tL and tH de-pending on pioneer species density at the begin-ning of each rotation.

This model is applied by mapping three expo-nential growth functions over the planning hori-zon at each rate. The function is applied as amultiplier to the initial species by product pricematrix (P0), with r depending on t.

4. Rotation analysis

With the non-renewable stand value specifica-tion of Eq. (28) and the price growth modelassumed in Eqs. (29)–(33), the analysis turns toestimating and comparing rotation lengths. Spe-cifically, the question of whether the benefitsfrom recovery in future harvest periods influencethe harvest timing decision in current periods isaddressed. Four harvest cycles are modeled. Apositive discount rate causes profits from harvestcycles beyond four periods to have a negligibleeffect on the choice of rotation lengths in earlierperiods.

The applied problem is to choose the rotationset that maximizes the present value of profitsover four harvest cycles. Again, timber cuttingcosts are reflected in the stumpage price paid tothe forest owner. High labor costs are also as-sumed to prohibit thinning pioneer species fromyoung dense stands. This can be solved as afour-stage dynamic programming problem.

to remain constant over a 250-year horizon, noadditional pioneer species are added, and onlysugar maple and total hardwood pulpwood val-ues are plotted. Initially the stand generatesmostly hardwood pulpwood. Below grade sugarmaple sawtimber rises steadily over time, sur-passed first by grade 3 lumber, and eventuallyby grade 2 and 1 as the stand matures.

To capture these dynamics, an exponentialmodel for stand profit growth with a shiftinggrowth rate is assumed. In the northeastern US,from 1961–1991, Sendak (1994) reports averagereal hardwood stumpage prices for sawtimberrose 4.3% per year, and for pulpwood rose 1.3%per year. As these rates are an average acrossall quality classes and species, the followingprice growth model is assumed to apply to theentire price matrix:

Pt=P0 er(t)t (29)

where:

r(t)=1% if t5 tL+DI (30)

r(t)=3% if tL+DiB t5 tH+DI (31)

r(t)=4% if t\ tH+DI (32)

and

DI=PS/250 (33)

The parameters tL and tH represent the num-ber of years since harvest when the growingforest stand shifts into higher quality productclasses. Following a clear-cut, the recoveringforest stand can only produce pulpwood, aproduct class where prices are growing slowly at

J.D. Erickson et al. / Ecological Economics 31 (1999) 91–106 101

Max

P=er(T1)T1 p(T1, PS0)

edT1

+er(T2)(T1+T2)

p(T2, f(T1, 1))ed(T1+T2) +…

+er(T4)(T1+T2+T3+T4)

p(T4, f(T3, 3))ed(T1+T2+T3+T4) (34)

Max

P=e(r(T1)−d)T1 p(T1, PS0)

+e(r(T2)−d)(T1+T2)p(T2, f(T1, 1))+…

+e(r(T4)−d)(T1+T2+T3+T4)p(T4, f(T3, 3)) (35)

4.1. Risk and choosing an economic optimum

The difficulty in solving Eq. (35) over fourperiods is that as r varies within each rotation cycle(from 1 to 3 to 4%), the possibility of multipleoptimums arises. To illustrate, take the case ofmaximizing profit over a single rotation. AssumingPS=100, d=5%, tL=30, and tH=100, two localoptimums emerge, at T=70 and T=196. Underthe model assumptions, a 196-year global optimumstabilizes the pioneer species seed bank at ‘natural’background levels, and thus the profit-maximizingsingle rotation is potentially a Kimmins’ ecologicalrotation.

However, is this a realistic rotation length?Indeed, at a discount rate of 5% a rotation lengthof T=70 is perhaps more characteristic of the endof most commercial rotations for large landownersin the northern hardwood forest.

Are managers behaving irrationally? Not whenrisk and uncertainty are taken into account. Alandholder will not face a profit maximizationproblem with perfectly forecasted profits. Risk anduncertainty increase in later periods through mar-ket, government, and environmental variability.For example, as a forest matures its potential foryielding high quality wood increases, but so doesthe likelihood of disease, aging effects, or blow-down. Furthermore, given the public’s preferencefor old growth forests, there may be a risk of strictercutting regulations as a stand ages. As present value

declines during the interval 63BTB100 at aconstant r=3%, the landowner must also evaluatethe expectation that prices will jump (in this caseto a growth rate of r=4%) at some age tH.

In the single rotation example, consider the effectof simply raising the landowner’s discount rate inthe high quality timber phase (T\100 years) bytwo percentage points.7 One optimum at T=70results. A product phase-dependent discount ratehas some intuitive appeal, and is helpful in solvingthe multirotation problem.

4.2. The optimal rotation set with risk, and themarginal benefit of reco6ery

Assume that because of risk and uncertainty thehypothetical landowner will maximize profits ineither the low quality timber or pulpwood pricephases. The task is to solve Eq. (35) for T1, T2, T3

and T4. Parameter values are as follows: d=5%,PS0=100, tL=30, and tH=100.

Table 2 outlines the optimal rotation set undertwo cases. The first is the successional retrogressionhypothesis with p(Ti, f(Ti−1, i−1)). The second isthe traditional perfectly renewable growth hypoth-

Table 2Four harvest period solution with 2% long-run risk factor

Renewable growthRotation-dependentRotationspecification mis-specificationp(TN, f(TN−1, N p(TN, PSN=100)

−1)

Years

T1 58 4068T2 44

51T3 7083 70T4

US$400.3/acre US$470.2/acreNet presentvalue

7 If stochastic growth was carried through, or stochasticprice growth introduced, risk could be modeled with optionvalue methodology by including growth or price variance.Clarke and Reed (1989) found an optimal stopping frontierassuming brownian motion for age-dependent growth andgeometric brownian motion for price evolution, and an opti-mal stopping rule under deterministic growth.

J.D. Erickson et al. / Ecological Economics 31 (1999) 91–106102

esis with p(Ti, PSi=100). The sum of presentvalue over four periods reveals a 17.5% overesti-mate of stand profits in the traditional specifica-tion. Rotation lengths differ by as much as 24years in the second cycle, and become longer infuture cycles as prices continue to grow exponen-tially and profit from future rotations goes tozero. The rotation length for T4 simply maximizesprofits in this cycle.

Compare the first cycle rotation lengths withthat of the single rotation problem, where Tequaled 70 years. The effect of considering profitsin cycles 2, 3 and 4 at considerably higher pricesand identical growth conditions reduces T1 from70 to 40 years. This is the result of consideringthree period future profits. When successional ret-rogression is assumed, the shift from 40 to 58years is the result of including a marginal benefitof recovery.

Differentiating Eq. (35) by T1 yields the first-or-der condition for T1. The terms can be arrangedso the marginal benefit of waiting another periodequals the marginal cost of delaying first cycleprofits plus the marginal cost of delaying all fu-ture profits (site value), as was the case in thetraditional Faustmann formula, and the additionof a marginal benefit of recovery in the secondcycle:

eRT1(p(T1, PS0)(T1

=R eRT1 p(T, PS0)

+R f+eR(T1+T2)

×(p(T2, f(T1, 1))(f(T1, 1)

(f(T1, 1)(T1

(36)

where:

r(T1)=r(T2)=r(T3)=r(T4)=r, (37)

R=r−d, (38)

f=eR(T1+T2)p(T2, f(T1, 1))

+eR(T1+T2+T3)p(T3, f(T2, 2))

+eR(T1+T2+T3+T4) p(T4, f(T3, 3)) (39)

At the optimal first cycle rotation (T1=58) themarginal benefit of waiting another year untilharvest is US$7.50. It equals the marginal cost ofdelaying first cycle profits of US$6.30, the mar-

Table 3The optimal four-cycle rotation set and long-run economicand ecological health, varying the discount rate

d (%)5 10 15

Optimal rotation setT1

a 8311013751107T2

a

48 30T3a 108

T4a 30115 51

Economic indicators48.6 24.01122.6Net present valueb

1592 500Undiscounted profit @ year 105b 5909Ecological indicators

2224 6538f(T3, 3)c 527739tL+D3

a 51 56109 126tH+D3

a 121

a Unit: years.b Unit: US$/acre.c Unit: stems/acre.

ginal cost of delaying the next three harvests (sitevalue) of US$1.70, and the marginal benefit ofrecovery in future cycles of US$0.50. Site valuewell exceeds MBR because of the effect of expo-nential price growth.

4.3. Economic and ecological indicators under6arious discount rates

The discount rate measures the landowner’sopportunity cost. A relatively low opportunitycost of d=5% may be characteristic of a largelandowner with many sources of income. Forinstance, the highest return for a pulp and papermill in the northern hardwood forest is in makingpaper. As long as their mill is fed with a continu-ous, inexpensive supply of fiber, management canhold onto timber stands for speculation in thehigher return sawtimber markets, particularlywhen land is drawing income between rotations,for instance, through recreational leasing.Medium opportunity cost in the range of d=10%may be more characteristic of a small primaryforest product industry or small woodlot owner.A discount rate of 15% may be characteristic of alandowner not necessarily in the timber industry.In this case it may be more profitable to use theland for an activity with a shorter investmenthorizon, for instance, housing development.

J.D. Erickson et al. / Ecological Economics 31 (1999) 91–106 103

Table 3 lists the results of the four-cycle opti-mization when the discount rate is varied, assum-ing no risk factor. In the case of high opportunitycost (d=15%), four pulpwood rotations are opti-mal at interior solutions of 8, 37, 30 and 30 yearswith a total present value of US$24/acre. Atd=10% the optimal rotation set occurs in the lowquality sawtimber phase at rotations of 31, 51, 48and 51 years, all of which are corner solutionssince tL=30, D1=21, D2=18 and D3=21. Atd=5%, the solution occurs at the corner of thehigh quality sawtimber phase.

The sum of present value over four cycles indi-cates the effect on profit of both shorter rotationswith lower quality products and a higher discountrate. A second economic indicator, summarizingstand profit at year 105 (the end of the fourthcycle under d=15%) with no discounting, indi-cates only the effect of shorter rotations andlower quality products on future profits. Underthis second indication, just over one rotation ofhigh quality sawtimber (at T1=101 and T2=4)produces 2.7 times more undiscounted profitsthan three and a half rotations under the lowquality management case, and 10.8 times moreundiscounted profits than four full pulpwoodrotations.

The ecological indicators of the three manage-ment scenarios reflect the important ecologicalbenefits to longer rotations. At the beginning ofthe fourth harvest cycle, pioneer species density is2224 stems under long rotations, 5277 stems un-der medium length rotations, and 6538 stemsunder short rotations. In the pulpwood harvestingcase, entrance into both sawtimber phases is de-layed a full 27 years by the fourth harvest cycle.The cases where d=10% and d=15% demon-strate the declining trend in successional integrityas suggested by the Kimmins’ successional retro-gression hypothesis, while the case where d=5%perhaps approaches a set of ecological rotations(i.e. both scenarios outlined in Fig. 2).

4.4. Single period management under decliningforest health

Another method to solving the multiple rota-tions problem is to assume the values for PSi over

subsequent rotations are forest health endow-ments to new generations of owners or managers.In other words, a different owner during eachcycle solves a single rotation problem, withoutconsideration of site value or benefits to recovery.Here, the first-order condition within each cyclebecomes:

d−r(T1)=p %(Ti, PSi−1)p(Ti, PSi−1)

(40)

Again, assuming the landowner will manageeither in the low quality sawtimber or pulpwoodprice growth phases, the four-cycle interior solu-tion for T is 70, 80, 80, and 83. Future landown-ers must wait longer to maximize profits due topoorer forest health endowments. Profits continueto increase in later cycles due to exponential pricegrowth, however, timber quality and quantity areconstrained by a degrading resource. By thefourth cycle, the pulpwood price phase is 70 yearslong, and initial pioneer species density is 3979stems/acre.

4.5. Ecological rotations and 6aluing non-timberamenities

As was evident in the single period problem,given low constant discount rates ecological rota-tions may be economically optimal. Solving Eq.(35) with a constant discount rate of 5% yieldedthe rotation length set of 101, 107, 108 and 115years with a total present value of US$1122.6/acre. Assuming constant initial pioneer speciesdensity across all future harvest cycles (i.e. PS=100 and Di=0), the optimal set becomes 101, 101,101 and 116 with a total present value ofUS$1200.2/acre. Not accounting for non-re-newability results in a negligible 2% overestimateof total present value.

These rotation lengths are approaching whatmight be considered ecological rotations as de-scribed by Kimmins. Ignoring risk considerationand assuming preventively high forest mainte-nance costs, such lengths may also be economi-cally optimal. Therefore, the question remainsunder what conditions will landowners rotateforests at 100+years?

J.D. Erickson et al. / Ecological Economics 31 (1999) 91–106104

Perhaps including the value of non-timberamenities would justify ecological rotation lengthsas socially optimal. Amenity values that exhibitincreasing returns to rotation length might includerecreation value, provision for certain habitats,and watershed protection. For example, referringto Table 3, consider the low discount rate solution(d=5%) and middle discount rate solution (d=10%) as the social and private optimal rotationsets. Next, evaluating the social rotation set at theprivate discount rate of 10% results in a totalpresent value of just US$5/acre. If a landownerwas forced to rotate at these lengths, this wouldresult in a private loss of US$41/acre. However, ifthe sum of non-timber amenities exceeds this lossand the landowner experiences these benefits di-rectly (for example, hunting or recreational use),then there may be a private incentive to lengthenrotations.

Alternatively, if the amenity values are of astrictly social nature (for example, watershed pro-tection or biodiversity preservation) then an op-portunity may exist for the government or anenvironmental group to accommodate the landowner’s loss in timber profits through a paymentor incentive mechanism (for example, paying for aconservation easement or providing tax relief). Inaddition, alternative silviculture practices such asselective cutting may strike common ground be-tween the interplay of social and private benefits.Furthermore, assessing the ecological impact ofeconomic decisions contributes to the definitionand assessment of ‘new’ or ‘sustainable’ forestry,which embraces management practices derivedfrom ecological principles (Franklin, 1989; Gillis,1990; Gane, 1992; Fiedler, 1992; Maser, 1994).

5. Concluding remarks

Accounting for the ecological recovery of thenorthern hardwood forest over a series of harvestswas shown to increase rotation lengths over thetraditional Faustmann result. A positive marginalbenefit of recovery offsets the marginal costs ofdelaying current and future rotations, creating abenefit to delaying rotations under a non-renew-able stand value growth specification.

The model presented in this paper is limited tothe specific ecological dynamic of pioneer speciesintroduction and interspecies competition for lightand other resources. In the context of forest man-agement, this approach can be generalized tomany intertemporal dynamics such as other suc-cessional sequences, alteration of nutrient cycles,or disturbance from anthropogenic climatechange. Knowledge of benefits to ecosystem re-covery can help define both ecological and eco-nomic rotation lengths under various scenarios ofKimmins’ ecosystem retrogression. At one ex-treme, given low discount rates and risk, relativelylong ecological rotations may be economicallyoptimal. At the other extreme, a site managedwith short rotations motivated by short-termprofits and a high discount rate may result indegraded forest stands with low value species—adetriment to long-run ecological health and socialbenefits.

When considering social welfare and themaintenance of ecosystem processes from multi-ple-use management, many non-timber benefitshave increasing returns in rotation length anddecreasing returns to harvest intensity. The benefitof recovery was shown to have a market value,and its inclusion more accurately estimates theoptimal rotation set. Including this benefit, how-ever, may not completely provide the private in-centive to move from ecologically unsustainableto sustainable rotation lengths and practices, par-ticularly when the net private cost of doing so ishigh. However, this net private cost can be com-pared with benefits from non-timber amenitiesand alternative management practices, or to costsof forest maintenance (i.e. thinning undesirablespecies), providing rationale for socialmanagement.

Questions of where to manage along ecologicaleconomic dimensions in a forest will ultimatelydepend on a region’s spatial ownership pattern,land holder motivations, policy variables, man-agement costs, timber markets, and ecosystemcharacteristics. These modeling results suggestvery different economic and ecological outcomesby varying opportunity cost and ecosystem recov-ery assumptions, and suggest a positive benefit torecovery. Estimating economic benefits across

J.D. Erickson et al. / Ecological Economics 31 (1999) 91–106 105

ecological gradients, rather than through non-market valuation techniques, could capture non-timber amenity value and contribute tostewardship policies aimed at managing multiple,spatial benefits of a forest ecosystem.

Acknowledgements

The authors would like to thank Jon Conrad,Bill Schultze, Neha Khanna and three reviewersfor invaluable comments and suggestions.

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