ON SOLVING THE MULTIROTATIONAL TIMBER … · ON SOLVING THE MULTIROTATIONAL TIMBER HARVESTING PROBLEM WITH STOCHASTIC PRICES: ALINEAR COMPLEMENTARITY FORMULATION MARGARET INSLEY AND

ON SOLVING THE MULTIROTATIONAL TIMBER

HARVESTING PROBLEM WITH STOCHASTIC PRICES:A LINEAR COMPLEMENTARITY FORMULATION

MARGARET INSLEY AND KIMBERLY ROLLINS

This article develops a two-factor real options model of the harvesting decision over infinite rotationsassuming a known stochastic price process and using a rigorous Hamilton–Jacobi–Bellman method-ology. The harvesting problem is formulated as a linear complementarity problem that is solved nu-merically using a fully implicit finite difference method. This approach is contrasted with the Markovdecision process models commonly used in the literature. The model is used to estimate the value ofa representative stand in Ontario’s boreal forest, both when there is complete flexibility regardingharvesting time and when regulations dictate the harvesting date.

Key words: linear complementarity problem, Markov decision process, mean reversion, optimal har-vesting, real options.

The forest economics literature has long dealtwith the problem of optimal harvesting underuncertainty. An overview is provided in re-cent bibliographies by Newman, and Brazeeand Newman. A thirty-year-long strand ofthis literature emphasizes the importance ofvaluing managerial flexibility in the contextof irreversible harvesting decisions when for-est product prices are volatile relative to har-vesting costs (Hool; Lembersky and Johnson).Failure to include the value of managementoptions where they exist will result in an in-correct valuation of a forestry investment.Because the formulation and modeling of tim-ber harvesting problems to accurately incor-porate the value of managerial flexibility isunlikely to result in closed-form solutions, for-

Margaret Insley is assistant professor in the Department of Eco-nomics, University of Waterloo, Waterloo, Ontario, Canada. Kim-berly Rollins is associate professor in the Department of ResourceEconomics, University of Nevada, Reno, Nevada.

The authors acknowledge with gratitude the financial support ofthe Living Legacy Trust (Ontario), Tembec, Inc., the SustainableForest Management Network, the Upper Great Lakes Environ-mental Research Network, and the Ministry of Natural Resourcesof Ontario. The opinions expressed herein represent the views ofthe authors, and not of the agencies that provided financial support.

The authors would also like to thank numerous individuals whoprovided useful information and comments on an earlier versionof this article. These include Margaret Penner (Forest AnalysisLtd.), George Bruemmer, Al Stinson, and Paul Krabbe (Tembec),Wayne Bell (Ontario Ministry of Natural Resources), Glenn Fox(University of Guelph), and Peter Forsyth (University of Water-loo). Thanks also to Tony Wirjanto of the University of Waterloofor assistance with statistical testing. Any shortcomings or errorsare solely the responsibility of the authors.

est economics models have used a number ofapproaches to solve complex optimal harvest-ing problems under stochastic prices, includingMarkov decision process (MDP) models andsimulation. More recently, these approacheshave increasingly drawn from the burgeon-ing finance literature on the valuation of fi-nancial and real options (Dixit and Pindyck;Trigeorgis). These approaches explicitly incor-porate into the value of the resource any op-portunities for managers to adjust harvestingplans in response to stochastic events as theyunfold—opportunities that may be thought ofas embedded options.

The contributions of the real options litera-ture are in the development of powerful deci-sion models and solution techniques that canoffer improvements in accuracy over methodsbased on MDP models, a standard in the foresteconomics literature; and in the formulationand interpretation of the decision problem byexplicitly recognizing the parallels between fi-nancial options, such as call options on a stock,and real options, which refer to the opportu-nities to acquire real assets. We can view theopportunity to harvest a stand of trees as areal option, similar to an American call option,which can be exercised at any time. The exer-cise price is the cost of harvesting the trees andtransporting them to the point of sale. The op-tion to choose the optimal harvest time, basedon wood volume and price, and the option toabandon the investment if wood prices are too

Amer. J. Agr. Econ. 87(3) (August 2005): 735–755Copyright 2005 American Agricultural Economics Association

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low are embedded in the tree harvesting op-portunity. A real options approach focuses onthe importance of options embedded in theharvesting decision.1

In their 1999 review article, Brazee and New-man refer to the rather slow development ofthe options approach to forestry due to itsmathematically demanding nature. In this ar-ticle, we make a methodological contributionto implementing real options approaches bydemonstrating the numerical solution of themultirotational optimal harvesting problem toa specified degree of accuracy using a tech-nique that can handle a fairly general class ofspecifications for price uncertainty. The mul-tirotational harvesting problem represents apath-dependent option, which is significantlymore complex than the single rotation prob-lem. Thus, this article extends the single rota-tion analysis of Insley in a nontrivial manner.

An important benefit of the approach pre-sented in this article is the assurance that thesolution obtained is accurate given the cho-sen inputs. The particular numerical procedureused in solving the option value problem canhave a large impact on the estimated value ofan uncertain investment—a point emphasizedin the finance literature (Wilmott; Wilmott,Dewynne, and Howison). In instances whenaccuracy is important, the approach presentedin this article will represent a significant im-provement over other techniques in the liter-ature for dealing with price uncertainty. In apolicy-making context, accuracy will be impor-tant in decisions that involve tradeoffs—suchas in an evaluation of the benefits versus costsof a regulation that will restrict forest owners’ability to freely determine the optimal harvestschedule, but that would provide environmen-tal benefits.

We develop a model of the harvesting de-cision at the stand level assuming stochasticprices and deterministic volume. The objectiveis to maximize the net present value of the in-vestment over an infinite stream of future ro-tations, where price, P, is assumed to follow anIto process,

d P = a(P, t) dt + b(P, t) dz.(1)

1 Another innovation in the finance literature relevant to foresteconomics is contingent claims analysis, which allows one to avoidthe conceptual difficulties of specifying an exogenous discount rate.However, for a storable commodity it is still necessary to estimatea “convenience yield” or market price of risk (Trigeorgis, Dixit andPindyck). We do not use contingent claims analysis in this article.

In equation (1), a(P, t) and b(P, t) representknown functions and dz is the increment ofa Wiener process. For this article, we assumemean-reverting prices (first-order autoregres-sive), but our technique could easily handlegeometric Brownian motion or some otherprocess with alternate specifications of a(P, t)and b(P, t), such as a table of discrete param-eter values.

The harvesting problem is specified as a lin-ear complementarity problem (LCP), and issolved numerically using a fully implicit fi-nite difference approach (Wilmott, Dewynne,and Howison; Tavella). This is a rigorous tech-nique for which the existence and uniquenessof solutions follow from a large mathemat-ics literature. It has guaranteed convergencewith easily determined error bounds. Wilmott,Dewynne, and Howison discusses Americanoptions in terms of variational inequalities andlinear complementarity formulations. Proofsof the existence and uniqueness of solutionsare available in the literature (Elliott and Ock-endon; Friedman; Kinderlehrer and Stampac-chia). To our knowledge this technique hasnot been used previously to solve a multirota-tion optimal harvesting problem with a generalstochastic price process (as in equation (1))and when land value is determined endoge-nously.

One of the purposes of this article is to dis-tinguish among different approaches that canbe used to incorporate managerial flexibilityinto optimal harvest decisions, and to describethe contributions of real options approachesto solving forestry economics problems. Wederive theoretically the relationship betweenthe MDP and LCP approaches used to solvethe stochastic, optimal harvesting problem. Wedemonstrate that the numerical solution of theLCP provides guaranteed error bounds andpermits the use of a finer level of resolutionthan is typically used in solving MDP models.We show that the improvement in accuracy of-fered by this finer level of resolution can bevery significant. We apply the model to datafrom Ontario, Canada to estimate the oppor-tunity costs of harvest restrictions such as thosethat are imposed to maintain an even annualflow of timber from public forest lands.

In the next section we provide a fairly de-tailed literature review that traces develop-ments in forestry economics in dealing withoptimal harvesting with uncertain prices. Wesituate our article within the literature andspecify in more detail its particular contribu-tions.

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Insley and Rollins Timber Harvesting with Stochastic Prices 737

Literature Review and Modeling Approaches

Almost thirty years ago, Lembersky and John-son illustrated that a simplistic Faustmann-type approach to estimating stand valueignores managerial flexibility and hence un-dervalues the resource. Speaking in the con-text of optimal actions for a forest managerfaced with uncertainty in product prices, theseauthors point out that “it is not advisable topredetermine the specific management actionto carry out at each of the future time pointsat which a decision will be made. A predeter-mined action can turn out to be inappropriatefor stand and market conditions at implemen-tation” (Lembersky and Johnson, p. 109). Thedecision maker who has the ability to makedecisions based on observed outcomes of ran-dom variables such as price and growth overtime, instead of having to follow a prescribedrule based on expected values, can decide toharvest early to take advantage of an upswingin prices or delay harvesting if prices are de-pressed.2

Analytical Models and Closed-Form Solutions

A number of earlier studies focused on generaltheoretical implications of the problem of un-certainty in harvesting decisions, using stylizedanalytical models with closed-form solutions.These models necessarily greatly abstract fromthe reality of most forest harvesting problemsin order to obtain analytical solutions.

For example, Brock, Rothschild, andStiglitz; Brock and Rothschild; and Millerand Voltaire consider the harvesting problemin the general context of stochastic capitaltheory and optimal stopping problems. Thisapproach focuses on deriving analyticalsolutions and comparative static results todetermine how the value of the asset varieswith the parameters that describe the stochas-tic growth process of the state variables. Themajority of these is single rotation models,while Miller and Voltaire extend the Brock,Rothschild, and Stiglitz capital theory modelto the multirotation case and develop a barrierrule for stochastic revenue. These models treatrevenue as the stochastic variable; the priceand quantity state variables are not separatelydistinguished. Lohmander (1988a) derives

2 This is true if prices follow a stochastic process that reverts tosome mean over time. If prices follow geometric Brownian mo-tion, then the value of flexibility comes from avoiding uneconomicharvests.

comparative static results for continuousharvesting under price and volume growthuncertainty.

Willassen uses the theory of stochastic im-pulse control to derive an explicit solution tothe stochastic multirotational optimal harvest-ing problem with revenue as the state variable.Drift and diffusion parameters of the stochas-tic process are independent of time. Sødal alsoderives the closed-form solution to the sameproblem using a simplified approach based onDixit, Pindyck, and Sødal. While allowing forclosed-form solutions, these models are notpracticable for more applied problems whereprice and quantity must be separately mod-eled.

Recent research in this vein focuses onadding different sources of uncertainty suchas stochastic forest growth (Alvarez) and astochastic interest rate (Alvarez and Koskela).

Models Where Prices Follow GeometricBrownian Motion

Earlier articles to model price and volumeseparately typically assumed price could becharacterized by geometric Brownian motion.Clarke and Reed, and Reed and Clarke de-rive an optimal harvesting rule when price andvolumetric growth are random variables. Pricefollows geometric Brownian motion whereasgrowth in volume is a function of stand age plusrandom Brownian motion. Under their (my-opic look-ahead) rule the age at which a standis harvested becomes a random variable thatis independent of the absolute level of timberprices. This is a consequence of assuming geo-metric Brownian motion and ignoring harvest-ing costs. Insley, and Yin and Newman (1995)demonstrate that incorporating harvesting andmanagement costs into the model would implythat the optimal harvest time is no longer in-dependent of price.

Using the Clarke and Reed results, but in-corporating land rent costs as deterministic,Yin and Newman (1997) compare the opti-mal harvest time where growth and price arestochastic to what would be proscribed byFaustmann or maximum sustained yield rules.In reality land rent should reflect the valueof the bare land, which would equal the ex-pected discounted net benefit from optimallymanaging the timber stand forever. Thus landrent should be endogenous, determined jointlywith the value of the harvesting opportunity,although this considerably complicates theanalysis.

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Thomson provides an example of one ofthe earlier uses of a real options approach ina forestry application. Thomson determinesland rent endogenously assuming stumpageprices follow geometric Brownian motion. Hecompares stand value and rotation ages (as afunction of price) with a fixed price Faustmannmodel. Thomson solves his model using a lat-tice method (a binomial tree), which is com-monly used in finance and is in fact a simpleversion of an explicit finite difference scheme.Wilmott discusses the advantages of finite dif-ference methods, such as the one used in thisarticle, over the binomial tree. In general, fi-nite difference schemes offer more flexibilitythan binomial trees to handle complex prob-lems. For example, with mean-reverting pricesa trinomial tree with nonstandard branchingwould be employed, as described in Hull. Aswith other explicit methods, the binomial treeapproach suffers from stability constraints thatrestrict the timestep size used in the numeri-cal solution and can make solution very slow.Finally, Coleman, Li, and Verma demonstratethat the smooth pasting condition is only ap-proximately satisfied with binomial trees.3

Morck, Schwartz, and Strangeland moreexplicitly bring to their model insights fromfinancial real options methods. They use acontingent claims approach to determine theoptimal harvesting rate for a firm with a ten-year lease on a mature forest. This is a problemof inventory management where growth in in-ventory is assumed to follow Brownian motionwith a drift, and timber prices are assumed tofollow geometric Brownian motion.

Models That Incorporate Alternate PriceProcesses

In general, the assumption of geometric Brow-nian motion makes solution of the tree harvest-ing problem more tractable. If managementand harvesting costs are ignored, the problemcan be solved analytically. However, the as-sumption of geometric Brownian motion maynot be realistic for many commodities over thelong term because of the implication that theexpected price level and variance will rise overtime without bound. Alternatively, price maybe modeled as some sort of stationary process.The precise specification of the price process

3 The smooth pasting condition must be satisfied at the early ex-ercise of an American-type option—in our case when it is optimalto harvest a stand of trees. Dixit and Pindyck explain the smoothpasting condition.

can have a large effect on the estimated valueand optimal timing of a resource investment(Insley, Sarkar).

Some researchers of the late 1980s andearly 1990s modeled price as an identicallydistributed random variable. For example,Lohmander (1988b) investigated the effect ofstochastic prices on optimal harvesting in a sin-gle rotation problem assuming price could bemodeled as a random draw from a uniformdistribution. Brazee and Mendelsohn solveda multirotation model with price representedas a random draw from a normal distribution.Under this assumption the timber price in anygiven period is statistically independent of itslevel in any other period. Haight (1990, 1991)presents two other examples in this vein. Al-though the assumption of serially uncorrelatedprices is not realistic, these articles provide use-ful intuition of the impact of stochastic, sta-tionary prices. More recently stationarity ormean reversion is typically captured by as-suming price follows an Ornstein–Uhlenbecksort of process (see Dixit and Pindyck). AnOrnstein–Uhlenbeck process (as specified inSection 3, equation (2)) has independent in-crements, meaning that the change in price be-tween two consecutive periods is independentof the change between any two other consec-utive periods.

With mean-reverting prices, the optimal ro-tation problem must be solved numerically.Models based on MDPs represent one possibleapproach. MDPs have become a standard inthe forest economics literature for incorporat-ing ecological and market risks into harvest de-cisions. This approach models stochastic statevariables in discrete time and computes matri-ces of transition probabilities that reflect theprobability of moving from one state to an-other, conditional on management decisions.The transition matrix is typically estimated bysimulation, and various techniques are usedto determine optimal decisions based on thepossibilities offered by the transition matrix.These techniques include the policy improve-ment algorithm, linear programming, and suc-cessive approximation, and are described inthe operations research literature (e.g., Hillierand Lieberman).

The MDP approach is somewhat limited bythe use of the transition matrix, which expandsdramatically as the number of possible out-comes of a stochastic variable increases. This istypically handled by grouping stochastic out-comes, such as prices, into a small numberof categories, such as “high,” “medium,” and

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“low.” As will be shown below, the LCP ap-proach used in this article does not requiresimulations to calculate transition probabili-ties, and stochastic variables can be specified inmuch finer detail. This allows for finer degreesof resolution, and therefore greater accuracyin the determination of decision criteria. How-ever, it may be noted that if there are manystochastic factors, MDP models (or simulation,discussed below) may be the only computa-tionally feasible method.

Examples of MDP models are numerous(Lembersky and Johnson; Norstrom; Kao;Teeter and Caulfield; Kaya and Buongiorno;Lin and Buongiorno; Buongiorno). Lin andBuongiorno incorporate natural catastrophesas well as diversity of tree species and tree size.Buongiorno provides useful insight by inter-preting Faustmann’s formula as a special caseof a MDP model in which the probability ofmoving from one state to another is equal tounity.

Plantinga,4 Haight and Holmes, and Gongsolve the optimal harvesting problem withmean-reverting prices using a Markov transi-tion matrix and a discrete stochastic dynamicprogramming algorithm. For simplicity, thesearticles treat the value of the bare land as deter-ministic. Brazee, Amacher, and Conway, in ex-amining the benefits of adaptive managementwhen prices are mean reverting, note that thegains vary directly with the level of mean re-version.

Gong compares a stochastic dynamic pro-gramming approach based on the use of aMDP, with a simulation method for findingthe optimal harvest policy. He notes the diffi-culty of determining accurate stand values withsimulation. Longstaff and Schwartz, and An-dersen have recently proposed two methodsto adapt a simulation approach to valuing anAmerican option. Hull provides a useful sum-mary of these methods. Wilmott discusses inmore detail the difficulties with using simula-tion approaches to value an American type op-tion, that is, one for which early exercise maybe optimal.

The real options literature provides an al-ternative to MDP approaches and simulationin formulating the problem in terms of a par-tial differential equation, which can be solvednumerically using techniques from the largeliterature on numerical analysis. Saphores,

4 Plantinga describes the conceptual relationship between thenotion of option value and the previous literature on harvestingunder uncertainty.

Khalaf, and Pelletier demonstrate the use ofGalerkin’s method (a finite-element method)to solve the problem of whether to preserveor harvest a stand of old growth forest whenlumber prices follow geometric Brownian mo-tion with jumps. Insley formulates the harvest-ing problem with mean-reverting prices for asingle rotation as an LCP, and demonstrates anumerical solution using a fully implicit finitedifference scheme.

Characterizing the Price Process for Timber

There is a large literature examining thetime path of commodity prices, and this con-tinues to be an area of active research. Ithas been suggested that some sort of mean-reverting process provides a better descriptionof the price path of many commodities (seeLund, Bessembinder et al. Hassett and Met-calf, Schwartz.) As noted by Schwartz, in anequilibrium setting we would expect that whenprices are relatively high, supply will increasesince higher-cost producers of the commoditywill enter the market putting downward pres-sure on prices. Conversely, when prices are rel-atively low, the higher cost producers will exitthe market putting upward pressure on prices.Schwartz examines the spot prices of oil, cop-per, and gold. He finds strong mean reversionfor oil and copper. Bessembinder et al. use theterm structure of futures prices to test whetherinvestors anticipate mean reversion in spot as-set prices. They find a large degree of mean re-version in crude oil and agricultural commodi-ties, and a lesser degree of mean reversion inmetals.

Unfortunately, it is difficult to concludedefinitively that the price of any particularcommodity exhibits mean reversion or pos-sesses a unit root and hence is nonstationary.Many different tests exist, none of which hasbeen shown analytically to be uniformly mostpowerful (e.g., Ahrens and Sharma).

A number of studies have examined the sta-tistical properties of stumpage prices in vari-ous markets and have obtained mixed results.Several researchers have examined pine saw-timber stumpage prices in the southern UnitedStates. Haight and Holmes, Hultkrantz (1993)and Yin and Newman (1995, 1996, 1997) allfind evidence that supports stationary, autore-gressive models. Prestemon extends the dataseries used in Hultkrantz (1993), and Yin andNewman (1996) and improves upon the sta-tistical tests used. He finds that most of themonthly series contain nonstationary as well as

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stationary components and that quarterlyprices are closer to pure nonstationary pro-cesses. Brazee, Amacher, and Conway testprice series for pine and hardwood in Virginiaand finds the unit root hypothesis is rejectedfor the former, but not the latter. Hultkrantz(1995) examines the behavior of timber rentsin Sweden over a seventy-nine year time spanand accounts for a structural break in the pricelevel using a Perron test. He rejects the unitroot hypothesis for his data series.5 Saphores,Khalaf, and Pelletier find evidence of bothjumps and ARCH effects in stumpage pricesin the U.S. Pacific Northwest.

One theoretical argument that has beenused in favor of a random walk process is thatit implies prices that are consistent with an in-formationally efficient timber market. How-ever, McGough, Plantinga, and Provenchershow that stationary serially correlated pricescan arise in an informationally efficient timbermarket even when market shocks are indepen-dent and identically distributed.

The choice of price process in modeling theoptimal harvesting decision will continue to bethe subject of ongoing research. A challengeof resource economists is to develop modelsand solution algorithms that handle various as-sumptions regarding price, depending on thecircumstances of a particular market.

A Multirotational Real Options Model

This article extends the model of Insley to amultirotation framework with the bare landvalue determined endogenously. Much of theprevious cited literature may be thought ofin terms of different approaches to solvingthe LCP. Unlike the single rotation prob-lem, the multirotation case represents a “path-dependent option.” A path-dependent optionis one whose value depends on the history ofan underlying state variable, not just on its fi-nal value. For the multirotational optimal har-vesting problem, the value of the stand todaydepends on the quantity of lumber which it-self depends on when the stand was last har-vested. Path dependency significantly compli-cates the solution of the valuation problem(see Wilmott).

The importance of solving the complete mul-tirotational problem will vary on a case-by-case basis. In areas where trees are fast growing

5 Perron found that the exclusion of a break in trend can bias theaugmented Dickey–Fuller test and the Leybourne and McCabetest toward acceptance of a unit root.

and rotations are fairly short, the harvest-ing decision proscribed by the multirotationalproblem will be expected to be quite differ-ent from the single rotation case. As anotherexample, if recreational value of the stand-ing forest depends on population growth, thenthe value of the harvesting opportunity will betime dependent, making the multirotationalspecification important. We will examine theempirical significance of solving the multi- ver-sus single rotation problem in an empirical ex-ample.

The solution of the LCP can be directlylinked to the transition probability matrix es-timated in MDP models. We will show that theMDP approach is, in fact, an indirect methodof solving the LCP and that it is unnecessaryto use simulation to calculate transition proba-bilities. The transition probability densities arethe solutions of the forward Kolmogorov equa-tions, which are embedded in the solution ofthe LCP. The explicit connection between theMDP model and the numerical solution of theLCP is discussed in more detail in Appendix B.

Our empirical application examines the costof harvesting restrictions on the value of har-vesting a stand in Ontario’s boreal forests. Itis only by correctly modeling the impact ofstochastic prices on the optimal harvesting de-cision that the impact of regulatory restrictionscan be fully described.

Formulation of the Model

The tree harvesting decision applicable to apublicly owned forest is modeled from thepoint of view of a social planner. Hence, taxesand stumpage payments are ignored. The priceof timber sold to the mill is assumed to followa known stochastic process. The value of thestand of trees is estimated assuming the har-vesting decision will be determined optimallyin the future whatever the price path turns outto be. In a world without taxes and stumpagepayments, the estimated value at the beginningof the first rotation is the maximum amountthat a private firm would be willing to pay forthe right to harvest the trees at some time in thefuture, providing the firm has complete flexi-bility to determine the harvest date and thatmarkets exist for the logs.

The mean-reverting price process is a specialcase of the general Ito process (equation (1)).We specify the process as follows:

d P = �(P − P) dt + � P dz(2)

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where P is the price of saw logs, � is the meanreversion parameter, � is the constant variancerate, and dz is an increment of a Wiener pro-cess. According to equation (2), price revertsto a long-run average of P . The variance rategrows with P, so that the variance is zero if Pis zero.6

The age of the stand, or time since the lastharvest, � is given as

� = t − th(3)

where t is the current time and th is the time ofthe last harvest. Wood volume is assumed tobe a deterministic function of age

Q = g(�).(4)

Age is used as a state variable, along with price,P. It follows that

d� = dt.(5)

The decision to harvest the stand of trees canbe formulated as an optimal stopping problemwhere the owner must decide in each periodwhether it is better to harvest immediately ordelay until the next period. This decision pro-cess can be expressed as a Hamilton–Jacobi–Bellman equation,

V (t, P, �) = max{(P − C)Q

+ V (t, P, 0);

A�t + (1 + ��t)−1

× E[V (t + �t,

P + �P, � + ��)]}

(6)

where E is the expectation operator, V is thevalue of the opportunity to harvest, C is theper unit harvesting cost, A is the per periodamenity value of standing forest less any man-agement costs, and � is the annual discountrate.

The first expression in the braces ({ }) repre-sents the return if harvesting occurs in the cur-rent period, t. It includes the net revenue fromharvesting the trees plus the value of the landafter harvesting, V(t, P, 0). This is the value thatcould be attained if the land were sold subse-quent to the harvest, assuming that the land

6 This format is more appealing than the simple Ornstein–Uhlenbeck process in which the variance rate is � dz. In the simpleOrnstein–Uhlenbeck process, as price becomes small, the constantvolatility could cause prices to become negative.

will remain in forestry. This value is ignored inthe single rotation problem.

The second expression in the braces is thecontinuation region and represents the valueof delaying the decision to harvest for anotherperiod. It includes any amenity value of thestanding forest, such as its value as a recreationarea, less any forest management costs, A. Italso includes the expected value of the optionto harvest in the next period, discounted to thecurrent period.

In the empirical application that follows,amenity value is not considered. However, itwould be easy to include some expression foramenity value as a function of stand age, pop-ulation growth, or some other variable. Invest-ment values and harvesting rules would bechanged accordingly. If A were expressed asa function of population growth, then V wouldbe dependent on calendar time, as well as standage, and we would no longer be characterizinga steady-state solution.

Following standard arguments (Dixit andPindyck; Wilmott, Dewynne, and Howison) wecan derive a partial differential equation thatdescribes V in the continuation region

Vt + 12 �2 P2VP P + �(P − P)VP

− � V + A + V� = 0.

(7)

In contrast to the single rotation problem, wenow have the term V� in the partial differentialequation.

The full optimal stopping problem can beformulated as an LCP, which is equivalent tothe optimal stopping problem of equation (6)(Wilmott, Dewynne, and Howison; Tavella). Tdenotes the terminal time. Let � be definedas time remaining in the option’s life, that is,� ≡ T − t. Rearranging equation (7) and sub-stituting � for t, we define an expression, HV,as follows:

H V ≡ � V −[

12

�2 P2VP P + �(P − P)

× VP + A + V� − V�

].

(8)

In equation (8), �V represents the return re-quired on the investment opportunity for therational investor to continue to hold the op-tion. The expression within brackets ([ ]) rep-resents the actual return over the infinitesimaltime interval dt. The actual return has terms re-flecting how V changes with changes in P and�. It also includes the flow of amenity valueless management costs, A.

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Then, the LCP is given as

(i) H V ≥ 0

(i i) V (�, P, �) − [(P − C)Q

+ V (�, P, 0)] ≥ 0

(i i i) H V [V (�, P, �) − [(P − C)Q

+ V (�, P, 0)]] = 0.

(9)

The LCP expresses the rational individual’sstrategy with regard to holding versus killingthe option to harvest the stand of trees. Part(i) of equation (9) states that the required re-turn for holding the option must be at least asgreat as the actual return. We would not ex-pect a situation in which the required returnis less than the actual return to persist in com-petitive markets. Part (ii) states that the valueof the option, V, must be at least as great asthe return from harvesting immediately. Thereturn from harvesting immediately is the sumof the net revenue from selling the logs (P −C)Q plus the value of the land immediately af-ter harvesting, V(t, P, 0). V would never dropbelow the return from harvesting immediatelybecause the rational investor would harvest be-fore that could happen. Finally, part (iii) statesthat at least one of statements (i) or (ii) musthold as a strict equality. If HV = 0, it is worth-while to continue to hold the asset. If V − (P −C)Q − V(t, P, 0) = 0, it is worthwhile harvest-ing the asset. If both expressions hold as strictequalities, then the investor is indifferent be-tween harvesting and continuing to hold theasset.

The LCP is solved numerically as isdescribed in Appendix A.7 This involvesdiscretizing the relevant partial differentialequation including a penalty term that en-forces the American constraint (equation (9),(ii)). We are left with a series of nonlinearalgebraic equations that must be solved iter-atively. The implied Markov matrix can befound through manipulation of the discretizedversion of equation (9). Details are providedin Appendix B.

We can contrast the LCP for the multirota-tion case with that for the single rotation as inInsley. There are two state variables: � and P,as opposed to only P in the single rotation case.In general, having more than one state variableconsiderably complicates the estimation of V.However, through the “method of character-

7 The pseudocode is available on request from the authors.

istics” described in Appendix A we are ableto simplify the solution. The other differenceof note with the multirotation problem is thatpart (ii) of equation (9), the so-called Ameri-can constraint, now contains the value of thebare land after harvesting. Of course, it is thevalue of the bare land that is being solved forin the first place. Hence, solving the LCP willrequire an iterative procedure that starts withan initial guess for V(t, P, 0) and then updatesthat guess through successive iterations. In thesingle rotation case, V(t, P, 0) is set to zero,ignoring the value of the land after harvesting.

Boundary conditions for the problem arespecified as follows:

1. As P → 0, we need no special boundarycondition to prevent negative prices. Re-ferring back to equation (2), we see that asP → 0, d P → �P , which is positive.

2. As P → ∞, we follow Wilmott and setVPP = 0.

3. As � → 0, we require no boundary condi-tion since the partial differential equationis first-order hyperbolic in the � direction,with outgoing characteristic in the negative� direction.

4. As � → ∞, we assume V�→ 0, and henceno boundary condition is required. Thismeans that as stand age gets very large,the value of the option to harvest, V, doesnot change with �. In essence, we are pre-suming the wood volume in the stand hasreached some sort of steady state.

5. Terminal condition. As T gets large, it is as-sumed that V = 0. T is made large enoughthat this assumption has a negligible effecton V today.

We use this model to examine the opportu-nity costs of harvesting restrictions in Ontario’sprovincially owned forests. Almost 90% of thetimber volume consumed by Ontario mills in2000–2001 was from public land. Firms with li-censes to harvest in Ontario’s public forests areconstrained by allowable cut regulations thataim to maintain constant annual wood flowswithin a region, and hence are limited in theirability to manage for price risk.8 Such con-straints reduce the return to the firm holding a

8 At one point in Ontario’s history, a firm risked forfeiting itsharvesting license if it failed to harvest the allowable cut. In recentyears it appears that firms are not penalized for harvesting less thanthe allowable cut, and there is little documented evidence of firmsexceeding the allowable cut. There appears to be a trend towardallowing firms holding forestry licenses more flexibility, within lim-its. Firms are also affected by industry structure, developed overyears of forestry regulation, which tends to favor even wood flows.

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harvesting license and thus affect a firm’s will-ingness to make investments in forest manage-ment. The real options model developed hereis used to compare the value of a license to har-vest a stand of trees with and without allowablecut restrictions on harvesting time. The differ-ence between the two gives some indication ofthe cost of current allowable cut restrictions.

To fully address the efficiency and othereconomic implications of allowable cut reg-ulations would require consideration of allbenefits and costs of these policies, includ-ing environmental impacts and possible effectson employment in logging communities. Thiswould also require modeling the optimal har-vesting decisions at the forest level, rather thanthe stand level as is done in this article. Such acomplete analysis is beyond the scope of thisarticle. Using a stand-level model and a realoptions approach, we estimate the value of thecommercial harvest and the opportunity costof policies to maintain annual sustained yields.

Data and Parameter Estimates

The case examined is for a stand of Jack Pine(Site Class 1) in the Romeo Malette ForestUnit, which is managed under a SustainableForest License by Tembec, Inc. The RomeoMalette forest consists of 477,109 hectares ofproductive forestland and is located northeastof the town of Timmins, Ontario. For this ar-ticle we will examine the economics of a so-called basic level of silvicultural investmentthat represents the current level of spendingon many stands in Ontario’s boreal forest. Ba-sic management involves assisted natural andartificial regeneration, including site prepara-tion and removal of competing species. The fo-cus is on manipulating species composition andachieving full site occupancy. Silvicultural costsestimated by Tembec (in Canadian$/hectare)are $200 for site preparation and $360 to pur-chase nursery stock in year 1, $360 for plantingin year 2, $120 for tending in year 5, and finally$10 for monitoring in year 35. Amenity valueis assumed to be zero, so that A in equation (8)reflects only silvicultural costs.

Yield curves for Jack Pine saw logs andpulp consistent with basic management in theRomeo Malette Forest Unit were provided byTembec.9 Yield of the most valuable class of

9 The yield curves were estimated by M. Penner of Forest Anal-ysis Ltd., Huntsville, Ontario, for Tembec. These yield curves areavailable from the authors on request.

saw logs peaks at about 300 cubic meters perhectare at around an age of 100 years.

We did not have a historical time se-ries for mill gate lumber prices, and insteadused monthly data (1980–2003) for the priceof spruce-pine-fir random length 2 × 4’s inToronto.10 The data, converted to Canadiandollars, deflated by the consumer price in-dex, and seasonally adjusted, are shown infigure 1. We performed an augmented Dickey–Fuller (ADF) test on the price series to in-vestigate whether the data-generating processappears to be a random walk (and hence non-stationary). As in Prestemon, the lag lengthfor the ADF test was chosen by minimizingthe Schwartz information criterion. The op-timal lag length by this criterion is zero. TheLjung–Box Q-statistic at twelve lags was 12.8,which is not significant (p-value of 0.38), mean-ing that we do not reject the null of no serialcorrelation in the residuals. The results of theDickey–Fuller test, given in table 1, indicatethat we can reject the null hypothesis of a unitroot at the 1% significance level.

Two other tests of stationarity were car-ried out, and the results are also reported intable 1. Using the Leybourne–McCabe test(Leybourne and McCabe), we do not rejectthe null hypothesis at the 5% level that the se-ries is stationary. Using the variance ratio test(Lo and MacKinlay), we are able to reject thenull of a unit root but only at the 10% level.

It may also be noted that an ARCH LMtest was done on the residuals of the regres-sion: Pt − Pt−1 = c(1) + c(2)Pt−1 and therewas evidence of the ARCH effect. When wemove from discrete to continuous time, ARCHand GARCH models translate into complexstochastic volatility models, which are beyondthe scope of this article.11 However, the modeland solution algorithm presented in this arti-cle could easily be adapted to time-dependentvolatility.

Based on these test results it seems reason-able to adopt a mean-reverting stochastic pro-cess for lumber price in our optimal harvestingmodel. Of course, as noted above, none of thetests for a unit root is considered definitive.However, the purpose of this article is not toexamine in detail the price path of timber, butrather to demonstrate a methodology that can

10 Data were purchased from Madison’s Canadian Lumber Re-porter Ltd., P.O. Box 2486, Vancouver, British Columbia V6B 3W7Canada, Phone: 604-984-6838.

11 Chuan Duan derives the diffusion limit for a class ofGARCH(1,1) models and also extends to the GARCH(p,q) spec-ification.

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0

50

100

150

200

250

300

350

400

1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002

Figure 1. Real price of softwood lumber, Toronto, Ontario, 1993 Canadian dollars per cubicmeter (Data source: Madison’s Canadian Lumber Reporter, Monthly data from January 1980to December 2003, First Friday of each month, Eastern Spruce-Pine-Fir Std #2& Better, Kiln-dried, Random Length = 2 × 4, Deflated by the Canadian consumer price index, converted toCanadian dollars, and seasonally adjusted.)

solve the optimal harvesting problem under anautoregressive or other chosen stochastic pricepath.

We assume lumber prices follow a mean-reverting process as is described in equation(2). A discrete time approximation is

Pt − Pt−1 = �P�t − ��t Pt−1

+ � Pt−1√

�t�t

(10)

where �t is N(0, 1). Dividing through by Pt−1and using the notation

c(1) ≡ −��t ; c(2) ≡ ��t P ;

et ≡ �√

�t�t ,

(11)

the relevant parameters can be estimated byordinary least squares on the following equa-tion:

Pt − Pt−1

Pt−1= c(1) + c(2)

1Pt−1

+ et .(12)

Regression estimates are shown in table 2.From the definitions of c(1), c(2), and et in

Table 1. Tests for Stationarity of the Ontario Softwood Lumber Price Series

Test Statistic Critical Value (Signif. Level) Conclusion

Dickey–Fuller −3.88 −3.45 (1%) Reject H0 of unit rootLeybourne–McCabe 0.075 0.148 (5%) Do not reject H0 of stationarityVariance ratio (6 lags) 1.77 1.645 (10%) Reject H0 of unit root

equation (11) and given that �t is one month,the parameter estimates are � = 0.8, P =$230/m3, and � = 0.27.

Harvesting costs and product prices wereprovided to the authors on a confidential basis.Representative harvesting costs for Ontarioare reported in Rollins et al. as $31/m3. Theprice of saw logs at the millgate is approxi-mately $50/m3. The P estimate given aboverefers to the real price at Toronto. This hadto be translated into the price of raw logs pur-chased by the mill. The value of spruce-pine-firlumber in Toronto in 2003 is close to this valuefor P . Hence, the 2003 price going into the millreported (in confidence) by Tembec was cho-sen as P for the analysis. We use 3% and 5%real discount rates for the analysis.

Empirical Results

Flexible Harvesting Time

This section presents the estimated value ofthe option to harvest the representative standof trees on public forest land (V in equation(9)) given that the goal is to maximize the net

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Table 2. Parameter Estimates of Equation (12)

Variable Coefficient t-statistic

c(1) −.069 −2.95 Sample: 1980:02 to 2003:12c(2) 15.9 3.10 Number of observations: 287

R2 = 0.03 SE of regression: 0.077

present value of the commercial value of thetimber. Consumer surplus is ignored, implyingthe wood harvested is destined for the exportmarket. We assume that adequate log marketsexist, and ignore any values other than com-mercial timber value. The decision variable iswhether to harvest the stand given the mar-ket price for any given time period. This im-plies that there are no restrictions (regulatoryor otherwise) on firms as to when a stand couldbe harvested. This is called the social perspec-tive to distinguish it from the perspective of aregulated firm, which is constrained as to har-vesting times.

Solving equation (9) using the prices, costs,and discount rates given in the previous sec-tion, we estimate, for any given stand age, thethreshold price above which it is optimal toharvest the stand. Appendix A provides de-tails about the solution of the numerical model.Figure 2 illustrates these results for a real dis-count rate of 3%. Each individual graph in the

0 100 200

-5

0

5

10

15

20

25p*= 90

Age= 36 NMV=168V

in $

(000

)/ha

0 100 200

-5

0

5

10

15

20

25 p*= 86

Age= 40 NMV=197

0 100 200

-5

0

5

10

15

20

25 p*= 82

Age= 50NMV=254

0 100 200

-5

0

5

10

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20

25 p*= 80

Age= 60NMV=286

P, $/cubic meter

V in

$(0

00)/

ha

0 100 200

-5

0

5

10

15

20

25 p*= 79

Age= 70NMV=307

P, $/cubic meter

0 100 200

-5

0

5

10

15

20

25 p*= 77

Age= 85NMV=325

P, $/cubic meter

Figure 2. Value of the opportunity to harvest a stand at different stand ages, discount rate =3% (Heavy solid line: value of the option to harvest. Heavy dashed line: payout from harvest-ing immediately. P∗: critical price at which harvesting is worthwhile. NMV: net merchantablevolume, m3/ha.)

figure represents a different stand age, and in-dicates the net merchantable volume in cubicmeters per hectare (NMV) achieved by thatage, as well as the critical price, P∗, at whichit is worthwhile harvesting. The solid curvein each of the graphs represents the value ofthe opportunity to harvest the stand of trees,V, after the silvicultural treatments are com-pleted. The dashed line represents the payoutfrom harvesting immediately. When the valueof the opportunity to harvest, V, is above thepayout line, the value of delaying the harvestexceeds the value of harvesting immediately,and it is worthwhile waiting. Once V touchesthe payout line, it is worthwhile harvesting im-mediately. The point of tangency determinesthe critical price (and demonstrates the smoothpasting condition.)

Figure 2 indicates that it would be worth har-vesting a thirty-six-year-old stand if the price ofSPF1 reached $90/m3. Although this is a fairlyyoung age by boreal forest standards, with a

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0 20 40 60 80 100 120 14076

78

80

82

84

86

88

90

92

Stand age

pri

ce, $

/m3

Figure 3. Critical price versus stand age, discount rate = 3%

mean-reverting price, it makes sense to harvestearly to take advantage of a temporarily highprice. By the age of forty years the critical pricehas dropped to $86. As the stand ages, the crit-ical price continues to drop as the opportunitycost of delaying the harvest is reduced.

Figure 3 depicts the whole spectrum of criti-cal prices for stand ages from zero to 130 years.Critical prices begin at the age of thirty-fiveyears; harvesting is not allowed in the modelprior to that date, by which time all silviculturalexpenditures have been incurred. The criticalprice drops rapidly to about the age of eighty-five years and then reaches a steady state of$77.

Of significant interest is the value of theopportunity to harvest the stand of trees atthe beginning of the first rotation shown intable 3 (second column). This value representsthe maximum amount that a firm would be will-ing to pay for harvesting rights to the stand,ignoring taxes and other charges, and assum-ing the firm has complete flexibility as to thetiming of the harvest and that markets existfor the logs. For comparison the value using asingle rotation analysis is also shown, as wellthe value calculated using the simple Faust-mann formula, assuming a constant price equalto the long-run mean-reverting level used inthe stochastic model. The magnitude by whichvaluation estimates from the real options ap-proach exceed those from the Faustmann anal-ysis reflects the value of having the flexibilityto optimally manage in the face of price volatil-

ity. The analysis is very sensitive to the discountrate chosen, since the silvicultural costs occurin the first thirty-five years, and the benefits, interms of higher volumes, occur after thirty-fiveyears.

In table 3, we show a single amount for landvalue that is unrelated to price. This contrastswith stands of the age of thirty-six years andabove, shown in figure 2, for which value in-creases with the current price. Given our mean-reverting price process, no matter what theprice is at the beginning of a rotation, by thetime the stand achieves harvestable volumes,we expect the price to have reverted towardthe long-run mean. Thus, land values at thebeginning of a rotation are insensitive to to-day’s product prices given the parameters wehave chosen for our mean-reverting process. Ifwe had assumed that price follows a process ofgeometric Brownian motion, then the value ofthe bare land would be dependent on today’sprice.

Table 3. Value of the Land at the Beginningof the First Rotation, $/Hectare

RealReal Options:

Discount Options: SingleRate Multirotation Rotation Faustmann

3% $1,978 $1,520 $3055% $61 $54 −$512

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0 100 200

-5

0

5

10

15

20

25

Age= 36NMV=168

V in

$(0

00)/

ha

0 100 200

-5

0

5

10

15

20

25

Age= 40NMV=197

0 100 200

-5

0

5

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25

Age= 45 NMV=234

0 100 200

- 5

0

5

10

15

20

25p*= 75

Age= 50Vol=254

P, $/cubic meter

V in

$(0

00)/

ha

0 100 200

-5

0

5

10

15

20

25p*= 64

Age= 55Vol=273

P, $/cubic meter0 100 200

-5

0

5

10

15

20

25

Age= 70NMV=307

P, $/cubic meter

Figure 4. Value of the opportunity to harvest when harvesting must occur between ages 50and 55, 3% discount rate (Heavy solid line: value of the option to harvest. Heavy dashed line:Payout from harvesting immediately. P∗: critical price at which harvesting is worthwhile. NMV:net merchantable volume, m3/ha.)

Harvesting Restrictions

Wood flow in Ontario’s public forests is typ-ically determined over an entire region (for-est unit) consisting of many individual stands.With our stand-level model we cannot fullydescribe the impact of harvesting restrictions;however, we can mimic their impact to someextent. In particular we consider the effectof minimum harvesting requirements, whichwould force a firm to harvest a certain quantityof wood even in times of depressed markets. Inthe stand-level model we mimic this restrictionby requiring harvesting at a particular standage.

Figure 4 shows the value of the option toharvest when the restriction is imposed thatharvesting must occur when the stand is be-tween the ages of fifty and fifty-five years. Ifharvesting does not occur within that period itis assumed the firm loses its rights to harvest.Comparing figure 4 with figure 2, we note thatin the restricted case, value is now fairly insen-sitive to price at the age of forty years, since it isstill ten years before harvesting can occur. Wealso note that at the ages of fifty and fifty-fiveyears the critical prices are lower than for theunrestricted case. By the age of fifty-five yearsthe critical price has dropped to $64 (comparedto $80 for the unrestricted case), reflecting thefact that if harvesting does not occur in that

year the land will be worthless to the firm as itwill have to give up its license. Under this re-striction, the value of the land at the beginningof the rotation is $960/ha, significantly lowerthan in the unrestricted case of $1,978/ha.

It is already well known in the forestry liter-ature that the pursuit of an even flow of timbercan significantly change the economics of com-mercial forestry due to the impact on the abil-ity of the forest manager to respond to pricevolatility. Our modeling approach offers an im-proved ability to estimate the magnitude of thecosts of these types of restrictions.

Accuracy of Results and MDP Approaches

The value estimates reported above are com-puted using a numerical solution methodol-ogy. As with any numerical method, we mustbe concerned with truncation error in the dis-cretizations of time, age, and price (Tavella andRandall). The finer the grid used in the numer-ical solution, the more accurate will be our es-timated value for V. In Appendix C, we showthat the grid size with which we have computedthe results of table 3 gives us results to an ac-ceptable degree of accuracy. Refining the gridfurther changes the solution by only 0.4%.

We do not do a direct comparison with MDPmodels. However, we do show that we can get

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Table 4. Comparing Solutions for Very CoarseGrid and Medium Grid. Value of the Land atthe Beginning of the First Rotation, $/Hectare,3% Discount Rate

Very MediumCoarse Grid Grid

Price $/m3 11 nodes 73 nodesP = [0, . . . , 277]

Quantity $/ha 13 nodes 107 nodes� = [0, . . . , 125]

Timestep size 0.25 years 0.125 yearsValue at t = 0 $4,123 $ 1,978

a very large change in our answer when we usea grid size that is much coarser than that usedto compute our base case results. This is an im-portant point when comparing with the MDPmodels. We demonstrate in Appendix B thatin theory the MDP model and the numericalsolution of the LCP are equivalent. However,a numerical solution of the LCP using a finitedifference approach permits use of a finer gridscheme. In an effort to handle more stochasticvariables and keep the solution tractable, MDPmodels are often solved with a very coarse grid.In addition, convergence studies are typicallynot reported with MDP models.

In table 4 we demonstrate how our value es-timate changes when we significantly reducethe number of nodes at which a solution is es-timated. We observe a huge change in our es-timated V values as we move from a mediumto a very coarse grid. In any given example, itis clearly important to determine whether lim-iting the number of nodes has resulted in largeinaccuracies.

Conclusion

This article has presented a two-factor mul-tirotation model of the tree harvesting de-cision. The problem is specified as an LCP,which is solved using a fully implicit fi-nite difference approach—an approach that iscommonly used in the finance literature forvaluing real and financial options. We con-trast our methodology with other approachesused in the forestry literature to handle op-timal rotation with stochastic prices, such asMDP models. We note that the LCP and theMDP models are in theory equivalent. An im-portant benefit of the LCP approach is that weare assured that the solution will converge to

the correct answer (based on a large numericalanalysis literature), and we can easily check theaccuracy by solving for successively finer grids.We demonstrated that the value of the harvest-ing option varies widely when a coarser solu-tion grid is used. We have not solved an MDPmodel for comparison. However, our analysisdoes suggest that care should be taken whenreporting results without carrying out numer-ical convergence studies, since the value ofthe option to harvest can be very sensitive tothe number of discrete levels of the stochasticvariables.

We used our model to address a policy is-sue in the Ontario boreal forest. This articlehas shown that the value of an investmentin forestry can be significantly affected whena firm’s ability to react to volatile prices isconstrained. Constraints may be due to gov-ernment regulations, such as allowable cutrequirements, or may reflect the structural re-alities of an industry in which vertically inte-grated firms having invested in mill capacitywant to maintain a reasonable capacity utiliza-tion. There are, no doubt, costs to a mill if in-put is highly variable, but these costs should bebalanced with the benefits of being able to re-act optimally to price swings. The value of theoption to harvest a stand of trees should bean important consideration in any review offorest management regulations, with the goalof designing regulations that continue to meetenvironmental constraints, but offer firms themaximum flexibility to manage license areasin the face of price risk. The true costs of reg-ulations that limit flexibility can only be fullyunderstood using a model that correctly valuesthe option to harvest under uncertainty.

A direction for future research is to ex-tend the methodologies from the finance lit-erature to consider harvesting and other con-straints at a regional forest unit level understochastic prices. This would involve a mul-tistand approach, as well as consideration ofincremental mill costs with swings in capacityutilization. Biological and catastrophic risk areother avenues of research using a real optionsapproach.

[Received February 2004;accepted November 2004.]

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Perron, P. “The Great Crash, the Oil Price Shock,and the Unit Root Hypothesis.” Econometrica57(1989): 1361–1401.

Plantinga, A.J. “The Optimal Timber Rotation:An Option Value Approach.” Forest Science44(1998): 192–202.

Prestemon, J.P. “Evaluation of U.S. Southern PineStumpage Market Informational Efficiency.”Canadian Journal of Forest Research 33(2003):561–72.

Reed, W., and H. Clarke. “Harvest Decisions andAsset Valuation for Biological Resources Ex-hibiting Size-Dependent Stochastic Growth.”International Economic Review 31(1990): 147–69.

Rollins, K., M. Forsyth, S. Bonti-Ankomah, and B.Amoah. “A Financial Analysis of a White PineImprovement Cut in Ontario.” Forestry Chron-icle 71(1995): 466–71.

Saphores, J.-D., L. Khalaf, and D. Pelletier. “OnJump and ARCH Effects in Natural ResourcePrices: An Application to Pacific NorthwestStumpage Prices.” American Journal of Agri-cultural Economics 84(2002): 387–400.

Sarkar, S. “The Effect of Mean Reversion on Invest-ment under Uncertainty.” Journal of EconomicDynamics & Control 28(2003): 377–96.

Schwartz, E.S. “The Stochastic Behaviour of Com-modity Prices: Implications for Valuation andHedging.” Journal of Finance 52(1997): 923–73.

Sødal, S. “The Stochastic Rotation Problem: AComment.” Journal of Economic Dynamicsand Control 26(2002): 509–15.

Tavella, D.A. Quantitative Methods in DerivativesPricing. New York: Wiley, 2002.

Tavella, D.A., and C. Randall. Pricing Financial In-struments, The Finite Difference Method. NewYork: Wiley, 2000.

Teeter, L.D., and J.P. Caulfield. “Stand DensityManagement Strategies under Risk: Effects ofStochastic Prices.” Canadian Journal of ForestResearch 21(1991): 1373–79.

Thomson, T. “Optimal Forest Rotation WhenStumpage Prices Follow a Diffusion Process.”Land Economics 68(1992): 329–42.

Trigeorgis, L. Real Options, Managerial Flexibil-ity and Strategy in Resource Allocations. Cam-bridge, MA: MIT Press, 1996.

Varga, R.S. Matrix Iterative Analysis. New York:Springer Verlag, 2000.

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Wilmott, P., J. Dewynne, and S. Howison. OptionPricing, Mathematical Models and Computa-tion. Oxford: Oxford Financial Press, 1993.

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Appendix A: Numerical Solution of theLinear Complementarity Problem

Method of Characteristics

The solution of equation (9) is accomplished by dis-cretizing the term V� − V� in HV by the method ofcharacteristics (Morton and Mayers).

Consider some function U(X, �). � refers to timeto expiry of the option, or T − t. Then, we can write

dU

d�= UX X� + U� .(A.1)

If U satisfies the equation

U� + a(X, �)UX = 0(A.2)

then, from equation (A.1), if we let a(X, �) = X� ,then dU = 0 along the characteristic curves definedby

d X

d�= a(X, �).(A.3)

If we consider the simple case where a(X, �) =constant = a, then the solution to equation (A.2)is

U(X, �) = U(X − a�, 0).(A.4)

This can be verified by taking the total derivative ofU(X − a�, �) and observing that dU = 0 when X� =a and � = 0. In the case when a(X, �) �= constant,we can still approximate equation (A.4) in discretetime by

U(Xi , � n+1) − U(Xi − a(Xi , � n+1)��, � n)��

+ O(��) = 0.

(A.5)

The basic PDE in the continuation regions forthe tree harvesting problem, equation (7), can bewritten in terms of � as follows:

V� − V� = 12

�2 P2VP P

+ �(P − P)VP − � V + A

(A.6)

where the left-hand side is a function of � and � fora fixed P, and the right-hand side is a function of Pfor a fixed � and � .

The left hand side of equation (A.6) looks likethe left-hand side of equation (A.2), if a(X, ��) =− 1, recalling that d�/dt = 1, and replacing X with� and U with V. As will be shown below, this ob-servation allows us to approximate the two-factorproblem by solving a set of one-dimensional PDEsand employing an interpolation operation at eachtime step to exchange information between the one-dimensional PDEs.

We now consider the numerical solution of theLCP, equation (9), using the characteristic approachand a fully implicit differencing scheme. Definenodes on the axes for P, �, and � by

P = [P1, P2, . . . , Pi , . . . , PM ]

� = [�1, �2, . . . , � j , . . . , �J ]

� = [�1, �2, . . . , �n, . . . �N ].

(A.7)

We concern ourselves with the value of the op-tion to harvest at points defined by the three-dimensional grid (P, �, �) = (Pi, �j, � n). At any pointon the grid the value of the option is V = V(Pi, �j,� n) ≡ Vn

ij.To impose the conditions of the LCP we define a

function �(V) to be a penalty term that will preventthe value of the option V from ever falling below thepayout from harvesting immediately, (P − C)Q +V(� , P, 0). Zvan, Forsyth, and Vetzal discuss thepenalty method. The penalty term, �, equals 0 in thecontinuation region (i.e., when HV = 0 and V > (P −C)Q + V(� , P, 0)) and � > 0 when it is optimal toharvest (i.e., when HV > 0 and V = (P − C)Q + V(� ,P, 0)). If we include the penalty term in equation(A.6), equation (9), the LCP, can be approximatedby

V� − V� = 12 �2 P2VP P + �(P − P)VP

− � V + A + �(V ).

(A.8)

Using equation (A.5), our difference scheme forequation (A.8) can be written as

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V (Pi , � j , � n+1) − V (Pi , � j + ��, � n)��

=[

12

�2 P2VP P + �(P − P)VP

− � V + A + �(V )

]n+1

i j

.

(A.9)

Let

V ∗(Pi , � j )n ≡ V (Pi , � j + ��, � n) .(A.10)

Then equation (A.9) can be written as (Bermejo)

V (Pi , � j )n+1 − V ∗(Pi , � j )n

��

=[

12

�2 P2VP P + �(P − P)VP

− � V + A + �(V )

]n+1

i j

.

(A.11)

Note that the right-hand side of equation (A.11)has derivatives with respect to P only. Therefore,equation (A.11) defines a set of discretized one-dimensional partial differential equations, one foreach �j. These can be solved independently withineach time step. After a time step is completed,these one-dimensional PDEs exchange informationthrough the interpolation operation,

V ∗(Pi , � j )n = V (Pi , � j + ��, � n) .(A.12)

We used linear interpolation.Within each time step the LCP, equation (9), is

solved for fixed �j using a fully implicit finite differ-ence method and the penalty method, as describedin Insley. The right-hand side of equation (A.11)is discretized using a central, forward, or backwarddifference scheme as appropriate. Note that if wesolve first for V(Pi, � = 0)n+1 at each time step, thenwe have the information required to determine thevalue of the penalty term implicitly. Hence, we donot need to iterate within each time step to deter-mine the value of the bare land.

Discretization for i = [2, . . . , M − 1]

We now concern ourselves with the discretizationof the RHS of equation (A.11) and make use of thefollowing definitions:

�i ≡ �2 P2i

�Pi+1/2 + �Pi−1/2;

�i ≡ �(P − Pi )�Pi+1/2 + �Pi−1/2

(A.13)

where �Pi+1/2 ≡ Pi+1 − Pi and �Pi−1/2 ≡ Pi − Pi−1.Using a fully implicit approach and central differ-encing along the P axis, our difference scheme wheni = 2, . . . , M − 1 for Pi and j = 1, . . . , J for �j is

V(Pi , � j , � n+1

) − V (Pi , � j + ��, � n)��

=�2 P2

2

Vi+1, j −Vi j

�Pi+1/2− Vi j −Vi−1, j

�Pi−1/2

�Pi+1/2−�Pi−1/22

+ �(P − P)

[Vi+1, j − Vi−1, j

�Pi+1/2 + �Pi−1/2

]

− � Vi j + A + �i j

��[(Pi − C)Q j

+ Vi0 − Vi j ]

n+1

.

(A.14)

The superscript n + 1 on the right-hand side meansthat all variables within the braces are evaluated at� = n + 1.

Extensive manipulation of the right-hand side ofequation (A.14) results in

RHS ≡[

ai Vi−1, j + bi Vi+1, j − (ai + bi + )Vi j + A

+ �i j

��[(Pi − C)Q j + Vi0 − Vi j ]

]n+1

(A.15)

where

ai ≡ �i

�Pi−1/2− �i ; bi ≡ �i

�Pi+1/2+ �i .(A.16)

The last term on the right-hand side of equa-tion (A.15) is the discretized penalty function �(V),where

�n+1i j = 0 if V n+1

i j ≥ (Pi − C)Q j + Vi0

�n+1i j = L if V n+1

i j < (Pi − C)Q j + Vi0

(A.17)

where L is a suitably large number. In this article, Lis set to 106.

To avoid an oscillatory solution it is necessary thata and b from equation (A.16) both be nonnegative.Since � can be either positive or negative, a and bcan be either positive or negative, but if one is pos-itive then the other is negative. If a < 0, a forwarddifference scheme for P is used to avoid oscillationsrather than the central scheme shown in equation(A.14). If b is negative, then a backward differencescheme must be used for P (Zvan, Forsyth, and Vet-zal).

Using a forward difference scheme a and b inequation (A.15) are defined as

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ai ≡ �i

�Pi−1/2; ≡ �i

�Pi+1/2+ i

where i = �(P − P)�Pi+1/2

.

(A.18)

Using a backward difference scheme a and b inequation (A.15) are defined as

ai ≡ �i

�Pi−1/2− �i ; bi ≡ �i

�Pi+1/2

where �i = �(P − P)�Pi−1/2

.

(A.19)

Central differencing is used as much as possibleto ensure the most accurate solution. Forward orbackward differencing is used only at nodes wherenegative values of a or b are obtained using centraldifferencing.

Discretization of Boundary Conditions for P

When i = 1 no special boundary condition is re-quired for P, and a forward discretization schemecan be used. As P → 0, equation (A.11) becomes

V (Pi , � j )n+1 − V ∗(Pi , � j )n

��

= [�PVP − � V + A + �(V )]n+1i=1, j .

(A.20)

A forward discretization is then

V (Pi , � j )n+1 − V ∗(Pi , � j )n

��

=[

bi Vi+1 − (bi + �)Vi + A + �i j

��

× [(Pi − C)Q j + Vi0 − Vi j ]]n+1

1, j

(A.21)

with b defined as in equation (A.18). In equation(A.18) as P → 0 we see that � = 0 and hence a = 0and also, b = .

When i = M we set VPP = 0 in equation (A.11)and use a backward difference scheme that gives

V (Pi , � j )n+1 − V ∗(Pi , � j )n

��

=[

�(P − P)�Pi−1/2

Vi−1 + (�(P − P)�Pi−1/2

− �)Vi + A

+ �i j

��[(Pi − C)Q j + Vi0 − Vi j ]

]n+1

M, j

.

(A.22)

Iterative Solution

Equations (A.14), (A.17), (A.21), and (A.24) repre-sent a system of nonlinear algebraic equations, dueto the penalty term and hence, must be solved it-eratively. A description of the iterative solution fora single rotation problem is given in Insley. The it-erative solution can best be understood if we writethe system of equations in matrix form. To this end,define the vectors

Vn+1 ≡

V n+11

V n+12...

V n+1M

; V∗n ≡

V ∗n1

V ∗n2...

V ∗nM

;

Pn ≡

Pn1

Pn2

...Pn

M

; An ≡

An1

An2

...An

M

.

(A.23)

Also, define a diagonal matrix as

[Q(Vn+1)]i i = L if V n+1i <

(Pn+1

i − C)

Q j − V n+1i0

= 0 otherwise

[Q(Vn+1)]i j = 0 if i �= j.

(A.24)

Let B be an M × M matrix. Multiplying B by thevector Vn+1 gives a vector BVn+1 with the followingelements:

(A.25)

[��P

�Pi+1/2+ �

]V n+1

1 −[

��P�Pi+1/2

]V n+1

2

−��b2V n+11 + ��(a2 + b2 + �)V n+1

2 − ��a2V n+13

...

−��bi Vn+1

i−1 + ��(ai + bi + �)V n+1i − ��ai V

n+1i+1

...

−��bM−1V n+1M−2 + ��(aM−1 + bM−1 + �M−1)V n+1

M−1 − ��aM−1V n+1M[

�� (P−PM )�PM−1/2

]V n+1

M−1 −[�� (P−P)

�PM−1/2− �

]V n+1

M

.

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Equations (A.14), (A.17), (A.21), and (A.24) canbe written as

BVn+1 + [I + Q(Vn+1)]Vn+1

= V∗n + �� An + Q(Vn+1)

(A.26)

where I is the identity matrix. We can express V∗n

as V∗n = FVn where F is an interpolation matrix.For linear interpolation, F has the properties thatits entries are nonnegative and all row sums are 1.Equation (A.26) is solved iteratively as described inInsley.

Appendix B: Markov Decision ProcessModels and the Linear ComplementarityProblem

The key to a Markov decision model is the Markovmatrix, or transition probability matrix of themodel. (Karlin and Taylor, and Hillier and Lieber-man discuss Markov transition probability matri-ces.) Consider a random variable X. We refer toXn as the outcome of the nth trial. The state spaceis labeled by nonnegative integers (0, 1, 2, 3, . . .).The probability of Xn+1 being in state j, given thatXn is in state i (a one-step transition probability) isdenoted by Gn,n+1

ij , that is

Gn,n+1i j = Pr{Xn+1 = j | Xn = i}.(B.1)

When the one-step transition probabilities are in-dependent of time, the Markov process is said tohave stationary transition probabilities. In this caseGn,n+1

ij =Gij. Gij is the probability that the state valuegoes from state i to j in one trial. All the Gij can be ar-ranged in a matrix referred to as the Markov matrixor the transition probability matrix of the process,denoted by P. An example of G for four differentstates is

G =

G00 G01 G02 G03

G10 G11 G12 G13

G20 G21 G22 G23

G30 G31 G32 G33

.(B.2)

The (i + 1)st row of G is the probability distribu-tion of the values of Xn+1 under the condition Xn =

i. The order of the matrix (number of rows) is equalto the number of states. Given four different statesthe quantities Gij satisfy

Gi j ≥ 0, i, j = 0, 1, 2, 33∑

j=0

Gi j = 1, i = 0, 1, 2, 3.

(B.3)

We can show how a matrix with all the propertiesof the Markov matrix is derived in the solution ofthe LCP. For convenience we define a variable Wsuch that V = e−�� W. Thus,

V� = −e−�� W + e−�� W� .(B.4)

Substituting these expressions into equation(A.8) and setting A = 0, we get

W� − W� = 12 �2 P2WPP

+ �(P − P)WP + e�� (e−�� W ).

(B.5)

We can discretize equation (B.5) as shown in Ap-pendix A. We end up with an equation similar toequation (A.26). We are ignoring the penalty termhere for simplicity.

[B + I]Wn+1 = W∗n(B.6)

The matrix [B] has the following elements:

(B.7) B =

��P�Pi+1/2

−��P�Pi+1/2

0 0 0 . . . 0 0

−��b2 ��(a2 + b2) −��a2 0 0 . . . 0 0

. . . . . . . . . . . . . . . . . . . . . . . .

0 0 0 . . . 0 −��bM−1 ��(aM−1 + bM−1) −��aM−1

0 0 0 . . . 0 0 ��(P−PM )�PM−1/2

−��(P−PM )�PM−1/2

.

It follows that

Wn+1 = [B + I]−1W∗n .(B.8)

We know that Vn+1 = e−�� Wn+1. It follows that

Vn+1 = e−�� [B + I]−1V∗n .(B.9)

Recall that V∗n = FVn where F is the interpolationmatrix. We note that [B + I] will have positive diag-onals, nonpositive off-diagonals, and rows that sumto 1.

LEMMA 1. The row sums of [B + I]−1 F are unity.

Proof : Let e be an [ I × 1] column vector of 1’s.We can observe that [B + I]Fe = Fe (since Be = 0and Fe = e). This implies that Fe = [B + I]−1 Fe.The lemma follows since

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rowsumi [B + I]−1 F = [[B + I]−1 Fe]i

= [Fe]i = 1. �(B.10)

LEMMA 2. All elements of [B + I]−1 F arenonnegative.

Proof : [B + I] has positive diagonals, nonposi-tive off-diagonals, and is diagonally dominant. Itmay therefore be classified as an M-matrix, whichhas the property that all elements of its inverse arenonnegative (Varga). We stated previously that theelements of F are all nonnegative. �THEOREM 1: [B + I]−1 F satisfies (B.4) and henceis a Markov matrix.

Proof : By Lemma 2, all elements of [B + I]−1 Fare nonnegative. By Lemma 1, the row sums of [B +I]−1 F are all unity. Hence, all elements must be lessthan or equal to 1. �

We can see that [B + I]−1 F is equivalent to aMarkov transition matrix showing the probability ofmoving from one state at � n to another at � n+1. Fromequation (B.9), the value of the option at period n+ 1 is equal to the value in period n multiplied bythe Markov transition matrix and a discount factor.

Solving the Markov chain decision model is analternative method to solving the partial differen-tial equation, equation (A.8). The Markov matrixis frequently estimated through simulation. This isunnecessary with the approach used in this article.Instead, we discretize the PDE and solve the re-sulting system of equations iteratively using well-established numerical techniques. There is no needto directly estimate the Markov matrix. Note thatthe elements of (B + I)−1 F are to O(�� 2) equal tothe discretized Green’s function that must solve thePDE,

V� − V� = 12 �2 P2VPP + �(P − P)VP − � V

+ A + (P − P ′)(� − � ′)

(B.11)

where (� − � ′) represents the delta function (seeWilmott, Dewynne, and Howison). The Green’sfunction of this PDE is simply the transition prob-ability density function, P(P, t, P′, t′) relating theprobability of a transition from state (P, t) → (P′, t′).This is also the solution of the forward Kolmogorovequation.

Although the Green’s function (hence the tran-sition density function) can be determined in somesimple cases, it is not in general possible to obtain

Table C.1. Convergence of Solution as Grid Is Refined for the Basic Regime, 3% Discount Rate

Grid No. of � Nodes No. of P Nodes Time Step Size (Years) V ($/ha) Change in V Ratio

Coarse 55 37 0.25 1,967.5Medium 110 73 0.125 1,978.6 11.1Fine 220 145 0.0625 1,986.1 7.5 1.5

an analytic solution. However, we can always dis-cretize the PDE directly and hence obtain an ap-proximate transition matrix to O(�� 2). Note thatsince we solve the LCP at each time step we aredirectly enforcing the optimal control, which en-sures that the solution satisfies the smooth pastingcondition. The usual stochastic dynamic program-ming approach applies a control in explicit fashion,hence the solution is in an inconsistent state aftereach timestep. This will be an issue only in nonau-tonomous problems, but will not matter when weare solving for a steady state, as in this article.

Note that we do not compute (B + I)−1, whichis a dense matrix, but solve (B + I)Vn+1 = V∗n ,which is considerably more efficient, since (B + I)is tridiagonal.

Appendix C: Convergence of the FiniteDifference Scheme

The accuracy of the numerical solution to thePDE depends critically on the number of nodesused for each factor: price (P), age (�), and time(�) (Wilmott, Tavella and Randall). The more thenodes, the more accurate the solution, but the longerthe solution time. An upper limit on the number ofnodes will eventually be reached based on the mem-ory limits of the computer. It is important to checkwhether an acceptable number of nodes has beenincluded by noting how much the solution changesas the grid is refined. Specifically, we observe thechange in the answer as we double the total numberof nodes.

More formally, let �� = c1h, �P = c2h, �� =c3h, where c1, c2, c3 are constants independent ofh. Since first-order timestepping is being used, andthe cumulative effect of linear interpolation in the� direction will be O( ��2

��) = O(h), we expect that

the computed solution Vcomp is related to the exactsolution Vexact by

V comph = V exact + c4h, h → 0(C.1)

where c4 is a constant independent of h. This impliesthat

Vh − Vh/2

Vh/2 − Vh/4∼ 2, h → 0.(C.2)

In table C.1 we see that this ratio is about 1.5, indi-cating that we are near the asymptotic convergencerange. In particular, we can be confident that thesolution on the finest grid is 1,986 ± 7.5 or accurateto within about 0.4%.

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ON SOLVING THE MULTIROTATIONAL TIMBER … · ON SOLVING THE MULTIROTATIONAL TIMBER HARVESTING PROBLEM WITH STOCHASTIC PRICES: ALINEAR COMPLEMENTARITY FORMULATION MARGARET INSLEY AND

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