carbon sequestration through afforestation under uncertainty

7
Carbon sequestration through afforestation under uncertainty Jan Lewandrowski a, , C.S. Kim b , Marcel Aillery b a Climate Change Program Ofce, U.S. Department of Agriculture, 1400 Independence Ave SW, Washington DC 20250, USA b Economic Research Service, U.S. Department of Agriculture, 355 E Street SW, Washington DC 20024, USA abstract article info Article history: Received 31 July 2012 Received in revised form 5 June 2013 Accepted 13 June 2013 Available online 22 July 2013 Keywords: Afforestation Carbon sequestration Uncertainty Fire risk Pest risk Dynamic optimal control Economic studies have demonstrated that agricultural landowners could mitigate signicant quantities of greenhouse gas (GHG) emissions through afforestation. The associated carbon, however, must remain stored in soils or biomass for several decades to achieve substantial mitigation benets. Policies and programs to en- hance carbon sequestration in forest systems must accommodate the possibility of premature carbon re- leases. We develop a dynamic nested optimal-control model of carbon sequestration through afforestation given uncertainties associated with re and pest hazards. Our framework highlights a number of factors that affect landowner decisions to invest in re or pest prevention measures. For re, we show the net inu- ence of these factors is to encourage investment in prevention measures when the probability of re occur- ring is less than the ratio of expected net economic benets to expected gross economic benets of adopting re prevention measures. For pests, we show that landowners will invest in prevention measures when the probability of re is less than the ratio of the difference between net benets before and after the discovery of tree pests to the difference between gross economic benets before and after the discovery of pests. For both risks, landowners will over-invest in prevention if the other risk is ignored. Published by Elsevier B.V. 1. Introduction Numerous economic studies have demonstrated that agricultural landowners could mitigate signicant quantities of greenhouse gas (GHG) emissions through afforestation the shifting of cropland and pasture into trees (McCarl and Schneider, 2001; Lewandrowski et al., 2004; Lubowski et al., 2006; U.S. EPA, 2005). To realize the GHG mitiga- tion potential of afforestation, however, farmers must be able to convert increases in carbon stored in soils and biomass to income. Several policy approaches could incentivize carbon capture through afforestation in- cluding the establishment of a carbon market (such as would occur under a state, regional, or national cap-and-trade program), the crea- tion of a direct government payment explicitly for adoption of carbon sequestering practices (analogous to payments farmers receive under USDA's Conservation Reserve Program), and the development of volun- tary carbon-related contracts between two or more private parties. For afforestation to result in signicant GHG mitigation, the asso- ciated carbon must remain stored in soils or biomass for an extended time (viewpoints range from 20 to over 100 years). As an example, the forest project protocol developed by the Climate Action Reserve for use in the California climate program requires: 1) afforestation/ reforestation projects target lands not in forest cover during the pre- vious 10 years; and 2) project lands remain in forest for 100 years (Climate Action Reserve, 2010). In policy and scientic settings this is referred to as the permanenceissue. Regardless of the policy approach, permanence has important implications for the design of carbon sequestration incentives. Specif- ically, incentives must accommodate both the possibility and uncer- tainty that carbon sequestered and credited within a mitigation framework may be prematurely released at some point in the future. Such releases could be unintentional (as in the case of a future re event or a pest/disease outbreak) or deliberate (as in the case of a landowner decision to harvest timber prior to a previously agreed on date). The premature release of carbon from a parcel of afforested land would likely create an obligation to either replace the released carbon or compensate the buyer since it would already have been pur- chased and, presumably, used to meet a GHG mitigation commitment on the part of the buyer. Conceptually, the replacement obligation could rest with either the buyer or seller. For two reasons, we assume that it rests with the seller. First, sellers will typically be landowners and as such will have direct control over how afforested lands are ac- tually managed. Putting the replacement obligation on landowners then creates an economic incentive for them to take specic land management actions that reduce the probability of a premature re- lease. Second, buyers would likely be entities trying to meet specic emission reduction targets (for example, a private sector rm, indus- try or trade organization, or municipality that has made a public com- mitment to reduce its carbon footprint). If carbon sequestered through afforestation came with signicant uncertainty regarding Forest Policy and Economics 38 (2014) 9096 Corresponding author. Tel.: +1 202 720 6699; fax: +1 202 401 1176. E-mail addresses: [email protected] (J. Lewandrowski), [email protected] (C.S. Kim), [email protected] (M. Aillery). 1389-9341/$ see front matter. Published by Elsevier B.V. http://dx.doi.org/10.1016/j.forpol.2013.06.014 Contents lists available at ScienceDirect Forest Policy and Economics journal homepage: www.elsevier.com/locate/forpol

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Page 1: Carbon sequestration through afforestation under uncertainty

Forest Policy and Economics 38 (2014) 90–96

Contents lists available at ScienceDirect

Forest Policy and Economics

j ourna l homepage: www.e lsev ie r .com/ locate / fo rpo l

Carbon sequestration through afforestation under uncertainty

Jan Lewandrowski a,⁎, C.S. Kim b, Marcel Aillery b

a Climate Change Program Office, U.S. Department of Agriculture, 1400 Independence Ave SW, Washington DC 20250, USAb Economic Research Service, U.S. Department of Agriculture, 355 E Street SW, Washington DC 20024, USA

⁎ Corresponding author. Tel.: +1 202 720 6699; fax:E-mail addresses: [email protected] (J. L

[email protected] (C.S. Kim), [email protected] (M

1389-9341/$ – see front matter. Published by Elsevier Bhttp://dx.doi.org/10.1016/j.forpol.2013.06.014

a b s t r a c t

a r t i c l e i n f o

Article history:Received 31 July 2012Received in revised form 5 June 2013Accepted 13 June 2013Available online 22 July 2013

Keywords:AfforestationCarbon sequestrationUncertaintyFire riskPest riskDynamic optimal control

Economic studies have demonstrated that agricultural landowners could mitigate significant quantities ofgreenhouse gas (GHG) emissions through afforestation. The associated carbon, however, must remain storedin soils or biomass for several decades to achieve substantial mitigation benefits. Policies and programs to en-hance carbon sequestration in forest systems must accommodate the possibility of premature carbon re-leases. We develop a dynamic nested optimal-control model of carbon sequestration through afforestationgiven uncertainties associated with fire and pest hazards. Our framework highlights a number of factorsthat affect landowner decisions to invest in fire or pest prevention measures. For fire, we show the net influ-ence of these factors is to encourage investment in prevention measures when the probability of fire occur-ring is less than the ratio of expected net economic benefits to expected gross economic benefits of adoptingfire prevention measures. For pests, we show that landowners will invest in prevention measures when theprobability of fire is less than the ratio of the difference between net benefits before and after the discovery oftree pests to the difference between gross economic benefits before and after the discovery of pests. For bothrisks, landowners will over-invest in prevention if the other risk is ignored.

Published by Elsevier B.V.

1. Introduction

Numerous economic studies have demonstrated that agriculturallandowners could mitigate significant quantities of greenhouse gas(GHG) emissions through afforestation — the shifting of cropland andpasture into trees (McCarl and Schneider, 2001; Lewandrowski et al.,2004; Lubowski et al., 2006; U.S. EPA, 2005). To realize the GHGmitiga-tion potential of afforestation, however, farmersmust be able to convertincreases in carbon stored in soils and biomass to income. Several policyapproaches could incentivize carbon capture through afforestation in-cluding the establishment of a carbon market (such as would occurunder a state, regional, or national cap-and-trade program), the crea-tion of a direct government payment explicitly for adoption of carbonsequestering practices (analogous to payments farmers receive underUSDA's Conservation Reserve Program), and the development of volun-tary carbon-related contracts between two or more private parties.

For afforestation to result in significant GHG mitigation, the asso-ciated carbon must remain stored in soils or biomass for an extendedtime (viewpoints range from 20 to over 100 years). As an example,the forest project protocol developed by the Climate Action Reservefor use in the California climate program requires: 1) afforestation/reforestation projects target lands not in forest cover during the pre-vious 10 years; and 2) project lands remain in forest for 100 years

+1 202 401 1176.ewandrowski),. Aillery).

.V.

(Climate Action Reserve, 2010). In policy and scientific settings thisis referred to as the “permanence” issue.

Regardless of the policy approach, permanence has importantimplications for the design of carbon sequestration incentives. Specif-ically, incentives must accommodate both the possibility and uncer-tainty that carbon sequestered and credited within a mitigationframework may be prematurely released at some point in the future.Such releases could be unintentional (as in the case of a future fireevent or a pest/disease outbreak) or deliberate (as in the case of alandowner decision to harvest timber prior to a previously agreedon date).

The premature release of carbon from a parcel of afforested landwould likely create an obligation to either replace the released carbonor compensate the buyer — since it would already have been pur-chased and, presumably, used to meet a GHG mitigation commitmenton the part of the buyer. Conceptually, the replacement obligationcould rest with either the buyer or seller. For two reasons, we assumethat it rests with the seller. First, sellers will typically be landownersand as such will have direct control over how afforested lands are ac-tually managed. Putting the replacement obligation on landownersthen creates an economic incentive for them to take specific landmanagement actions that reduce the probability of a premature re-lease. Second, buyers would likely be entities trying to meet specificemission reduction targets (for example, a private sector firm, indus-try or trade organization, or municipality that has made a public com-mitment to reduce its carbon footprint). If carbon sequesteredthrough afforestation came with significant uncertainty regarding

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91J. Lewandrowski et al. / Forest Policy and Economics 38 (2014) 90–96

its permanence, these entities would likely look either to alternativesuppliers of GHG mitigation or opportunities to reduce emissionswithin their own operation.

In this paper we examine the issue of permanence in the contextof sequestering carbon through afforestation. We develop a dynamicnested optimal-control model of carbon sequestration associatedwith the decision to establish forest cover on a tract of land giventhe inherent uncertainties associated with fire and insect/diseasehazards.1 Conceptually, these hazards are similar in that their occur-rence at any time t is uncertain and landowners can take specificactions — although generally different actions — to reduce the proba-bility of sustaining related losses. The hazards differ, however, in thatfire represents a large loss in carbon at a moment in time, whileinsect/disease (hereafter, “pest”) infestations are more likely to dis-play a period of gradual to significant slowing in the anticipatedrate of carbon accumulation followed by a sustained period of steadycarbon losses. The nature and uncertainties associated with these po-tential losses will influence: 1) the design of sequestration incentivesunder any GHG mitigation policy that requires premature carbon re-leases to be replaced or compensated; and 2) the set of actions land-owners adopt to reduce the probability of such releases occurring.

The paper proceeds as follows: Section 2 briefly reviews timberproduction and riskmanagement. An optimal-control model of timberproduction and carbon sequestration through afforestation under riskand uncertainty is presented in Section 3. Theoretical properties of op-timal solutions are then discussed. Section 4 discusses optimal policiesfor budget allocation between fire preventive measures and pest pre-ventive activities. Effects of adopting fire and pest preventive mea-sures on carbon sequestration are discussed in Section 5. Section 6provides summary and concluding remarks.

2. Timber production and risk management

Existing dynamic forest product models typically focus on produc-tion of timber and assume a point-of-input and point-of-output struc-ture. Some more recent forest product models have added productionof sequestered carbon with annual payments to landowners, butthese generally assume a risk-free environment (van Kooten et al.,1995). More rigorous treatments that consider carbon sequestrationand timber as jointly produced products need to reflect the inherentrisks associated with forest fire and pest outbreaks.

Historically, response to the threat of forest fire has consisted ofboth preventive measures before forest fires occur and suppressionactivities once fires are detected. Preventive measures include variousforms of monitoring (e.g., manned fire observation towers, aircraft,and satellite imagery) and activities that remove or reduce the quan-tity of combustible material on the forest floor and understory (e.g.,removal of dead wood, controlled burns, and thinning). Suppressionmeasures include a host of ground- and aerial-based fire-fightingsystems. To model carbon capture in forest production under uncer-tainty, we use a modified hazard function approach to reflectrisks and uncertainties associated with the timing of forest fires(Kamien and Schwartz, 1971; Kieffer, 1988; Kim et al., 2010). DefineM(t) to be the probability that forest fire occurs by time t, withM(t = 0) = 0, as:

M tð Þ ¼ 1− exp −αm F tð Þð Þ½ �t; m F t ¼ 0ð Þð Þ ¼ 0;∂m∂F b 0; α ¼ 1

1þ σ;

ð1Þ

where, m(t) is the hazard rate, representing the conditional probabil-ity of a forest fire occurring during the next time period given that

1 Previous studies in forest management considered either fire hazards (Amacher etal., 2005; Johnson and Wagner, 1985) or pest hazard (Kim et al., 2007), but not bothfire and pest hazards simultaneously.

one has not occurred at time t, F is the preventive measures beforethe forest fire occurs, where ∂F

∂t ≥ 0; and σ = tm

� � ∂m∂t

� �≤ 0, is the

time elasticity of the conditional probability of forest fire. In Eq. (1),both the probability that forest fire occurs by time t and the condi-tional probability that forest fire will occur during the next time peri-od, t + Δt, decline as fire prevention measures are adopted.

Using Eq. (1), the probability density function of the time for for-est fire occurrence, ∂M tð Þ

∂t , can be presented as the state equation:

∂M tð Þ∂t ¼ m F tð Þð Þ 1–M tð Þ½ �; where m F t ¼ 0ð Þð Þ ¼ 0;

∂m∂F b 0; ð2Þ

Insect and disease pests are part of all forest ecosystems. However,landowners can take actions that reduce the likelihood of a pest out-break occurring, and the damage done to standing trees if an outbreakdoes occur (e.g., selecting pest/disease resistant seedlings, prophylac-tic spraying, and other treatments to discourage pests and diseasesfrom taking hold). Define N(t) to be the probability that a forestpest (tree damaging disease or insect) is discovered by time t, withN(t = 0) = 0, as:

N tð Þ ¼ 1–exp –βn Eb tð Þð Þ½ �t;n Eb t ¼ 0ð Þð Þ ¼ 0;∂n∂Eb

b 0;β ¼ 11þ γ

; ð3Þ

where, n(Eb) represents the conditional probability that discovery ofthe pest will occur during the next time interval (t + Δt) given thatone has not occurred at time t, Eb represents the preventive measuresadopted before the first discovery of the pest, where ∂Eb

∂t ≥0; andγ ¼ t

n

� � ∂n∂t

� �≤ 0, is the time elasticity of the conditional probability

of discovering the pest. Eq. (3) states that the probability of discover-ing a forest pest at time t, and the conditional probability of discover-ing a pest during the next year, declines as the adoption of preventivemeasures increases.

Eq. (3) can be rewritten as the state equation:

∂N tð Þ∂t ¼ n Eb tð Þð Þ 1–N tð Þ½ �; n Eb t ¼ 0ð Þð Þ ¼ 0;

∂n∂Eb

b 0; ð4Þ

where ∂N tð Þ∂t is the probability density function of the time for initial

discovery of a pest.Once a pest is discovered in a tract of forest, landowners can imple-

ment control measures, (i.e., Ea(t)), to reduce the associated damages(e.g., more aggressive spraying and removal of infected and nearbytrees). Populations of pest species are assumed to follow a logisticgrowth function (Eiswerth and Johnson, 2002; Huffaker and Cooper,1995).When control measures are implemented, we adjust the popula-tion growth function to (Kim et al., 2007):

∂a tð Þ∂t ¼ g 1−k Ea tð Þð Þ½ �a tð Þ 1− 1þ k Ea tð Þð Þð Þa tð Þ

A

� �; ð5Þ

where a is acres impacted by the pest in time t, A is total acres, g is therate of tree pest spread, and Ea is control measures after discovery oftree pests such that ∂k

∂Ea tð Þ N 0, where k is a fractional coefficient reflectingthe technical effectiveness of the control measures.

Traditionally, forest product models have employed timbergrowth functions that are based on the age-structure of the trees inthe forest area of interest (Amacher et al., 2005, 2009; Chang, 1984;van Kooten et al., 1995). The logic being that trees grow at relativelypredictable rates so if one knows the species mix and age-structure ofthe trees in a given tract, one can assess with reasonable accuracy thevolume of timber — and sequestered carbon — in the tract at anypoint time. Age-structured growth models are particularly suited toafforestation where forest is established on land previously in a lesscarbon intense use — typically cropland or grass. This means thatthe species mix will be known and all trees will be of the same age.An alternative approach would be to use a size-structured growth

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92 J. Lewandrowski et al. / Forest Policy and Economics 38 (2014) 90–96

function. While these functions can better describe the dynamics oftimber production in some forest situations, there is no standard op-timization technique for such models and their investigation is highlynontrivial (Hritonenko et al., 2008). Given the above considerations,we consider general forms of the age-structured timber growth func-tion, Y(t), that had been used in earlier studies, including Exponentialfunction (Amacher et al., 2005; Chang, 1984; Crowley et al., 2009),Weibull function (Bailey and Dell, 1973; Cao, 2004), and a generalflexible form (van Kooten et al., 1995).2

The timber growth function after discovery of a disease or insectpest is then specified as: 3

y tð Þ ¼ 1− a tð ÞA

� Y tð Þ: ð6Þ

where Y(t) ≥ y(t)

3. The model

In the case where risks associated with forest fire and tree pestsare not considered, the present value of the net economic benefitsresulting from timber production and carbon sequestration throughafforestation is represented by modifying the earlier study by vanKooten et al. (1995) as follows4:

NPV ¼ e−rT Py−Cy−Cc

h iY Tð Þ þ ∫

T

0

e−rtPc θY ′ tð Þ þ S′ tð Þ �dt

¼ e−rT Py þ θPc−Cy−Cc

� �Y Tð Þ þ S tð Þ

h iþ ∫

T

0

e−rtrPc θY tð Þ þ S tð Þ½ �dt;

ð7Þ

where r is the rate of discount, T is terminal time, Y(t) is the volume oftimber in year t, S(t) is the volume of carbon sequestered in soil in yeart, Cy is the cost of per unit timber harvest, Cc is the social cost of carbonrelease at the time of per unit timber harvest, Py is per unit price of tim-ber, Pc is carbon payment per unit of carbon sequestered, and θ is conver-sion factor relating timber volume to carbon sequestered in trees.5

The formulation of NPV described in Eq. (8) embodies two assump-tions that should be made explicit. First, it has been common in forestcarbon assessments to assume that 50% of tree biomass is carbon.Thomas andMartin (2012) show that this percent actually varies some-what by species. The stand-level look-up tables in Smith et al. (2005)(see Footnote 4), however, show that in a given forest carbon poolscan be related to timber volumes but that this relationship changesover time. In a carbon market then, the coefficient θ would likely varyacross forest tracks to reflect differences in the species mix of treesplanted and the anticipated time horizon that the trees will stand. Sec-ond, organic carbon levels in soils are maintained through a dynamicprocess that includes sequestration (in the form of carbon containedin decaying plant matter) and emissions (in the form of CO2 released

2 van Kooten et al. chose a flexible timber growth function Y(t) = αtβe− γt for its ro-bustness and for practical reasons such that it is easily estimable, twice differentiable,and integrable (van Kooten et al., 1995, p367).

3 In this paper, we assume that pests reduce the quantity of timber but do not affectits quality.

4 We consider Hartman rotation for optimal rotation age (T), which maximizes thepresent value of timber and carbon uptake benefits. See van Kooten et al. (1995) fordetail.

5 Soil carbon sequestration is included in our model for completeness, as suggestedby a reviewer. In practice, soil carbon stocks change relatively slowly in afforested sys-tems. Smith et al. (2005) develop stand-level look-up tables showing timber volumesand carbon stocks by pool (i.e., live tree, standing dead tree, understory, down deadtree, forest floor, and soil) as a function of stand age for 51forest types in 10 regionsof the United States. Soil carbon stocks typically increase less than 0.25 and 0.50 mtper acre per year up to ages 25 and 55 years, respectively. Hence, the inclusion of soilcarbon in a carbon market would likely require relatively low-cost protocols for mea-suring and monitoring.

to the atmosphere through the oxidation of carbon in soils and deadplant matter). Over time and under relatively constant environmentaland management conditions, rates of carbon additions to, and CO2

emissions from, soils tend to equilibrate and the amount of soil organiccarbon stabilizes at a constant (or equilibrium) level. To simplify ourmodel and its discussion, we assume that lands that get afforestedhave been in their previous use (i.e., cropland or grass) long enoughto have achieved soil carbon equilibrium. This means that any increasein soil carbon related to afforestation is not partially offset by foregonesequestration associated with the previous land use.

Using Eqs. (1) through (7), the value function associated with tim-ber growth and carbon sequestration to be maximized under risksand uncertainties associated with forest fire and tree pests is:

V M tð Þ;N tð Þ; a tð Þ; t0; Tð Þ

¼ Sup∫T

0

e−rtn

1−M tð Þð Þh1−N tð Þð Þ rPc θY tð Þ þ S tð Þð Þ−Cb F tð Þ; Eb tð Þð Þ½ �

þN tð Þ rPc εy a tð Þð Þ þ s a tð Þð Þð Þ−Ca F tð Þ; Ea tð Þð Þ½ �i

−M tð Þ 1–N tð Þð ÞCb F tð Þ; Eb tð Þð Þ þ N tð ÞCa F tð Þ; Ea tð Þð Þ½ �odt

þ e−rTn

Py þ Pc−Cy−Cc

� �1−M Tð Þð Þ½ 1−N Tð Þð Þ θY Tð Þ þ S Tð Þð Þ

þN Tð Þ εy Tð Þ þ s a Tð Þð Þð Þ�o

ð8ÞWhere, ε is a factor for converting a unit of pest-infested tree volume

to a quantity of sequestered carbon, s(a(t)) is the volume of carbon se-questered in soil when trees are pest-infested, and the subscripts b anda represent “before” and “after” discovery of the pest, respectively.

The Hamiltonian equation is stated as:

H ¼ e−rtn

1−M tð Þð Þh1−N tð Þð Þ rPc θY tð Þ þ S tð Þð Þ−Cb F tð Þ; Eb tð Þð Þ½ �

þ N tð Þ rPc εy a tð Þð Þ þ s a tð Þð Þð Þ−Ca F tð Þ; Ea tð Þð Þ½ �i

−M tð Þ 1−N tð Þð ÞCb F tð Þ; Eb tð Þð Þ þ N tð ÞCa F tð Þ; Ea tð Þð Þ½ �o

þ e−rTn

Py þ Pc−Cy−Cc

� �1−M Tð Þð Þ

h1−N Tð Þð Þ θY Tð Þ þ S Tð Þð Þ

þN Tð Þ εy Tð Þ þ s a Tð Þð Þð Þio

þ λ1m F tð Þð Þ 1–M tð Þ½ � þ λ2n Eb tð Þð Þ 1−N tð Þ½ �þλ3:g 1−k Ea tð Þð Þð Þa tð Þ 1− 1þ k Ea tð Þð Þð Þa tð Þ

A

� �:

ð9Þ

In Eq. (9), the control variables include prevention measuresadopted to avert forest fire (F(t)), prevention measures adopted toavert forest pests before the discovery of forest pests (Eb(t)), and con-trol measures adopted to limit pest damages after the initial discovery(Ea(t)). The state variables are the probability of forest fire occurrence(M(t)), the probability of discovering a pest species (N(t)), and theacres impacted by the pest (a). Finally, λi(t)(i = 1, 2, 3) are adjointvariables associated withM(t), N(t), and a(t), respectively. The neces-sary conditions for optimum are presented in Appendix (A).

The economic properties of the optimal conditions for control var-iables (see Eqs. (A1) through (A3) in Appendix A) are better servedby investigating the adjoint variables λi (i = 1, 2, 3). From Eqs. (A1)through (A3), the adjoint variables are represented as:

λ1 tð Þ ¼e−rt 1−Nð Þ ∂Cb

∂F

� �þ N ∂Ca

∂F

� �h i1−Mð Þ ∂m

∂F

� � b 0: ð10� 1Þ

λ2 tð Þ ¼e−rt ∂Cb

∂Eb

� �∂n∂Eb

� � b 0: ð10� 2Þ

λ3 tð Þ ¼ −e−rtN ∂Ca

∂Ea

� �ga 1− 2ka

A

� � ∂k∂Ea

� � b 0: ð10� 3Þ

Page 4: Carbon sequestration through afforestation under uncertainty

93J. Lewandrowski et al. / Forest Policy and Economics 38 (2014) 90–96

Following previous authors (Kamien and Schwartz, 1971; Arrow,1968; Seierstad and Sydsæter, 1977) the adjoint variables λi(t0),where i = 1, 2, 3 and 0 b t0 b T, can be viewed as the shadow valuesassociated with small changes in the state variables, M(t0), N(t0), anda(t0), respectively, on the value function in Eq. (8).

Eq. (10-1) shows that as the probability of a forest fire occurring inyear t increases, the net economic benefits associated with the treesdeclines. The adjoint variable λ1(t) is thus the shadow cost of an in-crease in the probability of fire. Similarly, λ2(t) represents the shadowcost of an increase in the probability of discovering a pest. That is,Eq. (10-2) shows that as the probability of discovering acres infestedwith a tree pest increases, the net economic benefits of standing treesdeclines. Finally, Eq. (10-3) shows that the net economic benefits ofthe forest decrease as the number of infested acres increase. Soλ3(t) is the shadow cost of an additional infested acre. UsingEqs. (10-1) through (10-3) we can derive the economic propertiesof the optimal conditions presented in Appendix (A).

Optimality condition (A1) indicates that the expected marginalbenefits resulting from a reduction in the forest-fire hazard rate byadopting fire prevention measures equals the present values of theexpected marginal costs of adopting forest-fire prevention measuresbefore and after the discovery of acres with tree pests. Eq. (A2) indi-cates that the marginal benefits resulting from reducing the pest haz-ard rate by adopting preventive measures equal the present values ofthe expected marginal costs of adopting preventive measures beforetree pests spread. Meanwhile, Eq. (A3) explains that the presentvalues of the expected marginal costs of adopting control measuresafter discovery of a pest species equal the marginal benefits resultingfrom the reduction in impacted acres.

4. Optimal budget allocation/optimal policy

Inserting Eqs. (10-1) through (10-3) into Eqs. (A4) through (A6),respectively, results in Eqs. (11-1), (11-2), and (11-3) below:

∂λ1 tð Þ∂t ¼ e−rt

(1−Nð Þ rPc θY þ Sð Þ þ Cb

1−M

� ηCbFηmF

� � �

þ N rPc εyþ sð Þ þ Ca

1−M

� ηCaFηmF

� � �)N 0;

ð11� 1Þ

where ηCbF =∂Cb∂F

� �FCb

� �N 0, ηmF = ∂m

∂F

� �Fm

� �b 0, ηCaF =

∂Ca∂F

� �FCa

� �N 0,

and M b1−Nð Þ rPc θYþSð ÞþCb

ηCbFηmF

� �h iþN rPc εyþsð ÞþCa

ηCaFηmF

� �h i1−Nð ÞrPc θYþSð ÞþNrPc εyþsð Þ .

∂λ2 tð Þ∂t ¼ e−rt 1−Mð Þ rPc θY þ Sð Þ−rPc εyþ sð Þð Þ− Cb−Cað Þ½ � þ ηbEb

ηnEb

� Cb

� N 0;

ð11� 2Þ

where ηbEb = ∂Cb∂Eb

� �EbCb

� �N 0, ηnEb = ∂n

∂Eb

� �Ebn

� �b 0, and M b

rPc θYþSð Þ−Cb 1−ηbEbηnEb

� �h i− rPc εyþsð Þ−Ca½ �

rPc θYþSð Þ−rPc εyþsð Þ .

∂λ3 tð Þ∂t ¼ − e−rtN 1−Mð ÞrPc ε

∂y∂a

� þ ∂s∂a

� − 1−kð Þ

akA−2a 1þ kð Þ

A−2ak

� ηaEaηkEa

� � �Ca

� N 0;

ð11� 3Þ

where ηaEa = ∂Ca∂Ea

� �EaCa

� �N 0 and ηkEa = ∂k

∂Ea

� �Eak

� �N 0.

Eq. (11-1) shows how the cost of an increase in the probability of aforest fire changes (specifically, increases) over time. Note that thepresent values of the expected gross economic benefits are expressedin terms that reflect the probabilities of not discovering and discover-ing a tree pest, (1 − N) and N, respectively.

The expected total costs are weighted somewhat differently. Spe-cifically, the elasticity ηmF is a measure of the technical efficiency of

fire prevention activities, while the elasticities ηCbF and ηCaF reflect, re-spectively, the costs of these activities before and after the discoveryof a forest pest. The ratios ηCbF/ηmF and ηCaF/ηmF therefore, reflectthe relative cost effectiveness of fire prevention activities (i.e., beforeand after the discovery of a pest species, respectively) with smallerratios implying more cost effective activities to undertake. Thepresent value of the expected costs before the discovery of a pestare weighted by the probability of not discovering a pest, the relativecost effectiveness of fire prevention activities, and the inverse ofthe probability of a fire not occurring (since, by assumption, thereis no economic decision regarding the trees in the case of a fire)(i.e., 1−Nð Þ

1−Mð ÞηCbFηmF

). Similarly, the present value of the expected costsafter the discovery of a pest are weighted by the probability of discov-ering a pest, the relative cost effectiveness of fire prevention activitiesgiven that discovery, and the inverse of the probability of a fire notoccurring (i.e., N

1−Mð ÞηCaFηmF

).Mathematically, Eq. (11-1) could be positive or negative. Land-

owners, however, will only be interested in maintaining land intrees (i.e., sequestering carbon) within the positive range since a neg-ative value would imply that the value of standing trees is decreasing.To ensure ∂λ1 tð Þ

∂t N 0, requires that;

M tð Þ b1−Nð Þ rPc θY þ Sð Þ þ Cb

ηCbFηmF

� �h iþ N rPc εyþ sð Þ þ Ca

ηCaFηmF

� �h i1−Nð ÞrPc θY þ Sð Þ þ NrPc εyþ sð Þ :

ð12Þ

Eq. (12) explains that the forest fire preventive measures areadopted as long as the probability of forest fire occurrence is lessthan the ratio of the expected net economic benefits to the expectedgross economic benefits of adopting fire preventive measures, whereall expected benefits are measured with the probabilities of discover-ing and not discovering forest diseases.

Given that the condition in Eq. (12) is met, Eq. (11-1) says that alandowner will adopt more fire prevention measures, before andafter the discovery of a pest species, as t increases (i.e., the standingtrees and carbon sequestered increase in net value), the probabilityof fire occurring decreases or the cost efficiency of fire preventionmeasures increase (these discount the costs of fire prevention mea-sures), the probability of a pest infestation decreases (since expectedcarbon sequestration is closer to θY + S than εy + s), and if a pest isdiscovered, it is discovered closer to terminal time T (again, becausethe expected carbon sequestration is closer to θY + S than εy + s).In the form of a decision rule, Eq. (12) explains that the forest fire pre-ventive measures are adopted as long as the probability of forest fireoccurrence is less than the ratio of the expected net economic bene-fits to the expected gross economic benefits, where the expected ben-efits are measured with the probabilities of discovering forestdiseases. Additionally, setting N = 0 in Eq. (12) makes clear thatlandowners will over-allocate resources to fire prevention in caseswhere they do not consider real risks of incurring pest damages.This is because they act to protect θY + S carbon on all acres wheninfested acres actually have εy + s carbon (and εy + s b θY + S).

Eq. (11-2) describes how the value of an increase in the probabil-ity of discovering a tree pest changes over time. Following the discus-sion of Eq. (11-1), the elasticity ηnEb reflects the technical efficiency ofpest prevention activities before discover, the elasticity ηbEb reflectsthe costs of those activities, and the ratio ηbEb/ηnEb is a measure oftheir cost-effectiveness with smaller ratios reflecting more cost effec-tive activities. Eq. (11-2) then, states that landowners are more likelyto adopt prevention measures for pest threats: 1) the lower the prob-ability of a fire occurring (since, by assumption, a fire destroys all eco-nomic value associated carbon sequestered in standing trees); 2) thelarger the expected threat to terminal period (time T) carbon seques-tration (i.e., the smaller (rPcθY − rPcεy)), and the more cost-effectivepest prevention activities are (i.e., the smaller the ratio ηbEb/ηnEb).

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94 J. Lewandrowski et al. / Forest Policy and Economics 38 (2014) 90–96

Finally, as with Eq. (11-1), landowners will only adopt pest pre-vention measures if the expected net economic benefits of doing soare positive — that is, as long as ∂λ2 tð Þ

∂t N 0. For this to happen,Eq. (13) below must hold.

M tð Þ brPc θY þ Sð Þ−Cb 1− ηbEb

ηnEb

� �h i− rPc εyþ sð Þ−Ca½ �

rPc θY þ Sð Þ−rPc εyþ sð Þ : ð13Þ

Viewed as a decision rule, Eq. (13) states that pest preventionmeasures are adopted if the probability of forest fire occurrence isless than the ratio of the difference between the net benefits beforeand after the discovery of tree pests to the difference between thegross economic benefits before and after the discovery of pests. Paral-lel with Eq. (12) above, setting M = 0 in Eq. (13) shows that land-owners will over-allocate resources to pest prevention when theydo not consider real risks of fire losses (since M = 0 is the lowestpossible value that could trigger investments in pest prevention mea-sures). Eq. (13) can be also be rewritten to highlight the comparisonof costs and benefits of pest prevention over time in the landownerdecision framework. Specifically;

1−Mð Þ Pc θY þ S−εy−sð Þ½ � N 1r

� Cb 1þ ηbEb

ηnEb

� −Ca

� �;

where ηbEb N 0 and ηnEb b 0:

ð14Þ

Eq. (14) explains that pest preventive measures are adopted if theexpected benefit differences between before and after the discoveryof tree pests are greater than cost differences between before andafter the discovery of tree pests.

Eq. (11-3) explains how the costs associated with increasing thenumber of pest-infested acres changes over time. Here, the elasticityηkEa is the technical efficiency of pest control activities (includingeradication and containment measures), the elasticity ηaEa reflectsthe costs of adopting control activities, and the ratio ηaEa/ηkEa de-scribes the relative cost effectiveness of these activities. Note thatin valuing pest control measures, landowners are only concernedwith the possible outcomes that a forest fire does not occur (forthe reason stated above), and that a pest is discovered (since thereare only prevention activities prior to a pest discovery). Given theprobabilities of both outcomes, landowners will increase the adop-tion of pest control measures when the carbon released by a pest(i.e., ε ∂y

∂a þ ∂s∂a) increases and/or the cost effectiveness of pest control

activities increases.Eqs. (11-2) and (11-3) can be compared to evaluate the effi-

cient budget allocation between pest prevention measures beforediscovery of a pest and control measures to limit infested acresafter its discovery. However, it is well documented in previousstudies that it is economically efficient to allocate a larger shareof expenditures for preventive measures before the initial discov-ery than for control measures after discovery of tree pest (Kim etal., 2006).

5. Effects of adopting preventive measures for carbon sequestration

Assuming the existence of a framework through which land-owners can afforest cropland or pasture and get paid for the carbonsequestered in the standing trees, the previous section developedoptimality conditions to guide landowners in decisions regardingthe adoption measures that reduce the uncertainties associatedwith premature releases of carbon due to forest fire and pestinfestations. There remains, however, the important question re-garding the expectation of how much carbon will be sequesteredat terminal time (T) under various scenarios. The quantity of carbon

sequestration (CS) during the planning horizon in a risky environ-ment is given by:

CS ¼ ∫T

0

1−M tð Þð Þ 1−N tð Þð Þ θY ′ tð Þ þ S′ tð Þ �þ N tð Þ εy′ tð Þ þ s′ a tð Þð Þ½ � �dt;

ð15Þ

where Y′(t) = dY tð Þdt , S′(t) = dS tð Þ

dt , y′(t) = dy tð Þdt , and s′(a(t)) = ds tð Þ

dt .To evaluate the effects on carbon sequestration of preventive mea-

sures associated with forest fire and tree pests, we insert M(t) andN(t) in Eqs. (1) and (3), respectively, into Eq. (15) and then applystandard integration by equation component to give the following:

S ¼ θY Tð Þ þ S Tð Þ½ �e− αmþβnð ÞT

þ ∫T

0

αmþ αm′� �

t� �

þ βnþ βn′� �

t� �

θY tð Þ þ S tð Þ½ �e− αmþβnð Þtdt

þ εy Tð Þ þ s a Tð Þð Þ½ �e−αmT þ ∫T

0

αmþ αm′ð Þtð Þ εy tð Þ þ s a tð Þð Þ½ �e−αmtdt

−hεy Tð Þ þ s a Tð Þð Þe− αmþβnð ÞT

−∫T

0

αmþ αm′ð Þtð Þ þ βnþ βn′� �

t� �

εy tð Þ þ s a tð Þð Þ½ �e− αmþβnð Þtdt

ð16Þ

The first two terms of the right-hand side of the equality inEq. (16) explain the expected carbon sequestration when there areno occurrences of forest fires or tree pests. The rest of Eq. (16) repre-sents the expected carbon sequestration when there are no fires, buttree pests are discovered. Since αm + (αm′)t = m and βn + (βn′)t = n, as shown in Appendix (C), Eq. (16) can be rewritten as follows:

CS ¼ θY Tð Þ þ S Tð Þ−εy Tð Þ−s a Tð Þð Þ½ �e− αmþβnð ÞT

þ∫T

0

mþ nð Þ θY tð Þ þ S tð Þ−εy tð Þ−s a tð Þð Þ½ �e− αmþβnð Þtdt

þ εy Tð Þ þ s a Tð Þð Þ½ �e−αmT þ ∫T

0

m εy tð Þ þ s a tð Þð Þ½ �e−αmtdt:

ð17Þ

Note that when the conditional probabilities, m(t) and n(t) equalto zero, the amount of carbon sequestration in Eq. (17) equals[θY(T) + S(T)], which is the amount of carbon sequestration under anenvironment free of fire and pest risks. Hence, ignoring these risks willresult in an over-expectation of the amount of carbon that will be se-questered if, in fact, the risks are real. To evaluate carbon sequestrationimplications of adopting the preventivemeasures associatedwith forestfires and tree pests, differentiate Eq. (17) with respect to F(t) and Eb(t).The results are presented in Eqs. (18-1) and (18-2), respectively.

∂CS∂F tð Þ ¼ − αT

∂m∂F

� θY Tð Þ þ S Tð Þ−εy Tð Þ−s a Tð Þð Þ½ �e− αmþβnð ÞT þ εy Tð Þ þ s a Tð Þð Þð Þe−αmT

n o

þ∫T

0

θY tð Þ þ S tð Þ−εy tð Þ−s a tð Þð Þ½ � ∂m∂F

� 1−α mþ nð Þt½ �e− αmþβnð Þtdt

þ∫T

0

εy tð Þ þ s a tð Þð Þð Þ ∂m∂F

� 1−αmt½ �e−αmtdt:

ð18� 1Þ

∂CS∂Eb tð Þ ¼ − βT

∂n∂Eb

� θY Tð Þ þ S Tð Þ−εy Tð Þ−s a Tð Þð Þ½ �e− αmþβnð ÞT

þ∫T

0

θY tð Þ þ S tð Þ−εy tð Þ−s a tð Þð Þ½ � ∂n∂Eb

� 1−β mþ nð Þt½ �e− αmþβnð Þtdt:

ð18� 2Þ

While Eqs. (18-1) and (18-2) describe, respectively, the marginaleffects of fire and pest prevention measures on the quantity of carbonsequestered through afforestation, it is difficult to compare these

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95J. Lewandrowski et al. / Forest Policy and Economics 38 (2014) 90–96

risk-reducing measures more thoroughly without the parameters ofthe hazard functions described in Eqs. (1) and (3).

6. Conclusions

The view that greenhouse gas emissions can be mitigated by in-creasing the quantity of carbon stored in forest systems hinges onthe condition that the associated carbon remains stored in soils andbiomass for at least several decades. This means that any policy orprogram designed to enhance carbon sequestration in forest systemswill need to accommodate the possibility of premature carbon re-leases. Two important sources of potential carbon releases from forestsystems are fires and pest infestations.

This paper has laid out a dynamic nested optimal-control model ofcarbon sequestration through afforestation given uncertainties associ-ated with fire and tree pest hazards. Previous authors have consideredeach hazard individually but not the two simultaneously. Fire and pesthazards, while differing in nature, are similar in that landowners can,at a cost, take actions to reduce the probability of sustaining relatedlosses. We embody these actions and costs into decision rules forlandowner investments in fire and pest prevention In the case offire, we show that prevention measures are undertaken when theprobability of fire occurring is less than the ratio of the expectednet economic benefits to the expected gross economic benefits ofadopting fire prevention measures. For pests, we show that land-owners will invest in prevention measures when the probability offire is less than the ratio of the difference between net benefits beforeand after the discovery of tree pests to the difference between grosseconomic benefits before and after the discovery of pests. For eithersource of risk, landowners will over-invest in prevention if the othersource is ignored. Ignoring both sources of risk will result in a gener-ally over-optimistic assessment of the quantity of carbon the forestwill sequester over time.

Disclaimer

The views expressed in this article are those of the authors anddo not necessarily reflect the views of the U.S. Department ofAgriculture.

Acknowledgments

The authors gratefully acknowledge the time and effort that GregAmacher and an anonymous reviewer dedicated to this paper. Address-ing their comments and suggestions significantly improved our paperand we wish to thank these individuals explicitly.

Appendix A. Necessary conditions for optimality

Necessary condition for fire prevention measures:

∂H∂F ¼ 0 implies

− e−rt 1–Nð Þ ∂Cb

∂F

� þ N

∂Ca

∂F

� � �þ λ1 1−Mð Þ ∂m

∂F

� ¼ 0:

ðA1Þ

Necessary condition for pest prevention measures (before pestdiscovery):

∂H∂Eb

¼ 0 implies

− e−rt 1–Nð Þ ∂Cb

∂Eb

� þ λ2 1−Nð Þ ∂n

∂Eb

� ¼ 0:

ðA2Þ

Necessary condition for pest controlmeasures (after pest discovery):

∂H∂Ea

¼ 0 implies

− e−rtN∂Ca

∂Ea

� −λ3ga 1−2ka

A

� ∂k∂Ea

� ¼ 0:

ðA3Þ

Other necessary conditions:

− ∂H∂M ¼ ∂λ1

∂t implies

e−rt 1−Nð ÞrPc θY þ Sð Þ þ NrPc εyþ sð Þ½ � þ λ1m ¼ ∂λ1

∂t :

ðA4Þ

− ∂H∂N ¼ ∂λ2

∂t implies

e−rt 1−Mð Þ rPc θY þ Sð Þ−rPc εyþ sð Þ½ �− Cb−Cað Þf g þ λ2n ¼ ∂λ2

∂t :

ðA5Þ

− ∂H∂a ¼ ∂λ3

∂t implies

−e−rt 1−Mð ÞN rPc ε∂y∂a

� þ ∂s

∂a

� � �−λ3g 1−kð Þ 1−2a 1þ kð Þ

A

� �¼ ∂λ3

∂tðA6Þ

∂H∂λ1

¼ ∂M∂t implies

∂M tð Þ∂t ¼ m F tð Þð Þ 1−M tð Þ½ �

ðA7Þ

∂H∂λ2

¼ ∂N∂t implies

∂N tð Þ∂t ¼ n Eb tð Þð Þ 1−N tð Þ½ �

ðA8Þ

∂H∂λ3

¼ ∂a∂t implies

∂a∂t ¼ g 1−kð Þa 1− 1þ kð Þa

A

� �:

ðA9Þ

limt→T

λ1 tð Þ≤ 0; limt→T

λ2 tð Þ≤ 0; limt→T

λ3 tð Þ≤ 0: ðA10Þ

Appendix B. The approximated marginal contribution of the statevariables at time t0 to the optimal value function

∂V0

∂M t0ð Þ ¼ −∫T

t0

h1−N t0ð Þð ÞrPc θY t0ð Þ þ S t0ð Þð Þ

þN t0ð ÞrPc εy t0ð Þ þ s a t0ð Þð Þð Þidt ≡ λ1 t0ð Þ b 0:

ðB1Þ

∂V0

∂N t0ð Þ ¼ −∫T

t0

h1−M t0ð Þð ÞrPc θY t0ð Þ þ S t0ð Þð Þ

− rPc εy t0ð Þ þ s a t0ð Þð Þð Þidt ≡ λ2 t0ð Þ b 0:

ðB2Þ

∂V0

∂a t0ð Þ ¼ ∫T

t0

1−M t0ð Þð ÞN t0ð Þ rPc ε∂y t0ð Þ∂a t0ð Þ

� þ ∂s a t0ð Þð Þ

∂a t0ð Þ� � �

dt ≡ λ3 t0ð Þ b 0:

ðB3Þ

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96 J. Lewandrowski et al. / Forest Policy and Economics 38 (2014) 90–96

Appendix C. Proffs of [αm + (αm′)t] = m and [βn + (βn′)t] = n

Using the hazard function in Eq. (1), [αm + (αm′)t] can be rewrit-ten as follows:

αmþ αm′ð Þt½ � ¼mþ ∂m

∂t

� �t

1þ ∂m∂t

� �tm

� � ¼ m 1þ σð Þ1þ σ

¼ m:

Similarly, using the hazard function in Eq. (3), one can show that:

βnþ βn′� �

t � ¼ n 1þ γð Þ

1þ γ¼ n: Q :E:D:

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