The optimal harvesting problem with price uncertaintythe results developed for the portfolio problem. For example in Blomvall and Shapiro [2] the authors use similar techniques to

The optimal harvesting problem with price

uncertainty∗

Bernardo K. Pagnoncelli

Escuela de Negocios, Universidad Adolfo Ibanez

Adriana Piazza

Departamento de Matematica, Universidad Tecnica Federico SantaMarıa

July 1, 2011

Abstract

In this paper we study the exploitation of a one species forest plan-tation when timber price is governed by a stochastic process. The workfocuses on providing closed expressions for the optimal harvesting policyin terms of the parameters of the price process and the discount factor.We assume that harvest is restricted to mature trees older than a certainage and that the growth after maturity is neglected as well as the naturalmortality. We use stochastic dynamic programming techniques to char-acterize the optimal policy for two important cases: when prices follow ageometric Brownian motion we completely characterize the optimal policyfor all possible choices of drift and discount factor. If prices are governedby a mean-reverting (Ornstein-Uhlenbeck) process we provide sufficientconditions, based on explicit expressions for reservation prices at everytime period above which harvesting everything available is optimal. Inboth cases we solve the problem for every initial condition and the bestpolicy is obtained endogenously, that is, without imposing any ad hoc re-strictions such as maximum sustained yield or convergence to a predefinedfinal state.

∗This research was partially supported by Project Anillo ACT-88 and Programa Basal PFB03, CMM. Adriana Piazza also acknowledges the financial support of Fondecyt under Project11090254. The authors thank Marcos Goycoolea, Alexander Shapiro and Andres Weintraubfor fruitful conversation and encouragement.

1

1 Introduction

Forest resources exploitation with the goal of obtaining maximal economic ben-efits dates back many centuries. In his seminal paper of 1849, Faustmann [10]proposed the correct formula to valuate a standing forest plantation and to de-termine its optimal rotation length. The simplicity of Faustmann result comesfrom the fact that all the trees are considered identical and prices are assumed tobe known and constant over time. A wide literature has developed since then innumerous directions that go from dealing with more sophisticated growth mod-els to considering non-timber forest products such as tourism and environmentalservices or introducing uncertainty in forest growth, prices and costs. Optimalharvesting policies are sometimes studied numerically with different types ofeven-flow constraints, or requiring convergence to a predetermined steady state.In this work, we generalize Faustmann work considering stochastic prices at thesame time that we allow for trees of different ages in the forest. Even more, weaim to determine the optimal harvesting policy without imposing any ad hocrestrictions.

The generalization of Faustmann problem to one where different age classesare allowed was already known before Faustmann’ times but its complete res-olution remains open even today. The first approaches to solve the problemcomprised the additional assumption that the forest converged to a predefinedfinal state or the imposition of constraints on timber flow [8, 22]. Later on, linearprogramming was applied to the resolution of the optimal harvesting problem.The objective function is either the timber yield or the net economic benefitobtained over time, and the constraints specify the growth dynamics and theimpact of harvesting in the forest population, see for example [8, 13].

Many other growth models have been developed with different levels of detail,describing growth rate as a linear or non-linear function of age, basal area, treedensity and site index among other variables. Additional information on forestgrowth models related to mathematical optimization can be obtained throughthe following surveys [17, 21, 22, 23, 25] and the references therein. For anexample of a stochastic growth model, we refer the reader to [14].

The choice of the most appropriate model depends strongly on the intendeduse. Since our main goal is to investigate the effect of random prices, we assumethat growth is a purely aging process and adopt a simple deterministic growthmodel in discrete time, where the forest population is divided into age classes 1.

The model we use was proposed by Rapaport et al. [20] and represents aforest plantation where harvest is forbidden before a pre-defined maturity ageand growth after maturity is neglected as well as natural mortality. Rapaportet al. define a greedy policy as one in which every tree is harvested as soon asit reaches maturity. They show that, in the deterministic setting with lineartimber price, every optimal trajectory is greedy and the optimization problemcan be trivially solved. In a more general case, when the benefit obtained fromthe timber is represented by a concave function of the harvested timber volume,

1For a complete discussion of the implications of using continuous or discrete time variableswhen using age class models we refer the reader to [22].

2

the authors prove that every optimal trajectory becomes greedy in a finite timethat is bounded above by twice the maturity age. This implies that the infinitetime horizon problem can be stated as a finite dimensional optimization problemand can be numerically solved.

An extension of the Rapaport et al. model to a mixed forest composed byseveral species of different maturity ages is considered in [6]. The authors provethe existence and uniqueness of a steady state and the fact that an optimallymanaged forest converges towards the steady state under a mild additionalcondition on the maturity ages, with no restriction on the discount factor 2.

Both works [20] and [6] describe the structure and behavior of the optimalpolicy but ignore any source of uncertainty. In real-life timber prices are un-certain and the need for the inclusion of some random price process has beenhighlighted by several authors ([5, 15, 18]). In the static setting, Reeves andHaight [19] proposed a mean-variance optimization model and estimated re-turns using time series on historical data. In a dynamic setting, the question ofwhich process best represents the dynamics of timber prices is still the objectof discussion inside the resource economists community. Some authors assumenormality with respect to prices fluctuation and suggest that a geometric Brow-nian motion (GBM) should be used as it is consistent with an informationallyefficient market ([4, 5, 24]). However, other authors ([1, 12]) point out that amean-reverting process, or Ornstein-Uhlenbeck (O-U) is a better description ofthe timber price path due to empirical data that has been collected for severalspecies. It is not our intention to go any further into this discussion, we referthe reader to [16] and references therein. As the issue seems far from beingsettled, we decided to consider both stochastic processes most commonly usedto describe natural resource prices: GBM and O-U.

When we model the price dynamics as a GBM process, we completely char-acterize the optimal harvesting policy as a function of the discount factor andthe drift of the price process. We prove that the greedy policy is optimal for aregion in the parameters’ space defined by the drift of the GBM and the discountfactor. Outside this region, we also characterize the optimal policy and provethat it is not greedy, showing an important difference with the deterministiccase.

However, when the prices follow an O-U process we are not always able tocharacterize the optimal policy. We obtain a sufficient condition on the discountfactor and the parameters of the price process assuring that the optimal policyis greedy. In addition, when this sufficient condition is not fulfilled, we providea weaker sufficient condition depending also on the current price level. At everytime step, if the current price is greater than a predefined threshold we can assurethat it is optimal to harvest everything available, regardless of the future 3. Evenif we cannot characterize what the optimal policy is when the condition is notsatisfied, the result proved to be extremely useful in the numerical experiments.

2The fact that global convergence is obtained in a discounted utility framework with norestriction on the discount factor is a distinguishing feature of this work as most convergencetheorems are either local or assume discount factors close to one.

3This threshold depends of the discount factor and the parameters of the price process.

3

We used the threshold as a reservation price, harvesting everything if the currentprice is greater than the threshold, and postponing the harvest if it is not. Theobtained benefits were within 2% of the optimum (see Table 3.2.1).

Hence, our result can be viewed as a very simple approach to derive a reser-vation price and provide a closed formula depending only on the discount factorand the parameters of the price process. A significant difference of our approachwith that of [4, 11], where an asset sale model for forestry is proposed, is that wearrive at a closed formula of the reservation price as a function of the discountfactor and the parameters of the price process; whereas in [4, 11] the equationsderived must be solved numerically 4.

It is interesting to note that although the problem we will consider has simi-larities with multi-period portfolio selection problems, the unique age structureof the forest and the fact that the harvesting decisions ultimately determinehow much timber will be available in the future make it impossible to applythe results developed for the portfolio problem. For example in Blomvall andShapiro [2] the authors use similar techniques to characterize the optimal pol-icy in a multistage stochastic portfolio selection problem under logarithmic andexponential utility. In their model assets can be bought and sold at anytimeand in any quantity, which is a reasonable assumption for such problems. Suchassumption does not make sense for harvesting problems since one can onlyharvest what is available at a given time period.

The paper is organized as follows: Section 2 presents the forest growth modeland the price processes we are considering as well as the optimization problemto be solved. Section 3 is the main contribution of this work and starts with thederivation of the stochastic dynamic programming equations that are the maintools to tackle the optimal harvesting problem. It also comprises the derivationof our theoretical results characterizing the optimal harvesting policies and somenumerical experiments for self-generated instances. Finally, Section 4 concludesthis work and brings up some futures lines of research. Some technical proofsare relegated to the Appendix.

2 Model formulation

Let us consider a forest of total area S occupied by one species forest withmaturity age of n years. In contrast with the case of wild forests, the stateof a forest plantation may be described by specifying the areas occupied bytrees of different ages, making the assumption that tress are planted within apre-specified and constant distance of each other.

For each period t ∈ N we denote xa(t) ≥ 0 the area of trees of age a = 1, . . . , nin year t, and x(t) ≥ 0 the area occupied by trees beyond maturity (older thann). Using a single state variable to represent the over-mature trees conveys theunderlying assumption that the growth of trees is negligible beyond maturity.Each period we must decide how much land c(t) ≥ 0 to harvest. Assuming that

4In [4, 11] the problem is proposed in continuous time and with harvesting allowed forevery age class.

4

only mature trees can be harvested we must have

0 ≤ c(t) ≤ x(t) + xn(t), (1)

and then the area not harvested in that period will comprise the over-maturetrees at the next step, namely

x(t+ 1) = x(t) + xn(t)− c(t). (2)

We neglect natural mortality at every age, again an assumption valid in managedforest plantations but not in wild forests. Hence, the transition between ageclasses is given by

xa+1(t+ 1) = xa(t) ∀ a = 1, . . . , n− 1. (3)

The harvested area is immediately allocated to new seedlings that will comprisethe 1 year old tress in the following year:

x1(t+ 1) = c(t). (4)

We represent the state of the tree population by the vector state

X = (x, xn, . . . , x1)

and the dynamics described by equations (2), (3) and (4) can be represented asfollows:

X(t+ 1) = AX(t) +Bc(t), (5)

where

A =

1 1

1. . .

10 . . . . . . . . . 0

and B =

−10...01

.

The expression of constraint (1) in terms of the defined matrices is

0 ≤ c(t) ≤ CAX(t) where C =(1 0 . . . . . . 0

). (6)

We will study two different dynamics of prices: Geometric Brownian Mo-tion (GBM) and mean-reverting Ornstein-Uhlenbeck process (O-U). The firsthas been extensively used to model asset prices in financial markets and there-fore represents a natural choice for timber prices. Nevertheless, it is well-knowthat GBM does not capture some behaviors of price movement such as mean-reversion, which is best emulated by the O-U process. Since both processes arewell-known, we simply define them without further detail.

dpt = µptdt+ σptdWt (GBM), (7)

dpt = η(p− pt)dt+ σdWt (O-U), (8)

5

where µ ∈ R is the drift of the GBM, σ > 0 is the constant variance, η > 0is the rate of mean-reversion to an equilibrium p and Wt denotes the Wienerprocess.

In this paper we will work with discrete time. We define two stochasticdifference equations that approximate GBM and O-U:

p(t+ 1) = ξ(t+ 1)p(t) for all t (GBM), (9)

p(t+ 1) = p(1− e−η) + p(t)e−η + ξ(t+ 1) for all t (O-U), (10)

where ξt is a random variable for every t. Following the binomial approxima-tion proposed by Cox, Ross and Rubinstein (CRR) [7], the price evolution (9)converges to (7) by assuming that ξt can only take two discrete values, u withprobability q and d with probability 1− q, with u = 1/d. CRR gives us

q =eµ − du− d

, E[p(t+ 1)|p(t)] = p(t)eµ, for all t, (11)

where E[· | ·] denotes the conditional expectation of a random variable.The dynamics described in (10) represents an autoregressive AR(1) model. It

can be shown ([9]) that the continuous time O-U process (8) is the limiting caseof (10) as ∆t goes to zero. We will often compute the conditional expectation ofthe O-U process, which can be explicitly calculated since a closed-form solutionfor the process is known.

E[p(t+ 1) | p(t)] = p(t)e−η + p(1− e−η), for all t.

Assuming that there is a delay of one period between the harvesting decisionand the moment that the timber is finally sold in the market, the objectivefunction is defined by the expected sum of the instantaneous benefits (p(t +1)c(t)) discounted by the factor δ ∈ (0, 1),

E

[T∑t=0

δt+1p(t+ 1)c(t)

],

where T is the time horizon of the problem. The control c(t) in each periodwill depend only on the current state of the population and price through thedecision function c(t) = πt(p(t),X(t)). A sequence Π = (π1, . . . , πT ) is called apolicy. Of course, a policy is feasible if X(t) and c(t) = πt(p(t),X(t)) satisfy (5)and (6) for every possible value of X(t) and p(t) at every instant t = 1, . . . , T .

We now define the expected discounted benefit of a given policy Π from aparticular initial state as

Q0(p(0),X(0),Π) = E

[T∑t=0

δt+1p(t+ 1)πt(p(t),X(t))

]. (12)

We stress here that, as we assumed that the timber harvested at time t issold in the market at period t + 1, the price is unknown at the moment thedecision is taken.

6

The problem is then to find a feasible policy that maximizes (12),

V0(p(0),X(0)) =

{MaxΠ Q0(p(0),X(0),Π)

s.t. Π is a feasible policy.(13)

Observe that the non-negativity of the state variables x(t) and xa(t) for a =1, . . . , n is assured by (5) and (6).

In the sequel, we will also use the expected discounted benefit from an in-termediate step

Qt′(p(t′),X(t′),Π) = E

[T∑t=t′

δt+1p(t+ 1)πt(p(t),X(t))

]and the corresponding value function

Vt′(p(t′),X(t′)) =

{MaxΠ Qt′(p(t

′),X(t′),Π)

s.t. Π is a feasible policy.(14)

3 Finding optimal policies

Let us start this section by deriving the stochastic dynamic programming equa-tions for our problem. From the definitions of V0 and Q0 it is straightforwardto see that:

V0(p(0),X(0)) = MaxΠ

Q0(p(0),X(0),Π)

= MaxΠ

E[δp(1)c(0) +

T∑t=1

δt+1p(t+ 1)c(t)],

where c(t) stands for πt(p(t),X(t)). Using simple properties of the expectedvalue we get:

V0(p(0),X(0)) = MaxΠ

Ep(1)

[Ep(2),...,p(T+1)

[δp(1)c(0)

+

T∑t=1

δt+1p(t+ 1)c(t)∣∣p(1) = p(1)

]]where we use p(1) to differentiate the random variable price from its realizationp(1) . Observing that the first term does not depend on p(t) for t ≥ 2 or πt fort ≥ 1 we can write,

V0(p(0),X(0)) = Maxπ0

Ep(1)

[δp(1)c(0)

+ Maxπ1,...,πT

Ep(2),...,p(T+1)

[ T∑t=1

δt+1p(t+ 1)c(t)∣∣p(1) = p(1)

]]

7

And finally using (14) we get,

V0(p(0),X(0)) = Maxπ0

Ep(1)

[δp(1)c(0) + δ V1(p(1),X(1))

]We can repeat this process for any other period t, hence the dynamic pro-

gramming equations for the problem for t = 0, . . . , T − 1 are

Vt(p(t),X(t)) =Maxπt

Ep(t+1)[δp(t+ 1)c(t) + δ Vt+1(p(t+ 1),X(t+ 1))|p(t) = p(t)]

s.t. X(t+ 1) = AX(t) +Bc(t),0 ≤ c(t) ≤ CAX(t).

(15)

3.1 Optimal policies for GBM

Before showing rigorously which is the optimal policy, we propose an intuitiverule of thumb to decide when the optimal policy is greedy and when it is not. Itis natural to think that if discounted expected prices in the future are smallerthan the present price, the decision maker should harvest everything availablesince on average prices will drop. Given that in our model there is a delaybetween the harvesting decision and the sell in the market, the comparisonmust be performed between the expected price for the timber obtained todayand the expected price that future harvests may have. If the price process is aGBM, and assuming that the future is just one period ahead, this condition canbe translated by

δ2E[p(t+ 2)|p(t)] ≤ δE[p(t+ 1)|p(t)], t = 1, . . . , T − 1,

δ2p(t)e2µ ≤ δeµ, t = 1, . . . , T − 1,

which is equivalent toδeµ ≤ 1. (16)

The reader has no reason to believe that if condition (16) is satisfied thenthe optimal policy is to harvest everything available. First because we are onlyconsidering expected prices. Secondly, and most important, because we are onlylooking one period ahead: it could be the case that after two or more periodsprices would rise and so harvesting everything available now is not the bestoption.

Let us now move to the formal derivation of conditions for the optimal policyand see if there is any relationship with condition (16).

3.1.1 When is the Greedy policy optimal?

Theorem 1 Consider problem (13) and assume prices evolve according to (9).If condition (16) holds, the greedy policy is optimal.

8

Proof. For the last period t = T it is easy to solve the underlying optimizationproblem: just consume the maximum possible amount, subject to the dynamicconstraints. Denoting by c∗(T ) the optimal solution of the problem of the laststage and assuming prices follow (9), we have

VT (p(T ),X(T )) = Ep(T+1) [δc∗(T )p(T + 1) | p(T ) = p(T )] = δCAX(T )p(T )eµ,(17)

where in the last equality we have used (11).We now proceed by backwards induction on t. Let us assume that the greedy

policy is optimal for every s > t, (t ≤ T ). We claim that it is also optimal fort. We recall the stochastic dynamic programming equation at t,

Vt(p(t),X(t)) =Maxc

Ep(t+1)[δp(t+ 1)c+ δ Vt+1(p(t+ 1), AX(t) +Bc)|p(t) = p(t)]

s.t. 0 ≤ c ≤ CAX(t).(18)

Let us compute explicitly X(t+ 1) in terms of X(t) and c,

X(t) =

x(t)xn(t)xn−1(t)

...

...x1(t)

−→ X(t+ 1) = AX(t) +Bc =

x(t) + xn(t)− cxn−1(t)xn−2(t)

...

...c

Knowing that for s > t the greedy policy is optimal, we can calculate the

coefficient of c in the expression of Vt+1(p(t + 1),X(t + 1)). We know that thefraction of surface x(t) +xn(t)− c+xn−1(t) will be harvested at t+ 1 and everyn steps afterwards, until reaching T . The total benefit delivered by this fractionof surface from t+ 1 to T , actualized to its value at t will be:

(δ2p(t+2)+δn+2p(t+n+2)+· · ·+δkn+2p(t+kn+2))(x(t)+xn(t)−c+xn−1(t)),

where k is the integer such that t + kn + 1 ≤ T and t + kn + 1 > T − n. Inaddition to this, the area of trees that are one year old at t+1 is c. This fractionsurface will be harvested at t+n and every n afterwards, until reaching T . Thetotal benefit delivered by this fraction of surface from t+ 1 to T , actualized tovalue at t will be:

(δn+1p(t+ n+ 1) + · · ·+ δk′n+1p(t+ k′n+ 1))c,

where k′ is such that t + k′n ≤ T and t + k′n > T − n. All the other possiblebenefits do not depend on c and will be represented by a generic function

g(p(t+ 1), . . . , p(T + 1),X(t)).

9

Observe that either k′ = k or k′ = k + 1. In the case that k′ = k, thecoefficient of c at (18) is

δp(t+ 1) + (δn+1p(t+ n+ 1) + · · ·+ δkn+1p(t+ kn+ 1))

−(δ2p(t+ 2) + δn+2p(t+ n+ 2) + · · ·+ δkn+2p(t+ kn+ 2))

= δ(p(t+ 1)− δp(t+ 2)) + δn+1(p(t+ n+ 1)− δp(t+ n+ 2)) + . . .(19)

· · ·+ δkn+1(p(t+ kn+ 1)− δp(t+ kn+ 2))

Hence, using (18), Vt(p(t),X(t)) can be written as

Maxc∈[0,CAX(t)]

E[c(δ[p(t+ 1)− δp(t+ 2)] + δn+1[p(t+ n+ 1)−

δp(t+ n+ 2)] + · · ·+ δkn+1[p(t+ kn+ 1)− δp(t+ kn+ 2)])

+g(p(t+ 1), . . . , p(T + 1),X(t))|p(t) = p(t)]

= Maxc∈[0,CAX(t)]

c[δp(t)(eµ − δe2µ) + δn+1p(t)(e(n+1)µ − δe(n+2)µ) + . . .

· · ·+ δkn+1p(t)(e(kn+1)µ − δe(kn+2)µ)]

+E[g(p(t+ 1), . . . , p(T + 1)X(t))|p(t) = p(t)

]= Maxc∈[0,CAX(t)]

[cp(t)(1− δeµ)(δeµ + δn+1e(n+1)µ + δkn+1e(kn+1)µ)

]+E[g(p(t+ 1), . . . , p(T + 1)X(t))|p(t) = p(t)

].

It is evident that the coefficient affecting c will be positive whenever (16)holds with strict inequality, and the maximum will be attained at c = CAX(t).If condition (16) holds with equality, there is no influence of c in the value ofVt(p(t),X(t)) and we can freely chose the value of c provided that it is feasible.We impose c(t) = CAX(t).

We consider now the case k′ = k + 1. The coefficient of c at (18) is

δ(p(t+ 1)− δp(t+ 2)) + δn+1(p(t+ n+ 1)− δp(t+ n+ 2)) + . . .

· · ·+ δkn+1(p(t+ kn+ 1)− δp(t+ kn+ 2)) + δ(k+1)n+1p(t+ (k + 1)n+ 1)

Observe that the coefficient is the same expression we obtained in (19) plus thepositive term δ(k+1)n+1p(t + (k + 1)n + 1). It is evident that if condition (16)holds, the coefficient of c is positive also in this case, and then the maximumwill be attained at c = CAX(t). As a consequence, the greedy policy is optimalat t.

Surprisingly, the intuitive condition (16) derived with a very simple reasoningis exactly what it takes to make the formal proof of the optimality of the greedypolicy. The economic intuition for the GBM case is that if condition (16) issatisfied then the discounted future price of time t+1 is smaller than the presentprice p(t) for all t and therefore the optimal policy is to harvest everythingavailable.

10

Another immediate consequence of condition (16) is that if the drift of GBMµ is equal to zero then the condition is satisfied for all δ. The intuition behindthis result is that because of (11), µ = 0 implies that the expected value of theprices is simply the price at the present. Therefore, it is better to harvest assoon as timber becomes available to avoid heavier discounts In Figure 1, theshaded area shows the values of δ and µ for which the sufficient condition holds.

d0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

m

0

1

2

3

4

5

6

Figure 1: (δ, µ) values for GBM.

3.1.2 Another optimal policy

When condition (16) does not hold then it is natural to think that the deci-sion maker should postpone the harvest as much as possible since discountedexpected prices are greater than the present price. Hence, before the final timeT , harvesting should be stopped altogether in order to have the maximum sur-face available at time T . However, observe that every land plot harvested andplanted n or more time steps before T , will contain mature tress available forharvesting at time T . Hence, it is convenient to harvest every mature tree attime T − n, since there is enough time for seedlings to mature before reachingT . Repeating this reasoning we can conjecture that the only time steps whenharvesting is allowed are T − kn for k = 0, 1, . . . , bT/nc and that everythingavailable then should be harvested. The proof of this intuitive claim is pre-sented in Theorem 2, but, let us present a toy example before getting into theproof. Let n = 3, S = 6, T = 7 and suppose the initial state is X(0) = (0, 1, 2, 3).If we apply the policy detailed above, then it is only possible to harvest at times

11

1, 4 and 7, and the evolution of the state is0123

→

1230

→

0303

→

3030

→

3300

→

0006

→

0060

→

0600

,

and the corresponding harvests are

c(t) =

3 t = 16 t = 46 t = 70 else.

The value function at t = 0 is

V0(p(0),X(0)) = E[δ2p(2)[x(0) + x3(0) + x2(0)] + δ5p(5)S + δ8p(8)S | p(0)

]= δ2e2µp(0)[x(0) + x3(0) + x2(0)] + δ5e5µp(0)S + δ8e8µp(0)S.

Theorem 2 If condition (16) does not hold, the optimal harvest for any real-ization of the price process is

c(t) =

{CAX(t) if t = T − kn0 else

and the corresponding value function is

for t = T − kn+ 1:

Vt(p(t),X(t)) = Sp(t)

k∑l=1

(δnenµ)l,

and for t = T − (kn+ j) + 1 with j ∈ [1, n− 1]

Vt(p(t),X(t)) = δjejµSp(t)

k∑l=1

(δnenµ)l+δjejµp(t)(x(t)+xn(t)+· · ·+xn−j+1(t)

).

Proof. For t = T , the proof follows directly since c(T ) = CAX(T ) and VT =δeµp(T )(x(T ) + xn(T )) (see (17)).

For the rest of the proof we will use backwards induction. The proof is notdifficult but the calculus is cumbersome. Let us divide the proof into three cases:(a) t = T −kn+1, (b) t = T −kn and (c) t = T − (kn+ j) with j = 2, . . . , n−1.

12

(a) Case t = T − kn+ 1.

Vt(·, ·) = maxc(t)

Ep(t)[δp(t+ 1)c(t) +

δVT−kn+2(·, ·)|p(t) = p(t)]

= maxc(t)

Ep(t)[δp(t+ 1)c(t) +

δVT−((k−1)n+n−1)+1(·, ·)|p(t) = p(t)]

= maxc(t)

Ep(t)[δp(t+ 1)c(t) + δδn−1e(n−1)µSp(t+ 1)

k−1∑l=1

(δnenµ)l

+δδn−1e(n−1)µp(t+ 1)(x(t+ 1) + xn(t+ 1) + . . .

· · ·+ x2(t+ 1))|p(t) = p(t)].

The last term of the rhs can be written in terms of X(t) and c(t) as:

δne(n−1)µp(t+ 1)(x(t) + xn(t)− c(t) + xn−1(t) + · · ·+ x1(t)) =

δne(n−1)µp(t+ 1)(S − c(t)),

and hence we obtain

Vt(·, ·) = maxc(t)

Ep(t)[δp(t+ 1)(1− δn−1e(n−1)µ)c(t)

+δne(n−1)µSp(t+ 1)

k−1∑l=1

(δnenµ)l +

δne(n−1)µSp(t+ 1)|p(t) = p(t)]

= maxc(t)

[δeµp(t)(1− δn−1e(n−1)µ)c(t)

+δnenµSp(t)

k−1∑l=1

(δnenµ)l + δnenµSp(t)]

= maxc(t)

[δeµp(t)(1− δn−1e(n−1)µ)c(t) + Sp(t)

k∑l=1

(δnenµ)l].

Now it is easy to see that the coefficient of c(t) is

δeµp(t)(1− δeµ)

n−2∑l=0

(δeµ)l < 0

whenever condition (16) is not satisfied, which implies that the optimal harvestis c∗(t) = 0 and finally

Vt(·, ·) = Sp(t)

k∑l=1

(δnenµ)l.

13

The proof of cases (b) and (c) is completely analogous and we leave thedetails to the reader.

We would like to point out that the characterization of the optimal policy iscomplete. If condition (16) is satisfied, then it is optimal to harvest everythingavailable at every t, which corresponds to the greedy policy. Otherwise, Theorem2 tells us that the optimal policy consists in harvesting only at periods t =T −kn, k = 0, 1, . . .. Therefore, optimal decisions for the whole interval 0, . . . , Tcan be taken at t = 0 by checking if condition (16) is satisfied or not.

3.1.3 Numerical experiments

To close this section we provide some numerical computation comparing thegreedy policy with the policy described in Theorem 2, which we will refer toas transient greedy (TG). When condition (16) is not satisfied, we know fromTheorem 2 that the greedy policy is suboptimal. Table 1 shows statistics ofthe present value of both policies for δ = 0.8, µ = 0.3 and S = 4, with initialprice p(0) = 1. Note that in this case δeµ = 1.08 and therefore condition (16) isnot satisfied. For some pairs n and T we compute the optimal value for everypossible initial state of the forest and calculate the average under greedy andTG policies. We also report the percentage by which the average TG is greaterthen the greedy. The results are described in columns 2, 3 and 4 of Table 1.

To further illustrate the differences between the two policies, we comparetheir performance when the initial state represents the youngest possible forest,that is, X(0) = (0, 0, . . . , S). The comparison is interesting from a managerialviewpoint: if a manager starts with a young forest he/she wants to know whatare the implications of adopting a greedy policy instead of a transient greedyone. The percentage change is shown on column 5 of Table 1. The differencesin revenue are almost 50% for n = 6 and T = 40. For n = 4 and T = 31 the twopolicies coincide, which can be immediately checked by noticing the structureof the TG policy given by Theorem 2. The fourth column of Table 1 shows thatthe TG policy is clearly different from the greedy policy, yielding up to 22%larger average percentage returns.

n, T Avg greedy Avg TG GAP GAP for (0, 0, . . . , S)

n = 2, T = 10 37.0811 37.9458 2% 8%n = 3, T = 10 24.2735 26.0861 7% 16%n = 4, T = 31 141.3661 161.6831 14% 0 %n = 5, T = 32 125.9261 145.9399 15% 26%n = 6, T = 40 198.2520 242.0839 22% 47%

Table 1: Comparison between the greedy and the transient greedy policies.

14

3.2 Optimal policies for O-U

Similarly to what we did for GBM, we can derive intuitive conditions underwhich the optimal policy would be greedy:

δ2E[p(t+ 2)|p(t)] ≤ δE[p(t+ 1)|p(t)],δ2[p(t)e−2η + p(1− e−2η)] ≤ δ[p(t)e−η + p(1− e−η)],

which is equivalent to

p(t)

p≥ 1− 1− δ

e−η(1− δe−η). (20)

It remains to be seen if condition (20) can be proven to lead to greedy policies.Let the right hand side of (20) be denoted by r, we note that (20) suggeststhat if p(t) is greater than the threshold rp then one should harvest everythingavailable. We point out the threshold is smaller than p, hence the statementmay not seem obvious. In Theorem 4 we prove the validity of this assertion.Furthermore, we provide some numerical experiments where the threshold isused as a reservation price, i.e., we harvest everything available if p(t) is aboverp and we postpone the harvest if it is below. The results obtained are verysatisfactory.

Note that, unlike (16) condition (20) depends on t. We derive now a coarsersufficient condition independent of t, observing that if the right hand side aboveis negative then the inequality is automatically satisfied since prices are alwayspositive. We have

1− 1− δe−η(1− δe−η)

≤ 0⇔

e−η(1− δe−η) ≤ 1− δ ⇔δ(1− e−2η) ≤ 1− e−η ⇔

δ ≤ 1

(1 + e−η). (21)

If the condition above is satisfied, then it is always optimal to harvest everythingthat is available. This is intuitive since smaller values of δ imply that futureutilities will be more heavily discounted and it is optimal to obtain the benefitas soon as possible.

Another interesting interpretation can be derived by observing that a largemean reversion speed η implies that future prices will have smaller stationary(long-term) variance since

VaR[pt] =σ2

2η.

Therefore, prices will not exhibit significant variation, remaining close to p, andwith constant prices it is always optimal to harvest everything available [20].We see from (21) that large values of η imply that the inequality is satisfied for

15

a wide range of δ and the optimal policy will be to harvest everything availableas well.

Finally, observe that the right hand side of the inequality above is alwaysgreater than 1/2, meaning that for values of δ smaller than 1/2, it is optimal toharvest everything available regardless of the value of η. In Figure 2 the shadedarea shows the values of δ and η for which the sufficient condition holds.

d0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

h

0

1

2

3

4

5

6

7

Figure 2: (δ, η) values for O-U.

When the sufficient condition (21) holds, the optimal policy can be charac-terized.

Theorem 3 Consider problem (13) and assume prices evolve according to (10).If condition (21) holds, the greedy policy is optimal.

Proof. The proof follows the lines of Theorem 1. For the last period t = T ,the optimal solution is to harvest everything available and, hence, the valuefunction is

VT (p(T ),X(T )) = δCAX(T )[e−ηp(T ) + (1− e−η)p(T )].

For the rest of the proof we proceed by backwards induction on t. Let usassume that the greedy policy is optimal for s > t, we will show that it is alsooptimal for t. Following the same steps as in Theorem 1, we can see that thecoefficient affecting c in the dynamic programming equation (15) is equal to orgreater than

16

A = δp(t+ 1) + (δn+1p(t+ n+ 1) + · · ·+ δt+kn+2p(t+ kn+ 1))

−(δ2p(t+ 2) + δn+2p(t+ n+ 2) + · · ·+ δt+kn+2p(t+ kn+ 2))

= δ(p(t+ 1)− δp(t+ 2)) + δn+1(p(t+ n+ 1)− δp(t+ n+ 2)) + . . .

· · ·+ δkn+1(p(t+ kn+ 1)− δp(t+ kn+ 2))

Again following the procedure of the proof of Theorem 1, the coefficientaffecting c will be E[A|p(t) = p(t)]. Nevertheless, since all the terms of Ainvolve differences of the form p(t+ j)− δp(t+ j + 1), let us first pre-computethese quantities.

E[p(t+ j)− δp(t+ j + 1)|p(t) = p(t)] =

e−jηp(t) + (1− e−jη)p− [e−(j+1)ηp(t) + (1− e−(j+1)η)p] =

p(t)e−jη(1− δe−η) + p[1− δ − e−jη(1− δe−η)] ≥ 0

⇐⇒ p(t)

p≥ 1− 1− δ

e−jη(1− δe−η)(22)

A simple computation shows that

1− 1− δe−η(1− δe−η)

≥ 1− 1− δe−jη(1− δe−η)

,

and so, condition (21) implies that the right hand side of inequality (22) issmaller than zero, implying that (22) is automatically satisfied. Having disposedof this preliminary step, it is now straightforward to conclude that E[A|p(t) =p(t)] ≥ 0. Thus, the optimal solution for all t is to harvesting everythingavailable if condition (21) holds. This concludes the proof.

Although Theorem 3 provides a sufficient condition for optimality of thegreedy policy under O-U prices, we believe it is not totally satisfactory because,unlike condition (16), it does not include the drift of the process. Since thedrift of the O-U process is not constant and depends on the current value of theprocess, an analogous of condition (16) should include p(t), p and the discountfactor δ. The following theorem claims that if (20) holds at t, then it is optimalto harvest everything available at that particular time step. We cannot affirmthat the greedy policy is optimal from t on, but we claim that even if we do notknow what the future will be, we are able to assure that harvesting everythingavailable at t is optimal if (20) holds for p(t). The condition is sufficient butnot necessary: if (20) do not hold, we are not able to discard the greedy policy.

Theorem 4 If condition (20) holds at time t, then c(t) = CAX(t) is optimal.

The proof, being quite technical is presented in the Appendix.

17

3.2.1 Numerical experiments

We use the recombining binomial tree proposed by Bastian-Pinto et al. [3] tomodel the mean reversion stochastic process followed by the prices 5. Using thisapproximation, we numerically obtain the optimal trajectory for every initialstate.

We also implement an alternative harvesting policy, using the threshold rpderived in (20) as a reservation price: we harvest everything available if p(t) isabove rp and nothing if it is below. Theorem 4 assures that if (20) holds, then itis optimal to harvest everything available but we have no theoretical justificationfor postponing the harvest if it does not. We call this the alternative policy Π0

and compare the present value it provides with that of the optimal policy.Table 2 shows statistics of the present value of both policies for δ = 0.9, η =

0.8, σ = 0.27 and S = 4, with initial price p(0) = 1.2 and p = 1. Note that in thiscase 1/(1 + e−η) = 0.69 and therefore condition (21) is not satisfied. For everypair n and T we compute the average over all initial states of the optimal presentvalue and the present value obtained with the alternative policy Π0. We reportthese values together with the relative optimality gap (1−Avg Π0/Avg optimal)that is never above 1.5%. As in the numerical experiments presented in §3.1.3,we also report the percentage change for the particular state (0, . . . , 0, S).

n, T Avg optimal Avg Π0 GAP GAP for (0, . . . , 0, S)

n = 2, T = 10 11.90 11.98 1.0% 0.51%n = 3, T = 10 8.23 8.11 1.4% 1.8%n = 4, T = 30 9.01 8.93 0.80% 0.24%n = 5, T = 30 7.21 7.18 0.54% 1.5%n = 6, T = 40 6.21 6.19 0.37% 0.006%

Table 2: Comparison between the optimal and the alternative Π0 policies.

4 Conclusions

We study a harvest scheduling problem under price uncertainty. At every perioda decision maker must decide the amount of timber that is going to be harvestedand subsequently sold in the following period at a random price. In this workwe considered the two most common stochastic processes used to model timberprices: the geometric Brownian motion (GBM) and the Ornstein-Uhlenbeckprocess (O-U).

In the deterministic case, when prices are constant, Rapaport et al. [20]showed that the greedy policy is always optimal. We adopted the model de-scribed in Rapaport et al., in which harvest can only occur at a maturity agen and beyond maturity. The most important contribution of our work is the

5In [3], the authors prove that the logarithms of prices following O-U can be generated bythe recombining binomial tree they propose.

18

characterization of the optimal policy under stochastic prices. Put it simply,when prices are random it is unclear if one should harvest everything availableat once or postpone the decision in the hope that prices will rise. The questionswe posed ourselves were the following: given a stochastic process, is it truethat the greedy policy is always optimal? If not, are there conditions on theparameters of the process that assure optimality of the greedy policy?

When prices follow GBM, we obtained a closed expression (16) characterizingoptimality of the greedy policy that depends only on the drift of the processand on the discount factor. Furthermore, when the condition is not satisfiedwe also characterized the optimal policy and proved it is not greedy, showing aclear depart from the deterministic case.

When prices follow O-U we first found a condition (21) on the mean reversionspeed of the process and on the discount factor that guarantees optimality of thegreedy policy. Nevertheless, such condition has some shortcomings: first, it failsto characterize the optimal policy for a wide region of the state space definedby the speed of reversion and the discount factor. In addition, condition (21)does not depend on the price of timber at every time period t, which is a crucialinformation in a mean-reverting type of process. By carefully analyzing theharvesting decisions and their consequences in subsequent periods, we were ableto obtain a new theoretical condition which implies optimality of the greedypolicy. It is interesting to note that such condition depends on timber priceat every period t and can be interpreted as a reservation price. When thecondition is not satisfied, we do not know what is the optimal policy but ournumerical experiments showed that a policy based on such reservation pricesachieves benefits within 2% of the optimal solution.

Although several simplifications were made mainly for tractability, we be-lieve our results provide important managerial insights in the harvest schedulingproblem. The majority of works that incorporate uncertain elements presentnumerical results that provide intuition about such problems but do not offercomplete descriptions of the optimal policy. Our theoretical results give a com-plete understanding of the GBM case and characterize the optimal policy for theO-U case by providing a closed expression for a reservation price. In addition,there are very few benchmark problems in stochastic harvest scheduling and wehope our results will help future applied research in the field. For instance, thecalibration of more complex models could benefit from our theoretical resultsin the sense that the optimal policy obtained numerically for simpler versionsof some model should be greedy if the appropriate condition on the parametersis satisfied.

Future work includes a more complex forest growth model. In this article itis assumed that prices only are stochastic, treating stand growth as determin-istic. In fact, the stand growth dynamics are very simply modeled, consideringonly the age of the trees and neglecting natural mortality or the occurrence ofrare catastrophic events that may affect the tree population. In addition, it isassumed that there is no natural regeneration, and that every bit of liberatedsurface is planted immediately with seedlings of the same species. To take intoaccount natural mortality a very simple modification in the matrix that defines

19

the growth dynamics is needed.We could also work with a stand model described in terms of size classes,

instead of age classes, in order to give a representation of the population thatcould be more useful to estimate the volume of timber contained in the stand.Again modifications in the matrix could accommodate such changes. Althoughsuch modifications are simple, the mathematical proofs presented here need athorough revision and we leave this extension for future work.

Another interesting avenue of research is the incorporation of risk measuresin the objective function. The maximization of expected value does not pro-tect the decision maker against the possibility of extreme events such as low-probability scenario with extremely low prices. There are several risk measuresavailable in the literature such as the Value-at-Risk, Conditional Value-at-Risk,Mean Semi-deviation and others and theoretical advances have made possiblethe incorporation of such measures within the optimization context. It wouldbe interesting to analyze the structure of the optimal policy if the objectivefunction was for instance, a weighted combination of the expected value andsome risk measure.

5 Appendix

Before presenting the proof of Theorem 4 we give a technical proposition.

Proposition 1 Let rt,j = 1 − 1−δje−tη(1−δje−jη) for t, j ∈ N with δ ∈ (0, 1) and

η > 0. Then rt,j ≥ rt′,j′ for all t ≥ t′ and j ≥ j′, with equality iff t = t′ andj = j′. Consequently r1,1 ≥ rt,j for all t, j ∈ N.

Proof.

rt,j ≥ rt′,j′

1− 1− δj

e−tη(1− δje−jη)≥ 1− 1− δj′

e−t′η(1− δj′e−j′η)

⇐⇒ 1− δj

(1− δje−jη)≤ e(t′−t)η 1− δj′

(1− δj′e−j′η)

As t′ ≥ t we know that e(t′−t)η ≥ 1 (with equality if only if t = t′). As aresult, to prove our claim it is enough to show that

20

1− δj

(1− δje−jη)≤ 1− δj′

(1− δj′e−j′η)

⇐⇒ 1− δ(1− δe−η)

∑j−1i=0 δ

i∑j−1i=0 (δe−η)i

≤ 1− δ(1− δe−η)

∑j′−1k=0 δ

k∑j′−1k=0 (δe−η)k

⇐⇒∑j−1i=0 δ

i∑j−1i=0 (δe−η)i

≤∑j−1k=0 δ

k +∑j′−1k=j δ

k∑j−1k=0(δe−η)k +

∑j′−1k=j (δe−η)k

⇐⇒j−1∑i=0

δi

j−1∑k=0

(δe−η)k +

j′−1∑k=j

(δe−η)k

≤

j−1∑k=0

δk +

j′−1∑k=j

δk

j−1∑i=0

(δe−η)i

⇐⇒j−1∑i=0

δij′−1∑k=j

(δe−η)k ≤j′−1∑k=j

δkj−1∑i=0

(δe−η)i

⇐⇒ 0 ≤j′−1∑k=j

j−1∑i=0

δi+k(e−iη − e−kη)

where the last inequality holds because i < k implies e−iη − e−kη > 0. Theequality holds if and only if j = j′.

Proof of Theorem 4. We present a proof under the assumption that at every stepwe either harvest nothing at all or everything available. The result is still validwithout this assumption, but we decided not to present the general proof for tworeasons: (i) the general proof follows the same lines that the one presented here,but the calculi are more cumbersome; (ii) due to the linearity of the forestrymodel, this assumption is equivalent to requiring that the coefficient of c in (15)is never zero, but having a zero coefficient is an event with zero probability.

Let us now proceed with the proof. Thanks to Theorem 3 we know thatat the last step it is optimal to harvest everything available, regardless of theprice, so we only need to prove the result for t < T .

The main idea of the proof is to consider the role played by c in all thepossible expressions of Vt(·, ·). Of course, characterizing completely every pos-sible expression of Vt(·, ·) is a titanic task, but we will only be interested in thecoefficient affecting c.

As before, we express X(t+ 1) and state equation (15) in terms of X(t) andc as follows.

21

X(t) =

x(t)xn(t)xn−1(t)

...

...x1(t)

−→ X(t+ 1) = AX(t) +Bc =

x(t) + xn(t)− cxn−1(t)xn−2(t)

...

...c

,

Vt(p(t),X(t)) = maxc

E[δp(t+ 1)c+ δVt+1(p(t+ 1),X(t+ 1))|p(t)].

Let us consider first the case where t > T − n. This case is easy to studybecause the one year old trees at t + 1 (covering an area equal to c) will notreach maturity before the end of the interval, i.e., they will never be availablefor harvesting. Hence, to characterize the coefficient of c in Vt+1(·, ·) we onlyneed to consider the harvest of the over mature trees. Theorem 3 tells us that itis optimal to harvest everything available at T , hence, if the over mature treesare not harvested before T , they will be harvested then.

Let us write as t + j0 the time step where the first harvest after t occurs,with 1 ≤ j0 ≤ T − t. The state will be

X(t+ j0) =

x+ xn(t) + · · ·+ xn−j0+1(t)− cxn−j0(t)xn−j0−1(t)

...x1(t)c0...0

.

And we write (15) as

Vt(p(t),X(t)) = maxc

E[δp(t+ 1)c+ δj0+1p(t+ j0 + 1)(x+ xn(t) + . . .

· · ·+ xn−j0(t)− c) + δj0+1Vt+j0+1(p(t+ j0 + 1),X(t+ j0 + 1))|p(t)],

22

where X(t+ j0 + 1) is

X(t+ j0 + 1) =

0xn−j0−1(t)

...x1(t)c0...0

x+ xn(t) + · · ·+ xn−j0(t)− c

. (23)

We do not know the complete expression of Vt+j0+1 but as the term c willnot be available for harvesting before the end of the interval it is evident thatit does not have terms comprising c. Hence, the coefficient affecting c in (15)will be simply,

E[δp(t+ 1)− δj0+1p(t+ j0 + 1)|p(t)]. (24)

Let us now consider the case t ≤ T − n. It may happen that after t nothingelse is harvested in the next (n− 1) steps and hence at t+n the state would be

X(t+ n) =

S − cc0......0

At t + n we either harvest S or zero and the influence of c extinguishes. Wedo not know the complete expression of Vt+1(·, ·) but we do know that thecoefficient of c in (15) will be simply

E[δp(t+ 1)|p(t)].

It remains to study the case where there is harvest at time t + j0 with1 ≤ j0 < n. In this case, we have c(t+ j0) = x+ xn(t) + · · ·+ xn−j0(t)− c andthe state at t+ j0 + 1 is of the form (23).

Equation (15) is now,

Vt(p(t),X(t)) = maxc

E[(δp(t+ 1)− δj0+1p(t+ j0 + 1))c (25)

+δj0+1p(t+ j0 + 1)(x(t) + xn(t) + · · ·+ xn−j0(t)− c)+δj0+1Vt+j0+1(·, ·)|p(t)]

23

We are left with the task of characterizing the coefficient of c in Vt+j0+1(·, ·).From now on, we will omit the state variables and will represent the state as

X(t+ j0 + 1) =

0∗...∗∗+ c∗...∗∗ − c

(26)

The coefficient of c will not be affected by the actions taken until the term“∗+ c” is available for harvesting at time t+ n. The state will be

X(t+ n) =

∗∗+ c∗...∗∗ − c

...∗

.

From t + n on we have to consider two cases: either t > T − n − j0 (i.e.,T < t+j0 +n) or t ≤ T −n−j0. The time step t+j0 +n is an important bench-mark as it is the moment when the term “∗−c” becomes available for harvesting.

Let us first consider the case t > T − n − j0. This case is easy to studybecause we know that the term “∗ − c” will not be available for harvesting,and then we only have to take care of the harvest of the area covered withover mature trees: “∗ + c”. We write t + m1 (n ≤ m1 ≤ T − t) the time stepwhen the first harvest after t + j0 takes place. We do not know the completeexpression of Vt+j0+1(·, ·) in (25) but we know that the only term comprising cwill be E[δm1+1p(t+m1 + 1)|p(t)]. Hence, the coefficient affecting c in (15) willbe simply the expression in (24) plus the added term:

E[δp(t+ 1)− δj0+1p(t+ j0 + 1) + δm1+1p(t+m1 + 1)|p(t)].

We consider now the case t ≤ T − j0 − n. It may happen that nothing isharvested until the term “∗− c” is also available for harvesting, at that moment

24

the state would be

X(t+ j0 + n) =

∗+ c∗ − c∗......∗

Observe that if we reach such a state, the influence of c extinguishes as CAX(·)does not depend on c. Hence, we obtain that there are no terms comprising cin Vt+j0+1 and the coefficient of c in (15) will be simply the one stated in (24).

It may also happen that there is harvest at t + m1 with n ≤ m1 < j0 + n.Then, the term E[δt+m1+1p(t + m1 + 1)|p(t)] is added to the coefficient of cexpressed in (24) and we are left with a state of the form

X(t+m1 + 1) =

0∗...∗ − c∗...∗∗+ c

.

After the harvest of “∗ + c”, the coefficient of c will not be affected by theactions taken until the term “∗− c” is available for harvesting; the state at thattime would be in this case

X(t+ j0 + n) =

∗∗ − c∗...∗∗+ c

...∗

.

From t+ j0 + n on, we have to divide again the study into two cases: eithert > T −m1 − n (i.e., T < t+m1 + n) or t ≤ T −m1 − n.

Repeating the previous study we know that in the former case the end ofthe period will be reached before “∗+c” is available for harvesting and only onemore term will be added to the coefficient of c. This term will be of the form

25

E[−δm1+j1p(t+m1 + j1)|p(t)], where t+m1 + j1 is the time step when “∗ − c”is harvested (with 1 ≤ j1 < n). The coefficient of c in (15) will be

E[(δp(t+ 1)− δj0+1p(t+ j0 + 1)

)+(

δm1+1p(t+m1 + 1)− δm1+j1+1p(t+m1 + j1 + 1))|p(t)].

In the latter case, it may happen that nothing is harvested until the term“∗+ c” is also available for harvesting, at that moment the state would be

X(t+m1 + n) =

∗ − c∗+ c∗......∗

Observe that if we reach such a state, the influence of c extinguishes as CAX(·)does not depend on c. In this case the coefficient of c in (15) will be

E[(δp(t+ 1)− δj0+1p(t+ j0 + 1)) + δm1+1p(t+m1 + 1)|p(t)].

The other case that we have to consider is that “∗ − c” is harvested att+m1 +j1, with 1 ≤ j1 < n. Then, the term E[−δm1+j1+1p(t+m1 +j1 +1)|p(t)]is added to the coefficient of c and we are left with a state of the form:

X(t+m1 + j1 + 1) =

0∗...∗+ c∗...∗∗ − c

.

Observe that the state at t+m1 + j1 + 1 is of the same form that the stateat t+ j0 + 1 (see (26)) and the same reasoning can be applied over and over.

In each completed cycle first the term “∗ + c” is harvested, the index mi

is generated, with mi ≥ mi−1 + n, and the term E[δmi+1p(t + mi + 1)|p(t)]is added to the coefficient of c. Second, if t ≤ T − mi − n then the term“∗ − c” is harvested, the index ji is generated, with 1 ≤ ji < n and the termE[−δmi+ji+1p(t + mi + ji + 1)|p(t)] is added to the coefficient of c. If t ≤T −mi − ji − n a new cycle begins. The algorithm ends as soon as one of the

26

following situations occurs. Either we reach a state of the form

X(t+mi + n) =

∗ − c∗+ c

...

...

...

or X(t+mi + ji + n) =

∗+ c∗ − c

...

...

...

and the influence of c extinguishes as CAX(·) is independent of c. Or one ofthe following conditions holds t > T − mi − n or t > T − mi − ji − n. Aslimi→∞mi = ∞ and T is finite, the algorithm must end in a finite number ofsteps.

For each completed cycle two terms are added to the coefficient of c in (15):

E[(δmi+1p(t+mi + 1)− δmi+ji+1p(t+mi + ji + 1)

)|p(t)]. (27)

Definingm0 = 0, we can affirm that the coefficient of c is a finite sum of termslike the one above, and possibly one positive term of the form E[δmi+1p(t+mi+1)|p(t)]. The proof is completed by showing that the coefficient of c is positive.To this end, we study the sign of expression (27) and its relation with condition(20),

0 ≤ E[(δmi+1p(t+mi + 1)− δmi+ji+1p(t+mi + ji + 1)

)|p(t)]

⇐⇒0 ≤ E[p(t+mi + 1)− δjip(t+mi + ji + 1)|p(t)]

= e−(mi+1)ηp(t) + (1− e−(mi+1)η)(1− p)−δji [e−(mi+ji+1)ηp(t) + (1− e−(mi+ji+1)η)(1− p)]

= e−(mi+1)ηp(t)(1− δjie−jiη)

+ p[1− δji − e−(mi+1)η(1− δjie−jiη)]

⇐⇒p(t)

p≥ 1− 1− δji

e−(mi+1)η(1− δjie−jiη)

But the right hand side above is exactly rmi+1,j of Proposition 1. Thisproposition together with Condition (20) imply

p(t)

p≥ r1,1 ≥ rmi+1,j ,

which proves that (27) is non-negative. Furthermore, we know that the secondinequality holds strictly unless mi = 0 and j = 1. Hence, the positivity of the

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coefficient of c in (15) follows for every case, except only for the case where itconsists exclusively of one term of the form (27) with mi = 0 and j = 1, i.e.,

E[(δp(t+ 1)− δ2p(t+ 2)

)|p(t)] (28)

From the previous calculus it is direct to see that (28) is positive if (20)holds strictly. If, on the contrary, (20) holds with equality, there is no influenceof c in the value of Vt(p(t),X(t)) and we can freely chose the value of c providedthat it is feasible. We impose c(t) = CAX(t), which concludes the proof.

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