ENTRY AND EXIT DECISIONS PROBLEM: A SURVEY

In this survey, the entry-exit decisions problem is studied. In its simplest version, the problem concerns to the investment and disinvestment decisions process faced by a firm, when its output price is considered being stochastic. This is essentially the problem faced by a firm exploiting natural resources (i.e. oil, copper, etc.) whose output prices are daily traded in the commodities markets.

ENTRY AND EXIT DECISIONS

PROBLEM: A SURVEY

Andrea Girometti∗Master in Advanced Studies in Finance University of Z¨ urich - ETH Z¨ urich andreagirometti@yahoo.it

by Dixit in 1989 and it is based on the contingent claim theory Inparticular, the values of a firm in both activity and inactivity statesare determined by using the theory of real options pricing Thanks to

the so-called value-matching and smooth-pasting conditions, it is

pos-sible to interlink these two firm’s values and, therefore, to determine apair of trigger prices, giving an optimal decision policy The optimalpolicy fixes the levels of the output prices at which it is economicallyconvenient either to start the production or to abandon the market Inaddition, some interesting extensions of the basic model are presentedand, finally, the two most recent approaches are explained The firstone is based on the ”mark-up” concept, the second one is based onthe optimal stopping time theory

∗ The author is grateful to Ente Luigi Einaudi for the support He is also grateful to

Paolo Casini, Paolo Verzella and Antonio Sciarretta for useful discussions.

1 Introduction

In this survey, the entry-exit decisions problem is studied In its simplestversion, the problem concerns to the investment and disinvestment decisionsprocess faced by a firm, when its output price is considered being stochastic.This is essentially the problem faced by a firm exploiting natural resources(i.e oil, copper, etc.) whose output prices are daily traded in the commodi-ties markets

As a general framework, a single firm having access to a single goodproduction opportunity is considered If{Z t } t ≥0is a stochastic process taking

discrete values 0, 1 and indicating the current firm’s state, at time t such a monopolistic firm is supposed to be either inactive Z t = 0 or already active

Z t = 1 In the inactive state there is no production at all, and the firm iswaiting the best conditions to enter in the market and to produce If the firm

is already active, it has full capacity utilization, the resource is infinite and

it can abandon the market if its profitability is not satisfying Suppose that

the firm, changing its state Z t, switches production on or off instantaneously

Switching production on, the firm can invest a lump-sum cost K I in order tostart the production and, in the opposite case, the firm has to pay a lump-

sum cost K E in order to exit to the market It is assumed that K I + K E > 0,

otherwise a firm could have an infinite profit switching continuously The

firm’s activity implies also a variable production cost w o per each unit ofoutput flow This is the operating cost Moreover, it is supposed that the

firm has to pay again the lump-sum entry cost K I if it exits and wants tore-entry later All these costs are supposed constant over time The riskless

cost of capital, at which all values will be discounted, is r and it is constant.

The uncertainty comes uniquely from the market price of the output Thisprice is represented by the stochastic process {P t } t ≥0, and it is assumed that

it is driven by a geometric Brownian motion1:

where α and σ are constant and strictly positive, and {B t } t ≥0 is a standard

Brownian motion The output price represents the profit flow per unit of

production, and its expected value grows at the rate α2 We will identify these

hypotheses as standard assumptions Later, some of them will be relaxed.

1The case of a geometric Brownian motion is the simplest case because of the existence

and uniqueness of a solution for the stochastic differential equation (1) However, the prices of commodities do not seem to be well described by such a process In fact, it is known that, for example, the oil price seems to be better represented by a mean-reverting process Obviously, this complicates much more the analysis and, for the moment, nobody should have presented any interesting result.

2As we can see in Dixit and Pindyck [DP], an interesting interpretation of the drift can

The aim of a firm is to determine, jointly, optimal bounds (P L , P H) Thispair represents the trigger prices for entry and exit into the market At a

price level P H it begins to be profitable for a firm starting the production If

the output price is lower than P L, producing is not profitable anymore andthe firm leaves the market

Various approaches to the problem and several possible extensions arepresented in this survey All these methods are alternative to the standardcapital budgeting process based on the NPV rule The inadequacy of thisapproach is widely acknowledged, because of total neglect of the stochasticnature of the output prices In general, the real option valuation theoryseems to be more suitable and more useful

The first and simplest model was pioneered by Mossin [Mo] in 1968, butthe first formal discussion on the entry decisions problem was proposed byMcDonald and Siegel [MS] in 1985 The combined problem concerning theentry and the exit was faced by Brennan and Schwartz [BS] in 1985 Theyapplied, for the fist time, the well known options pricing theory developed byBlack, Merton and Scholes in 1973 in order to evaluate active and inactivefirms, and they defined the concepts of option to enter and option to abandon

as part of the firm’s value A formal and complete discussion was presented byDixit [D2] in 1989 In particular, he focused on entry and exit trigger prices

as fundamental indicators for firm’s decision policies Moreover, he madecomparisons between the standard Marshallian theory and the new optiontheory-based approach, introducing the hysteresis concept, and he proposed

a numerical analysis with a comparative statics analysis of the results hefound3 During the ’90s, many authors developed the Dixit’s basic modelconsidering various possible extensions like adding taxes or an investment’slag, considering a restricted number of switches or the possibility of laying-up

or scrapping the production Other authors discussed the possibility and theeffects of an interest rate risk or a currency rate risk, or focused on studying

be done According to the CAPM model, the appropriate risk-adjusted discount rate for the firm’s cash-flows should be:

µ = r + βσρpm,

where β is the market price of the risk and ρ pm is the correlation between the output

price and the market portfolio µ is the total risk-adjusted expected rate of return that investors would require if they are to own the project α is viewed as the expected capital gain Assuming α < µ, the difference δ = µ − α represents a kind of dividend In case of

a storable commodity like oil, copper etc., δ is called net marginal convenience yield from

storage and represents the benefit coming from last stored unit This benefit is given, for

example, by the possibility of smoothing production.

3These economic results will not be presented in this survey The interest reader can

refer to the Dixit’s paper [D2].

the economic equilibrium between firms using the Dixit decision’s rules (seerespectively, for example, Ingersoll and Ross [IR], Kogut and Kulatilaka [KK],Leahy [Le]) Gauthier [GL] applied the theory of exotic options (in particular

parisienne options) in order to study the case of the delay (always) existing

during the capital budgeting process However, a definitive and rigoroustreatment of the problem was proposed by Brekke and Øksendal in 1994[BØ] They analyzed the entry-exit decisions problem applying both theoption pricing theory and the dynamic programming theory, gave a formalproof of the existence of a solution, and extended the classical approachconsidering the case of a finite resource In 2001 Sødal [Sø] proposed a totallynew approach, based on the ”mark-up” concept Finally, in 2003, Chesneyand Hamza [CH] proposed a probabilistic approach

The survey is organized as follows In section 2 the basic model proposed

by Dixit will be examined In sections 3, 4, 5 the most interesting extensions

of the basic model are presented They concern respectively the presence of

an investment’s lag, of a restricted number of possible switches, of the bility of either laying-up or scrapping the project Section 6 will be devoted

possi-to the problem viewed as a sequential optimal spossi-topping problem, as Brekkeand Øksendal did, and the case of finite resource will be introduced Finally,sections 7 and 8 will present the new approaches based on the mark-up con-cept developed by Sødal and on probabilistic tools proposed by Chesney andHamza

Through this paragraph we will refer to the fundamental Dixit’s paper [D2]

Consider the standard assumptions At level price P t, the firm could be either

active Z t = 1 or inactive Z t= 0 Then, the first step is determining the firm’svalue in both states

In state (P t , 0), suppose that the inactive firm has an expected net present value V0(P t) Such a firm can observe the current output price and then, fol-lowing optimal policies4, decides whether to continue being inactive or toenter in to the market Assume, for the moment, that this decision is irre-

versible The firm has simply an option to invest and its value is completely

represented by the value of this option If the option is exercised, the firmstarts the production

Analogously, in state (P t , 1), the active firm has an expected net present value V1(P t) In this case, the firm decides whether to continue being active

or to exit Suppose that this decision is again irreversible In this case,

4In general, the firm maximizes its expected net present value.

the firm’s value is given by the value of the current profit and the value of

an option to abandon If this option is exercised, the firm goes back to the

inactive state

Viewed as options to enter and to abandon, the two expected net present

values V1(P t ) and V0(P t) can be determined by using the contingent claimanalysis Therefore, the market must supposed to be complete, i.e everytraded asset is supposed to be spanned by other assets in the economy Inparticular, both the values of the active and inactive firms are supposed to bepositively correlated with either a traded asset or a basket of traded assets,

in order to make possible the replication the firms values

Once the two values V1(P t ) and V0(P t) are determined, it is possible to

determine the two trigger prices (P L , P H ) considering the so-called matching conditions and the smooth-pasting conditions, as we will see in the

value-following sub-section 2.3

Consider an inactive firm having only one possibility to enter in its market.Intuition suggests that such firm finds optimal to remain inactive as long as

the output price is lower a certain threshold P H and it will invest as soon as

the price reaches P H Therefore, over the range (0, P H) the inactive firm’sinvestment opportunity, i.e the value of the inactive firm, is equivalent to

a perpetual call option, where the strike is the entry cost K I The decision

to invest corresponds to the decision of exercising the option Therefore, inorder to obtain the inactive firm’s value, it can be applied the contingentclaim theory, as McDonald and Siegel [MS] have done Applying the Itˆo’s

Lemma to dV0(P t) and substituting the price dynamic (1), we have:

The term in square brackets is the expected value of dV0(P t), and there being

no operating profit, it is the only expected capital gain In a risk-neutralworld5 it has to be equal to rV0(P )dt, the riskless return Thus, we obtain

5Here the fundamental assumption regarding the completeness of the market plays a

central role Thanks to this assumption, in fact, it is possible to replicate the expected

value of dV0(P t).

the following differential equation:

1

2σ

2P2

t V0 (P t ) + αP t V0 (P t)− rV0(P t ) = 0, (2)

with boundary condition limP t →0 V0(P t) = 0

Equation (2) is a second order differential equation and is homogeneous

and linear It is known that its general solution is given by substituting P ξ.One can obtain:

Analogous considerations can be made in order to determine the net presentvalue of the active firm Consider, thus, for the moment, an active firmhaving only one possibility to exit to its market An active firm persists in

this state as long as the output price is higher than a certain threshold P L and it will abandon the market as soon as the price falls down P L Then,

in the interval [P L , ∞) an active firm holds its option to abandon, and the

firm’s value is given by both the operating profit and the option to abandon

Considering the value of an active firm V1(P ) and applying the Itˆo’s Lemma,

The term in square brackets is the expected value of dV1(P ) and, as before, in

a risk-neutral world it has to be equal to the riskless portfolio value rV1(P )dt.

Note that, this time, there exists a dividend, namely the operating flow of

operating profit (P t − w o) in addition to the expected capital gain comingfrom the option to abandon Therefore, under the fundamental assumptionregarding the completeness of the market, we have the following differentialequation:

1

2V



1(P t )σ2P t2+ V1 (P t )αP t − rV1(P t ) + (P t − w o ) = 0. (4)The boundary condition is now limP →+∞ V1(P ) = lim P →+∞ r−α P − w o

r tion (4) is not homogeneous anymore For the homogeneous part, it is pos-sible to proceed as before For the non-homogeneous part one should try alinear form and solving for the coefficients6 Finally, we get the followinggeneral solution:



for P ∈ [P L , ∞), where coefficients B1 and B2 have to be determined cause the homogeneous part is identical in equations (2) and (4), coefficients

Be-β1 and β2 are the same as before.7

Our problem is, now, to determine the constants A1, A2, B1, B2 in

equa-tions (3) and (5) and the two thresholds (P L , P H) It is possible to simplifythe situation, considering the endpoints conditions When the output price

is very small, the probability of raising to P H is small for any fixed horizon Therefore, the option to invest becomes out-of-the-money and thus,

time-6For further explanations, see Dixit and Pindyck [DP], p.187.

7As we said, the value of the active firm is given by the sum of the current profit and

the option to abandon We can see that the terms in parentheses can be written as:

This is just the expected present value of a project that lives forever, starting from an

initial price P Therefore, the remaining part, B1P t β1+ B2P t β2, can be considered as the value of the option to abandon.

the coefficient A2 must to be zero Similarly, as P → ∞ the probability that the output price raises to P L is small for any fixed time-horizon Therefore,the option to abandon becomes out-of-the-money and, thus, the coefficient

B1 must to be zero Finally, the problem simplifies considering the followingtwo equations:

abandoning an asset of value V0(P ) and getting another one of value V1(P ) Therefore, at level P H it must be satisfied, as feasibility conditions, the fol-lowing condition:

V0(P H ) = V1(P H)− K I , = ⇒ V 

0(P H ) = V1 (P H ).

Similarly, at the threshold P L the firm pays K E to exercise the option to

disinvest, abandoning an asset of value V1(P ) and getting another one of value V0(P ) Therefore, at level P L the feasibility conditions are:

These four equations determine the four unknowns of the entry-exit

prob-lem A, B, P L and P H They are non-linear in P L and P H, so that an analyticsolution in a closed form is impossible However, it can be proved that a so-lution exists9 The thresholds satisfy 0 < P L < P H < ∞, and the coefficients

of the option value terms, A and B, are non-negative Further results require

a numerical solution

With this model it can be possible to make a comparison with the shallian approach, and making a comparative statics analysis The interestedreader should see the Dixit and Pindyck book [DP] In the book is also pre-sented a useful example regarding the application of the model in the copperindustry

So far, the model assumed that the project is brought on line immediatelyafter the decision to invest is made However, many investments take timeand the lag between the decision to invest and the start of the productioncan be quite long For example, McRee noted that, on average, the lag is 6years in case of building a power generating plant10 In this paragraph, we

intend to extend the basic model considering a so-called time-to-build dt ≥ 0.

This problem was faced for the first time by Bar-Ilan and Strange [BaS]

in 1992 Starting from the Dixit’s basic model and considering the standard assumptions, they supposed that there exists an investment lag with time- to-build equal to dt ≥ 0 This implies that a project started at time t will begin generating revenues and incurring marginal costs at time t + dt The entry cost K I is supposed to be paid at the end of construction, but thecommitment is irreversible once the decision is made This is equivalent to

say that at time t the entry cost is given by e −rdt K I

Therefore, instead of two, we have three state of nature In fact, the firm

can be inactive, Z t = 0, active Z t = 1 and ”under construction” Z t = c, where c = 0, 1 is an arbitrary constant.

Proceeding as before, we have to determine the value of a firm in the threestates of nature If a firm is inactive, it has the opportunity of exercising the

9A formal proof was given by Brekke and Øksendal in 1994 See [BØ].

10See McRee K.M., Critical issues in electric power planning in 1990s, Canadian Energy

Research Institute, 1989.

option to invest It is sure that in the range (0, P H) it will remain inactive

and its value is given by V0(P t) This value can be obviously written as:

V0(P t ) = e −rdt V0(P t +dt) =⇒ dV0(P t ) = e −rdt dV0(P t +dt ).

The value of the inactive firm at time t is equal to the value of the inactive firm

at time t + dt discounted for the lag dt It is possible to apply to dV0(P t +dt)the same analysis of sub-section 2.1 Moreover, because the discount factor

is not affected by the output price, taking the limit as dt → 0+, we obtain

the same differential equation (2) with the same boundary condition, andtherefore the same solution:

where the constant A has to be determined.

Analogously, in the range (P L , ∞) an active firm will not switch to the

inactive state and its value is given by:

This means that V1(P t) is given by the sum of the discounted future value

V1(P t +dt) of the firm and the expected cash flow coming from the operating

gain in the interval [t, t + dt]. Applying the Itˆo’s Lemma to dV1(P t +dt),

substituting the geometric Brownian motion to P t +dt and taking the limit

as dt → 0+, we obtain the same differential equation (4) with the same

boundary condition, and therefore the solution is always the following:



where the constant B has to be determined.

To complete the solution we must determine the value function V c (P t , θ), giving the value of a project under construction, where θ ∈ [0, dt] is the

remaining time until completion of the investment This value is given by:

V c (P t , θ) = e −rdt V c (P t +dt , θ − dt) =⇒ dV c (P t , θ) = e −rdt dV c (P t +dt , θ − dt).

This equation can be explained as follows Once the firm’s project is underconstruction, the firm can not enter in the market (because is already in)and does not produce (since the construction is not finished) Moreover, thefirm will not abandon the project because the model assumes that it has to

pay an exit cost K E ≥ 0 Since the costs of the investment are sunk and

the discounted cost of abandonment can be reduced by delaying, it would berational not abandoning the project

Applying, as usual, the Itˆo’s Lemma to dV c (P t , θ), substituting dP t, and

taking the limit as dt → 0+, we obtain the following partial differential

Although in this case the decision to invest is already undertaken, the firm

is not producing yet Therefore, in a risk neutral world the expected value

of the firm has to be equal to the riskless value rV c (P t , θ), as the case of an

inactive firm The boundary conditions are:

i) limP →0 V c (P t , θ) = −e −rθ K

E,ii) limP →∞ V c (P t , θ) = lim P →∞ e −(r−α)θ r −α P − e −rθ w o

to 0, it will not rise above both P L and P H during the construction periodand over any finite horizon Then, the project will be abandoned and the firmwill suffer an exit cost −e −rθ K

E Similarly, the second boundary condition

says that a very high price will not fall below P L and the value of the firmwill be given by the expected value of the future profit of a project that will

live forever, discounted by the period θ The third boundary condition states

that near the end of construction, the firm will be either active or abandon

the investment, depending on whether the price is above or below P L.The solution of the partial differential equation is given by:

where Φ(·) is the standard normal distribution function and u is defined as:

u = u(P, θ) = log P L − log P − ( α −σ2

Having determined the three values (9), (10) and (11), we need to specify nowthe value-matching conditions and their respective smooth-pasting conditions

in order to complete the solution to the entry-exit decisions problem with aninvestment’s lag They are specified by the following system:

construction and therefore it must to be discounted for the lag dt gously, in (13) we can see that at the price level P Ltriggering the abandon ofthe project, the value of the firm must to be equal to the value of an inactivefirm minus the exit cost, as before

Analo-Considering the three values (9), (10) and (11) and substituting them

in the value-matching and smooth-pasting conditions, we find a system of

four equations with the four unknowns A, B, P L and P H The full systemcan be find in Bar-Ilan and Strange, [BaS] Again the system has no closed

solution because the equations are not linear in the thresholds (P L , P H) and

it is necessary to employ numerical methods in order to find a solution andanalyze its properties

Bar-Ilan and Strange presented both an analytic solution, comparing theirsolution with the Dixit’s solution, and a numerical solution The numericalresults show that investment lags change the effect of price uncertainty oninvestment Considering a lag of 6 years, they showed that, for low levels

of variance, uncertainty has a smaller effect on investment when there are

investment lags, i.e the entry price P H is lower than the case with no vestment lags Instead, for higher levels of variance, they show not only that

in-the entry price is greater than in-the entry price in in-the case of no investment

lags, but also it is P H is grater than the entry price in the case of certainty.Contrary to the expectations, this means that an increase in uncertainty doesnot delay the investment They explain this more surprising result by thefact that, with investment lags, an increase in uncertainty raises the benefit

of waiting but not its opportunity cost The opportunity cost is given by theprofit during the period of inaction and it is independent of uncertainty

In general, the literature focused on two extremal cases: complete ity and unlimited reversibility The hypothesis of a complete irreversibilitysupposes that the firm has only one possibility to switch to the other state.With an unlimited reversibility the firm can switch infinite many times (eventhough switching is costly) However, both these cases are not representative

irreversibil-of the reality In particular, many entries or exits may destroy the projectprofitability or, for example, the firm’s reputation Therefore, it can be in-teresting to assume a restricted reversibility, i.e a finite number of switching

possibilities Using the Dixit’s approach and, then, under the standard sumptions, in 1993 Ekern [Ek] examined the entry-exit decisions problem

as-considering a countable and arbitrary finite number of switching

opportuni-ties n = 0, 1, 2, , representing the degree of flexibility He supposed that the maximum possible number of switches is N , and then N − n is the number

of the remaining number of switching possibilities for the firm If N = 1, the

decision is completely irreversible and the firm can switch just once When

N = ∞, switches are ever allowed In the special case N = 0, he supposed

that, if the firm does not change its status immediately, its current

produc-tion state will remain forever This is the so-called last chance and, in this

case, the trigger prices are easily determined In general, a firm survives if itsmarginal profit equals its marginal costs In case of an active firm having nopossibilities to switch to the inactive state (i.e the firm produces forever),this means that:

K E =− P L(0)

w o r

More in general, for n ∈ [0, N] let P H (n) be the optimal entry price for

an inactive firm at the n-th possibility to switch Analogously, let P L (n) be

the optimal exit price for an active firm at the n-th possibility to switch Therefore, the optimal strategy at the n-th possibility to switch is given by

τ n = {(P L (n) , P H (n))} Then, the model is defined by a sequence of optimal strategies τ = {τ n } n ≤N given the sequences of critical values {P L (n) } n ≤N and

{P H (n) } n ≤N : at each point n, the firm should regard the optimal policy τ n

This means that at each point n the value of the firm is given by the usual

equations (6) and (7), depending on the current firm’s state However, now

both the coefficients A and B depend on both the current firm’s state Z t = z, where z = 0, 1, and on the number of possible switches n Then, through this section, we will indicate both the coefficients with A(z, n) and suppose, for

simplicity, that at each switching point the lump-sum entry and exit costs

K I and K E remain constant and do not depend on the flexibility

Therefore, the value of an active and an inactive firm are, respectively:

We want to find the values of coefficients A(0, n) and A(1, n) Consider

a currently active firm with N − n remaining switching opportunities, and its relative optimal exit value P L (n) In this case, the value-matching andsmooth-pasting conditions are:

This means that, at price P L (n) the value of an active firm having N − n

possibilities to switch has to be equal to the value of an inactive firm having

(N − n) − 1 possibilities to switch, minus the exit cost K E Substituting inthese feasibility conditions the respective firm’s values, we obtain the follow-ing system:

second equation by P L (n) , the coefficient A 1,n will be eliminated, yielding theimplicit exit equation:

φ L (P L (n) ) = (β1− β2)A 0,n−1 (P L (n))β1 + (β2− 1) P L (n)

r − α − β2

w o

r + β2K E = 0, where the optimal exit price P L (n) is the unique unknown Analogously, mul-

tiplying again the first equation by β1 and the second equation by P L (n),

re-organizing and solving for the unknown coefficient A 0,n, we obtain:

(n)

L − β1(P L (n) − P L(0))

(β1− β2)(r − α)(P L (n))β2 Consider, now, an inactive firm with N − n remaining switching opportu- nities and its relative optimal entry value P H (n) In this case, the feasibilityconditions are:

At stage n, the coefficient A 1,n−1 is known, but the coefficient A 0,n has to be

determined together with P H (n) By multiplying the first equation by β1 and

the second equation by P H (n), we obtain the following implicit entry equation:

φ H (P H (n) ) = (β1− β2)A 1,n−1 (P H (n))β2 + (β1− 1) P H (n)

r − α − β1

w o

r − β1K I = 0,

and the optimal entry price P H (n) can be found Moreover, multiplying the

first equation by β2 and the second equation by P H (n), re-organizing, and

solving for the unknown coefficient A 0,n, it can be obtained:

should refer directly to the paper Moreover, his numerical results seems

to be more interesting In fact, he showed that the series {P H (n) } n of entry

trigger prices is decreasing, as n → ∞ This is rational because with a

re-stricted number of switching possibilities the risk faced by the firm is greater.Moreover, the series of exit trigger prices {P L (n) } n is increasing, as n tends to

infinity Also this seems rational: consider, for example, that the active firmhas just one possibility to exit from the market In this case, the firm shouldwait more to abandon the market, i.e should wait a lower exit trigger price

In addition, he showed that with n = 7 switching possibilities both series verge to the trigger prices with unlimited flexibility, i.e τ ∞ ={P L (∞) , P H (∞) } With n = 4 switching possibilities one already has sufficient conditions for

con-the convergence to con-the unlimited flexibility case

An other interesting extension of the basic model is given by considering asuspension of the activity A firm could find the output price too low forcontinuing the production, but also too high for a definitive abandonment ofthe market This case was studied by Dixit, [D1], and Dixit and Pindyck,

[DP] Assuming the standard assumptions, they proposed an extension of the

basic model where it is supposed that a firm can temporarily suspend and

mothball the activity, paying a lump-sum sunk cost K M The productioncould be reactivated in the future (if the market conditions will allow it)

paying a further lump-sum sunk cost K R Obviously, both mothballing sunk

cost K M and reactivation sunk cost K R must to be lower than the initial

investment cost K I, otherwise the suspension is not economically profitable

In addition, in case of suspension, it is supposed that the firm affords a

maintenance cost flow w M per unit of the capital The cost of maintainance

w M has to be less than the actual operating cost flow w o, in order to makethe suspension economically significant

As in the basic model, the aim is now to determine the value of theopportunity to invest in such a project, and the output prices thresholds forinvestment, mothballing, reactivation and scrapping Hence, in this case,the thresholds giving the firm’s optimal decisions rules are four In fact,

an inactive firm will invest in a project once the output price rises to the

threshold P H Became active, the firm has the possibility of suspending

the activity, if the price falls to the threshold P M If the project will bemothballed, the firm can either reactivate the production when the price

achieves a third threshold P R, or definitely abandon the market, if the output

price reaches the forth threshold P S = P L Note that an economic significance

requires P R < P H, i.e the price level of reactivation must to be lower thanthe entry price, because of the maintenance cost paid

To keep the exposition simple, Dixit an Pindyck assumed that the total

exit cost K E is given by the sum of the cost of mothballing an operating

project K M and the cost of scrapping a project already mothballed K S11

On the contrary, it can not be economically acceptable considering the total

entry investment K I as the sum of an imaginary sunk cost of mothballing K M and the reactivation cost K R, because for a firm it is never optimal investing

in a mothballed project12

Then, there are now three possible states: active Z t = 1, mothballed

Z t = m, and inactive/scrapped Z t = 0 where m = 0, 1 is a constant and,

po-tentially, there are six possible switches However, we exclude the possibility

to go from state Z t = 0 to state Z t = m because this is not economically

efficient

We already know that over the interval [0, P H) the firm remains inactiveand its value is given uniquely by the option to invest Over the interval

[P M , ∞) the firm is active and its net present value is given, this time, by the

sum of discounted profit and the value of the option to suspend the activity

11This means that going from an operational project directly to total scrapping is as

costly as passing by the mothballed production state.

12Obviously, if the firm postpones the investment until the time of operation, it will

postpone the payment of K M and save the maintenance cost w M.

Therefore, these value are:

Note that this solution is true only over the interval (P S , P R) The special

cases P S = 0 (never scrap) and P R = ∞ (never reactivate) are obviously excluded, and this implies that neither C1P β1 = 0 nor C2P β2 = 0 The firstcomponent of the solution is the value of the option to reactivate, the secondcomponent is the value of the option to scrap the project and w M