ENTRY AND EXIT DECISIONS PROBLEM: A SURVEY

ENTRY AND EXIT DECISIONS
PROBLEM: A SURVEY
Andrea Girometti∗
Master in Advanced Studies in Finance
University of Ză
urich - ETH Ză
urich

January 2004

Abstract
In this survey, the entry-exit decisions problem is studied. The
problem concerns to the investment and disinvestment decisions process
of a firm, when its output price is stochastic. Typically, this is the
problem faced by an oil company or by a firm involved in the commodities markets. The first approach, here analyzed in detail, was proposed
by Dixit in 1989 and it is based on the contingent claim theory. In
particular, the values of a firm in both activity and inactivity states
are determined by using the theory of real options pricing. Thanks to
the so-called value-matching and smooth-pasting conditions, it is possible to interlink these two firm’s values and, therefore, to determine a
pair of trigger prices, giving an optimal decision policy. The optimal
policy fixes the levels of the output prices at which it is economically
convenient either to start the production or to abandon the market. In
addition, some interesting extensions of the basic model are presented
and, finally, the two most recent approaches are explained. The first
one is based on the ”mark-up” concept, the second one is based on
the 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


1

Introduction

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.
As a general framework, a single firm having access to a single good
production opportunity is considered. If {Zt }t≥0 is 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 Zt = 0 or already active
Zt = 1. In the inactive state there is no production at all, and the firm is
waiting 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 Zt , switches production on or off instantaneously.
Switching production on, the firm can invest a lump-sum cost KI in order to
start the production and, in the opposite case, the firm has to pay a lumpsum cost KE in order to exit to the market. It is assumed that KI + KE > 0,
otherwise a firm could have an infinite profit switching continuously. The
firm’s activity implies also a variable production cost wo per each unit of
output flow. This is the operating cost. Moreover, it is supposed that the
firm has to pay again the lump-sum entry cost KI if it exits and wants to
re-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. This
price is represented by the stochastic process {Pt }t≥0 , and it is assumed that
it is driven by a geometric Brownian motion1 :
dPt = αPt dt + σPt dBt ,

(1)

where α and σ are constant and strictly positive, and {Bt }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.
1

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

2


The aim of a firm is to determine, jointly, optimal bounds (PL , PH ). This
pair represents the trigger prices for entry and exit into the market. At a
price level PH it begins to be profitable for a firm starting the production. If
the output price is lower than PL , producing is not profitable anymore and
the firm leaves the market.

Various approaches to the problem and several possible extensions are
presented in this survey. All these methods are alternative to the standard
capital budgeting process based on the NPV rule. The inadequacy of this
approach is widely acknowledged, because of total neglect of the stochastic
nature of the output prices. In general, the real option valuation theory
seems to be more suitable and more useful.
The first and simplest model was pioneered by Mossin [Mo] in 1968, but
the first formal discussion on the entry decisions problem was proposed by
McDonald and Siegel [MS] in 1985. The combined problem concerning the
entry and the exit was faced by Brennan and Schwartz [BS] in 1985. They
applied, for the fist time, the well known options pricing theory developed by
Black, Merton and Scholes in 1973 in order to evaluate active and inactive
firms, 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 by
Dixit [D2] in 1989. In particular, he focused on entry and exit trigger prices
as fundamental indicators for firm’s decision policies. Moreover, he made
comparisons between the standard Marshallian theory and the new option
theory-based approach, introducing the hysteresis concept, and he proposed
a numerical analysis with a comparative statics analysis of the results he
found3 . During the ’90s, many authors developed the Dixit’s basic model
considering various possible extensions like adding taxes or an investment’s
lag, considering a restricted number of switches or the possibility of laying-up
or scrapping the production. Other authors discussed the possibility and the
effects 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 ρpm 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.
3
These economic results will not be presented in this survey. The interest reader can
refer to the Dixit’s paper [D2].

3


the economic equilibrium between firms using the Dixit decision’s rules (see
respectively, 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 rigorous
treatment of the problem was proposed by Brekke and Øksendal in 1994
[BØ]. They analyzed the entry-exit decisions problem applying both the
option pricing theory and the dynamic programming theory, gave a formal
proof of the existence of a solution, and extended the classical approach
considering the case of a finite resource. In 2001 Sødal [Sø] proposed a totally
new approach, based on the ”mark-up” concept. Finally, in 2003, Chesney
and 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 possibility of either laying-up or scrapping the project. Section 6 will be devoted
to the problem viewed as a sequential optimal stopping problem, as Brekke
and 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 concept developed by Sødal and on probabilistic tools proposed by Chesney and

Hamza.

2

The basic model

Through this paragraph we will refer to the fundamental Dixit’s paper [D2].
Consider the standard assumptions. At level price Pt , the firm could be either
active Zt = 1 or inactive Zt = 0. Then, the first step is determining the firm’s
value in both states.
In state (Pt , 0), suppose that the inactive firm has an expected net present
value V0 (Pt ). Such a firm can observe the current output price and then, following optimal policies4 , decides whether to continue being inactive or to
enter in to the market. Assume, for the moment, that this decision is irreversible. 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 firm
starts the production.
Analogously, in state (Pt , 1), the active firm has an expected net present
value V1 (Pt ). In this case, the firm decides whether to continue being active
or to exit. Suppose that this decision is again irreversible. In this case,
4

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

4


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 (Pt ) and V0 (Pt ) can be determined by using the contingent claim

analysis. Therefore, the market must supposed to be complete, i.e. every
traded asset is supposed to be spanned by other assets in the economy. In
particular, both the values of the active and inactive firms are supposed to be
positively 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 (Pt ) and V0 (Pt ) are determined, it is possible to
determine the two trigger prices (PL , PH ) considering the so-called valuematching conditions and the smooth-pasting conditions, as we will see in the
following sub-section 2.3.

2.1

Value of an inactive firm

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 PH and it will invest as soon as
the price reaches PH . Therefore, over the range (0, PH ) the inactive firm’s
investment opportunity, i.e. the value of the inactive firm, is equivalent to
a perpetual call option, where the strike is the entry cost KI . The decision
to invest corresponds to the decision of exercising the option. Therefore, in
order to obtain the inactive firm’s value, it can be applied the contingent
claim theory, as McDonald and Siegel [MS] have done. Applying the Itˆo’s
Lemma to dV0 (Pt ) and substituting the price dynamic (1), we have:
1
dV0 (Pt ) = V0 (Pt )dPt + V0 (dPt )2 =
2



1 


2 2
= V0 (Pt )αPt + V0 (Pt )σ Pt dt + σPt dBt .
2
The term in square brackets is the expected value of dV0 (Pt ), and there being
no operating profit, it is the only expected capital gain. In a risk-neutral
world5 it has to be equal to rV0 (P )dt, the riskless return. Thus, we obtain
5

Here 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 (Pt ).

5


the following differential equation:
1 2 2 
σ Pt V0 (Pt ) + αPt V0 (Pt ) − rV0 (Pt ) = 0,
2

(2)

with boundary condition limPt →0 V0 (Pt ) = 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:
1 2
σ ξ(ξ − 1) + αξ − r = 0
2

φ(ξ) = ξ 2 − (1 − m)ξ − ρ = 0,

or

and ρ = σ2r2 . The convergence condition is r > m, and
where m = 2α
σ2
φ(0) = −ρ < 0 and φ(1) − (ρ − m) < 0. Since φ (ξ) = 2 > 0, one root must
be greater than 1 and the other must be less than 0:

(1 − m) + (1 − m)2 + 4ρ
> 1,
β1 =
2

β2 =

(1 − m) −



(1 − m)2 + 4ρ
< 0.
2

Finally, the general solution of (2) is given by:
V0 (P ) = A1 P β1 + A2 P β2 ,

(3)

where coefficients A1 and A2 have to be determined. As we said, this solution
is valid for P ∈ [0, PH ].

2.2

Value of an active firm

Analogous considerations can be made in order to determine the net present
value of the active firm. Consider, thus, for the moment, an active firm
having 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 PL
and it will abandon the market as soon as the price falls down PL . Then,
in the interval [PL , ∞) 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,
we obtain:
1
dV1 (Pt ) = V1 (Pt )dPt + V1 (dPt )2 =
2
6





1 

2 2
= V1 (Pt )αPt + V1 (Pt )σ Pt dt + σPt dBt .
2
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 (Pt − wo ) in addition to the expected capital gain coming
from the option to abandon. Therefore, under the fundamental assumption
regarding the completeness of the market, we have the following differential
equation:
1 
V (Pt )σ 2 Pt2 + V1 (Pt )αPt − rV1 (Pt ) + (Pt − wo) = 0.
2 1

(4)

P
− wro . EquaThe boundary condition is now limP →+∞ V1 (P ) = limP →+∞ r−α
tion (4) is not homogeneous anymore. For the homogeneous part, it is possible to proceed as before. For the non-homogeneous part one should try a
linear form and solving for the coefficients6 . Finally, we get the following
general solution:


wo
P
β1
β2
−
,
(5)
V1 (P ) = B1 P + B2 P +
r−α
r

for P ∈ [PL , ∞), where coefficients B1 and B2 have to be determined. Because the homogeneous part is identical in equations (2) and (4), coefficients
β1 and β2 are the same as before.7 .

2.3

The trigger prices

Our problem is, now, to determine the constants A1 , A2 , B1 , B2 in equations (3) and (5) and the two thresholds (PL , PH ). It is possible to simplify
the situation, considering the endpoints conditions. When the output price
is very small, the probability of raising to PH is small for any fixed timehorizon. Therefore, the option to invest becomes out-of-the-money and thus,
6

For further explanations, see Dixit and Pindyck [DP], p.187.
As 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:

 ∞

wo
Pt
−ru
−
=E
e
(Pu − wo )du .
r−α
r
0
7

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

initial price P . Therefore, the remaining part, B1 Ptβ1 + B2 Ptβ2 , can be considered as the
value of the option to abandon.

7


the coefficient A2 must to be zero. Similarly, as P → ∞ the probability that
the output price raises to PL 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 following
two equations:
V0 (P ) = AP β1 ,

V1 (P ) = BP

β2

+

P
wo
−
r−α
r

(6)

.

(7)

Moreover, economically meaningful solution for V0 (P ) and V1 (P ) requires
their non-negativity and, therefore, both coefficients A and B have to be
non-negative.
Those two solutions have to be linked by the so-called value-matching
condition (or high-contact condition) and its relative smooth-pasting condition 8 At the threshold PH , the firm pays KI to exercise the option to invest,
abandoning an asset of value V0 (P ) and getting another one of value V1 (P ).
Therefore, at level PH it must be satisfied, as feasibility conditions, the following condition:
V0 (PH ) = V1 (PH ) − KI , =⇒ V0 (PH ) = V1 (PH ).
Similarly, at the threshold PL the firm pays KE to exercise the option to
disinvest, abandoning an asset of value V1 (P ) and getting another one of
value V0 (P ). Therefore, at level PL the feasibility conditions are:
V1 (PL ) = V0 (PL ) − KE , =⇒ V1 (PL ) = V0 (PL ).
Substituting equations (6) and (7) in the feasibility constraints we obtain the
following system:


BPLβ2 +








β2


BPH +

PL
r−α

−

wo
r

= APLβ1 − KE

PH
r−α

−

wo
r

= APHβ1 + KI



β2 −1

+

β2 BPL







β BP β2 −1 +
2
H
8

(8)
β1 APLβ1−1

1
r−α

=

1
r−α

= β1 APHβ1−1 .

In the following, we can call these conditions also feasibility conditions.

8


These four equations determine the four unknowns of the entry-exit problem A, B, PL and PH . They are non-linear in PL and PH , so that an analytic
solution in a closed form is impossible. However, it can be proved that a solution exists9 . The thresholds satisfy 0 < PL < PH < ∞, 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 Marshallian approach, and making a comparative statics analysis. The interested
reader should see the Dixit and Pindyck book [DP]. In the book is also presented a useful example regarding the application of the model in the copper
industry.

3

Investment’s Lag

So far, the model assumed that the project is brought on line immediately
after the decision to invest is made. However, many investments take time
and the lag between the decision to invest and the start of the production
can be quite long. For example, McRee noted that, on average, the lag is 6
years 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 timeto-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 KI is supposed to be paid at the end of construction, but the
commitment is irreversible once the decision is made. This is equivalent to
say that at time t the entry cost is given by e−rdt KI .
Therefore, instead of two, we have three state of nature. In fact, the firm
can be inactive, Zt = 0, active Zt = 1 and ”under construction” Zt = c,
where c = 0, 1 is an arbitrary constant.

3.1

The value of the firm in the three states

Proceeding as before, we have to determine the value of a firm in the three
states of nature. If a firm is inactive, it has the opportunity of exercising the
9

A formal proof was given by Brekke and Øksendal in 1994. See [BØ].
See McRee K.M., Critical issues in electric power planning in 1990s, Canadian Energy
Research Institute, 1989.
10

9


option to invest. It is sure that in the range (0, PH ) it will remain inactive
and its value is given by V0 (Pt ). This value can be obviously written as:
V0 (Pt ) = e−rdt V0 (Pt+dt )

=⇒

dV0 (Pt ) = e−rdt dV0 (Pt+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 (Pt+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, and
therefore the same solution:
V0 (P ) = AP β1 ,

(9)

where the constant A has to be determined.
Analogously, in the range (PL , ∞) an active firm will not switch to the
inactive state and its value is given by:

 dt

−rdt
−ru
V1 (Pt ) = e
V1 (Pt+dt ) + E
e (Pu − wo )du =⇒
0

−rdt

=⇒ dV1 (Pt ) = e


dV1 (Pt+dt ) +

0

dt

e−ru (Pu − wo )du.

This means that V1 (Pt ) is given by the sum of the discounted future value
V1 (Pt+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 (Pt+dt ),
substituting the geometric Brownian motion to Pt+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:


wo
Pt
β2
−
V1 (P ) = BP +
,
(10)
r−α
r
where the constant B has to be determined.
To complete the solution we must determine the value function Vc (Pt , 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:
Vc (Pt , θ) = e−rdt Vc (Pt+dt , θ − dt)

=⇒
10

dVc (Pt , θ) = e−rdt dVc (Pt+dt , θ − dt).


This equation can be explained as follows. Once the firm’s project is under
construction, the firm can not enter in the market (because is already in)
and does not produce (since the construction is not finished). Moreover, the
firm will not abandon the project because the model assumes that it has to
pay an exit cost KE ≥ 0. Since the costs of the investment are sunk and
the discounted cost of abandonment can be reduced by delaying, it would be

rational not abandoning the project.
Applying, as usual, the Itˆo’s Lemma to dVc (Pt , θ), substituting dPt , and
taking the limit as dt → 0+ , we obtain the following partial differential
equation:
∂Vc (Pt , θ)
∂Vc (Pt , θ)
1 2 2 ∂ 2 Vc (Pt , θ)
+ αP
σ P
− rVc (Pt , θ) −
= 0.
2
2
∂P
∂P
∂θ
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 rVc (Pt , θ), as the case of an
inactive firm. The boundary conditions are:
i) limP →0 Vc (Pt , θ) = −e−rθ KE ,
−(r−α)θ

−rθ

ii) limP →∞ Vc (Pt , θ) = limP →∞ e r−α P − e r wo ,

P
wo
β2


BP + r−α − r for P > PL ,
iii) limθ→0 Vc (Pt , θ) =


AP β1 − KE
for P ≤ PL .
The first boundary condition says us that when the output price is very close
to 0, it will not rise above both PL and PH during the construction period
and over any finite horizon. Then, the project will be abandoned and the firm
will suffer an exit cost −e−rθ KE . Similarly, the second boundary condition
says that a very high price will not fall below PL and the value of the firm
will 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 PL .
The solution of the partial differential equation is given by:
Vc (P, θ) = [1 − Φ(u − β2 σ)]AP β2 +
+[1 − Φ(u − σ)]

e−(r−α)θ P
e−rθ wo
− [1 − Φ(u)]
,
r−α
r
11

(11)

where Φ(·) is the standard normal distribution function and u is defined as:
2

)θ
log PL − log P − ( α−σ
2
√
.
u = u(P, θ) =
σ θ
This solution can be found in the Appendix of the paper of Bar-Ilan and
Strange [BaS].

3.2

The trigger prices

Having determined the three values (9), (10) and (11), we need to specify now
the value-matching conditions and their respective smooth-pasting conditions
in order to complete the solution to the entry-exit decisions problem with an
investment’s lag. They are specified by the following system:
V0 (PH ) = Vc (PH , dt) − e−rdt KI ,
V1 (PL ) = Vc (PL , dt) − KE ,

V0 (PH ) =

∂Vc (PH , dt)
,
∂P

V1 (PL ) = V0 (PL ).

(12)
(13)

We can see in (12) that at the price level PH triggering the entry in market
(and starting the construction of the project), the value of an inactive firm
must to be equal to the value of a firm that has already started the construction minus the entry cost. This entry cost will be paid at the end of the
construction and therefore it must to be discounted for the lag dt. Analogously, in (13) we can see that at the price level PL triggering the abandon of
the project, the value of the firm must to be equal to the value of an inactive
firm minus the exit cost, as before.
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, PL and PH . The full system
can be find in Bar-Ilan and Strange, [BaS]. Again the system has no closed
solution because the equations are not linear in the thresholds (PL, PH ) and
it is necessary to employ numerical methods in order to find a solution and
analyze its properties.
Bar-Ilan and Strange presented both an analytic solution, comparing their
solution with the Dixit’s solution, and a numerical solution. The numerical
results show that investment lags change the effect of price uncertainty on
investment. 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 PH is lower than the case with no investment lags. Instead, for higher levels of variance, they show not only that
12


the entry price is greater than the entry price in the case of no investment
lags, but also it is PH is grater than the entry price in the case of certainty.

Contrary to the expectations, this means that an increase in uncertainty does
not delay the investment. They explain this more surprising result by the
fact that, with investment lags, an increase in uncertainty raises the benefit
of waiting but not its opportunity cost. The opportunity cost is given by the
profit during the period of inaction and it is independent of uncertainty.

4

Restricting number of switching

In general, the literature focused on two extremal cases: complete irreversibility and unlimited reversibility. The hypothesis of a complete irreversibility
supposes that the firm has only one possibility to switch to the other state.
With an unlimited reversibility the firm can switch infinite many times (even
though switching is costly). However, both these cases are not representative
of the reality. In particular, many entries or exits may destroy the project
profitability or, for example, the firm’s reputation. Therefore, it can be interesting to assume a restricted reversibility, i.e. a finite number of switching
possibilities. Using the Dixit’s approach and, then, under the standard assumptions, in 1993 Ekern [Ek] examined the entry-exit decisions problem
considering a countable and arbitrary finite number of switching opportunities 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 production 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 its
marginal profit equals its marginal costs. In case of an active firm having no
possibilities to switch to the inactive state (i.e. the firm produces forever),
this means that:
(0)

w
PH
wo
o
(0)
−
= KI =⇒ PH = (r − α)
+ KI .
r−α
r
r
In case of an inactive firm having no possibilities to switch to the active state,
the profit is represented by the saving of the exit cost, and the cost is the
opportunity cost lost from the production:


(0)


w
wo
PL
o
(0)
KE = −
−
=⇒ PL = (r − α)
− KE .
r−α
r

r
13


(n)

More in general, for n ∈ [0, N] let PH be the optimal entry price for
(n)
an inactive firm at the n-th possibility to switch. Analogously, let PL 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
(n)
(n)
τn = {(PL , PH )}. Then, the model is defined by a sequence of optimal
(n)
strategies τ = {τn }n≤N given the sequences of critical values {PL }n≤N and
(n)
{PH }n≤N : at each point n, the firm should regard the optimal policy τn .
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 Zt = 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
KI and KE remain constant and do not depend on the flexibility.
Therefore, the value of an active and an inactive firm are, respectively:


P
wo

(n)
(n)
(n) β2
−
V1 (P ) = A1,n [P ] +
V0 (P ) = A0,n [P (n) ]β1 .
r−α
r
As usual, the unknown coefficients and the unknown optimal strategy are
determined simultaneously, using the usual feasibility conditions.
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
(n)
its relative optimal exit value PL . In this case, the value-matching and
smooth-pasting conditions are:
(n)

(n)

(n−1)

V1 (PL ) = V0

d

(n)

(PL ) − KE ,

(n)

d

(n)

V (PL ) =
(n) 1

dPL

(n)
dPL

(n−1)

V0

(n)

(PL ).

(n)

This means that, at price PL 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 KE . Substituting in
these feasibility conditions the respective firm’s values, we obtain the following system:

(n)
PL

(n) β
(n)

− wro = A0,n−1 [PL ]β1 − KE
A1,n [PL ] 2 + r−α


(n)
β2 A1,n [PL ]β2 −1 +

1
r−α

(n)

= β1 A0,n−1 [PL ]β1 −1

The system is given by two equations in two unknown: the coefficient A1,n
(n)
and the optimal exit price PL . Multiplying the first equation by β2 and the
14


(n)

second equation by PL , the coefficient A1,n will be eliminated, yielding the
implicit exit equation:
(n)

(n)

(n)

φL (PL ) = (β1 − β2 )A0,n−1 (PL )β1 + (β2 − 1)

PL
wo
− β2
+ β2 KE = 0,
r−α
r

(n)

where the optimal exit price PL is the unique unknown. Analogously, mul(n)
tiplying again the first equation by β1 and the second equation by PL ,
re-organizing and solving for the unknown coefficient A0,n , we obtain:
(n)

(n)

ψL (A1,n ) = (β1 − β2 )A1,n (PL )β2 + (β1 − 1)

wo
PL
− β1
+ β1 KE = 0.
r−α
r

(n)

Solving the system {φL (PL ), ψL (A1,n )}, and recalling the last chance exit


(0)
value PL = (r − α) wro − KE , we finally obtain:
(n)

A1,n =

(n)

(0)

PL − β1 (PL − PL )
(n)

(β1 − β2 )(r − α)(PL )β2

.

Consider, now, an inactive firm with N − n remaining switching opportu(n)
nities and its relative optimal entry value PH . In this case, the feasibility
conditions are:
(n)

(n)

(n−1)

V0 (PH ) = V1

d

(n)

(PL ) − KI ,

(n)
dPL

(n)

d

(n)

V0 (PH ) =

(n)
dPH

(n−1)

V1

(n)

(PL ).

(n)

At price PH the value of an inactive firm having N −n possibilities to switch
has to be equal to the value of an active firm having (N −n)−1 possibilities to
switch, minus the entry cost KI . Substituting in these feasibility conditions
the respective firm’s values, we obtain the following system:

(n)
PH
(n)
(n)

− wro − KI
A0,n [PH ]β1 = A1,n−1 [PH ]β2 + r−α



(n)

(n)

β1 A0,n [PH ]β1 −1 = β2 A1,n−1 [PH ]β2 −1 +

1
.
r−α

At stage n, the coefficient A1,n−1 is known, but the coefficient A0,n has to be
(n)

determined together with PH . By multiplying the first equation by β1 and
(n)
the second equation by PH , we obtain the following implicit entry equation:
(n)

(n)

(n)

φH (PH ) = (β1 − β2 )A1,n−1 (PH )β2 + (β1 − 1)
15

wo
PH
− β1
− β1 KI = 0,
r−α
r


(n)

and the optimal entry price PH can be found. Moreover, multiplying the
(n)
first equation by β2 and the second equation by PH , re-organizing, and
solving for the unknown coefficient A0,n , it can be obtained:
(n)

(n)

ψH (A0,n ) = (β1 − β2 )A0,n (PH )β1 + (β2 − 1)

wo
PH
− β2
− β2 KI = 0.
r−α
r

(n)

Solving the system {φH (PH ), ψH (A0,n )} and recalling the last chance entry


(0)
value PH = (r − α) wro + KI , we finally obtain:
(n)

A0,n =

(n)

(0)

PH − β2 (PH − PH )
(n)

(β1 − β2 )(r − α)(PH )β1

.

In conclusion, for a fixed maximum number of switching possibilities N and
for a general n ∈ [0, N], Ekern determined the optimal strategy in term of
(n)
(n)
the two trigger prices PH and PL , as well as the coefficients A0,n and A1,n
(n)
required for computing the values for the active firm V1 (P ) and for the
(n)
inactive firm V0 (P ). He also analyzed the special cases N = 1 (complete
irreversibility) and N = ∞ (unlimited reversibility), but the interest reader
should refer directly to the paper. Moreover, his numerical results seems
(n)
to be more interesting. In fact, he showed that the series {PH }n of entry
trigger prices is decreasing, as n → ∞. This is rational because with a restricted number of switching possibilities the risk faced by the firm is greater.
(n)
Moreover, the series of exit trigger prices {PL }n is increasing, as n tends to
infinity. Also this seems rational: consider, for example, that the active firm
has just one possibility to exit from the market. In this case, the firm should
wait 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 con(∞)
(∞)
verge to the trigger prices with unlimited flexibility, i.e. τ∞ = {PL , PH }.
With n = 4 switching possibilities one already has sufficient conditions for
the convergence to the unlimited flexibility case.

5

Laying-Up and Scrapping

An other interesting extension of the basic model is given by considering a
suspension of the activity. A firm could find the output price too low for
continuing the production, but also too high for a definitive abandonment of
the 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
16


mothball the activity, paying a lump-sum sunk cost KM . The production
could be reactivated in the future (if the market conditions will allow it)
paying a further lump-sum sunk cost KR . Obviously, both mothballing sunk
cost KM and reactivation sunk cost KR must to be lower than the initial
investment cost KI , otherwise the suspension is not economically profitable.
In addition, in case of suspension, it is supposed that the firm affords a
maintenance cost flow wM per unit of the capital. The cost of maintainance
wM has to be less than the actual operating cost flow wo , in order to make
the suspension economically significant.
As in the basic model, the aim is now to determine the value of the
opportunity to invest in such a project, and the output prices thresholds for
investment, 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 PH . Became active, the firm has the possibility of suspending
the activity, if the price falls to the threshold PM . If the project will be
mothballed, the firm can either reactivate the production when the price
achieves a third threshold PR , or definitely abandon the market, if the output
price reaches the forth threshold PS = PL . Note that an economic significance
requires PR < PH , i.e. the price level of reactivation must to be lower than
the entry price, because of the maintenance cost paid.

To keep the exposition simple, Dixit an Pindyck assumed that the total
exit cost KE is given by the sum of the cost of mothballing an operating
project KM and the cost of scrapping a project already mothballed KS 11 .
On the contrary, it can not be economically acceptable considering the total
entry investment KI as the sum of an imaginary sunk cost of mothballing KM
and the reactivation cost KR , because for a firm it is never optimal investing
in a mothballed project12 .
Then, there are now three possible states: active Zt = 1, mothballed
Zt = m, and inactive/scrapped Zt = 0 where m = 0, 1 is a constant and, potentially, there are six possible switches. However, we exclude the possibility
to go from state Zt = 0 to state Zt = m because this is not economically
efficient.
We already know that over the interval [0, PH ) the firm remains inactive
and its value is given uniquely by the option to invest. Over the interval
[PM , ∞) 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.
11
This means that going from an operational project directly to total scrapping is as
costly as passing by the mothballed production state.
12
Obviously, if the firm postpones the investment until the time of operation, it will
postpone the payment of KM and save the maintenance cost wM .

17


Therefore, these value are:
V0 (P ) = AP β1

V1 (P ) = BP β2 +

wo
P
−
.
r−α
r

(14)

Note that the term BPtβ2 is now the value of the option to suspend the activity and it is representative of the two possibilities, reactivation or scrapping.
It remains to determine the value of a mothballed project Vm (P ). The firm
temporarily suspends its activities just over the range (PS , PR ) . Therefore,
by applying the Itˆo’s Lemma, substituting the price’s dynamic and replicating the obtained portfolio, the value of a firm with such a project is given by
the following differential equation13 :
1 2 2 
σ P Vm (P ) + αP Vm (P ) − rVm (P ) = wM .
2
with boundary conditions:
i) limP →PS Vm (P ) = V0 (PS ) − KS ,
ii) limP →PR Vm (P ) = V1 (PR ) − KR .
Applying the same method as in sub-section 2.2, the solution of this nonhomogeneous differential equation is given by:
Vm (P ) = C1 P β1 + C2 P β2 −

wM
.
r

(15)

Note that this solution is true only over the interval (PS , PR ). The special

cases PS = 0 (never scrap) and PR = ∞ (never reactivate) are obviously
excluded, and this implies that neither C1 P β1 = 0 nor C2 P β2 = 0. The first
component of the solution is the value of the option to reactivate, the second
component is the value of the option to scrap the project and wrM is the value
of a perpetual mothballed project. The value-matching and smooth-pasting
conditions are now:
a) Original investment
V0 (PH ) = V1 (PH ) − KI ,

V0 (PH ) = V1 (PH ).

b) Mothballing
V1 (PM ) = Vm (PM ) − KM ,
13

V1 (PM ) = Vm (PM ).

In this case we have a negative dividend represented by the maintenance cost wM that
has to be paid during the suspension.

18


c) Reactivation
Vm (PR ) = V1 (PR ) − KR ,

Vm (PR ) = V1 (PR ).

Vm (PS ) = V0 (PS ) − KS ,

Vm (PS ) = V0 (PS ).

d) Scrapping

It is now clear that, at price PR , the value of a mothballed project must
equal the value of an active firm minus the reactivation cost KR , and at
price PM , the value of an active firm must equal the value of a mothballed
project minus the cost of mothballing KM . Finally, the firm’s values (14)
and (15) in the feasibility conditions, we obtain the system of eight equation
for eight unknowns, i.e. the four option value coefficients A, B, C1 and C2 ,
and the four thresholds PH , PM , PR and PS . In spite of solving this system,
it is possible to decompose it. Considering conditions b) and c), that is the
interaction between the mothballed state and the reactivation state, we have
the following sub-system:

PR
M)

−C1 PRβ1 + (B − C2 )PRβ2 + r−α
− (wo −w
= KR

r








β1 −1
1


+ β2 (B − C2 )PRβ2−1 + r−α
=0
−β1 C1 PR


β1
β2
PR
M)

−C1 PM
+ (B − C2 )PM
+ r−α
− (wo −w
= −KM

r







−β C P β1 −1 + β (B − C )P β2−1 + 1 = 0.
1 1 M

2
2
M
r−α
This is a sub-system of four equations in four unknowns C1 , (B − C2 ), PR ,
PM and it is possible to solve it by numerical procedures. This sub-system,
in fact, has the same form as system (8).
Considering conditions a) and d), that is the interaction between the
initial investment state and the scrapping state, we have the following subsystem:

PH

−APHβ1 + BPHβ2 + r−α
− wro = KI








β1 −1
β2 −1
1


−β1 APH + β2 BPH + r−α = 0




(C1 − A)PSβ1 + C2 PSβ2 + wrM = −KS








β (C − A)P β1−1 + β C P β2 −1 = 0.
1
1
2 2 S
S
19


These equations have six unknowns: the thresholds PH , PS , and the coefficients A, B, (D1 − A) and C2 . However, from the solution of the first
sub-system we know C1 = (C1 − A) + A and (B − D2 ). Therefore we can
complete the solution.
Obviously, this model is more general than the basic model and, taking
the limits of the various parameters, we can easily obtain the specific case of
the basic model.

6

Entry-exit decisions and the impulse control problem: the case of finite resource

Starting from the simplest model proposed by Brennan and Schwartz, and

Dixit, in 1994 Brekke and Øksendal [BØ] gave a formal and rigorous proof of
the existence of a solution. In particular, looking at the problem as special
case of sequential optimal stopping problem, they analyzed it as a generalized impulse control problem, solved using optimal stochastic control calculus. Moreover, referring directly to the extraction problem of an oil company,
they involved into the discussion the resource depletion.
Under standard assumptions, they studied the problem in a general way
and, then, they analyzed the case of a geometric Brownian motion for the
output price.
Let Zt = 0, 1 be the indicator random variable at time t, and let Zt ∈ Z,
where Z is the set of all possible indicator functions of the state of the system.
Moreover, assume Zt right-continuous with left limits (cad lag). If Qt denotes
the stock of remaining resources in the field, Brekke and Øksendal considered
a constant λ > 0 as the extraction rate, proportional to the amount of
remaining reserves:
dQt = −λZt Qt dt.
Therefore, the whole economic system at time t can be represented by the
stochastic process Xt :


t
 Pt 

Xt = 
 Qt  .
Zt
If s is the starting time, the process Xt takes starting value X0 = x = (s, p, z),
where Ps = p and Zs = z. The probability law of Xt given X0 is denoted by
Px and the expectation with respect to Px is denoted by Ex .
20