Kwon Lippman2011
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OPERATIONS R ESEARCHVol. 59, No. 5, September–October 2011, pp. 1119–1130
issn 0030-364X eissn 1526-5463 11 5905 1119 http://dx.doi.org/10.1287/opre.1110.0949
© 2011 INFORMS
Acquisition of Project-Specific
Assets with Bayesian Updating
H. Dharma KwonDepartment of Business Administration, University of Illinois at Urbana–Champaign, Champaign, Illinois 61820,
Steven A. LippmanUCLA Anderson School of Management, Los Angeles, California 90095, [email protected]
We study the impact of learning on the optimal policy and the time-to-decision in an infinite-horizon Bayesian sequentialdecision model with two irreversible alternatives: exit and expansion. In our model, a firm undertakes a small-scale pilotproject to learn, via Bayesian updating, about the project’s profitability, which is known to be in one of two possible states.The firm continuously observes the project’s cumulative profit, but the true state of the profitability is not immediatelyrevealed because of the inherent noise in the profit stream. The firm bases its exit or expansion decision on the posterior
probability distribution of the profitability. The optimal policy is characterized by a pair of thresholds for the posteriorprobability. We find that the time-to-decision does not necessarily have a monotonic relation with the arrival rate of newinformation.
Subject classifications : decision analysis: sequential; finance: investment criteria; probability: diffusion; Bayesiansequential decision; project-specific investment; time to decision; Brownian motion.
Area of review: Stochastic Models. History : Received October 2009; revision received September 2010; accepted December 2010.
1. Introduction
Launching a new project entails enormous uncertainty. To
learn whether it is advisable to launch a new project, firms
often perform small-scale experiments before making an
irreversible investment in project-specific assets. One perti-
nent well-known example is U.S. steel maker Nucor’s deci-
sion to adopt the world’s first continuous thin-slab casting
technology. Recognizing the huge uncertainty in the prof-
itability of the new technology, Nucor built a pilot plant in
1989 (Ghemawat and Stander 1998). After the pilot plant
proved to be a success, Nucor expanded the use of the
new technology by building several other thin-slab cast-
ing plants, beginning in 1992. In this context of launching
a new project, our paper explores the central question of
interest to the decision-maker: how uncertainty and learn-
ing impact (1) the optimal expansion and exit policy and
(2) the length of time before an expansion or exit decisionis made.
Our paper addresses three main difficulties with launch-
ing a new project that requires project-specific investment.
First, the prospect of success is highly uncertain. Moreover,
the firm undertaking the project might have unproven capa-
bility. In the case of Nucor’s adoption of the thin-slab tech-
nology, there was uncertainty in the technical feasibility as
well as in the demand for the new product, and both fac-
tors contributed to uncertainty in the profit. Second, when
the uncertainty in the profit is project-specific, it can be
resolved only through experimentation. Sources of uncer-tainty that are not project-specific can be resolved by other
means. Third, because the salvage value of the project-
specific assets can be very small, the expansion decision
often is irreversible; Nucor’s adoption of thin-slab cast-ing technology, which is useless outside the industry, was
irreversible. Hence, the decision to acquire project-specific
assets should be made only if the firm is sufficiently confi-
dent of the project’s success.
In this paper, we study a stylized model of a firm exper-imenting with an unproven project. As the experimentation
proceeds, the firm must decide when to cease the exper-
iment and whether to expand or stop the project. In our
model, the profitability of the project is either high or low,
and it does not change in time. The firm knows the prior
(initial) probability that the new project’s profitability is in
the high state. The firm experiments by launching a small-
scale enterprise. Because the firm observes the profit ateach point in time, it continuously updates its belief regard-ing the profitability of the new project. The firm cannot
immediately determine the state of profitability because the
realized profit contains noise. At each point in time after
the launch, the firm can (1) continue the current experi-
mentation, (2) stop and exit the project, or (3) make an
irreversible investment to expand the project by acquiring
project-specific assets.
The cumulative profit is modeled as a Brownian motion
with constant drift and constant volatility . The drift of
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the Brownian motion is known to be in either a high stateor a low state, but the true state of the drift is unknown. Weformulate our model as a continuous-time infinite-horizonoptimal stopping problem.
Employing the methods of optimal stopping, we obtainthe following results: (1) The optimal policy of expan-
sion and exit is characterized by two thresholds withrespect to the posterior probability P t that the state is high:expand (abandon) the project if the posterior probabilityP t hits the upper (lower) threshold; otherwise, continue thepilot project. (2) The upper (lower) threshold decreases(increases) in the volatility of the cumulative profit.(3) The expected time-to-decision has nonmonotone depen-dence on .
Shiryaev (1967) and Peskir and Shiryaev (2006) stud-ied the Bayesian problem of minimizing the undiscounted
cost of incorrect decisions in sequential hypothesis test-ing in a setting in which the true state is slowly revealedvia a Brownian motion, and they obtained elegant analyt-
ical solutions. In their model, the true state can be eithergood or bad, and the decision to choose one hypothesisis irreversible. They showed that the optimal stopping pol-icy is characterized by a pair of thresholds with respectto the posterior probability of being in the good state. Asexplained later, they (and others) obtain a stochastic dif-ferential equation for the posterior probability of being in
the good state at time t given the history of the Brownianmotion up to time t. This stochastic differential equationis the starting point of our analysis. Our model extendstheirs by incorporating discounting (i.e., the time value of money). In applying the model to real options problems, itis essential to add discounting to the model because models
with discounting carry considerably more economic inter-est and realism. In considering the generic issue of utiliz-ing a general purpose asset versus a specialized asset, werealized that if the project turns out to be unprofitable on
an operating basis, the purchaser of the specialized assetwould have bought a white elephant, an asset that has noeconomic use. However, even if the project turns out to beunprofitable, the purchaser might have the option to exitthe project, albeit having sunk funds into the purchase of a specialized asset that no longer has any economic value.For this reason, we also modify Shiryaev’s model by incor-porating an embedded exit option after expansion (see §4).Our principal goals are to study the comparative statics of
the time-to-decision in our Bayesian real options model aswell as to characterize the optimal policy.
Although ours is an extension of Shiryaev’s model,our model addresses a different economic problem, andits comparative statics analysis is different from that of Shiryaev’s model. Our model does not have a zero-discountrate counterpart because it allows perpetual operation of aproject after expansion, whence it does not simply reduceto Shiryaev’s model as the discount rate goes to zero. Also,due to the difference in the economic context, Shiryaev’s
model does not allow for an embedded option to stop after
expansion, as our model does. Furthermore, the compara-
tive statics analysis of our model with respect to the volatil-
ity is more complex because of the nonzero discount rate,
which adds an additional model parameter, and the large-
volatility results are different because of the difference in
the economic problem. For example, the expected time-to-
decision is never monotonically increasing in the volatilityin Shiryaev’s model, whereas there are parameter values in
our model under which the relationship is monotone (see
Figure 2).
Our paper contributes to the literature on real options
theory, in particular, in the context of the comparative stat-
ics with respect to the volatility. The theory of real options
has shown that there is value to waiting before making
an irreversible decision when the value of such a decision
is uncertain. Moreover, both the asset value and the value
of waiting increase in the volatility (Dixit 1992, Alvarez
2003). In our model, the volatility of P t decreases in :
as increases, the arrival of new information slows down
as shown by Bergemann and Välimäki (2000). The assetvalue, that is, the value of the project and the associated
option to expand or exit, decreases in because the arrival
of information about the true state slows down. Likewise,
the value of waiting also decreases in . A decrease in
the value of waiting also decreases the time-to-decision.
There is, however, a countervailing effect of an increase
in . In our model, constitutes noise in the observed
profit stream, so the arrival rate of information is higher
when is low. If increases, then the decision-maker
might want to wait a very long time to collect enough
information to increase the likelihood of making the cor-
rect decision. To our knowledge, our paper is the first work that studies the resulting effect of an increase in on the
time-to-decision, which depends on the relative magnitudes
of these two countervailing effects.
We study the dependence of the expected time-to-
decision on and obtain the following results. For
sufficiently small values of , the effect of the rate of
information arrival dominates the effect of the value of
waiting, so the expected time-to-decision increases in .
In contrast, for sufficiently large values of , the resulting
comparative statics of the expected time-to-decision with
respect to depends on the model parameters.
Our results also have practical implications for firms
faced with expansion and exit decisions. In most busi-ness decisions, the project’s lifetime is a significant fac-
tor. However, there is a paucity of work on the length of
time-to-decision; our paper begins to fill this gap in the
literature.
The paper is organized as follows. We review related
literature in §2. In §3, we solve and analyze the infinite-
horizon model of a firm learning from a pilot project
with an exit and expansion decision. In §4, we allow the
decision-maker to exit even after the expansion, and we
study the effect of the post-expansion exit option. The
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Kwon and Lippman: Acquisition of Project-Specific Assets with Bayesian UpdatingOperations Research 59(5), pp. 1119–1130, © 2011 INFORMS 1121
implications of having an additional source of informa-tion are discussed in §5. Conclusions are given in §6. Anelectronic companion to this paper is available as part of the online version that can be found at http://or.journal.informs.org.
2. Related LiteratureOur paper is closely related to the literature on the valueof acquiring information. The seminal work on the subjectof learning-by-doing is Arrow (1962). In particular, thereis a strand of papers that analyzed optimal stopping mod-els to study the value of experimentation under incompleteinformation. Bernardo and Chowdhry (2002) used numer-ical methods to study the real option value of a firm thatinvests in either a specialized or generalized project, learnsabout its own capabilities, and then decides either to scaleup the specialized project or to expand into a multiseg-ment business. McCardle (1985) and Ulu and Smith (2009)studied the adoption and abandonment of new technologies
under incomplete information regarding profits when theinformation acquisition cost per unit time is deterministic;they characterized the optimal adoption policy, but they didnot study the time-to-decision. In our model, the implicitcost of acquiring information is uncertain.
Our paper is an extension of the classic sequentialhypothesis testing problem, the objective of which is tominimize the expected cost of errors (cost of choosing theincorrect hypothesis) when the probability distribution of the hypotheses is updated in a Bayesian manner. The sub- ject of sequential hypothesis testing was pioneered by Wald(1945), whose work spawned a vast literature on this sub- ject. See Poor and Hadjiliadis (2008) and Lai (2001) for
references.Sequential hypothesis testing problems can be solved
as stopping time problems. There is a vast literature onoptimal stopping problems, and the standard expositionof the optimal stopping theory in continuous time can befound in Shiryaev (1978), Oksendal (2003), and Peskir andShiryaev (2006). The solution method consists of solvinga characteristic partial differential equation and applyingsmooth-pasting conditions; the existence of such a solutionautomatically ensures an optimal solution (Dayanik andKaratzas 2003, Chapter 10 of Oksendal 2003, and Chap-ter IV of Peskir and Shiryaev 2006). (The continuous-timeoptimal stopping theory has been applied to study mathe-matical properties of real options models; see, for example,Alvarez 2001 and Wang 2005.) We solve our model by for-mulating it as an optimal stopping problem. In solving it,we build upon the work of Shiryaev (1967) and Peskir andShiryaev (2006). Shiryaev (1967) studied the problem of minimizing the cost of error with two simple hypotheseson the drift of a one-dimensional Brownian motion, and heobtained an analytical solution. The basic element of hismodel is the stochastic differential equation for the poste-rior probability distribution. This is the starting point of ouranalysis.
This two-drift Brownian motion model has been appliedin many economics papers concerning the optimal level of experimentation in several different contexts. (See Kellerand Rady 1999, Moscarini and Smith 2001, Bolton andHarris 1999, and Bergemann and Välimäki 2000.) For themost part, these papers focus upon decision-making that
leads to more rapid learning. In contrast, our paper focusesupon the optimal stopping decision.In an optimal stopping problem that applied Shiryaev’s
framework, Ryan and Lippman (2003) considered when tostop (abandon) a project with unknown profitability. In theirproblem, a firm seeks to maximize the discounted cumula-tive profit from a project. The firm can abandon the projectat any point in time, and the cumulative profit is modeled asa Brownian motion in which the drift takes either a knownpositive value or a known negative value. Employing the
methods of stochastic analysis, they obtained an optimalpolicy that is stationary and characterized by a thresholdon the posterior probability. Decamps et al. (2005) also
employed Shiryaev’s framework to study the optimal timeto invest in an asset with an unknown underlying value.In their model, the reward from stopping is the Brownianmotion itself rather than the expected value of the time-integral of a Brownian motion. In both of these papers,there is a single alternative to continuing, and the optimalpolicies are characterized by a single threshold.
In our model, there are two alternatives to continuing,and the optimal policy entails two thresholds. This resultis similar to the two-threshold policy obtained by Shiryaev(1967) and Peskir and Shiryaev (2006). Policies with twothresholds also occur in many models under complete infor-mation. See, for example, Alvarez and Stenbacka (2004)
and Decamps et al. (2006).Finally, the comparative statics properties of optimal
stopping policies with respect to the volatility are of partic-ular interest. In the economics literature, it has been shown
that the value of waiting before decision has a monotonicityproperty with respect to the volatility (Dixit 1992). In aneconomics model of a firm that has an option to enter andexit an industry, Dixit (1989) employed numerical exam-ples to depict the comparative statics of the optimal entryand exit thresholds with respect to the volatility of the profitstream. In their Bayesian decision model, Bernardo andChowdhry (2002) demonstrated by numerical illustrationthat the volatility of the observed profit shrinks the contin-
uation region. Alvarez (2003) obtained general comparativestatics result for the optimal policy and the optimal returnwith respect to the volatility for a class of optimal stoppingproblems that arise in economic decisions. Kwon (2010)showed that the comparative statics of the optimal policywith respect to the volatility is nontrivial when there is anembedded option. All these papers focus on the effect of the volatility on the optimal policy or the optimal return;none of these papers studied the impact of volatility onthe time-to-decision. In the classical sequential hypothesis
testing problem, Wald (1973, Ch. 3) studied the expected
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number of observations before choosing a hypothesis, butits dependence on the noise level of the observation wasnot explored.
3. Base Model
Consider the decision problem of a firm launching a new
project. The state S of the project lies in HL: the projectwill bring in either a high profit stream if it is in state H or a low profit stream if it is in state L. The state is notknown, but the firm knows p0, the prior probability that thestate is H . Before launching the project on a large scale,the firm can collect information about the profitability of the project by operating a pilot project. During the pilotphase, the mean profit per unit time is either h if S = H orl if S = L. We assume that h > 0 and l < 0 so the projectis profitable only if it is in the high state. At each pointin time during the pilot phase, the firm can (1) expandthe project by acquiring project-specific assets (equipment)at cost k or (2) permanently exit the project. Once the
project is expanded, the mean profit stream is changed byh† in state H and l† in state L so that the mean profit rateafter acquisition is hA ≡ h + h† or lA ≡ l + l†. Let > 0denote the continuous time discount rate. In this section,we limit consideration to the case in which exit is neveroptimal after expansion due to a prohibitively high exitcost. In similar settings with investment in project-specificassets, the assumption of a prohibitively high exit cost wasalso made by Lippman and Rumelt (1992) and Bernardoand Chowdhry (2002). We relax this assumption in §4 andinvestigate its effect.
The firm observes the profit stream without error. How-ever, because of noise, the firm can never perfectly deter-mine the underlying profit rate (i.e., state of the project).We model the cumulative profit stream X t t 0 duringthe pilot phase as a Brownian motion with unknown drift:
X t = t + Bt (1)
where the unknown drift ∈ hl is the mean profit perunit time, is the volatility of the cumulative profit, andB = Bt t 0 is a one-dimensional standard Brownianmotion. After the firm expands the project, the new profitstream Y t t 0 is given by Y t = At + Bt , where A ∈hA lA; if the firm exits, then the profit stream is zero.
We use a continuous-time approach and model the profit
stream as a one-dimensional Brownian motion because of the analytical tractability of the resulting model. Likewise,numerous papers in economics have modeled cumulativeprofits under incomplete information as Brownian motionwith unknown drift and obtained useful insights from ana-lytically tractable models (Bolton and Harris 1999, Kellerand Rady 1999, Bergemann and Välimäki 2000, Bernardoand Chowdhry 2002, Ryan and Lippman 2003, Decampset al. 2005).
This model is general enough to be applicable in manycontexts.
Example 1 (Expansion). Consider a firm testing a new,unproven technology by operating a single pilot productionplant. If the plant proves to be a success, n more plants willbe built at cost k. Alternatively, n can be regarded as anincrease in the production capacity. Suppose that the profitper plant is not changed by the expansion. The expected
profit per unit time from the pilot plant is h > 0 or l < 0,and the expected profit per unit time from the n + 1 plantsafter expansion will be hA = n + 1h or lA = n + 1l.
Example 2 (Acquisition of a Dedicated Facility).Consider a firm testing a new project by outsourcing themain task or by renting a facility for the core task of the project. By outsourcing or renting, the firm can easilyterminate the project, but the profit stream is diminishedbecause it has to pay a price for outsourcing or renting.The firm can acquire a dedicated facility of its own toavoid paying the price for renting or outsourcing. Beforethe acquisition of the dedicated facility, the expected profitper unit time is h or l; after the acquisition, the expected
profit per unit time improves by h† and l†, which are theexpected cost of outsourcing or renting in the high and lowstates, respectively. The increase h† or l† in expected profitper unit time might be state-dependent because the cost of renting might be on a per-unit-sold basis.
To simplify our presentation, we will focus on expan-sion decisions (Example 1) with the understanding that ourmodel fully addresses Example 2 as well. For consistency,we shall speak in terms of project expansion rather thanacquisition of project-specific assets.
3.1. Posterior Probability
At time zero, the firm’s prior probability that S = H isp0. Our first task is to compute the posterior probabilitythat the project is in the high state at time t. Subsequently,we utilize the posterior probability process to ascertain theoptimal expansion and exit decisions.
Let X t, , and Bt of Equation (1) be defined on aprobability space so that X t, , and Bt aremeasurable with respect to , and let t t 0 bethe natural filtration with respect to the observable pro-cess X t t 0. (The unknown drift and the unobserv-able process Bt t 0 are not adapted to the filtration t.) Here the probability measure is denoted by . LetP t ≡ = h t = E p0 1=h t denote the posterior
probability of the event = h at time t, where E p0 ·denotes the expectation conditional on the initial condi-tion P 0 = p0, and 1A is the indicator function of a set A.The posterior probability process P is a continuous mar-tingale with respect to t t 0, a strong Markov pro-cess that is homogeneous in time, and the unique solutionto the following stochastic differential equation (Liptserand Shiryaev 1974, p. 371; Peskir and Shiryaev 2006,pp. 288–289; Ryan and Lippman 2003, pp. 252–254):
dP t =h − l
P t1 − P tdBt (2)
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where B = Bt t t 0 is a one-dimensional standardBrownian motion defined by
Bt ≡1
X t −
t
0E s ds
=1
X t −
t
0
P sh + 1 − P slds
Unlike Bt which is unobservable, the new Brownian motionBt is observable because it can be completely constructedfrom the observed value of X t over time.
Equation (2) implies that the posterior process undergoesa more rapid change if the coefficient h − l/P t1−P t increases. The factor h − l/ , which is the uncertaintyin the drift normalized by the volatility, has the interpreta-tion of the signal-to-noise ratio. Hence, h − l/ measuresthe rate of arrival of new information, and it follows thatthe posterior evolves more rapidly with a high value of h − l/ (Bergemann and Välimäki 2000). The additional
factor P t1−
P t, which peaks at P t=
05, results from thefact that the posterior process requires more information(and hence a longer time) to change if P t is closer to strongbeliefs (either 0 or 1).
Using the strong Markov property of the process P andBayes rule (Peskir and Shiryaev 2006, pp. 288–289), wehave
P t ≡ E p0 1=h t = E p0 1=h X t
=
p0 exp
−
X t − ht2
2 2t
·
p0 exp
−
X t − ht2
2 2t
+ 1 − p0 exp−X t − lt2
2 2
t
−1
=
1 +
1 − p0
p0
exp
−
h − l
2
·
−
h + l
2
t + Bt
−1
(3)
Because − h + l/2 = h − l/2 if = h and −h + l/2 = l − h/2 if = l, Equation (3) reflects the factthat P t tends to increase if = h and to decrease if = l.
3.2. The Objective Function
The firm seeks to maximize its expected discounted cumu-
lative profit where > 0 is the discount rate. Suppose thefirm decides to expand at the stopping time . Then thediscounted reward over the time interval discountedback to time is given by
E p0
e−t− dY t
= E P
0e−t dY t
= E P
0Ae−t dt
=1
P h
A + 1 − P lA
where the first equality follows from P being a strongMarkov process and the second from E x
0
e−t dBt = 0.If the firm decides to exit, then the associated reward fromexit is zero. Thus, the optimal reward r · from stoppingat time is the greater of the reward from expansion andthe reward from exit
rP = max
1
P hA + 1 − P lA − k 0
where k is the cost of expansion.The objective function of the firm is the time-integral of
the discounted profit stream up to time plus the rewardfrom stopping at time
V p0 = E p0
0e−t dX t + e− r P
= E p0
1 − e− + e− rP
= 1
p0h + 1 − p0l + E p0 e− gP (4)
where
gx=1
maxxh† +1−xl† −k−xh −1−xl (5)
The reward function gx = V 0x − V x is the differencebetween the return from immediate stopping ( = 0) andthe return from never stopping ( = ).
Because the only -dependence of V p0 is in the termE p0 e− gP , the stopping problem of Equation (4) isequivalent to maximizing
R p0 = E p0 e− gP (6)
For convenience, we regard R p0 as the objective func-tion for the remainder of this section.
3.3. Optimal Policy
As revealed in Equation (4), the firm seeks a stoppingtime ∗, termed optimal, that maximizes R p0 defined inEquation (6). Because our infinite horizon stopping prob-lem is stationary, it comes as no surprise that it suffices tofocus on the class of stopping times that are time-invariant(Oksendal 2003, p. 220). Each time-invariant stopping timecan be defined through a set A as follows:
A ≡ inf t 0 P t ∈ A
We call A the exit time from the set A. For notational con-venience, we let R∗p ≡ R ∗ p denote the optimal returnfunction whenever the optimal stopping time ∗ exists.
The existence of an optimal stopping time is not guaran-teed in general, but we can prove that our model satisfiesthe sufficient conditions for the existence of an optimalstopping time. Before proving the sufficient conditions, we
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need to lay out some technical preliminaries. Clearly, if
P t = p at some time t, then it is optimal to continue if the
return R∗p from continuing exceeds the return R0p =
gp from stopping. Accordingly, we refer to p R∗p>
R0p as the continuation region. Let
≡ − + 12
h − l
2
p21 − p22p
be the characteristic differential operator for P t; plays a
fundamental role in the solution of stopping time problems.
Here t is replaced by − because our solution is time-
invariant except for the discount factor e−t. One of the
sufficient conditions for the optimality of ∗ is that R∗ ·
satisfy the differential equation R∗p = 0 if p belongs to
the continuation region. There are two fundamental solu-
tions, and , to the second-order linear ordinary differ-
ential equation fp = 0:
p = p1/21+ 1 − p1/21−
p = p1/21− 1 − p1/21+
where
≡
1 +
8 2
h − l2 (7)
Note that · is convex increasing and · is convex
decreasing.
If the firm stops at time t with P t = p, then the
firm’s (expected discounted) return over the interval t
is gp. From Equation (5), note that gp is the maxi-
mum of two functions g1p ≡ ph†/+1−pl†/−k and
g2p ≡ −ph/ − 1 − pl/, each of which is linear in
the posterior probability P ∗ = p at the stopping time. The
first function g1p gives the return associated with expan-
sion and is increasing in p. The second function g2p is
associated with the return to exit and is decreasing in p.
If k − lA/hA − lA ∈ 0 1, then the two linear func-
tions cross at p where
p ≡k − lA
hA − lA
and g · is minimized at p. Note that g1 p > g2p if and
only if p > p, so it is optimal to expand if P ∗ p and
optimal to exit if P ∗ < p.
First, we consider the cases k − lA/hA − lA 1 or
k − lA/hA − lA 0.
Proposition 1. (i) Suppose k − lA/hA − lA 1. Then
the optimal policy is to exit when P t hits a lower threshold
=− − 1l
+ 1h − − 1l
and to continue operation otherwise. The optimal return is
given by
R∗p =
−2lh/ 2 − 1−lh
− 1
+ 1
−l
h
/2
· p
if p >
gp otherwise
(ii) Suppose k − lA/hA − lA 0. Then the optimal
policy is to expand the project when P t hits an upper
threshold
=− + 1l† − k
− 1h† − k − + 1l† − k
and to continue operation otherwise. The optimal return is
given by
R∗p =
−2l† − kh† − k/ 2 − 1h† − kk − l†
·
− 1
+ 1
h† − k
k − l†
/2
· p if p <
gp otherwise
If k − lA/hA − lA 1 or k hA, the exit option is
always better than the expansion option because the expan-
sion cost is too high, so expansion is never optimal. If
k − lA/hA − lA 0 or k lA, then the expansion
option is always better than the exit option because theprofit improvement, even in the low state, exceeds the cost
of investment. In these two cases, the optimal policy is
characterized by only one threshold. The single-threshold
solutions are studied in detail by Ryan and Lippman (2003)
and Kwon et al. (2011). Henceforth, we focus on the more
interesting case of p = k − lA/hA − lA ∈ 0 1 when
the optimal policy is characterized by two thresholds.
Theorem 1. Assume k − lA/hA − lA ∈ 0 1. (i) The
optimal policy to Equation (6) always exists, and there is
a pair of thresholds and which satisfy 0 < < < 1
such that the optimal stopping time is given by
∗ = inf t > 0 P t ∈
The optimal return function is given by
R∗p =
c1p + c2p if p ∈
gp otherwise(8)
where c1 and c2 are the unique positive numbers such that
R∗ · is continuously differentiable.
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(ii) The time-to-decision ∗ is finite with probability one,and
E p ∗ =p −
− T +
− p
− T − T p (9)
where Tp ≡ 2 2/h − l22p − 1 lnp/1 − p.
The proof of Theorem 1 is given in the online appendix.The proof of the existence of an optimal policy consistsof constructing a function f · (a candidate for the opti-mal return function) that satisfies the differential equa-
tion fp = 0, the boundary conditions fp = gp forp ∈ , and continuous differentiability (smooth past-ing conditions). The expected time-to-decision formula inEquation (9) was also proved in Poor and Hadjiliadis (2008,
p. 87) using the standard property of martingales; we repro-duce its proof in the online appendix.
The optimal policy is intuitively straightforward: expand
the project if the posterior probability P t is high enough,
and exit if it is low enough; otherwise, continue thepilot project. The expected time-to-decision E p ∗ is finitebecause the optimal policy is characterized by two thresh-
olds, either of which is reached by P t eventually. By con-trast, in exit-only or investment-only models as in the caseof Proposition 1 or as in Ryan and Lippman (2003), theexpected time-to-decision is infinite because there is a pos-
itive probability that the threshold is never reached in afinite amount of time.
The optimal return R∗ · given in Equation (8) is strictly
convex and larger than gp for p ∈ , which is clearfrom the fact that and are convex and c1 and c2 arepositive.
3.4. Comparative Statics
The analytical solution obtained in the previous subsection
enables us to obtain comparative statics with respect to .
Proposition 2. The optimal return function V ∗ p is non-
increasing in , and the upper (lower) threshold decreases(increases) in .
As the noise level of the observed profit increases, thearrival of new information slows down, and it takes longerto accumulate information about the profit state. It fol-
lows that it is optimal to set a smaller continuation region
(smaller values of the upper threshold and larger values of the lower threshold) as increases. Thus, the upper thresh-old decreases and the lower threshold increases in .
Proposition 2 is entirely consistent with the results fromconventional real options theory. It is well known that thevalue of a real option increases in the volatility of the assetvalue process and that the continuation region grows with
the volatility. In our Bayesian stopping problem, by Equa-tion (2), h − l/P t1 − P t is the volatility of the pos-terior process P t t 0. As per Proposition 2, an increase
in the volatility h − l/P t1 − P t (i.e., an increase in
the rate of information arrival) leads to an increase in thevalue of the optimal return V ∗ · .
Because we do not have closed-form expressions for and , Equation (9) does not provide a closed-form expres-sion for the expected time-to-decision as a function of .However, asymptotic results can be derived from Equa-
tion (9). Theorem 2 provides the asymptotic properties of the thresholds as well as the expected time-to-decision forsmall and large values of .
Theorem 2 (Expected Time-to-Decision). (i) As → 0, 0, 1, and E p ∗ 0.
(ii) Suppose gp 0. As → , p, p, and E p ∗ 0.
(iii) Suppose gp < 0. As → , −l/h − l, −l† + k/h† − l†, E p ∗ → if p ∈ −l/h − l
−l† + k/h† − l†, E p ∗ → 0 if p ∈ −l/h − l−l† + k/h† − l†, and E p ∗ = O if p ∈ −l/h − l−l† + k/h† − l†.1
Note that gp< 0 if and only if −l/h−l<p<−l† +k/h† −l†. If gp < 0, the interval −l/h − l−l† + k/h† − l† always remains a subset of thecontinuation region, so < −l/h − l and −l† + k/h† − l† < because an immediate stop in the interval−l/h − l−l† + k/h† − l† would result in a nega-tive reward.
From Proposition 2 and Theorem 2, we can deducethe global behavior of , , and E p ∗ as a function of the volatility . As increases from zero to very largenumbers, increases from zero to a limiting value, and decreases from 1 to another limiting value. (See Fig-ures 1 and 2.) The large- behavior comports with intu-
ition. When is very large, arrival of new information is soslow that the firm should proceed as if P t will never change.In particular, suppose gp 0. Then take almost imme-diate action: exit if P t < p and expand if P t p. Instead,
suppose gp < 0 Then exit if P t < −l/h − l and expandif P t −l† + k/h† − l†; otherwise, the firm can expectto wait a very long time until it makes an expansion or exitdecision.
For small values of , the expected time-to-decisionE p ∗ is very small because it takes a very short time tolearn the true state. The behavior of E p ∗ for large depends on the initial probability and the sign of gp. If gp 0, then gp 0 for all p ∈ 0 1 because g ·
takes its minimum value at p. In this case, if is verylarge, due to slow arrival of information, it is not worthspending much time on the pilot project, and the firm isbetter off making a quick expansion or exit decision. Con-sequently, E p ∗ is very small for large values of . Asillustrated in Figure 1 the expected time-to-decision ini-tially increases in , achieves a maximum at an inter-mediate value of , and then approaches zero for largevalues of . If gp < 0, then gp < 0 if and onlyif p ∈ −l/h − l−l† + k/h† − l†. In this case, as
shown in Figure 2, E p ∗ keeps increasing for large
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Figure 1. The thresholds and E p ∗ in case gp > 0.
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1.0
P r o b a b i l i t y t h
r e s h o l d s
0
0.1
0.2
0.3
E x p e c t e d t i m e
t o d e c i s i o n
0 2 4 6 8 10
Base model
Extension
Notes. = 1, k = 025, h = 1, l = −1, hA = 15, and lA = −05. The dashed lines are the results from the model with a post-expansion exit option.
because it takes a long time for P t to exit the interval−l/h − l−l† + k/h† − l† due to very low speed
of evolution of P t .
The case gp < 0 arises, for instance, when hA/h =lA/l > 0. The relation hA/h = lA/l > 0 applies to Exam-ple 1. If the production plant is expanded n + 1-fold,then the profit per unit time is also expanded n + 1-fold:hA = n + 1h and lA = n + 1l, so hA/lA = h/l.
4. Exit Option After Expansion
From both a legal and a practical perspective, the ability toabandon a project, before or after an expansion, is alwaysavailable, although there are costs associated with aban-donment. Such an embedded exit option is known to result
in unconventional comparative statics of the optimal policy
with respect to the volatility (Alvarez and Stenbacka 2004,Kwon 2010). Hence, we are motivated to study the effectof the post-expansion exit option. In this section, we inves-tigate the effect of the post-expansion exit option upon theoptimal policy and the time-to-decision, and we scrutinizethe robustness of our comparative statics results of Theo-
rem 2. We show that the comparative statics in the limiting
Figure 2. The thresholds and E p ∗ in case gp < 0.
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1.0
P r o b a b i l i t y t h r e s h o l d
s
0
0.2
0.4
0.6
0.8
1.0
1.2
E x p e c t e d t i m e t o d e c i s i o n
0 2 4 6 8 10
Base model
Extension
Notes. = 1, k = 025, h = 1, l = −1, hA = 10, and lA = −10. The dashed lines are the results from the model with a post-expansion exit option.
cases of small and large values of are robust against the
embedded exit option.
4.1. The Objective Function
In this model with an embedded option, the decision-
makers policy determines two stopping times: A (time to
acquire assets) and E (time to exit). The cumulative profit
process is X t given by Equation (1) until ≡ min A E .
After acquisition at time A, the cumulative profit process
is represented by X At , where
dX At = Adt + AdBt
Here, A = n + 1 for Example 1 (expansion) and
A = for Example 2 (acquisition of a dedicated facility).Then the objective function is given by
V A E p0
= E p0
0e−t dX t + 1 A< E
−ke− A +
E
A
e−t dX At
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From the strong Markov property of Bt t 0, we find
that
V p0 ≡ sup A E
V A E p0
= sup
E p0
0
e−t dX t + e− maxV AP − k 0
where
V Ax = sup
E x
0e−t dX At
In analogy with Equation (4), we re-express V · as
Vx =1
xh + 1 − xl + sup
R x
where
R x = E xe− gP
gx =1
max
V Ax − k − xh − 1 − xl
− xh − 1 − xl
If we assume that hA − lA/ A = h − l/ , which is
the case of Examples 1 and 2, then the form of V A · is
obtained by Ryan and Lippman (2003) as follows:
V A
x =
1
xhA + 1 − xlA
+ C · x1/21− 1 − x1/21+ if x > A
0 otherwise
where
A =− − 1lA
+ 1hA − − 1lA
C =2
2 −1
−1
+1
/2
hA1/21− −lA1/21+ (10)
For the remainder of this section, we assume that
V A
1 > k so that it is profitable to expand when the pos-terior probability p is sufficiently high.
4.2. Optimal Policy
In this subsection, we obtain the optimal stopping time ∗
that maximizes R · . We do so by establishing the exis-tence of ∗ and the necessary conditions for the continua-
tion region.
In analogy with §3.3, we define p as the point at which
V Ap − k = 0, and we focus on the interesting case of
p ∈ 0 1.
Theorem 3. Assume p ∈ 0 1. The optimal policy tomaximize R · always exists. Moreover, the continuation
region includes a component such that 0 < p < 1. The optimal return function R∗ · ≡ R ∗ · in theinterval is given by
R∗p =c1p + c2p if p ∈
¯
gp if p ∈ (11)
where c1 and c2 are positive numbers such that R∗ · is
continuously differentiable.
Note that this theorem is an analog of Theorem 1(i)
and that Theorem 1(ii) also applies to this model withoutmodification. Theorem 3 does not preclude the existence
of other components of the continuation region discon-nected from because it is technically difficult to pre-clude them. Even if other components of the continuation
region exist, the interval is the only region where thedichotomous (exit vs. expansion) decision takes place, so
we will focus only on the region .
4.3. Comparative Statics
In this subsection, we scrutinize the robustness of the com-parative statics results of §3.4 and investigate the effectof the exit option after expansion. The post-expansion exit
option has the following effect on .
Proposition 3. The post-expansion exit option induces to weakly decrease.
If the post-expansion exit option is available, then the
expansion option becomes more attractive to the decision-maker, and it is optimal to wait longer before making apermanent exit from the pilot project. In contrast, the effect
of the post-expansion exit option on the upper threshold is not mathematically straightforward. Hence, the effecton the expected time-to-decision is even less clear because
it depends on both and . Although a general analyti-cal result is unavailable regarding this issue, we can obtainpartial answers in the limits of small and large . We also
obtain the comparative statics of and in the same lim-iting cases.
Lemma 1. In the small- limit,
= 22
h − l2
−l
hA − k+ o 2 (12)
= 1 − 22h† − k
kh − l2+ o 2 (13)
We also note that the upper threshold is 1 − 2 ·2h† − k/k − lA/h − l2 + o 2 if the post-
expansion exit option is absent.2 In contrast, the lowerthreshold is not affected by the post-expansion exit option
up to o 2. Hence, the following proposition obtains.
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Proposition 4. In the small- limit, the post-expansionexit option causes both and the expected time-to-decisionto decrease.
With a post-expansion exit option, the expansion optionis more attractive to the decision-maker, so it makes sensethat the threshold for expansion is lower. The expected
time-to-decision is also smaller because the decision-makermakes an earlier decision to expand the project due to ahigher value of the expansion option. It is also understand-able that the effect of Proposition 3 (smaller value of )is a secondary effect on the expected time-to-decision: theincreased value of the expansion option does not makean exit option too much less attractive to the decision-maker when the posterior probability P t is low (when theprospect of exercising the expansion option is low). Numer-ical examples are displayed in Figures 1 and 2, which showthat the extra exit option indeed decreases , , and theexpected time-to-decision.
In the large- limit, we obtain the following result.
Proposition 5. In the large- limit, the post-expansionexit option does not have any effect on , , or the expected time-to-decision up to O −n for any finite positive n.
The above proposition merely states that there is essen-tially no effect of the post-expansion exit option in thelarge- limit. This is consistent with the intuition that theoption value from the learning opportunity is minimal if is very large due to the slow arrival of information, andhence the effect of the post-expansion exit option is alsominimal.
Lastly, we are in a position to inspect the comparativestatics of the expected time-to-decision with respect to .The next theorem follows immediately from Lemma 1 andProposition 5.
Theorem 4 (Expected Time-to-Decision). Theorem 2(i),(ii), and (iii) holds for the model with a post-expansion exit option.
Figures 1 and 2 also illustrate a numerical example of Theorem 4. Therefore, we established the robustness of ourresult to the additional exit option.
5. Reducing the Volatility
In some business situations, the firm has an external source
of information regarding the profitability of the project thateffectively reduces . Therefore, the comparative staticswith respect to can provide a useful prediction of howthe additional source of information impacts the optimalpolicy and the duration ∗ of the pilot project.
Suppose that the firm observes another firm’s profitstream from a similar project subject to the same uncer-tainty of the state. Assume that the other firm’s cumulativeprofit process is
X et = t + eBet
where the volatility e of the external information processis known to the firm but not necessarily the same as . HereBe
t is an unobservable one-dimensional Brownian motion.Moreover, assume that X et is independent of X t except thatthey share the same drift .
Define the reduced volatility r ≡ e/ 2 + e2,
a new one-dimensional standard Brownian motion B
r
t ≡ r Bt/ + Bet / e, and a new process X r
t ≡ r 2X t / 2 +
X et / e2 = t + r Br t . The new Brownian motion Br is
unobservable, but the process X r is constructed fromobserving both X and X e, so X r is an observable process.Let r
t t 0 be the natural filtration with respect tothe process X r . Then we can construct the posterior pro-cess P r
t ≡ = h r t , which is adapted to r
t t 0.Using a Bayes rule, we can show that the posterior belief is given by
P r t ≡
p = h r t
=
p exp h
r 2 X r t −
1
2 h
r 2
t
·
p exp
h
r 2X r
t −1
2
h
r
2
t
+ 1 − p exp
l
r 2X r
t −1
2
l
r
2
t
−1
(14)
which is clearly constructed from the observable pro-cess X r . As before, the posterior process satisfies thestochastic differential equations
dP r t =
h − l
r
P r t · 1 − P r
t dW t
where W t is another one-dimensional standard Brownianmotion that is observable. Last, the objective function forthis problem is
V r p = E p
0e−t dX t + e− rP r
=1
ph + 1 − pl + E pe− gP r
(15)
where p is the initial prior P r 0 ≡ = h r
0 and is astopping time for the filtration r
t . The objective function
has the same form as Equation (4), except that the posteriorprocess P r t in Equation (15) is constructed from X r with
volatility r via Equation (14). Therefore, the problem withexternal information reduces to that of §3 with new volatil-
ity r such that r < min e. The only effect of theadditional source of information is to reduce the volatilityof the cumulative profit.
6. Conclusions
Prior to expanding a new project, it behooves the firm
to learn more about the projects profitability. Firms
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undertaking a pilot project can learn about profitability and
subsequently make expansion and exit decisions. In this
paper, we studied the expansion and exit problem under
incomplete information. Our results show that there is a
nonmonotonic relationship between the expected time-to-
decision and the size of the continuation region for an irre-
versible decision: as the volatility of the cumulative profitincreases (so the rate of information arrival decreases),
the optimal continuation region shrinks, but the time-to-
decision does not monotonically decrease.
Our paper focused on extracting useful insights from a
simple and tractable model. Hence, we have not addressed
practical applications of our findings. To model realistic
business expansion decisions, a number of extensions need
to be undertaken. For example, our model assumes that
the noise in the cumulative profit is a Wiener process with
constant volatility. Relaxing this assumption would sacri-
fice the analytical tractability. More significantly, our model
assumes that the number of possible states of the project
is two. For practical applications, we need to provide ananalytical or numerical solution to problems with a larger
number of states.
As shown by Decamps et al. (2005), our model can
generalize to a problem in which the number of possi-
ble states of the project is larger than two but finite. (See
also Olsen and Stensland 1992 and Hu and Oksendal 1998for multidimensional stopping time problems.) In such a
multi-state extension, it is straightforward to show that
much of §3 generalizes to multi-state analogs. In particu-
lar, there is a multi-state analog of Theorem 2 regarding
the asymptotic behavior of the expected time-to-decision.
However, analytical solutions such as in Equation (8) are
not available for a multi-state model. Hence, for practi-
cal purposes, we need to establish numerical procedures to
obtain the optimal policy and the solution. The difficulty
is that a numerical procedure would require continuous
differentiability (smooth-pasting condition) of the optimal
solution (Muthuraman and Kumar 2008), but the multi-
state model does not necessarily result in continuously dif-
ferentiable solutions.3 An even more difficult problem is
a model with a continuous distribution of states of the
project. Finding appropriate numerical solution procedures
for multi-state extensions would constitute a major research
program.
7. Electronic Companion
An electronic companion to this paper is available as part
of the online version that can be found at http://or.journal
.informs.org/.
Endnotes
1. A function f is said to be O if there is a positive
constant M such that f < M for all sufficiently
large.
2. A function f is said to be o 2 if f/ 2 → 0 in
the limit as → 0.3. Continuous differentiability over the boundary points
requires regularity of boundary points of the continua-
tion region, but the regularity is sometimes violated for adegenerate diffusive process such as a multi-dimensional
generalization of the posterior process P t governed by thedynamics of one-dimensional Brownian motion Bt. The
authors thank Renming Song for pointing this out.
Acknowledgments
The authors thank three anonymous referees and the asso-ciate editor for their helpful suggestions that considerably
improved the manuscript.
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